├── storjv3.pdf ├── images ├── graphics.ai ├── uptime.png ├── graphics.afdesign ├── header.eps └── front.eps ├── README.md ├── listings-protobuf.sty ├── Makefile ├── calc.py ├── RS-appendix-files ├── 10000nodes.tex ├── conc.tex ├── model.tex ├── numexp.tex ├── example_cdf.eps └── example_pmf.eps ├── listings-golang.sty ├── .gitignore ├── audit-success ├── main.tex ├── uniform_prior.eps ├── jeffrey_prior.eps └── jeffrey_estimate.eps ├── structure.tex └── LICENSE /storjv3.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/storj/whitepaper/HEAD/storjv3.pdf -------------------------------------------------------------------------------- /images/graphics.ai: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/storj/whitepaper/HEAD/images/graphics.ai -------------------------------------------------------------------------------- /images/uptime.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/storj/whitepaper/HEAD/images/uptime.png -------------------------------------------------------------------------------- /images/graphics.afdesign: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/storj/whitepaper/HEAD/images/graphics.afdesign -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Storj Whitepaper V3 2 | 3 | Available at [https://storj.io/whitepaper](https://storj.io/whitepaper) 4 | -------------------------------------------------------------------------------- /listings-protobuf.sty: -------------------------------------------------------------------------------- 1 | \RequirePackage{listings} 2 | 3 | \lstdefinelanguage{protobuf}% 4 | {morekeywords=[1]{message,enum,bytes,string,int32,int64,repeated},% 5 | sensitive=true,% 6 | morecomment=[l]{//},% 7 | } 8 | -------------------------------------------------------------------------------- /Makefile: -------------------------------------------------------------------------------- 1 | all: storjv3.pdf 2 | 3 | clean: 4 | rm -f *.4ct *.4tc *.aux *.bbl *.blg *.css *.dvi *.epub *.fdb_latexmk *.fls *.html *.idv *.lg *.log *.mobi *.out *.pdf *.tmp *.xref storj-whitepaper-v3*x.png *.ptc *.toc images/*.pdf 5 | 6 | %.pdf: *.tex *.bib */*.tex 7 | latexmk -pdf $< 8 | 9 | .PHONY: all clean 10 | -------------------------------------------------------------------------------- /calc.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env python 2 | 3 | import decimal, math 4 | 5 | decimal.getcontext().prec = 20 6 | 7 | def P(p, k, n): 8 | p, n, k = map(decimal.Decimal, (p, n, k)) 9 | l = p*n 10 | return (-l).exp() * sum((l ** i)/math.factorial(i) for i in xrange(n-k+1)) 11 | 12 | for k, n in ( 13 | (2, 4), (4, 8), (8, 16), (16, 32), (20, 40), (32, 64), 14 | (1, 1), (1, 2), (1, 3), (1, 10), (1, 16), 15 | (4, 6), (4, 12), (20, 30), (20, 50), (100, 150)): 16 | print "%d\t%d\t%s\t%0.015f" % (k, n, n/float(k), P(.1, k, n) * 100) 17 | -------------------------------------------------------------------------------- /RS-appendix-files/10000nodes.tex: -------------------------------------------------------------------------------- 1 | \begin{table}[!htpb]\centering 2 | 3 | \begin{tabular}{| c | c c c | c | c|}\hline 4 | MTTF (months) &$k$& $n$ & $m$ &Repair Bandwidth Ratio&Durability (\# nines) \\\hline 5 | 1 &20& 40 & 35 & 9.36 & 0.9999 (8) \\ 6 | 6 &20& 40 & 30 & 0.87 & 0.9999 (17) \\ 7 | 12 &20& 40 & 25 & 0.31 & 0.9999 (13)\\\hline 8 | 9 | 1 &30& 60 & 35 & 3.40 &0.9999 (4)\\ 10 | 6 &30& 70 & 40 & 0.60 &0.9999 (15)\\ 11 | 12 &30& 80 & 45 & 0.31 &0.9999 (25) \\\hline 12 | 13 | 1 &40& 80 & 60 & 5.21 &0.9999 (4)\\ 14 | 6 &40&120&50 & 0.52 &0.9999(14)\\ 15 | 12 &40&120&45 & 0.24 &0.9999 (11)\\\hline 16 | 17 | \end{tabular} 18 | \caption{Decision tables showing the relationship between churn (MTTF), 19 | Reed-Solomon parameters ($k$, $n$, $m$), repair bandwidth ratio, and durability} 20 | \label{rs:decision-tables} 21 | \end{table} 22 | -------------------------------------------------------------------------------- /listings-golang.sty: -------------------------------------------------------------------------------- 1 | %% Golang definition for listings 2 | %% Copyright Julien Chaumont, 2017 3 | %% Licensed MIT, https://github.com/julienc91/listings-golang 4 | %% 5 | \RequirePackage{listings} 6 | 7 | \lstdefinelanguage{Golang}% 8 | {morekeywords=[1]{package,import,func,type,struct,return,defer,panic,% 9 | recover,select,var,const,iota,},% 10 | morekeywords=[2]{string,uint,uint8,uint16,uint32,uint64,int,int8,int16,% 11 | int32,int64,bool,float32,float64,complex64,complex128,byte,rune,uintptr,% 12 | error,interface},% 13 | morekeywords=[3]{map,slice,make,new,nil,len,cap,copy,close,true,false,% 14 | delete,append,real,imag,complex,chan,},% 15 | morekeywords=[4]{for,break,continue,range,goto,switch,case,fallthrough,if,% 16 | else,default,},% 17 | morekeywords=[5]{Println,Printf,Error,Print,},% 18 | sensitive=true,% 19 | morecomment=[l]{//},% 20 | morecomment=[s]{/*}{*/},% 21 | morestring=[b]',% 22 | morestring=[b]",% 23 | morestring=[s]{`}{`},% 24 | } 25 | -------------------------------------------------------------------------------- /RS-appendix-files/conc.tex: -------------------------------------------------------------------------------- 1 | \subsection{Conclusion} 2 | 3 | We conclude by observing that these models may be tuned to target specific network scenarios and requirements. One network may require one set of Reed-Solomon parameters, while a different network may require another. In general, the closer $m/n$ is to 1, the more rebuilds per month should be expected under a fixed churn rate. While having a larger ratio for $m/n$ increases file durability for any given churn rate, it comes at the expense of more bandwidth used since repairs are triggered more often. To maintain a low mean rebuilds/month value while also maintaining a higher file durability, the aim should be to increase the value of $n$ as much as feasible given other network conditions (latency, download speed, etc.), which allows for a lower relative value of $m$ while still not jeopardizing file durability. 4 | 5 | Informally, it takes longer to lose more pieces under a given fixed network size and churn rate. Therefore, to maximize durability while minimizing repair bandwidth usage, $n$ should be as large as existing network conditions allow. This allows for a value of $m$ that is relatively closer to $k$, reducing the mean rebuilds/month value, which in turn lowers the amount of repair bandwidth used. 6 | 7 | For example, assume we have a network with a mean time to failure of six months. 8 | Suppose we consider the same file encoded with two different RS parameters: 9 | one under a $(20,40)$ schema and the other as an $(30,80)$ schema. If we set $m$ so that $m=k+10$ for both cases, we observe from the above table 10 | that the bandwidth repair ratio is $0.87$ in the $(20,40)$ case and is $0.60$ in the $(40,80)$ case. Both encoding schemes have similar durability, as a repair in both cases is triggered when there are $k+10$ pieces left; even though the mean of rebuilds per month 11 | is empirically and theoretically lower for the $(40,80)$ case using $m=k+10$. 12 | -------------------------------------------------------------------------------- /images/header.eps: -------------------------------------------------------------------------------- 1 | %!PS-Adobe-3.0 EPSF-3.0 2 | %%Creator: cairo 1.14.8 (http://cairographics.org) 3 | %%CreationDate: Sun Sep 2 23:52:33 2018 4 | %%Pages: 1 5 | %%DocumentData: Clean7Bit 6 | %%LanguageLevel: 2 7 | %%BoundingBox: 0 3 1600 805 8 | %%EndComments 9 | %%BeginProlog 10 | save 11 | 50 dict begin 12 | /q { gsave } bind def 13 | /Q { grestore } bind def 14 | /cm { 6 array astore concat } bind def 15 | /w { setlinewidth } bind def 16 | /J { setlinecap } bind def 17 | /j { setlinejoin } bind def 18 | /M { setmiterlimit } bind def 19 | /d { setdash } bind def 20 | /m { moveto } bind def 21 | /l { lineto } bind def 22 | /c { curveto } bind def 23 | /h { closepath } bind def 24 | /re { exch dup neg 3 1 roll 5 3 roll moveto 0 rlineto 25 | 0 exch rlineto 0 rlineto closepath } bind def 26 | /S { stroke } bind def 27 | /f { fill } bind def 28 | /f* { eofill } bind def 29 | /n { newpath } bind def 30 | /W { clip } bind def 31 | /W* { eoclip } bind def 32 | /BT { } bind def 33 | /ET { } bind def 34 | /pdfmark where { pop globaldict /?pdfmark /exec load put } 35 | { globaldict begin /?pdfmark /pop load def /pdfmark 36 | /cleartomark load def end } ifelse 37 | /BDC { mark 3 1 roll /BDC pdfmark } bind def 38 | /EMC { mark /EMC pdfmark } bind def 39 | /cairo_store_point { /cairo_point_y exch def /cairo_point_x exch def } def 40 | /Tj { show currentpoint cairo_store_point } bind def 41 | /TJ { 42 | { 43 | dup 44 | type /stringtype eq 45 | { show } { -0.001 mul 0 cairo_font_matrix dtransform rmoveto } ifelse 46 | } forall 47 | currentpoint cairo_store_point 48 | } bind def 49 | /cairo_selectfont { cairo_font_matrix aload pop pop pop 0 0 6 array astore 50 | cairo_font exch selectfont cairo_point_x cairo_point_y moveto } bind def 51 | /Tf { pop /cairo_font exch def /cairo_font_matrix where 52 | { pop cairo_selectfont } if } bind def 53 | /Td { matrix translate cairo_font_matrix matrix concatmatrix dup 54 | /cairo_font_matrix exch def dup 4 get exch 5 get cairo_store_point 55 | /cairo_font where { pop cairo_selectfont } if } bind def 56 | /Tm { 2 copy 8 2 roll 6 array astore /cairo_font_matrix exch def 57 | cairo_store_point /cairo_font where { pop cairo_selectfont } if } bind def 58 | /g { setgray } bind def 59 | /rg { setrgbcolor } bind def 60 | /d1 { setcachedevice } bind def 61 | %%EndProlog 62 | %%BeginSetup 63 | %%EndSetup 64 | %%Page: 1 1 65 | %%BeginPageSetup 66 | %%PageBoundingBox: 0 3 1600 805 67 | %%EndPageSetup 68 | q 0 3 1600 802 rectclip q 69 | 0.054902 0.137255 0.298039 rg 70 | 0 805 1599.75 -800.25 re f 71 | Q Q 72 | showpage 73 | %%Trailer 74 | end restore 75 | %%EOF 76 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | ## Core latex/pdflatex auxiliary files: 2 | *.aux 3 | *.lof 4 | *.log 5 | *.lot 6 | *.fls 7 | *.out 8 | *.toc 9 | *.fmt 10 | *.fot 11 | *.cb 12 | *.cb2 13 | 14 | ## Intermediate documents: 15 | *.dvi 16 | *-converted-to.* 17 | # these rules might exclude image files for figures etc. 18 | # *.ps 19 | # *.eps 20 | # *.pdf 21 | *.epub 22 | *.mobi 23 | *.ptc 24 | 25 | ## htlatex 26 | *.4ct 27 | *.4tc 28 | *.css 29 | *.html 30 | *.idv 31 | *.lg 32 | *.tmp 33 | *.xref 34 | storj-whitepaper-v3*x.png 35 | 36 | ## Bibliography auxiliary files (bibtex/biblatex/biber): 37 | *.bbl 38 | *.bcf 39 | *.blg 40 | *-blx.aux 41 | *-blx.bib 42 | *.brf 43 | *.run.xml 44 | 45 | ## Build tool auxiliary files: 46 | *.fdb_latexmk 47 | *.synctex 48 | *.synctex.gz 49 | *.synctex.gz(busy) 50 | *.pdfsync 51 | 52 | ## Auxiliary and intermediate files from other packages: 53 | # algorithms 54 | *.alg 55 | *.loa 56 | 57 | # achemso 58 | acs-*.bib 59 | 60 | # amsthm 61 | *.thm 62 | 63 | # beamer 64 | *.nav 65 | *.snm 66 | *.vrb 67 | 68 | # cprotect 69 | *.cpt 70 | 71 | # fixme 72 | *.lox 73 | 74 | #(r)(e)ledmac/(r)(e)ledpar 75 | *.end 76 | *.?end 77 | *.[1-9] 78 | *.[1-9][0-9] 79 | *.[1-9][0-9][0-9] 80 | *.[1-9]R 81 | *.[1-9][0-9]R 82 | *.[1-9][0-9][0-9]R 83 | *.eledsec[1-9] 84 | *.eledsec[1-9]R 85 | *.eledsec[1-9][0-9] 86 | *.eledsec[1-9][0-9]R 87 | *.eledsec[1-9][0-9][0-9] 88 | *.eledsec[1-9][0-9][0-9]R 89 | 90 | # glossaries 91 | *.acn 92 | *.acr 93 | *.glg 94 | *.glo 95 | *.gls 96 | *.glsdefs 97 | 98 | # gnuplottex 99 | *-gnuplottex-* 100 | 101 | # hyperref 102 | *.brf 103 | 104 | # knitr 105 | *-concordance.tex 106 | # TODO Comment the next line if you want to keep your tikz graphics files 107 | *.tikz 108 | *-tikzDictionary 109 | 110 | # listings 111 | *.lol 112 | 113 | # makeidx 114 | *.idx 115 | *.ilg 116 | *.ind 117 | *.ist 118 | 119 | # minitoc 120 | *.maf 121 | *.mlf 122 | *.mlt 123 | *.mtc 124 | *.mtc[0-9] 