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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One of the most influential recent results in network analysis is that many natural networks exhibit a power-law or log-normal degree distribution. This has inspired numerous generative models that match this property. However, more recent work has shown that while these generative models do have the right degree distribution, they are not good models for real-life networks due to their differences on other important metrics like conductance. We believe this is, in part, because many of these real-world networks have very different joint degree distributions, that is, the probability that a randomly selected edge will be between nodes of degree k and l. Assortativity is a sufficient statistic of the joint degree distribution, and it has been previously noted that social networks tend to be assortative, while biological and technological networks tend to be disassortative. We suggest understanding the relationship between network structure and the joint degree distribution of graphs is an interesting avenue of further research. An important tool for such studies are algorithms that can generate random instances of graphs with the same joint degree distribution. This is the main topic of this article, and we study the problem from both a theoretical and practical perspective. We provide an algorithm for constructing simple graphs from a given joint degree distribution, and a Monte Carlo Markov chain method for sampling them. We also show that the state space of simple graphs with a fixed degree distribution is connected via endpoint switches. We empirically evaluate the mixing time of this Markov chain by using experiments based on the autocorrelation of each edge. These experiments show that our Markov chain mixes quickly on these real graphs, allowing for utilization of our techniques in practice. 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. Represented as graphs, real networks are intricate combinations of order and disorder. Fixing some of the structural properties of network models to their values observed in real networks, many other properties appear as statistical consequences of these fixed observables, plus randomness in other respects. Here we employ the dk-series, a complete set of basic characteristics of the network structure, to study the statistical dependencies between different network properties. We consider six real networks—the Internet, US airport network, human protein interactions, technosocial web of trust, English word network, and an fMRI map of the human brain—and find that many important local and global structural properties of these networks are closely reproduced by dk-random graphs whose degree distributions, degree correlations and clustering are as in the corresponding real network. We discuss important conceptual, methodological, and practical implications of this evaluation of network randomness, and release software to generate dk-random graphs.