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.

💡 Research Summary

The paper introduces a macroscopic framework for analyzing temporal networks by extending the concept of accessibility graphs, which in static networks indicate whether a path of any length exists between two nodes, to the time‑ordered domain. The authors propose an “unfolding” procedure: starting with an accessibility graph that contains only length‑one temporal paths (links active in a single time step), they iteratively add longer temporal paths, increasing the maximum allowed path length by one at each step. After k iterations, the resulting accessibility graph captures all node pairs that can be connected by a causal path of up to k time steps. By monitoring how the density (fraction of reachable node pairs) evolves with each unfolding step, one can infer the distribution of shortest‑path durations and identify characteristic time scales at which the network rapidly becomes globally reachable.

Two quantitative measures are defined. First, the distribution of shortest‑path durations records, for every node pair, the minimal number of time steps required for a causal connection; peaks in this distribution correspond to rapid connectivity transitions. Second, “causal fidelity” quantifies how well a static representation (the aggregated graph that ignores temporal ordering) reproduces the temporal accessibility structure. It is computed as the ratio of the static graph’s connectivity density to that of the full temporal accessibility graph; values close to one indicate that the static model preserves the causal constraints, while lower values reveal substantial over‑ or under‑estimation of reachable pairs.

The methodology is applied to three empirical datasets. (1) A high‑resolution human contact network collected on a university campus shows that after roughly six hours (≈12 time steps) the accessibility graph becomes almost fully connected, indicating a very short characteristic time for potential disease spread. The causal fidelity of the aggregated network is 0.78, suggesting that static analysis captures most but not all temporal constraints. (2) A European livestock trade network exhibits seasonal trading patterns; the unfolding reveals a smoother increase in reachability with an average shortest‑path duration of about three days (≈72 steps). Here causal fidelity reaches 0.85, supporting the use of static trade networks for epidemiological risk assessment. (3) A sexual contact network displays a much slower unfolding, with a long‑tailed distribution of shortest‑path durations, implying prolonged and gradual transmission potential. The causal fidelity is only 0.62, indicating that static aggregation substantially overestimates the speed and extent of spread, and static models would be unreliable for this context.

Overall, the unfolding accessibility approach provides a clear, computationally tractable way to extract macroscopic temporal dynamics—such as typical transmission times and the effectiveness of static approximations—from complex time‑varying interaction data. The introduced causal fidelity metric offers a practical criterion for deciding when static network analyses are justified and when full temporal modeling is indispensable. By demonstrating the method across diverse domains, the authors argue that unfolding accessibility can become a standard tool for researchers studying diffusion, contagion, and information flow in temporal networks.