Extremal properties of evolving networks: local dependence and heavy tails

Markovich, Natalia M.

doi:10.1007/s10479-023-05175-y

Cited by 1 publication

(17 citation statements)

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“…To estimate the extremal index in random graphs, we use the interval estimator [20] that is modified in [16,21]. Nonparametric estimators of the extremal index require , as a rule, the selection of a threshold and/or another declustering parameter, e.g., a block size [17].…”

Section: Methodsmentioning

confidence: 99%

“…Our survey concerns random graphs evolved using preferential or clustering attachment of new nodes. Considering extremes in directed graphs, recent results in [14,15] concerning the tail and extremal indices of non-stationary sums and maxima of regularly varying random variables (r.v.s) are represented in the new context of evolving directed random graphs [16]. The relation between the tail and extremal indices of the latter sums and maxima and of PageRanks and the Max-linear models is discussed in [16].…”

Section: Introductionmentioning

confidence: 99%

“…Considering extremes in directed graphs, recent results in [14,15] concerning the tail and extremal indices of non-stationary sums and maxima of regularly varying random variables (r.v.s) are represented in the new context of evolving directed random graphs [16]. The relation between the tail and extremal indices of the latter sums and maxima and of PageRanks and the Max-linear models is discussed in [16]. This novel approach allows us to find the tail and extremal indices of PageRanks and the Max-linear models of random graphs for attachment methods providing that the distribution tails of the latter measures are regularly varying, e.g., the preferential attachment.…”

Section: Introductionmentioning

confidence: 99%

“…It is used to determine a limit distribution of the maximum of a stationary sequence of r.v.s [17,18]. To determine the extremal index of such node influence indices as PageRank and the Max-linear model, it is proposed in [16] to consider the nodes as roots of elementary trees with the nearest neighbor nodes. Then PageRanks and the Max-linear models of the roots are calculated using sums and maxima of PageRanks of their nearest neighbors that have in-coming edges to the roots.…”

Section: Introductionmentioning

confidence: 99%

“…Then PageRanks and the Max-linear models of the roots are calculated using sums and maxima of PageRanks of their nearest neighbors that have in-coming edges to the roots. Using Theorem 1 in [16], the tail and extremal indices of PageRanks and the Max-linear models are determined by the latter indices of the neighbor nodes belonging to the most heavy-tailed communities of the network.…”

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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