├── Data ├── ABCDo │ ├── Noise_30 │ │ ├── cs.dat │ │ ├── config.toml │ │ ├── deg.dat │ │ └── com.dat │ ├── Noise_40 │ │ ├── cs.dat │ │ ├── config.toml │ │ ├── deg.dat │ │ └── com.dat │ └── Noise_50 │ │ ├── cs.dat │ │ ├── config.toml │ │ ├── deg.dat │ │ └── com.dat ├── SBM_5000.pkl └── Football │ ├── football.txt │ ├── football.community │ ├── football.ecg │ ├── football.readme │ ├── football.edgelist │ └── football.gml ├── LICENSE ├── README.md ├── example_with_sknetwork.ipynb ├── example_with_igraph.ipynb ├── example_with_networkx.ipynb ├── partition_networkx.py ├── partition_igraph.py └── partition_sknetwork.py /Data/ABCDo/Noise_30/cs.dat: -------------------------------------------------------------------------------- 1 | 50 2 | 192 3 | 177 4 | 161 5 | 142 6 | 101 7 | 69 8 | 58 9 | 50 10 | -------------------------------------------------------------------------------- /Data/ABCDo/Noise_40/cs.dat: -------------------------------------------------------------------------------- 1 | 50 2 | 192 3 | 177 4 | 161 5 | 142 6 | 101 7 | 69 8 | 58 9 | 50 10 | -------------------------------------------------------------------------------- /Data/ABCDo/Noise_50/cs.dat: -------------------------------------------------------------------------------- 1 | 50 2 | 192 3 | 177 4 | 161 5 | 142 6 | 101 7 | 69 8 | 58 9 | 50 10 | -------------------------------------------------------------------------------- /Data/SBM_5000.pkl: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/ftheberge/graph-partition-and-measures/HEAD/Data/SBM_5000.pkl -------------------------------------------------------------------------------- /Data/Football/football.txt: -------------------------------------------------------------------------------- 1 | The file football.gml contains the network of American football games 2 | between Division IA colleges during regular season Fall 2000, as compiled 3 | by M. Girvan and M. Newman. The nodes have values that indicate to which 4 | conferences they belong. The values are as follows: 5 | 6 | 0 = Atlantic Coast 7 | 1 = Big East 8 | 2 = Big Ten 9 | 3 = Big Twelve 10 | 4 = Conference USA 11 | 5 = Independents 12 | 6 = Mid-American 13 | 7 = Mountain West 14 | 8 = Pacific Ten 15 | 9 = Southeastern 16 | 10 = Sun Belt 17 | 11 = Western Athletic 18 | 19 | If you make use of these data, please cite M. Girvan and M. E. J. Newman, 20 | Community structure in social and biological networks, 21 | Proc. Natl. Acad. Sci. USA 99, 7821-7826 (2002). 22 | 23 | Correction: Two edges were erroneously duplicated in this data set, and 24 | have been removed (21 SEP 2014) 25 | -------------------------------------------------------------------------------- /Data/Football/football.community: -------------------------------------------------------------------------------- 1 | 7 2 | 0 3 | 2 4 | 3 5 | 7 6 | 3 7 | 2 8 | 8 9 | 8 10 | 7 11 | 3 12 | 10 13 | 6 14 | 2 15 | 6 16 | 2 17 | 7 18 | 9 19 | 6 20 | 1 21 | 9 22 | 8 23 | 8 24 | 7 25 | 10 26 | 0 27 | 6 28 | 9 29 | 11 30 | 1 31 | 1 32 | 6 33 | 2 34 | 0 35 | 6 36 | 1 37 | 5 38 | 0 39 | 6 40 | 2 41 | 3 42 | 7 43 | 5 44 | 6 45 | 4 46 | 0 47 | 11 48 | 2 49 | 4 50 | 11 51 | 10 52 | 8 53 | 3 54 | 11 55 | 6 56 | 1 57 | 9 58 | 4 59 | 11 60 | 10 61 | 2 62 | 6 63 | 9 64 | 10 65 | 2 66 | 9 67 | 4 68 | 11 69 | 8 70 | 10 71 | 9 72 | 6 73 | 3 74 | 11 75 | 3 76 | 4 77 | 9 78 | 8 79 | 8 80 | 1 81 | 5 82 | 3 83 | 5 84 | 11 85 | 3 86 | 6 87 | 4 88 | 9 89 | 11 90 | 0 91 | 5 92 | 4 93 | 4 94 | 7 95 | 1 96 | 9 97 | 9 98 | 10 99 | 3 100 | 6 101 | 2 102 | 1 103 | 3 104 | 0 105 | 7 106 | 0 107 | 2 108 | 3 109 | 8 110 | 0 111 | 4 112 | 8 113 | 4 114 | 9 115 | 11 116 | -------------------------------------------------------------------------------- /Data/Football/football.ecg: -------------------------------------------------------------------------------- 1 | 6 2 | 2 3 | 3 4 | 8 5 | 6 6 | 8 7 | 3 8 | 0 9 | 0 10 | 6 11 | 8 12 | 1 13 | 5 14 | 3 15 | 5 16 | 3 17 | 6 18 | 11 19 | 5 20 | 7 21 | 11 22 | 0 23 | 0 24 | 6 25 | 1 26 | 2 27 | 5 28 | 11 29 | 1 30 | 7 31 | 7 32 | 5 33 | 3 34 | 2 35 | 5 36 | 7 37 | 4 38 | 2 39 | 5 40 | 3 41 | 8 42 | 6 43 | 5 44 | 5 45 | 10 46 | 2 47 | 9 48 | 3 49 | 10 50 | 9 51 | 1 52 | 0 53 | 8 54 | 9 55 | 5 56 | 7 57 | 11 58 | 10 59 | 4 60 | 4 61 | 3 62 | 5 63 | 11 64 | 4 65 | 3 66 | 11 67 | 10 68 | 9 69 | 0 70 | 1 71 | 11 72 | 5 73 | 8 74 | 9 75 | 8 76 | 10 77 | 11 78 | 0 79 | 0 80 | 7 81 | 7 82 | 8 83 | 7 84 | 9 85 | 8 86 | 5 87 | 10 88 | 11 89 | 9 90 | 2 91 | 1 92 | 10 93 | 10 94 | 6 95 | 7 96 | 11 97 | 11 98 | 4 99 | 8 100 | 5 101 | 3 102 | 7 103 | 8 104 | 2 105 | 6 106 | 2 107 | 3 108 | 8 109 | 0 110 | 2 111 | 9 112 | 0 113 | 10 114 | 11 115 | 9 116 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | MIT License 2 | 3 | Copyright (c) 2020 François Théberge 4 | 5 | Permission is hereby granted, free of charge, to any person obtaining a copy 6 | of this software and associated documentation files (the "Software"), to deal 7 | in the Software without restriction, including without limitation the rights 8 | to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 9 | copies of the Software, and to permit persons to whom the Software is 10 | furnished to do so, subject to the following conditions: 11 | 12 | The above copyright notice and this permission notice shall be included in all 13 | copies or substantial portions of the Software. 14 | 15 | THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 16 | IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 17 | FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE 18 | AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 19 | LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 20 | OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE 21 | SOFTWARE. 22 | -------------------------------------------------------------------------------- /Data/Football/football.readme: -------------------------------------------------------------------------------- 1 | Dataset -- American College Football Graph 2 | 3 | [REF]: "Community structure in social and biological networks", M. Girvan and M. E. J. Newman PNAS June 11, 2002 99 (12) 7821-7826; https://doi.org/10.1073/pnas.122653799 4 | 5 | Teams are part of 12 conferences (the 'communities'): 6 | 0 = Atlantic Coast 7 | 1 = Big East 8 | 2 = Big Ten 9 | 3 = Big Twelve 10 | 4 = Conference USA 11 | 5 = Independents 12 | 6 = Mid-American 13 | 7 = Mountain West 14 | 8 = Pacific Ten 15 | 9 = Southeastern 16 | 10 = Sun Belt 17 | 11 = Western Athletic 18 | 19 | 14 teams out of 115 appear as anomalies as can be seen in Figure 5 of [REF], namely: 20 | 21 | 5 teams in #5 conference (Independent) play teams in other conferences (green triangles) 22 | 7 teams in #10 conference (Sun Belt) are broken in 2 clumps (pink triangles) 23 | 2 teams from #11 conference play mainly with #10 conference (red triangles) 24 | 25 | Here, we try to recover those anomalous teams by running several embeddings (we use node2vec): 26 | 27 | for each embedding: 28 | compute divergence using our framework 29 | also compute entropy of b-vector for each node (probability distribution of edges w.r.t. every community in the geometric Chung-Lu model) 30 | plot entropy vs divergence 31 | for some good/bad embedding, boxplot entropy of anomalous vs other nodes 32 | -------------------------------------------------------------------------------- /Data/ABCDo/Noise_30/config.toml: -------------------------------------------------------------------------------- 1 | seed = "42" # RNG seed, use "" for no seeding 2 | n = "1000" # number of vertices in graph 3 | t1 = "2.5" # power-law exponent for degree distribution 4 | d_min = "5" # minimum degree 5 | d_max = "50" # maximum degree 6 | d_max_iter = "1000" # maximum number of iterations for sampling degrees 7 | t2 = "1.5" # power-law exponent for cluster size distribution 8 | c_min = "50" # minimum cluster size 9 | c_max = "200" # maximum cluster size 10 | c_max_iter = "1000" # maximum number of iterations for sampling cluster sizes 11 | # Exactly one of xi and mu must be passed as Float64. Also if xi is provided islocal must be set to false or omitted. 12 | xi = "0.3" # fraction of edges to fall in background graph 13 | #mu = "0.2" # mixing parameter 14 | islocal = "false" # if "true" mixing parameter is restricted to local cluster, otherwise it is global 15 | isCL = "false" # if "false" use configuration model, if "true" use Chung-Lu 16 | degreefile = "./deg.dat" # name of file do generate that contains vertex degrees 17 | communitysizesfile = "./cs.dat" # name of file do generate that contains community sizes 18 | communityfile = "./com.dat" # name of file do generate that contains assignments of vertices to communities 19 | networkfile = "./edge.dat" # name of file do generate that contains edges of the generated graph 20 | nout = "50" # number of vertices in graph that are outliers; optional parameter 21 | # if nout is passed and is not zero then we require islocal = "false", 22 | # isCL = "false", and xi (not mu) must be passed 23 | # if nout > 0 then it is recommended that xi > 0 24 | -------------------------------------------------------------------------------- /Data/ABCDo/Noise_40/config.toml: -------------------------------------------------------------------------------- 1 | seed = "42" # RNG seed, use "" for no seeding 2 | n = "1000" # number of vertices in graph 3 | t1 = "2.5" # power-law exponent for degree distribution 4 | d_min = "5" # minimum degree 5 | d_max = "50" # maximum degree 6 | d_max_iter = "1000" # maximum number of iterations for sampling degrees 7 | t2 = "1.5" # power-law exponent for cluster size distribution 8 | c_min = "50" # minimum cluster size 9 | c_max = "200" # maximum cluster size 10 | c_max_iter = "1000" # maximum number of iterations for sampling cluster sizes 11 | # Exactly one of xi and mu must be passed as Float64. Also if xi is provided islocal must be set to false or omitted. 12 | xi = "0.4" # fraction of edges to fall in background graph 13 | #mu = "0.2" # mixing parameter 14 | islocal = "false" # if "true" mixing parameter is restricted to local cluster, otherwise it is global 15 | isCL = "false" # if "false" use configuration model, if "true" use Chung-Lu 16 | degreefile = "./deg.dat" # name of file do generate that contains vertex degrees 17 | communitysizesfile = "./cs.dat" # name of file do generate that contains community sizes 18 | communityfile = "./com.dat" # name of file do generate that contains assignments of vertices to communities 19 | networkfile = "./edge.dat" # name of file do generate that contains edges of the generated graph 20 | nout = "50" # number of vertices in graph that are outliers; optional parameter 21 | # if nout is passed and is not zero then we require islocal = "false", 22 | # isCL = "false", and xi (not mu) must be passed 23 | # if nout > 0 then it is recommended that xi > 0 24 | -------------------------------------------------------------------------------- /Data/ABCDo/Noise_50/config.toml: -------------------------------------------------------------------------------- 1 | seed = "42" # RNG seed, use "" for no seeding 2 | n = "1000" # number of vertices in graph 3 | t1 = "2.5" # power-law exponent for degree distribution 4 | d_min = "5" # minimum degree 5 | d_max = "50" # maximum degree 6 | d_max_iter = "1000" # maximum number of iterations for sampling degrees 7 | t2 = "1.5" # power-law exponent for cluster size distribution 8 | c_min = "50" # minimum cluster size 9 | c_max = "200" # maximum cluster size 10 | c_max_iter = "1000" # maximum number of iterations for sampling cluster sizes 11 | # Exactly one of xi and mu must be passed as Float64. Also if xi is provided islocal must be set to false or omitted. 12 | xi = "0.5" # fraction of edges to fall in background graph 13 | #mu = "0.2" # mixing parameter 14 | islocal = "false" # if "true" mixing parameter is restricted to local cluster, otherwise it is global 15 | isCL = "false" # if "false" use configuration model, if "true" use Chung-Lu 16 | degreefile = "./deg.dat" # name of file do generate that contains vertex degrees 17 | communitysizesfile = "./cs.dat" # name of file do generate that contains community sizes 18 | communityfile = "./com.dat" # name of file do generate that contains assignments of vertices to communities 19 | networkfile = "./edge.dat" # name of file do generate that contains edges of the generated graph 20 | nout = "50" # number of vertices in graph that are outliers; optional parameter 21 | # if nout is passed and is not zero then we require islocal = "false", 22 | # isCL = "false", and xi (not mu) must be passed 23 | # if nout > 0 then it is recommended that xi > 0 24 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Graph Partition and Measures 2 | 3 | Python3 code implementing 11 graph-aware measures (**gam**) for comparing graph partitions as well as a stable ensemble-based graph partition algorithm (**ECG**). 4 | This code is pip installable for **igraph**, **networkx** and **sknetwork**: 5 | 6 | * PyPI (igraph): https://pypi.org/project/partition-igraph/ 7 | * PyPI (networkx): https://pypi.org/project/partition-networkx/ 8 | * PyPI (sknetwork): https://pypi.org/project/partition-sknetwork/ 9 | 10 | Illustrative examples can be found in the following supplied notebooks: 11 | * [example_with_igraph](https://github.com/ftheberge/graph-partition-and-measures/blob/master/example_with_igraph.ipynb) 12 | * [example_with_networkx](https://github.com/ftheberge/graph-partition-and-measures/blob/master/example_with_networkx.ipynb) 13 | * [example_with_sknetwork](https://github.com/ftheberge/graph-partition-and-measures/blob/master/example_with_sknetwork.ipynb) 14 | 15 | ## Graph aware measures (gam) 16 | 17 | The measures are respectively: 18 | * 'rand': the RAND index 19 | * 'jaccard': the Jaccard index 20 | * 'mn': pairwise similarity normalized with the mean function 21 | * 'gmn': pairwise similarity normalized with the geometric mean function 22 | * 'min': pairwise similarity normalized with the minimum function 23 | * 'max': pairwise similarity normalized with the maximum function 24 | 25 | Each measure can be adjusted (recommended) or not, except for 'jaccard'. 