DiamS

diamS is a graph-theoretic metric used in network science to quantify distance heterogeneity in a finite graph. It combines the concept of the graph’s diameter with the dispersion of all-pairs shortest-path distances to produce a single, dimensionless index of distance variability. The idea is to capture not only how far the farthest nodes are, but how evenly distributed the pairwise distances are across the network.

To compute diamS for a connected graph G, compute all-pairs shortest-path distances d(u,v). Let D be the diameter, the maximum distance between any two nodes. Let sigma be the standard deviation of the set of all pairwise distances {d(u,v) | u ≠ v}. Then diamS(G) is defined as diamS(G) = sigma / D. For graphs with multiple components, calculations are typically performed within each component, and diamS can be reported per component or after an appropriate aggregation. In all cases, diamS is a dimensionless quantity that ranges toward smaller values for more homogeneous distance distributions and toward larger values for highly heterogeneous distance patterns.

DiamS is used to compare network geometries, assess navigability and robustness, and study how distance structure correlates with function in social, transportation, and biological networks. It complements traditional measures such as diameter and average path length by emphasizing variability in distances rather than central tendency alone. Limitations include sensitivity to outliers and the need to decide how to handle disconnected graphs. Computationally, all-pairs shortest paths dominate the cost, with options including Dijkstra from each node, Floyd-WWarshall, or sampling-based approximations for large networks.