Higher-order structure and epidemic dynamics in clustered networks

Abstract

Clustering is typically measured by the ratio of triangles to all triples regardless of whether open or closed. Generating clustered networks, and how clustering affects dynamics on networks, is reasonably well understood for certain classes of networks (Volz et al., 2011; Karrer and Newman, 2010), e.g. networks composed of lines and non-overlapping triangles. In this paper we show that it is possible to generate networks which, despite having the same degree distribution and equal clustering, exhibit different higher-order structure, specifically, overlapping triangles and other order-four (a closed network motif composed of four nodes) structures. To distinguish and quantify these additional structural features, we develop a new network metric capable of measuring order-four structure which, when used alongside traditional network metrics, allows us to more accurately describe a network׳s topology. Three network generation algorithms are considered: a modified configuration model and two rewiring algorithms. By generating homogeneous networks with equal clustering we study and quantify their structural differences, and using SIS (Susceptible-Infected-Susceptible) and SIR (Susceptible-Infected-Recovered) dynamics we investigate computationally how differences in higher-order structure impact on epidemic threshold, final epidemic or prevalence levels and time evolution of epidemics. Our results suggest that characterising and measuring higher-order network structure is needed to advance our understanding of the impact of network topology on dynamics unfolding on the networks.

Impact of higher-order network structure on emergent cortical activity

“…Consistent with previous definitions of higher-order network structure (Benson et al, 2016), our definition includes degree distributions of neurons, which are not constrained by the pairwise statistics of neurons alone. Nevertheless, reducing higher-order network structure while keeping degree distributions fixed is possible, and such higher-order structure can have an impact on network dynamics (Ritchie, Berthouze, House, & Kiss, 2014). To better understand the respective impact of changes in in- and out-degrees on one side, and clustering and high-dimensional motifs on the other side, it will be necessary to create a more refined control model that conserves degree distributions on top of first-order structure.…”

Relaxation dynamics of maximally clustered networks

“…We have also limited ourselves to studying rewiring that consistently destroys triangles, but what about rewiring with a view to increasing the number of triangles? A number of greedy as well as equilibrium algorithms exist and are widely applied to model highly clustered networks [28,41], but it is unclear how they cover the space of all networks and can lead to interesting behavior such as hysteresis loops [28]. Finally, there is a growing literature on generalized network structures such as simplicial complexes which allow for more general types of connections between nodes [42][43][44][45].…”

Mapping Structural Diversity in Networks Sharing a Given Degree Distribution and Global Clustering: Adaptive Resolution Grid Search Evolution with Diophantine Equation-Based Mutations

“…It is important to stress at the outset that the algorithm is not exact: first, Figure 1: First five panels: Histograms of the prevalence of each subgraph for 1000 networks generated by CMA for specification ( :35, :128, :277, :35, :42). The subgraphs were counted using the method in [25]. The number of denotes the number of not involved in any other clustering-inducing subgraphs.…”