A Proof Of The Block Model Threshold Conjecture

Mossel, Elchanan; Neeman, Joe; Sly, Allan

doi:10.48550/arxiv.1311.4115

Cited by 80 publications

(173 citation statements)

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“…When δ = 0, this matches the boundary for weak consistency in (Mossel et al, 2013b;Massoulié, 2014). In addition, SNR > 1 + 2 log n implies Err( Â) < 1/n → 0, which means strong consistency (recovery) in the regular tree case (C ≡ 1).…”

Section: The Transition Boundary For P-sbm Depends On the Valuesupporting

confidence: 68%

“…Sharp phase transitions for weak consistency have been thoroughly investigated in (Coja-Oghlan, 2010;Mossel et al, 2012Mossel et al, , 2013aMassoulié, 2014). In particular, spectral algorithms on the non-backtracking matrix have been studied in (Massoulié, 2014) and the non-backtracking walk in (Mossel et al, 2013b). Spectral algorithms as initialization and belief propagation as further refinement to achieve optimal recovery was established in (Mossel et al, 2013a).…”

Section: Prior Workmentioning

confidence: 99%

“…Most of the literature studies the standard SBM with no side labeling information. Here, many algorithms that achieve the sharp phase boundary are either global algorithms, or a combination of global and local algorithms, see (Mossel et al, 2013b;Massoulié, 2014;Hajek et al, 2014;Abbe et al, 2014). However, from the theoretical perspective, it is not clear how to distinguish the limitation for global v.s.…”

Section: Our Contributionsmentioning

confidence: 99%

“…This also matches the well-known Kesten-Stigum bound. For the standard SBM in k = 2 case, the Kesten-Stigum bound is proved to be sharp (even for global algorithms), see (Mossel et al, 2013b;Massoulié, 2014).…”

Section: The Transition Boundary For P-sbm Depends On the Valuementioning

confidence: 99%

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Inference via Message Passing on Partially Labeled Stochastic Block Models

Cai,

Liang,

Rakhlin

2016

Preprint

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We study the community detection and recovery problem in partially-labeled stochastic block models (SBM). We develop a fast linearized message-passing algorithm to reconstruct labels for SBM (with n nodes, k blocks, p, q intra and inter block connectivity) when δ proportion of node labels are revealed. The signal-to-noise ratio SNR(n,k, p, q, δ) is shown to characterize the fundamental limitations of inference via local algorithms. On the one hand, when SNR > 1, the linearized messagepassing algorithm provides the statistical inference guarantee with mis-classification rate at most exp(−(SNR−1)/2), thus interpolating smoothly between strong and weak consistency. This exponential dependence improves upon the known error rate (SNR − 1) −1 in the literature on weak recovery.On the other hand, when SNR < 1 (for k = 2) and SNR < 1/4 (for general growing k), we prove that local algorithms suffer an error rate at least 1 2 − δ • SNR, which is only slightly better than random guess for small δ.

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Section: The Transition Boundary For P-sbm Depends On the Valuesupporting

confidence: 68%

Section: Prior Workmentioning

confidence: 99%

Section: Our Contributionsmentioning

confidence: 99%

Section: The Transition Boundary For P-sbm Depends On the Valuementioning

confidence: 99%

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Inference via Message Passing on Partially Labeled Stochastic Block Models

Cai,

Liang,

Rakhlin

2016

Preprint

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“…To be precise, we show that under certain conditions on α n and B, this method achieves Err(ψ, ψ ) = o(n). In the block model terminology (Mossel et al, 2013), this statement implies that the algorithm is weakly consistent. Furthermore, if the hypergraph is dense (α n = 1), then we show that TTM can exactly recover the partitions, i.e., Err(ψ, ψ ) = o(1), and hence, exhibits strong consistency properties.…”

Section: Formal Description Of the Problemsmentioning

confidence: 99%

Uniform Hypergraph Partitioning: Provable Tensor Methods and Sampling Techniques

Ghoshdastidar,

Dukkipati

2016

Preprint

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In a series of recent works, we have generalised the consistency results in the stochastic block model literature to the case of uniform and non-uniform hypergraphs. The present paper continues the same line of study, where we focus on partitioning weighted uniform hypergraphs-a problem often encountered in computer vision. This work is motivated by two issues that arise when a hypergraph partitioning approach is used to tackle computer vision problems: (i) The uniform hypergraphs constructed for higher-order learning contain all edges, but most have negligible weights. Thus, the adjacency tensor is nearly sparse, and yet, not binary. (ii) A more serious concern is that standard partitioning algorithms need to compute all edge weights, which is computationally expensive for hypergraphs. This is usually resolved in practice by merging the clustering algorithm with a tensor sampling strategy-an approach that is yet to be analysed rigorously.We build on our earlier work on partitioning dense unweighted uniform hypergraphs (Ghoshdastidar and Dukkipati, ICML, 2015), and address the aforementioned issues by proposing provable and efficient partitioning algorithms. Our analysis justifies the empirical success of practical sampling techniques. We also complement our theoretical findings by elaborate empirical comparison of various hypergraph partitioning schemes.

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Structured Networks and Coarse‐Grained Descriptions

Delvenne

Lambiotte

Barahona

2019

Advances in Network Clustering and Blockmodeling

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This chapter discusses the interplay between structure and dynamics in complex networks. Given a particular network with an endowed dynamics, our goal is to find partitions aligned with the dynamical process acting on top of the network. We thus aim to gain a reduced description of the system that takes into account both its structure and dynamics.In the first part, we introduce the general mathematical setup for the types of dynamics we consider throughout the chapter. We provide two guiding examples, namely consensus dynamics and diffusion processes (random walks), motivating their connection to social network analysis, and provide a brief discussion on the general dynamical framework and its possible extensions.In the second part, we focus on the influence of graph structure on the dynamics taking place on the network, focussing on three concepts that allow us to gain insight into this notion. First, we describe how time scale separation can appear in the dynamics on a network as a consequence of graph structure. Second, we discuss how the presence of particular symmetries in the network give rise to invariant dynamical subspaces that can be precisely described by graph partitions. Third, we show how this dynamical viewpoint can be extended to study dynamics on networks with signed edges, which allow us to discuss connections to concepts in social network analysis, such as structural balance.In the third part, we discuss how to use dynamical processes unfolding on the network to detect meaningful network substructures. We then show how such dynamical measures can be related to seemingly different algorithm for community detection and coarse-graining proposed in the literature. We conclude with a brief summary and highlight interesting open future directions.

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A Proof Of The Block Model Threshold Conjecture

Cited by 80 publications

References 0 publications

Inference via Message Passing on Partially Labeled Stochastic Block Models

Inference via Message Passing on Partially Labeled Stochastic Block Models

Uniform Hypergraph Partitioning: Provable Tensor Methods and Sampling Techniques

Structured Networks and Coarse‐Grained Descriptions

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