125 | *.mtc[1-9][0-9] 126 | 127 | # minted 128 | _minted* 129 | *.pyg 130 | 131 | # morewrites 132 | *.mw 133 | 134 | # mylatexformat 135 | *.fmt 136 | 137 | # nomencl 138 | *.nlo 139 | 140 | # sagetex 141 | *.sagetex.sage 142 | *.sagetex.py 143 | *.sagetex.scmd 144 | 145 | # sympy 146 | *.sout 147 | *.sympy 148 | sympy-plots-for-*.tex/ 149 | 150 | # pdfcomment 151 | *.upa 152 | *.upb 153 | 154 | # pythontex 155 | *.pytxcode 156 | pythontex-files-*/ 157 | 158 | # thmtools 159 | *.loe 160 | 161 | # TikZ & PGF 162 | *.dpth 163 | *.md5 164 | *.auxlock 165 | 166 | # todonotes 167 | *.tdo 168 | 169 | # xindy 170 | *.xdy 171 | 172 | # xypic precompiled matrices 173 | *.xyc 174 | 175 | # endfloat 176 | *.ttt 177 | *.fff 178 | 179 | # Latexian 180 | TSWLatexianTemp* 181 | 182 | ## Editors: 183 | # WinEdt 184 | *.bak 185 | *.sav 186 | 187 | # Texpad 188 | .texpadtmp 189 | 190 | # Kile 191 | *.backup 192 | 193 | # KBibTeX 194 | *~[0-9]* 195 | 196 | # macOS 197 | .DS_Store 198 | -------------------------------------------------------------------------------- /RS-appendix-files/model.tex: -------------------------------------------------------------------------------- 1 | In the context of storing an erasure-coded segment on a decentralized network, we consider the loss of a {\em piece} from two different perspectives. 2 | 3 | \subsection{Direct piece loss} 4 | With direct piece loss, we assume that for a specific segment, its erasure pieces are lost according to a certain rate. We point out that modeling this is straightforward: if pieces are lost at a rate $0\frac{\ln(m/n)}{a\ln(1-p/a)}$. Thus it becomes clear, given parameters $n,m,a$ and $p$, how long we expect a segment to last between repairs. 7 | 8 | \subsection{Indirect piece loss} 9 | 10 | When modeling indirect piece loss, we suppose that a fixed rate of nodes 11 | drop out of the network each month,\footnote{Though the rate may be taken over 12 | any desired time interval.} whether or not they are holding pieces of the segment 13 | under consideration. To describe the probability that $d$ of the dropped nodes 14 | were each storing one of the $n$ pieces of a specific segment, we turn to the 15 | hypergeometric probability distribution. Suppose $c$ nodes are replaced per 16 | month out of $C$ total nodes on the network. Then the probability that $d$ 17 | nodes were each storing a piece of the segment is given by 18 | \begin{align} 19 | && P(X=d)&=\frac{\binom{n}{d}\binom{C-n}{c-d}}{\binom{C}{c}}\label{hgeom} && 20 | \end{align} 21 | which has mean $nc/C$. We then determine how long it will take for the number of pieces to fall below the desired threshold $m$ by iterating, holding the overall churn $c$ fixed but reducing the number of existing pieces by the distribution's mean in each iteration and counting the number of iterations required. For example, after one iteration, the number of existing pieces is reduced by $nc/C$, so instead of $n$ pieces on the network (as the parameter in \eqref{hgeom}), there are $n-nc/C$ pieces, changing both the parameter and the mean for \eqref{hgeom} in iteration 2. 22 | 23 | We may extend this model by considering multiple checks per month (as in the direct piece loss case), assuming that $c/a$ nodes are lost every $1/a$-th of a month instead of assuming that $c$ nodes are lost per month, where $a$ is the number of checks per month. This yields an initial hypergeometric probability distribution with mean $nc/aC$. 24 | 25 | In either of these two cases (single or multiple segment integrity checks per month), we track the number of iterations until the number of available pieces fall below the repair threshold. This number may then be used to determine the expected number of rebuilds per month for any given segment. 