├── LICENSE ├── README.md ├── __init__.py ├── email-enron.txt ├── hypergraph.py ├── read.py ├── requirements.txt └── tutorial.ipynb /LICENSE: -------------------------------------------------------------------------------- 1 | MIT License 2 | 3 | Copyright (c) 2019 Phil Chodrow 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 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Configuration Models of Random Hypergraphs 2 | 3 | This is the software repository accompanying the [paper](https://arxiv.org/abs/1902.09302) "Configuration Models of Random Hypergraphs" by Phil Chodrow. It includes the `hypergraph` class used in conducting all analyses presented in the main text. Python 3.5+ is required, as well as the packages listed in `requirements.txt`. See `tutorial.ipynb` for an illustration of how to use the software to conduct significance tests against hypergraph null models. 4 | 5 | The illustration data is derived from the `email-Enron` corpus shared by Austin Benson on his data [website](https://www.cs.cornell.edu/~arb/data/). 6 | 7 | 8 | -------------------------------------------------------------------------------- /__init__.py: -------------------------------------------------------------------------------- 1 | 2 | -------------------------------------------------------------------------------- /email-enron.txt: -------------------------------------------------------------------------------- 1 | 0 3 2 | 0 112 123 3 | 0 46 4 | 0 46 5 | 0 36 6 | 0 112 7 | 0 36 8 | 0 60 9 | 0 36 10 | 0 102 11 | 0 102 12 | 0 123 13 | 0 117 14 | 0 46 15 | 0 46 16 | 0 36 17 | 0 26 117 18 | 0 26 57 60 117 127 19 | 0 112 20 | 0 3 21 | 0 60 22 | 0 60 23 | 0 46 24 | 0 36 46 25 | 0 60 26 | 0 36 60 27 | 0 46 28 | 0 66 29 | 0 112 30 | 0 112 123 31 | 0 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4641 | 50 109 4642 | 22 50 4643 | 22 42 50 99 4644 | 22 50 4645 | 22 42 50 99 4646 | 50 126 4647 | 50 126 4648 | 22 42 50 99 4649 | 22 42 50 99 4650 | 22 42 50 99 4651 | 22 42 50 99 4652 | 4 50 109 126 4653 | 22 42 50 99 4654 | 22 42 50 99 4655 | 50 56 4656 | 4 50 109 126 4657 | 22 42 50 99 4658 | 50 126 4659 | 50 56 4660 | 22 42 50 99 4661 | 4 50 109 126 4662 | 4 50 109 126 4663 | 4 50 109 126 4664 | 4 50 4665 | 22 42 50 99 4666 | 22 50 4667 | 50 56 4668 | 22 42 50 99 4669 | 4 50 109 126 4670 | 22 42 50 99 4671 | 50 109 126 4672 | 4 50 4673 | 22 42 50 99 4674 | 4 50 109 126 4675 | 22 42 50 99 4676 | 22 42 50 99 4677 | 22 42 50 99 4678 | 50 56 4679 | 50 56 4680 | 50 56 4681 | 22 42 50 99 4682 | 22 42 50 99 4683 | 50 109 4684 | 4 50 109 126 4685 | 50 126 4686 | 22 42 50 99 4687 | 50 124 4688 | 4 49 50 109 126 4689 | 50 109 4690 | 50 99 4691 | 22 42 50 99 4692 | 22 42 50 99 4693 | 50 126 4694 | 22 50 4695 | 42 50 4696 | 22 42 50 99 4697 | 50 126 4698 | 4 50 109 126 4699 | 22 42 50 99 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4820 | 22 42 50 99 4821 | 4 50 82 109 126 4822 | 22 50 4823 | 22 50 4824 | 4 50 4825 | 22 42 50 99 4826 | 22 42 50 99 4827 | 50 109 4828 | 4 50 82 109 126 4829 | 50 126 4830 | 22 42 50 99 4831 | 50 82 4832 | 22 42 50 4833 | 42 50 4834 | 22 42 50 4835 | 22 42 50 4836 | 22 42 50 4837 | 4 50 82 109 126 4838 | 22 42 50 4839 | 22 42 50 4840 | 4 50 82 109 126 4841 | 22 42 50 4842 | 50 126 4843 | 22 42 50 4844 | 50 126 4845 | 22 42 50 4846 | 50 109 126 4847 | 22 42 50 4848 | 4 50 82 109 126 4849 | 50 126 4850 | 22 42 50 4851 | 22 50 4852 | 1 50 88 110 4853 | 22 42 50 4854 | 49 50 4855 | 22 42 50 4856 | 49 50 4857 | 50 126 4858 | 22 42 50 4859 | 22 42 50 4860 | 18 50 88 124 142 4861 | 22 42 50 4862 | 42 50 4863 | 42 50 4864 | 22 42 50 4865 | 22 42 50 4866 | 22 42 50 4867 | 50 126 4868 | 50 109 126 4869 | 4 50 4870 | 22 42 50 4871 | 22 42 50 4872 | 22 42 50 4873 | 50 126 4874 | 42 50 4875 | 42 50 4876 | 50 126 4877 | 50 109 4878 | 50 109 4879 | 50 109 4880 | 22 42 50 4881 | 22 42 50 4882 | 42 50 4883 | 22 42 50 4884 | 50 109 4885 | 50 109 126 4886 | 22 42 50 4887 | 50 109 4888 | 22 42 50 4889 | 4 50 82 109 126 4890 | 4 50 82 109 126 4891 | 50 126 4892 | 22 42 50 4893 | 4 50 82 109 126 4894 | 4 50 82 4895 | 50 109 4896 | 4 50 82 109 126 4897 | 50 109 4898 | 22 42 50 4899 | 22 42 50 4900 | 4 50 82 109 126 4901 | 42 50 4902 | 22 42 50 4903 | 22 42 50 4904 | 50 109 4905 | 22 42 50 4906 | 22 42 50 4907 | 18 50 124 142 4908 | 22 42 50 4909 | 4 50 4910 | 42 50 4911 | 22 42 50 4912 | 42 50 4913 | 22 42 50 4914 | 4 50 82 109 126 4915 | 50 109 126 4916 | 50 126 4917 | 22 50 126 4918 | 4 50 82 109 126 4919 | 22 42 50 4920 | 22 42 50 4921 | 50 126 4922 | 22 42 50 4923 | 4 50 82 109 126 4924 | 42 50 4925 | 42 50 4926 | 50 109 4927 | 50 126 4928 | 50 56 4929 | 50 56 4930 | 22 42 50 4931 | 22 42 50 4932 | 4 50 82 109 126 4933 | 50 109 126 4934 | 50 109 126 4935 | 22 42 50 4936 | 50 126 4937 | 50 109 4938 | 4 50 82 109 126 4939 | 4 50 82 4940 | 4 50 109 4941 | 22 42 50 