Complex Network Geometry and Frustrated Synchronization

Abstract

The dynamics of networks of neuronal cultures has been recently shown to be strongly dependent on the network geometry and in particular on their dimensionality. However, this phenomenon has been so far mostly unexplored from the theoretical point of view. Here we reveal the rich interplay between network geometry and synchronization of coupled oscillators in the context of a simplicial complex model of manifolds called Complex Network Manifold. The networks generated by this model combine small world properties (infinite Hausdorff dimension) and a high modular structure with finite and tunable spectral dimension. We show that the networks display frustrated synchronization for a wide range of the coupling strength of the oscillators, and that the synchronization properties are directly affected by the spectral dimension of the network.

Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplexes

“…Such interactions are called "simplicial", where an n-simplex represents an interaction between n + 1 units, so 2-simplices describe three-way interactions, etc [9]. Recent advances suggest that simplicial interactions may be vital in general oscillator systems [10][11][12] and may play an important role in brain dynamics [13][14][15] and other complex systems phenomena such as, the dynamics of collaborations [16] or social contagion [17]. In particular, interactions in 2-simplices (named holes or cavities) are important because they can describe correlations in neuronal spiking activity (that can be mapped to phase oscillators [18]) in the brain [19] providing a missing link between structure and function.…”

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].…”

Synchronization in network geometries with finite spectral dimension

“…Only very recently a few works have pointed out that network geometry can have a profound effect on sychronization dynamics [16,25,48]. In particular, it has been found that neuronal cultures have synchronization properties strongly affected by their dimensionality, so that 2d neuronal cultures display weaker synchronization properties than neuronal cultures grown in 3d scaffolds [48].…”

“…Rather they can be very localized on a small fraction of nodes, reflecting the symmetries present in the network. Therefore, here we characterize the spectral properties of Complex Network Manifolds and study the effect of these properties on the entrained phase synchronization, which is known to display strong spatio-temporal fluctuations of the order parameter [25]. This phase, also called frustruated synchornization [43,44], has a very rich structure and can be interpreted as an extended critical region to be related to the smeared phase observed in critical phenomena on hyperbolic networks, such as percolation [51,52].…”