26 | -------------------------------------------------------------------------------- /RS-appendix-files/numexp.tex: -------------------------------------------------------------------------------- 1 | 2 | \subsection{Numerical simulations for indirect piece loss}\label{sec:RS-sim} 3 | 4 | We produce decision tables (Table \ref{rs:decision-tables}) showcasing 5 | worst-case mean segment rebuild outcomes based on simulating piece loss for segments encoded with varying Reed-Solomon parameters. 6 | We assume a $(k,n)$ RS encoding scheme, where $n$ pieces are generated, with 7 | $k$ pieces needed for reconstruction, using three different values for $n$. 8 | We also assume that a segment undergoes the process of repair when less than $m$ pieces remain on the network, using three different values of $m$ for each $n$. 9 | For the initial table, we use a simplifying assumption that pieces on the network are lost at a constant rate per month,\footnote{This constant rate may be viewed as the mean of the Poisson distribution modeling piece loss per month.} which may be due to node churn, data corruption, or other problems. 10 | 11 | To arrive at the value for mean rebuilds per month, we consider a single segment that is encoded with $n$ pieces which are distributed uniformly randomly to nodes on the network. To simulate conditions leading to a rebuild, we uniformly randomly select a subset of nodes from the total population and designate them as failed. We do this multiple times per (simulated) month, scaling the piece loss rate linearly according to the number of segment integrity checks per month.\footnote{ 12 | For example, if the monthly network piece loss rate is assumed to be 0.1 of the network size (or 10\%), and if 10 segment integrity checks are performed per month, we assume that, on average, 1\% of pieces are lost between checks.} 13 | 14 | Once enough nodes have failed to bring the number of pieces above the repair threshold $m$, the segment is rebuilt, and we track the number of rebuilds over the course of 24 months. 15 | We repeat this simulation for 1000 iterations, simulating 1000 two-year periods for a single segment. We then take the number of rebuilds at the 99th percentile (or higher) of the number of rebuilds occurring over these 1000 iterations. In other words, we choose the value for which the value of the observed cumulative distribution function (CDF), describing the number of rebuilds over this two-year period, is at least 0.99. This value is then divided by the number of months to arrive at the mean rebuilds/month value. An example of the approach is shown in Figure \ref{fig:sim_method}. We perform the experiment on a network of 10,000 nodes, observing that the network size will not directly impact the mean rebuilds/month value for a single segment under our working assumption of a constant rate of loss per month.\footnote{We represent piece loss as a proportion of nodes selected uniformly randomly from the total network. The proportion scales directly with network size, so the overall number of pieces lost stays the same for networks of different sizes.} 16 | 17 | \begin{figure}[!htbp] 18 | \centering 19 | \includegraphics[scale=0.5]{RS-appendix-files/example_pmf.eps} 20 | \includegraphics[scale=0.5]{RS-appendix-files/example_cdf.eps} 21 | \caption{Left: Density for the number of rebuilds over a 24 month period, repeated for 1000 iterations. Right: CDF of the number of rebuilds. In this case, the mean rebuilds/month value would be taken as $26/24\approx1.083$, with there being a 99.7\% chance that a segment is rebuilt at most 26 times over the course of 24 months.} 22 | \label{fig:sim_method} 23 | \end{figure} 24 | 25 | In forming the decision tables, we consider as part of our calculations how 26 | different choices of $k$, $n$, $m$, and mean time to failure affect durability and repair bandwidth. What we are looking for is the lowest repair bandwidth that also meets our 27 | durability requirements. 