The shape of memory in temporal networks

An international team of researchers, with participation of IFISC (UIB-CSIC), have proposed a mathematical framework to estimate the memory structure of temporal networks. These findings are published in the latest issue Nature Communications.

Temporal Networks (those whose structure changes over time) are a very useful tool for studying complex systems. But the properties of a temporal network not only depend on the patterns of activities of each of its links, but also on the ways in which these activities influence each other across the network. This rich dynamic gives rise to the notion of network memory, this is, the dependence of a temporal network’s structure on its past.

To this objective, an interdisciplinary and international team of researchers, with the participation of IFISC (UIB-CSIC), have proposed a mathematical framework to unveil the microscopic structure of a network’s memory. Their results suggest that, contrary to previously assumed, it is not possible to reduce the structure of such memory to a number: memory is inherently high-dimensional, it is not just one number, it is more like a shape. As Richard Feynman famously said, “there is more room at the bottom”, and this is true for network’s memory.

Furthermore, they showed that inside such memory shape, weird things happen. Like the emergence of so-called virtual loops, consisting in resonance effects that happen when the future of a link depends on the past of a second link whose future, in turn, depends on the past of the first one. While a priori only a mathematical curiosity, it turns out that these virtual loops are physically felt by the system: the outcome of an epidemic process running on top of a network is very different if virtual loops emerge.

The researchers applied this framework to characterize the memory of a large number of real-world networks, including urban transport networks in European cities (such as bus, train or metro), cortical brain networks, communication networks (SMS or email) and even contact networks between university students. Studying these systems, they found, among other results, that network memory is much higher in offline networks. In other words, a current state has a greater capacity to affect future configurations since offline social interactions are more mediated by tight schedules, which facilitate the emergence of various orders of memory. The analysis of real-world temporal networks also revealed that there are asymmetries in the contribution of links to the evolution of the network in terms of followers/influencers. All together provides evidence that memory shapes can very heterogeneous in real-world systems.

The manuscript also provides implementations of the algorithms proposed hoping that their framework will prompt further studies and applications in other areas of complex systems.

Williams, O.E., Lacasa, L., Millán, A.P. et al. The shape of memory in temporal networks. Nat Commun 13, 499 (2022). Doi: https://doi.org/10.1038/s41467-022-28123-z