Formulating effective resistance in temporal networks: Models and empirical insights

Effective resistance is a fundamental metric for quantifying connectivity and transport efficiency in static networks, yet its generalization to temporal networks, where connectivity evolves over time, remains an open challenge. This paper addresses this gap by proposing and systematically investigating a suite of definitions for temporal effective resistance (TER). We introduce four distinct formulations based on averaging and aggregation, generalized multi-path costs, random walk commute times, and a principled energy minimization framework derived from electrical circuit theory. A key argument within our models is a “retention” mechanism that allows flow to be carried over between time steps at an energetic cost, explicitly analogous to buffering or storage. Through extensive numerical experiments on a diverse set of networks, we demonstrate that these TER definitions capture distinct and non-equivalent aspects of spatio-temporal connectivity. Our results show that while the metrics provide convergent assessments in well-connected networks, their values diverge significantly in sparse or fragmented systems. Proposing a conduction efficiency metric to assess the network’s overall transmission capability, we show that the retention factor is critical for performance in temporally fragmented networks. Our analysis reveals a fundamental trade-off between patience (waiting for further connections) and progress (traversing existing paths), where an optimal waiting strategy could maximize conduction efficiency. This work provides a versatile and principled toolkit for analyzing flow, diffusion, and resilience in time-varying networked systems.