Noise-Induced Desynchronization and Stochastic Escape from Equilibrium in Complex Networks

Complex physical systems are unavoidably subjected to external environments not accounted for in the set of differential equations that models them. The resulting perturbations are standardly represented by noise terms. We derive conditions under which such noise terms perturb the dynamics strongly enough that they lead to stochastic escape from the initial basin of attraction of an initial stable equilibrium state of the unperturbed system. Focusing on Kuramoto-like models we find in particular that, quite counterintuitively, systems with inertia leave their initial basin faster than or at the same time as systems without inertia, except for strong white-noise perturbations.Comment: Main text: 5 pages, 4 figures. Supplemental material: 6 pages, 7 figure

Network Inference using Sinusoidal Probing

The aim of this manuscript is to present a non-invasive method to recover the network structure of a dynamical system. We propose to use a controlled probing input and to measure the response of the network, in the spirit of what is done to determine oscillation modes in large electrical networks. For a large class of dynamical systems, we show that this approach is analytically tractable and we confirm our findings by numerical simulations of networks of Kuramoto oscillators. Our approach also allows us to determine the number of agents in the network by probing and measuring a single one of them.Comment: 5 pages, 4 figure

The Kuramoto model on oriented and signed graphs

Many real-world systems of coupled agents exhibit directed interactions, meaning that the influence of an agent on another is not reciprocal. Furthermore, interactions usually do not have identical amplitude and/or sign. To describe synchronization phenomena in such systems, we use a generalized Kuramoto model with oriented, weighted and signed interactions. Taking a bottom-up approach, we investigate the simplest possible oriented networks, namely acyclic oriented networks and oriented cycles. These two types of networks are fundamental building blocks from which many general oriented networks can be constructed. For acyclic, weighted and signed networks, we are able to completely characterize synchronization properties through necessary and sufficient conditions, which we show are optimal. Additionally, we prove that if it exists, a stable synchronous state is unique. In oriented, weighted and signed cycles with identical natural frequencies, we show that the system globally synchronizes and that the number of stable synchronous states is finite.Comment: 20 pages, 9 figure

Reconstructing Network Structures from Partial Measurements

The dynamics of systems of interacting agents is determined by the structure of their coupling network. The knowledge of the latter is, therefore, highly desirable, for instance, to develop efficient control schemes, to accurately predict the dynamics, or to better understand inter-agent processes. In many important and interesting situations, the network structure is not known, however, and previous investigations have shown how it may be inferred from complete measurement time series on each and every agent. These methods implicitly presuppose that, even though the network is not known, all its nodes are. Here, we investigate the different problem of inferring network structures within the observed/measured agents. For symmetrically coupled dynamical systems close to a stable equilibrium, we establish analytically and illustrate numerically that velocity signal correlators encode not only direct couplings, but also geodesic distances in the coupling network within the subset of measurable agents. When dynamical data are accessible for all agents, our method is furthermore algorithmically more efficient than the traditional ones because it does not rely on matrix inversion.Comment: 10 pages, 5 figure

Locating line and node disturbances in networks of diffusively coupled dynamical agents

A wide variety of natural and human-made systems consist of a large set of dynamical units coupled into a complex structure. Breakdown of such systems can have dramatic impact, as for instance neurons in the brain or lines in an electric grid. Preventing such catastrophic events requires in particular to be able to detect and locate the source of disturbances as fast as possible. We propose a simple method to identify and locate disturbances in networks of coupled dynamical agents, relying only on time series measurements and on the knowledge of the (Kron-reduced) network structure. The strength and the appeal of the present approach lies in its simplicity paired with the ability to precisely locate disturbances and even to differentiate between line and node disturbances. If we have access to measurement at only a subset of nodes, our method is still able to identify the location of the disturbance if the disturbed nodes are measured. If not, we manage to identify the region of the network where the disturbance occurs.Comment: 15 pages, 5 figure

On the Robustness of Democratic Electoral Processes to Computational Propaganda

There is growing evidence of systematic attempts to influence democratic elections by controlled and digitally organized dissemination of fake news. This raises the question of the intrinsic robustness of democratic electoral processes against external influences. Particularly interesting is to identify the social characteristics of a voter population that renders it more resilient against opinion manipulation. Equally important is to determine which of the existing democratic electoral systems is more robust to external influences. Here we construct a mathematical electoral model to address these two questions. We find that electorates are more resilient against opinion manipulations (i) if they are less polarized and (ii) when voters interact more with each other, regardless of their opinion differences, and that (iii) electoral systems based on proportional representation are generally the most robust. Our model qualitatively captures the volatility of the US House of Representatives elections. We take this as a solid validation of our approach.Comment: Main text: 26 pages, 6 figures. Supplementary information: 14 pages, 9 figure