Unfolding Accessibility Provides a Macroscopic Approach to Temporal Networks

An accessibility graph of a network contains a link wherever there is a path of arbitrary length between two nodes. We generalize the concept of accessibility to temporal networks. Building an accessibility graph by consecutively adding paths of growing length (unfolding), we obtain information about the distribution of shortest path durations and characteristic time scales in temporal networks. Moreover, we define causal fidelity to measure the goodness of their static representation. The practicability of our proposed methods is demonstrated for three examples: networks of social contacts, livestock trade, and sexual contacts.

Figure 1

Unfolding of an accessibility graph. Starting with a network (left) we unfold its accessibility graph by adding direct edges for paths of length 2 (center) and so forth. The fully unfolded accessibility graph is shown on the right.

Figure 2

Density of the accessibility graph (black line) and distribution of shortest path durations (red line with circles) for a network of face-to-face contacts at a conference.

Figure 3

Density of the accessibility graph (black line) and distribution of shortest path durations (red line with circles) for a network of pig trade in Germany.

Figure 4

Density of the accessibility graph (black line) and distribution of shortest path durations (red line with circles) for a network of sexual contacts.