4942 | 22 42 50 4943 | 50 56 141 4944 | 4 50 82 109 126 4945 | 22 42 50 4946 | 22 50 4947 | 50 126 4948 | 50 126 4949 | 22 42 50 4950 | 22 42 50 4951 | 50 126 4952 | 22 42 50 126 4953 | 50 56 141 4954 | 22 42 50 4955 | 50 56 141 4956 | 22 42 50 4957 | 22 42 50 4958 | 22 42 50 4959 | 4 50 82 109 126 4960 | 22 42 50 4961 | 42 50 4962 | 50 82 4963 | 22 42 50 4964 | 42 50 4965 | 4 50 82 109 126 4966 | 42 50 4967 | 22 42 50 4968 | 22 42 50 4969 | 4 50 82 4970 | 50 56 141 4971 | 22 42 50 4972 | 42 50 4973 | 42 50 4974 | 4 50 82 109 126 4975 | 22 42 50 4976 | 50 56 141 4977 | 22 42 50 4978 | 22 42 50 4979 | 4 50 82 109 126 4980 | 50 126 4981 | 22 42 50 4982 | 22 42 50 4983 | 50 56 141 4984 | 4 50 4985 | 22 42 50 4986 | 22 42 50 4987 | 1 50 67 90 4988 | 42 50 126 4989 | 22 42 50 4990 | 50 56 141 4991 | 50 56 141 4992 | 4 50 82 109 126 4993 | 50 56 141 4994 | 22 42 50 4995 | 42 50 126 4996 | 42 50 4997 | 42 50 4998 | 4 50 82 109 126 4999 | 4 50 82 5000 | 4 50 82 109 126 5001 | 42 50 126 5002 | 22 42 50 5003 | 4 50 5004 | 50 109 5005 | 50 109 5006 | 22 42 50 5007 | 42 50 5008 | 50 109 5009 | 42 50 5010 | 42 50 126 5011 | 4 50 5012 | 4 50 5013 | 50 56 141 5014 | 50 126 5015 | 22 42 50 5016 | 22 42 50 5017 | 50 56 141 5018 | 50 56 141 5019 | 50 109 126 5020 | 22 42 50 5021 | 42 50 5022 | 50 82 5023 | 4 50 82 109 126 5024 | 50 109 126 5025 | 50 109 5026 | 4 50 82 5027 | 50 109 5028 | 4 50 82 109 126 5029 | 50 126 5030 | 22 42 50 5031 | 4 50 5032 | 22 42 50 5033 | 50 126 5034 | 22 42 50 5035 | 22 42 50 5036 | 22 42 50 5037 | 42 50 5038 | 42 50 5039 | 22 42 50 5040 | 50 99 126 5041 | 4 50 5042 | 50 126 5043 | 50 56 141 5044 | 4 50 82 109 126 5045 | 50 109 5046 | 50 109 5047 | 22 42 50 5048 | 50 126 5049 | 50 109 5050 | 22 42 50 5051 | 50 126 5052 | 4 50 82 109 126 5053 | 50 56 141 5054 | 50 56 141 5055 | 42 50 5056 | 50 56 141 5057 | 42 50 5058 | 42 50 5059 | 42 50 5060 | 42 50 5061 | 4 50 82 5062 | 42 50 5063 | 42 50 5064 | 42 50 5065 | 42 50 5066 | 42 50 5067 | 50 103 5068 | 50 56 141 5069 | 50 126 5070 | 42 50 5071 | 42 50 5072 | 4 50 82 109 126 5073 | 50 126 5074 | 4 50 82 109 126 5075 | 4 50 82 5076 | 42 50 5077 | 42 50 5078 | 42 50 5079 | 50 56 141 5080 | 42 50 5081 | 42 50 5082 | 42 50 5083 | 42 50 5084 | 42 50 5085 | 42 50 5086 | 42 50 5087 | 50 56 5088 | 50 82 5089 | 50 109 5090 | 50 126 5091 | 42 50 5092 | 42 50 5093 | 50 126 5094 | 50 126 5095 | 4 50 82 109 126 5096 | 50 109 5097 | 42 50 5098 | 50 82 5099 | 50 82 5100 | 4 50 82 109 126 5101 | 50 126 5102 | 50 137 5103 | 4 50 82 5104 | 42 50 5105 | 42 50 5106 | 50 109 5107 | 42 50 5108 | 42 50 5109 | 42 50 5110 | 50 126 5111 | 42 50 5112 | 4 50 82 109 126 5113 | 42 50 5114 | 1 50 90 5115 | 4 50 82 109 126 5116 | 42 50 5117 | 4 50 82 109 126 5118 | 42 50 5119 | 22 50 126 5120 | 42 50 5121 | 50 126 5122 | 22 50 126 5123 | 50 126 5124 | 42 50 5125 | 42 50 5126 | 42 50 5127 | 42 50 5128 | 22 50 126 5129 | 4 50 82 109 126 5130 | 50 126 5131 | 42 50 5132 | 42 50 5133 | 4 50 82 109 126 5134 | 50 109 5135 | 50 68 5136 | 4 50 82 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5205 | 42 50 5206 | 42 50 5207 | 42 50 5208 | 1 50 90 5209 | 50 82 5210 | 4 50 82 109 126 5211 | 42 50 5212 | 42 50 5213 | 50 126 5214 | 42 50 5215 | 42 50 5216 | 42 50 5217 | 42 50 5218 | 50 126 5219 | 50 126 5220 | 50 126 5221 | 42 50 5222 | 42 50 5223 | 50 109 126 5224 | 42 50 5225 | 50 126 5226 | 42 50 5227 | 50 82 5228 | 42 50 5229 | 50 126 5230 | 42 50 5231 | 50 137 5232 | 50 109 5233 | 50 137 5234 | 4 42 50 82 109 126 5235 | 42 50 5236 | 42 50 126 5237 | 42 50 5238 | 50 141 5239 | 42 50 5240 | 42 50 5241 | 42 50 5242 | 42 50 5243 | 42 50 5244 | 4 42 50 82 109 126 5245 | 42 50 5246 | 1 50 90 5247 | 42 50 5248 | 42 50 5249 | 4 50 82 109 126 5250 | 42 50 5251 | 1 50 90 5252 | 42 50 5253 | 42 50 5254 | 42 50 5255 | 1 50 90 5256 | 4 42 50 82 109 126 5257 | 42 50 5258 | 42 50 5259 | 42 50 5260 | 42 50 5261 | 42 50 5262 | 4 50 82 109 126 5263 | 42 50 5264 | 50 126 5265 | 42 50 5266 | 42 50 5267 | 4 50 5268 | 4 42 50 82 5269 | 50 109 5270 | 4 50 5271 | 4 50 82 109 126 5272 | 50 126 5273 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10705 | 19 37 62 114 116 125 138 10706 | 125 138 10707 | 19 37 62 114 116 125 138 10708 | 125 138 10709 | 138 10710 | 138 10711 | 138 10712 | 108 138 10713 | 15 138 10714 | 19 37 62 114 116 125 138 10715 | 125 138 10716 | 19 37 62 114 116 125 138 10717 | 19 37 62 114 116 125 138 10718 | 125 138 10719 | 19 37 62 114 116 125 138 10720 | 19 37 62 114 116 125 138 10721 | 19 37 62 114 116 125 138 10722 | 19 37 62 114 116 125 138 10723 | 19 37 62 114 116 125 138 10724 | 19 37 62 114 116 125 138 10725 | 138 10726 | 138 10727 | 125 138 10728 | 125 138 10729 | 125 138 10730 | 19 37 62 114 116 125 138 10731 | 19 37 62 114 116 125 138 10732 | 125 138 10733 | 138 10734 | 125 138 10735 | 138 10736 | 19 37 62 114 116 125 138 