Local topological moves determine global diffusion properties of hyperbolic higher-order networks

Abstract

From social interactions to the human brain, higher-order networks are key to describe the underlying network geometry and topology of many complex systems. While it is well known that network structure strongly affects its function, the role that network topology and geometry has on the emerging dynamical properties of higher-order networks is yet to be clarified. In this perspective, the spectral dimension plays a key role since it determines the effective dimension for diffusion processes on a network. Despite its relevance, a theoretical understanding of which mechanisms lead to a finite spectral dimension, and how this can be controlled, represents nowadays still a challenge and is the object of intense research. Here we introduce two non-equilibrium models of hyperbolic higher-order networks and we characterize their network topology and geometry by investigating the interwined appearance of small-world behavior, δ-hyperbolicity and community structure. We show that different topological moves determining the non-equilibrium growth of the higher-order hyperbolic network models induce tunable values of the spectral dimension, showing a rich phenomenology which is not displayed in random graph ensembles. In particular, we observe that, if the topological moves used to construct the higher-order network increase the area/volume ratio, the spectral dimension continuously decreases, while the opposite effect is observed if the topological moves decrease the area/volume ratio. Our work reveals a new link between the geometry of a network and its diffusion properties, contributing to a better understanding of the complex interplay between network structure and dynamics.

What are higher-order networks?

“…One possible approach to gain analytical insights into these dynamics are mean-field approaches that capture the higher-order interactions: For example, these can uncover how critical transitions in such systems change from subcritical to supercritical and vice versa [162]. Other examples of higher-order dynamical processes include opinion formation and consensus processes on hypergraphs [81,200,201], diffusion processes [49,50,188,224,234] (see also Subsection 3.4), and replicator dynamics [7].…”

Dynamics on higher-order networks: A review

“…Moreover, this higher-order structure affects the diffusion dynamics taking place on this, with the spectral di-mensions having meaningful influence on the return-time probability of the concerned diffusion process. Furthermore, relation between the geometry of a network and diffusion dynamics is unraveled [125] based on the investigation of two family of models, namely NGF and 'Short-Range Triadic Closure (STC)'. Thus far, many generalizations of different random walk models for higher-order interactions have been put forward, as discussed above.…”

Higher-order percolation processes on multiplex hypergraphs

“…Notably multilayer networks are characterized by very relevant correlations [15,16] that have the ability to modify the critical properties of the dynamics defined on these structures. On their turn, higher-order networks reveal unexpected phenomena in the context of synchronization transitions [17][18][19][20][21], diffusion [22][23][24][25] and spreading processes [10][11][12][26][27][28].…”