28 | 29 | \input{RS-appendix-files/10000nodes.tex} 30 | 31 | 32 | -------------------------------------------------------------------------------- /audit-success/main.tex: -------------------------------------------------------------------------------- 1 | We rely on a Bayesian approach to determine the probability 2 | that a storage node is maintaining stored pieces faithfully. 3 | At a high level, we seek to answer the following question: 4 | how do consecutive successful audits change our estimate of 5 | the probability that a node will continue to return successful audits? 6 | 7 | We model the audit process as being a binomial random variable 8 | with an unknown probability of success $p\in[0,1]$, with each audit 9 | being an independent Bernoulli trial. 10 | It is well-known that the conjugate prior of the binomial distribution 11 | is the beta distribution $\beta(a,b)$, 12 | and that the posterior also follows the beta distribution. 13 | As in \cite{tumor-occurrence}, 14 | we use the mean of the posterior distribution as our Bayes 15 | estimator, which is given by $P=(a+x)/(a+b+n)$ where $a,b$ are the parameters of 16 | the prior distribution, and $x$ is the number of successes observed in $n$ audits. 17 | Under our assumption that each audit is successful, 18 | we arrive at the Bayes estimate of the success probability $P=(a+n)/(a+b+n)$. 19 | 20 | \begin{figure}[!htbp] 21 | \centering 22 | \includegraphics[height=.22\textheight]{audit-success/jeffrey_prior.eps} 23 | \includegraphics[height=.22\textheight]{audit-success/jeffrey_estimate.eps} 24 | \caption{In Jeffrey's prior, we see the estimate for audit success probability is heavily weighted to be near 0 or near 1.} 25 | \label{fig:jeff_prior} 26 | \end{figure} 27 | 28 | \begin{figure}[!htbp] 29 | \centering 30 | \includegraphics[height=.22\textheight]{audit-success/uniform_prior.eps} 31 | \includegraphics[height=.22\textheight]{audit-success/uniform_estimate.eps} 32 | \caption{Using a Uniform prior, there is no assumption placed on the estimated 33 | audit success probability, and all probabilities are assumed to be equally likely.} 34 | \label{fig:unif_prior} 35 | \end{figure} 36 | 37 | We now choose a prior to derive a numerical estimate of the audit success probability 38 | based on the number of audits performed. 39 | There are many reasonable choices of Bayesian priors, but we restrict our attention to 40 | two popular choices: the Uniform prior and Jeffrey's prior \cite{jeffrey}. 41 | Using the Uniform prior $\beta(1,1)$ initializes the experiment 42 | by assigning an equal probability to 43 | all possible outcomes; 44 | that is, the probability of success is drawn from the uniform 45 | distribution on $(0,1)$. 46 | Under Jeffrey's prior $\beta(0.5,0.5)$, 47 | it is assumed that the probability of 48 | success falls towards either extreme, so that a node will return a successful audit 49 | either with probability near 0 or with probability near 1. 50 | 51 | \begin{table}[!htbp] 52 | \centering 53 | \begin{tabulary}{\linewidth}{| C | C | C |}\hline 54 | Number of audits & Audit success estimate given uniform prior & Audit success estimate given Jeffrey's prior\\\hline 55 | 0 & 0.5 & 0.5 \\ 56 | 20 & 0.9545 & 0.9762 \\ 57 | 40 & 0.9762 & 0.9878 \\ 58 | 80 & 0.9878 & 0.9938 \\ 59 | 200 & 0.99505 & 0.99751 \\ 60 | \hline 61 | \end{tabulary} 62 | \caption{Estimate of audit success probability by 63 | number of audits, each assumed to be successful. 64 | We find that the estimated probability of success begins at 0.5 when there is 65 | no information known about the node (no audits have been performed), 66 | with the estimate quickly jumping to above 99\% in as few as 80 audits using Jeffrey's prior.} 67 | \label{table:audit-success-estimates} 68 | \end{table} 69 | 70 | In Table \ref{table:audit-success-estimates}, 71 | we present results obtained from using both priors. 72 | We remark that the well-established Bayesian approach 73 | allows us to rapidly gain more confidence 74 | in a node's ability to return a successful audit, 75 | given that the success probability estimate tends closer to 1 76 | with each consecutive audit success. 