26 | Details can be found in: 27 | 28 | * V. Poulin and F. Theberge, "Comparing Graph Clusterings: Set partition measures vs. Graph-aware measures," in IEEE Transactions on Pattern Analysis and Machine Intelligence, https://doi.org/10.1109/TPAMI.2020.3009862. Pre-print: https://arxiv.org/abs/1806.11494 29 | 30 | ## Ensemble clustering for graphs (ECG) 31 | 32 | This is a good, stable graph partitioning algorithm. Description and applications of ECG can be found in: 33 | 34 | * Valérie Poulin and François Théberge, Ensemble clustering for graphs: comparisons and applications, Network Science (2019) 4:51 https://doi.org/10.1007/s41109-019-0162-z or https://rdcu.be/bLn9i. Pre-print: https://arxiv.org/abs/1903.08012 35 | * Valérie Poulin and François Théberge, Ensemble Clustering for Graphs. in: Aiello L., Cherifi C., Cherifi H., Lambiotte R., Lió P., Rocha L. (eds) Complex Networks and Their Applications VII. COMPLEX NETWORKS 2018. Studies in Computational Intelligence, vol 812. Springer (2019), https://doi.org/10.1007/978-3-030-05411-3_19. Pre-print: https://arxiv.org/abs/1809.05578 36 | 37 | ### new experiment 38 | 39 | We added a new experiment over SBM (stochastic block model) graphs, and comparing the results of several clustering algorithms, including ECG, with respect to the **spectral detectability threshold**. Results are summmarized in this [wiki](https://github.com/ftheberge/graph-partition-and-measures/wiki/Spectral-Threshold-Experiment) and code to run this experiment is provided in this [notebook](https://github.com/ftheberge/graph-partition-and-measures/blob/master/ECG_spectral_threshold.ipynb). 40 | 41 | ## ECG Extras 42 | 43 | Beside providing a good, stable graph clustering method, ECG can be useful for a few other tasks: 44 | 45 | * We can define some **refuse to cluster scores** to rank the nodes in decreasing order, from the most likely to be an outlier (i.e. not part of a community) to the least likely. This can be useful for tasks such as **outlier detection**, but also to improve the **robustness** of the clustering results by leaving out nodes that are not stongly memeber of any community. 46 | * Another use for the derived ECG edge weights is to obtain better **layouts** for community graphs. 47 | * ECG also returns a Community strength indidcator (CSI), where values close to 1 are indicative of strong communities in the graph. 48 | 49 | Those *extra* features are illustrated in the supplied notebook: [ECG_extras](https://github.com/ftheberge/graph-partition-and-measures/blob/master/ECG_extras.ipynb), 50 | as well as in this [wiki](https://github.com/ftheberge/graph-partition-and-measures/wiki/ECG-Extras). 51 | 52 | 53 | -------------------------------------------------------------------------------- /example_with_sknetwork.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "code", 5 | "execution_count": 2, 6 | "id": "af2b1a5e", 7 | "metadata": {}, 8 | "outputs": [], 9 | "source": [ 10 | "import numpy as np\n", 11 | "from sklearn.metrics import adjusted_rand_score as ARI\n", 12 | "\n", 13 | "## pip install partition-sknetwork\n", 14 | "import sknetwork as sn\n", 15 | "import partition_sknetwork as ps" 16 | ] 17 | }, 18 | { 19 | "cell_type": "markdown", 20 | "id": "ebf4b115", 21 | "metadata": {}, 22 | "source": [ 23 | "## Simple block model graph\n", 24 | "\n", 25 | "We generate a simple 1000-node graph with 10 communities of expected size 100 where\n", 26 | "* p_in = 0.1, edge probability for pairs of nodes in the same community\n", 27 | "* p_out = 0.025, edge probability for pairs of nodes in different communities" 28 | ] 29 | }, 30 | { 31 | "cell_type": "code", 32 | "execution_count": 3, 33 | "id": "b62f4050", 34 | "metadata": {}, 35 | "outputs": [], 36 | "source": [ 37 | "block_sizes = [100 for _ in range(10)]\n", 38 | "g = sn.data.models.block_model(block_sizes, 0.1, 0.025, seed=42)\n", 39 | "\n", 40 | "# Store the ground truth communities\n", 41 | "labels = np.array([i for i,block_size in enumerate(block_sizes) for _ in range(block_size)])" 42 | ] 43 | }, 44 | { 45 | "cell_type": "markdown", 46 | "id": "bb60aa14", 47 | "metadata": {}, 48 | "source": [ 49 | "## Example using partition_sknetwork\n" 50 | ] 51 | }, 52 | { 53 | "cell_type": "code", 54 | "execution_count": 4, 55 | "id": "22958e1b", 56 | "metadata": {}, 57 | "outputs": [], 58 | "source": [ 59 | "## run Louvain and ECG:\n", 60 | "louvain = sn.clustering.Louvain(shuffle_nodes=True, random_state=42).fit_predict(g)\n", 61 | "ecg = ps.ECG(random_state=42).fit_predict(g)" 62 | ] 63 | }, 64 | { 65 | "cell_type": "code", 66 | "execution_count": 5, 67 | "id": "a3e2f36d", 68 | "metadata": {}, 69 | "outputs": [ 70 | { 71 | "name": "stdout", 72 | "output_type": "stream", 73 | "text": [ 74 | "Modularity with Louvain: 0.15900220102498056\n", 75 | "Modularity with ECG: 0.20854700246394825\n" 76 | ] 77 | } 78 | ], 79 | "source": [ 80 | "print('Modularity with Louvain:',sn.clustering.get_modularity(g, louvain))\n", 81 | "print('Modularity with ECG:',sn.clustering.get_modularity(g, ecg))" 82 | ] 83 | }, 84 | { 85 | "cell_type": "code", 86 | "execution_count": 6, 87 | "id": "88510eac", 88 | "metadata": {}, 89 | "outputs": [ 90 | { 91 | "name": "stdout", 92 | "output_type": "stream", 93 | "text": [ 94 | "Adjusted Graph-Aware Rand Index for Louvain: 0.18624818643002058\n", 95 | "Adjusted Graph-Aware Rand Index for ECG: 0.8406463276687145\n", 96 | "\n", 97 | "Jaccard Graph-Aware for Louvain: 0.2759559300064809\n", 98 | "Jaccard Graph-Aware for ECG: 0.8015787585217079\n" 99 | ] 100 | } 101 | ], 102 | "source": [ 103 | "## compute some graph-aware measure given ground truth communities\n", 104 | "print('Adjusted Graph-Aware Rand Index for Louvain:',ps.gam(g, labels, louvain))\n", 105 | "print('Adjusted Graph-Aware Rand Index for ECG:',ps.gam(g, labels, ecg))\n", 106 | "\n", 107 | "print('\\nJaccard Graph-Aware for Louvain:',ps.gam(g, labels, louvain, method=\"jaccard\", adjusted=False))\n", 108 | "print('Jaccard Graph-Aware for ECG:',ps.gam(g, labels, ecg, method=\"jaccard\", adjusted=False))" 109 | ] 110 | }, 111 | { 112 | "cell_type": "code", 113 | "execution_count": 7, 114 | "id": "b3ad1200", 115 | "metadata": {}, 116 | "outputs": [ 117 | { 118 | "name": "stdout", 119 | "output_type": "stream", 120 | "text": [ 121 | "Adjusted non-Graph-Aware Rand Index for Louvain: 0.1394572171773798\n", 122 | "Adjusted non-Graph-Aware Rand Index for ecg: 0.7958650479244012\n" 123 | ] 124 | } 125 | ], 126 | "source": [ 127 | "## compute the adjusted RAND index \n", 128 | "print(\"Adjusted non-Graph-Aware Rand Index for Louvain:\",ARI(labels, louvain))\n", 129 | "print(\"Adjusted non-Graph-Aware Rand Index for ecg:\",ARI(labels, ecg))" 130 | ] 131 | } 132 | ], 133 | "metadata": { 134 | "kernelspec": { 135 | "display_name": "complexnetworks", 136 | "language": "python", 137 | "name": "complexnetworks" 138 | }, 139 | "language_info": { 140 | "codemirror_mode": { 141 | "name": "ipython", 142 | "version": 3 143 | }, 144 | "file_extension": ".py", 145 | "mimetype": "text/x-python", 146 | "name": "python", 147 | "nbconvert_exporter": "python", 148 | "pygments_lexer": "ipython3", 149 | "version": "3.12.11" 150 | } 151 | }, 152 | "nbformat": 4, 153 | "nbformat_minor": 5 154 | } 155 | -------------------------------------------------------------------------------- /example_with_igraph.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "code", 5 | "execution_count": 1, 6 | "metadata": {}, 7 | "outputs": [], 8 | "source": [ 9 | "import numpy as np\n", 10 | "from sklearn.metrics import adjusted_rand_score as ARI\n", 11 | "\n", 12 | "## pip install partition-igraph\n", 13 | "import igraph as ig\n", 14 | "import partition_igraph\n" 15 | ] 16 | }, 17 | { 18 | "cell_type": "markdown", 19 | "metadata": {}, 20 | "source": [ 21 | "## Simple block model graph\n", 22 | "\n", 23 | "We generate a simple 1000-node graph with 10 communities of expected size 100 where\n", 24 | "* p_in = 0.1, edge probability for pairs of nodes in the same community\n", 25 | "* p_out = 0.025, edge probability for pairs of nodes in different communities\n" 26 | ] 27 | }, 28 | { 29 | "cell_type": "code", 30 | "execution_count": 2, 31 | "metadata": {}, 32 | "outputs": [], 33 | "source": [ 34 | "# Graph generation with 10 communities of expected size 100\n", 35 | "P = np.full((10,10),.025)\n", 36 | "np.fill_diagonal(P,.1)\n", 37 | "g = ig.Graph.Preference(n=1000, type_dist=list(np.repeat(.1,10)),\n", 38 | " pref_matrix=P.tolist(),attribute='class')\n", 39 | "## the 'ground-truth' communities\n", 40 | "tc = {k:v for k,v in enumerate(g.vs['class'])}" 41 | ] 42 | }, 43 | { 44 | "cell_type": "markdown", 45 | "metadata": {}, 46 | "source": [ 47 | "## Example using partition_igraph\n" 48 | ] 49 | }, 50 | { 51 | "cell_type": "code", 52 | "execution_count": 3, 53 | "metadata": {}, 54 | "outputs": [], 55 | "source": [ 56 | "## run Louvain and ECG (with Leiden):\n", 57 | "Louvain = g.community_multilevel()\n", 58 | "Ecg = g.community_ecg(ens_size=16, final='leiden', resolution=1.0)\n" 59 | ] 60 | }, 61 | { 62 | "cell_type": "code", 63 | "execution_count": 4, 64 | "metadata": {}, 65 | "outputs": [ 66 | { 67 | "name": "stdout", 68 | "output_type": "stream", 69 | "text": [ 70 | "Modularity with Louvain: 0.19438468869260003\n", 71 | "Modularity with ECG: 0.21222510890778576\n" 72 | ] 73 | } 74 | ], 75 | "source": [ 76 | "## modularity (w.r.t. original weights for ECG)\n", 77 | "print('Modularity with Louvain:',Louvain.modularity)\n", 78 | "print('Modularity with ECG:',Ecg.modularity)\n" 79 | ] 80 | }, 81 | { 82 | "cell_type": "code", 83 | "execution_count": 5, 84 | "metadata": {}, 85 | "outputs": [ 86 | { 87 | "name": "stdout", 88 | "output_type": "stream", 89 | "text": [ 90 | "Adjusted Graph-Aware Rand Index for Louvain: 0.5838260201782259\n", 91 | "Adjusted Graph-Aware Rand Index for ECG: 0.8300281824636697\n", 92 | "\n", 93 | "Jaccard Graph-Aware for Louvain: 0.5628111273792094\n", 94 | "Jaccard Graph-Aware for ECG: 0.7907469539113544\n" 95 | ] 96 | } 97 | ], 98 | "source": [ 99 | "## compute some graph-aware measure given ground truth communities\n", 100 | "\n", 101 | "# for 'gam' partition are either 'igraph.clustering.VertexClustering' or 'dict'\n", 102 | "print('Adjusted Graph-Aware Rand Index for Louvain:',g.gam(Louvain, tc))\n", 103 | "print('Adjusted Graph-Aware Rand Index for ECG:',g.gam(Ecg, tc))\n", 104 | "\n", 105 | "print('\\nJaccard Graph-Aware for Louvain:',g.gam(Louvain, tc, method=\"jaccard\", adjusted=False))\n", 106 | "print('Jaccard Graph-Aware for ECG:',g.gam(Ecg, tc, method=\"jaccard\", adjusted=False))\n" 107 | ] 108 | }, 109 | { 110 | "cell_type": "code", 111 | "execution_count": 6, 112 | "metadata": {}, 113 | "outputs": [ 114 | { 115 | "name": "stdout", 116 | "output_type": "stream", 117 | "text": [ 118 | "Adjusted non-Graph-Aware Rand Index for Louvain: 0.45885976004310497\n", 119 | "Adjusted non-Graph-Aware Rand Index for ecg: 0.7782542844437367\n" 120 | ] 121 | } 122 | ], 123 | "source": [ 124 | "## compute the adjusted RAND index \n", 125 | "print(\"Adjusted non-Graph-Aware Rand Index for Louvain:\",ARI(g.vs['class'], Louvain.membership))\n", 126 | "print(\"Adjusted non-Graph-Aware Rand Index for ecg:\",ARI(g.vs['class'], Ecg.membership))\n" 127 | ] 128 | } 129 | ], 130 | "metadata": { 131 | "kernelspec": { 132 | "display_name": "graphmining", 133 | "language": "python", 134 | "name": "graphmining" 135 | }, 136 | "language_info": { 137 | "codemirror_mode": { 138 | "name": "ipython", 139 | "version": 3 140 | }, 141 | "file_extension": ".py", 142 | "mimetype": "text/x-python", 143 | "name": "python", 144 | "nbconvert_exporter": "python", 145 | "pygments_lexer": "ipython3", 146 | "version": "3.10.9" 147 | } 148 | }, 149 | "nbformat": 4, 150 | "nbformat_minor": 2 151 | } 152 | -------------------------------------------------------------------------------- /example_with_networkx.