Dynamical complexity as a proxy for the network degree distribution

Does the brain behave like a (complex) network? I. Dynamics

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.

Unveiling the Connectivity of Complex Networks Using Ordinal Transition Methods

Ordinal measures provide a valuable collection of tools for analyzing correlated data series. However, using these methods to understand information interchange in the networks of dynamical systems, and uncover the interplay between dynamics and structure during the synchronization process, remains relatively unexplored. Here, we compare the ordinal permutation entropy, a standard complexity measure in the literature, and the permutation entropy of the ordinal transition probability matrix that describes the transitions between the ordinal patterns derived from a time series. We find that the permutation entropy based on the ordinal transition matrix outperforms the rest of the tested measures in discriminating the topological role of networked chaotic Rössler systems. Since the method is based on permutation entropy measures, it can be applied to arbitrary real-world time series exhibiting correlations originating from an existing underlying unknown network structure. In particular, we show the effectiveness of our method using experimental datasets of networks of nonlinear oscillators.

Asymmetry-induced order in multilayer networks

Symmetries naturally occur in real-world networks and can significantly influence the observed dynamics. For instance, many synchronization patterns result from the underlying network symmetries, and high symmetries are known to increase the stability of synchronization. Yet here we find that general macroscopic features of network solutions such as regularity can be induced by breaking their symmetry of interactions. We demonstrate this effect in an ecological multilayer network where the topological asymmetries occur naturally. These asymmetries rescue the system from chaotic oscillations by establishing stable periodic orbits and equilibria. We call this phenomenon asymmetry-induced order and uncover its mechanism by analyzing both analytically and numerically the absence of dynamics on the system's synchronization manifold. Moreover, the bifurcation scenario describing the route from chaos to order is also disclosed. We demonstrate that this result also holds for generic node dynamics by analyzing coupled paradigmatic Rössler and Lorenz systems.

Node differentiation dynamics along the route to synchronization in complex networks

Synchronization has been the subject of intense research during decades mainly focused on determining the structural and dynamical conditions driving a set of interacting units to a coherent state globally stable. However, little attention has been paid to the description of the dynamical development of each individual networked unit in the process towards the synchronization of the whole ensemble. In this paper we show how in a network of identical dynamical systems, nodes belonging to the same degree class, differentiate in the same manner, visiting a sequence of states of diverse complexity along the route to synchronization independently on the global network structure. In particular, we observe, just after interaction starts pulling orbits from the initially uncoupled attractor, a general reduction of the complexity of the dynamics of all units being more pronounced in those with higher connectivity. In the weak-coupling regime, when synchronization starts to build up, there is an increase in the dynamical complexity, whose maximum is achieved, in general, first in the hubs due to their earlier synchronization with the mean field. For very strong coupling, just before complete synchronization, we found a hierarchical dynamical differentiation with lower degree nodes being the ones exhibiting the largest complexity departure. We unveil how this differentiation route holds for several models of nonlinear dynamics, including toroidal chaos and how it depends on the coupling function. This study provides insights to understand better strategies for network identification or to devise effective methods for network inference.

Complex networks exhibit intermittent synchronization

The path toward the synchronization of an ensemble of dynamical units goes through a series of transitions determined by the dynamics and the structure of the connections network. In some systems on the verge of complete synchronization, intermittent synchronization, a time-dependent state where full synchronization alternates with non-synchronized periods, has been observed. This phenomenon has been recently considered to have functional relevance in neuronal ensembles and other networked biological systems close to criticality. We characterize the intermittent state as a function of the network topology to show that the different structures can encourage or inhibit the appearance of early signs of intermittency. In particular, we study the local intermittency and show how the nodes incorporate to intermittency in hierarchical order, which can provide information about the node topological role even when the structure is unknown.