View source for Stochastic block model

{{Network science}}
The '''stochastic block model''' is a [[Generative model|generative model]] for random [[Graph (discrete mathematics)|graphs]]. This model tends to produce graphs containing ''communities'', subsets of nodes characterized by being connected with one another with particular edge densities. For example, edges may be more common within communities than between communities. Its mathematical formulation has been firstly introduced in 1983 in the field of social network by [[Biography:Paul W. Holland|Paul W. Holland]] et al.<ref name="hol" /> The stochastic block model is important in [[Statistics|statistics]], [[Machine learning|machine learning]], and [[Network science|network science]], where it serves as a useful benchmark for the task of recovering [[Community structure|community structure]] in graph data.

== Definition == 
The stochastic block model takes the following parameters:
* The number <math>n</math> of vertices;
* a partition of the vertex set <math>\{1,\ldots,n\}</math> into disjoint subsets <math>C_1,\ldots,C_r</math>, called ''communities'';
* a symmetric <math>r \times r</math> matrix <math>P</math> of edge probabilities.

The edge set is then sampled at random as follows: any two vertices <math>u \in C_i</math> and <math>v \in C_j</math> are connected by an edge with probability <math>P_{ij}</math>. An example problem is: given a graph with  <math>n</math> vertices, where the edges are sampled as described, recover the groups  <math>C_1,\ldots,C_r</math>.

== Special cases ==
[[File:Assortative Case of the SBM.png|thumb|149x149px|An example of the assortative case for the stochastic block model.]]
If the probability matrix is a constant, in the sense that <math>P_{ij} = p</math> for all <math>i,j</math>, then the result is the [[Erdős–Rényi model]] <math>G(n,p)</math>. This case is degenerate—the partition into communities becomes irrelevant—but it illustrates a close relationship to the Erdős–Rényi model.

The ''planted partition model'' is the special case that the values of the probability matrix <math>P</math> are a constant <math>p</math> on the diagonal and another constant <math>q</math> off the diagonal. Thus two vertices within the same community share an edge with probability <math>p</math>, while two vertices in different communities share an edge with probability <math>q</math>. Sometimes it is this restricted model that is called the stochastic block model. The case where <math>p > q</math> is called an ''assortative'' model, while the case <math>p < q</math> is called ''disassortative''.

Returning to the general stochastic block model, a model is called ''strongly assortative'' if <math>P_{ii} > P_{jk}</math> whenever <math>j \neq k</math>: all diagonal entries dominate all off-diagonal entries. A model is called ''weakly assortative'' if <math>P_{ii} > P_{ij}</math> whenever <math>i \neq j</math>: each diagonal entry is only required to dominate the rest of its own row and column.<ref name="al14" /> ''Disassortative'' forms of this terminology exist, by reversing all inequalities. For some algorithms, recovery might be easier for block models with assortative or disassortative conditions of this form.<ref name="al14" />

== Typical statistical tasks ==
Much of the literature on algorithmic community detection addresses three statistical tasks: detection, partial recovery, and exact recovery.

=== Detection ===
The goal of detection algorithms is simply to determine, given a sampled graph, whether the graph has latent community structure. More precisely, a graph might be generated, with some known prior probability, from a known stochastic block model, and otherwise from a similar Erdos-Renyi model. The algorithmic task is to correctly identify which of these two underlying models generated the graph.<ref name="mns12" />

=== Partial recovery ===
In partial recovery, the goal is to approximately determine the latent partition into communities, in the sense of finding a partition that is correlated with the true partition significantly better than a random guess.<ref name="mas13" />

=== Exact recovery ===
In exact recovery, the goal is to recover the latent partition into communities exactly. The community sizes and probability matrix may be known<ref name="as15a" /> or unknown.<ref name="as15b" />

== Statistical lower bounds and threshold behavior ==
Stochastic block models exhibit a sharp threshold effect reminiscent of [[Percolation threshold|percolation threshold]]s.<ref name="decelle11"/><ref name="mns12" /><ref name="abh14" /> Suppose that we allow the size <math>n</math> of the graph to grow, keeping the community sizes in fixed proportions. If the probability matrix remains fixed, tasks such as partial and exact recovery become feasible for all non-degenerate parameter settings. However, if we scale down the probability matrix at a suitable rate as <math>n</math> increases, we observe a sharp phase transition: for certain settings of the parameters, it will become possible to achieve recovery with probability tending to 1, whereas on the opposite side of the parameter threshold, the probability of recovery tends to 0 no matter what algorithm is used.