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "code", 5 | "execution_count": 1, 6 | "metadata": {}, 7 | "outputs": [], 8 | "source": [ 9 | "import numpy as np\n", 10 | "from sklearn.metrics import adjusted_rand_score as ARI\n", 11 | "\n", 12 | "## pip install partition-networkx\n", 13 | "## also 'pip install python-louvain' to get 'community' \n", 14 | "import networkx as nx\n", 15 | "import partition_networkx\n", 16 | "import community\n" 17 | ] 18 | }, 19 | { 20 | "cell_type": "markdown", 21 | "metadata": {}, 22 | "source": [ 23 | "## Simple block model graph\n", 24 | "\n", 25 | "We generate a simple 1000-node graph with 10 communities of size 100 where\n", 26 | "* p_in = 0.1, edge probability for pairs of nodes in the same community\n", 27 | "* p_out = 0.025, edge probability for pairs of nodes in different communities\n" 28 | ] 29 | }, 30 | { 31 | "cell_type": "code", 32 | "execution_count": 2, 33 | "metadata": {}, 34 | "outputs": [], 35 | "source": [ 36 | "# Graph generation with 10 communities of size 100\n", 37 | "commSize = 100\n", 38 | "numComm = 10\n", 39 | "g = nx.generators.planted_partition_graph(l=numComm, k=commSize, p_in=0.1, p_out=0.025)\n", 40 | "\n", 41 | "## store groud truth communities as 'iterables of sets of vertices'\n", 42 | "true_comm = [set(list(range(commSize*i, commSize*(i+1)))) for i in range(numComm)]\n" 43 | ] 44 | }, 45 | { 46 | "cell_type": "markdown", 47 | "metadata": {}, 48 | "source": [ 49 | "## Generate partitions with Louvain and ECG" 50 | ] 51 | }, 52 | { 53 | "cell_type": "code", 54 | "execution_count": 3, 55 | "metadata": {}, 56 | "outputs": [], 57 | "source": [ 58 | "## run Louvain and ECG:\n", 59 | "Louvain = community.best_partition(g)\n", 60 | "Ecg = community.ecg(g, ens_size=16, resolution=1.0)" 61 | ] 62 | }, 63 | { 64 | "cell_type": "code", 65 | "execution_count": 4, 66 | "metadata": {}, 67 | "outputs": [ 68 | { 69 | "name": "stdout", 70 | "output_type": "stream", 71 | "text": [ 72 | "Modularity with Louvain: 0.16761826944320063\n", 73 | "Modularity with ECG: 0.2022198878300814\n" 74 | ] 75 | } 76 | ], 77 | "source": [ 78 | "## modularity (w.r.t. original weights for ECG)\n", 79 | "print('Modularity with Louvain:',community.modularity(Louvain, g))\n", 80 | "print('Modularity with ECG:',community.modularity(Ecg.partition, g))" 81 | ] 82 | }, 83 | { 84 | "cell_type": "code", 85 | "execution_count": 5, 86 | "metadata": {}, 87 | "outputs": [ 88 | { 89 | "name": "stdout", 90 | "output_type": "stream", 91 | "text": [ 92 | "Adjusted Graph-Aware Rand Index for Louvain: 0.1633348341602583\n", 93 | "Adjusted Graph-Aware Rand Index for ECG: 0.7066922428045695\n", 94 | "\n", 95 | "Jaccard Graph-Aware for Louvain: 0.26434583014537105\n", 96 | "Jaccard Graph-Aware for ECG: 0.6664522354454808\n" 97 | ] 98 | } 99 | ], 100 | "source": [ 101 | "## compute some graph-aware measure given ground truth communities\n", 102 | "\n", 103 | "# for 'gam' partition are either iterables of sets of vertices or 'dict'\n", 104 | "print(\"Adjusted Graph-Aware Rand Index for Louvain:\",g.gam(true_comm, Louvain))\n", 105 | "print(\"Adjusted Graph-Aware Rand Index for ECG:\",g.gam(true_comm, Ecg.partition))\n", 106 | "\n", 107 | "print(\"\\nJaccard Graph-Aware for Louvain:\",g.gam(true_comm, Louvain, method=\"jaccard\",adjusted=False))\n", 108 | "print(\"Jaccard Graph-Aware for ECG:\",g.gam(true_comm, Ecg.partition, method=\"jaccard\",adjusted=False))\n" 109 | ] 110 | }, 111 | { 112 | "cell_type": "code", 113 | "execution_count": 6, 114 | "metadata": {}, 115 | "outputs": [ 116 | { 117 | "name": "stdout", 118 | "output_type": "stream", 119 | "text": [ 120 | "Adjusted non-Graph-Aware Rand Index for Louvain: 0.1262778991500082\n", 121 | "Adjusted non-Graph-Aware Rand Index for ecg: 0.600695798951731\n" 122 | ] 123 | } 124 | ], 125 | "source": [ 126 | "## compute the adjusted RAND index \n", 127 | "# it requires iterables over the vertices:\n", 128 | "tc = {val:idx for idx,part in enumerate(true_comm) for val in part}\n", 129 | "# compute ARI\n", 130 | "print(\"Adjusted non-Graph-Aware Rand Index for Louvain:\",ARI(list(tc.values()), list(Louvain.values())))\n", 131 | "print(\"Adjusted non-Graph-Aware Rand Index for ecg:\",ARI(list(tc.values()), list(Ecg.partition.values())))\n" 132 | ] 133 | } 134 | ], 135 | "metadata": { 136 | "kernelspec": { 137 | "display_name": "graphmining", 138 | "language": "python", 139 | "name": "graphmining" 140 | }, 141 | "language_info": { 142 | "codemirror_mode": { 143 | "name": "ipython", 144 | "version": 3 145 | }, 146 | "file_extension": ".py", 147 | "mimetype": "text/x-python", 148 | "name": "python", 149 | "nbconvert_exporter": "python", 150 | "pygments_lexer": "ipython3", 151 | "version": "3.10.9" 152 | } 153 | }, 154 | "nbformat": 4, 155 | "nbformat_minor": 2 156 | } 157 | -------------------------------------------------------------------------------- /partition_networkx.py: -------------------------------------------------------------------------------- 1 | # coding=utf-8 2 | import numpy as np 3 | import networkx 4 | 5 | ## Graph-aware measures (igraph version) 6 | def gam(self, u, v, method="rand", adjusted=True): 7 | """ 8 | Compute one of 11 graph-aware measures to compare graph partitions. 9 | 10 | Parameters 11 | ---------- 12 | self: object of type 'networkx.classes.graph.Graph' 13 | Graph on which the partitions are defined. 14 | 15 | u: iterable of sets of nodes in 'self' where each set of node is a community, or a dictionary of node:community 16 | 17 | v: iterable of sets of nodes in 'self' where each set of node is a community, or a dictionary of node:community 18 | 19 | method: str 20 | one of 'rand', 'jaccard', 'mn', 'gmn', 'min' or 'max' 21 | 22 | adjusted: bool 23 | if True, return adjusted measure (preferred). All measures can be adjusted except 'jaccard'. 24 | 25 | Returns 26 | ------- 27 | A graph-aware similarity measure between vertex partitions u and v. 28 | 29 | Examples 30 | -------- 31 | >>> import networkx as nx 32 | >>> import GAM_networkx 33 | >>> ## two complete graphs connectedby a path 34 | >>> g = nx.barbell_graph(10,3) 35 | >>> ## Girvan-Newman returns a sequence of partitions 36 | >>> gn = list(nx.algorithms.community.girvan_newman(g)) 37 | >>> ## compare the partitions with 2 or 3 parts 38 | >>> g.GAM(gn[0], gn[1], method='rand', adjusted=True) 39 | 40 | Reference 41 | --------- 42 | Valérie Poulin and François Théberge, "Comparing Graph Clusterings: Set partition measures vs. Graph-aware measures", https://arxiv.org/abs/1806.11494. 43 | """ 44 | if(type(u) is dict): 45 | d1 = u 46 | else: 47 | d1 = {val:idx for idx,part in enumerate(u) for val in part} 48 | if(type(v) is dict): 49 | d2 = v 50 | else: 51 | d2 = {val:idx for idx,part in enumerate(v) for val in part} 52 | bu = np.array([d1[e[0]] == d1[e[1]] for e in self.edges()]) 53 | bv = np.array([d2[e[0]] == d2[e[1]] for e in self.edges()]) 54 | su = np.sum(bu) 55 | sv = np.sum(bv) 56 | suv = np.sum(bu*bv) 57 | m = len(bu) 58 | ## all adjusted measures 59 | if adjusted: 60 | if method=="jaccard": 61 | print("no adjusted jaccard measure, set adjusted=False") 62 | return None 63 | if method=="rand" or method=="mn": 64 | return((suv-su*sv/m)/(np.average([su,sv])-su*sv/m)) 65 | if method=="gmn": 66 | return((suv-su*sv/m)/(np.sqrt(su*sv)-su*sv/m)) 67 | if method=="min": 68 | return((suv-su*sv/m)/(np.min([su,sv])-su*sv/m)) 69 | if method=="max": 70 | return((suv-su*sv/m)/(np.max([su,sv])-su*sv/m)) 71 | else: 72 | print('Wrong method!') 73 | 74 | ## all non-adjusted measures 75 | else: 76 | if method=="jaccard": 77 | union_b = sum((bu+bv)>0) 78 | return(suv/union_b) 79 | if method=="rand": 80 | return(1-(su+sv)/m+2*suv/m) 81 | if method=="mn": 82 | return(suv/np.average([su,sv])) 83 | if method=="gmn": 84 | return(suv/np.sqrt(su*sv)) 85 | if method=="min": 86 | return(suv/np.min([su,sv])) 87 | if method=="max": 88 | return(suv/np.max([su,sv])) 89 | else: 90 | print('Wrong method!') 91 | 92 | return None 93 | 94 | networkx.classes.graph.Graph.gam = gam 95 | 96 | from networkx.algorithms.core import core_number 97 | import community 98 | from collections import namedtuple 99 | 100 | def community_ecg(self, weight='weight', ens_size = 16, min_weight = 0.05, resolution=1.0): 101 | """ 102 | Stable ensemble-based graph clustering; 103 | the ensemble consists of single-level randomized Louvain; 104 | each member of the ensemble gets a "vote" to determine if the edges 105 | are intra-community or not; 106 | the votes are aggregated into ECG edge-weights in range [0,1]; 107 | a final (full depth) Louvain is run using those edge weights; 108 | 109 | Parameters 110 | ---------- 111 | self: graph of type 'networkx.classes.graph.Graph' 112 | Graph to define the partition on. 113 | weight : str, optional 114 | the key in graph to use as weight. Default to 'weight' 115 | ens_size: int, optional 116 | the size of the ensemble of single-level Louvain 117 | min_weight: float in range [0,1], optional 118 | the ECG edge weight for edges with zero votes from the ensemble 119 | resolution: positive float, optional 120 | resolution parameter; larger values favors smaller communities 121 | 122 | Returns 123 | ------- 124 | an object of type 'partition_networkx.Partition' with: 125 | 126 | Partition.partition: 127 | The final partition as a dictionary on the vertices 128 | Partition.W 129 | The ECG edge weights s a dictionary on the edges 130 | Partition.CSI 131 | The community strength index (float) 132 | 133 | Notes 134 | ----- 135 | The ECG edge weight function is defined as: 136 | 137 | min_weight + ( 1 - min_weight ) x (#votes_in_ensemble) / ens_size 138 | 139 | Edges outside the 2-core are assigned 'min_weight'. 140 | 141 | Examples 142 | -------- 143 | >>> g = nx.karate_club_graph() 144 | >>> P = community.ecg(g) 145 | >>> part = P.partition 146 | >>> print(P.CSI) 147 | 148 | Reference 149 | --------- 150 | Valérie Poulin and François Théberge, "Ensemble clustering for graphs: comparisons and applications", Appl Netw Sci 4, 51 (2019). 151 | https://doi.org/10.1007/s41109-019-0162-z 152 | """ 153 | W = {k:0 for k in self.edges()} 154 | ## Ensemble of level-1 Louvain 155 | for i in range(ens_size): 156 | d = community.generate_dendrogram(self, weight=weight, randomize=True) 157 | l = community.partition_at_level(d,0) 158 | for e in self.edges(): 159 | W[e] += int(l[e[0]] == l[e[1]]) 160 | ## vertex core numbers 161 | core = core_number(self) 162 | ## set edge weights 163 | for e in self.edges(): 164 | m = min(core[e[0]],core[e[1]]) 165 | if m > 1: 166 | W[e] = min_weight + (1-min_weight)*W[e]/ens_size 167 | else: 168 | W[e] = min_weight 169 | 170 | networkx.set_edge_attributes(self, W, 'W') 171 | part = community.best_partition(self, weight='W', resolution=resolution) 172 | P = namedtuple('Partition', ['partition', 'W', 'CSI']) 173 | w = list(W.values()) 174 | CSI = 1-2*np.sum([min(1-i,i) for i in w])/len(w) 175 | p = P(part,W,CSI) 176 | return p 177 | 178 | community.ecg = community_ecg 179 | 180 | -------------------------------------------------------------------------------- /Data/Football/football.edgelist: -------------------------------------------------------------------------------- 1 | 0 1 2 | 2 3 3 | 0 4 4 | 4 5 5 | 3 5 6 | 2 6 7 | 6 7 8 | 7 8 9 | 8 9 10 | 0 9 11 | 4 9 12 | 5 10 13 | 10 11 14 | 5 11 15 | 3 11 16 | 12 13 17 | 2 13 18 | 2 14 19 | 12 14 20 | 14 15 21 | 13 15 22 | 2 15 23 | 4 16 24 | 9 16 25 | 0 16 26 | 16 17 27 | 12 17 28 | 12 18 29 | 18 19 30 | 17 20 31 | 20 21 32 | 8 21 33 | 7 21 34 | 9 22 35 | 7 22 36 | 21 22 37 | 8 22 38 | 22 23 39 | 9 23 40 | 4 23 41 | 16 23 42 | 0 23 43 | 11 24 44 | 24 25 45 | 1 25 46 | 3 26 47 | 12 26 48 | 14 26 49 | 26 27 50 | 17 27 51 | 1 27 52 | 4 28 53 | 11 28 54 | 24 28 55 | 19 29 56 | 29 30 57 | 19 30 58 | 18 31 59 | 31 32 60 | 21 32 61 | 15 32 62 | 13 32 63 | 6 32 64 | 0 33 65 | 1 33 66 | 25 33 67 | 19 33 68 | 31 34 69 | 26 34 70 | 12 34 71 | 18 34 72 | 34 35 73 | 0 35 74 | 29 35 75 | 19 35 76 | 30 35 77 | 18 36 78 | 12 36 79 | 20 36 80 | 19 36 81 | 36 37 82 | 1 37 83 | 25 37 84 | 33 37 85 | 18 38 86 | 16 38 87 | 28 38 88 | 26 38 89 | 14 38 90 | 12 38 91 | 38 39 92 | 6 39 93 | 32 39 94 | 13 39 95 | 15 39 96 | 7 40 97 | 3 40 98 | 40 41 99 | 8 41 100 | 4 41 101 | 23 41 102 | 9 41 103 | 0 41 104 | 16 41 105 | 34 42 106 | 29 42 107 | 18 42 108 | 26 42 109 | 42 43 110 | 36 43 111 | 26 43 112 | 31 43 113 | 38 43 114 | 12 43 115 | 14 43 116 | 19 44 117 | 35 44 118 | 30 44 119 | 44 45 120 | 13 45 121 | 33 45 122 | 1 45 123 | 37 45 124 | 25 45 125 | 21 46 126 | 46 47 127 | 22 47 128 | 6 47 129 | 15 47 130 | 2 47 131 | 39 47 132 | 32 47 133 | 44 48 134 | 48 49 135 | 32 49 136 | 46 49 137 | 30 50 138 | 24 50 139 | 11 50 140 | 28 50 141 | 50 51 142 | 40 51 143 | 8 51 144 | 22 51 145 | 21 51 146 | 3 52 147 | 40 52 148 | 5 52 149 | 52 53 150 | 25 53 151 | 48 53 152 | 49 53 153 | 46 53 154 | 39 54 155 | 31 54 156 | 38 54 157 | 14 54 158 | 34 54 159 | 18 54 160 | 54 55 161 | 31 55 162 | 6 55 163 | 35 55 164 | 29 55 165 | 19 55 166 | 30 55 167 | 27 56 168 | 56 57 169 | 1 57 170 | 42 57 171 | 44 57 172 | 48 57 173 | 3 58 174 | 6 58 175 | 17 58 176 | 36 58 177 | 36 59 178 | 58 59 179 | 59 60 180 | 10 60 181 | 39 60 182 | 6 60 183 | 47 60 184 | 13 60 185 | 15 60 186 | 2 60 187 | 43 61 188 | 47 61 189 | 54 61 190 | 18 61 191 | 26 61 192 | 31 61 193 | 34 61 194 | 61 62 195 | 20 62 196 | 45 62 197 | 17 62 198 | 27 62 199 | 56 62 200 | 27 63 201 | 58 63 202 | 59 63 203 | 42 63 204 | 63 64 205 | 9 64 206 | 32 64 207 | 60 64 208 | 2 64 209 | 6 64 210 | 47 64 211 | 13 64 212 | 0 65 213 | 27 65 214 | 17 65 215 | 63 65 216 | 56 65 217 | 20 65 218 | 65 66 219 | 59 66 220 | 24 66 221 | 44 66 222 | 48 66 223 | 16 67 224 | 41 67 225 | 46 67 226 | 53 67 227 | 49 67 228 | 67 68 229 | 15 68 230 | 50 68 231 | 21 68 232 | 51 68 233 | 7 68 234 | 22 68 235 | 8 68 236 | 4 69 237 | 24 69 238 | 28 69 239 | 50 69 240 | 11 69 241 | 69 70 242 | 43 70 243 | 65 70 244 | 20 70 245 | 56 70 246 | 62 70 247 | 27 70 248 | 60 71 249 | 18 71 250 | 14 71 251 | 34 71 252 | 54 71 253 | 38 71 254 | 61 71 255 | 31 71 256 | 71 72 257 | 2 72 258 | 10 72 259 | 3 72 260 | 40 72 261 | 52 72 262 | 7 73 263 | 49 73 264 | 53 73 265 | 67 73 266 | 46 73 267 | 73 74 268 | 2 74 269 | 72 74 270 | 5 74 271 | 10 74 272 | 52 74 273 | 3 74 274 | 40 74 275 | 20 75 276 | 66 75 277 | 48 75 278 | 57 75 279 | 44 75 280 | 75 76 281 | 27 76 282 | 59 76 283 | 20 76 284 | 70 76 285 | 66 76 286 | 56 76 287 | 62 76 288 | 73 77 289 | 22 77 290 | 7 77 291 | 51 77 292 | 21 77 293 | 8 77 294 | 77 78 295 | 23 78 296 | 50 78 297 | 28 78 298 | 22 78 299 | 8 78 300 | 68 78 301 | 7 78 302 | 51 78 303 | 31 79 304 | 43 79 305 | 30 79 306 | 19 79 307 | 29 79 308 | 35 79 309 | 55 79 310 | 79 80 311 | 37 80 312 | 29 80 313 | 16 81 314 | 5 81 315 | 40 81 316 | 10 81 317 | 72 81 318 | 3 81 319 | 81 82 320 | 74 82 321 | 39 82 322 | 77 82 323 | 80 82 324 | 30 82 325 | 29 82 326 | 7 82 327 | 53 83 328 | 81 83 329 | 69 83 330 | 73 83 331 | 46 83 332 | 67 83 333 | 49 83 334 | 83 84 335 | 24 84 336 | 49 84 337 | 52 84 338 | 3 84 339 | 74 84 340 | 10 84 341 | 81 84 342 | 5 84 343 | 6 85 344 | 14 85 345 | 38 85 346 | 43 85 347 | 80 85 348 | 12 85 349 | 26 85 350 | 31 85 351 | 44 86 352 | 53 86 353 | 75 86 354 | 57 86 355 | 48 86 356 | 80 86 357 | 66 86 358 | 86 87 359 | 17 87 360 | 62 87 361 | 56 87 362 | 24 87 363 | 20 87 364 | 65 87 365 | 49 88 366 | 58 88 367 | 83 88 368 | 69 88 369 | 46 88 370 | 53 88 371 | 73 88 372 | 67 88 373 | 88 89 374 | 1 89 375 | 37 89 376 | 25 89 377 | 33 89 