Overlapping community detection in networks based on link partitioning and partitioning around medoids

Ponomarenko, Alexander; Pitsoulis, Leonidas; Shamshetdinov, Marat

doi:10.48550/arxiv.1907.08731

Cited by 2 publications

(4 citation statements)

References 11 publications

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“…We have proposed an auxiliary field representation in the spirit of statistical field theory for the resistance distance distribution in large Erdős-Rényi graphs of fixed mean degree c. Using this representation, a saddle point estimation of this distribution becomes possible at large c in terms of the saddle point equation (19), producing the analytic curve (30), which at progressively larger values of c can be further simplified to (34) or even to the Gaussian form (35). We have furthermore identified the subleading peaks observed at large resistance distances in numerical simulations with contributions of vertices of low degrees, and developed an analytic estimate (53) for the rightmost prominent peak of this sort, coming from vertices of degree 1.…”

Section: Discussionmentioning

confidence: 99%

“…Resistance distances thus in principle capture properties of all possible paths, though there are important qualifications to this statement [11], see below. In view of these appealing properties, resistance distances have surfaced in research on subjects as diverse as theoretical physics [12,13], chemistry and bioinformatics [14][15][16][17][18][19], mathematical graph theory [20][21][22], data analysis and computer science [23][24][25][26][27][28][29][30][31][32][33][34][35]. Studying resistance distance can also be useful for understanding nutrient transport in leaf vascular networks, as hydraulic conductance of laminar flows in this setting can be equivalently studied in terms of electrical conductance in resistor networks [36][37][38][39].…”

Section: Introductionmentioning

confidence: 99%

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Resistance distance distribution in large sparse random graphs

Akara-pipattana¹,

Chotibut²,

Evnin³

2022

J. Stat. Mech.

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We consider an Erdős–Rényi random graph consisting of N vertices connected by randomly and independently drawing an edge between every pair of them with probability c/N so that at N → ∞ one obtains a graph of finite mean degree c. In this regime, we study the distribution of resistance distances between the vertices of this graph and develop an auxiliary field representation for this quantity in the spirit of statistical field theory. Using this representation, a saddle point evaluation of the resistance distance distribution is possible at N → ∞ in terms of an 1/c expansion. The leading order of this expansion captures the results of numerical simulations very well down to rather small values of c; for example, it recovers the empirical distribution at c = 4 or 6 with an overlap of around 90%. At large values of c, the distribution tends to a Gaussian of mean 2/c and standard deviation 2 / c 3 . At small values of c, the distribution is skewed toward larger values, as captured by our saddle point analysis, and many fine features appear in addition to the main peak, including subleading peaks that can be traced back to resistance distances between vertices of specific low degrees and the rest of the graph. We develop a more refined saddle point scheme that extracts the corresponding degree-differentiated resistance distance distributions. We then use this approach to recover analytically the most apparent of the subleading peaks that originates from vertices of degree 1. Rather intuitively, this subleading peak turns out to be a copy of the main peak, shifted by one unit of resistance distance and scaled down by the probability for a vertex to have degree 1. We comment on a possible lack of smoothness in the true N → ∞ distribution suggested by the numerics.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Resistance distance distribution in large sparse random graphs

Akara-pipattana¹,

Chotibut²,

Evnin³

2022

J. Stat. Mech.

View full text Add to dashboard Cite

show abstract

“…Using this representation, a saddle point estimation of this distribution becomes possible at large c in terms of the saddle point equation (17), producing the analytic curve (30), which at progressively larger values of c can be further simplified to (34) or even to the Gaussian form (35). We have furthermore identified the subleading peaks observed at large resistance distances in numerical simulations with contributions of vertices of low degrees, and developed an analytic estimate (53) for the rightmost prominent peak of this sort, coming from vertices of degree 1.…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Resistance distance distribution in large sparse random graphs

Akara-pipattana,

Chotibut,

Evnin

2021

Preprint

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We consider an Erdős-Rényi random graph consisting of N vertices connected by randomly and independently drawing an edge between every pair of them with probability c/N so that at N → ∞ one obtains a graph of finite mean degree c. In this regime, we study the distribution of resistance distances between the vertices of this graph and develop an auxiliary field representation for this quantity in the spirit of statistical field theory. Using this representation, a saddle point evaluation of the resistance distance distribution is possible at N → ∞ in terms of an 1/c expansion. The leading order of this expansion captures the results of numerical simulations very well down to rather small values of c; for example, it recovers the empirical distribution at c = 4 or 6 with an overlap of around 90%. At large values of c, the distribution tends to a Gaussian of mean 2/c and standard deviation 2/c 3 . At small values of c, the distribution is skewed toward larger values, as captured by our saddle point analysis, and many fine features appear in addition to the main peak, including subleading peaks that can be traced back to resistance distances between vertices of specific low degrees and the rest of the graph. We develop a more refined saddle point scheme that extracts the corresponding degree-differentiated resistance distance distributions. We then use this approach to recover analytically the most apparent of the subleading peaks that originates from vertices of degree 1. Rather intuitively, this subleading peak turns out to be a copy of the main peak, shifted by one unit of resistance distance and scaled down by the probability for a vertex to have degree 1. We comment on a possible lack of smoothness in the true N → ∞ distribution suggested by the numerics.

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Overlapping community detection in networks based on link partitioning and partitioning around medoids

Cited by 2 publications

References 11 publications

Resistance distance distribution in large sparse random graphs

Resistance distance distribution in large sparse random graphs

Resistance distance distribution in large sparse random graphs

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