Fixed Degree Sequence Research Articles

Impossibility result for Markov chain Monte Carlo sampling from microcanonical bipartite graph ensembles.

Markov Chain Monte Carlo (MCMC) algorithms are commonly used to sample from graph ensembles. Two graphs are neighbors in the state space if one can be obtained from the other with only a few modifications, e.g., edge rewirings. For many common ensembles, e.g., those preserving the degree sequences of bipartite graphs, rewiring operations involving two edges are sufficient to create a fully connected state space, and they can be performed efficiently. We show that, for ensembles of bipartite graphs with fixed degree sequences and number of butterflies (k_{2,2} bicliques), there is no universal constant c such that a rewiring of at most c edges at every step is sufficient for any such ensemble to be fully connected. Our proof relies on an explicit construction of a family of pairs of graphs with the same degree sequences and number of butterflies, with each pair indexed by a natural c, and such that any sequence of rewiring operations transforming one graph into the other must include at least one rewiring operation involving at least c edges. Whether rewiring this many edges is sufficient to guarantee the full connectivity of the state space of any such ensemble remains an open question. Our result implies the impossibility of developing efficient, graph-agnostic, MCMC algorithms for these ensembles, as the necessity to rewire an impractically large number of edges may hinder taking a step on the state space.

Facets of Random Symmetric Edge Polytopes, Degree Sequences, and Clustering

Symmetric edge polytopes are lattice polytopes associated with finite simple graphs that are of interest in both theory and applications. We investigate the facet structure of symmetric edge polytopes for various models of random graphs. For an Erd\H{o}s-Renyi random graph, we identify a threshold probability at which with high probability the symmetric edge polytope shares many facet-supporting hyperplanes with that of a complete graph. We also investigate the relationship between the average local clustering, also known as the Watts-Strogatz clustering coefficient, and the number of facets for graphs with either a fixed number of edges or a fixed degree sequence. We use well-known Markov Chain Monte Carlo sampling methods to generate empirical evidence that for a fixed degree sequence, higher average local clustering in a connected graph corresponds to higher facet numbers in the associated symmetric edge polytope.

Extremal trees with fixed degree sequence for $\sigma$-irregularity

Extremal trees with fixed degree sequence for $\sigma$-irregularity

Fastball: a fast algorithm to randomly sample bipartite graphs with fixed degree sequences

Abstract Many applications require randomly sampling bipartite graphs with fixed degrees or randomly sampling incidence matrices with fixed row and column sums. Although several sampling algorithms exist, the ‘curveball’ algorithm is the most efficient with an asymptotic time complexity of $O(n~log~n)$ and has been proven to sample uniformly at random. In this article, we introduce the ‘fastball’ algorithm, which adopts a similar approach but has an asymptotic time complexity of $O(n)$. We show that a C$\texttt{++}$ implementation of fastball randomly samples large bipartite graphs with fixed degrees faster than curveball, and illustrate the value of this faster algorithm in the context of the fixed degree sequence model for backbone extraction.

Fast mixing via polymers for random graphs with unbounded degree

The polymer model framework is a classical tool from statistical mechanics that has recently been used to obtain approximation algorithms for spin systems on classes of bounded-degree graphs; examples include the ferromagnetic Potts model on expanders and on the grid. One of the key ingredients in the analysis of polymer models is controlling the growth rate of the number of polymers, which has been typically achieved so far by invoking the bounded-degree assumption. Nevertheless, this assumption is often restrictive and obstructs the applicability of the method to more general graphs. For example, sparse random graphs typically have bounded average degree and good expansion properties, but they include vertices with unbounded degree, and therefore are excluded from the current polymer-model framework.We develop a less restrictive framework for polymer models that relaxes the standard bounded-degree assumption, by reworking the relevant polymer models from the edge perspective. The edge perspective allows us to bound the growth rate of the number of polymers in terms of the total degree of polymers, which in turn can be related more easily to the expansion properties of the underlying graph. To apply our methods, we consider random graphs with unbounded degrees from a fixed degree sequence (with minimum degree at least 3) and obtain approximation algorithms for the ferromagnetic Potts model, which is a standard benchmark for polymer models. Our techniques also extend to more general spin systems.