378 | 55 89 379 | 45 89 380 | 5 90 381 | 8 90 382 | 23 90 383 | 0 90 384 | 11 90 385 | 50 90 386 | 24 90 387 | 69 90 388 | 28 90 389 | 29 91 390 | 48 91 391 | 66 91 392 | 69 91 393 | 44 91 394 | 86 91 395 | 57 91 396 | 80 91 397 | 91 92 398 | 35 92 399 | 15 92 400 | 86 92 401 | 48 92 402 | 57 92 403 | 61 92 404 | 66 92 405 | 75 92 406 | 0 93 407 | 23 93 408 | 80 93 409 | 16 93 410 | 4 93 411 | 82 93 412 | 91 93 413 | 41 93 414 | 9 93 415 | 34 94 416 | 19 94 417 | 55 94 418 | 79 94 419 | 80 94 420 | 29 94 421 | 30 94 422 | 82 94 423 | 35 94 424 | 70 95 425 | 69 95 426 | 76 95 427 | 62 95 428 | 56 95 429 | 27 95 430 | 17 95 431 | 87 95 432 | 37 95 433 | 48 96 434 | 17 96 435 | 76 96 436 | 27 96 437 | 56 96 438 | 65 96 439 | 20 96 440 | 87 96 441 | 5 97 442 | 86 97 443 | 58 97 444 | 11 97 445 | 59 97 446 | 63 97 447 | 97 98 448 | 77 98 449 | 48 98 450 | 84 98 451 | 40 98 452 | 10 98 453 | 5 98 454 | 52 98 455 | 81 98 456 | 89 99 457 | 34 99 458 | 14 99 459 | 85 99 460 | 54 99 461 | 18 99 462 | 31 99 463 | 61 99 464 | 71 99 465 | 99 100 466 | 82 100 467 | 13 100 468 | 2 100 469 | 15 100 470 | 32 100 471 | 64 100 472 | 47 100 473 | 39 100 474 | 6 100 475 | 51 101 476 | 30 101 477 | 94 101 478 | 1 101 479 | 79 101 480 | 58 101 481 | 19 101 482 | 55 101 483 | 35 101 484 | 29 101 485 | 100 102 486 | 74 102 487 | 52 102 488 | 98 102 489 | 72 102 490 | 40 102 491 | 10 102 492 | 3 102 493 | 102 103 494 | 33 103 495 | 45 103 496 | 25 103 497 | 89 103 498 | 37 103 499 | 1 103 500 | 70 103 501 | 72 104 502 | 11 104 503 | 0 104 504 | 93 104 505 | 67 104 506 | 41 104 507 | 16 104 508 | 87 104 509 | 23 104 510 | 4 104 511 | 9 104 512 | 89 105 513 | 103 105 514 | 33 105 515 | 62 105 516 | 37 105 517 | 45 105 518 | 1 105 519 | 80 105 520 | 25 105 521 | 25 106 522 | 56 106 523 | 92 106 524 | 2 106 525 | 13 106 526 | 32 106 527 | 60 106 528 | 6 106 529 | 64 106 530 | 15 106 531 | 39 106 532 | 88 107 533 | 75 107 534 | 98 107 535 | 102 107 536 | 72 107 537 | 40 107 538 | 81 107 539 | 5 107 540 | 10 107 541 | 84 107 542 | 4 108 543 | 9 108 544 | 7 108 545 | 51 108 546 | 77 108 547 | 21 108 548 | 78 108 549 | 22 108 550 | 68 108 551 | 79 109 552 | 30 109 553 | 63 109 554 | 1 109 555 | 33 109 556 | 103 109 557 | 105 109 558 | 45 109 559 | 25 109 560 | 89 109 561 | 37 109 562 | 67 110 563 | 13 110 564 | 24 110 565 | 80 110 566 | 88 110 567 | 49 110 568 | 73 110 569 | 46 110 570 | 83 110 571 | 53 110 572 | 23 111 573 | 64 111 574 | 46 111 575 | 78 111 576 | 8 111 577 | 21 111 578 | 51 111 579 | 7 111 580 | 108 111 581 | 68 111 582 | 77 111 583 | 52 112 584 | 96 112 585 | 97 112 586 | 57 112 587 | 66 112 588 | 63 112 589 | 44 112 590 | 92 112 591 | 75 112 592 | 91 112 593 | 28 113 594 | 20 113 595 | 95 113 596 | 59 113 597 | 70 113 598 | 17 113 599 | 87 113 600 | 76 113 601 | 65 113 602 | 96 113 603 | 83 114 604 | 88 114 605 | 110 114 606 | 53 114 607 | 49 114 608 | 73 114 609 | 46 114 610 | 67 114 611 | 58 114 612 | 15 114 613 | 104 114 -------------------------------------------------------------------------------- /partition_igraph.py: -------------------------------------------------------------------------------- 1 | # coding=utf-8 2 | import numpy as np 3 | import igraph 4 | 5 | ## Graph-aware measures (igraph version) 6 | def gam(self, u, v, method="rand", adjusted=True): 7 | """ 8 | Compute one of 11 graph-aware measures to compare graph partitions. 9 | 10 | Parameters 11 | ---------- 12 | self: Graph of type 'igraph.Graph' on which the partitions are defined. 13 | 14 | u: Partition of type 'igraph.clustering.VertexClustering' on 'self', or a dictionary of node:community. 15 | 16 | v: Partition of type 'igraph.clustering.VertexClustering' on 'self', or a dictionary of node:community. 17 | 18 | method: 'str' 19 | one of 'rand', 'jaccard', 'mn', 'gmn', 'min' or 'max' 20 | 21 | adjusted: 'bool' 22 | if True, return adjusted measure (preferred). All measures can be adjusted except 'jaccard'. 23 | 24 | Returns 25 | ------- 26 | A graph-aware similarity measure between vertex partitions u and v. 27 | 28 | Examples 29 | -------- 30 | >>> g = ig.Graph.Famous('Zachary') 31 | >>> part1 = g.community_multilevel() 32 | >>> part2 = g.community_label_propagation() 33 | >>> print(g.GAM(part1, part2)) 34 | 35 | Reference 36 | --------- 37 | Valérie Poulin and François Théberge, "Comparing Graph Clusterings: Set partition measures vs. Graph-aware measures", https://arxiv.org/abs/1806.11494. 38 | """ 39 | if(type(u) is dict): 40 | d1 = u 41 | else: 42 | d1 = {val:idx for idx,part in enumerate(u) for val in part} 43 | if(type(v) is dict): 44 | d2 = v 45 | else: 46 | d2 = {val:idx for idx,part in enumerate(v) for val in part} 47 | bu = np.array([(d1[x.tuple[0]]==d1[x.tuple[1]]) for x in self.es]) 48 | bv = np.array([(d2[x.tuple[0]]==d2[x.tuple[1]]) for x in self.es]) 49 | su = np.sum(bu) 50 | sv = np.sum(bv) 51 | suv = np.sum(bu*bv) 52 | m = len(bu) 53 | ## all adjusted measures 54 | if adjusted: 55 | if method=="jaccard": 56 | print("no adjusted jaccard measure, set adjusted=False") 57 | return None 58 | if method=="rand" or method=="mn": 59 | return((suv-su*sv/m)/(np.average([su,sv])-su*sv/m)) 60 | if method=="gmn": 61 | return((suv-su*sv/m)/(np.sqrt(su*sv)-su*sv/m)) 62 | if method=="min": 63 | return((suv-su*sv/m)/(np.min([su,sv])-su*sv/m)) 64 | if method=="max": 65 | return((suv-su*sv/m)/(np.max([su,sv])-su*sv/m)) 66 | else: 67 | print('Wrong method!') 68 | 69 | ## all non-adjusted measures 70 | else: 71 | if method=="jaccard": 72 | union_b = sum((bu+bv)>0) 73 | return(suv/union_b) 74 | if method=="rand": 75 | return(1-(su+sv)/m+2*suv/m) 76 | if method=="mn": 77 | return(suv/np.average([su,sv])) 78 | if method=="gmn": 79 | return(suv/np.sqrt(su*sv)) 80 | if method=="min": 81 | return(suv/np.min([su,sv])) 82 | if method=="max": 83 | return(suv/np.max([su,sv])) 84 | else: 85 | print('Wrong method!') 86 | 87 | return None 88 | 89 | igraph.Graph.gam = gam 90 | 91 | import igraph 92 | import numpy as np 93 | 94 | def community_ecg(self, weights=None, ens_size = 16, min_weight = 0.05, 95 | final='louvain', resolution=1.0, refuse_score=False): 96 | """ 97 | Stable ensemble-based graph clustering; 98 | the ensemble consists of single-level randomized Louvain; 99 | each member of the ensemble gets a "vote" to determine if the edges 100 | are intra-community or not; 101 | the votes are aggregated into ECG edge-weights in range [0,1]; 102 | a final (full depth) Louvain is run using those edge weights; 103 | 104 | Parameters 105 | ---------- 106 | self: graph of type 'igraph.Graph' 107 | Graph to define the partition on. 108 | weights: list of double, optional 109 | the edge weights 110 | ens_size: int, optional 111 | the size of the ensemble of single-level Louvain 112 | min_weight: double in range [0,1], optional 113 | the ECG edge weight for edges with zero votes from the ensemble 114 | final: 'louvain' (default) or 'leiden' 115 | the algorithm to run on the final re-weighted graph 116 | resolution: positive float, optional 117 | resolution parameter; larger values favors smaller communities 118 | 119 | Returns 120 | ------- 121 | partition 122 | The final partition, of type 'igraph.clustering.VertexClustering' 123 | partition.W 124 | The ECG edge weights 125 | partition.CSI 126 | The community strength index 127 | partition.original_modularity 128 | The modularity with respect to the original edge weights 129 | 130 | Notes 131 | ----- 132 | The ECG edge weight function is defined as: 133 | 134 | min_weight + ( 1 - min_weight ) x (#votes_in_ensemble) / ens_size 135 | 136 | Edges outside the 2-core are assigned 'min_weight'. 137 | 138 | Examples 139 | -------- 140 | >>> g = igraph.Graph.Famous('Zachary') 141 | >>> part = g.community_ecg(ens_size=25, min_weight = .1) 142 | >>> print(part.CSI) 143 | 144 | Reference 145 | --------- 146 | Valérie Poulin and François Théberge, "Ensemble clustering for graphs: comparisons and applications", Appl Netw Sci 4, 51 (2019). 147 | https://doi.org/10.1007/s41109-019-0162-z 148 | """ 149 | W = [0]*self.ecount() 150 | ## Ensemble of level-1 Louvain 151 | for i in range(ens_size): 152 | p = np.random.permutation(self.vcount()).tolist() 153 | g = self.permute_vertices(p) 154 | l1 = g.community_multilevel(weights=weights, return_levels=True)[0].membership 155 | b = [l1[p[x.tuple[0]]]==l1[p[x.tuple[1]]] for x in self.es] 156 | W = [W[i]+b[i] for i in range(len(W))] 157 | W = [min_weight + (1-min_weight)*W[i]/ens_size for i in range(len(W))] 158 | ## Force min_weight outside 2-core 159 | core = self.shell_index() 160 | ecore = [min(core[x.tuple[0]],core[x.tuple[1]]) for x in self.es] 161 | w = [W[i] if ecore[i]>1 else min_weight for i in range(len(ecore))] 162 | if final=='leiden': 163 | part = self.community_leiden(weights=w, objective_function='modularity', resolution=resolution) 164 | else: 165 | part = self.community_multilevel(weights=w, resolution=resolution) 166 | part.W = w 167 | part.CSI = 1-2*np.sum([min(1-i,i) for i in w])/len(w) 168 | part._modularity_params['weights'] = weights 169 | part.recalculate_modularity() 170 | 171 | ## experimental - "refuse to cluster" scores 172 | if refuse_score: 173 | self.vs['_deg'] = self.degree() 174 | self.es['_W'] = part.W 175 | self.vs['_ecg'] = part.membership 176 | for v in self.vs: 177 | scr = 0 178 | my_comm = v['_ecg'] 179 | good = 0 180 | bad = 0 181 | for e in v.incident(): 182 | scr += e['_W'] 183 | if self.vs[e.source]['_ecg'] == self.vs[e.target]['_ecg']: 184 | good += e['_W'] 185 | else: 186 | bad += e['_W'] 187 | v['_overall'] = ((v['_deg']-scr)/v['_deg']) 188 | v['_community'] = (bad/(bad+good)) 189 | part.refuse_overall = self.vs['_overall'] 190 | part.refuse_community = self.vs['_community'] 191 | del(self.vs['_deg']) 192 | del(self.es['_W']) 193 | del(self.vs['_ecg']) 194 | del(self.vs['_overall']) 195 | del(self.vs['_community']) 196 | ## end experimental scores 197 | 198 | return part 199 | 200 | igraph.Graph.community_ecg = community_ecg 201 | -------------------------------------------------------------------------------- /Data/ABCDo/Noise_30/deg.dat: -------------------------------------------------------------------------------- 1 | 50 2 | 48 3 | 48 4 | 48 5 | 45 6 | 45 7 | 44 8 | 44 9 | 43 10 | 42 11 | 41 12 | 41 13 | 40 14 | 40 15 | 40 16 | 39 17 | 39 18 | 38 19 | 38 20 | 38 21 | 37 22 | 37 23 | 36 24 | 35 25 | 35 26 | 35 27 | 34 28 | 33 29 | 32 30 | 32 31 | 31 32 | 31 33 | 30 34 | 30 35 | 29 36 | 29 37 | 29 38 | 28 39 | 28 40 | 28 41 | 28 42 | 28 43 | 28 44 | 27 45 | 27 46 | 27 47 | 27 48 | 26 49 | 26 50 | 26 51 | 26 52 | 26 53 | 25 54 | 25 55 | 25 56 | 25 57 | 25 58 | 25 59 | 24 60 | 24 61 | 24 62 | 24 63 | 24 64 | 24 65 | 24 66 | 24 67 | 24 68 | 23 69 | 23 70 | 23 71 | 23 72 | 23 73 | 23 74 | 23 75 | 23 76 | 23 77 | 22 78 | 22 79 | 22 80 | 22 81 | 22 82 | 22 83 | 21 84 | 21 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5 952 | 5 953 | 5 954 | 5 955 | 5 956 | 5 957 | 5 958 | 5 959 | 5 960 | 5 961 | 5 962 | 5 963 | 5 964 | 5 965 | 5 966 | 5 967 | 5 968 | 5 969 | 5 970 | 5 971 | 5 972 | 5 973 | 5 974 | 5 975 | 5 976 | 5 977 | 5 978 | 5 979 | 5 980 | 5 981 | 5 982 | 5 983 | 5 984 | 5 985 | 5 986 | 5 987 | 5 988 | 5 989 | 5 990 | 5 991 | 5 992 | 5 993 | 5 994 | 5 995 | 5 996 | 5 997 | 5 998 | 5 999 | 5 1000 | 5 1001 | -------------------------------------------------------------------------------- /partition_sknetwork.py: -------------------------------------------------------------------------------- 1 | import numpy as np 2 | import numba 3 | import scipy.sparse as sp 4 | import sknetwork as sn 5 | from sknetwork.clustering import BaseClustering, Louvain, Leiden 6 | from sknetwork.utils.check import check_format, check_random_state, get_probs 7 | from sknetwork.topology import get_core_decomposition 8 | 9 | 10 | 11 | @numba.njit 12 | def _internal_edge(g_indptr, g_indices, partition): 13 | is_internal_edge = np.empty(len(g_indices), dtype="bool") 14 | for n1 in range(len(g_indptr)-1): 15 | for data_offset, n2 in enumerate(g_indices[g_indptr[n1]:g_indptr[n1+1]]): 16 | is_internal_edge[g_indptr[n1]+data_offset] = partition[n1] == partition[n2] 17 | return is_internal_edge 18 | 19 | 20 | def gam(g, u, v, method="rand", adjusted=True): 21 | """ 22 | Compute one of 11 graph-aware measures to compare graph partitions. 23 | 24 | Parameters 25 | ---------- 26 | g: adjaceny matrix of the graph on which the partitions are defined. 27 | 28 | u: Partiton of nodes. A numpy array of length n where u[i] = j means node i is in part j. Parts must be labeled 0-n_parts. 29 | 30 | v: Partiton of nodes. A numpy array of length n where u[i] = j means node i is in part j. Parts must be labeled 0-n_parts. 31 | 32 | method: 'str' 33 | one of 'rand', 'jaccard', 'mn', 'gmn', 'min' or 'max' 34 | 35 | adjusted: 'bool' 36 | if True, return adjusted measure (preferred). All measures can be adjusted except 'jaccard'. 