10737 | 19 37 62 114 116 125 138 10738 | 125 138 10739 | 125 138 10740 | 125 138 10741 | 108 138 10742 | 108 138 10743 | 125 138 10744 | 37 116 138 10745 | 19 37 62 114 116 125 138 10746 | 19 37 62 114 116 125 138 10747 | 125 138 10748 | 104 138 10749 | 125 138 10750 | 114 138 10751 | 138 10752 | 138 10753 | 138 10754 | 125 138 10755 | 37 138 10756 | 125 138 10757 | 19 37 62 114 116 125 138 10758 | 125 138 10759 | 19 37 62 114 116 125 138 10760 | 19 37 62 114 116 125 138 10761 | 19 37 62 77 114 116 125 138 10762 | 125 138 10763 | 125 138 10764 | 125 138 10765 | 125 138 10766 | 125 138 10767 | 125 138 10768 | 125 138 10769 | 125 138 10770 | 125 138 10771 | 138 10772 | 138 10773 | 138 10774 | 108 138 10775 | 15 138 10776 | 114 138 10777 | 19 37 76 77 114 116 125 138 10778 | 19 37 76 77 114 116 125 138 10779 | 19 37 77 114 116 138 10780 | 19 138 10781 | 114 138 10782 | 19 138 10783 | 19 37 77 114 116 138 10784 | 19 37 77 114 116 138 10785 | 116 138 10786 | 19 37 77 114 116 138 10787 | 121 138 10788 | 77 138 10789 | 138 10790 | 19 37 77 101 114 116 138 10791 | 26 36 60 139 10792 | 26 139 10793 | 26 139 10794 | 60 139 10795 | 60 139 10796 | 140 10797 | 140 10798 | 40 140 10799 | 24 140 10800 | 40 63 132 140 10801 | 24 132 140 10802 | 40 140 10803 | 132 140 10804 | 91 140 10805 | 24 140 10806 | 58 141 10807 | 58 141 10808 | 56 134 141 10809 | 56 141 10810 | 56 141 10811 | 70 141 10812 | 2 141 10813 | 70 141 10814 | 134 141 10815 | 134 141 10816 | 75 141 10817 | 1 141 10818 | 75 141 10819 | 75 141 10820 | 56 141 10821 | 75 141 10822 | 75 141 10823 | 56 141 10824 | 56 141 10825 | 75 141 10826 | 35 141 10827 | 1 141 10828 | 75 141 10829 | 75 141 10830 | 56 141 10831 | 75 141 10832 | 75 141 10833 | 70 141 10834 | 75 141 10835 | 1 141 10836 | 1 141 10837 | 75 141 10838 | 75 141 10839 | 75 141 10840 | 70 141 10841 | 70 141 10842 | 106 141 10843 | 134 141 10844 | 106 141 10845 | 75 141 10846 | 35 141 10847 | 106 141 10848 | 1 141 10849 | 70 141 10850 | 56 58 134 141 10851 | 58 141 10852 | 75 141 10853 | 93 142 10854 | 58 142 10855 | 58 142 10856 | 74 142 10857 | 74 142 10858 | 93 142 10859 | 58 142 10860 | 58 142 10861 | 58 142 10862 | 93 142 10863 | 93 142 10864 | 93 142 10865 | 93 142 10866 | 36 142 10867 | 93 142 10868 | 93 142 10869 | 25 142 10870 | 93 142 10871 | 58 142 10872 | 93 142 10873 | 58 142 10874 | 58 142 10875 | 93 142 10876 | 93 142 10877 | 58 142 10878 | 56 142 10879 | 93 142 10880 | 25 93 142 10881 | 25 142 10882 | 134 142 10883 | 93 142 10884 | 25 142 10885 | 25 93 142 10886 | 10887 | -------------------------------------------------------------------------------- /hypergraph.py: -------------------------------------------------------------------------------- 1 | ''' 2 | An object of class hypergraph is a list of tuples on a specified node set, which can be implicit. 3 | It is equipped with methods for computing hypergraph moments of interest and running Markov Chain Monte Carlo. 4 | ''' 5 | 6 | import numpy as np 7 | import networkx as nx 8 | from collections import Counter 9 | from itertools import accumulate 10 | from bisect import bisect 11 | import random 12 | import itertools 13 | from scipy.special import binom 14 | import random 15 | 16 | class hypergraph: 17 | 18 | def __init__(self, C, n_nodes = None, node_labels = None): 19 | 20 | 21 | self.C = [tuple(sorted(f)) for f in C] # edge list 22 | 23 | self.nodes = list(set([v for f in self.C for v in f])) # node list 24 | self.n = len(self.nodes) # number of nodes 25 | 26 | # optional node labels -- not really used for anything at the moment 27 | if node_labels is not None: 28 | self.node_labels = node_labels 29 | 30 | # number of edges 31 | self.m = len(self.C) 32 | 33 | # node degree vector 34 | D = np.zeros(self.n) 35 | for f in self.C: 36 | for v in f: 37 | D[v] += 1 38 | self.D = D 39 | 40 | # edge dimension sequence 41 | K = np.array([len(f) for f in self.C]) 42 | self.K = K 43 | 44 | # bookkeeping for Monte Carlo 45 | self.MH_rounds = 0 46 | self.MH_steps = 0 47 | self.acceptance_rate = 0 48 | 49 | def node_degrees(self, by_dimension = False): 50 | ''' 51 | Return a np.array() of node degrees. If by_dimension, return a 2d np.array() 52 | in which each entry gives the number of edges of each dimension incident upon the given node. 53 | ''' 54 | if not by_dimension: 55 | return(self.D) 56 | else: 57 | D = np.zeros((len(self.D), max(self.K))) 58 | for f in self.C: 59 | for v in f: 60 | D[v, len(f)-1] += 1 61 | return(D) 62 | 63 | def edge_dimensions(self): 64 | ''' 65 | Return an np.array() of edge dimensions. 66 | ''' 67 | return(self.K) 68 | 69 | def node_dimension_matrix(self): 70 | ''' 71 | Return a matrix in which the i,j entry gives the number of dimension j edges incident on node i. 72 | ''' 73 | A = np.zeros((self.n, max([len(f) for f in self.C])+1)) 74 | for f in self.C: 75 | for v in f: 76 | A[v, len(f)] += 1 77 | return(A) 78 | 79 | 80 | def line_graph(self): 81 | ''' 82 | Return a networkx Graph() in which each node corresponds to a hyperedge 83 | and two nodes are linked if the corresponding edges intersect in the primal hypergraph. 