Edmilson Roque

The identification of the minimal set of nodes that maximizes the propagation of information is one of the most relevant problems in network science. In this paper, we introduce a new method to find the set of initial spreaders to maximize the information propagation in complex networks. We evaluate this method in assortative networks and verify that degree-degree correlation plays a fundamental role in the spreading dynamics. Simulation results show that our algorithm is statistically similar, regarding the average size of outbreaks, to the greedy approach in real-world networks. However, our method is much less time consuming than the greedy algorithm.

Nature Physics 13, 431 (2017). doi:10.1038/nphys4035

Authors: Juan Carrasquilla & Roger G. Melko

Condensed-matter physics is the study of the collective behaviour of infinitely complex assemblies of electrons, nuclei, magnetic moments, atoms or qubits. This complexity is reflected in the size of the state space, which grows exponentially with the number of particles, reminiscent of the ‘curse of dimensionality’ commonly encountered in machine learning. Despite this curse, the machine learning community has developed techniques with remarkable abilities to recognize, classify, and characterize complex sets of data. Here, we show that modern machine learning architectures, such as fully connected and convolutional neural networks, can identify phases and phase transitions in a variety of condensed-matter Hamiltonians. Readily programmable through modern software libraries, neural networks can be trained to detect multiple types of order parameter, as well as highly non-trivial states with no conventional order, directly from raw state configurations sampled with Monte Carlo.

From critical infrastructure, to physiology and the human brain, complex systems rarely occur in isolation. Instead, the functioning of nodes in one system often promotes or suppresses the functioning of nodes in another. Despite advances in structural interdependence, modeling interdependence and other interactions between dynamic systems has proven elusive. Here we define a broadly applicable dynamic dependency link and develop a general framework for interdependent and competitive interactions between general dynamic systems. We apply our framework to studying interdependent and competitive synchronization in multi-layer oscillator networks and cooperative/competitive contagions in an epidemic model. Using a mean-field theory which we verify numerically, we find explosive transitions and rich behavior which is absent in percolation models including hysteresis, multi-stability and chaos. The framework presented here provides a powerful new way to model and understand many of the interacting complex systems which surround us.

Author(s): Ivan Kryven

In the infinite configuration network the links between nodes are assigned randomly with the only restriction that the degree distribution has to match a predefined function. This work presents a simple equation that gives for an arbitrary degree distribution the corresponding size distribution of c…

[Phys. Rev. E 95, 052303] Published Tue May 02, 2017

Author(s): Liyue Zhang, Adilson E. Motter, and Takashi Nishikawa

In previously identified forms of remote synchronization between two nodes, the intermediate portion of the network connecting the two nodes is not synchronized with them but generally exhibits some coherent dynamics. Here we report on a network phenomenon we call incoherence-mediated remote synchro...

[Phys. Rev. Lett. 118, 174102] Published Fri Apr 28, 2017

We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.

We study the extinction of long-lived epidemics on finite complex networks induced by intrinsic noise. Applying analytical techniques to the stochastic Susceptible-Infected-Susceptible model, we predict the distribution of large fluctuations, the most probable, or optimal path through a network that leads to a disease-free state from an endemic state, and the average extinction time in general configurations. Our predictions agree with Monte-Carlo simulations on several networks, including synthetic weighted and degree-distributed networks with degree correlations, and an empirical high school contact network. In addition, our approach quantifies characteristic scaling patterns for the optimal path and distribution of large fluctuations, both near and away from the epidemic threshold, in networks with heterogeneous eigenvector centrality and degree distributions.

We investigate the scaling properties of the order parameter and the largest nonvanishing Lyapunov exponent for the fully locked state in the Kuramoto model with a finite number $N$ of oscillators. We show that, for any finite value of $N$, both quantities scale as $(K-K_L)^{1/2}$ with the coupling strength $K$ sufficiently close to the locking threshold $K_L$. We confirm numerically these predictions for oscillator frequencies evenly spaced in the interval $[-1, 1]$ and additionally find that the coupling range $\delta K$ over which this scaling is valid shrinks like $\delta K \sim N^{-\alpha}$ with $\alpha\approx1.5$ as $N \rightarrow \infty$. Away from this interval, the order parameter exhibits the infinite-$N$ behavior $r-r_L \sim (K-K_L)^{2/3}$ proposed by Paz\'o [Phys. Rev. E 72, 046211 (2005)]. We argue that the crossover between the two behaviors occurs because at the locking threshold, the upper bound of the continuous part of the spectrum of the fully locked state approaches zero as $N$ increases. Our results clarify the convergence to the $N \rightarrow \infty$ limit in the Kuramoto model.