37 | 38 | Returns 39 | ------- 40 | float: A graph-aware similarity measure between vertex partitions u and v. 41 | 42 | Examples 43 | -------- 44 | >>> g = sn.data.karate_club() 45 | >>> part1 = sn.clustering.Louvain().fit_predict(g) 46 | >>> part2 = sn.clustering.PropagationClustering().fit_predict(g) 47 | >>> print(gam(g, part1, part2)) 48 | 49 | Reference 50 | --------- 51 | Valérie Poulin and François Théberge, "Comparing Graph Clusterings: Set Partition Measures vs. Graph-aware Measures", 52 | IEEE Transactions on Pattern Analysis and Machine Intelligence 43, 6 (2021) https://doi.org/10.1109/TPAMI.2020.3009862 53 | """ 54 | g = sp.triu(g).tocsr() 55 | bu = _internal_edge(g.indptr, g.indices, u) 56 | bv = _internal_edge(g.indptr, g.indices, v) 57 | su = np.sum(bu) 58 | sv = np.sum(bv) 59 | suv = np.sum(bu*bv) 60 | m = len(bu) 61 | ## all adjusted measures 62 | if adjusted: 63 | if method=="jaccard": 64 | raise ValueError("no adjusted jaccard measure, set adjusted=False") 65 | elif method=="rand" or method=="mn": 66 | return((suv-su*sv/m)/(np.average([su,sv])-su*sv/m)) 67 | elif method=="gmn": 68 | return((suv-su*sv/m)/(np.sqrt(su*sv)-su*sv/m)) 69 | elif method=="min": 70 | return((suv-su*sv/m)/(np.min([su,sv])-su*sv/m)) 71 | elif method=="max": 72 | return((suv-su*sv/m)/(np.max([su,sv])-su*sv/m)) 73 | else: 74 | raise ValueError(f"Method not found. Should be one of ['jaccard', 'rand', 'gmn', 'min', 'max']. Got {method}") 75 | ## all non-adjusted measures 76 | else: 77 | if method=="jaccard": 78 | union_b = np.sum((bu+bv)>0) 79 | return(suv/union_b) 80 | elif method=="rand": 81 | return(1-(su+sv)/m+2*suv/m) 82 | elif method=="mn": 83 | return(suv/np.average([su,sv])) 84 | elif method=="gmn": 85 | return(suv/np.sqrt(su*sv)) 86 | elif method=="min": 87 | return(suv/np.min([su,sv])) 88 | elif method=="max": 89 | return(suv/np.max([su,sv])) 90 | else: 91 | raise ValueError(f"Method not found. Should be one of ['jaccard', 'rand', 'gmn', 'min', 'max']. Got {method}") 92 | 93 | 94 | @numba.njit 95 | def _ecg_weights(g_indptr, g_indices, g_data, partitions): 96 | for n1 in range(len(g_indptr)-1): 97 | for data_offset, n2 in enumerate(g_indices[g_indptr[n1]:g_indptr[n1+1]]): 98 | g_data[g_indptr[n1]+data_offset] = np.sum(partitions[n1, :] == partitions[n2, :]) 99 | return g_data 100 | 101 | 102 | @numba.njit 103 | def _compute_refuse_scores(weights_indptr, weights_indices, weights_data, labels): 104 | overall_refuse = np.empty(len(weights_indptr)-1) 105 | community_refuse = np.empty(len(weights_indptr)-1) 106 | for node in range(len(weights_indptr)-1): 107 | degree = weights_indptr[node+1] - weights_indptr[node] 108 | edge_weights = weights_data[weights_indptr[node]:weights_indptr[node+1]] 109 | weighted_degree = np.sum(edge_weights) 110 | neighbor_in_community = labels[weights_indices[weights_indptr[node]:weights_indptr[node+1]]] == labels[node] 111 | in_community_weighted_degree = np.sum(edge_weights[neighbor_in_community]) 112 | overall_refuse[node] = (degree - weighted_degree)/degree 113 | community_refuse[node] = in_community_weighted_degree / weighted_degree 114 | return overall_refuse, community_refuse 115 | 116 | 117 | class ECG(BaseClustering): 118 | """ 119 | Stable ensemble-based graph clustering; 120 | the ensemble consists of single-level randomized Louvain; 121 | each member of the ensemble gets a "vote" to determine if the edges are intra-community or not; 122 | the votes are aggregated into ECG edge-weights in range [0,1]; 123 | a final (full depth) Leiden is run using those edge weights; 124 | 125 | Parameters 126 | ---------- 127 | resolution : 128 | Resolution parameter. 129 | ens_size : 130 | Number of Louvain runs in the ensemble 131 | min_weight : 132 | Minimum edge weight 133 | sort_clusters : 134 | If ``True``, sort labels in decreasing order of cluster size. 135 | return_probs : 136 | If ``True``, return the probability distribution over clusters (soft clustering). 137 | return_aggregate : 138 | If ``True``, return the adjacency matrix of the graph between clusters. 139 | random_state : 140 | Random number generator or random seed. If None, numpy.random is used. 141 | rng : 142 | numpy Generator object to use. If passed random_state is not used 143 | 144 | Attributes 145 | ---------- 146 | labels_ : np.ndarray, shape (n_labels,) 147 | Label of each node. 148 | probs_ : sparse.csr_matrix, shape (n_row, n_labels) 149 | Probability distribution over labels. 150 | labels_row_, labels_col_ : np.ndarray 151 | Labels of rows and columns, for bipartite graphs. 152 | probs_row_, probs_col_ : sparse.csr_matrix, shape (n_row, n_labels) 153 | Probability distributions over labels for rows and columns (for bipartite graphs). 154 | aggregate_ : sparse.csr_matrix 155 | Aggregate adjacency matrix or biadjacency matrix between clusters. 156 | 157 | Notes 158 | ----- 159 | The ECG edge weight function is defined as: 160 | min_weight + ( 1 - min_weight ) x (#votes_in_ensemble) / ens_size 161 | Edges outside the 2-core are assigned 'min_weight'. 162 | 163 | Example 164 | -------- 165 | >>> g = sn.data.karate_club() 166 | >>> part = ECG().fit_predict(g) 167 | 168 | Reference 169 | --------- 170 | Valérie Poulin and François Théberge, "Ensemble clustering for graphs: comparisons and applications", 171 | Appl Netw Sci 4, 51 (2019). https://doi.org/10.1007/s41109-019-0162-z 172 | """ 173 | def __init__( 174 | self, 175 | ens_size:int=16, 176 | min_weight:float=0.05, 177 | final:str="leiden", 178 | resolution:float=1.0, 179 | sort_clusters:bool=True, 180 | return_probs:bool=False, 181 | refuse_score:bool=False, 182 | random_state=None, 183 | rng=None, 184 | return_aggregate:bool=False 185 | ): 186 | super(ECG, self).__init__(sort_clusters=sort_clusters, return_probs=return_probs, return_aggregate=return_aggregate) 187 | if ens_size <= 0 or not float(ens_size).is_integer(): 188 | raise ValueError(f"ens_size must be a positive integer. Got {ens_size}") 189 | self.ens_size = ens_size 190 | if min_weight < 0: 191 | raise ValueError(f"min_weight must be non-negative. Got {min_weight}") 192 | self.min_weight = min_weight 193 | if final not in ["louvain", "leiden"]: 194 | raise ValueError(f"final must be one of 'louvain' or 'leiden'. Got {final}") 195 | self.final = final 196 | if resolution < 0: 197 | raise ValueError(f"resolution must be non-negative. Got {resolution}") 198 | self.resolution = resolution 199 | self.refuse_score = refuse_score 200 | if rng is not None: 201 | self.rng = rng 202 | elif random_state is not None: 203 | self.rng = np.random.default_rng(random_state) 204 | else: 205 | self.rng = np.random.default_rng() 206 | 207 | 208 | def fit(self, g): 209 | g = check_format(g) 210 | # Stage one, compute weights 211 | self.weights = g.copy().astype("float64") 212 | partitions = np.empty((g.shape[0], self.ens_size), dtype="int32") 213 | for i in range(self.ens_size): 214 | louvain = Louvain(resolution=self.resolution, n_aggregations=0, shuffle_nodes=True, random_state=self.rng.choice(2**32)) 215 | partitions[:, i] = louvain.fit_predict(g) 216 | _ecg_weights(self.weights.indptr, self.weights.indices, self.weights.data, partitions) 217 | self.weights.data = self.weights.data/self.ens_size 218 | self.weights.data = self.min_weight + (1-self.min_weight)*self.weights.data 219 | # Force min_weight outside 2-core 220 | core = get_core_decomposition(g) 221 | for i, core_num in enumerate(core): 222 | if core_num < 2: 223 | self.weights.data[self.weights.indptr[i]:self.weights.indptr[i+1]] = self.min_weight 224 | 225 | # Stage two, cluster weighted graph 226 | if self.final == "louvain": 227 | clusterer = Louvain(resolution=self.resolution, shuffle_nodes=True, sort_clusters=self.sort_clusters, random_state=self.rng.choice(2**32)) 228 | else: 229 | clusterer = Leiden(resolution=self.resolution, shuffle_nodes=True, sort_clusters=self.sort_clusters, random_state=self.rng.choice(2**32)) 230 | 231 | self.labels_ = clusterer.fit_predict(self.weights) 232 | self.CSI = 1 - np.mean(np.minimum(self.weights.data, 1-self.weights.data)) 233 | self._secondary_outputs(g) 234 | 235 | if self.refuse_score: 236 | self.refuse_overall_, self.refuse_community_ = _compute_refuse_scores(self.weights.indptr, self.weights.indices, self.weights.data, self.labels_) 237 | 238 | return self 239 | 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-------------------------------------------------------------------------------- /Data/Football/football.gml: -------------------------------------------------------------------------------- 1 | Creator "Mark Newman on Sat Jul 22 05:32:16 2006" 2 | graph 3 | [ 4 | directed 0 5 | node 6 | [ 7 | id 0 8 | label "BrighamYoung" 9 | value 7 10 | ] 11 | node 12 | [ 13 | id 1 14 | label "FloridaState" 15 | value 0 16 | ] 17 | node 18 | [ 19 | id 2 20 | label "Iowa" 21 | value 2 22 | ] 23 | node 24 | [ 25 | id 3 26 | label "KansasState" 27 | value 3 28 | ] 29 | node 30 | [ 31 | id 4 32 | label "NewMexico" 33 | value 7 34 | ] 35 | node 36 | [ 37 | id 5 38 | label "TexasTech" 39 | value 3 40 | ] 41 | node 42 | [ 43 | id 6 44 | label "PennState" 45 | value 2 46 | ] 47 | node 48 | [ 49 | id 7 50 | label "SouthernCalifornia" 51 | value 8 52 | ] 53 | node 54 | [ 55 | id 8 56 | label "ArizonaState" 57 | value 8 58 | ] 59 | node 60 | [ 61 | id 9 62 | label "SanDiegoState" 63 | value 7 64 | ] 65 | node 66 | [ 67 | id 10 68 | label "Baylor" 69 | value 3 70 | ] 71 | node 72 | [ 73 | id 11 74 | label "NorthTexas" 75 | value 10 76 | ] 77 | node 78 | [ 79 | id 12 80 | label "NorthernIllinois" 81 | value 6 82 | ] 83 | node 84 | [ 85 | id 13 86 | label "Northwestern" 87 | value 2 88 | ] 89 | node 90 | [ 91 | id 14 92 | label "WesternMichigan" 93 | value 6 94 | ] 95 | node 96 | [ 97 | id 15 98 | label "Wisconsin" 99 | value 2 100 | ] 101 | node 102 | [ 103 | id 16 104 | label "Wyoming" 105 | value 7 106 | ] 107 | node 108 | [ 109 | id 17 110 | label "Auburn" 111 | value 9 112 | ] 113 | node 114 | [ 115 | id 18 116 | label "Akron" 117 | value 6 118 | ] 119 | node 120 | [ 121 | id 19 122 | label "VirginiaTech" 123 | value 1 124 | ] 125 | node 126 | [ 127 | id 20 128 | label "Alabama" 129 | value 9 130 | ] 131 | node 132 | [ 133 | id 21 134 | label "UCLA" 135 | value 8 136 | ] 137 | node 138 | [ 139 | id 22 140 | label "Arizona" 141 | value 8 142 | ] 143 | node 144 | [ 145 | id 23 146 | label "Utah" 147 | value 7 148 | ] 149 | node 150 | [ 151 | id 24 152 | label "ArkansasState" 153 | value 10 154 | ] 155 | node 156 | [ 157 | id 25 158 | label "NorthCarolinaState" 159 | value 0 160 | ] 161 | node 162 | [ 163 | id 26 164 | label "BallState" 165 | value 6 166 | ] 167 | node 168 | [ 169 | id 27 170 | label "Florida" 171 | value 9 172 | ] 173 | node 174 | [ 175 | id 28 176 | label "BoiseState" 177 | value 11 178 | ] 179 | node 180 | [ 181 | id 29 182 | label "BostonCollege" 183 | value 1 184 | ] 185 | node 186 | [ 187 | id 30 188 | label "WestVirginia" 189 | value 1 190 | ] 191 | node 192 | [ 193 | id 31 194 | label "BowlingGreenState" 195 | value 6 196 | ] 197 | node 198 | [ 199 | id 32 200 | label "Michigan" 201 | value 2 202 | ] 203 | node 204 | [ 205 | id 33 206 | label "Virginia" 207 | value 0 208 | ] 209 | node 210 | [ 211 | id 34 212 | label "Buffalo" 213 | value 6 214 | ] 215 | node 216 | [ 217 | id 35 218 | label "Syracuse" 219 | value 1 220 | ] 221 | node 222 | [ 223 | id 36 224 | label "CentralFlorida" 225 | value 5 226 | ] 227 | node 228 | [ 229 | id 37 230 | label "GeorgiaTech" 231 | value 0 232 | ] 233 | node 234 | [ 235 | id 38 236 | label "CentralMichigan" 237 | value 6 238 | ] 239 | node 240 | [ 241 | id 39 242 | label "Purdue" 243 | value 2 244 | ] 245 | node 246 | [ 247 | id 40 248 | label "Colorado" 249 | value 3 250 | ] 251 | node 252 | [ 253 | id 41 254 | label "ColoradoState" 255 | value 7 256 | ] 257 | node 258 | [ 259 | id 42 260 | label "Connecticut" 261 | value 5 262 | ] 263 | node 264 | [ 265 | id 43 266 | label "EasternMichigan" 267 | value 6 268 | ] 269 | node 270 | [ 271 | id 44 272 | label "EastCarolina" 273 | value 4 274 | ] 275 | node 276 | [ 277 | id 45 278 | label "Duke" 279 | value 0 280 | ] 281 | node 282 | [ 283 | id 46 284 | label "FresnoState" 285 | value 11 286 | ] 287 | node 288 | [ 289 | id 47 290 | label "OhioState" 291 | value 2 292 | ] 293 | node 294 | [ 295 | id 48 296 | label "Houston" 297 | value 4 298 | ] 299 | node 300 | [ 301 | id 49 302 | label "Rice" 303 | value 11 304 | ] 305 | node 306 | [ 307 | id 50 308 | label "Idaho" 309 | value 10 310 | ] 311 | node 312 | [ 313 | id 51 314 | label "Washington" 315 | value 8 316 | ] 317 | node 318 | [ 319 | id 52 320 | label "Kansas" 321 | value 3 322 | ] 323 | node 324 | [ 325 | id 53 326 | label "SouthernMethodist" 327 | value 11 328 | ] 329 | node 330 | [ 331 | id 54 332 | label "Kent" 333 | value 6 334 | ] 335 | node 336 | [ 337 | id 55 338 | label "Pittsburgh" 339 | value 1 340 | ] 341 | node 342 | [ 343 | id 56 344 | label "Kentucky" 345 | value 9 346 | ] 347 | node 348 | [ 349 | id 57 350 | label "Louisville" 351 | value 4 352 | ] 353 | node 354 | [ 355 | id 58 356 | label "LouisianaTech" 357 | value 11 358 | ] 359 | node 360 | [ 361 | id 59 362 | label "LouisianaMonroe" 363 | value 10 364 | ] 365 | node 366 | [ 367 | id 60 368 | label "Minnesota" 369 | value 2 370 | ] 371 | node 372 | [ 373 | id 61 374 | label "MiamiOhio" 375 | value 6 376 | ] 377 | node 378 | [ 379 | id 62 380 | label "Vanderbilt" 381 | value 9 382 | ] 383 | node 384 | [ 385 | id 63 386 | label "MiddleTennesseeState" 387 | value 10 388 | ] 389 | node 390 | [ 391 | id 64 392 | label "Illinois" 393 | value 2 394 | ] 395 | node 396 | [ 397 | id 65 398 | label "MississippiState" 399 | value 9 400 | ] 401 | node 402 | [ 403 | id 66 404 | label "Memphis" 405 | value 4 406 | ] 407 | node 408 | [ 409 | id 67 410 | label "Nevada" 411 | value 11 412 | ] 413 | node 414 | [ 415 | id 68 416 | label "Oregon" 417 | value 8 418 | ] 419 | node 420 | [ 421 | id 69 422 | label "NewMexicoState" 423 | value 10 424 | ] 425 | node 426 | [ 427 | id 70 428 | label "SouthCarolina" 429 | value 9 430 | ] 431 | node 432 | [ 433 | id 71 434 | label "Ohio" 435 | value 6 436 | ] 437 | node 438 | [ 439 | id 72 440 | label "IowaState" 441 | value 3 442 | ] 443 | node 444 | [ 445 | id 73 446 | label "SanJoseState" 447 | value 11 448 | ] 449 | node 450 | [ 451 | id 74 452 | label "Nebraska" 453 | value 3 454 | ] 455 | node 456 | [ 457 | id 75 458 | label "SouthernMississippi" 459 | value 4 460 | ] 461 | node 462 | [ 463 | id 76 464 | label "Tennessee" 465 | value 9 466 | ] 467 | node 468 | [ 469 | id 77 470 | label "Stanford" 471 | value 8 472 | ] 473 | node 474 | [ 475 | id 78 476 | label "WashingtonState" 477 | value 8 478 | ] 479 | node 480 | [ 481 | id 79 482 | label "Temple" 483 | value 1 484 | ] 485 | node 486 | [ 487 | id 80 488 | label "Navy" 489 | value 5 490 | ] 491 | node 492 | [ 493 | id 