Synchronization in network geometries with finite spectral dimension

Abstract

Recently there is a surge of interest in network geometry and topology. Here we show that the spectral dimension plays a fundamental role in establishing a clear relation between the topological and geometrical properties of a network and its dynamics. Specifically we explore the role of the spectral dimension in determining the synchronization properties of the Kuramoto model. We show that the synchronized phase can only be thermodynamically stable for spectral dimensions above four and that phase entrainment of the oscillators can only be found for spectral dimensions greater than two. We numerically test our analytical predictions on the recently introduced model of network geometry called Complex Network Manifolds which displays a tunable spectral dimension.

Abrupt phase transition of epidemic spreading in simplicial complexes

“…Recently, the network science community has turned its attention to network geometry [6-9] to better represent the kinds of interactions that one can find beyond typical pairwise interactions.These higher-order interactions are encoded in geometrical structures that describe the different kinds of simplex structure present in the network: a filled clique of m + 1 nodes is known as an m-simplex, and together a set of 1-simplexes (links), 2-simplexes (filled triangles), etc., comprise the simplicial complex. While simplicial complexes have been proven to be very useful for the analysis and computation in high dimensional data sets, e.g., using persistent homologies [10][11][12][13][14], little is understood about their role in shaping dynamical processes, save for a handful of examples [15][16][17][18].A more accurate description of dynamical processes on complex systems necessarily requires a new paradigm where the network structure representation helps to include higher-order interactions [19]. Simplicial geometry of complex networks is a natural way to extend manybody interactions in complex systems.…”

“…These higher-order interactions are encoded in geometrical structures that describe the different kinds of simplex structure present in the network: a filled clique of m + 1 nodes is known as an m-simplex, and together a set of 1-simplexes (links), 2-simplexes (filled triangles), etc., comprise the simplicial complex. While simplicial complexes have been proven to be very useful for the analysis and computation in high dimensional data sets, e.g., using persistent homologies [10][11][12][13][14], little is understood about their role in shaping dynamical processes, save for a handful of examples [15][16][17][18].…”

Simplicial complexes: higher-order spectral dimension and dynamics

“…The recently proposed non-equilibrium growing simplicial complex model called 'network geometry with flavor' (NGF) [16] is able to display emergent hyperbolic network geometry [17] together with the major universal properties of complex networks including scale-free degree distribution, small-word distance property, high clustering coefficient and significant modular structure. Interestingly, the simplicial complexes generated by the NGF model display also a finite spectral dimension [18,19,30,31]. The coexistence of a finite spectral dimension and the small-world properties might be strongly related to the hyperbolicity of the simplicial complexes.…”

“…Being built by geometrical building blocks, simplicial complexes represent an ideal setting to investigate the properties of emergent network geometry and topology in complex systems [1,[15][16][17]. Moreover they reveal the rich interplay between network geometry and dynamics [18][19][20][26][27][28][29].…”

Critical dynamics of the Kuramoto model on sparse random networks

“…Using linearization of the dynamical equations of the Kuramoto oscillators, which is justified by the system being in the synchronized phase, consistency, i.e. the existence of synchronization, has been found in regular lattices for dimensions d > 4 [8], while, for general networks, for d s > 4, where d s is the spectral dimension of the network [15]. Another question is the nature of the synchronization transition, provided it exists, for which numerical results are available, mainly on the scaling of the order parameter in the stationary state.…”