Percolation on Random Graphs with a Fixed Degree Sequence

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 23 June 2020Accepted: 30 August 2021Published online: 04 January 2022Keywordsrandom graphs with given degrees, bond percolation, giant component, power lawAMS Subject Headings05C80, 05C82Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial and Applied MathematicsCODEN: sjdmec

The principal Erdős–Gallai differences of a degree sequence

The Erdős–Gallai criteria for recognizing degree sequences of simple graphs involve a system of inequalities. Given a fixed degree sequence, we consider the list of differences of the two sides of these inequalities. These differences have appeared in varying contexts, including characterizations of the split and threshold graphs, and we survey their uses here. Then, enlarging upon properties of these graph families, we show that both the last term and the maximum term of the principal Erdős–Gallai differences of a degree sequence are preserved under graph complementation and are monotonic under the majorization order and Rao's order on degree sequences.

Comparing alternatives to the fixed degree sequence model for extracting the backbone of bipartite projections

Projections of bipartite or two-mode networks capture co-occurrences, and are used in diverse fields (e.g., ecology, economics, bibliometrics, politics) to represent unipartite networks. A key challenge in analyzing such networks is determining whether an observed number of co-occurrences between two nodes is significant, and therefore whether an edge exists between them. One approach, the fixed degree sequence model (FDSM), evaluates the significance of an edge’s weight by comparison to a null model in which the degree sequences of the original bipartite network are fixed. Although the FDSM is an intuitive null model, it is computationally expensive because it requires Monte Carlo simulation to estimate each edge’s p value, and therefore is impractical for large projections. In this paper, we explore four potential alternatives to FDSM: fixed fill model, fixed row model, fixed column model, and stochastic degree sequence model (SDSM). We compare these models to FDSM in terms of accuracy, speed, statistical power, similarity, and ability to recover known communities. We find that the computationally-fast SDSM offers a statistically conservative but close approximation of the computationally-impractical FDSM under a wide range of conditions, and that it correctly recovers a known community structure even when the signal is weak. Therefore, although each backbone model may have particular applications, we recommend SDSM for extracting the backbone of bipartite projections when FDSM is impractical.

Binomial ideals of domino tilings

In this paper, we consider the set of all domino tilings of a cubiculated region. The primary question we explore is: How can we move from one tiling to another? Tiling spaces can be viewed as spaces of subgraphs of a fixed graph with a fixed degree sequence. Moves to connect such spaces have been explored in algebraic statistics. Thus, we approach this question from an applied algebra viewpoint, making new connections between domino tilings, algebraic statistics, and toric algebra. Using results from toric ideals of graphs, we are able to describe moves that connect the tiling space of a given cubiculated region of any dimension. This is done by studying binomials that arise from two distinct domino tilings of the same region. Additionally, we introduce tiling ideals and flip ideals and use these ideals to restate what it means for a tiling space to be flip connected. Finally, we show that if R is a 2-dimensional simply connected cubiculated region, any binomial arising from two distinct tilings of R can be written in terms of quadratic binomials. As a corollary to our main result, we obtain an alternative proof to the fact that the set of domino tilings of a 2-dimensional simply connected region is connected by flips.

Extremal Trees with Fixed Degree Sequence

The greedy tree $\mathcal{G}(D)$ and the $\mathcal{M}$-tree $\mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which $\mathcal{G}(D)$ or $\mathcal{M}(D)$ is extremal. Many known results, for example on the Wiener index, the number of subtrees, the number of independent subsets and the number of matchings follow as corollaries, as do some new results on invariants such as the number of rooted spanning forests, the incidence energy and the solvability. We also extend our results on trees with fixed degree sequence $D$ to the set of trees whose degree sequence is majorised by a given sequence $D$, which also has a number of applications.

