Switch chain mixing times and triangle counts in simple random graphs with given degrees

Bannink, Tom; Hofstad, Remco van der; Stegehuis, Clara

doi:10.1093/comnet/cny013

Cited by 8 publications

(11 citation statements)

References 39 publications

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“…Note that 1 − e −x x/(1 + x) for all x > 0. Interestingly, this implies that the erased configuration model contains more triangles than the uniform random graph with the same degree sequence, even though edges are removed in the erased configuration model, due to the presence of multiple edges and loops in the configuration model, which was empirically observed in [1]. This is because the large-degree vertices in the uniform random graph model have more low-degree neighbours than in the erased configuration model.…”

Section: Number Of Trianglesmentioning

confidence: 87%

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Counting Triangles in Power-Law Uniform Random Graphs

Gao

Hofstad²,

Southwell

et al. 2020

Electron. J. Combin.

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We count the asymptotic number of triangles in uniform random graphs where the degree distribution follows a power law with degree exponent $\tau\in(2,3)$. We also analyze the local clustering coefficient $c(k)$, the probability that two random neighbors of a vertex of degree $k$ are connected. We find that the number of triangles, as well as the local clustering coefficient, scale similarly as in the erased configuration model, where all self-loops and multiple edges of the configuration model are removed. Interestingly, uniform random graphs contain more triangles than erased configuration models with the same degree sequence. The number of triangles in uniform random graphs is closely related to that in a version of the rank-1 inhomogeneous random graph, where all vertices are equipped with weights, and the probabilities that edges are present are moderated by asymptotically linear functions of the products of these vertex weights.

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Section: Number Of Trianglesmentioning

confidence: 87%

“…Then the number of 2-paths from any specified vertex is bounded by 1) by (1.4). Since τ ∈ (2, 3) the above is o(n).…”

Section: Connection Probability Estimatesmentioning

confidence: 99%

Counting Triangles in Power-Law Uniform Random Graphs

Gao

Hofstad²,

Southwell

et al. 2020

Electron. J. Combin.

Self Cite

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“…We also note some additional works that are lessdirectly related to ours. Network-rewiring and MCMC algorithms are widely used to sample static networks [156][157][158][159][160]; in stationarity, these can be viewed as temporal networks satisfying the Equilibrium Property, with a level of persistence tunable via the number of iterations between adjacent snapshots. Adaptive network models (for instance, SIS-dynamics [161] alongside contactswitching [23,24]), have dynamic node-properties that evolve with time and guide network evolution, a commonality with THVMs.…”

Section: Related Workmentioning

confidence: 99%

Dynamic Hidden-Variable Network Models

Hartle,

Papadopoulos,

Krioukov

2021

Preprint

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Models of complex networks often incorporate node-intrinsic properties abstracted as hidden variables. The probability of connections in the network is then a function of these variables. Realworld networks evolve over time, and many exhibit dynamics of node characteristics as well as of linking structure. Here we introduce and study natural temporal extensions of static hidden-variable network models with stochastic dynamics of hidden variables and links. The rates of the hidden variable dynamics and link dynamics are controlled by two parameters, and snapshots of networks in the dynamic models may or may not be equivalent to a static model, depending on the location in the parameter phase diagram. We quantify deviations from static-like behavior, and examine the level of structural persistence in the considered models. We explore temporal versions of popular static models with community structure, latent geometry, and degree-heterogeneity. We do not attempt to directly model real networks, but comment on interesting qualitative resemblances, discussing possible extensions, generalizations, and applications.

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“…Due to this feature the configuration model is widely used to analyze the influence of degrees on other properties or processes on networks [11,14,15,25,32,38]. An important property that many networks share is that their degree distributions are regularly varying, with the exponent γ of the degree distribution satisfying γ ∈ (1,2), so that the degrees have infinite variance. In this regime of degrees, the configuration model results in a simple graph with vanishing probability.…”

Section: Motivationmentioning

confidence: 99%

“…This is important because very commonly in the literature, various quantities measured in real-world networks are compared to null-models with same degrees but random rewiring. These rewired null-models are similar to a version of the inhomogeneous random graph [2,12]. Without knowing the scaling of these quantities in the inhomogeneous random graph, it is not possible to asses how similar a small measured value on the real network is to that of the null model.…”

Section: Motivationmentioning

confidence: 99%

Limit theorems for assortativity and clustering in null models for scale-free networks

et al. 2020

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An important problem in modeling networks is how to generate a randomly sampled graph with given degrees. A popular model is the configuration model, a network with assigned degrees and random connections. The erased configuration model is obtained when self-loops and multiple edges in the configuration model are removed. We prove an upper bound for the number of such erased edges for regularly-varying degree distributions with infinite variance, and use this result to prove central limit theorems for Pearson’s correlation coefficient and the clustering coefficient in the erased configuration model. Our results explain the structural correlations in the erased configuration model and show that removing edges leads to different scaling of the clustering coefficient. We prove that for the rank-1 inhomogeneous random graph, another null model that creates scale-free simple networks, the results for Pearson’s correlation coefficient as well as for the clustering coefficient are similar to the results for the erased configuration model.

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Switch chain mixing times and triangle counts in simple random graphs with given degrees

Cited by 8 publications

References 39 publications

Counting Triangles in Power-Law Uniform Random Graphs

Counting Triangles in Power-Law Uniform Random Graphs

Dynamic Hidden-Variable Network Models

Limit theorems for assortativity and clustering in null models for scale-free networks

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