303 research outputs found
Community detection thresholds and the weak Ramanujan property
Decelle et al.\cite{Decelle11} conjectured the existence of a sharp threshold
for community detection in sparse random graphs drawn from the stochastic block
model. Mossel et al.\cite{Mossel12} established the negative part of the
conjecture, proving impossibility of meaningful detection below the threshold.
However the positive part of the conjecture remained elusive so far. Here we
solve the positive part of the conjecture. We introduce a modified adjacency
matrix that counts self-avoiding paths of a given length between
pairs of nodes and prove that for logarithmic , the leading eigenvectors
of this modified matrix provide non-trivial detection, thereby settling the
conjecture. A key step in the proof consists in establishing a {\em weak
Ramanujan property} of matrix . Namely, the spectrum of consists in two
leading eigenvalues , and eigenvalues of a lower
order for all , denoting
's spectral radius. -regular graphs are Ramanujan when their second
eigenvalue verifies . Random -regular graphs have
a second largest eigenvalue of (see
Friedman\cite{friedman08}), thus being {\em almost} Ramanujan.
Erd\H{o}s-R\'enyi graphs with average degree at least logarithmic
() have a second eigenvalue of (see Feige and
Ofek\cite{Feige05}), a slightly weaker version of the Ramanujan property.
However this spectrum separation property fails for sparse ()
Erd\H{o}s-R\'enyi graphs. Our result thus shows that by constructing matrix
through neighborhood expansion, we regularize the original adjacency matrix to
eventually recover a weak form of the Ramanujan property
Projected Power Iteration for Network Alignment
The network alignment problem asks for the best correspondence between two
given graphs, so that the largest possible number of edges are matched. This
problem appears in many scientific problems (like the study of protein-protein
interactions) and it is very closely related to the quadratic assignment
problem which has graph isomorphism, traveling salesman and minimum bisection
problems as particular cases. The graph matching problem is NP-hard in general.
However, under some restrictive models for the graphs, algorithms can
approximate the alignment efficiently. In that spirit the recent work by Feizi
and collaborators introduce EigenAlign, a fast spectral method with convergence
guarantees for Erd\H{o}s-Reny\'i graphs. In this work we propose the algorithm
Projected Power Alignment, which is a projected power iteration version of
EigenAlign. We numerically show it improves the recovery rates of EigenAlign
and we describe the theory that may be used to provide performance guarantees
for Projected Power Alignment.Comment: 8 page
Non-Backtracking Spectrum of Degree-Corrected Stochastic Block Models
Motivated by community detection, we characterise the spectrum of the
non-backtracking matrix in the Degree-Corrected Stochastic Block Model.
Specifically, we consider a random graph on vertices partitioned into two
equal-sized clusters. The vertices have i.i.d. weights
with second moment . The intra-cluster connection probability for
vertices and is and the inter-cluster
connection probability is .
We show that with high probability, the following holds: The leading
eigenvalue of the non-backtracking matrix is asymptotic to . The second eigenvalue is asymptotic to when , but asymptotically bounded by
when . All the remaining eigenvalues are
asymptotically bounded by . As a result, a clustering
positively-correlated with the true communities can be obtained based on the
second eigenvector of in the regime where
In a previous work we obtained that detection is impossible when meaning that there occurs a phase-transition in the sparse regime of the
Degree-Corrected Stochastic Block Model.
As a corollary, we obtain that Degree-Corrected Erd\H{o}s-R\'enyi graphs
asymptotically satisfy the graph Riemann hypothesis, a quasi-Ramanujan
property.
A by-product of our proof is a weak law of large numbers for
local-functionals on Degree-Corrected Stochastic Block Models, which could be
of independent interest
Inference on graphs via semidefinite programming
Inference problems on graphs arise naturally when trying to make sense of network data. Oftentimes, these problems are formulated as intractable optimization programs. This renders the need for fast heuristics to find adequate solutions and for the study of their performance. For a certain class of problems, Javanmard et al. (1) successfully use tools from statistical physics to analyze the performance of semidefinite programming relaxations, an important heuristic for intractable problems.National Science Foundation (U.S.) (Grant DMS- 1317308
Community detection and stochastic block models: recent developments
The stochastic block model (SBM) is a random graph model with planted
clusters. It is widely employed as a canonical model to study clustering and
community detection, and provides generally a fertile ground to study the
statistical and computational tradeoffs that arise in network and data
sciences.
This note surveys the recent developments that establish the fundamental
limits for community detection in the SBM, both with respect to
information-theoretic and computational thresholds, and for various recovery
requirements such as exact, partial and weak recovery (a.k.a., detection). The
main results discussed are the phase transitions for exact recovery at the
Chernoff-Hellinger threshold, the phase transition for weak recovery at the
Kesten-Stigum threshold, the optimal distortion-SNR tradeoff for partial
recovery, the learning of the SBM parameters and the gap between
information-theoretic and computational thresholds.
The note also covers some of the algorithms developed in the quest of
achieving the limits, in particular two-round algorithms via graph-splitting,
semi-definite programming, linearized belief propagation, classical and
nonbacktracking spectral methods. A few open problems are also discussed
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