Overview of metrics and their correlation patterns for multiple- metric topology analysis on heterogeneous graph ensembles

This study is an overview of network topology metrics and a computational approach to analyzing graph topology via multiple-metric analysis on graph ensembles. The paper cautions against studying single metrics or combining disparate graph ensembles from different domains to extract global patterns. This is because there often exists considerable diversity among graphs that share any given topology metric, patterns vary depending on the underlying graph construction model, and many real data sets are not actual statistical ensembles. As real data examples, we present five airline ensembles, comprising temporal snapshots of networks of similar topology. Wikipedia language networks are shown as an example of a nontemporal ensemble. General patterns in metric correlations, as well as exceptions, are discussed by representing the data sets via hierarchically clustered correlation heat maps. Most topology metrics are not independent and their correlation patterns vary across ensembles. In general, density-related metrics and graph distance-based metrics cluster and the two groups are orthogonal to each other. Metrics based on degree-degree correlations have the highest variance across ensembles and cluster the different data sets on par with principal component analysis. Namely, the degree correlation, the s metric, their elasticities, and the rich club moments appear to be most useful in distinguishing topologies.