77 | \FloatBarrier 78 | -------------------------------------------------------------------------------- /structure.tex: -------------------------------------------------------------------------------- 1 | %% 2 | %% Theme originally by Andrea Hidalgo, licensed 3 | %% LaTex Project Public License 1.3c 4 | %% Available from https://www.overleaf.com/articles/clustering-the-interstellar-medium/mtthgyyfrdkn 5 | %% 6 | 7 | \definecolor{internallinkcolor}{RGB}{0,0,0} 8 | \colorlet{urllinkcolor}{defaultcolor} 9 | 10 | %---------------------------------------------------------------------------------------- 11 | % VARIOUS REQUIRED PACKAGES 12 | %---------------------------------------------------------------------------------------- 13 | 14 | \usepackage{titlesec} % Allows customization of titles 15 | 16 | \usepackage{enumitem} % Customize lists 17 | \setlist{nolistsep} % Reduce spacing between bullet points and numbered lists 18 | 19 | \usepackage{booktabs} % Required for nicer horizontal rules in tables 20 | 21 | \usepackage{eso-pic} % Required for specifying an image background in the title page 22 | 23 | %---------------------------------------------------------------------------------------- 24 | % MAIN TABLE OF CONTENTS 25 | %---------------------------------------------------------------------------------------- 26 | 27 | \usepackage{titletoc} % Required for manipulating the table of contents 28 | 29 | \contentsmargin{0cm} % Removes the default margin 30 | % Chapter text styling 31 | \titlecontents{chapter}[1.25cm] % Indentation 32 | {\addvspace{1pt}\large\sffamily\bfseries} % Spacing and font options for chapters 33 | 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%---------------------------------------------------------------------------------------- 54 | % MINI TABLE OF CONTENTS IN CHAPTER HEADS 55 | %---------------------------------------------------------------------------------------- 56 | 57 | % Section text styling 58 | \titlecontents{lsection}[0em] % Indendating 59 | {\footnotesize\sffamily} % Font settings 60 | {} 61 | {} 62 | {} 63 | 64 | % Subsection text styling 65 | \titlecontents{lsubsection}[.5em] % Indentation 66 | {\normalfont\footnotesize\sffamily} % Font settings 67 | {} 68 | {} 69 | {} 70 | 71 | %---------------------------------------------------------------------------------------- 72 | % PAGE HEADERS 73 | %---------------------------------------------------------------------------------------- 74 | 75 | \usepackage{fancyhdr} % Required for header and footer configuration 76 | 77 | \pagestyle{fancy} 78 | \renewcommand{\chaptermark}[1]{\markboth{\sffamily\normalsize\bfseries\chaptername\ \thechapter.\ #1}{}} % Chapter text 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\vspace*{\fill} 94 | \thispagestyle{empty} 95 | \newpage 96 | \fi} 97 | 98 | %---------------------------------------------------------------------------------------- 99 | % SECTION NUMBERING IN THE MARGIN 100 | %---------------------------------------------------------------------------------------- 101 | 102 | \makeatletter 103 | \renewcommand{\@seccntformat}[1]{\llap{\textcolor{defaultcolor}{\csname the#1\endcsname}\hspace{1em}}} 104 | \renewcommand{\section}{\@startsection{section}{1}{\z@} 105 | {-4ex \@plus -1ex \@minus -.4ex} 106 | {1ex \@plus.2ex } 107 | {\normalfont\large\sffamily\bfseries}} 108 | \renewcommand{\subsection}{\@startsection {subsection}{2}{\z@} 109 | {-3ex \@plus -0.1ex \@minus -.4ex} 110 | {0.5ex \@plus.2ex } 111 | {\normalfont\sffamily\bfseries}} 112 | \renewcommand{\subsubsection}{\@startsection {subsubsection}{3}{\z@} 113 | {-2ex \@plus -0.1ex \@minus -.2ex} 114 | {.2ex \@plus.2ex } 115 | {\normalfont\small\sffamily\bfseries}} 116 | 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manually adapted to the overall margin page septup controlled by the geometry package loaded in the main.tex document. It