84 | ''' 85 | H = nx.Graph() 86 | 87 | counts = Counter(self.C) 88 | d = {v : counts[v] for v in counts} 89 | 90 | H.add_nodes_from(d.keys()) 91 | nx.set_node_attributes(H, values = d, name = 'm') 92 | 93 | node_list = list(H.nodes()) 94 | n_nodes = len(node_list) 95 | for u in range(n_nodes): 96 | for v in range(u+1, n_nodes): 97 | j = len(set(node_list[u]).intersection(set(node_list[v]))) 98 | if j > 0: 99 | H.add_edge(node_list[u],node_list[v], weight = j) 100 | return(H) 101 | 102 | def get_edges(self, node): 103 | ''' 104 | Return a list of edges incident upon a specified node. 105 | ''' 106 | return([f for f in self.C if node in f]) 107 | 108 | def remove_degeneracy(self, verbose = True): 109 | ''' 110 | Use pairwise reshuffles to remove degenerate edges, as may be generated in stub-matching. 111 | ''' 112 | m_degenerate = self.check_degeneracy() 113 | while self.check_degeneracy() > 0: 114 | for i in range(len(self.C)): 115 | while is_degenerate(self.C[i]): 116 | j = np.random.choice(range(len(self.C))) 117 | f1, f2 = self.C[i], self.C[j] 118 | self.C[i], self.C[j] = pairwise_reshuffle(f1, f2, True) 119 | if verbose: 120 | print(str(m_degenerate) + ' degeneracies removed, ' + str(self.check_degeneracy()) + ' remain.') 121 | 122 | def MH(self, n_steps = 1000, verbose = True, label = 'edge', n_clash = 1, detailed = False, **kwargs): 123 | ''' 124 | Conduct Markov Chain Monte Carlo in order to approximately sample from the space of appropriately-labeled graphs. 125 | n_steps: number of steps to perform 126 | verbose: if True, print a finishing message with descriptive summaries of the algorithm run. 127 | label: the label space to use. Can take values in ['vertex' , 'stub', 'edge']. 128 | n_clash: the number of clashes permitted when updating the edge counts in vertex-labeled MH. n_clash = 0 will be exact but very slow. n_clash >= 2 may lead to performance gains at the cost of decreased accuracy. 129 | detailed: if True, preserve the number of edges of given dimension incident to each node 130 | **kwargs: additional arguments passed to sample_fun 131 | ''' 132 | if (label == 'edge') or (label == 'stub'): 133 | self.stub_edge_MH(n_steps = n_steps, verbose = verbose, label = label, detailed = detailed, **kwargs) 134 | elif label == 'vertex': 135 | self.vertex_labeled_MH(n_steps = n_steps, verbose = verbose, n_clash = n_clash, detailed = detailed, **kwargs) 136 | else: 137 | print('not implemented') 138 | 139 | def stub_edge_MH(self, n_steps = 1000, verbose = True, label = 'edge', detailed = False, message = True, **kwargs): 140 | ''' 141 | See description of self.MH() 142 | ''' 143 | 144 | C_new = [list(c) for c in self.C] 145 | m = len(C_new) 146 | 147 | proposal = proposal_generator(m, detailed) 148 | 149 | def MH_step(label = 'edge'): 150 | i, j, f1, f2, g1, g2 = proposal(C_new) 151 | C_new[i] = sorted(g1) 152 | C_new[j] = sorted(g2) 153 | 154 | n = 0 155 | n_rejected = 0 156 | 157 | while n < n_steps: 158 | MH_step() 159 | n += 1 160 | 161 | self.C = [tuple(sorted(f)) for f in C_new] 162 | self.MH_steps += n 163 | self.MH_rounds += 1 164 | 165 | if message: 166 | print(str(n_steps) + ' steps completed.') 167 | 168 | def vertex_labeled_MH(self, n_steps = 10000, sample_every = 500, sample_fun = None, verbose = False, n_clash = 0, message = True, detailed = False, **kwargs): 169 | ''' 170 | See description of self.MH() 171 | ''' 172 | 173 | rand = np.random.rand 174 | randint = np.random.randint 175 | 176 | k = 0 177 | done = False 178 | c = Counter(self.C) 179 | 180 | epoch_num = 0 181 | n_rejected = 0 182 | 183 | m = sum(c.values()) 184 | 185 | while not done: 186 | # initialize epoch 187 | 188 | l = list(c.elements()) 189 | 190 | add = [] 191 | remove = [] 192 | 193 | end_epoch = False 194 | num_clash = 0 195 | 196 | epoch_num += 1 197 | 198 | # within each epoch 199 | 200 | k_rand = 20000 # generate many random numbers at a time 201 | 202 | k_ = 0 203 | IJ = randint(0, m, k_rand) 204 | A = rand(k_rand) 205 | while True: 206 | if k_ >= k_rand/2.0: 207 | IJ = randint(0, m, k_rand) 208 | A = rand(k_rand) 209 | k_ = 0 210 | i,j = (IJ[k_],IJ[k_+1]) 211 | k_ += 2 212 | 213 | f1, f2 = l[i], l[j] 214 | while f1 == f2: 215 | i,j = (IJ[k_],IJ[k_+1]) 216 | k_ += 2 217 | f1, f2 = l[i], l[j] 218 | if detailed: 219 | while len(f1) != len(f2): 220 | i,j = (IJ[k_],IJ[k_+1]) 221 | k_ += 2 222 | f1, f2 = l[i], l[j] 223 | while f1 == f2: 224 | i,j = (IJ[k_],IJ[k_+1]) 225 | k_ += 2 226 | f1, f2 = l[i], l[j] 227 | 228 | inter = 2**(-len((set(f1).intersection(set(f2))))) 229 | if A[k_] > inter /(c[f1] * c[f2]): 230 | n_rejected += 1 231 | k += 1 232 | else: # if proposal was accepted 233 | g1, g2 = pairwise_reshuffle(f1, f2, True) 234 | num_clash += remove.count(f1) + remove.count(f2) 235 | if (num_clash >= n_clash) & (n_clash >=1): 236 | break 237 | else: 238 | remove.append(f1) 239 | remove.append(f2) 240 | add.append(g1) 241 | add.append(g2) 242 | k += 1 243 | if n_clash == 0: 244 | break 245 | 246 | add = Counter(add) 247 | add.subtract(Counter(remove)) 248 | 249 | c.update(add) 250 | done = k - n_rejected>=n_steps 251 | if message: 252 | print(str(epoch_num) + ' epochs completed, ' + str(k - n_rejected) + ' steps taken, ' + str(n_rejected) + ' steps rejected.') 