Moments of Uniform Random Multigraphs with Fixed Degree Sequences

We study the expected adjacency matrix of a uniformly random multigraph with fixed degree sequence $\mathbf{d} \in \mathbb{Z}_+^n$. This matrix arises in a variety of analyses of networked data sets, including modularity-maximization and mean-field theories of spreading processes. Its structure is well understood for large, sparse, simple graphs: the expected number of edges between nodes $i$ and $j$ is roughly $\frac{d_id_j}{\sum_\ell{d_\ell}}$. Many network data sets are neither large, sparse, nor simple, and in these cases the standard approximation no longer applies. We derive a novel estimator using a dynamical approach: the estimator emerges from the stationarity conditions of a class of Markov Chain Monte Carlo algorithms for graph sampling. We derive error bounds for this estimator and provide an efficient scheme with which to compute it. We test the estimator on synthetic and empirical degree sequences, finding that it enjoys relative error against ground truth a full order of magnitude smaller than the standard approximation. We then compare modularity maximization techniques using both the standard and novel estimators, finding that the qualitative structure of the optimization landscape depends significantly on the estimator choice. Our results emphasize the importance of using carefully specified random graph models in data scientific applications.

A Limit Theorem for the Six-length of Random Functional Graphs with a Fixed Degree Sequence

We obtain results on the limiting distribution of the six-length of a random functional graph, also called a functional digraph or random mapping, with given in-degree sequence. The six-length of a vertex $v\in V$ is defined from the associated mapping, $f:V\to V$, to be the maximum $i\in V$ such that the elements $v, f(v), \ldots, f^{i-1}(v)$ are all distinct. This has relevance to the study of algorithms for integer factorisation.

The graph spectra and spectral moments of random graphs

The spectra of some specific classes of random graphs have received considerable interest in the literature. Here, we investigate the spectra for two random graph models: the FDSM model and the G(n,p) model in which every possible edge in a graph with n vertices occurs with probability p. We determine that under some conditions, the k-th spectral moment of the G(n,p) model is in O(nkpk). Moreover, we give results for the expected number of common neighbors (or cooccurrence) and, more generally, the expected number of walks of length ℓ for the fixed degree sequence model.

Percolation and the Effective Structure of Complex Networks

Analytical approaches to model the structure of complex networks can be distinguished into two groups according to whether they consider an intensive (e.g., fixed degree sequence and random otherwise) or an extensive (e.g., adjacency matrix) description of the network structure. While extensive approaches---such as the state-of-the-art Message Passing Approach---typically yield more accurate predictions, intensive approaches provide crucial insights on the role played by any given structural property in the outcome of dynamical processes. Here we introduce an intensive description that yields almost identical predictions to the ones obtained with MPA for bond percolation. Our approach distinguishes nodes according to two simple statistics: their degree and their position in the core-periphery organization of the network. Our near-exact predictions highlight how accurately capturing the long-range correlations in network structures allows to easily and effectively compress real complex network data.

The degree-wise effect of a second step for a random walk on a graph

AbstractIn this paper we consider the degree-wise effect of a second step for a random walk on a graph. We prove that under the configuration model, for any fixed degree sequence the probability of exceeding a given degree threshold is smaller after two steps than after one. This builds on recent work of Krameret al.(2016) regarding the friendship paradox under random walks.

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

`Double edge swaps' transform one graph into another while preserving the graph's degree sequence, and have thus been used in a number of popular Markov chain Monte Carlo (MCMC) sampling techniques. However, while double edge-swaps can transform, for any fixed degree sequence, any two graphs inside the classes of simple graphs, multigraphs, and pseudographs, this is not true for graphs which allow self-loops but not multiedges (loopy graphs). Indeed, we exactly characterize the degree sequences where double edge swaps cannot reach every valid loopy graph and develop an efficient algorithm to determine such degree sequences. The same classification scheme to characterize degree sequences can be used to prove that, for all degree sequences, loopy graphs are connected by a combination of double and triple edge swaps. Thus, we contribute the first MCMC sampler that uniformly samples loopy graphs with any given sequence.