81 494 | label "TexasA&M" 495 | value 3 496 | ] 497 | node 498 | [ 499 | id 82 500 | label "NotreDame" 501 | value 5 502 | ] 503 | node 504 | [ 505 | id 83 506 | label "TexasElPaso" 507 | value 11 508 | ] 509 | node 510 | [ 511 | id 84 512 | label "Oklahoma" 513 | value 3 514 | ] 515 | node 516 | [ 517 | id 85 518 | label "Toledo" 519 | value 6 520 | ] 521 | node 522 | [ 523 | id 86 524 | label "Tulane" 525 | value 4 526 | ] 527 | node 528 | [ 529 | id 87 530 | label "Mississippi" 531 | value 9 532 | ] 533 | node 534 | [ 535 | id 88 536 | label "Tulsa" 537 | value 11 538 | ] 539 | node 540 | [ 541 | id 89 542 | label "NorthCarolina" 543 | value 0 544 | ] 545 | node 546 | [ 547 | id 90 548 | label "UtahState" 549 | value 5 550 | ] 551 | node 552 | [ 553 | id 91 554 | label "Army" 555 | value 4 556 | ] 557 | node 558 | [ 559 | id 92 560 | label "Cincinnati" 561 | value 4 562 | ] 563 | node 564 | [ 565 | id 93 566 | label "AirForce" 567 | value 7 568 | ] 569 | node 570 | [ 571 | id 94 572 | label "Rutgers" 573 | value 1 574 | ] 575 | node 576 | [ 577 | id 95 578 | label "Georgia" 579 | value 9 580 | ] 581 | node 582 | [ 583 | id 96 584 | label "LouisianaState" 585 | value 9 586 | ] 587 | node 588 | [ 589 | id 97 590 | label "LouisianaLafayette" 591 | value 10 592 | ] 593 | node 594 | [ 595 | id 98 596 | label "Texas" 597 | value 3 598 | ] 599 | node 600 | [ 601 | id 99 602 | label "Marshall" 603 | value 6 604 | ] 605 | node 606 | [ 607 | id 100 608 | label "MichiganState" 609 | value 2 610 | ] 611 | node 612 | [ 613 | id 101 614 | label "MiamiFlorida" 615 | value 1 616 | ] 617 | node 618 | [ 619 | id 102 620 | label "Missouri" 621 | value 3 622 | ] 623 | node 624 | [ 625 | id 103 626 | label "Clemson" 627 | value 0 628 | ] 629 | node 630 | [ 631 | id 104 632 | label "NevadaLasVegas" 633 | value 7 634 | ] 635 | node 636 | [ 637 | id 105 638 | label "WakeForest" 639 | value 0 640 | ] 641 | node 642 | [ 643 | id 106 644 | label "Indiana" 645 | value 2 646 | ] 647 | node 648 | [ 649 | id 107 650 | label "OklahomaState" 651 | value 3 652 | ] 653 | node 654 | [ 655 | id 108 656 | label "OregonState" 657 | value 8 658 | ] 659 | node 660 | [ 661 | id 109 662 | label "Maryland" 663 | value 0 664 | ] 665 | node 666 | [ 667 | id 110 668 | label "TexasChristian" 669 | value 4 670 | ] 671 | node 672 | [ 673 | id 111 674 | label "California" 675 | value 8 676 | ] 677 | node 678 | [ 679 | id 112 680 | label "AlabamaBirmingham" 681 | value 4 682 | ] 683 | node 684 | [ 685 | id 113 686 | label "Arkansas" 687 | value 9 688 | ] 689 | node 690 | [ 691 | id 114 692 | label "Hawaii" 693 | value 11 694 | ] 695 | edge 696 | [ 697 | source 1 698 | target 0 699 | ] 700 | edge 701 | [ 702 | source 3 703 | target 2 704 | ] 705 | edge 706 | [ 707 | source 4 708 | target 0 709 | ] 710 | edge 711 | [ 712 | source 5 713 | target 4 714 | ] 715 | edge 716 | [ 717 | source 5 718 | target 3 719 | ] 720 | edge 721 | [ 722 | source 6 723 | target 2 724 | ] 725 | edge 726 | [ 727 | source 7 728 | target 6 729 | ] 730 | edge 731 | [ 732 | source 8 733 | target 7 734 | ] 735 | edge 736 | [ 737 | source 9 738 | target 8 739 | ] 740 | edge 741 | [ 742 | source 9 743 | target 0 744 | ] 745 | edge 746 | [ 747 | source 9 748 | target 4 749 | ] 750 | edge 751 | [ 752 | source 10 753 | target 5 754 | ] 755 | edge 756 | [ 757 | source 11 758 | target 10 759 | ] 760 | edge 761 | [ 762 | source 11 763 | target 5 764 | ] 765 | edge 766 | [ 767 | source 11 768 | target 3 769 | ] 770 | edge 771 | [ 772 | source 13 773 | target 12 774 | ] 775 | edge 776 | [ 777 | source 13 778 | target 2 779 | ] 780 | edge 781 | [ 782 | source 14 783 | target 2 784 | ] 785 | edge 786 | [ 787 | source 14 788 | target 12 789 | ] 790 | edge 791 | [ 792 | source 15 793 | target 14 794 | ] 795 | edge 796 | [ 797 | source 15 798 | target 13 799 | ] 800 | edge 801 | [ 802 | source 15 803 | target 2 804 | ] 805 | edge 806 | [ 807 | source 16 808 | target 4 809 | ] 810 | edge 811 | [ 812 | source 16 813 | target 9 814 | ] 815 | edge 816 | [ 817 | source 16 818 | target 0 819 | ] 820 | edge 821 | [ 822 | source 17 823 | target 16 824 | ] 825 | edge 826 | [ 827 | source 17 828 | target 12 829 | ] 830 | edge 831 | [ 832 | source 18 833 | target 12 834 | ] 835 | edge 836 | [ 837 | source 19 838 | target 18 839 | ] 840 | edge 841 | [ 842 | source 20 843 | target 17 844 | ] 845 | edge 846 | [ 847 | source 21 848 | target 20 849 | ] 850 | edge 851 | [ 852 | source 21 853 | target 8 854 | ] 855 | edge 856 | [ 857 | source 21 858 | target 7 859 | ] 860 | edge 861 | [ 862 | source 22 863 | target 9 864 | ] 865 | edge 866 | [ 867 | source 22 868 | target 7 869 | ] 870 | edge 871 | [ 872 | source 22 873 | target 21 874 | ] 875 | edge 876 | [ 877 | source 22 878 | target 8 879 | ] 880 | edge 881 | [ 882 | source 23 883 | target 22 884 | ] 885 | edge 886 | [ 887 | source 23 888 | target 9 889 | ] 890 | edge 891 | [ 892 | source 23 893 | target 4 894 | ] 895 | edge 896 | [ 897 | source 23 898 | target 16 899 | ] 900 | edge 901 | [ 902 | source 23 903 | target 0 904 | ] 905 | edge 906 | [ 907 | source 24 908 | target 11 909 | ] 910 | edge 911 | [ 912 | source 25 913 | target 24 914 | ] 915 | edge 916 | [ 917 | source 25 918 | target 1 919 | ] 920 | edge 921 | [ 922 | source 26 923 | target 3 924 | ] 925 | edge 926 | [ 927 | source 26 928 | target 12 929 | ] 930 | edge 931 | [ 932 | source 26 933 | target 14 934 | ] 935 | edge 936 | [ 937 | source 27 938 | target 26 939 | ] 940 | edge 941 | [ 942 | source 27 943 | target 17 944 | ] 945 | edge 946 | [ 947 | source 27 948 | target 1 949 | ] 950 | edge 951 | [ 952 | source 28 953 | target 4 954 | ] 955 | edge 956 | [ 957 | source 28 958 | target 11 959 | ] 960 | edge 961 | [ 962 | source 28 963 | target 24 964 | ] 965 | edge 966 | [ 967 | source 29 968 | target 19 969 | ] 970 | edge 971 | [ 972 | source 30 973 | target 29 974 | ] 975 | edge 976 | [ 977 | source 30 978 | target 19 979 | ] 980 | edge 981 | [ 982 | source 31 983 | target 18 984 | ] 985 | edge 986 | [ 987 | source 32 988 | target 31 989 | ] 990 | edge 991 | [ 992 | source 32 993 | target 21 994 | ] 995 | edge 996 | [ 997 | source 32 998 | target 15 999 | ] 1000 | edge 1001 | [ 1002 | source 32 1003 | target 13 1004 | ] 1005 | edge 1006 | [ 1007 | source 32 1008 | target 6 1009 | ] 1010 | edge 1011 | [ 1012 | source 33 1013 | target 0 1014 | ] 1015 | edge 1016 | [ 1017 | source 33 1018 | target 1 1019 | ] 1020 | edge 1021 | [ 1022 | source 33 1023 | target 25 1024 | ] 1025 | edge 1026 | [ 1027 | source 33 1028 | target 19 1029 | ] 1030 | edge 1031 | [ 1032 | source 34 1033 | target 31 1034 | ] 1035 | edge 1036 | [ 1037 | source 34 1038 | target 26 1039 | ] 1040 | edge 1041 | [ 1042 | source 34 1043 | target 12 1044 | ] 1045 | edge 1046 | [ 1047 | source 34 1048 | target 18 1049 | ] 1050 | edge 1051 | [ 1052 | source 35 1053 | target 34 1054 | ] 1055 | edge 1056 | [ 1057 | source 35 1058 | target 0 1059 | ] 1060 | edge 1061 | [ 1062 | source 35 1063 | target 29 1064 | ] 1065 | edge 1066 | [ 1067 | source 35 1068 | target 19 1069 | ] 1070 | edge 1071 | [ 1072 | source 35 1073 | target 30 1074 | ] 1075 | edge 1076 | [ 1077 | source 36 1078 | target 18 1079 | ] 1080 | edge 1081 | [ 1082 | source 36 1083 | target 12 1084 | ] 1085 | edge 1086 | [ 1087 | source 36 1088 | target 20 1089 | ] 1090 | edge 1091 | [ 1092 | source 36 1093 | target 19 1094 | ] 1095 | edge 1096 | [ 1097 | source 37 1098 | target 36 1099 | ] 1100 | edge 1101 | [ 1102 | source 37 1103 | target 1 1104 | ] 1105 | edge 1106 | [ 1107 | source 37 1108 | target 25 1109 | ] 1110 | edge 1111 | [ 1112 | source 37 1113 | target 33 1114 | ] 1115 | edge 1116 | [ 1117 | source 38 1118 | target 18 1119 | ] 1120 | edge 1121 | [ 1122 | source 38 1123 | target 16 1124 | ] 1125 | edge 1126 | [ 1127 | source 38 1128 | target 28 1129 | ] 1130 | edge 1131 | [ 1132 | source 38 1133 | target 26 1134 | ] 1135 | edge 1136 | [ 1137 | source 38 1138 | target 14 1139 | ] 1140 | edge 1141 | [ 1142 | source 38 1143 | target 12 1144 | ] 1145 | edge 1146 | [ 1147 | source 39 1148 | target 38 1149 | ] 1150 | edge 1151 | [ 1152 | source 39 1153 | target 6 1154 | ] 1155 | edge 1156 | [ 1157 | source 39 1158 | target 32 1159 | ] 1160 | edge 1161 | [ 1162 | source 39 1163 | target 13 1164 | ] 1165 | edge 1166 | [ 1167 | source 39 1168 | target 15 1169 | ] 1170 | edge 1171 | [ 1172 | source 40 1173 | target 7 1174 | ] 1175 | edge 1176 | [ 1177 | source 40 1178 | target 3 1179 | ] 1180 | edge 1181 | [ 1182 | source 41 1183 | target 40 1184 | ] 1185 | edge 1186 | [ 1187 | source 41 1188 | target 8 1189 | ] 1190 | edge 1191 | [ 1192 | source 41 1193 | target 4 1194 | ] 1195 | edge 1196 | [ 1197 | source 41 1198 | target 23 1199 | ] 1200 | edge 1201 | [ 1202 | source 41 1203 | target 9 1204 | ] 1205 | edge 1206 | [ 1207 | source 41 1208 | target 0 1209 | ] 1210 | edge 1211 | [ 1212 | source 41 1213 | target 16 1214 | ] 1215 | edge 1216 | [ 1217 | source 42 1218 | target 34 1219 | ] 1220 | edge 1221 | [ 1222 | source 42 1223 | target 29 1224 | ] 1225 | edge 1226 | [ 1227 | source 42 1228 | target 18 1229 | ] 1230 | edge 1231 | [ 1232 | source 42 1233 | target 26 1234 | ] 1235 | edge 1236 | [ 1237 | source 43 1238 | target 42 1239 | ] 1240 | edge 1241 | [ 1242 | source 43 1243 | target 36 1244 | ] 1245 | edge 1246 | [ 1247 | source 43 1248 | target 26 1249 | ] 1250 | edge 1251 | [ 1252 | source 43 1253 | target 31 1254 | ] 1255 | edge 1256 | [ 1257 | source 43 1258 | target 38 1259 | ] 1260 | edge 1261 | [ 1262 | source 43 1263 | target 12 1264 | ] 1265 | edge 1266 | [ 1267 | source 43 1268 | target 14 1269 | ] 1270 | edge 1271 | [ 1272 | source 44 1273 | target 19 1274 | ] 1275 | edge 1276 | [ 1277 | source 44 1278 | target 35 1279 | ] 1280 | edge 1281 | [ 1282 | source 44 1283 | target 30 1284 | ] 1285 | edge 1286 | [ 1287 | source 45 1288 | target 44 1289 | ] 1290 | edge 1291 | [ 1292 | source 45 1293 | target 13 1294 | ] 1295 | edge 1296 | [ 1297 | source 45 1298 | target 33 1299 | ] 1300 | edge 1301 | [ 1302 | source 45 1303 | target 1 1304 | ] 1305 | edge 1306 | [ 1307 | source 45 1308 | target 37 1309 | ] 1310 | edge 1311 | [ 1312 | source 45 1313 | target 25 1314 | ] 1315 | edge 1316 | [ 1317 | source 46 1318 | target 21 1319 | ] 1320 | edge 1321 | [ 1322 | source 47 1323 | target 46 1324 | ] 1325 | edge 1326 | [ 1327 | source 47 1328 | target 22 1329 | ] 1330 | edge 1331 | [ 1332 | source 47 1333 | target 6 1334 | ] 1335 | edge 1336 | [ 1337 | source 47 1338 | target 15 1339 | ] 1340 | edge 1341 | [ 1342 | source 47 1343 | target 2 1344 | ] 1345 | edge 1346 | [ 1347 | source 47 1348 | target 39 1349 | ] 1350 | edge 1351 | [ 1352 | source 47 1353 | target 32 1354 | ] 1355 | edge 1356 | [ 1357 | source 48 1358 | target 44 1359 | ] 1360 | edge 1361 | [ 1362 | source 49 1363 | target 48 1364 | ] 1365 | edge 1366 | [ 1367 | source 49 1368 | target 32 1369 | ] 1370 | edge 1371 | [ 1372 | source 49 1373 | target 46 1374 | ] 1375 | edge 1376 | [ 1377 | source 50 1378 | target 30 1379 | ] 1380 | edge 1381 | [ 1382 | source 50 1383 | target 24 1384 | ] 1385 | edge 1386 | [ 1387 | source 50 1388 | target 11 1389 | ] 1390 | edge 1391 | [ 1392 | source 50 1393 | target 28 1394 | ] 1395 | edge 1396 | [ 1397 | source 51 1398 | target 50 1399 | ] 1400 | edge 1401 | [ 1402 | source 51 1403 | target 40 1404 | ] 1405 | edge 1406 | [ 1407 | source 51 1408 | target 8 1409 | ] 1410 | edge 1411 | [ 1412 | source 51 1413 | target 22 1414 | ] 1415 | edge 1416 | [ 1417 | source 51 1418 | target 21 1419 | ] 1420 | edge 1421 | [ 1422 | source 52 1423 | target 3 1424 | ] 1425 | edge 1426 | [ 1427 | source 52 1428 | target 40 1429 | ] 1430 | edge 1431 | [ 1432 | source 52 1433 | target 5 1434 | ] 1435 | edge 1436 | [ 1437 | source 53 1438 | target 52 1439 | ] 1440 | edge 1441 | [ 1442 | source 53 1443 | target 25 1444 | ] 1445 | edge 1446 | [ 1447 | source 53 1448 | target 48 1449 | ] 1450 | edge 1451 | [ 1452 | source 53 1453 | target 49 1454 | ] 1455 | edge 1456 | [ 1457 | source 53 1458 | target 46 1459 | ] 1460 | edge 1461 | [ 1462 | source 54 1463 | target 39 1464 | ] 1465 | edge 1466 | [ 1467 | source 54 1468 | target 31 1469 | ] 1470 | edge 1471 | [ 1472 | source 54 1473 | target 38 1474 | ] 1475 | edge 1476 | [ 1477 | source 54 1478 | target 14 1479 | ] 1480 | edge 1481 | [ 1482 | source 54 1483 | target 34 1484 | ] 1485 | edge 1486 | [ 1487 | source 54 1488 | target 18 1489 | ] 1490 | edge 1491 | [ 1492 | source 55 1493 | target 54 1494 | ] 1495 | edge 1496 | [ 1497 | source 55 1498 | target 31 1499 | ] 1500 | edge 1501 | [ 1502 | source 55 1503 | target 6 1504 | ] 1505 | edge 1506 | [ 1507 | source 55 1508 | target 35 1509 | ] 1510 | edge 1511 | [ 1512 | source 55 1513 | target 29 1514 | ] 1515 | edge 1516 | [ 1517 | source 55 1518 | target 19 1519 | ] 1520 | edge 1521 | [ 1522 | source 55 1523 | target 30 1524 | ] 1525 | edge 1526 | [ 1527 | source 56 1528 | target 27 1529 | ] 1530 | edge 1531 | [ 1532 | source 57 1533 | target 56 1534 | ] 1535 | edge 1536 | [ 1537 | source 57 1538 | target 1 1539 | ] 1540 | edge 1541 | [ 1542 | source 57 1543 | target 42 1544 | ] 1545 | edge 1546 | [ 1547 | source 57 1548 | target 44 1549 | ] 1550 | edge 1551 | [ 1552 | source 57 1553 | target 48 1554 | ] 1555 | edge 1556 | [ 1557 | source 58 1558 | target 3 1559 | ] 1560 | edge 1561 | [ 1562 | source 58 1563 | target 6 1564 | ] 1565 | edge 1566 | [ 1567 | source 58 1568 | target 17 1569 | ] 1570 | edge 1571 | [ 1572 | source 58 1573 | target 36 1574 | ] 1575 | edge 1576 | [ 1577 | source 59 1578 | target 36 1579 | ] 1580 | edge 1581 | [ 1582 | source 59 1583 | target 58 1584 | ] 1585 | edge 1586 | [ 1587 | source 60 1588 | target 59 1589 | ] 1590 | edge 1591 | [ 1592 | source 60 1593 | target 10 1594 | ] 1595 | edge 1596 | [ 1597 | source 60 1598 | target 39 1599 | ] 1600 | edge 1601 | [ 1602 | source 60 1603 | target 6 