Abstract We examine two greedy heuristics — wiring and rewiring — for constructing maximum assortative graphs over all simple connected graphs with a target degree sequence. Counterexamples show that natural greedy rewiring heuristics do not necessarily return a maximum assortative graph, even though it is known that the meta-graph of all simple connected graphs with given degree is connected under rewiring. Counterexamples show an elegant greedy graph wiring heuristic from the literature may fail to achieve the target degree sequence or may fail to wire a maximally assortative graph.
Joint Degree Matrix (JDM). The generation of random graphs with a particular JDM (also called a 2K-series) has been the subject of a number of recent papers. Stanton @cite_19 and Orsini @cite_6 have proposed random edge rewiring as a method of sampling graphs with a given JDM, while Gjorka @cite_22 has introduced a random wiring method for constructing these graphs. However, there is no means known to us by which JDMs may be efficiently enumerated, and therefore there is no easy means to maximize assortativity, which is a statistic of the JDM, short of enumerating all (in our case, simple and connected) graphs with a given degree sequence.
@cite_19 @cite_22 @cite_6
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A joint degree matrix (JDM) specifies the number of connections between nodes of given degrees in a graph, for all degree pairs, and uniquely determines the degree sequence of the graph. We consider the space of all balanced realizations of an arbitrary JDM, realizations in which the links between any two fixed-degree groups of nodes are placed as uniformly as possible. We prove that a swap Markov chain Monte Carlo algorithm in the space of all balanced realizations of an arbitrary graphical JDM mixes rapidly, i.e., the relaxation time of the chain is bounded from above by a polynomial in the number of nodes @math . To prove fast mixing, we first prove a general factorization theorem similar to the Martin--Randall method for disjoint decompositions (partitions). This theorem can be used to bound from below the spectral gap with the help of fast mixing subchains within every partition and a bound on an auxiliary Markov chain between the partitions. Our proof of the general factorization theorem is direct and ... Random graphs with a given degree sequence are a useful model capturing several features absent in the classical Erd˝os-Renyi model, such as dependent edges and non-binomial degrees. In this paper, we use a characterization due to Erd˝os and Gallai to develop a sequential algorithm for generating a random labeled graph with a given degree sequence. The algorithm is easy to implement and allows surprisingly ecient sequential importance sampling. Applications are given, in- cluding simulating a biological network and estimating the number of graphs with a given degree sequence. 1. Introduction. Random graphs with given vertex degrees have recently attracted great interest as a model for many real-world complex networks, including the World Wide Web, peer-to-peer networks, social networks, and biological networks. Newman (58) contains an excellent survey of these networks, with extensive references. A common approach to simulating these systems is to study (empirically or theoretically) the degrees of the vertices in instances of the network, and then to generate a random graph with the appropriate degrees. Graphs with prescribed degrees also appear in random matrix theory and string theory, which can call for large simulations based on random k-regular graphs. Throughout, we are concerned with generating simple graphs, i.e., no loops or multiple edges are allowed (the problem becomes considerably easier if loops and multiple edges are allowed). The main result of this paper is a new sequential importance sampling algorithm for generating random graphs with a given degree sequence. The idea is to build up the graph sequentially, at each stage choosing an edge from a list of candidates with probability proportional to the degrees. Most previously studied algorithms for this problem sometimes either get stuck or produce loops or multiple edges in the output, which is handled by starting over and trying again. Often for such algorithms, the probability of a restart being needed on a trial rapidly approaches 1 as the degree parameters grow, resulting in an enormous number of trials being needed on average to obtain a simple graph. A major advantage of our algorithm is that it never gets stuck. This is achieved using the Erd˝os- Gallai characterization, which is explained in Section 2, and a carefully chosen order of edge selection. In networking research, it is often desirable to generate synthetic graphs with certain properties. In this paper, we present a new algorithm, 2K_Simple, for exact construction of simple graphs with a target joint degree matrix (JDM). We prove that the algorithm constructs exactly the target JDM and that its running time is linear in the number of edges. Furthermore, we show that the algorithm poses less constraints on the graph structure than previous state-of-the-art construction algorithms. We exploit this flexibility to extend 2K_Simple and design two algorithms that achieve additional network properties on top of the exact target JDM. In particular, 2K_Simple_Clustering produces simple graphs with a target JDM and average clustering coefficient close to a target, while 2K_Simple_Attributes produces exactly simple graphs with a target JDM and joint occurrence of node attribute pairs. We exhaustively evaluate our algorithms through simulation for small graphs, and we also demonstrate their benefits in generating graphs that resemble real-world social networks in terms of accuracy and speed; we reduce the running time by orders of magnitudes compared to previous approaches that rely on Monte Carlo Markov Chains. In network modeling of complex systems one is often required to sample random realizations of networks that obey a given set of constraints, usually in form of graph measures. A much studied class of problems targets uniform sampling of simple graphs with given degree sequence or also with given degree correlations expressed in the form of a joint degree matrix. One approach is to use Markov chains based on edge switches (swaps) that preserve the constraints, are irreducible (ergodic) and fast mixing. In 1999, Kannan, Tet ali and Vempala (KTV) proposed a simple swap Markov chain for sampling graphs with given degree sequence and conjectured that it mixes rapidly (in poly-time) for arbitrary degree sequences. While the conjecture is still open, it was proven for special degree sequences, in particular, for those of undirected and directed regular simple graphs, of half-regular bipartite graphs, and of graphs with certain bounded maximum degrees. Here we prove the fast mixing KTV conjecture for novel, exponentially large classes of irregular degree sequences. Our method is based on a canonical decomposition of degree sequences into split graph degree sequences, a structural theorem for the space of graph realizations and on a factorization theorem for Markov chains. After introducing bipartite splitted degree sequences, we also generalize the canonical split graph decomposition for bipartite and directed graphs.