Temporal Networks

Temporal networks are networks whose edges exist only at specific times, making connectivity a time-dependent property rather than a static structural feature. In a temporal network, a path from node A to node B requires not merely that a sequence of edges connects them, but that each edge in the sequence exists at a time no earlier than the previous edge — a constraint known as time-respecting paths. This transforms the percolation question from 'is there a path?' to 'can information or disease traverse the network given the temporal order of edge availability?'

The study of temporal networks has revealed that temporal structure can fundamentally alter spreading dynamics. In a static network, the epidemic threshold depends on the largest eigenvalue of the adjacency matrix. In a temporal network, the threshold depends on the temporal correlation structure — the burstiness, periodicity, and memory of edge appearances — in ways that static measures cannot predict. A network with the same aggregate degree distribution can have dramatically different spreading properties depending on whether edges appear randomly, periodically, or in correlated bursts.

Temporal networks appear in communication systems (email, phone calls), transportation (flight schedules, public transit), epidemiology (contact patterns), and biology (protein interactions). In each case, ignoring the temporal dimension produces systematic errors: overestimating connectivity when edges are bursty, underestimating cascade speed when edges are periodic, and missing entirely the existence of time-respecting paths that have no static equivalent.