A family of tractable graph metrics

Abstract

Important data mining problems such as nearest-neighbor search and clustering admit theoretical guarantees when restricted to objects embedded in a metric space. Graphs are ubiquitous, and clustering and classification over graphs arise in diverse areas, including, e.g., image processing and social networks. Unfortunately, popular distance scores used in these applications, that scale over large graphs, are not metrics and thus come with no guarantees. Classic graph distances such as, e.g., the chemical distance and the Chartrand-Kubiki-Shultz distance are arguably natural and intuitive, and are indeed also metrics, but they are intractable: as such, their computation does not scale to large graphs. We define a broad family of graph distances, that includes both the chemical and the Chartrand-Kubiki-Shultz distances, and prove that these are all metrics. Crucially, we show that our family includes metrics that are tractable. Moreover, we extend these distances by incorporating auxiliary node attributes, which is important in practice, while maintaining both the metric property and tractability.

Network comparison and the within-ensemble graph distance

“…they satisfy properties such as triangle inequality, etc.) [13], whereas others have not. These are not exhaustive descriptions of every graph distance in use today, but they represent coarse similarities between the various methods.…”

“…They do not restrict the large possibility of measures we might use, while also providing a clean separation between how we choose to describe graphs and how we calculate the differences between those descriptions. A common property when considering distance measures is the triangle inequality ; however, we have not included this in the list above as not all commonly used graph distances obey this property [13]. As in the case of pseudometrics, d ( x , y ) = 0 does not always imply x = y [7].…”

Network comparison and the within-ensemble graph distance

“…Some graph distances have been shown to be metrics (i.e., they satisfy properties such as triangle inequality, etc.) [12], whereas others have not. These are not exhaustive descriptions of every graph distance in use today, but they represent coarse similarities between the various methods.…”

“…The properties listed in this definition are general, and they do not restrict the large possibility of measures we might use, while also providing a clean separation between how we choose to describe graphs and how we calculate the differences between those descriptions. A common property when considering distance measures is the triangle inequality; however we have not included this in the list above as not all commonly used graph distances obey this property [12]. As in the case of pseudometrics, d(x, y) = 0 does not always imply x = y [7] [13].…”

Clustering of heterogeneous populations of networks

“…While mathematically elegant, however, this approach has some shortcomings. For example, the model is not easily estimated when the distance between networks is difficult to compute [32]. The approach of La Rosa et al also does not differentiate between false-positive and false-negative rates, which may differ substantially and change the composition of the corresponding clusters of networks (see Appendix A).…”