253 | self.C = [tuple(sorted(f)) for f in list(c.elements())] 254 | self.MH_steps += k - n_rejected 255 | self.MH_rounds += 1 256 | self.acceptance_rate = (1.0*(k - n_rejected)) / (k) 257 | 258 | 259 | def check_degeneracy(self): 260 | ''' 261 | Find the number of degeneracies in self.C 262 | ''' 263 | return np.sum([is_degenerate(f) for f in self.C]) 264 | def choose_nodes(self, n_samples, choice_function = 'uniform'): 265 | ''' 266 | Utility function for choosing pairs of nodes from self.C, used in assortativity calculations. 267 | ''' 268 | D = self.node_degrees() 269 | 270 | def uniform(x): 271 | i = np.random.randint(len(x)) 272 | j = i 273 | while i == j: 274 | j = np.random.randint(len(x)) 275 | return(np.array([x[i],x[j]])) 276 | 277 | def top_2(x): 278 | ind = np.argpartition(D[x,], -2)[-2:] 279 | y = np.array(x)[ind] 280 | random.shuffle(y) 281 | return(y) 282 | 283 | def top_bottom(x): 284 | top = np.argmax(D[x,]) 285 | bottom = np.argmin(D[x,]) 286 | y = np.array(x)[[bottom, top]] 287 | random.shuffle(y) 288 | return(y) 289 | 290 | choice_functions = { 291 | 'uniform': uniform, 292 | 'top_2' : top_2, 293 | 'top_bottom' : top_bottom, 294 | 'NA' : uniform 295 | } 296 | 297 | n = 0 298 | v = [] 299 | while True: 300 | edge = self.C[np.random.randint(self.m)] 301 | if len(edge) < 2: 302 | continue 303 | x = choice_functions[choice_function](edge) 304 | v.append(x) 305 | n+=1 306 | if n > n_samples: 307 | break 308 | return(v) 309 | 310 | def assortativity(self, n_samples = 10, choice_function = 'uniform', method = 'pearson'): 311 | ''' 312 | Compute the approximate degree assortativity of a hypergraph using the specified choice_function and method in ['spearman', 'pearson'] 313 | ''' 314 | D = self.node_degrees() 315 | arr = np.array(self.choose_nodes(n_samples, choice_function)) 316 | arr = D[arr] 317 | 318 | if method == 'spearman': 319 | order = np.argsort(arr, axis = 0) 320 | arr = np.argsort(order, axis = 0) 321 | elif method == 'pearson': 322 | arr = arr - 1 323 | 324 | return(np.corrcoef(arr.T))[0,1] 325 | 326 | is_degenerate = lambda x: len(set(x)) < len(x) 327 | 328 | def proposal_generator(m, detailed = False): 329 | ''' 330 | Propose a transition in stub- and edge-labeled MH. 331 | ''' 332 | def proposal(edge_list): 333 | i,j = np.random.randint(0,m,2) 334 | f1, f2 = edge_list[i], edge_list[j] 335 | if detailed: 336 | while len(f1) != len(f2): 337 | i,j = np.random.randint(0,m,2) 338 | f1, f2 = edge_list[i], edge_list[j] 339 | g1, g2 = pairwise_reshuffle(f1, f2, True) 340 | return(i, j, f1, f2, g1, g2) 341 | return(proposal) 342 | 343 | def pairwise_reshuffle(f1, f2, preserve_dimensions = True): 344 | ''' 345 | Randomly reshuffle the nodes of two edges while preserving their sizes. 346 | ''' 347 | 348 | f = list(f1) + list(f2) 349 | s = set(f) 350 | 351 | intersection = set(f1).intersection(set(f2)) 352 | ix = list(intersection) 353 | 354 | g1 = ix.copy() 355 | g2 = ix.copy() 356 | 357 | for v in ix: 358 | f.remove(v) 359 | f.remove(v) 360 | 361 | for v in f: 362 | if (len(g1) < len(f1)) & (len(g2) < len(f2)): 363 | if np.random.rand() < .5: 364 | g1.append(v) 365 | else: 366 | g2.append(v) 367 | elif len(g1) < len(f1): 368 | g1.append(v) 369 | elif len(g2) < len(f2): 370 | g2.append(v) 371 | if len(g1) != len(f1): 372 | print('oops') 373 | print(f1, f2, g1, g2) 374 | return (tuple(sorted(g1)), tuple(sorted(g2))) 375 | 376 | def projected_graph(C, weighted = False, as_hyper = False, multi = True): 377 | ''' 378 | Compute the projected (clique) graph corresponding to a given hypergraph. Can be slow when many high-dimensional edges are present. 