Dynamical complexity as a proxy for the network degree distribution

Abstract

We explore the relation between the topological relevance of a node in a complex network and the individual dynamics it exhibits. When the system is weakly coupled, the effect of the coupling strength against the dynamical complexity of the nodes is found to be a function of their topological role, with nodes of higher degree displaying lower levels of complexity. We provide several examples of theoretical models of chaotic oscillators, pulse-coupled neurons and experimental networks of nonlinear electronic circuits evidencing such a hierarchical behavior. Importantly, our results imply that it is possible to infer the degree distribution of a network only from individual dynamical measurements.

Connectivity Influences on Nonlinear Dynamics in Weakly-Synchronized Networks: Insights From Rössler Systems, Electronic Chaotic Oscillators, Model and Biological Neurons

“…Another study simulating a network of Stuart-Landau oscillators instanced over a realistic human connectome reported a correspondence of the node degree with the relative phases and amplitudes of oscillations [34]. Recently, a more comprehensive analysis revealed an association between topological role and dynamics, such that high degree appears to be associated with lower levels of complexity across diverse systems [35]. Together, these results point to a possibly general relationship that seems worthy of additional consideration.…”

“…After mean subtraction, all local maxima points, given byẋ(t) = 0,ẍ(t) < 0 and x(t) > x(t ± δt) with δt = 10 interpolated points, were extracted, yielding a corresponding step-wise amplitude time-seriesx max (i) having length l. Based on this representation, previously used to summarize chaotic dynamics, e.g., in Refs. [35] and [55], we computed another measure of complexity, based on information-theoretical rather than dynamical notions, namely, the permutation entropy. This is a robust non-parametric method, based on a purely ordinal representation of the data.…”

“…This is deemed acceptable therein, given that the systems under consideration are wholly or strongly deterministic, and non-triviality is thus considered only in reference to deviation from order and not from randomness. Other works consider measures of statistical complexity explicitly grounded on the thermodynamical notion of disequilibrium [35], [59], [60]. Some comparisons with the measure of statistical complexity S introduced in Ref.…”

“…Some comparisons with the measure of statistical complexity S introduced in Ref. [35] (therein, C) are nevertheless provided in Sec. III and V. As regards the coefficient of variation, by construction it is insensitive to the temporal order of samples, thus, it only serves as an adjunct measure and cannot be formally considered a complexity index.…”

“…At this point, to allow explicit comparison with the results in Ref. [35], we also calculated the statistical complexity introduced therein and defined as S m = h m Q m , where S m ∈ [0, 1], h m is the normalized permutation entropy, and Q m is a normalized measure of disequilibrium, i.e., distance from an equiprobable distribution. The corresponding trends are different because this index by construction assumes low values when the system appears close to equilibrium, which is the case for low orders m that cannot fully represent the dynamics.…”