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The spreading of initially localized perturbations across networks impacts the functionality of a wide range of systems, from social to biological networks, where ideas, nutrients or diseases spread, and to our infrastructure networks such as power grids that supply us with electricity.
Despite its ubiquity, there is currently no general theory of dynamic spreading on networks, in part because it constitutes a transient phenomenon emerging in driven multi-dimensional nonlinear dynamical systems.
Here, we present two complementary approaches to make progress. First, even for linear networked systems it is generally impossible to analytically determine time and amplitude of the maximal response of a unit to a perturbation impinging from some other unit. We propose to extract approximate peak times and amplitudes from effective expectation values used to
characterize the typical time and magnitude of the response of a unit by interpreting the system’s deterministic response as a probability distribution over time. We derive analytic estimators for the peak response based on these expectation values. Further analysis suggests that these estimators become exact in the limit of weak coupling.
Second, we disentangle universal from system-specific features of the transient spreading dynamics of networks driven by fluctuating signals. We first analytically identify a purely topological factor encoding the structure and strengths of network interactions, and second, numerically estimate a master function characterizing the universal scaling of the perturbation arrival times across topologically different networks. The proposed approach thereby provides intuitive insights into complex propagation patterns as well as accurate predictions for the
perturbation arrival times.
We illustrate application to models of power grids and highlight topical questions for future research.