Assortativity in Chung Lu Random Graph Models

Abstract

Due to the widespread interest in networks as a representation to investigate the properties of complex systems, there has been a great deal of interest in generative models of graph structure that can capture the properties of networks observed in the real world. Recent models have focused primarily on accurate characterization of sparse networks with skewed degree distributions, short path lengths, and local clustering. While assortativity-degree correlation among linked nodes-is used as a measure to both describe and evaluate connectivity patterns in networks, there has been little effort to explicitly incorporate patterns of assortativity into model representations. This is because many graph models are edge-based (modeling whether a link should be placed between a pair of nodes i and j) and assortativity is a second-order characteristic that depends on the global properties of the graph (i.e., the final degree of i and j). As such, it is difficult to incorporate direct optimization of assortativity into edge-based generative models.One exception is the BTER method [5], which generates graphs with positive assortativity (e.g., high degree nodes link to each other). However, BTER does not directly estimate assortativity and also is not applicable for networks with negative assortativity (e.g, high degree nodes link primarily to low degree nodes). In this work, we present a novel approach to directly model observed assortativity (both positive and negative) via accept-reject sampling. Our key observation is to use a coarse approximation of the observed joint degree distribution and modify the likelihood that two nodes i, j should link based on the output properties of the original model. We implement our approach as an augmentation of Chung-Lu models and refer to it as Binning Chung Lu (BCL). We apply our method to six network datasets and show that it captures assortativity significantly more accurately than other methods while maintaining other graph properties of the original CL models. Also, our BCL approach is efficient (linear in the number of observed edges), thus it scales easily to large networks.

The Perceived Assortativity of Social Networks: Methodological Problems and Solutions

“…The processes that create dissortative networks with is a large and active research area (e.g. Yang 2014;Mussmann et al 2014) and so we will not investigate this result further here. We also note that many of the direct social networks in our table were based on interactions online.…”

Attributed Graph Modeling with Vertex Replacement Grammars

“…However, most graph generation literature focuses on non-attributed graphs. The few exceptions [37][38][39] adapt Chung and Lu's model for attributed graphs; yet, they ignore local graph topology, which is highly diverse across nodes in real graphs [40] with implications for tasks like anomaly detection [41], social contagion and diffusion [42,43], and engagement prediction [44].…”

The Infinity Mirror Test for Analyzing the Robustness of Graph Generators

“…However, on many graphs, the clustering coefficients and assortativity metrics of the output graphs do not match the original graph. Extensions of the Chung-Lu (CL) model, such as Transitive CL (TCL) [14], Binning CL (BCL) [11] and Block Two-Level Erdős-Rényi Model (BTER) [7], have been developed to further improve performance.…”

“…The Kronecker Model, for example, can only represent graphs with a power law degree distribution. Both Kronecker and the Chung-Lu models ignore local subnetwork properties, giving rise to more complex models like Transitive Chung-Lu for better clustering coefficient results [14] or Chung-Lu with Binning for better assortativity results [11,12]. Exponential Random Graph Models (ERGMs) take into consideration the local substructures of a given graph.…”

“…Like in the case with the clustering coefficient, the standard Chung Lu model was found to not accurately model the assortativity of real world graphs. Mussmann et al developed a Chung Lu with Binning adaptation that was shown to generate graphs with appropriate assortativity [11]. Even better is that the transitive and binning models can be combined to create a Transitive Chung Lu with Binning generator that models the degree distribution, clustering coefficient and assortativity of some input graph.…”