Fig. 1: Shape of memory and emergence of virtual loops in temporal networks with correlated link dynamics.

A temporal fully connected network \({{{{{{{\mathcal{G}}}}}}}}\) with N = 10 nodes and L = 45 links whose dynamics are both autocorrelated and heterogeneously cross-correlated, generated from an eCDARN(p) model with parameters q = 0.9, y = 0.5, c = 0.7 and a set of memory lengths p randomly sampled with uniform (panel (a)) or a bimodal (panel (b)) probability from {0, 1, …, 6} (see SI section IVA for details). a The 45 × 45 entries of the co-memory matrix \({\mathbb{M}}\) (shown with a colour code) display the shape of the network memory at the microscopic scale of pairs of links. In this specific case the eCDARN(p) model is chosen such that the causal structure of link dependencies is restricted in a Bayesian ring topology of L = 45 nodes, so that when link α samples its activity from the past history of other links, it randomly samples from α ± 1. The scalar memory of the network is \({{\Omega }}({{{{{{{\mathcal{G}}}}}}}})=6\). Pairs of neighbouring or close links in the Bayesian ring exhibit high memory co-order, often above \({{\Omega }}({{{{{{{\mathcal{G}}}}}}}})\), due to the onset of virtual loops (see the text), whereas distant links are seldom cross-correlated and thus display low co-order memory. b Similar to (a), but where the link’s causal structure is given by a different Bayesian graph (see SI section IVA for full details). A notably different memory shape emerges, however the scalar memory of the network is still \({{\Omega }}({{{{{{{\mathcal{G}}}}}}}})=6\). c Distribution of memory co-orders in both examples, showing different heterogeneous profiles which in both cases are not well characterised by \({{\Omega }}({{{{{{{\mathcal{G}}}}}}}})\).