Author(s): Liyue Zhang, Adilson E. Motter, and Takashi Nishikawa

In previously identified forms of remote synchronization between two nodes, the intermediate portion of the network connecting the two nodes is not synchronized with them but generally exhibits some coherent dynamics. Here we report on a network phenomenon we call incoherence-mediated remote synchro…

[Phys. Rev. Lett. 118, 174102] Published Fri Apr 28, 2017

Author(s): Sanjiv K. Dwivedi, Murilo S. Baptista, and Sarika Jalan

The mathematical framework of multiplex networks has been increasingly realized as a more suitable framework for modeling real-world complex systems. In this work, we investigate the optimization of synchronizability in multiplex networks by evolving only one layer while keeping other layers fixed. …

[Phys. Rev. E 95, 040301(R)] Published Thu Apr 27, 2017

Author(s): Duane Q. Nykamp, Daniel Friedman, Sammy Shaker, Maxwell Shinn, Michael Vella, Albert Compte, and Alex Roxin

The emergent dynamics in networks of recurrently coupled spiking neurons depends on the interplay between single-cell dynamics and network topology. Most theoretical studies on network dynamics have assumed simple topologies, such as connections that are made randomly and independently with a fixed …

[Phys. Rev. E 95, 042323] Published Thu Apr 27, 2017

Abstract

We study the dynamics in a system of coupled oscillators when Arnold Tongues overlap. By varying the initial conditions, the deterministic system can be attracted to different limit cycles. Adding noise, the mode hopping between different states become a dominating part of the dynamics. We simplify the system through a Poincare section, and derive a 1D model to describe the dynamics. We explain that for some parameter values of the external oscillator, the time distribution of occupancy in a state is exponential and thus memoryless. In the general case, on the other hand, it is a sum of exponential distributions characteristic of a system with time correlations.

In this paper we develop conditions for various types of stability in social networks governed by Imitation of Success principle. Considering so-called Prisoner's Dilemma as the base of node-to-node game in the network we obtain well-known Hopfield neural network model. Asymptotic behavior of the original model and dynamic Hopfield model has a certain correspondence. To obtain more general results, we consider Hopfield model dynamic system on time scales. Developed stability conditions combine main parameters of network structure such as network size and maximum relative nodes' degree with the main characteristics of time scale, nodes' inertia and resistance, rate of input-output response.

Delay-coupled networks are investigated with nonidentical delay times and the effects of such heterogeneity on the emergent dynamics of complex systems are characterized. A simple decomposition method is presented that decouples the dynamics of the network into node-size modal equations in the vicinity of equilibria. The resulting independent components contain distributed delays that map the spatiotemporal complexity of the system to the time domain. We demonstrate that this new approach can be used to reveal new physical phenomena in heterogenous vehicular traffic when vehicles are linked via vehicle-to-vehicle (V2V) communication.

Stochastic dynamical systems with continuous symmetries arise commonly in nature and often give rise to coherent spatio-temporal patterns. However, because of their random locations, these patterns are not well captured by current order reduction techniques and a large number of modes is typically necessary for an accurate solution. In this work, we introduce a new methodology for efficient order reduction of such systems by combining (i) the method of slices, a symmetry reduction tool, with (ii) any standard order reduction technique, resulting in efficient mixed symmetry-dimensionality reduction schemes. In particular, using the Dynamically Orthogonal (DO) equations in the second step, we obtain a novel nonlinear Symmetry-reduced Dynamically Orthogonal (SDO) scheme. We demonstrate the performance of the SDO scheme on stochastic solutions of the 1D Korteweg-de Vries and 2D Navier-Stokes equations.

Exploratory adaptation in large random networks

Nature Communications, Published online: 21 April 2017; doi:10.1038/ncomms14826

Recent works suggest that cellular networks may respond to novel challenges on the time-scale of cellular lifetimes through large-scale perturbation of gene expression and convergence to a new state. Here, the authors demonstrate the theoretical feasibility of exploratory adaptation in cellular networks by showing that convergence to new states depends on known features of these networks.

