Network comparison and the within-ensemble graph distance

Abstract

Quantifying the differences between networks is a challenging and ever-present problem in network science. In recent years, a multitude of diverse, ad hoc solutions to this problem have been introduced. Here, we propose that simple and well-understood ensembles of random networks—such as Erdős–Rényi graphs, random geometric graphs, Watts–Strogatz graphs, the configuration model and preferential attachment networks—are natural benchmarks for network comparison methods. Moreover, we show that the expected distance between two networks independently sampled from a generative model is a useful property that encapsulates many key features of that model. To illustrate our results, we calculate this within-ensemble graph distance and related quantities for classic network models (and several parameterizations thereof) using 20 distance measures commonly used to compare graphs. The within-ensemble graph distance provides a new framework for developers of graph distances to better understand their creations and for practitioners to better choose an appropriate tool for their particular task.

Graph diffusion distance: Properties and efficient computation

“…We note here that the idea of computing intra- and inter- lineage distances is similar to recent work [ 43 ] computing distances between graph ensembles : certain classes of similarly-generated random graphs. Graph diffusion distance has been previously shown (in [ 43 ]) to capture key structural information about graphs; for example, GDD is known to be sensitive to certain critical transitions in ensembles of random graphs as the random parameters are varied. This is also true for our time dilated version of GDD.…”

Investigating Spatial Heterogeneity within Fracture Networks using Hierarchical Clustering and Graph Distance Metrics

“…-Fingerprint Distance (Louf and Barthelemy, 2014) -D-measure (Schieber et al, 2017) -NetLSD (Tsitsulin et al, 2018) -Portrait Divergence (Bagrow and Bollt, 2019) The performance of these similarity measures has been validated previously by Hartle et al (2020) and Tantardini et al…”

“…Several graph similarity measures exist within the graph theory literature to compare graphs (see Hartle et al, 2020;Tantardini et al, 2019;Emmert-Streib et al, 2016 for recent reviews). Graph comparisons are a challenging, non-trivial problem in terms of computing complexity (Schieber et al, 2017).…”

“…-On the choice of a graph distance metric We have restricted our investigation scope to four state-of-the-art graph similarity distances from the recent graph theory literature. Many more graph distances applicable to spatial graphs exist (Hartle et al, 2020;Tantardini et al, 2019), and the best means remains an open problem in network science research.…”

Untitled

“…We note here that the idea of computing intra-and inter-lineage distances is similar to recent work [43] computing distances between graph ensembles: certain classes of similarlygenerated random graphs. Graph diffusion distance has been previously shown (in [43]) to capture key structural information about graphs; for example, GDD is known to be sensitive to certain critical transitions in ensembles of random graphs as the random parameters are varied. This is also true for our time dilated version of GDD.…”