Large Communities in a Scale-Free Network

Alves, Caio; Ribeiro, Rodrigo; Sanchis, Rémy

doi:10.1007/s10955-016-1676-8

Cited by 10 publications

(28 citation statements)

References 20 publications

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“…Another difficulty in our setup is the degree of generality we work with. Our case replaces the parameter p ∈ (0, 1] in the models investigated in [1,12,13] by any non-negative real function f with f ∞ ≤ 1. The introduction of such function naturally increases the complexity of any analytical argument one may expect to rely on and makes it harder to discover threshold phenomena.…”

Section: 4mentioning

confidence: 99%

“…Our strategy to prove Theorem 1 will be to apply the results obtained for the function identically equal to p, with p ∈ (0, 1) (which we denote by cons p ) in [1] and propagate them to smaller functions. For the sake of the reader's convenience, we state and comment the aforementioned results here.…”

Section: Large Cliques: Proof Of Theoremmentioning

confidence: 99%

“…Theorem (Theorem 2 from [1] ). Given p ∈ (0, 1], let ξ ∈ (0, 2(2 − p) −1 − 1) and fix m ∈ Z + sufficiently large.…”

Section: Large Cliques: Proof Of Theoremmentioning

confidence: 99%

“…The kind of result we pursuit is to show that some graph properties hold asymptotically almost surely (a.a.s). Given a sequence of random graphs {G t } t∈N , we say that a graph property P holds a.a.s, if P (G t ∈ P) = 1 − o (1) i.e., the probability of observing such property increases to 1 as t goes to infinity. For instance, P may be the set of graphs having diameter less than the logarithm of the total number of vertices.…”

Section: Introductionmentioning

confidence: 99%

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Preferential Attachment Random Graphs with Edge-Step Functions

2019

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In this work we investigate a preferential attachment model whose parameter is a function f : N → [0, 1] that drives the asymptotic proportion between the numbers of vertices and edges of the graph. We investigate topological features of the graphs, proving general bounds for the diameter and the clique number. Our results regarding the diameter are sharp when f is a regularly varying function at infinity with strictly negative index of regular variation −γ. For this particular class, we prove a characterization for the diameter that depends only on −γ. More specifically, we prove that the diameter of such graphs is of order 1/γ with high probability, although its vertex set order goes to infinity polynomially. Sharp results for the diameter for a wide class of slowly varying functions are also obtained. The almost sure convergence for the properly normalized logarithm of the clique number of the graphs generated by slowly varying functions is also proved.

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Section: 4mentioning

confidence: 99%

Section: Large Cliques: Proof Of Theoremmentioning

confidence: 99%

“…Theorem (Theorem 2 from [1] ). Given p ∈ (0, 1], let ξ ∈ (0, 2(2 − p) −1 − 1) and fix m ∈ Z + sufficiently large.…”

Section: Large Cliques: Proof Of Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Preferential Attachment Random Graphs with Edge-Step Functions

2019

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“…[20] which is already recognized as a well-established method to analyze the structure of large scale graphs [21] is defined as the key property of agent in this paper. It means that we divide agents into different groups according to users' -Core value.…”

Section: Model Buildingmentioning

confidence: 99%

Based on Agent Model and K-Core Decomposition to Analyze the Diffusion of Mass Incident in Microblog

Pan¹,

Shen²,

Zhong³

2017

Complexity

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Mass incidents, which may influence the stability and security of the society in China, are getting more and more attentions not only from policy makers but also from Chinese social researchers. Catching the diffusion mechanism is believed to be critical to understand essential of these mass incidents since message dissemination plays important roles in every stage of mass incident. Recently, online social networks including Weibo (Chinese Twitter) become more and more popular in China. There are reports showing that Weibo discussion has accompanied the processes of most mass incidents happening in China in these few years. So, in this paper, we aim at introducing -Core decomposition method from complex network to the analysis on how to manage the diffusion of mass incident in Weibo based on agent based model which can simulate Weibo user's actions when mass incident happens. This work can help people understand how mass incident messages spread across the network. And then, people may have better strategy to manage the diffusion of mass incidents.

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Diameter of P.A. random graphs with edge‐step functions

Alves

Ribeiro²,

Sanchis

2020

Random Struct Algorithms

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In this paper, we investigate the global clustering coefficient (a.k.a transitivity) and clique number of graphs generated by a preferential attachment random graph model with an additional feature of allowing edge connections between existing vertices. Specifically, at each time step t, either a new vertex is added with probability f (t), or an edge is added between two existing vertices with probability 1 − f (t). We establish concentration inequalities for the global clustering and clique number of the resulting graphs under the assumption that f (t) is a regularly varying function at infinity with index of regular variation −γ, where γ ∈ [0, 1). We also demonstrate an inverse relation between these two statistics: the clique number is essentially the reciprocal of the global clustering coefficient.

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Large Communities in a Scale-Free Network

Cited by 10 publications

References 20 publications

Preferential Attachment Random Graphs with Edge-Step Functions

Preferential Attachment Random Graphs with Edge-Step Functions

Based on Agent Model and K-Core Decomposition to Analyze the Diffusion of Mass Incident in Microblog

Diameter of P.A. random graphs with edge‐step functions

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