Research Spotlights

This issue of SIAM Review features three articles in the Research Spotlights section. All three articles use novel analyses to provide computational tools for tackling their respective applications. The first of the three articles is “Configuring Random Graph Models with Fixed Degree Sequences,” by B. K. Fosdick, D. B. Larremore, J. Nishimura, and J. Ugander. Configuration graph models---uniform distributions over a space of graphs with fixed degree sequence---are used to generate ensembles that are then used to assess the significance of empirical network properties. This paper carefully and concisely argues that certain design choices can significantly influence conclusions drawn from the respective models. The authors first establish eight different graph spaces and detail what questions should be considered in selecting a graph space model for null sampling. Readers are then invited either to skip to the case studies or to read about new space-specific MCMC routines and alternatives. The application of the work to three real-world network problems makes a convincing argument of the soundness of the approach. Code is also provided. The second paper is authored by Sanghyeon Yu and Habib Ammari and is titled “Plasmonic Interaction between Nanospheres.” In the presence of incident visible light, the electromagnetic fields on the surfaces of certain nanoscale metallic particles resonate and oscillate. Characterizing this plasmonic interaction between nearly touching nanospheres, either analytically or numerically, has remained a challenge preventing wider use in the design of nanophotonic devices, biosensing, and spectroscopy. After detailing the sources of these analytic and computational difficulties, the authors endeavor to tackle both problems in turn: they develop a fully analytic solution for two plasmonic spheres which they then use in the development of a so-called hybrid numerical scheme for computing the field in the case of multiple, nearly touching spheres. Although the details of the analyses as well as the solution techniques are quite technical, the article is structured to deliver the key elements to the reader first, with many of the details left to the appendix. Numerical results illustrate that the hybrid method is capable of efficiently capturing desired behavior. The paper “SPECTRWM: Spectral Random Walk Method for the Numerical Solution of Stochastic Partial Differential Equations,” by Nawaf Bou-Rabee, rounds out this issue's Research Spotlights section. The emphasis of this paper is on development and testing of a new stochastic partial differential equation (SPDE) numerical solver that is, according to the author, more precisely tailored to the structure of the SPDE solutions in the case of parabolic, semilinear SPDEs driven by an additive space-time noise process. SPECTRWM is a generalization of the Markov chain method to SPDEs. Two versions of the algorithm are provided, with the first intended for effective illustration of basic concepts. The second elegantly builds upon the intuition provided by the first. The utility of the algorithms is demonstrated in four interesting examples, including the heat and Burgers' SPDEs. MATLAB script which illustrates the straightforward nature of the author's proposed approach is provided in the appendix for the first algorithm applied to Burgers' SPDE.

Configuring Random Graph Models with Fixed Degree Sequences

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, pr...

How to determine if a random graph with a fixed degree sequence has a giant component

For a fixed degree sequence $\mathcal{D}=(d_1,...,d_n)$, let $G(\mathcal{D})$ be a uniformly chosen (simple) graph on $\{1,...,n\}$ where the vertex $i$ has degree $d_i$. In this paper we determine whether $G(\mathcal{D})$ has a giant component with high probability, essentially imposing no conditions on $\mathcal{D}$. We simply insist that the sum of the degrees in $\mathcal{D}$ which are not 2 is at least $\lambda(n)$ for some function $\lambda$ going to infinity with $n$. This is a relatively minor technical condition, and when $\mathcal{D}$ does not satisfy it, both the probability that $G(\mathcal{D})$ has a giant component and the probability that $G(\mathcal{D})$ has no giant component are bounded away from $1$.

Graphs that locally maximize clustering coefficient in the space of graphs with a fixed degree sequence

This paper studies the problem of finding graphs that locally maximize the clustering coefficient in the space of graphs with a fixed degree sequence. Such a graph is characterized by the property that the clustering coefficient cannot be increased, no matter how a single 2-switch is applied. First, an explicit formula for the amount of change in the clustering coefficient of a graph caused by a single 2-switch is given. Next, some classes of graphs with the property stated above are presented. An example of such a graph is the one obtained from a tree by replacing its edges with cliques with the same order.