Distinguishing Power-Law Uniform Random Graphs from Inhomogeneous Random Graphs Through Small Subgraphs

Stegehuis, Clara

doi:10.1007/s10955-022-02884-9

Cited by 2 publications

(4 citation statements)

References 16 publications

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“…However, the Erdős-Rényi model is based on assumptions that are not met by typical real-life networks, many of which have been observed to posses power-law degree distribution [9]. Therefore, real-life networks are often modeled using power-law random graphs or, more generally, uniform random graphs with a given degree sequence [33]. Obtaining asymptotic estimates for clustering problems in such more realistic random graph models is another interesting direction for future research.…”

Section: Discussionmentioning

confidence: 99%

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Asymptotic bounds for clustering problems in random graphs

Lykhovyd,

Butenko,

Krokhmal

2023

Networks

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Graph clustering is an important problem in network analysis. This problem can be approached by first finding a large cluster subgraph (i.e., a subgraph in which every connected component is a complete graph), perhaps in a relaxed form (connected components may have missing edges), and then assigning each of the remaining vertices to one of the connected components of the cluster subgraph according to some optimization criteria. The more vertices can be included in the initial cluster subgraph (also referred to as independent union of clusters), the more “clusterable” the graph is. This paper proposes a framework for establishing asymptotic bounds on the cardinality of independent unions of clusters in Erdős‐Rényi random graphs with constant , referred to as uniform random graphs. In particular, sufficient conditions ensuring (where is the number of nodes) upper bounds with probability 1 are developed and shown to be applicable for the maximum independent union of cliques as well as some clique relaxations. In addition, it is shown that every graph must have an independent union of cliques of cardinality at least . Since this bound is asymptotically tight on uniform random graphs, this suggests that these graphs can be viewed as a “least clusterable” class of graphs.

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Section: Discussionmentioning

confidence: 99%

“…For r < 1 there exists b such that r < b < 1. The coefficient in parentheses in (33) tends to r when n → ∞, thus there exists…”

Section: Independent Union Of Cliquesmentioning

confidence: 99%

Asymptotic bounds for clustering problems in random graphs

Lykhovyd,

Butenko,

Krokhmal

2023

Networks

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“…Uniform random graphs with specified degree sequences are commonly used to model (power-law) real-world networks (see, Refs. [35,36] for a survey). In [37], a graph Hamiltonian is proposed as a method to model heterogeneous clustered graphs.…”

Section: Power-law Of In-and Out-degreesmentioning

confidence: 99%

“…, d n ), where ∑ n i=1 d i ≡ 0(mod2), the uniform random graph is a simple graph, uniformly sampled from the set of all simple graphs with degree sequence (d i ) i∈[n] , Refs. [35,36]. It is assumed that d is a realizable degree sequence, meaning that there exists a simple graph with degree sequence d. Let G(d) denote the ensemble of all simple graphs on degree sequence d, and let…”

Section: Triangle Counts and Local Clustering Coefficientsmentioning

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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Distinguishing Power-Law Uniform Random Graphs from Inhomogeneous Random Graphs Through Small Subgraphs

Cited by 2 publications

References 16 publications

Asymptotic bounds for clustering problems in random graphs

Asymptotic bounds for clustering problems in random graphs

Extreme Value Statistics for Evolving Random Networks

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