27,836 research outputs found
Concentration of the Stationary Distribution on General Random Directed Graphs
We consider a random model for directed graphs whereby an arc is placed from
one vertex to another with a prescribed probability which may vary from arc to
arc. Using perturbation bounds as well as Chernoff inequalities, we show that
the stationary distribution of a Markov process on a random graph is
concentrated near that of the "expected" process under mild conditions. These
conditions involve the ratio between the minimum and maximum in- and
out-degrees, the ratio of the minimum and maximum entry in the stationary
distribution, and the smallest singu- lar value of the transition matrix.
Lastly, we give examples of applications of our results to well-known models
such as PageRank and G(n, p).Comment: 14 pages, 0 figure
A spectral method for community detection in moderately-sparse degree-corrected stochastic block models
We consider community detection in Degree-Corrected Stochastic Block Models
(DC-SBM). We propose a spectral clustering algorithm based on a suitably
normalized adjacency matrix. We show that this algorithm consistently recovers
the block-membership of all but a vanishing fraction of nodes, in the regime
where the lowest degree is of order log or higher. Recovery succeeds even
for very heterogeneous degree-distributions. The used algorithm does not rely
on parameters as input. In particular, it does not need to know the number of
communities
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
Sparse random graphs: regularization and concentration of the Laplacian
We study random graphs with possibly different edge probabilities in the
challenging sparse regime of bounded expected degrees. Unlike in the dense
case, neither the graph adjacency matrix nor its Laplacian concentrate around
their expectations due to the highly irregular distribution of node degrees. It
has been empirically observed that simply adding a constant of order to
each entry of the adjacency matrix substantially improves the behavior of
Laplacian. Here we prove that this regularization indeed forces Laplacian to
concentrate even in sparse graphs. As an immediate consequence in network
analysis, we establish the validity of one of the simplest and fastest
approaches to community detection -- regularized spectral clustering, under the
stochastic block model. Our proof of concentration of regularized Laplacian is
based on Grothendieck's inequality and factorization, combined with paving
arguments.Comment: Added reference
Lack of Hyperbolicity in Asymptotic Erd\"os--Renyi Sparse Random Graphs
In this work we prove that the giant component of the Erd\"os--Renyi random
graph for c a constant greater than 1 (sparse regime), is not Gromov
-hyperbolic for any positive with probability tending to one
as . As a corollary we provide an alternative proof that the giant
component of when c>1 has zero spectral gap almost surely as
.Comment: Updated version with improved results and narrativ
Spectral density of random graphs with topological constraints
The spectral density of random graphs with topological constraints is
analysed using the replica method. We consider graph ensembles featuring
generalised degree-degree correlations, as well as those with a community
structure. In each case an exact solution is found for the spectral density in
the form of consistency equations depending on the statistical properties of
the graph ensemble in question. We highlight the effect of these topological
constraints on the resulting spectral density.Comment: 24 pages, 6 figure
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