TY - JOUR
T1 - Asymptotic mutual information for the balanced binary stochastic block model
AU - Deshpande, Yash
AU - Abbe, Emmanuel
AU - Montanari, Andrea
N1 - Funding Information:
Y.D. and A.M. were partially supported by NSF grants CCF-1319979 and DMS-1106627, and the AFOSR grant FA9550-13-1-0036. Part of this work was done while the authors were visiting Simons Institute for the Theory of Computing, UC Berkeley.
Publisher Copyright:
© The authors 2016.
PY - 2017
Y1 - 2017
N2 - We develop an information-theoretic viewof the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph G from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit 'single-letter' characterization of the per-vertex mutual information between the vertex labels and the graph, when the mean vertex degree diverges. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and -in particular-reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime, there exists a procedure that estimates the vertex labels better than random guessing.
AB - We develop an information-theoretic viewof the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph G from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit 'single-letter' characterization of the per-vertex mutual information between the vertex labels and the graph, when the mean vertex degree diverges. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and -in particular-reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime, there exists a procedure that estimates the vertex labels better than random guessing.
KW - Approximate message passing
KW - Community detection
KW - Mutual information
KW - Stochastic block model
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U2 - 10.1093/imaiai/iaw017
DO - 10.1093/imaiai/iaw017
M3 - Article
AN - SCOPUS:85048590456
SN - 2049-8772
VL - 6
SP - 125
EP - 170
JO - Information and Inference
JF - Information and Inference
IS - 2
ER -