  We study the temporal co-variation of network co-evolution via the cross-link
structure of networks, for which we take advantage of the formalism of
hypergraphs to map cross-link structures back to network nodes. We investigate
two sets of temporal network data in detail. In a network of coupled nonlinear
oscillators, hyperedges that consist of network edges with temporally
co-varying weights uncover the driving co-evolution patterns of edge weight
dynamics both within and between oscillator communities. In the human brain,
networks that represent temporal changes in brain activity during learning
exhibit early co-evolution that then settles down with practice, and subsequent
decreases in hyperedge size are consistent with emergence of an autonomous
subgraph whose dynamics no longer depends on other parts of the network. Our
results on real and synthetic networks give a poignant demonstration of the
ability of cross-link structure to uncover unexpected co-evolution attributes
in both real and synthetic dynamical systems. This, in turn, illustrates the
utility of analyzing cross-links for investigating the structure of temporal
networks.