is possible to implement below the dimensions used in the goemetry package (top,bottom,left,right)... TO BE DONE 142 | 143 | \newcommand{\thechapterimage}{} 144 | \newcommand{\chapterimage}[1]{\renewcommand{\thechapterimage}{#1}} 145 | 146 | % Numbered chapters with mini tableofcontents 147 | \def\thechapter{\arabic{chapter}} 148 | \def\@makechapterhead#1{ 149 | \thispagestyle{empty} 150 | {\centering \normalfont\sffamily 151 | \ifnum \c@secnumdepth >\m@ne 152 | \if@mainmatter 153 | \startcontents 154 | \begin{tikzpicture}[remember picture,overlay] 155 | \node at (current page.north west) 156 | {\begin{tikzpicture}[remember picture,overlay] 157 | \node[anchor=north west,inner sep=0pt] at (0,0) 158 | {\includegraphics[width=\paperwidth,height=4cm]{\thechapterimage}}; 159 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 160 | % Commenting the 3 lines below removes the small contents box in the chapter 161 | %heading 162 | \draw[anchor=east] (15cm,-2.2cm) node [rounded 163 | corners=4pt,align=left,fill=defaultcolor!10!white,text 164 | 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c h 237 | 402.98 1043.777 m f* 238 | Q Q 239 | showpage 240 | %%Trailer 241 | end restore 242 | %%EOF 243 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | Creative Commons Legal Code 2 | 3 | Attribution-ShareAlike 3.0 Unported 4 | 5 | CREATIVE COMMONS CORPORATION IS NOT A LAW FIRM AND DOES NOT PROVIDE 6 | LEGAL SERVICES. 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%%BoundingBox: 122 257 489 534 7 | %%EndComments 8 | %%BeginProlog 9 | /mpldict 8 dict def 10 | mpldict begin 11 | /m { moveto } bind def 12 | /l { lineto } bind def 13 | /r { rlineto } bind def 14 | /c { curveto } bind def 15 | /cl { closepath } bind def 16 | /box { 17 | m 18 | 1 index 0 r 19 | 0 exch r 20 | neg 0 r 21 | cl 22 | } bind def 23 | /clipbox { 24 | box 25 | clip 26 | newpath 27 | } bind def 28 | %!PS-Adobe-3.0 Resource-Font 29 | %%Title: DejaVu Sans 30 | %%Copyright: Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved. Copyright (c) 2006 by Tavmjong Bah. All Rights Reserved. DejaVu changes are in public domain 31 | %%Creator: Converted from TrueType to type 3 by PPR 32 | 25 dict begin 33 | /_d{bind def}bind def 34 | /_m{moveto}_d 35 | /_l{lineto}_d 36 | /_cl{closepath eofill}_d 37 | /_c{curveto}_d 38 | /_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d 39 | /_e{exec}_d 40 | /FontName /DejaVuSans def 41 | /PaintType 0 def 42 | /FontMatrix[.001 0 0 .001 0 0]def 43 | /FontBBox[-1021 -463 1793 1232]def 44 | /FontType 3 def 45 | /Encoding [ /space /quotesingle /period /zero /one /two /three /four /five /six /seven /eight /A /J /a /b /c /d /e /f /i /l /o /p /r /s /t /u /y ] def 46 | /FontInfo 10 dict dup begin 47 | /FamilyName (DejaVu Sans) def 48 | /FullName (DejaVu Sans) def 49 | /Notice (Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved. Copyright (c) 2006 by Tavmjong Bah. All Rights Reserved. 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All Rights Reserved. Copyright (c) 2006 by Tavmjong Bah. All Rights Reserved. DejaVu changes are in public domain 31 | %%Creator: Converted from TrueType to type 3 by PPR 32 | 25 dict begin 33 | /_d{bind def}bind def 34 | /_m{moveto}_d 35 | /_l{lineto}_d 36 | /_cl{closepath eofill}_d 37 | /_c{curveto}_d 38 | /_sc{7 -1 roll{setcachedevice}{pop pop pop pop pop pop}ifelse}_d 39 | /_e{exec}_d 40 | /FontName /DejaVuSans def 41 | /PaintType 0 def 42 | /FontMatrix[.001 0 0 .001 0 0]def 43 | /FontBBox[-1021 -463 1793 1232]def 44 | /FontType 3 def 45 | /Encoding [ /space /period /zero /one /two /four /five /six /eight /N /O /R /a /b /d /e /f /h /i /l /m /n /o /r /s /t /u /v /y ] def 46 | /FontInfo 10 dict dup begin 47 | /FamilyName (DejaVu Sans) def 48 | /FullName (DejaVu Sans) def 49 | /Notice (Copyright (c) 2003 by Bitstream, Inc. All Rights Reserved. Copyright (c) 2006 by Tavmjong Bah. All Rights Reserved. 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