Temporal Closeness Centrality

Static closeness uses geodesic (topological) shortest paths and ignores when edges exist. Temporal closeness uses foremost paths — routes that traverse edges in chronological order — so a path is only valid if each hop occurs after the previous one in time. This constraint can make some nodes unreachable and substantially reorder the centrality ranking.

The harmonic mean assigns zero contribution to unreachable pairs rather than making the whole score undefined. This is essential in temporal networks, where many pairs may be unreachable within a finite time window, and it is consistent with how closeness centrality is handled in disconnected static graphs.

The window should match the temporal scale of the process you are studying. For a disease outbreak, the window might span the outbreak period; for daily communications, a single day or week. Always report results for multiple windows and check whether rankings are stable, since sensitivity to the window is a known limitation.

All-pairs computation is expensive and typically feasible only up to tens of thousands of nodes and millions of events. For larger networks, approximation algorithms or sampling-based estimators are used, though these introduce additional uncertainty that must be reported.

It can be used, but coarse temporal resolution reduces the measure's advantage over static closeness. If all contacts within a day are treated as simultaneous, the analysis collapses to a static snapshot for that day. Finer time resolution yields more meaningful temporal centrality estimates.