The aim of this paper is to rigorously study dynamics of Heterogeneously Coupled Maps (HCM). Such systems are determined by a network with heterogeneous degrees. Some nodes, called hubs, are very well connected while most nodes interact with few others. The local dynamics on each node is chaotic, coupled with other nodes according to the network structure. Such high-dimensional systems are hard to understand in full, nevertheless we are able to describe the system over exponentially large time scales. In particular, we show that the dynamics of hub nodes can be very well approximated by a low-dimensional system. This allows us to establish the emergence of macroscopic behaviour such as coherence of dynamics among hubs of the same connectivity layer (i.e. with the same number of connections), and chaotic behaviour of the poorly connected nodes. The HCM we study provide a paradigm to explain why and how the dynamics of the network can change across layers.

We present an approach for reconstructing networks of pulse-coupled neuron-like oscillators from passive observation of pulse trains of all nodes. It is assumed that units are described by their phase response curves and that their phases are instantaneously reset by incoming pulses. Using an iterative procedure, we recover the properties of all nodes, namely their phase response curves and natural frequencies, as well as strengths of all directed connections.

Maintaining the synchronous motion of dynamical systems interacting on complex networks is often critical to their functionality. However, real-world networked dynamical systems operating synchronously are prone to random perturbations driving the system to arbitrary states within the corresponding basin of attraction, thereby leading to epochs of desynchronized dynamics with a priori unknown durations. Thus, it is highly relevant to have an estimate of the duration of such transient phases before the system returns to synchrony, following a random perturbation to the dynamical state of any particular node of the network. We address this issue here by proposing the framework of \emph{single-node recovery time} (SNRT) which provides an estimate of the relative time scales underlying the transient dynamics of the nodes of a network during its restoration to synchrony. We utilize this in differentiating the particularly \emph{slow} nodes of the network from the relatively \emph{fast} nodes, thus identifying the critical nodes which when perturbed lead to significantly enlarged recovery time of the system before resuming synchronized operation. Further, we reveal explicit relationships between the SNRT values of a network, and its \emph{global relaxation time} when starting all the nodes from random initial conditions. We employ the proposed concept for deducing microscopic relationships between topological features of nodes and their respective SNRT values. The framework of SNRT is further extended to a measure of resilience of the different nodes of a networked dynamical system. We demonstrate the potential of SNRT in networks of R\"{o}ssler oscillators on paradigmatic topologies and a model of the power grid of the United Kingdom with second-order Kuramoto-type nodal dynamics illustrating the conceivable practical applicability of the proposed concept.

This paper analyses the dynamics of infectious disease with a concurrent spread of disease awareness. The model includes local awareness due to contacts with aware individuals, as well as global awareness due to reported cases of infection and awareness campaigns. We investigate the effects of time delay in response of unaware individuals to available information on the epidemic dynamics by establishing conditions for the Hopf bifurcation of the endemic steady state of the model. Analytical results are supported by numerical bifurcation analysis and simulations.

Pedestrian evacuation is studied based on a modified lattice model. The payoff matrix in this model represents the complicated interactions between selfish individuals, and the mean force imposed on an individual is given by considering the impacts of neighbors, walls, and defector herding. Each passer-by moves to his selected location according to the Fermi function, and the average velocity of pedestrian flow is defined as a function of the motion rule. Two pedestrian types are included: cooperators, who adhere to the evacuation instructions; and defectors, who ignore the rules and act individually. It is observed that the escape time increases as fear degree increases, and the system remains smooth for a low fear degree, but exhibits three stages for a high fear degree. We prove that the fear degree determines the dynamics of this system, and the initial density of cooperators has a negligible impact. The system experiences three phases, a single phase of cooperator, a mixed ...

