
Recovering a Hidden Community in a Preferential Attachment Graph
A message passing algorithm (MP) is derived for recovering a dense subgr...
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Oriented and Degreegenerated Block Models: Generating and Inferring Communities with Inhomogeneous Degree Distributions
The stochastic block model is a powerful tool for inferring community st...
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Exploring Communities in Large Profiled Graphs
Given a graph G and a vertex q∈ G, the community search (CS) problem aim...
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A sparse stochastic block model with two unequal communities
We show posterior convergence for the community structure in the planted...
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Topological Effects on Attacks Against Vertex Classification
Vertex classification is vulnerable to perturbations of both graph topol...
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Estimating the Size of a Large Network and its Communities from a Random Sample
Most realworld networks are too large to be measured or studied directl...
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A Cluster Model for Growth of Random Trees
We first consider the growth of trees by probabilistic attachment of new...
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Preferential Attachment Graphs with Planted Communities
A variation of the preferential attachment random graph model of Barabási and Albert is defined that incorporates planted communities. The graph is built progressively, with new vertices attaching to the existing ones onebyone. At every step, the incoming vertex is randomly assigned a label, which represents a community it belongs to. This vertex then chooses certain vertices as its neighbors, with the choice of each vertex being proportional to the degree of the vertex multiplied by an affinity depending on the labels of the new vertex and a potential neighbor. It is shown that the fraction of halfedges attached to vertices with a given label converges almost surely for some classes of affinity matrices. In addition, the empirical degree distribution for the set of vertices with a given label converges to a heavy tailed distribution, such that the tail decay parameter can be different for different communities. Our proof method may be of independent interest, both for the classical Barabási Albert model and for other possible extensions.
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