For partial recovery, the appropriate scaling is to take <math>P_{ij} = \tilde P_{ij} / n</math> for fixed <math>\tilde P</math>, resulting in graphs of constant average degree. In the case of two equal-sized communities, in the assortative planted partition model with probability matrix
<math display="block"> P = \left(\begin{array}{cc} \tilde p/n & \tilde q/n \\ \tilde q/n & \tilde p/n \end{array} \right), </math>
partial recovery is feasible<ref name="mas13" /> with probability <math>1 - o(1)</math> whenever <math>(\tilde p - \tilde q)^2 > 2(\tilde p + \tilde q)</math>, whereas any [[Estimator|estimator]] fails<ref name="mns12" /> partial recovery with probability <math>1-o(1)</math> whenever <math>(\tilde p - \tilde q)^2 < 2(\tilde p + \tilde q)</math>.

For exact recovery, the appropriate scaling is to take <math>P_{ij} = \tilde P_{ij} \log n / n</math>, resulting in graphs of logarithmic average degree. Here a similar threshold exists: for the assortative planted partition model with <math>r</math> equal-sized communities, the threshold lies at <math>\sqrt{\tilde p} - \sqrt{\tilde q} = \sqrt{r}</math>. In fact, the exact recovery threshold is known for the fully general stochastic block model.<ref name="as15a" />
<!---
it'd be great to have plots of these thresholds --->

== Algorithms == 
In principle, exact recovery can be solved in its feasible range using maximum likelihood, but this amounts to solving a constrained or [[Regularization (mathematics)|regularized]] cut problem such as minimum bisection that is typically NP-complete. Hence, no known efficient algorithms will correctly compute the maximum-likelihood estimate in the worst case.

However, a wide variety of algorithms perform well in the average case, and many high-probability performance guarantees have been proven for algorithms in both the partial and exact recovery settings. Successful algorithms include [[Spectral clustering|spectral clustering]] of the vertices,<ref name="krzakala-pnas"/><ref name="mas13" /><ref name="as15a" /><ref name="lr15" /> [[Semidefinite programming|semidefinite programming]],<ref name="al14" /><ref name="abh14" /> forms of [[Belief propagation|belief propagation]],<ref name="decelle11"/><ref name="mns13" /> and community detection<ref name="fat19"/> among others.

== Variants ==
Several variants of the model exist. One minor tweak allocates vertices to communities randomly, according to a [[Categorical distribution|categorical distribution]], rather than in a fixed partition.<ref name="as15a" /> More significant variants include the degree-corrected stochastic block model,<ref name="ker" /> the hierarchical stochastic block model,<ref name="pei"/> the geometric block model,<ref name="gbm" /> censored block model and the mixed-membership block model.<ref name="ar1"/>

== Topic models ==
Stochastic block model have been recognised to be a [[Topic model|topic model]] on bipartite networks.<ref name="gerlachnetwork">{{cite journal
| author1 = Martin Gerlach |author2=Tiago Peixoto |author3=Eduardo Altmann
| title = A network approach to topic models 
| page=eaaq1360
| journal=Science Advances
|  date = 2018 |doi = 10.1126/sciadv.aaq1360 | volume=4 | issue=7
|pmid=30035215 |pmc=6051742 |arxiv=1708.01677 |bibcode=2018SciA....4.1360G }}</ref> In a network of documents and words, Stochastic block model can identify topics: group of words with a similar meaning.

== Extensions to signed graphs ==
Signed graphs allow for both favorable and adverse relationships and serve as a common model choice for various data analysis applications, e.g., correlation clustering. The stochastic block model can be trivially extended to signed graphs by assigning both positive and negative edge weights or equivalently using a difference of adjacency matrices of two stochastic block models.
<ref>{{cite book
| author1=Alyson Fox |author2=Geoffrey Sanders |author3=Andrew Knyazev
|title=2018 IEEE High Performance extreme Computing Conference (HPEC) |chapter=Investigation of Spectral Clustering for Signed Graph Matrix Representations |  date = 2018 |pages=1–7 |doi = 10.1109/HPEC.2018.8547575|osti=1476177 |isbn=978-1-5386-5989-2 |s2cid=54443034 }}</ref>

