Efficient and exact sampling of simple graphs with given arbitrary degree sequence

Uniform sampling from graphical realizations of a given degree sequence is a fundamental component in simulation-based measurements of network observables, with applications ranging from epidemics, through social networks to Internet modeling. Existing graph sampling methods are either link-swap based (Markov-Chain Monte Carlo algorithms) or stub-matching based (the Configuration Model). Both types are ill-controlled, with typically unknown mixing times for link-swap methods and uncontrolled rejections for the Configuration Model. Here we propose an efficient, polynomial time algorithm that generates statistically independent graph samples with a given, arbitrary, degree sequence. The algorithm provides a weight associated with each sample, allowing the observable to be measured either uniformly over the graph ensemble, or, alternatively, with a desired distribution. Unlike other algorithms, this method always produces a sample, without back-tracking or rejections. Using a central limit theorem-based reasoning, we argue, that for large N, and for degree sequences admitting many realizations, the sample weights are expected to have a lognormal distribution. As examples, we apply our algorithm to generate networks with degree sequences drawn from power-law distributions and from binomial distributions.Comment: 8 pages, 3 figure

Rao's Theorem for forcibly planar sequences revisited

We consider the graph degree sequences such that every realisation is a polyhedron. It turns out that there are exactly eight of them. All of these are unigraphic, in the sense that each is realised by exactly one polyhedron. This is a revisitation of a Theorem of Rao about sequences that are realised by only planar graphs. Our proof yields additional geometrical insight on this problem. Moreover, our proof is constructive: for each graph degree sequence that is not forcibly polyhedral, we construct a non-polyhedral realisation

On self-duality and unigraphicity for 33-polytopes

Recent literature posed the problem of characterising the graph degree sequences with exactly one $3$-polytopal (i.e. planar, $3$-connected) realisation. This seems to be a difficult problem in full generality. In this paper, we characterise the sequences with exactly one self-dual $3$-polytopal realisation. An algorithm in the literature constructs a self-dual $3$-polytope for any admissible degree sequence. To do so, it performs operations on the radial graph, so that the corresponding $3$-polytope and its dual are modified in exactly the same way. To settle our question and construct the relevant graphs, we apply this algorithm, we introduce some modifications of it, and we also devise new ones. The speed of these algorithms is linear in the graph order

Parallel enumeration of degree sequences of simple graphs. II.

Abstract In the paper we report on the parallel enumeration of the degree sequences (their number is denoted by G(n)) and zerofree degree sequences (their number is denoted by (Gz(n)) of simple graphs on n = 30 and n = 31 vertices. Among others we obtained that the number of zerofree degree sequences of graphs on n = 30 vertices is Gz(30) = 5 876 236 938 019 300 and on n = 31 vertices is Gz(31) = 22 974 847 474 172 374. Due to Corollary 21 in [52] these results give the number of degree sequences of simple graphs on 30 and 31 vertices.</jats:p

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\u27s order on degree sequences

Graph Realizability and Factor Properties Based on Degree Sequence

A graph is a structure consisting of a set of vertices and edges. Graph construction has been a focus of research for a long time, and generating graphs has proven helpful in complex networks and artificial intelligence. A significant problem that has been a focus of research is whether a given sequence of integers is graphical. Havel and Hakimi stated necessary and sufficient conditions for a degree sequence to be graphic with different properties. In our work, we have proved the sufficiency of the requirements by generating algorithms and providing constructive proof. Given a degree sequence, one crucial problem is checking if there is a graph realization with k-factors. For the degree sequence with a realizable k-factor, we analyze an algorithm that produces the realization and its k-factor. We then generate degree sequences having no realizations with connected k-factors. We also state the conditions for a degree sequence to have connected k-factors. In our work, we have also studied the necessary and sufficient conditions for a sequence of integer pairs to be realized as directed graphs. We have proved the sufficiency of the conditions by providing algorithms as constructive proofs for the directed graphs