Fig. 4: The shape of memory of real-world temporal networks.

For each temporal network, we estimate the shape of the memory \({\mathbb{M}}\) restricted to the 100 most frequently active links, and plot the respective heat maps (lighter colour means higher memory). From these we extract the distribution of memory co-orders (co-order histograms for the full set of networks studied, and the characterisation of their heterogeneities can be found in SI V-B,C). The effective network memory \({{{\Omega }}}_{{{{{\mathrm{eff}}}}}}({{{{{{{\mathcal{G}}}}}}}})\) is also highlighted by hollow circles, and the actual values are reported below the plots. Networks have been sampled at two different temporal resolutions Δt, namely every 1 and 10 min, or two different frequencies (gamma and theta bands) in the case of HB (heat maps only show the Δt = 1 min resolution and gamma band). In the two online social networks (EM and CM) the distributions of memory co-orders concentrate around zero and decay rapidly, indicating very short memory overall, except for a few pairs of links. In the offline university social network (RM) we find instead two clear peaks corresponding to the presence of memory at two timescales of about 5 and 40-50 min (corresponding to interactions during lecture room changes and during the lectures) respectively. Two peaks are also observed in the three engineered networks. However, both peaks are compatible with a timescale of 5–7 min in PT and PU, suggesting that such systems exhibit only one effective timescale, due to enforced planning and scheduling. The bus network (PB) in addition to the 5–7 min also shows a memory timescale of about 30 min, possibly due to external phenomena such as collective delays induced by traffic jams. In the human brain (HB), a peak emerges at memory order 1, and for the theta band only. The distribution of points in the (\({ \langle {{\Omega }} \rangle }_{{{{{\mathrm{in}}}}}},{ \langle {{\Omega }} \rangle }_{{{{{\mathrm{out}}}}}}\)) plane (shape projection, see the text for details) allows us to distinguish networks and classify links as influencers (above the diagonal) and followers (below diagonal) and to spot outliers (see SI V-D for statistics relating to the distribution of these points in the plane).