1604 | ] 1605 | edge 1606 | [ 1607 | source 60 1608 | target 47 1609 | ] 1610 | edge 1611 | [ 1612 | source 60 1613 | target 13 1614 | ] 1615 | edge 1616 | [ 1617 | source 60 1618 | target 15 1619 | ] 1620 | edge 1621 | [ 1622 | source 60 1623 | target 2 1624 | ] 1625 | edge 1626 | [ 1627 | source 61 1628 | target 43 1629 | ] 1630 | edge 1631 | [ 1632 | source 61 1633 | target 47 1634 | ] 1635 | edge 1636 | [ 1637 | source 61 1638 | target 54 1639 | ] 1640 | edge 1641 | [ 1642 | source 61 1643 | target 18 1644 | ] 1645 | edge 1646 | [ 1647 | source 61 1648 | target 26 1649 | ] 1650 | edge 1651 | [ 1652 | source 61 1653 | target 31 1654 | ] 1655 | edge 1656 | [ 1657 | source 61 1658 | target 34 1659 | ] 1660 | edge 1661 | [ 1662 | source 62 1663 | target 61 1664 | ] 1665 | edge 1666 | [ 1667 | source 62 1668 | target 20 1669 | ] 1670 | edge 1671 | [ 1672 | source 62 1673 | target 45 1674 | ] 1675 | edge 1676 | [ 1677 | source 62 1678 | target 17 1679 | ] 1680 | edge 1681 | [ 1682 | source 62 1683 | target 27 1684 | ] 1685 | edge 1686 | [ 1687 | source 62 1688 | target 56 1689 | ] 1690 | edge 1691 | [ 1692 | source 63 1693 | target 27 1694 | ] 1695 | edge 1696 | [ 1697 | source 63 1698 | target 58 1699 | ] 1700 | edge 1701 | [ 1702 | source 63 1703 | target 59 1704 | ] 1705 | edge 1706 | [ 1707 | source 63 1708 | target 42 1709 | ] 1710 | edge 1711 | [ 1712 | source 64 1713 | target 63 1714 | ] 1715 | edge 1716 | [ 1717 | source 64 1718 | target 9 1719 | ] 1720 | edge 1721 | [ 1722 | source 64 1723 | target 32 1724 | ] 1725 | edge 1726 | [ 1727 | source 64 1728 | target 60 1729 | ] 1730 | edge 1731 | [ 1732 | source 64 1733 | target 2 1734 | ] 1735 | edge 1736 | [ 1737 | source 64 1738 | target 6 1739 | ] 1740 | edge 1741 | [ 1742 | source 64 1743 | target 47 1744 | ] 1745 | edge 1746 | [ 1747 | source 64 1748 | target 13 1749 | ] 1750 | edge 1751 | [ 1752 | source 65 1753 | target 0 1754 | ] 1755 | edge 1756 | [ 1757 | source 65 1758 | target 27 1759 | ] 1760 | edge 1761 | [ 1762 | source 65 1763 | target 17 1764 | ] 1765 | edge 1766 | [ 1767 | source 65 1768 | target 63 1769 | ] 1770 | edge 1771 | [ 1772 | source 65 1773 | target 56 1774 | ] 1775 | edge 1776 | [ 1777 | source 65 1778 | target 20 1779 | ] 1780 | edge 1781 | [ 1782 | source 66 1783 | target 65 1784 | ] 1785 | edge 1786 | [ 1787 | source 66 1788 | target 59 1789 | ] 1790 | edge 1791 | [ 1792 | source 66 1793 | target 24 1794 | ] 1795 | edge 1796 | [ 1797 | source 66 1798 | target 44 1799 | ] 1800 | edge 1801 | [ 1802 | source 66 1803 | target 48 1804 | ] 1805 | edge 1806 | [ 1807 | source 67 1808 | target 16 1809 | ] 1810 | edge 1811 | [ 1812 | source 67 1813 | target 41 1814 | ] 1815 | edge 1816 | [ 1817 | source 67 1818 | target 46 1819 | ] 1820 | edge 1821 | [ 1822 | source 67 1823 | target 53 1824 | ] 1825 | edge 1826 | [ 1827 | source 67 1828 | target 49 1829 | ] 1830 | edge 1831 | [ 1832 | source 68 1833 | target 67 1834 | ] 1835 | edge 1836 | [ 1837 | source 68 1838 | target 15 1839 | ] 1840 | edge 1841 | [ 1842 | source 68 1843 | target 50 1844 | ] 1845 | edge 1846 | [ 1847 | source 68 1848 | target 21 1849 | ] 1850 | edge 1851 | [ 1852 | source 68 1853 | target 51 1854 | ] 1855 | edge 1856 | [ 1857 | source 68 1858 | target 7 1859 | ] 1860 | edge 1861 | [ 1862 | source 68 1863 | target 22 1864 | ] 1865 | edge 1866 | [ 1867 | source 68 1868 | target 8 1869 | ] 1870 | edge 1871 | [ 1872 | source 69 1873 | target 4 1874 | ] 1875 | edge 1876 | [ 1877 | source 69 1878 | target 24 1879 | ] 1880 | edge 1881 | [ 1882 | source 69 1883 | target 28 1884 | ] 1885 | edge 1886 | [ 1887 | source 69 1888 | target 50 1889 | ] 1890 | edge 1891 | [ 1892 | source 69 1893 | target 11 1894 | ] 1895 | edge 1896 | [ 1897 | source 70 1898 | target 69 1899 | ] 1900 | edge 1901 | [ 1902 | source 70 1903 | target 43 1904 | ] 1905 | edge 1906 | [ 1907 | source 70 1908 | target 65 1909 | ] 1910 | edge 1911 | [ 1912 | source 70 1913 | target 20 1914 | ] 1915 | edge 1916 | [ 1917 | source 70 1918 | target 56 1919 | ] 1920 | edge 1921 | [ 1922 | source 70 1923 | target 62 1924 | ] 1925 | edge 1926 | [ 1927 | source 70 1928 | target 27 1929 | ] 1930 | edge 1931 | [ 1932 | source 71 1933 | target 60 1934 | ] 1935 | edge 1936 | [ 1937 | source 71 1938 | target 18 1939 | ] 1940 | edge 1941 | [ 1942 | source 71 1943 | target 14 1944 | ] 1945 | edge 1946 | [ 1947 | source 71 1948 | target 34 1949 | ] 1950 | edge 1951 | [ 1952 | source 71 1953 | target 54 1954 | ] 1955 | edge 1956 | [ 1957 | source 71 1958 | target 38 1959 | ] 1960 | edge 1961 | [ 1962 | source 71 1963 | target 61 1964 | ] 1965 | edge 1966 | [ 1967 | source 71 1968 | target 31 1969 | ] 1970 | edge 1971 | [ 1972 | source 72 1973 | target 71 1974 | ] 1975 | edge 1976 | [ 1977 | source 72 1978 | target 2 1979 | ] 1980 | edge 1981 | [ 1982 | source 72 1983 | target 10 1984 | ] 1985 | edge 1986 | [ 1987 | source 72 1988 | target 3 1989 | ] 1990 | edge 1991 | [ 1992 | source 72 1993 | target 40 1994 | ] 1995 | edge 1996 | [ 1997 | source 72 1998 | target 52 1999 | ] 2000 | edge 2001 | [ 2002 | source 73 2003 | target 7 2004 | ] 2005 | edge 2006 | [ 2007 | source 73 2008 | target 49 2009 | ] 2010 | edge 2011 | [ 2012 | source 73 2013 | target 53 2014 | ] 2015 | edge 2016 | [ 2017 | source 73 2018 | target 67 2019 | ] 2020 | edge 2021 | [ 2022 | source 73 2023 | target 46 2024 | ] 2025 | edge 2026 | [ 2027 | source 74 2028 | target 73 2029 | ] 2030 | edge 2031 | [ 2032 | source 74 2033 | target 2 2034 | ] 2035 | edge 2036 | [ 2037 | source 74 2038 | target 72 2039 | ] 2040 | edge 2041 | [ 2042 | source 74 2043 | target 5 2044 | ] 2045 | edge 2046 | [ 2047 | source 74 2048 | target 10 2049 | ] 2050 | edge 2051 | [ 2052 | source 74 2053 | target 52 2054 | ] 2055 | edge 2056 | [ 2057 | source 74 2058 | target 3 2059 | ] 2060 | edge 2061 | [ 2062 | source 74 2063 | target 40 2064 | ] 2065 | edge 2066 | [ 2067 | source 75 2068 | target 20 2069 | ] 2070 | edge 2071 | [ 2072 | source 75 2073 | target 66 2074 | ] 2075 | edge 2076 | [ 2077 | source 75 2078 | target 48 2079 | ] 2080 | edge 2081 | [ 2082 | source 75 2083 | target 57 2084 | ] 2085 | edge 2086 | [ 2087 | source 75 2088 | target 44 2089 | ] 2090 | edge 2091 | [ 2092 | source 76 2093 | target 75 2094 | ] 2095 | edge 2096 | [ 2097 | source 76 2098 | target 27 2099 | ] 2100 | edge 2101 | [ 2102 | source 76 2103 | target 59 2104 | ] 2105 | edge 2106 | [ 2107 | source 76 2108 | target 20 2109 | ] 2110 | edge 2111 | [ 2112 | source 76 2113 | target 70 2114 | ] 2115 | edge 2116 | [ 2117 | source 76 2118 | target 66 2119 | ] 2120 | edge 2121 | [ 2122 | source 76 2123 | target 56 2124 | ] 2125 | edge 2126 | [ 2127 | source 76 2128 | target 62 2129 | ] 2130 | edge 2131 | [ 2132 | source 77 2133 | target 73 2134 | ] 2135 | edge 2136 | [ 2137 | source 77 2138 | target 22 2139 | ] 2140 | edge 2141 | [ 2142 | source 77 2143 | target 7 2144 | ] 2145 | edge 2146 | [ 2147 | source 77 2148 | target 51 2149 | ] 2150 | edge 2151 | [ 2152 | source 77 2153 | target 21 2154 | ] 2155 | edge 2156 | [ 2157 | source 77 2158 | target 8 2159 | ] 2160 | edge 2161 | [ 2162 | source 78 2163 | target 77 2164 | ] 2165 | edge 2166 | [ 2167 | source 78 2168 | target 23 2169 | ] 2170 | edge 2171 | [ 2172 | source 78 2173 | target 50 2174 | ] 2175 | edge 2176 | [ 2177 | source 78 2178 | target 28 2179 | ] 2180 | edge 2181 | [ 2182 | source 78 2183 | target 22 2184 | ] 2185 | edge 2186 | [ 2187 | source 78 2188 | target 8 2189 | ] 2190 | edge 2191 | [ 2192 | source 78 2193 | target 68 2194 | ] 2195 | edge 2196 | [ 2197 | source 78 2198 | target 7 2199 | ] 2200 | edge 2201 | [ 2202 | source 78 2203 | target 51 2204 | ] 2205 | edge 2206 | [ 2207 | source 79 2208 | target 31 2209 | ] 2210 | edge 2211 | [ 2212 | source 79 2213 | target 43 2214 | ] 2215 | edge 2216 | [ 2217 | source 79 2218 | target 30 2219 | ] 2220 | edge 2221 | [ 2222 | source 79 2223 | target 19 2224 | ] 2225 | edge 2226 | [ 2227 | source 79 2228 | target 29 2229 | ] 2230 | edge 2231 | [ 2232 | source 79 2233 | target 35 2234 | ] 2235 | edge 2236 | [ 2237 | source 79 2238 | target 55 2239 | ] 2240 | edge 2241 | [ 2242 | source 80 2243 | target 79 2244 | ] 2245 | edge 2246 | [ 2247 | source 80 2248 | target 37 2249 | ] 2250 | edge 2251 | [ 2252 | source 80 2253 | target 29 2254 | ] 2255 | edge 2256 | [ 2257 | source 81 2258 | target 16 2259 | ] 2260 | edge 2261 | [ 2262 | source 81 2263 | target 5 2264 | ] 2265 | edge 2266 | [ 2267 | source 81 2268 | target 40 2269 | ] 2270 | edge 2271 | [ 2272 | source 81 2273 | target 10 2274 | ] 2275 | edge 2276 | [ 2277 | source 81 2278 | target 72 2279 | ] 2280 | edge 2281 | [ 2282 | source 81 2283 | target 3 2284 | ] 2285 | edge 2286 | [ 2287 | source 82 2288 | target 81 2289 | ] 2290 | edge 2291 | [ 2292 | source 82 2293 | target 74 2294 | ] 2295 | edge 2296 | [ 2297 | source 82 2298 | target 39 2299 | ] 2300 | edge 2301 | [ 2302 | source 82 2303 | target 77 2304 | ] 2305 | edge 2306 | [ 2307 | source 82 2308 | target 80 2309 | ] 2310 | edge 2311 | [ 2312 | source 82 2313 | target 30 2314 | ] 2315 | edge 2316 | [ 2317 | source 82 2318 | target 29 2319 | ] 2320 | edge 2321 | [ 2322 | source 82 2323 | target 7 2324 | ] 2325 | edge 2326 | [ 2327 | source 83 2328 | target 53 2329 | ] 2330 | edge 2331 | [ 2332 | source 83 2333 | target 81 2334 | ] 2335 | edge 2336 | [ 2337 | source 83 2338 | target 69 2339 | ] 2340 | edge 2341 | [ 2342 | source 83 2343 | target 73 2344 | ] 2345 | edge 2346 | [ 2347 | source 83 2348 | target 46 2349 | ] 2350 | edge 2351 | [ 2352 | source 83 2353 | target 67 2354 | ] 2355 | edge 2356 | [ 2357 | source 83 2358 | target 49 2359 | ] 2360 | edge 2361 | [ 2362 | source 84 2363 | target 83 2364 | ] 2365 | edge 2366 | [ 2367 | source 84 2368 | target 24 2369 | ] 2370 | edge 2371 | [ 2372 | source 84 2373 | target 49 2374 | ] 2375 | edge 2376 | [ 2377 | source 84 2378 | target 52 2379 | ] 2380 | edge 2381 | [ 2382 | source 84 2383 | target 3 2384 | ] 2385 | edge 2386 | [ 2387 | source 84 2388 | target 74 2389 | ] 2390 | edge 2391 | [ 2392 | source 84 2393 | target 10 2394 | ] 2395 | edge 2396 | [ 2397 | source 84 2398 | target 81 2399 | ] 2400 | edge 2401 | [ 2402 | source 84 2403 | target 5 2404 | ] 2405 | edge 2406 | [ 2407 | source 85 2408 | target 6 2409 | ] 2410 | edge 2411 | [ 2412 | source 85 2413 | target 14 2414 | ] 2415 | edge 2416 | [ 2417 | source 85 2418 | target 38 2419 | ] 2420 | edge 2421 | [ 2422 | source 85 2423 | target 43 2424 | ] 2425 | edge 2426 | [ 2427 | source 85 2428 | target 80 2429 | ] 2430 | edge 2431 | [ 2432 | source 85 2433 | target 12 2434 | ] 2435 | edge 2436 | [ 2437 | source 85 2438 | target 26 2439 | ] 2440 | edge 2441 | [ 2442 | source 85 2443 | target 31 2444 | ] 2445 | edge 2446 | [ 2447 | source 86 2448 | target 44 2449 | ] 2450 | edge 2451 | [ 2452 | source 86 2453 | target 53 2454 | ] 2455 | edge 2456 | [ 2457 | source 86 2458 | target 75 2459 | ] 2460 | edge 2461 | [ 2462 | source 86 2463 | target 57 2464 | ] 2465 | edge 2466 | [ 2467 | source 86 2468 | target 48 2469 | ] 2470 | edge 2471 | [ 2472 | source 86 2473 | target 80 2474 | ] 2475 | edge 2476 | [ 2477 | source 86 2478 | target 66 2479 | ] 2480 | edge 2481 | [ 2482 | source 87 2483 | target 86 2484 | ] 2485 | edge 2486 | [ 2487 | source 87 2488 | target 17 2489 | ] 2490 | edge 2491 | [ 2492 | source 87 2493 | target 62 2494 | ] 2495 | edge 2496 | [ 2497 | source 87 2498 | target 56 2499 | ] 2500 | edge 2501 | [ 2502 | source 87 2503 | target 24 2504 | ] 2505 | edge 2506 | [ 2507 | source 87 2508 | target 20 2509 | ] 2510 | edge 2511 | [ 2512 | source 87 2513 | target 65 2514 | ] 2515 | edge 2516 | [ 2517 | source 88 2518 | target 49 2519 | ] 2520 | edge 2521 | [ 2522 | source 88 2523 | target 58 2524 | ] 2525 | edge 2526 | [ 2527 | source 88 2528 | target 83 2529 | ] 2530 | edge 2531 | [ 2532 | source 88 2533 | target 69 2534 | ] 2535 | edge 2536 | [ 2537 | source 88 2538 | target 46 2539 | ] 2540 | edge 2541 | [ 2542 | source 88 2543 | target 53 2544 | ] 2545 | edge 2546 | [ 2547 | source 88 2548 | target 73 2549 | ] 2550 | edge 2551 | [ 2552 | source 88 2553 | target 67 2554 | ] 2555 | edge 2556 | [ 2557 | source 89 2558 | target 88 2559 | ] 2560 | edge 2561 | [ 2562 | source 89 2563 | target 1 2564 | ] 2565 | edge 2566 | [ 2567 | source 89 2568 | target 37 2569 | ] 2570 | edge 2571 | [ 2572 | source 89 2573 | target 25 2574 | ] 2575 | edge 2576 | [ 2577 | source 89 2578 | target 33 2579 | ] 2580 | edge 2581 | [ 2582 | source 89 2583 | target 55 2584 | ] 2585 | edge 2586 | [ 2587 | source 89 2588 | target 45 2589 | ] 2590 | edge 2591 | [ 2592 | source 90 2593 | target 5 2594 | ] 2595 | edge 2596 | [ 2597 | source 90 2598 | target 8 2599 | ] 2600 | edge 2601 | [ 2602 | source 90 2603 | target 23 2604 | ] 2605 | edge 2606 | [ 2607 | source 90 2608 | target 0 2609 | ] 2610 | edge 2611 | [ 2612 | source 90 2613 | target 11 2614 | ] 2615 | edge 2616 | [ 2617 | source 90 2618 | target 50 2619 | ] 2620 | edge 2621 | [ 2622 | source 90 2623 | target 24 2624 | ] 2625 | edge 2626 | [ 2627 | source 90 2628 | target 69 2629 | ] 2630 | edge 2631 | [ 2632 | source 90 2633 | target 28 2634 | ] 2635 | edge 2636 | [ 2637 | source 91 2638 | target 29 2639 | ] 2640 | edge 2641 | [ 2642 | source 91 2643 | target 48 2644 | ] 2645 | edge 2646 | [ 2647 | source 91 2648 | target 66 2649 | ] 2650 | edge 2651 | [ 2652 | source 91 2653 | target 69 2654 | ] 2655 | edge 2656 | [ 2657 | source 91 2658 | target 44 2659 | ] 2660 | edge 2661 | [ 2662 | source 91 2663 | target 86 2664 | ] 2665 | edge 2666 | [ 2667 | source 91 2668 | target 57 2669 | ] 2670 | edge 2671 | [ 2672 | source 91 2673 | target 80 2674 | ] 2675 | edge 2676 | [ 2677 | source 92 2678 | target 91 2679 | ] 2680 | edge 2681 | [ 2682 | source 92 2683 | target 35 2684 | ] 2685 | edge 2686 | [ 2687 | source 92 2688 | target 15 2689 | ] 2690 | edge 2691 | [ 2692 | source 92 2693 | target 86 2694 | ] 2695 | edge 2696 | [ 2697 | source 92 2698 | target 48 2699 | ] 2700 | edge 2701 | [ 2702 | source 92 2703 | target 57 2704 | ] 2705 | edge 2706 | [ 2707 | source 92 2708 | target 61 2709 | ] 2710 | edge 2711 | [ 2712 | source 92 2713 | target 66 2714 | ] 2715 | edge 2716 | [ 2717 | source 92 2718 | target 75 2719 | ] 2720 | edge 2721 | [ 2722 | source 93 2723 | target 0 2724 | ] 2725 | edge 2726 | [ 2727 | source 93 2728 | target 23 2729 | ] 2730 | edge 2731 | [ 2732 | source 93 2733 | target 80 2734 | ] 2735 | edge 2736 | [ 2737 | source 93 2738 | target 16 2739 | ] 2740 | edge 2741 | [ 2742 | source 93 2743 | target 4 2744 | ] 2745 | edge 2746 | [ 2747 | source 93 2748 | target 82 2749 | ] 2750 | edge 2751 | [ 2752 | source 93 2753 | target 91 2754 | ] 2755 | edge 2756 | [ 2757 | source 93 2758 | target 41 2759 | ] 2760 | edge 2761 | [ 2762 | source 93 2763 | target 9 2764 | ] 2765 | edge 2766 | [ 2767 | source 94 2768 | target 34 2769 | ] 2770 | edge 2771 | [ 2772 | source 94 2773 | target 19 2774 | ] 2775 | edge 2776 | [ 2777 | source 94 2778 | target 55 2779 | ] 2780 | edge 2781 | [ 2782 | source 94 2783 | target 79 2784 | ] 2785 | edge 2786 | [ 2787 | source 94 2788 | target 80 2789 | ] 2790 | edge 2791 | [ 2792 | source 94 2793 | target 29 2794 | ] 2795 | edge 2796 | [ 2797 | source 94 2798 | target 30 2799 | ] 2800 | edge 2801 | [ 2802 | source 94 2803 | target 82 2804 | ] 2805 | edge 2806 | [ 2807 | source 94 2808 | target 35 2809 | ] 2810 | edge 2811 | [ 2812 | source 95 2813 | target 70 2814 | ] 2815 | edge 2816 | [ 2817 | source 95 2818 | target 69 2819 | ] 2820 | edge 2821 | [ 2822 | source 95 2823 | target 76 2824 | ] 2825 | edge 2826 | [ 2827 | source 95 2828 | target 62 2829 | ] 2830 | edge 2831 | [ 2832 | source 95 2833 | target 56 2834 | ] 2835 | edge 2836 | [ 2837 | source 95 2838 | target 27 2839 | ] 2840 | edge 2841 | [ 2842 | source 95 2843 | target 17 2844 | ] 2845 | edge 2846 | [ 2847 | source 95 2848 | target 87 2849 | ] 2850 | edge 2851 | [ 2852 | source 95 2853 | target 37 2854 | ] 2855 | edge 2856 | [ 2857 | source 96 2858 | target 48 2859 | ] 2860 | edge 2861 | [ 2862 | source 96 2863 | target 17 2864 | ] 2865 | edge 2866 | [ 2867 | source 96 2868 | target 76 2869 | ] 2870 | edge 2871 | [ 2872 | source 96 2873 | target 27 2874 | ] 2875 | edge 2876 | [ 2877 | source 96 2878 | target 56 2879 | ] 2880 | edge 2881 | [ 2882 | source 96 2883 | target 65 2884 | ] 2885 | edge 2886 | [ 2887 | source 96 2888 | target 20 2889 | ] 2890 | edge 2891 | [ 2892 | source 96 2893 | target 87 2894 | ] 2895 | edge 2896 | [ 2897 | source 97 2898 | target 5 2899 | ] 2900 | edge 2901 | [ 2902 | source 97 2903 | target 86 2904 | ] 2905 | edge 2906 | [ 2907 | source 97 2908 | target 58 2909 | ] 2910 | edge 2911 | [ 2912 | source 97 2913 | target 11 2914 | ] 2915 | edge 2916 | [ 2917 | source 97 2918 | target 59 2919 | ] 2920 | edge 2921 | [ 2922 | source 97 2923 | target 63 2924 | ] 2925 | edge 2926 | [ 2927 | source 98 2928 | target 97 2929 | ] 2930 | edge 2931 | [ 2932 | source 98 2933 | target 77 2934 | ] 2935 | edge 2936 | [ 2937 | source 98 2938 | target 48 2939 | ] 2940 | edge 2941 | [ 2942 | source 98 2943 | target 84 2944 | ] 2945 | edge 2946 | [ 2947 | source 98 2948 | target 40 2949 | ] 2950 | edge 2951 | [ 2952 | source 98 2953 | target 10 2954 | ] 2955 | edge 2956 | [ 2957 | source 98 2958 | target 5 2959 | ] 2960 | edge 2961 | [ 2962 | source 98 2963 | target 52 2964 | ] 2965 | edge 2966 | [ 2967 | source 98 2968 | target 81 2969 | ] 2970 | edge 2971 | [ 2972 | source 99 2973 | target 89 2974 | ] 2975 | edge 2976 | [ 2977 | source 99 2978 | target 34 2979 | ] 2980 | edge 2981 | [ 2982 | source 99 2983 | target 14 2984 | ] 2985 | edge 2986 | [ 2987 | source 99 2988 | target 85 2989 | ] 2990 | edge 2991 | [ 2992 | source 99 2993 | target 54 2994 | ] 2995 | edge 2996 | [ 2997 | source 99 2998 | target 18 2999 | ] 3000 | edge 3001 | [ 3002 | source 99 3003 | target 31 3004 | ] 3005 | edge 3006 | [ 3007 | source 99 3008 | target 61 3009 | ] 3010 | edge 3011 | [ 3012 | source 99 3013 | target 71 3014 | ] 3015 | edge 3016 | [ 3017 | source 100 3018 | target 99 3019 | ] 3020 | edge 3021 | [ 3022 | source 100 3023 | target 82 3024 | ] 3025 | edge 3026 | [ 3027 | source 100 3028 | target 13 3029 | ] 3030 | edge 3031 | [ 3032 | source 100 3033 | target 2 3034 | ] 3035 | edge 3036 | [ 3037 | source 100 3038 | target 15 3039 | ] 3040 | edge 3041 | [ 3042 | source 100 3043 | target 32 3044 | ] 3045 | edge 3046 | [ 3047 | source 100 3048 | target 64 3049 | ] 3050 | edge 3051 | [ 3052 | source 100 3053 | target 47 3054 | ] 3055 | edge 3056 | [ 3057 | source 100 3058 | target 39 3059 | ] 3060 | edge 3061 | [ 3062 | source 100 3063 | target 6 3064 | ] 3065 | edge 3066 | [ 3067 | source 101 3068 | target 51 3069 | ] 3070 | edge 3071 | [ 3072 | source 101 3073 | target 30 3074 | ] 3075 | edge 3076 | [ 3077 | source 101 3078 | target 94 3079 | ] 3080 | edge 3081 | [ 3082 | source 101 3083 | target 1 3084 | ] 3085 | edge 3086 | [ 3087 | source 101 3088 | target 79 3089 | ] 3090 | edge 3091 | [ 3092 | source 101 3093 | target 58 3094 | ] 3095 | edge 3096 | [ 3097 | source 101 3098 | target 19 3099 | ] 3100 | edge 3101 | [ 3102 | source 101 3103 | target 55 3104 | ] 3105 | edge 3106 | [ 3107 | source 101 3108 | target 35 3109 | ] 3110 | edge 3111 | [ 3112 | source 101 3113 | target 29 3114 | ] 3115 | edge 3116 | [ 3117 | source 102 3118 | target 100 3119 | ] 3120 | edge 3121 | [ 3122 | source 102 3123 | target 74 3124 | ] 3125 | edge 3126 | [ 3127 | source 102 3128 | target 52 3129 | ] 3130 | edge 3131 | [ 3132 | source 102 3133 | target 98 3134 | ] 3135 | edge 3136 | [ 3137 | source 102 3138 | target 72 3139 | ] 3140 | edge 3141 | [ 3142 | source 102 3143 | target 40 3144 | ] 3145 | edge 3146 | [ 3147 | source 102 3148 | target 10 3149 | ] 3150 | edge 3151 | [ 3152 | source 102 3153 | target 3 3154 | ] 3155 | edge 3156 | [ 3157 | source 103 3158 | target 102 3159 | ] 3160 | edge 3161 | [ 3162 | source 103 3163 | target 33 3164 | ] 3165 | edge 3166 | [ 3167 | source 103 3168 | target 45 3169 | ] 3170 | edge 3171 | [ 3172 | source 103 3173 | target 25 3174 | ] 3175 | edge 3176 | [ 3177 | source 103 3178 | target 89 3179 | ] 3180 | edge 3181 | [ 3182 | source 103 3183 | target 37 3184 | ] 3185 | edge 3186 | [ 3187 | source 103 3188 | target 1 3189 | ] 3190 | edge 3191 | [ 3192 | source 103 3193 | target 70 3194 | ] 3195 | edge 3196 | [ 3197 | source 104 3198 | target 72 3199 | ] 3200 | edge 3201 | [ 3202 | source 104 3203 | target 11 3204 | ] 3205 | edge 3206 | [ 3207 | source 104 3208 | target 0 3209 | ] 3210 | edge 3211 | [ 3212 | source 104 3213 | target 93 3214 | ] 3215 | edge 3216 | [ 3217 | source 104 3218 | target 67 3219 | ] 3220 | edge 3221 | [ 3222 | source 104 3223 | target 41 3224 | ] 3225 | edge 3226 | [ 3227 | source 104 3228 | target 16 3229 | ] 3230 | edge 3231 | [ 3232 | source 104 3233 | target 87 3234 | ] 3235 | edge 3236 | [ 3237 | source 104 3238 | target 23 3239 | ] 3240 | edge 3241 | [ 3242 | source 104 3243 | target 4 3244 | ] 3245 | edge 3246 | [ 3247 | source 104 3248 | target 9 3249 | ] 3250 | edge 3251 | [ 3252 | source 105 3253 | target 89 3254 | ] 3255 | edge 3256 | [ 3257 | source 105 3258 | target 103 3259 | ] 3260 | edge 3261 | [ 3262 | source 105 3263 | target 33 3264 | ] 3265 | edge 3266 | [ 3267 | source 105 3268 | target 62 3269 | ] 3270 | edge 3271 | [ 3272 | source 105 3273 | target 37 3274 | ] 3275 | edge 3276 | [ 3277 | source 105 3278 | target 45 3279 | ] 3280 | edge 3281 | [ 3282 | source 105 3283 | target 1 3284 | ] 3285 | edge 3286 | [ 3287 | source 105 3288 | target 80 3289 | ] 3290 | edge 3291 | [ 3292 | source 105 3293 | target 25 3294 | ] 3295 | edge 3296 | [ 3297 | source 106 3298 | target 25 3299 | ] 3300 | edge 3301 | [ 3302 | source 106 3303 | target 56 3304 | ] 3305 | edge 3306 | [ 3307 | source 106 3308 | target 92 3309 | ] 3310 | edge 3311 | [ 3312 | source 106 3313 | target 2 3314 | ] 3315 | edge 3316 | [ 3317 | source 106 3318 | target 13 3319 | ] 3320 | edge 3321 | [ 3322 | source 106 3323 | target 32 3324 | ] 3325 | edge 3326 | [ 3327 | source 106 3328 | target 60 3329 | ] 3330 | edge 3331 | [ 3332 | source 106 3333 | target 6 3334 | ] 3335 | edge 3336 | [ 3337 | source 106 3338 | target 64 3339 | ] 3340 | edge 3341 | [ 3342 | source 106 3343 | target 15 3344 | ] 3345 | edge 3346 | [ 3347 | source 106 3348 | target 39 3349 | ] 3350 | edge 3351 | [ 3352 | source 107 3353 | target 88 3354 | ] 3355 | edge 3356 | [ 3357 | source 107 3358 | target 75 3359 | ] 3360 | edge 3361 | [ 3362 | source 107 3363 | target 98 3364 | ] 3365 | edge 3366 | [ 3367 | source 107 3368 | target 102 3369 | ] 3370 | edge 3371 | [ 3372 | source 107 3373 | target 72 3374 | ] 3375 | edge 3376 | [ 3377 | source 107 3378 | target 40 3379 | ] 3380 | edge 3381 | [ 3382 | source 107 3383 | target 81 3384 | ] 3385 | edge 3386 | [ 3387 | source 107 3388 | target 5 3389 | ] 3390 | edge 3391 | [ 3392 | source 107 3393 | target 10 3394 | ] 3395 | edge 3396 | [ 3397 | source 107 3398 | target 84 3399 | ] 3400 | edge 3401 | [ 3402 | source 108 3403 | target 4 3404 | ] 3405 | edge 3406 | [ 3407 | source 108 3408 | target 9 3409 | ] 3410 | edge 3411 | [ 3412 | source 108 3413 | target 7 3414 | ] 3415 | edge 3416 | [ 3417 | source 108 3418 | target 51 3419 | ] 3420 | edge 3421 | [ 3422 | source 108 3423 | target 77 3424 | ] 3425 | edge 3426 | [ 3427 | source 108 3428 | target 21 3429 | ] 3430 | edge 3431 | [ 3432 | source 108 3433 | target 78 3434 | ] 3435 | edge 3436 | [ 3437 | source 108 3438 | target 22 3439 | ] 3440 | edge 3441 | [ 3442 | source 108 3443 | target 68 3444 | ] 3445 | edge 3446 | [ 3447 | source 109 3448 | target 79 3449 | ] 3450 | edge 3451 | [ 3452 | source 109 3453 | target 30 3454 | ] 3455 | edge 3456 | [ 3457 | source 109 3458 | target 63 3459 | ] 3460 | edge 3461 | [ 3462 | source 109 3463 | target 1 3464 | ] 3465 | edge 3466 | [ 3467 | source 109 3468 | target 33 3469 | ] 3470 | edge 3471 | [ 3472 | source 109 3473 | target 103 3474 | ] 3475 | edge 3476 | [ 3477 | source 109 3478 | target 105 3479 | ] 3480 | edge 3481 | [ 3482 | source 109 3483 | target 45 3484 | ] 3485 | edge 3486 | [ 3487 | source 109 3488 | target 25 3489 | ] 3490 | edge 3491 | [ 3492 | source 109 3493 | target 89 3494 | ] 3495 | edge 3496 | [ 3497 | source 109 3498 | target 37 3499 | ] 3500 | edge 3501 | [ 3502 | source 110 3503 | target 67 3504 | ] 3505 | edge 3506 | [ 3507 | source 110 3508 | target 13 3509 | ] 3510 | edge 3511 | [ 3512 | source 110 3513 | target 24 3514 | ] 3515 | edge 3516 | [ 3517 | source 110 3518 | target 80 3519 | ] 3520 | edge 3521 | [ 3522 | source 110 3523 | target 88 3524 | ] 3525 | edge 3526 | [ 3527 | source 110 3528 | target 49 3529 | ] 3530 | edge 3531 | [ 3532 | source 110 3533 | target 73 3534 | ] 3535 | edge 3536 | [ 3537 | source 110 3538 | target 46 3539 | ] 3540 | edge 3541 | [ 3542 | source 110 3543 | target 83 3544 | ] 3545 | edge 3546 | [ 3547 | source 110 3548 | target 53 3549 | ] 3550 | edge 3551 | [ 3552 | source 111 3553 | target 23 3554 | ] 3555 | edge 3556 | [ 3557 | source 111 3558 | target 64 3559 | ] 3560 | edge 3561 | [ 3562 | source 111 3563 | target 46 3564 | ] 3565 | edge 3566 | [ 3567 | source 111 3568 | target 78 3569 | ] 3570 | edge 3571 | [ 3572 | source 111 3573 | target 8 3574 | ] 3575 | edge 3576 | [ 3577 | source 111 3578 | target 21 3579 | ] 3580 | edge 3581 | [ 3582 | source 111 3583 | target 51 3584 | ] 3585 | edge 3586 | [ 3587 | source 111 3588 | target 7 3589 | ] 3590 | edge 3591 | [ 3592 | source 111 3593 | target 108 3594 | ] 3595 | edge 3596 | [ 3597 | source 111 3598 | target 68 3599 | ] 3600 | edge 3601 | [ 3602 | source 111 3603 | target 77 3604 | ] 3605 | edge 3606 | [ 3607 | source 112 3608 | target 52 3609 | ] 3610 | edge 3611 | [ 3612 | source 112 3613 | target 96 3614 | ] 3615 | edge 3616 | [ 3617 | source 112 3618 | target 97 3619 | ] 3620 | edge 3621 | [ 3622 | source 112 3623 | target 57 3624 | ] 3625 | edge 3626 | [ 3627 | source 112 3628 | target 66 3629 | ] 3630 | edge 3631 | [ 3632 | source 112 3633 | target 63 3634 | ] 3635 | edge 3636 | [ 3637 | source 112 3638 | target 44 3639 | ] 3640 | edge 3641 | [ 3642 | source 112 3643 | target 92 3644 | ] 3645 | edge 3646 | [ 3647 | source 112 3648 | target 75 3649 | ] 3650 | edge 3651 | [ 3652 | source 112 3653 | target 91 3654 | ] 3655 | edge 3656 | [ 3657 | source 113 3658 | target 28 3659 | ] 3660 | edge 3661 | [ 3662 | source 113 3663 | target 20 3664 | ] 3665 | edge 3666 | [ 3667 | source 113 3668 | target 95 3669 | ] 3670 | edge 3671 | [ 3672 | source 113 3673 | target 59 3674 | ] 3675 | edge 3676 | [ 3677 | source 113 3678 | target 70 3679 | ] 3680 | edge 3681 | [ 3682 | source 113 3683 | target 17 3684 | ] 3685 | edge 3686 | [ 3687 | source 113 3688 | target 87 3689 | ] 3690 | edge 3691 | [ 3692 | source 113 3693 | target 76 3694 | ] 3695 | edge 3696 | [ 3697 | source 113 3698 | target 65 3699 | ] 3700 | edge 3701 | [ 3702 | source 113 3703 | target 96 3704 | ] 3705 | edge 3706 | [ 3707 | source 114 3708 | target 83 3709 | ] 3710 | edge 3711 | [ 3712 | source 114 3713 | target 88 3714 | ] 3715 | edge 3716 | [ 3717 | source 114 3718 | target 110 3719 | ] 3720 | edge 3721 | [ 3722 | source 114 3723 | target 53 3724 | ] 3725 | edge 3726 | [ 3727 | source 114 3728 | target 49 3729 | ] 3730 | edge 3731 | [ 3732 | source 114 3733 | target 73 3734 | ] 3735 | edge 3736 | [ 3737 | source 114 3738 | target 46 3739 | ] 3740 | edge 3741 | [ 3742 | source 114 3743 | target 67 3744 | ] 3745 | edge 3746 | [ 3747 | source 114 3748 | target 58 3749 | ] 3750 | edge 3751 | [ 3752 | source 114 3753 | target 15 3754 | ] 3755 | edge 3756 | [ 3757 | source 114 3758 | target 104 3759 | ] 3760 | ] 3761 | --------------------------------------------------------------------------------