Variational principle for scale-free network motifs

Stegehuis, Clara; Hofstad, Remco van der; Leeuwaarden, Johan S. H. van

doi:10.1038/s41598-019-43050-8

Cited by 15 publications

(16 citation statements)

References 50 publications

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“…This can be observed, for instance, in Figure 2. Furthermore, Theorem 1 shows that in this setting, the number of dominating cliques scales as n k(3−τ )/2 for all k, which is the same scaling in n as in many non-geometric scale-free models, such as in the inhomogeneous random graph, the erased configuration model and the uniform random graph [18,17,11]. This seems to imply that that when τ < 7/3, we cannot distinguish geometric and non-geometric scale-free networks by counting the number of cliques, or by studying the clustering coefficient.…”

Section: Discussionmentioning

confidence: 69%

Cliques in geometric inhomogeneous random graph

Michielan,

Stegehuis

2021

Preprint

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Many real-world networks were found to be highly clustered, and contain a large amount of small cliques. We here investigate the number of cliques of any size k contained in a geometric inhomogeneous random graph: a scale-free network model containing geometry. The interplay between scale-freeness and geometry ensures that connections are likely to form between either high-degree vertices, or between close by vertices. At the same time it is rare for a vertex to have a high degree, and most vertices are not close to one another. This trade-off makes cliques more likely to appear between specific vertices. In this paper, we formalize this trade-off and prove that there exists a predominant type of clique in terms of the degrees and the positions of the vertices that span the clique. Moreover, we show that the asymptotic number of cliques as well as the predominant clique type undergoes a phase transition, in which only k and the degree-exponent τ are involved. Interestingly, this phase transition shows that for small values of τ , the underlying geometry of the model is irrelevant: the number of cliques scales the same as in a non-geometric network model.

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Section: Discussionmentioning

confidence: 69%

Cliques in geometric inhomogeneous random graph

Michielan,

Stegehuis

2021

Preprint

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“…In [10,15], similar theorems for random graphs with connection probability (2.11) and (2.10) were derived. The number of all induced subgraphs in the model with connection probability (2.10) has the same scaling in n as the number of induced subgraphs in the models with connection probability (2.11).…”

Section: Distinguishing Uniform Random Graphs From Rank-1 Inhomogeneo...mentioning

confidence: 99%

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

Stegehuis

2022

J Stat Phys

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We investigate the asymptotic number of induced subgraphs in power-law uniform random graphs. We show that these induced subgraphs appear typically on vertices with specific degrees, which are found by solving an optimization problem. Furthermore, we show that this optimization problem allows to design a linear-time, randomized algorithm that distinguishes between uniform random graphs and random graph models that create graphs with approximately a desired degree sequence: the power-law rank-1 inhomogeneous random graph. This algorithm uses the fact that some specific induced subgraphs appear significantly more often in uniform random graphs than in rank-1 inhomogeneous random graphs.

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“…(ii): we consider this to be the most interesting regime, both from a mathematical viewpoint and for applications (see Stegehuis et al (2019)). The degrees have bounded first moment in N, but unbounded variance, while the weights have infinite mean.…”

Section: Our Contributionmentioning

confidence: 99%

Scale-free percolation mixing time

Cipriani¹,

Salvi²

2021

Preprint

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Assign to each vertex of the one-dimensional torus i.i.d. weights with a heavy-tail of index τ − 1 > 0. Connect then each couple of vertices with probability roughly proportional to the product of their weights and that decays polynomially with exponent α > 0 in their distance. The resulting graph is called scale-free percolation. The goal of this work is to study the mixing time of the simple random walk on this structure. We depict a rich phase diagram in α and τ. In particular we prove that the presence of hubs can speed up the mixing of the chain. We use different techniques for each phase, the most interesting of which is a bootstrap procedure to reduce the model from a phase where the degrees have bounded averages to a setting with unbounded averages.

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Variational principle for scale-free network motifs

Cited by 15 publications

References 50 publications

Cliques in geometric inhomogeneous random graph

Cliques in geometric inhomogeneous random graph

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

Scale-free percolation mixing time

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