Spectral properties of modularity matrices

Bolla, Marianna; Bullins, Brian; Chaturapruek, Sorathan; Chen, Shiwen; Friedl, Katalin

doi:10.1016/j.laa.2014.10.039

Cited by 10 publications

(15 citation statements)

References 9 publications

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“…But now since i∈I |C i | ≥ |I| and whp |I| = n − m we have whp 2 i |C i | − |I| ≥ n − m. On the other hand, since i |C i | ≤ n, whp 2 i |C i | − |I| ≤ 2n − |I| ≤ n + m. This together with (5) implies that whp…”

Section: Hence Whpmentioning

confidence: 89%

Modularity of Erdős‐Rényi random graphs

McDiarmid

Skerman

2020

Random Struct Algorithms

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For a given graph G, each partition of the vertices has a modularity score, with higher values taken to indicate that the partition better captures community structure in G. The modularity q * (G) (where 0 ≤ q * (G) ≤ 1) of the graph G is defined to be the maximum over all vertex partitions of the modularity score. Modularity is at the heart of the most popular algorithms for community detection, so it is an important graph parameter to understand mathematically.In particular, we may want to understand the behaviour of modularity for the Erdős-Rényi random graph G n,p with n vertices and edge-probability p. Two key features which we find are that the modularity is 1 + o(1) with high probability (whp) for np up to 1 + o(1) (and no further); and when np ≥ 1 and p is bounded below 1, it has order (np) −1/2 whp, in accord with a conjecture by Reichardt and Bornholdt in 2006 (and disproving another conjecture from the physics literature).

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Section: Hence Whpmentioning

confidence: 89%

Modularity of Erdős‐Rényi random graphs

McDiarmid

Skerman

2020

Random Struct Algorithms

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“…While the upper bound q N Cut G ≤ µ G has been shown in (12) by simple arguments, a converse relation, bounding q N Cut G from below in terms of µ G , is not that easy. In fact, there it is possible that µ G > 0 while q N Cut G < 0, as shown experimentally in [2]. If 0 = λ 1 < λ 2 ≤ · · · ≤ λ n ≤ 2 are the eigenvalues of L, the Cheeger inequality states that…”

Section: Cheeger-type Inequalitiesmentioning

confidence: 95%

“…Already at this stage intuition suggests that a close relation should exists between m G and the cut-modularity (3), and that the subsets S ⊆ V having positive modularity should be related with positive eigenvalues of M . The following theorem summarizes some important eigenproperties of M that have been proven in recent literature, see in particular, [2,8,16].…”

Section: The Newman-girvan Modularity Matrixmentioning

confidence: 98%

Modularity bounds for clusters located by leading eigenvectors of the normalized modularity matrix

Fasino

Tudisco

2017

Journal of Mathematical Inequalities

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Abstract. Nodal theorems for generalized modularity matrices ensure that the cluster located by the positive entries of the leading eigenvector of various modularity matrices induces a connected subgraph. In this paper we obtain lower bounds for the modularity of that subgraph showing that, under certain conditions, the nodal domains induced by eigenvectors corresponding to highly positive eigenvalues of the normalized modularity matrix have indeed positive modularity, that is, they can be recognized as modules inside the network. Moreover we establish Cheeger-type inequalities for the cut-modularity of the graph, providing a theoretical support to the common understanding that highly positive eigenvalues of modularity matrices are related with the possibility of subdividing a network into communities.Mathematics subject classification (2010): 05C50, 15A18, 15B99.

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“…Because of tr(M) < 0, M must have at least one negative eigenvalue, and it is usually indefinite. In [11] we proved that the modularity matrix of a simple graph is negative semidefinite if and only if it is a complete multipartite graph. The same applies to the normalized modularity matrix, since it has the same inertia.…”

Section: Notation and Graph Based Matricesmentioning

confidence: 99%

“…. k) and puv = p ∈ (0, 1), then the graph Gn(P, P k ) has a so-called soft-core multipartite structure, defined in [11]. In the special case when p = 1, it is the complete k-partite graph Kn 1 ,...,n k over the independent vertex classes of P k , where …”

Section: Definitionmentioning

confidence: 99%

Matrix and discrepancy view of generalized random and quasirandom graphs

Bolla

Elbanna

2015

Special Matrices

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We will discuss how graph based matrices are capable to find classification of the graph vertices with small within-and between-cluster discrepancies. The structural eigenvalues together with the corresponding spectral subspaces of the normalized modularity matrix are used to find a block-structure in the graph. The notions are extended to rectangular arrays of nonnegative entries and to directed graphs. We also investigate relations between spectral properties, multiway discrepancies, and degree distribution of generalized random graphs. These properties are regarded as generalized quasirandom properties, and we conjecture and partly prove that they are also equivalent for certain deterministic graph sequences, irrespective of stochastic models.

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Spectral properties of modularity matrices

Cited by 10 publications

References 9 publications

Modularity of Erdős‐Rényi random graphs

Modularity of Erdős‐Rényi random graphs

Modularity bounds for clusters located by leading eigenvectors of the normalized modularity matrix

Matrix and discrepancy view of generalized random and quasirandom graphs

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