== DARPA/MIT/AWS Graph Challenge: streaming stochastic block partition ==
GraphChallenge<ref>[http://graphchallenge.mit.edu] {{Webarchive|url=https://web.archive.org/web/20230204160402/http://graphchallenge.mit.edu/|date=2023-02-04}} DARPA/MIT/AWS Graph Challenge</ref> encourages community approaches to developing new solutions for analyzing graphs and sparse data derived from social media, sensor feeds, and scientific data to enable relationships between events to be discovered as they unfold in the field. Streaming stochastic block partition is one of the challenges since 2017.
<ref>[http://graphchallenge.mit.edu/champions] {{Webarchive|url=https://web.archive.org/web/20230204160403/http://graphchallenge.mit.edu/champions|date=2023-02-04}} DARPA/MIT/AWS Graph Challenge Champions</ref> [[Spectral clustering]] has demonstrated outstanding performance compared to the original and even improved<ref>{{cite book
| author1 = A. J. Uppal |author2 = J. Choi |author3 = T. B. Rolinger |author4 = H. Howie Huang
|title = 2021 IEEE High Performance Extreme Computing Conference (HPEC) |chapter = Faster Stochastic Block Partition Using Aggressive Initial Merging, Compressed Representation, and Parallelism Control |  date = 2021 |pages = 1–7 |doi = 10.1109/HPEC49654.2021.9622836|isbn = 978-1-6654-2369-4 |s2cid = 244780210 }}</ref>
base algorithm, matching its quality of clusters while being multiple orders of magnitude faster.<ref>{{cite journal
| author1 = David Zhuzhunashvili |author2 = Andrew Knyazev
| title = Preconditioned spectral clustering for stochastic block partition streaming graph challenge
| journal=2017 IEEE High Performance Extreme Computing Conference (HPEC)
|  date = 2017 |doi = 10.1109/HPEC.2017.8091045|arxiv = 1708.07481
|s2cid = 19781504
}}</ref>
<ref>{{cite book
| author1 = Lisa Durbeck |author2 = Peter Athanas
|title = 2020 IEEE High Performance Extreme Computing Conference (HPEC)
|chapter = Incremental Streaming Graph Partitioning
|  date = 2020 |pages = 1–8
|doi = 10.1109/HPEC43674.2020.9286181|isbn = 978-1-7281-9219-2
|s2cid = 229376193
}}</ref>

==See also==
* [[Blockmodeling|blockmodeling]]
* {{annotated link|Girvan–Newman algorithm}}
* {{annotated link|Lancichinetti–Fortunato–Radicchi benchmark}} for generating benchmark networks with communities

