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One of the most influential recent results in network analysis is that manynatural networks exhibit a power-law or log-normal degree distribution. Thishas inspired numerous generative models that match this property. However, morerecent work has shown that while these generative models do have the rightdegree distribution, they are not good models for real life networks due totheir differences on other important metrics like conductance. We believe thisis, in part, because many of these real-world networks have very differentjoint degree distributions, i.e. the probability that a randomly selected edgewill be between nodes of degree k and l. Assortativity is a sufficientstatistic of the joint degree distribution, and it has been previously notedthat social networks tend to be assortative, while biological and technologicalnetworks tend to be disassortative.  We suggest understanding the relationship between network structure and thejoint degree distribution of graphs is an interesting avenue of furtherresearch. An important tool for such studies are algorithms that can generaterandom instances of graphs with the same joint degree distribution. This is themain topic of this paper and we study the problem from both a theoretical andpractical perspective. We provide an algorithm for constructing simple graphsfrom a given joint degree distribution, and a Monte Carlo Markov Chain methodfor sampling them. We also show that the state space of simple graphs with afixed degree distribution is connected via end point switches. We empiricallyevaluate the mixing time of this Markov Chain by using experiments based on theautocorrelation of each edge. These experiments show that our Markov Chainmixes quickly on real graphs, allowing for utilization of our techniques inpractice.

Constructing and Sampling Graphs with a Prescribed Joint Degree Distribution