Cross-linked structure of network evolution

Abstract

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. 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. Networks provide a useful framework for gaining insights into a wide variety of social, physical, technological, and biological phenomena.1 As time-resolved data become more widely available, it is increasingly important to investigate not only static networks but also temporal networks.2,3 It is thus critical to develop methods to quantify and characterize dynamic properties of nodes (which represent entities) and/or edges (which represent ties between entities) that vary in time. In the present paper, we describe methods for the identification of cross-link structures in temporal networks by isolating sets of edges with similar temporal dynamics. We use the formalism of hypergraphs to map these edge sets to network nodes, thereby describing the complexity of interaction dynamics in system components. We illustrate our methodology using temporal networks that we extracted from synthetic data generated from coupled nonlinear oscillators and empirical data generated from human brain activity.

Network neuroscience

“…Finally, the application of concepts from algebraic topology (Fig. 3) attempts to discern non-random structure in networks by going beyond dyadic relations (two nodes linked by an edge) and considering non-dyadic, higher order relations among network nodes 63–65 , a goal that is complementary to that motivating the use of graphs in which edges can link any number of nodes, or so-called hypergraphs 66 . This approach can identify non-random structure in structural connectivity of cortical microcircuits 67 , such as unexpectedly high numbers of directed ‘all-to-all’ connected cliques of neurons, or cavities in which edges are conspicuously absent 68 .…”

Multi-scale brain networks

“…Degree and path length, along with other local and global network measures, are useful for characterizing networks at their most extreme topological scales: at the level of a network’s most commonly studied fundamental units (its nodes; although see Giusti et al, 2016 and Bassett et al, 2014 for alternatives) and the level of the network as a collective. Between these two scales lies a mesoscale, an intermediate scale at which a network can be characterized not in terms of local and global properties, but also in terms of differently sized clusters of nodes that adopt different types of configurations.…”

A Network Neuroscience of Human Learning: Potential to Inform Quantitative Theories of Brain and Behavior

“…These reconfigurations are more broadly characteristic of a flexible brain network organization [85,86], and individual differences in this flexibility of modular architecture predicts future learning rate [1]. Interestingly, this flexibility is produced by sets of edges that change in strength with one another instead of independently of one another [87], collectively linking a relatively stable core of regions that are thought to be necessary for task performance and a relatively flexible periphery of regions thought to be only supportive of task performance [88]. …”