Algorithms for graph partitioning on the planted partition model

Condon, Anne; Karp, Richard M.

doi:10.1002/1098-2418(200103)18:2<116::aid-rsa1001>3.0.co;2-2

Cited by 284 publications

(115 citation statements)

References 15 publications

(21 reference statements)

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“…Since the joint condition (8) is necessary for weak recovery, and hence also for exact recovery, it suffices to prove (14) under the assumption that (8) holds, i.e., KD(P Q) → ∞, KD(P Q) ≥ (2 − ǫ 0 ) log(n/K)…”

Section: The Necessary Conditionmentioning

confidence: 99%

“…Thus if K = O(1), then (42) implies (14). Hence, we assume K → ∞ in the following without loss of generality.…”

Section: The Necessary Conditionmentioning

confidence: 99%

“…The idea is to gain independence: the outcome of estimation based on the reduced set of indices is independent of the data between the withheld indices and the reduced set of indices, and the withheld subset is sufficiently small so that we can still obtain sharp constants. This method is mentioned in [14], and variations of it have been used in [14], [35], and [34].…”

Section: Introductionmentioning

confidence: 99%

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Information limits for recovering a hidden community

Hajek

2016

2016 IEEE International Symposium on Information Theory (ISIT)

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We study the problem of recovering a hidden community of cardinality K from an n × n symmetric data matrix A, where for distinct indices i, j, A ij ∼ P if i, j both belong to the community and A ij ∼ Q otherwise, for two known probability distributions P and Q depending on n. If P = Bern(p) and Q = Bern(q) with p > q, it reduces to the problem of finding a densely-connected K-subgraph planted in a large Erdös-Rényi graph; if P = N (µ, 1) and Q = N (0, 1) with µ > 0, it corresponds to the problem of locating a K × K principal submatrix of elevated means in a large Gaussian random matrix. We focus on two types of asymptotic recovery guarantees as n → ∞: (1) weak recovery: expected number of classification errors is o(K); (2) exact recovery: probability of classifying all indices correctly converges to one. Under mild assumptions on P and Q, and allowing the community size to scale sublinearly with n, we derive a set of sufficient conditions and a set of necessary conditions for recovery, which are asymptotically tight with sharp constants. The results hold in particular for the Gaussian case, and for the case of bounded log likelihood ratio, including the Bernoulli case whenever p q and 1−p 1−q are bounded away from zero and infinity. An important algorithmic implication is that, whenever exact recovery is information theoretically possible, any algorithm that provides weak recovery when the community size is concentrated near K can be upgraded to achieve exact recovery in linear additional time by a simple voting procedure.

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Section: The Necessary Conditionmentioning

confidence: 99%

“…Thus if K = O(1), then (42) implies (14). Hence, we assume K → ∞ in the following without loss of generality.…”

Section: The Necessary Conditionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Information limits for recovering a hidden community

Hajek

2016

2016 IEEE International Symposium on Information Theory (ISIT)

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“…• an unweighted graph, simulated similarly as the "planted 3-partition graph" described in [60]. The nodes of the groups 1 to 4 and the nodes of the groups 5 to 8 could not be distinguished in the graph structure: the edges within these two sets of nodes were randomly generated with a probability equal to 0.3.…”

Section: Multiple Relational Som On Simulated Datamentioning

confidence: 99%

On-line relational and multiple relational SOM

Olteanu¹,

Villa-Vialaneix²

2015

Neurocomputing

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In some applications and in order to address real-world situations better, data may be more complex than simple numerical vectors. In some examples, data can be known only through their pairwise dissimilarities or through multiple dissimilarities, each of them describing a particular feature of the data set. Several variants of the Self Organizing Map (SOM) algorithm were introduced to generalize the original algorithm to the framework of dissimilarity data. Whereas median SOM is based on a rough representation of the prototypes, relational SOM allows representing these prototypes by a virtual linear combination of all elements in the data set, referring to a pseudo-euclidean framework. In the present article, an on-line version of relational SOM is introduced and studied. Similarly to the situation in the Euclidean framework, this on-line algorithm provides a better organization and is much less sensible to prototype initialization than standard (batch) relational SOM. In a more general case, this stochastic version allows us to integrate an additional stochastic gradient descent step in the algorithm which can tune the respective weights of several dissimilarities in an optimal way: the resulting multiple relational SOM thus has the ability to integrate several sources of data of different types, or to make a consensus between several dissimilarities describing the same data. The algorithms introduced in this manuscript are tested on several data sets, including categorical data and graphs. On-line relational SOM is currently available in the R package SOMbrero that can be downloaded at http://sombrero.r-forge.r-project.org/ or directly tested on its Web User Interface at http://shiny.nathalievilla.org/sombrero.

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“…Here we shall focus on the simplest one, where p rs = p in , ∀r = s and p rs = p out , ∀r = s. Here, if p in > p out , the expected number of neighbors of a node within its group exceeds the expected number of neighbors of the node in each of the remaining q − 1 groups, so the groups are communities according to the general intuition. This version coincides with the planted partition model by Condon and Karp [5], and has generated popular benchmark graphs that are regularly used to test the performance of community detection techniques, like the four-groups test [1] and the LFR benchmark [6].…”

Section: Constrained Versus Unconstrained Community Detectionmentioning

confidence: 99%

Improving the performance of algorithms to find communities in networks

2014

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Most algorithms to detect communities in networks typically work without any information on the cluster structure to be found, as one has no a priori knowledge of it, in general. Not surprisingly, knowing some features of the unknown partition could help its identification, yielding an improvement of the performance of the method. Here we show that, if the number of clusters were known beforehand, standard methods, like modularity optimization, would considerably gain in accuracy, mitigating the severe resolution bias that undermines the reliability of the results of the original unconstrained version. The number of clusters can be inferred from the spectra of the recently introduced non-backtracking and flow matrices, even in benchmark graphs with realistic community structure. The limit of such two-step procedure is the overhead of the computation of the spectra.

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Algorithms for graph partitioning on the planted partition model

Cited by 284 publications

References 15 publications

Information limits for recovering a hidden community

Information limits for recovering a hidden community

On-line relational and multiple relational SOM

Improving the performance of algorithms to find communities in networks

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