harmonic_diameter

harmonic_diameter(G, sp=None, *, weight=None)

Returns the harmonic diameter of the graph G.

The harmonic diameter of a graph is the harmonic mean of the distances between all pairs of distinct vertices. Graphs that are not strongly connected have infinite diameter and mean distance, making such measures not useful. Restricting the diameter or mean distance to finite distances yields paradoxical values (e.g., a perfect match would have diameter one). The harmonic mean handles gracefully infinite distances (e.g., a perfect match has harmonic diameter equal to the number of vertices minus one), making it possible to assign a meaningful value to all graphs.

Note that in [1] the harmonic diameter is called “connectivity length”: however, “harmonic diameter” is a more standard name from the theory of metric spaces. The name “harmonic mean distance” is perhaps a more descriptive name, but is not used in the literature, so we use the name “harmonic diameter” here.

Parameters:

GNetworkX graph

A graph

spdict of dicts, optional

All-pairs shortest path lengths as a dictionary of dictionaries

weightstring, function, or None (default=None)

If None, every edge has weight/distance 1. If a string, use this edge attribute as the edge weight. Any edge attribute not present defaults to 1. If a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number.

Returns:

hdfloat

Harmonic diameter of graph

References

[1]

Massimo Marchiori and Vito Latora, “Harmony in the small-world”. Physica A: Statistical Mechanics and Its Applications 285(3-4), pages 539-546, 2000. <https://doi.org/10.1016/S0378-4371(00)00311-3>