Causality-Driven Slow-Down and Speed-Up of Diffusion in Non-Markovian Temporal Networks

Recent research has highlighted limitations of studying complex systems with time-varying topologies from the perspective of static, time-aggregated networks. Non-Markovian characteristics resulting from the ordering of interactions in temporal networks were identified as one important mechanism that alters causality, and affects dynamical processes. So far, an analytical explanation for this phenomenon and for the significant variations observed across different systems is missing. Here we introduce a methodology that allows to analytically predict causality-driven changes of diffusion speed in non-Markovian temporal networks. Validating our predictions in six data sets, we show that - compared to the time-aggregated network - non-Markovian characteristics can lead to both a slow-down, or speed-up of diffusion which can even outweigh the decelerating effect of community structures in the static topology. Thus, non-Markovian properties of temporal networks constitute an important additional dimension of complexity in time-varying complex systems.

💡 Research Summary

The paper tackles a fundamental yet under‑explored aspect of temporal networks: how the ordering of contacts—i.e., the causal structure—affects diffusion processes. While previous studies have focused on static, time‑aggregated representations or on heavy‑tailed inter‑event time distributions (burstiness), they largely ignore that a sequence of two directed edges (a→b, b→c) only constitutes a valid path if the first event precedes the second. This non‑Markovian (memory‑bearing) property creates “time‑respecting paths” and can dramatically alter the speed at which a random walk (or any diffusive process) mixes on the network.

To capture this effect analytically, the authors introduce a second‑order time‑aggregated network. In this construction each node corresponds to an edge of the original aggregated graph, and a directed edge between two second‑order nodes represents a two‑step causal sequence observed in the data. The weight of a second‑order edge equals the empirical frequency of the corresponding two‑path. This is analogous to a line‑graph transformation but preserves the actual temporal statistics of the original contact sequence.

From the second‑order network they define a transition matrix (T^{(2)}) whose entries are proportional to the second‑order edge weights, thus yielding a Markov chain that reproduces the observed two‑path statistics. For comparison they also build the usual first‑order transition matrix (T^{(1)}) from the static, weighted aggregated graph. Both matrices are row‑stochastic, so their largest eigenvalue is 1. The second largest eigenvalue (in magnitude) (\lambda_2) determines the mixing speed: a larger spectral gap (1-|\lambda_2|) implies faster convergence of a random walk to its stationary distribution.

The authors define a slow‑down factor (S(\varepsilon)=t^{(2)}{\text{conv}}(\varepsilon)/t^{(1)}{\text{conv}}(\varepsilon)), where (t_{\text{conv}}) is the number of steps needed for the walk’s visitation probabilities to fall within a tolerance (\varepsilon) of the stationary distribution. By approximating the convergence time with the logarithm of the spectral gap they derive an analytical prediction

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