Memory in network flows and its effects on spreading dynamics and community detection

Abstract

Random walks on networks is the standard tool for modelling spreading processes in social and biological systems. This first-order Markov approach is used in conventional community detection, ranking and spreading analysis, although it ignores a potentially important feature of the dynamics: where flow moves to may depend on where it comes from. Here we analyse pathways from different systems, and although we only observe marginal consequences for disease spreading, we show that ignoring the effects of second-order Markov dynamics has important consequences for community detection, ranking and information spreading. For example, capturing dynamics with a second-order Markov model allows us to reveal actual travel patterns in air traffic and to uncover multidisciplinary journals in scientific communication. These findings were achieved only by using more available data and making no additional assumptions, and therefore suggest that accounting for higher-order memory in network flows can help us better understand how real systems are organized and function.

Causality-driven slow-down and speed-up of diffusion in non-Markovian temporal networks

“…In summary, interpreting T (2) as transition matrix of a random walker in the second-order aggregate network, we obtain a second-order Markov model generating contact sequences that preserve the relative weights in the first-order aggregate network, as well as the statistics of two-paths. In line with recent observations that one-step memory is often sufficient to characterize time-respecting paths in empirical temporal networks 29 , in the remainder of this article we focus on such second-order models. However, our findings can be generalized to n-th order networks G (n) and matrices T (n) that capture the statistics of time-respecting paths of any length n. From this perspective, the weighted first-order aggregate network can be seen as a first-order approximation where weights only capture the statistics of edges, that is, timerespecting paths of length one.…”

“…Representing the shortest possible time-ordered interaction sequence, two-paths are the simplest possible extension of edges (which can be viewed as 'one-path') that capture causality in temporal networks. As such, two-paths are a particularly simple abstraction that allows to study causality in temporal networks 28,29 . Figure 1b shows time-unfolded representations of two different temporal networks G T andG T consisting of four nodes and 27 time steps.…”

“…As such, both the order and timing of interactions affect time-respecting paths-and thus causality-in temporal networks. Compared with the rich literature on node activities, a relatively smaller number of studies empirically investigated effects of causality in temporal networks 22,[24][25][26][27][28][29][30] . Recent works have shown that order correlations in temporal networks lead to causality structures, which significantly deviate from what is expected based on paths in the corresponding time-aggregated networks [27][28][29] .…”

“…Compared with the rich literature on node activities, a relatively smaller number of studies empirically investigated effects of causality in temporal networks 22,[24][25][26][27][28][29][30] . Recent works have shown that order correlations in temporal networks lead to causality structures, which significantly deviate from what is expected based on paths in the corresponding time-aggregated networks [27][28][29] . Studying time-respecting paths a-b-c from the perspective of a contact sequence a,b,c passing through node b, it was shown that the next contact c not only depends on the current contact b but also on the previous one [28][29][30][31] .…”

“…Studying time-respecting paths a-b-c from the perspective of a contact sequence a,b,c passing through node b, it was shown that the next contact c not only depends on the current contact b but also on the previous one [28][29][30][31] . As a consequence, contact sequences in real-world temporal networks exhibit nonMarkovian characteristics that are in conflict with the Markovian assumption implicitly made when studying temporal networks from the perspective of time-aggregated networks, and which can neither be attributed to inter-event time distributions nor to the concurrency or duration of interactions [27][28][29][30] . Furthermore, it was shown that causality structures resulting from non-Markovian contact sequences influence both the speed of and the paths taken by dynamical processes 28,29 .…”