379 | ''' 380 | if not as_hyper: 381 | if multi: 382 | G = nx.MultiGraph() 383 | else: 384 | G = nx.Graph() 385 | G.add_nodes_from(C.nodes) 386 | for f in C.C: 387 | if weighted: 388 | if len(f) >= 2: 389 | G.add_edges_from(itertools.combinations(f, 2), weight = 1.0/(len(f) - 1)) 390 | else : 391 | G.add_edges_from(itertools.combinations(f, 2)) 392 | return(G) 393 | else: 394 | G = [f for F in C.C for f in itertools.combinations(F, 2)] 395 | return(hypergraph(G, n_nodes = len(C.nodes))) -------------------------------------------------------------------------------- /read.py: -------------------------------------------------------------------------------- 1 | import numpy as np 2 | from scipy.stats import rankdata 3 | 4 | def read_data(data, t_min = None, t_max = None): 5 | 6 | path = data + '/' + data + '-' 7 | # read in the data 8 | 9 | nverts = np.array([int(f.rstrip('\n')) for f in open(path + 'nverts.txt')]) 10 | times = np.array([float(f.rstrip('\n')) for f in open(path + 'times.txt')]) 11 | simplices = np.array([int(f.rstrip('\n')) for f in open(path + 'simplices.txt')]) 12 | 13 | times_ex = np.repeat(times, nverts) 14 | 15 | # time filtering 16 | 17 | t_ix = np.repeat(True, len(times)) 18 | if t_min is not None: 19 | simplices = simplices[times_ex >= t_min] 20 | nverts = nverts[times >= t_min] 21 | if t_max is not None: 22 | simplices = simplices[times_ex <= t_max] 23 | nverts = nverts[times <= t_max] 24 | 25 | # relabel nodes: consecutive integers from 0 to n-1 26 | 27 | unique = np.unique(simplices) 28 | mapper = {unique[i] : i for i in range(len(unique))} 29 | simplices = np.array([mapper[s] for s in simplices]) 30 | 31 | # format as list of lists 32 | 33 | l = np.split(simplices, np.cumsum(nverts)) 34 | C = [list(c) for c in l] 35 | return(C) -------------------------------------------------------------------------------- /requirements.txt: -------------------------------------------------------------------------------- 1 | backcall==0.1.0 2 | cycler==0.10.0 3 | decorator==4.3.2 4 | ipykernel==5.1.0 5 | ipython==7.3.0 6 | ipython-genutils==0.2.0 7 | jedi==0.13.2 8 | jupyter-client==5.2.4 9 | jupyter-core==4.4.0 10 | kiwisolver==1.1.0 11 | networkx==2.2 12 | numpy==1.16.1 13 | parso==0.3.4 14 | pexpect==4.6.0 15 | pickleshare==0.7.5 16 | pkg-resources==0.0.0 17 | prompt-toolkit==2.0.9 18 | ptyprocess==0.6.0 19 | Pygments==2.3.1 20 | pyparsing==2.4.0 21 | python-dateutil==2.8.0 22 | pyzmq==18.0.0 23 | scipy==1.2.1 24 | six==1.12.0 25 | tornado==5.1.1 26 | traitlets==4.3.2 27 | wcwidth==0.1.7 28 | -------------------------------------------------------------------------------- /tutorial.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "# Configuration Models of Random Hypergraphs\n", 8 | "*Phil Chodrow, MIT Operations Research Center and Laboratory for Information and Decision Systems*\n", 9 | "\n", 10 | "This notebook is a basic tutorial for conducting null hypothesis tests by randomizing polyadic data under hypergraph configuration models. The `hypergraph` class is implemented in `hypergraph.py`, and includes methods for some basic descriptives as well as the relevant Monte Carlo algorithms. \n", 11 | "\n", 12 | "We also illustrate the triadic closure analysis shown in the main text. " 13 | ] 14 | }, 15 | { 16 | "cell_type": "code", 17 | "execution_count": 1, 18 | "metadata": {}, 19 | "outputs": [], 20 | "source": [ 21 | "import numpy as np\n", 22 | "from hypergraph import *\n", 23 | "from matplotlib import pyplot as plt" 24 | ] 25 | }, 26 | { 27 | "cell_type": "code", 28 | "execution_count": 2, 29 | "metadata": {}, 30 | "outputs": [], 31 | "source": [ 32 | "def read_email_data():\n", 33 | " '''\n", 34 | " Read the email data from the supplied .txt, returning it as a list of tuples. \n", 35 | " '''\n", 36 | " with open('email-enron.txt', 'r') as f:\n", 37 | " F = f.read()\n", 38 | " L = F.split('\\n')\n", 39 | " L = [tuple([int(v) for v in g.split(' ') if v is not '']) for g in L]\n", 40 | " return L" 41 | ] 42 | }, 43 | { 44 | "cell_type": "code", 45 | "execution_count": 3, 46 | "metadata": {}, 47 | "outputs": [], 48 | "source": [ 49 | "L = read_email_data() # read the data\n", 50 | "G = hypergraph(L) # construct a hypergraph from the data" 51 | ] 52 | }, 53 | { 54 | "cell_type": "code", 55 | "execution_count": 4, 56 | "metadata": {}, 57 | "outputs": [ 58 | { 59 | "name": "stdout", 60 | "output_type": "stream", 61 | "text": [ 62 | "The mean node degree is 188.0, and the standard deviation is 238.0.\n", 63 | "The mean edge dimension is 2.5, and the standard deviation is 1.4.\n" 64 | ] 65 | } 66 | ], 67 | "source": [ 68 | "def print_moments(G):\n", 69 | " D = G.node_degrees()\n", 70 | " K = G.edge_dimensions()\n", 71 | " print('The mean node degree is ' + str(round(D.mean())) + ', and the standard deviation is '+ str(round(np.sqrt(np.var(D)))) + '.')\n", 72 | " print('The mean edge dimension is ' + str(round(K.mean(), 1)) + ', and the standard deviation is '+ str(round(np.sqrt(np.var(K)), 1)) + '.')