Extremal independence in discrete random systems

Isaev, Mikhail; Rodionov, I. V.; Zhang, Rui-Ray; Zhukovskii, Maksim

doi:10.48550/arxiv.2105.04917

Cited by 4 publications

(5 citation statements)

References 35 publications

(67 reference statements)

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“…The dependency graph model we consider requires variables in disjoint non-adjacent subgraphs to be independent. There are some newly introduced dependency graph models such as weighted dependency graphs [Dousse andFéray, 2019, Féray et al, 2018], and the combination of mixing coefficients and dependency graphs [Lampert et al, 2018, Isaev et al, 2021. It would be interesting to use the new dependency graph to obtain generalization bounds for learning under different dependent settings.…”

Section: Discussionmentioning

confidence: 99%

Generalization bounds for learning under graph-dependence: A survey

Zhang¹,

Amini²

2022

Preprint

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Traditional statistical learning theory relies on the assumption that data are identically and independently generated from a given distribution (i.i.d.). The independently distributed assumption, on the other hand, fails to hold in many real applications. In this survey, we consider learning settings in which examples are dependent and their dependence relationship can be characterised by a graph. We collect various graph-dependent concentration bounds, which are then used to derive Rademacher and stability generalization bounds for learning from graph-dependent data. We illustrate this paradigm with three learning tasks and provide some research directions for future work. To the best of our knowledge this is the first survey on this subject.

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Section: Discussionmentioning

confidence: 99%

Generalization bounds for learning under graph-dependence: A survey

Zhang¹,

Amini²

2022

Preprint

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“…However, in that case, all graph-dependent indicators are dependent, and the only valid dependency graph for them is the complete graph. In this case, we may need to modify the method by incorporating the notion of weak dependence, see, for example, [6].…”

Section: Discussionmentioning

confidence: 99%

Asymptotic linearity of binomial random hypergraphs via cluster expansion under graph-dependence

Zhang¹

2022

Preprint

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Let integer n3 and integer r = r(n) 3. Define the binomial random r-uniform hypergraph H r (n, p) to be the r-uniform graph on the vertex set [n] such that each rset is an edge independently with probability p. A hypergraph is linear if every pair of hyperedges intersects in at most one vertex. We study the probability of linearity of random hypergraphs H r (n, p) via cluster expansion and give more precise asymptotics of the probability in question, improving the asymptotic probability of linearity obtained by McKay and Tian, in particular, when r = 3 and p = o(n −7/5 ).

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“…The latter means, in fact ,the fulfillment of the max-stable property for the maximal component of X. As applications, distributions of various extremal characteristics of binomial random hypergraphs, such as a maximum codegree and a maximum number of cliques sharing a given vertex, are obtained in [11].…”

Section: Testing Of Dependence On Graphsmentioning

confidence: 99%

“…Among the recent applications, we find graphical models that are used to model extremal dependence or causality between events [8,9], including a bond percolation; see [10] among others. The distributions of various extremal characteristics of random discrete structures, such as the maximum number of common neighbors of a set of vertices in the graph, the maximum codegree, and the maximum number of cliques sharing a given vertex in binomial random hypergraphs, are derived in [11,12].…”

Section: Introductionmentioning

confidence: 99%

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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Extremal independence in discrete random systems

Cited by 4 publications

References 35 publications

Generalization bounds for learning under graph-dependence: A survey

Generalization bounds for learning under graph-dependence: A survey

Asymptotic linearity of binomial random hypergraphs via cluster expansion under graph-dependence

Extreme Value Statistics for Evolving Random Networks

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