Bipartite Degree Research Articles

On matching numbers of tree and bipartite degree sequences

We study the possible values of the matching number among all trees with a given degree sequence as well as all bipartite graphs with a given bipartite degree sequence. For tree degree sequences, we obtain closed formulas for the possible values. For bipartite degree sequences, we show the existence of realizations with a restricted structure, which allows to derive an analogue of the Gale–Ryser Theorem characterizing bipartite degree sequences. More precisely, we show that a bipartite degree sequence has a realization with a certain matching number if and only if a cubic number of inequalities similar to those in the Gale–Ryser Theorem are satisfied. For tree degree sequences as well as for bipartite degree sequences, the possible values of the matching number form intervals.

Sampling k-partite graphs with a given degree sequence

Abstract The authors in the paper [15] presented an algorithm that generates uniformly all the bipartite realizations and the other algorithm that generates uniformly all the simple bipartite realizations whenever A is a bipartite degree sequence of a simple graph. The running time of both algorithms is 𝒪(m),where m = 1 2 ∑ i = 1 n a i ${\rm{m}} = {1 \over 2}\sum\nolimits_{\rm {i} = 1}^n {{ \rm{a}_\rm {i}}}$ . Let A =(A1 : A2 : ... : Ak) be a k-partite degree sequence of a simple graph, where Ai has ni entries such that ∑ni=n. In the present article, we give a generalized algorithm that generates uniformly all the k-partite realizations of A and another algorithm that generates uniformly all the simple k-partite realizations of A. The running time of both algorithms is 𝒪(m), where m = 1 2 ∑ i = 1 n a i $m = {1 \over 2}\sum\nolimits_{i = 1}^n {{a_i}}$ .

Efficiently sampling the realizations of bounded, irregular degree sequences of bipartite and directed graphs.

Since 1997 a considerable effort has been spent on the study of the swap (switch) Markov chains on graphic degree sequences. All of these results assume some kind of regularity in the corresponding degree sequences. Recently, Greenhill and Sfragara published a breakthrough paper about irregular normal and directed degree sequences for which rapid mixing of the swap Markov chain is proved. In this paper we present two groups of results. An example from the first group is the following theorem: let be a directed degree sequence on n vertices. Denote by Δ the maximum value among all in- and out-degrees and denote by the number of edges in the realization. Assume furthermore that . Then the swap Markov chain on the realizations of is rapidly mixing. This result is a slight improvement on one of the results of Greenhill and Sfragara. An example from the second group is the following: let d be a bipartite degree sequence on the vertex set U ⊎ V, and let 0 < c1 ≤ c2 < |U| and 0 < d1 ≤ d2 < |V| be integers, where c1 ≤ d(v) ≤ c2: ∀v ∈ V and d1 ≤ d(u) ≤ d2: ∀u ∈ U. Furthermore assume that (c2 − c1 − 1)(d2 − d1 − 1) < max{c1(|V| − d2), d1(|U| − c2)}. Then the swap Markov chain on the realizations of d is rapidly mixing. A straightforward application of this latter result shows that when a random bipartite or directed graph is generated under the Erdős—Rényi G(n, p) model with mild assumptions on n and p then the degree sequence of the generated graph has, with high probability, a rapidly mixing swap Markov chain on its realizations.

Degree distributions of bipartite networks and their projections.

Bipartite (two-mode) networks are important in the analysis of social and economic systems as they explicitly show conceptual links between different types of entities. However, applications of such networks often work with a projected (one-mode) version of the original bipartite network. The topology of the projected network, and the dynamics that take place on it, are highly dependent on the degree distributions of the two different node types from the original bipartite structure. To date, the interaction between the degree distributions of bipartite networks and their one-mode projections is well understood for only a few cases, or for networks that satisfy a restrictive set of assumptions. Here we show a broader analysis in order to fill the gap left by previous studies. We use the formalism of generating functions to prove that the degree distributions of both node types in the original bipartite network affect the degree distribution in the projected version. To support our analysis, we simulate several types of synthetic bipartite networks using a configuration model where node degrees are assigned from specific probability distributions, ranging from peaked to heavy-tailed distributions. Our findings show that when projecting a bipartite network onto a particular set of nodes, the degree distribution for the resulting one-mode network follows the distribution of the nodes being projected on to, but only so long as the degree distribution for the opposite set of nodes does not have a heavier tail. Furthermore, we show that bipartite degree distributions are not the only feature driving topology formation of projected networks, in contrast to what is commonly described in the literature.

Half-regular factorizations of the complete bipartite graph

We consider a bipartite version of the color degree matrix problem. A bipartite graph G(U,V,E) is half-regular if all vertices in U have the same degree. We give necessary and sufficient conditions for a bipartite degree matrix (also known as demand matrix) to be the color degree matrix of an edge-disjoint union of half-regular graphs. We also give necessary and sufficient perturbations to transform realizations of a half-regular degree matrix into each other. Based on these perturbations, a Markov chain Monte Carlo method is designed in which the inverse of the acceptance ratios is polynomial bounded.Realizations of a half-regular degree matrix are generalizations of Latin squares, and they also appear in applied neuroscience.

Fast extraction of the backbone of projected bipartite networks to aid community detection

This paper introduces a computationally inexpensive method for extracting the backbone of one-mode networks projected from bipartite networks. We show that the edge weights in the one-mode projections are distributed according to a Poisson binomial distribution and that finding the expected weight distribution of a one-mode network projected from a random bipartite network only requires knowledge of the bipartite degree distributions. Being able to extract the backbone of a projection is highly beneficial in filtering out redundant information in large complex networks and narrowing down the information in the one-mode projection to the most relevant. We demonstrate that the backbone of a one-mode projection aids in the detection of communities.

Bipartite dimensions and bipartite degrees of graphs

A cover (bipartite) of a graph G is a family of complete bipartite subgraphs of G whose edges cover G's edges. G's bipartite dimension d(G) is the minimum cardinality of a cover, and its bipartite degree η(G) is the minimum over all covers of the maximum number of covering members incident to a vertex. We prove that d( G) equals the Boolean interval dimension of the irreflexive complement of G, identify the 21 minimal forbidden induced subgraphs for d ⩽ 2, and investigate the forbidden graphs for d ⩽ n that have the fewest vertices. We note that for complete graphs, d( K n ) = [log 2 n], η( K n ) = d( K n ) for n ⩽ 16, and η( K n ) is unbounded. The list of minimal forbidden induced subgraphs for η ⩽ 2 is infinite. We identify two infinite families in this list along with all members that have fewer than seven vertices.

Difference graphs

A graph G=( V, E) is said to be a difference graph if there exist real numbers a 1, a 2,…, a n associated with the vertices of G and a positive real number T such that 1. (1) ¦ a i ¦< T for i=1, 2,…, n; 2. (2) distinct vertices i and j are adjacent if and only if ¦ a i − a j ¦≥ T. It can be shown that difference graphs are bipartite. Properties of difference graphs are given in Section 2. It turns out that they are very similar to threshold graphs. Section 3 describes the polytope of bipartite degree sequences with a given bipartition, and shows that its extreme points are exactly the degree sequences of the difference graphs with the same bipartition. Moreover, these degree sequences are characterized by a certain nonmajorizability property. In Section 4 we give an algorithm to recognize potentially difference degree sequences when the bipartition is unknown.