The connectivity of graphs of graphs with self-loops and a given degree sequence

Nishimura, Joel

doi:10.1093/comnet/cny008

Cited by 7 publications

(15 citation statements)

References 22 publications

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“…Thus the two graphs in the space are not connected by any sequence of double edge swaps that remain in the space of loopy graphs, and this lack of connectivity applies to both the stub-and vertex-labeled spaces. Generalizing this observation, it is the case that the space of loopy graphs is connected for any degree sequence that can wire a simple graph and is neither the degree sequence of a path, {2, 2, ..., 2}, nor that of a clique, {n − 1, n − 1, ..., n − 1} [111]. Alternatively, if the Markov chain is modified to occasionally employ a three-edge triangle-loop swap (the swap (u, u), (w, w)), a basic modification of Algorithm 1 and Algorithm 2 suffices to sample uniformly from these spaces; see [111] for more details.…”

Section: Markov Chains For Sampling Other Spacesmentioning

confidence: 89%

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Configuring Random Graph Models with Fixed Degree Sequences

Fosdick

Larremore

Nishimura³

et al. 2018

SIAM Rev.

Self Cite

214

228

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Random graph null models have found widespread application in diverse research communities analyzing network datasets, including social, information, and economic networks, as well as food webs, protein-protein interactions, and neuronal networks. The most popular family of random graph null models, called configuration models, are defined as uniform distributions over a space of graphs with a fixed degree sequence. Commonly, properties of an empirical network are compared to properties of an ensemble of graphs from a configuration model in order to quantify whether empirical network properties are meaningful or whether they are instead a common consequence of the particular degree sequence. In this work we study the subtle but important decisions underlying the specification of a configuration model, and investigate the role these choices play in graph sampling procedures and a suite of applications. We place particular emphasis on the importance of specifying the appropriate graph labeling-stub-labeled or vertex-labeled-under which to consider a null model, a choice that closely connects the study of random graphs to the study of random contingency tables. We show that the choice of graph labeling is inconsequential for studies of simple graphs, but can have a significant impact on analyses of multigraphs or graphs with self-loops. The importance of these choices is demonstrated through a series of three in-depth vignettes, analyzing three different network datasets under many different configuration models and observing substantial differences in study conclusions under different models. We argue that in each case, only one of the possible configuration models is appropriate. While our work focuses on undirected static networks, it aims to guide the study of directed networks, dynamic networks, and all other network contexts that are suitably studied through the lens of random graph null models. * All authors contributed equally to this work.

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Section: Markov Chains For Sampling Other Spacesmentioning

confidence: 89%

“…Although the space of loopy graphs is not necessarily connected under double-edge swaps, it can be shown to be connected for the degree sequence of this collaboration network, allowing the use of Algorithm 2 or 3 from Section 2; see[111] for details.…”

mentioning

confidence: 99%

Configuring Random Graph Models with Fixed Degree Sequences

Fosdick

Larremore

Nishimura³

et al. 2018

SIAM Rev.

Self Cite

214

228

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“…We denote by G ⟳ the set of all graphs on n nodes, and by G ⊂ G ⟳ the set of graphs on n nodes without self-loops. Parallel edges are permitted in G. While it is indeed possible to define configuration models on G ⟳ , we do not do so here [16,39]. Rather we will formulate most of our results for elements of G, only discussing G ⟳ below in the context of certain technical issues.…”

Section: Definition 1 (Graph)mentioning

confidence: 99%

Configuration Models of Random Hypergraphs

Chodrow¹

2019

Preprint

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Many empirical networks are intrinsically polyadic, with interactions occurring within groups of agents of arbitrary size. There are, however, few flexible null models that can support statistical inference for such polyadic networks. We define a class of null random hypergraphs that hold constant both the node degree and edge dimension sequences, generalizing the classical dyadic configuration model. We provide a Markov Chain Monte Carlo scheme for sampling from these models, and discuss connections and distinctions between our proposed models and previous approaches. We then illustrate these models through a triplet of applications. We start with two classical network topics -triadic clustering and degree-assortativity. In each, we emphasize the importance of randomizing over hypergraph space rather than projected graph space, showing that this choice can dramatically alter statistical inference and study findings. We then define and study the edge intersection profile of a hypergraph as a measure of higher-order correlation between edges, and derive asymptotic approximations under the stub-labeled null. Our experiments emphasize the ability of explicit, statistically-grounded polyadic modeling to significantly enhance the toolbox of network data science. We close with suggestions for multiple avenues of future work.

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“…An alternative solution to drawing from the other seven graph space, e.g., that of simple graphs, is to sample them using a Markov chain Monte Carlo (MCMC) algorithm based on double-edge swaps. Although a number of such algorithms have been defined, the Fosdick et al MCMC is known to asymptotically converge on the target uniform distribution for each of the eight graph spaces (with rare exceptions in graph spaces that allow loops but not multi-edges [18]). However, practical guidance on the time required for convergence with finite-sized networks remains unknown.…”

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confidence: 99%

Sampling random graphs with specified degree sequences

Dutta¹,

Fosdick²,

Clauset³

2021

Preprint

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The configuration model is a standard tool for generating random graphs with a specified degree sequence, and is often used as a null model to evaluate how much of an observed network's structure is explained by its degrees alone. Except for networks with both self-loops and multi-edges, we lack a direct sampling algorithm for the configuration model, e.g., for simple graphs. A Markov chain Monte Carlo (MCMC) algorithm, based on a degree-preserving double-edge swap, provides an asymptotic solution to sample from the configuration model without bias. However, accurately detecting convergence of this Markov chain on its stationary distribution remains an unsolved problem.Here, we provide a concrete solution to detect convergence and sample from the configuration model without bias. We first develop an algorithm for estimating a sufficient gap between sampled MCMC states for them to be effectively independent. Applying this algorithm to a corpus of 509 empirical networks, we derive a set of computationally efficient heuristics, based on scaling laws, for choosing this sampling gap automatically. We then construct a convergence detection method that applies a Kolmogorov-Smirnov test to sequences of network assortativity values derived from the Markov chain's sampled states. Comparing this test to three generic Markov chain convergence diagnostics, we find that our method is both more accurate and more efficient at detecting convergence.

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The connectivity of graphs of graphs with self-loops and a given degree sequence

Cited by 7 publications

References 22 publications

Configuring Random Graph Models with Fixed Degree Sequences

Configuring Random Graph Models with Fixed Degree Sequences

Configuration Models of Random Hypergraphs

Sampling random graphs with specified degree sequences

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