\n", 73 | " \n", 74 | "print_moments(G)" 75 | ] 76 | }, 77 | { 78 | "cell_type": "markdown", 79 | "metadata": {}, 80 | "source": [ 81 | "## Monte Carlo Sampling and Triadic Closure\n", 82 | "\n", 83 | "We'll now illustrate the use of Monte Carlo sampling to conduct a simple hypothesis-test using the `hypergraph` class. We'll replicate the triadic closure study shown in the main text. Recall that our metric of interest is the proportion of 3-cycles in the projected graph that are \"filled in\" by a higher-dimensional edge. " 84 | ] 85 | }, 86 | { 87 | "cell_type": "code", 88 | "execution_count": 5, 89 | "metadata": {}, 90 | "outputs": [], 91 | "source": [ 92 | "def count_triangles(C):\n", 93 | " '''\n", 94 | " Count the number of triangles in the (unweighted) projected graph of hypergraph C\n", 95 | " '''\n", 96 | " G = projected_graph(C, as_hyper=False)\n", 97 | " G_ = nx.Graph(G)\n", 98 | " triangles = nx.triangles(G_)\n", 99 | " n_triangles = sum(triangles.values())/3\n", 100 | " return(n_triangles)\n", 101 | "\n", 102 | "def count_filled_triangles(C):\n", 103 | " '''\n", 104 | " Count the number of distinct sub-edges of size 3 in C. \n", 105 | " '''\n", 106 | " container = set()\n", 107 | " for f in C.C:\n", 108 | " for t in itertools.combinations(f, 3):\n", 109 | " container.add(tuple(sorted(t)))\n", 110 | " return(len(container))\n", 111 | "\n", 112 | "def analyze_closure(G):\n", 113 | " tri = count_triangles(G)\n", 114 | " filled = count_filled_triangles(G)\n", 115 | " print('The rate of simplicial closure in triangles is ' + str(round(filled / tri, 2)) + '.')" 116 | ] 117 | }, 118 | { 119 | "cell_type": "code", 120 | "execution_count": 9, 121 | "metadata": {}, 122 | "outputs": [ 123 | { 124 | "name": "stdout", 125 | "output_type": "stream", 126 | "text": [ 127 | "The rate of simplicial closure in triangles is 0.81.\n" 128 | ] 129 | } 130 | ], 131 | "source": [ 132 | "analyze_closure(G)" 133 | ] 134 | }, 135 | { 136 | "cell_type": "markdown", 137 | "metadata": {}, 138 | "source": [ 139 | "Now we can compare this value to a sample from a null model. Let's start with a stub-labeled null. " 140 | ] 141 | }, 142 | { 143 | "cell_type": "code", 144 | "execution_count": 13, 145 | "metadata": {}, 146 | "outputs": [ 147 | { 148 | "name": "stdout", 149 | "output_type": "stream", 150 | "text": [ 151 | "100000 steps completed.\n", 152 | "Task completed in 3.24 seconds\n" 153 | ] 154 | } 155 | ], 156 | "source": [ 157 | "import time\n", 158 | "\n", 159 | "tick = time.perf_counter()\n", 160 | "G.MH(n_steps = 100000, label = 'stub')\n", 161 | "tock = time.perf_counter()\n", 162 | "\n", 163 | "print('Task completed in ' + str(round(tock - tick, 2)) + ' seconds')" 164 | ] 165 | }, 166 | { 167 | "cell_type": "markdown", 168 | "metadata": {}, 169 | "source": [ 170 | "We can check that the randomized graph has the same degree sequence and edge-dimension sequence as the old: " 171 | ] 172 | }, 173 | { 174 | "cell_type": "code", 175 | "execution_count": 8, 176 | "metadata": {}, 177 | "outputs": [ 178 | { 179 | "name": "stdout", 180 | "output_type": "stream", 181 | "text": [ 182 | "The mean node degree is 188.0, and the standard deviation is 238.0.\n", 183 | "The mean edge dimension is 2.5, and the standard deviation is 1.4.\n" 184 | ] 185 | } 186 | ], 187 | "source": [ 188 | "print_moments(G)" 189 | ] 190 | }, 191 | { 192 | "cell_type": "markdown", 193 | "metadata": {}, 194 | "source": [ 195 | "How does the null value for simplicial closure in triangles compare to the observed value? " 196 | ] 197 | }, 198 | { 199 | "cell_type": "code", 200 | "execution_count": 9, 201 | "metadata": {}, 202 | "outputs": [ 203 | { 204 | "name": "stdout", 205 | "output_type": "stream", 206 | "text": [ 207 | "The rate of simplicial closure in triangles is 0.25.\n" 208 | ] 209 | } 210 | ], 211 | "source": [ 212 | "analyze_closure(G)" 213 | ] 214 | }, 215 | { 216 | "cell_type": "markdown", 217 | "metadata": {}, 218 | "source": [ 219 | "Simplicial closure in triangles occurs in the observed data three times as often as would be expected by the stub-labeled configuration model. Formally, we should repeatedly randomize and pull many samples, but the spread is not large. Take my word for it, or try it yourself. \n", 220 | "\n", 221 | "What about the vertex-labeled model? \n", 222 | "\n", 223 | "The vertex-labeled MH algorithm accepts an additional parameter, `n_clash`. Tracking the counts of each edge is very expensive, so instead we implement an approximation that resamples the edge counts only when a \"clash\" requires an update. We recommend setting `n_clash = 1` when using vertex-labeled randomization. " 224 | ] 225 | }, 226 | { 227 | "cell_type": "code", 228 | "execution_count": 14, 229 | "metadata": {}, 230 | "outputs": [ 231 | { 232 | "name": "stdout", 233 | "output_type": "stream", 234 | "text": [ 235 | "2373 epochs completed, 100010 steps taken, 477951 steps rejected.