Statistical Analysis of Networks

Avrachenkov, Konstantin; Dreveton, Maximilien

doi:10.1561/9781638280514

Cited by 6 publications

(13 citation statements)

References 194 publications

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“…Example 2 Let G 0 be the same rectangle as in the previous example but with diagonal connecting nodes 1 and 3. Under the same evolution as in the previous example, we have c 2,t = c 4,t = 1 for all t ≥ 0 by (3).…”

Section: Bounds For Increments Of the Mean Clustering Coefficientsmentioning

confidence: 87%

“…Network evolution attracts interest of researchers due to numerous applications (Avrachenkov & Dreveton, 2022;Ghoshal et al, 2013; van der Hofstad, 2017; Norros & Reittu, 2006;Wan et al, 2020). The popular mechanism to model growing real-world networks and to explain a power-law distribution of their node degrees is a linear preferential attachment (LPA).…”

Section: Introductionmentioning

confidence: 99%

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Tail Index Estimation of PageRanks in Evolving Random Graphs

2022

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Random graphs are subject to the heterogeneities of the distributions of node indices and their dependence structures. Superstar nodes to which a large proportion of nodes attach in the evolving graphs are considered. In the present paper, a statistical analysis of the extremal part of random graphs is considered. We used the extreme value theory regarding sums and maxima of non-stationary random length sequences to evaluate the tail index of the PageRanks and max-linear models of superstar nodes in the evolving graphs where existing nodes or edges can be deleted or not. The evolution is provided by a linear preferential attachment. Our approach is based on the analysis of maxima and sums of the node PageRanks over communities (block maxima and block sums), which can be independent or weakly dependent random variables. By an empirical study, it was found that tail indices of the block maxima and block sums are close to the minimum tail index of representative series extracted from the communities. The tail indices are estimated by data of simulated graphs.

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Section: Bounds For Increments Of the Mean Clustering Coefficientsmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Tail Index Estimation of PageRanks in Evolving Random Graphs

2022

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“…The Stochastic Block Model (SBM) generalizes ER graphs [4]. Using the SBM, which is parameterized by the block probability matrix B ∈ [0, 1] k×k , where k is the number of blocks, one can check the dependence between two communities [62].…”

Section: Testing Of Dependence On Graphsmentioning

confidence: 99%

“…Some other models to generate random graphs that may fit observed networks are represented in [4,96,97].…”

Section: Other Models Of Random Networkmentioning

confidence: 99%

“…The study of power-law real-world random networks attracts significant attention due to the many fields of their application, e.g., citation, phone-call, cellular, urban transport, and economic trade networks, among others [1][2][3]. Several monographs [4][5][6][7] have certainly contributed to basic concepts of random network theory and practice. "Real networks differ from random graphs in that often their degree distribution follows a power-law" [1].…”

Section: Introductionmentioning

confidence: 99%

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Extreme Value Statistics for Evolving Random Networks

Markovich

Vaičiulis

2023

Mathematics

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Our objective is to survey recent results concerning the evolution of random networks and related extreme value statistics, which are a subject of interest due to numerous applications. Our survey concerns the statistical methodology but not the structure of random networks. We focus on the problems arising in evolving networks mainly due to the heavy-tailed nature of node indices. Tail and extremal indices of the node influence characteristics like in-degrees, out-degrees, PageRanks, and Max-linear models arising in the evolving random networks are discussed. Related topics like preferential and clustering attachments, community detection, stationarity and dependence of graphs, information spreading, finding the most influential leading nodes and communities, and related methods are surveyed. This survey tries to propose possible solutions to unsolved problems, like testing the stationarity and dependence of random graphs using known results obtained for random sequences. We provide a discussion of unsolved or insufficiently developed problems like the distribution of triangle and circle counts in evolving networks, or the clustering attachment and the local dependence of the modularity, the impact of node or edge deletion at each step of evolution on extreme value statistics, among many others. Considering existing techniques of community detection, we pay attention to such related topics as coloring graphs and anomaly detection by machine learning algorithms based on extreme value theory. In order to understand how one can compute tail and extremal indices on random graphs, we provide a structured and comprehensive review of their estimators obtained for random sequences. Methods to calculate the PageRank and PageRank vector are shortly presented. This survey aims to provide a better understanding of the directions in which the study of random networks has been done and how extreme value analysis developed for random sequences can be applied to random networks.

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