Non-backtracking Spectrum of Random Graphs: Community Detection and Non-regular Ramanujan Graphs

Bordenave, Charles; Lelarge, Marc; Massoulié, Laurent

doi:10.1109/focs.2015.86

Cited by 158 publications

(351 citation statements)

References 26 publications

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“…A large literature exists on both the recovery algorithms and the theory establishing when a recovery is possible [14,33,34,32,1,29,8]. There are methods that perform better than a random guess (i.e.…”

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Concentration and regularization of random graphs

Levina

Vershynin

2017

Random Struct Algorithms

124

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Abstract. This paper studies how close random graphs are typically to their expectations. We interpret this question through the concentration of the adjacency and Laplacian matrices in the spectral norm. We study inhomogeneous Erdös-Rényi random graphs on n vertices, where edges form independently and possibly with different probabilities pij. Sparse random graphs whose expected degrees are o(log n) fail to concentrate; the obstruction is caused by vertices with abnormally high and low degrees. We show that concentration can be restored if we regularize the degrees of such vertices, and one can do this in various ways. As an example, let us reweight or remove enough edges to make all degrees bounded above by O(d) where d = max npij. Then we show that the resulting adjacency matrix A concentrates with the optimal rate:Similarly, if we make all degrees bounded below by d by adding weight d/n to all edges, then the resulting Laplacian concentrates with the optimal rate:Our approach is based on GrothendieckPietsch factorization, using which we construct a new decomposition of random graphs. We illustrate the concentration results with an application to the community detection problem in the analysis of networks.

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confidence: 99%

“…Several types of algorithms are known to succeed in this regume, including non-backtracking walks [33,29,8], spectral methods [12] and methods based on semidefinite programming [19,31].…”

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confidence: 99%

Concentration and regularization of random graphs

Levina

Vershynin

2017

Random Struct Algorithms

124

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“…There has been renewed interest in the sparse regime of the SBM following the paper of [12], which made number of striking conjectures on phase-transitions. Subsequently, some of them have been established with the most notable and relevant achievements to ours being that of [33], [31], [26] and [7]. These papers prove that both Community Detection and the distinguishability problem for the two community sparse SBM undergo a phase-transition at the same point which they characterize explicitly.…”

Section: Overview Of Our Results and Techniquesmentioning

confidence: 99%

Community Detection on Euclidean Random Graphs

Sankararaman

Baccelli

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study Community Detection (CD) on a class of sparse spatial random graphs embedded in the Euclidean space. Our random graph is the planted-partition version of the random connection model studied in Stochastic Geometry. Each node has two labels -an i.i.d. uniform {−1, +1} valued community label and a R d valued location label which form the support of a Poisson Point Process of intensity λ on R d . Conditional on the labels, edges are drawn independently at random depending both on the Euclidean distance between the nodes and the community labels on the nodes. The CD problem then consists in estimating the partition of nodes into communities better than at random, based on an observation of the random graph and the spatial location labels on nodes. We establish a non-trivial phase-transition for this problem in terms of λ. We show that for small λ, there exists no algorithm for CD, For large λ, our algorithm solves CD efficiently. We show that for small λ, there exists no algorithm for CD. For large λ, we propose an algorithm which solves CD efficiently.In certain special cases, we establish the exact threshold on λ which separates the existence of an algorithm which solves CD from the impossibility of any algorithm. We also establish a distinguishability result which says that one can always efficiently infer the existence of a partition given the graph and the spatial locations even when one cannot identify the partition better than at random. This is a new phenomenon not observed thus far in any non-spatial Erdős-Rényi based models.

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“…There has been renewed interest in the sparse regime of the SBM following the paper [14], which made a number of striking conjectures on phase-transitions. Subsequently, some of them have been established with the most notable and relevant achievements to ours being that of [37], [35], [29] and [9]. These papers prove that both Community Detection and the distinguishability problem for the two community sparse SBM undergo a phase-transition at the same point which they characterize explicitly.…”

Section: Related Workmentioning

confidence: 99%

Community detection on euclidean random graphs

Sankararaman

Baccelli

2017

2017 55th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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We study the problem of community detection on Euclidean random geometric graphs where each vertex has two latent variables: a binary community label and a R d valued location label which forms the support of a Poisson point process of intensity λ. A random graph is then drawn with edge probabilities dependent on both the community and location labels. In contrast to the stochastic block model (SBM) that has no location labels, the resulting random graph contains many more short loops due to the geometric embedding. We consider the recovery of the community labels, partial and exact, using the random graph and the location labels. We establish phase transitions for both sparse and logarithmic degree regimes, and provide bounds on the location of the thresholds, conjectured to be tight in the case of exact recovery. We also show that the threshold of the distinguishability problem, i.e., the testing between our model and the null model without community labels exhibits no phase-transition and in particular, does not match the weak recovery threshold (in contrast to the SBM).

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Non-backtracking Spectrum of Random Graphs: Community Detection and Non-regular Ramanujan Graphs

Cited by 158 publications

References 26 publications

Concentration and regularization of random graphs

Concentration and regularization of random graphs

Community Detection on Euclidean Random Graphs

Community detection on euclidean random graphs

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