\n", 236 | "Task completed in 25.98 seconds\n" 237 | ] 238 | } 239 | ], 240 | "source": [ 241 | "tick = time.perf_counter()\n", 242 | "G.MH(n_steps = 100000, label = 'vertex', n_clash = 1)\n", 243 | "tock = time.perf_counter()\n", 244 | "print('Task completed in ' + str(round(tock - tick, 2)) + ' seconds')" 245 | ] 246 | }, 247 | { 248 | "cell_type": "markdown", 249 | "metadata": {}, 250 | "source": [ 251 | "The degree and dimension sequences are again preserved: " 252 | ] 253 | }, 254 | { 255 | "cell_type": "code", 256 | "execution_count": 11, 257 | "metadata": {}, 258 | "outputs": [ 259 | { 260 | "name": "stdout", 261 | "output_type": "stream", 262 | "text": [ 263 | "The mean node degree is 188.0, and the standard deviation is 238.0.\n", 264 | "The mean edge dimension is 2.5, and the standard deviation is 1.4.\n" 265 | ] 266 | } 267 | ], 268 | "source": [ 269 | "print_moments(G)" 270 | ] 271 | }, 272 | { 273 | "cell_type": "markdown", 274 | "metadata": {}, 275 | "source": [ 276 | "What about simplicial closure? " 277 | ] 278 | }, 279 | { 280 | "cell_type": "code", 281 | "execution_count": 12, 282 | "metadata": {}, 283 | "outputs": [ 284 | { 285 | "name": "stdout", 286 | "output_type": "stream", 287 | "text": [ 288 | "The rate of simplicial closure in triangles is 0.21.\n" 289 | ] 290 | } 291 | ], 292 | "source": [ 293 | "analyze_closure(G)" 294 | ] 295 | }, 296 | { 297 | "cell_type": "markdown", 298 | "metadata": {}, 299 | "source": [ 300 | "# Preserving Node-Edge Dimension Matrices" 301 | ] 302 | }, 303 | { 304 | "cell_type": "markdown", 305 | "metadata": {}, 306 | "source": [ 307 | "In some cases, it may be desirable to consider a more restrictive class of nulls that preserve more detailed information about the data. A fairly natural definition is to require that each node preserve not only its total degree, but also the number of edges of each dimension incident to it. Class `hypergraph` implements this as \"detailed\" MCMC, as we demonstrate here. \n", 308 | "\n", 309 | "First, we'll read in the data and take a look at the \"node-dimension matrix,\" whose $ij$th entry is the number of edges of dimension $j$ incident on node $i$. " 310 | ] 311 | }, 312 | { 313 | "cell_type": "code", 314 | "execution_count": 13, 315 | "metadata": {}, 316 | "outputs": [], 317 | "source": [ 318 | "L = read_email_data() # read the data\n", 319 | "G = hypergraph(L) # construct a hypergraph from the data\n", 320 | "A0 = G.node_dimension_matrix().T[:10,]" 321 | ] 322 | }, 323 | { 324 | "cell_type": "markdown", 325 | "metadata": {}, 326 | "source": [ 327 | "Now we'll do MCMC, finding that this matrix is preserved when we specify the `detailed = True` in the MCMC call. You can do this with either stub- or vertex-labeling. " 328 | ] 329 | }, 330 | { 331 | "cell_type": "code", 332 | "execution_count": 14, 333 | "metadata": {}, 334 | "outputs": [ 335 | { 336 | "name": "stdout", 337 | "output_type": "stream", 338 | "text": [ 339 | "3132 epochs completed, 100011 steps taken, 1330856 steps rejected.\n" 340 | ] 341 | } 342 | ], 343 | "source": [ 344 | "G.MH(n_steps = 100000, label = 'vertex', detailed = True, n_clash = 1)\n", 345 | "A1 = G.node_dimension_matrix().T[:10,]" 346 | ] 347 | }, 348 | { 349 | "cell_type": "code", 350 | "execution_count": 15, 351 | "metadata": {}, 352 | "outputs": [ 353 | { 354 | "data": { 355 | "text/plain": [ 356 | "1.0" 357 | ] 358 | }, 359 | "execution_count": 15, 360 | "metadata": {}, 361 | "output_type": "execute_result" 362 | } 363 | ], 364 | "source": [ 365 | "np.mean(A0 == A1) # check that all entries are equal" 366 | ] 367 | }, 368 | { 369 | "cell_type": "markdown", 370 | "metadata": {}, 371 | "source": [ 372 | "# Closing Thoughts\n", 373 | "\n", 374 | "While our particular illustration is quite simple, the recipe is extremely general. To conduct null hypothesis tests, \n", 375 | "\n", 376 | "1. Measure the empirical value of your quantity of interest on the observed data. \n", 377 | "2. Repeatedly randomize under the appropriate configuration model, measuring the value at appropriate intervals. \n", 378 | "3. Compare the empirical value to the null distribution. \n", 379 | "\n", 380 | "Some care is required in choosing $n_\\mathrm{steps}$ to ensure that the chain is appropriately mixed (i.e. that the hypergraph is close to fully-randomized). To my knowledge, there are no general results on this. A practical approach is to track your quantity of interest over many iterations, and only use samples after they have \"leveled off,\" which suggests stationarity. " 381 | ] 382 | } 383 | ], 384 | "metadata": { 385 | "kernelspec": { 386 | "display_name": "Python 3", 387 | "language": "python", 388 | "name": "python3" 389 | }, 390 | "language_info": { 391 | "codemirror_mode": { 392 | "name": "ipython", 393 | "version": 3 394 | }, 395 | "file_extension": ".py", 396 | "mimetype": "text/x-python", 397 | "name": "python", 398 | "nbconvert_exporter": "python", 399 | "pygments_lexer": "ipython3", 400 | "version": "3.7.2" 401 | } 402 | }, 403 | "nbformat": 4, 404 | "nbformat_minor": 2 405 | } 406 | --------------------------------------------------------------------------------