ﻻ يوجد ملخص باللغة العربية
Community detection is considered for a stochastic block model graph of n vertices, with K vertices in the planted community, edge probability p for pairs of vertices both in the community, and edge probability q for other pairs of vertices. The main focus of the paper is on weak recovery of the community based on the graph G, with o(K) misclassified vertices on average, in the sublinear regime $n^{1-o(1)} leq K leq o(n).$ A critical parameter is the effective signal-to-noise ratio $lambda=K^2(p-q)^2/((n-K)q)$, with $lambda=1$ corresponding to the Kesten-Stigum threshold. We show that a belief propagation algorithm achieves weak recovery if $lambda>1/e$, beyond the Kesten-Stigum threshold by a factor of $1/e.$ The belief propagation algorithm only needs to run for $log^ast n+O(1) $ iterations, with the total time complexity $O(|E| log^*n)$, where $log^*n$ is the iterated logarithm of $n.$ Conversely, if $lambda leq 1/e$, no local algorithm can asymptotically outperform trivial random guessing. Furthermore, a linear message-passing algorithm that corresponds to applying power iteration to the non-backtracking matrix of the graph is shown to attain weak recovery if and only if $lambda>1$. In addition, the belief propagation algorithm can be combined with a linear-time voting procedure to achieve the information limit of exact recovery (correctly classify all vertices with high probability) for all $K ge frac{n}{log n} left( rho_{rm BP} +o(1) right),$ where $rho_{rm BP}$ is a function of $p/q$.
We study the problem of recovering a hidden community of cardinality $K$ from an $n times n$ symmetric data matrix $A$, where for distinct indices $i,j$, $A_{ij} sim P$ if $i, j$ both belong to the community and $A_{ij} sim Q$ otherwise, for two know
We study a semidefinite programming (SDP) relaxation of the maximum likelihood estimation for exactly recovering a hidden community of cardinality $K$ from an $n times n$ symmetric data matrix $A$, where for distinct indices $i,j$, $A_{ij} sim P$ if
Let $(Z_n,ngeq 0)$ be a supercritical Galton-Watson process whose offspring distribution $mu$ has mean $lambda>1$ and is such that $int x(log(x))_+ dmu(x)<+infty$. According to the famous Kesten & Stigum theorem, $(Z_n/lambda^n)$ converges almost sur
We consider learning of fundamental properties of communities in large noisy networks, in the prototypical situation where the nodes or users are split into two classes according to a binary property, e.g., according to their opinions or preferences
The community detection problem requires to cluster the nodes of a network into a small number of well-connected communities. There has been substantial recent progress in characterizing the fundamental statistical limits of community detection under