Chimera states, namely complex spatiotemporal patterns that consist of coexisting domains of spatially coherent and incoherent dynamics, are investigated in a network of coupled identical oscillators. These intriguing spatiotemporal patterns were first reported in nonlocally coupled phase oscillators, and it was shown that such mixed type behavior occurs only for specific initial conditions in nonlocally and globally coupled networks. The influence of initial conditions on chimera states has remained a fundamental problem since their discovery. In this report, we investigate the robustness of chimera states together with incoherent and coherent states in dependence on the initial conditions. For this, we use the basin stability method which is related to the volume of the basin of attraction, and we consider nonlocally and globally coupled time-delayed Mackey-Glass oscillators as example. Previously, it was shown that the existence of chimera states can be characterized by mean phase velocity and a statistical measure, such as the strength of incoherence, by using well prepared initial conditions. Here we show further how the coexistence of different dynamical states can be identified and quantified by means of the basin stability measure over a wide range of the parameter space.

Nature advance online publication 19 April 2017. doi:10.1038/nature22486

Author: Steven M. Phelps

Oldfield mice and deer mice differ in their parental care, most dramatically in the behaviour of fathers. A study reveals the genetic and neuronal contributions to variation in parental care.

We investigate global stability properties of a HIV/AIDS population model with constant recruitment rate, mass action incidence, and variable population size. Existence and uniqueness results for disease-free and endemic equilibrium points are proved. Global stability of the equilibria is obtained through Lyapunov's direct method and LaSalle's invariance principle.

Communication delays and multiplexing are ubiquitous features of real-world networked systems. We here introduce a simple model where these two features are simultaneously present, and report the rich phe- nomenology which is actually due to their interplay on cluster synchronization. A delay in one layer has non trivial impacts on the collective dynamics of the other layers, enhancing or suppressing synchronization. At the same time, multiplexing may also enhance cluster synchronization of delayed layers. We elucidate several non trivial (and anti-intuitive) scenarios, which are of interest and potential application in various real-world systems, where introduction of a delay may render synchronization of a layer robust against changes in the properties of the other layers.

We study ensembles of globally coupled, nonidentical phase oscillators subject to correlated noise, and we identify several important factors that cause noise and coupling to synchronize or desynchronize a system. By introducing noise in various ways, we find an estimate for the onset of synchrony of a system in terms of the coupling strength, noise strength, and width of the frequency distribution of its natural oscillations. We also demonstrate that noise alone can be sufficient to synchronize nonidentical oscillators. However, this synchrony depends on the first Fourier mode of a phase-sensitivity function, through which we introduce common noise into the system. We show that higher Fourier modes can cause desynchronization due to clustering effects, and that this can reinforce clustering caused by different forms of coupling. Finally, we discuss the effects of noise on an ensemble in which antiferromagnetic coupling causes oscillators to form two clusters in the absence of noise.

Intrinsically nonlinear coupled systems present different oscillating components that exchange energy among themselves. We present a new approach to deal with such energy exchanges and to investigate how it depends on the system control parameters. The method consists in writing the total energy of the system, and properly identifying the energy terms for each component and, especially, their coupling. To illustrate the proposed approach, we work with the bi-dimensional spring pendulum, which is a paradigm to study nonlinear coupled systems, and is used as a model for several systems. For the spring pendulum, we identify three energy components, resembling the spring and pendulum like motions, and the coupling between them. With these analytical expressions, we analyze the energy exchange for individual trajectories, and we also obtain global characteristics of the spring pendulum energy distribution by calculating spatial and time average energy components for a great number of trajectories (periodic, quasi-periodic and chaotic) throughout the phase space. Considering an energy term due to the nonlinear coupling, we identify regions in the parameter space that correspond to strong and weak coupling. The presented procedure can be applied to nonlinear coupled systems to reveal how the coupling mediates internal energy exchanges, and how the energy distribution varies according to the system parameters.

Ergodicity is a fundamental requirement for a dynamical system to reach a state of statistical equilibrium. On the other hand, it is known that in slow-fast systems ergodicity of the fast sub- system impedes the equilibration of the whole system due to the presence of adiabatic invariants. Here, we show that the violation of ergodicity in the fast dynamics effectively drives the whole system to equilibrium. To demonstrate this principle we investigate dynamics of the so-called springy billiards. These consist of a point particle of a small mass which bounces elastically in a billiard where one of the walls can move - the wall is of a finite mass and is attached to a spring. We propose a random process model for the slow wall dynamics and perform numerical experiments with the springy billiards themselves and the model. The experiments show that for such systems equilibration is always achieved; yet, in the adiabatic limit, the system equilibrates with a positive exponential rate only when the fast particle dynamics has more than one ergodic component for certain wall positions.