==References==
{{reflist|refs=
<ref name="decelle11">{{cite journal| 
last1 = Decelle| first1 = Aurelien |
last2 = Krzakala| first2 = Florent |
last3 = Moore| first3 = Cristopher |
last4 = Zdeborová | first4 = Lenka | author4-link = Lenka Zdeborová |
title = Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications |
arxiv = 1109.3041 | 
date = September 2011 | doi=10.1103/PhysRevE.84.066106 | volume=84 | journal=Physical Review E| issue = 6 |bibcode = 2011PhRvE..84f6106D | pmid=22304154 | page=066106| s2cid = 15788070 }}</ref>
<ref name="abh14">{{cite arXiv| last1 = Abbe| first1 = Emmanuel| last2 = Bandeira| first2 = Afonso S.| last3 = Hall| first3 = Georgina| title = Exact Recovery in the Stochastic Block Model|eprint= 1405.3267| date = May 2014  | class = cs.SI}}</ref>
<ref name="as15a">{{cite arXiv| last1 = Abbe| first1 = Emmanuel| last2 = Sandon| first2 = Colin| title = Community detection in general stochastic block models: fundamental limits and efficient recovery algorithms|eprint= 1503.00609| date = March 2015  | class = math.PR}}</ref>
<ref name="as15b">{{cite arXiv| last1 = Abbe| first1 = Emmanuel| last2 = Sandon| first2 = Colin| title = Recovering communities in the general stochastic block model without knowing the parameters|eprint= 1506.03729|  date = June 2015  | class = math.PR}}</ref>
<ref name="lr15">{{Cite journal| doi = 10.1214/14-AOS1274| issn = 0090-5364| volume = 43| issue = 1| pages = 215–237| last1 = Lei| first1 = Jing| last2 = Rinaldo| first2 = Alessandro| title = Consistency of spectral clustering in stochastic block models| journal = The Annals of Statistics| date = February 2015 | arxiv = 1312.2050| s2cid = 88519551}}</ref>
<ref name="gbm">{{Cite journal| last1 = Galhotra| first1 = Sainyam| last2 = Mazumdar| first2 = Arya| last3 = Pal| first3 = Soumyabrata| 
last4 = Saha| first4 = Barna|author4-link=Barna Saha| title = The Geometric Block Model| journal = AAAI| date = February 2018 | volume = 32| doi = 10.1609/aaai.v32i1.11905| arxiv = 1709.05510| s2cid = 19152144}}</ref>
<ref name="krzakala-pnas">{{cite journal| 
last1 = Krzakala| first1 = Florent| 
last2 = Moore| first2 = Cristopher|
last3 = Mossel| first3 = Elchanan| 
last4 = Neeman| first4 = Joe| 
last5 = Sly| first5 = Allan| 
last6 = Lenka| first6 = Lenka| 
last7 = Zhang| first7 = Pan| 
title = Spectral redemption in clustering sparse networks | journal = Proceedings of the National Academy of Sciences| date = October 2013 | doi = 10.1073/pnas.1312486110 |arxiv = 1306.5550|bibcode = 2013PNAS..11020935K | volume=110 | issue = 52| pages=20935–20940| pmid = 24277835| pmc = 3876200| doi-access = free}}</ref>
<ref name="mns12">{{cite arXiv| last1 = Mossel| first1 = Elchanan| last2 = Neeman| first2 = Joe| last3 = Sly| first3 = Allan| title = Stochastic Block Models and Reconstruction|eprint= 1202.1499|  date = February 2012  | class = math.PR}}</ref>
<ref name="mns13">{{cite journal| last1 = Mossel| first1 = Elchanan| last2 = Neeman| first2 = Joe| last3 = Sly| first3 = Allan| title = Belief Propagation, Robust Reconstruction, and Optimal Recovery of Block Models| journal = The Annals of Applied Probability|arxiv= 1309.1380| date = September 2013 | volume = 26| issue = 4| pages = 2211–2256| doi = 10.1214/15-AAP1145| bibcode = 2013arXiv1309.1380M| s2cid = 184446}}</ref>
<ref name="mas13">{{cite arXiv| last = Massoulie| first = Laurent| title = Community detection thresholds and the weak Ramanujan property|eprint= 1311.3085| date = November 2013 | class = cs.SI}}</ref>
<ref name="al14">{{cite arXiv| last1 = Amini| first1 = Arash A.| last2 = Levina| first2 = Elizaveta| title = On semidefinite relaxations for the block model|eprint= 1406.5647|  date = June 2014 | class = cs.LG}}</ref>
<ref name="ar1">{{cite journal| 
last1 = Airoldi| first1 = Edoardo|authorlink1=Edoardo Airoldi| 
last2 = Blei| first2 = David|
last3 = Feinberg| first3 = Stephen| 
last4 = Xing| first4 = Eric| 
title = Mixed membership stochastic blockmodels| journal = Journal of Machine Learning Research |arxiv = 0705.4485|  date = May 2007  | volume = 9| pages = 1981–2014| pmid = 21701698| pmc = 3119541| bibcode = 2007arXiv0705.4485A}}</ref>
<ref name="fat19">{{cite arXiv| last = Fathi| first = Reza| title = Efficient Distributed Community Detection in the Stochastic Block Model|eprint= 1904.07494| date = April 2019 | class = cs.DC}}</ref>
<ref name="hol">{{cite journal |title=Stochastic blockmodels: First steps |journal=Social Networks |year=1983 |last1=Holland |first1=Paul W |last2=Laskey |first2=Kathryn Blackmond |last3=Leinhardt |first3=Samuel |volume=5 |issue=2 |pages=109–137 |issn=0378-8733 |doi=10.1016/0378-8733(83)90021-7 |s2cid=34098453 |url=https://doi.org/10.1016/0378-8733(83)90021-7 |accessdate=2021-06-16 |archive-date=2023-02-04 |archive-url=https://web.archive.org/web/20230204160405/https://www.sciencedirect.com/science/article/abs/pii/0378873383900217?via%3Dihub |url-status=live }}</ref>
<ref name="ker">{{cite journal |title=Stochastic blockmodels and community structure in networks |journal=Physical Review E |year=2011 |last1=Karrer |first1=Brian |last2=Newman |first2=Mark E J |volume=83 |issue=1 |page=016107 |doi=10.1103/PhysRevE.83.016107 |pmid=21405744 |url=https://link.aps.org/doi/10.1103/PhysRevE.83.016107 |accessdate=2021-06-16 |arxiv=1008.3926 |bibcode=2011PhRvE..83a6107K |s2cid=9068097 |archive-date=2023-02-04 |archive-url=https://web.archive.org/web/20230204160406/https://journals.aps.org/pre/abstract/10.1103/PhysRevE.83.016107 |url-status=live }}</ref>
<ref name="pei">{{cite journal |title=Hierarchical block structures and high-resolution model selection in large networks |journal=Physical Review X |year=2014 |last=Peixoto |first=Tiago |volume=4 |issue=1 |page=011047 |doi=10.1103/PhysRevX.4.011047 |url=https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011047 |accessdate=2021-06-16 |arxiv=1310.4377 |bibcode=2014PhRvX...4a1047P |s2cid=5841379 |archive-date=2021-06-24 |archive-url=https://web.archive.org/web/20210624195430/https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.011047 |url-status=live }}</ref>
}}

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