Network compression with configuration models and the minimum description length

TL;DR

This work addresses how to parsimoniously represent empirical networks by comparing nested Configuration Models through the minimum description length (MDL) lens. It derives and assembles MDL-based description lengths for CM, CCM, Layered Configuration Model (LCM), and Layered Correlated Configuration Model (LCCM), then evaluates their compression on 100+ diverse networks. The key finding is that the classic CM often yields the best compression for dense networks, while the LCM provides the most compact representation for sparse networks; the LCCM, despite generating the smallest ensembles, is rarely the best due to a high description cost. Overall, layered models that incorporate centrality structure via onion decomposition offer a promising direction for compact, structurally informed network representations with broad implications for dynamics and inference on networks.

Abstract

Random network models, constrained to reproduce specific statistical features, are often used to represent and analyze network data and their mathematical descriptions. Chief among them, the configuration model constrains random networks by their degree distribution and is foundational to many areas of network science. However, configuration models and their variants are often selected based on intuition or mathematical and computational simplicity rather than on statistical evidence. To evaluate the quality of a network representation, we need to consider both the amount of information required to specify a random network model and the probability of recovering the original data when using the model as a generative process. To this end, we calculate the approximate size of network ensembles generated by the popular configuration model and its generalizations, including versions accounting for degree correlations and centrality layers. We then apply the minimum description length principle as a model selection criterion over the resulting nested family of configuration models. Using a dataset of over 100 networks from various domains, we find that the classic Configuration Model is generally preferred on networks with an average degree above ten, while a Layered Configuration Model constrained by a centrality metric offers the most compact representation of the majority of sparse networks.

Figures (6)

• Figure 1: Modeling a perfect tree. (A) The original dataset, a Cayley tree. We show random realizations of two models that do not preserve the structure of the tree: (B) The Configuration Model (CM fosdick2018configuring) preserving only the degree distribution (represented with different node colors) and (C) the Correlated Configuration Model (CCM vazquez2003resilience) preserving joint degree-degree correlations also using two node types plus a 2-by-2 edge matrix specifying frequency of connections between them. (D) The original network is preserved by the Layered Configuration Model (LCM hebert2016multi) using a node type for each of the 7 layers---the model can only generate networks with the same structure as the original tree up to relabeling. The Layered Correlated Configuration Model (LCCM allard2018percolation), which is not shown, would be indistinguishable from the LCM shown in panel (D) as there are no degree-degree correlations not captured by the layer structure.
• Figure 2: Projection of random network ensembles. We generated an ensemble of Erdős-Rényi (ER) random graphs with $N=250$ nodes and $E=311$ edges (on average). We then plotted the density of graphs in the space defined by the clustering coefficient $C$ and the mean shortest path length $\ell$ of each graph Watts1998. We then picked one unique graph at random (shown with a cross), and generated graphs from the corresponding CM, CCM and LCM ensembles, to see how the area covered by the ensembles in $(C,\ell)$-space would shrink with additional constraints. All ensembles are visualized with $1.4 \times 10^6$ random draws from each ensemble discretized into the bins indicated by grid cells in the figures.
• Figure 3: Inclusion relationships of our configuration models. The Configuration Model consists of all simple graphs with a fixed degree sequence and is a superset of all of the other models considered in this study. The Layered and Correlated Configuration Models (LCM and CCM respectively) include additional information of a different nature---layer centrality and degree-degree correlations respectively. By combining the constraints in both the LCM and CCM, the LCCM represents a subset of all previous models.
• Figure 4: Shuffling edges in random networks. (A-B) In the Configuration Model, we can tag stubs according to the node to which they are attached, shown in A. All random orderings of all $2E$ stubs can be turned to networks by connecting adjacent nodes pairwise, shown in B. We must however account for the fact that different orderings can lead to the same network: The order of the stubs of a given node does not matter, the ordering of edges does not matter and the order of stubs within an edge does not matter. (C-D) In the Correlated Configuration Model, we now separate the stub list into distinct lists containing the stubs attached to nodes of a given degree, shown in C for the network in D. (E-F) In the Onion Network Ensemble of Ref. hebert2016multi, the stub list is separated into different lists for stubs attached to nodes of different layers, and stubs from layer $l$ are also colored according to whether they point to $l'<l-1$ (green), $l'=l-1$ (black) and $l'>l-1$ (red). Panel E shows an example of stub lists for the network in F. The colors of the stubs in E are not related to the colors of the nodes in F, which correspond to the layer where the node is found. The Layered Correlated Configuration Model considered in the text extends this description by distinguishing edges not only by the layer they connect (i.e., $e(l,l')$) but by the joint degree-layer type of nodes they connect (i.e., $e(\{k,l\},\{k',l'\})$.
• Figure 5: Layered Configuration Models. In Layered Configuration Models, nodes are described by their joint layer-degree type. In the models considered here, edges follow constraints that preserve the layer of a node (i.e., its centrality) under the Onion Decomposition hebert2016multi. Following these local connection rules provides an edge shuffling mechanism that allows us to produce random networks with a fixed centrality structure based on the concept of $k$-cores and onion layers.