Author(s): Mobolaji Williams
Ordered chains (such as chains of amino acids) are ubiquitous in biological cells, and these chains perform specific functions contingent on the sequence of their components. Using the existence and general properties of such sequences as a theoretical motivation, we study the statistical physics of…
[Phys. Rev. E 95, 042126] Published Mon Apr 17, 2017
We analyze the properties of order parameters measuring synchronization and phase locking in complex oscillator networks. First, we review network order parameters previously introduced and reveal several shortcomings: none of the introduced order parameters capture all transitions from incoherence over phase locking to full synchrony for arbitrary, finite networks. We then introduce an alternative, universal order parameter that accurately tracks the degree of partial phase locking and synchronization, adapting the traditional definition to account for the network topology and its influence on the phase coherence of the oscillators. We rigorously proof that this order parameter is strictly monotonously increasing with the coupling strength in the phase locked state, directly reflecting the dynamic stability of the network. Furthermore, it indicates the onset of full phase locking by a diverging slope at the critical coupling strength. The order parameter may find applications across systems where different types of synchrony are possible, including biological networks and power grids.
It has been known that noise can suppress multistability by dynamically connecting coexisting attractors in the system which are otherwise in separate basins of attraction. The purpose of this mini-review is to argue that quasiperiodic driving can play a similar role in suppressing multistability. A concrete physical example is provided where quasiperiodic driving was demonstrated to eliminate multistability completely to generate robust chaos in a semiconductor superlattice system.
A fundamental property of complex networks is the tendency for edges to cluster. The extent of the clustering is typically quantified by the clustering coefficient, which is the probability that a length-2 path is closed, i.e., induces a triangle in the network. However, higher-order structures beyond triangles are crucial to understanding complex networks, and the clustering behavior with respect to such higher-order patterns is not well understood. Here we introduce higher-order clustering coefficients that measure the closure probability of higher-order network structures and provide a more comprehensive view of how the edges of complex networks cluster. Our higher-order clustering coefficients are a natural generalization of the traditional clustering coefficient. We derive several properties about higher-order clustering coefficients and analyze them under common random graph models. Finally, we use higher-order clustering coefficients to gain new insights into the structure of real-world networks from several domains.
Author(s): Giovanni Russo
In this paper we investigate how so-called quorum-sensing networks can be desynchronized. Such networks, which arise in many important application fields, such as systems biology, are characterized by the fact that direct communication between network nodes is superimposed to communication with a sh…
[Phys. Rev. E 95, 042312] Published Fri Apr 14, 2017
Assuming a general distribution for the sojourn time in the in- fectious class, we consider an SIS type epidemic model formulated as a scalar integral equation. We prove that the endemic equilibrium of the model is globally asymptotically stable whenever it exists, solving the conjecture of Hethcote and van den Driessche (1995) for the case of nonfatal diseases.
Author(s): Tommaso Coletta, Robin Delabays, and Philippe Jacquod
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−KL)1/2 with the coupling strength K…
[Phys. Rev. E 95, 042207] Published Thu Apr 13, 2017
Optimization of the stability of synchronized states between a pair of symmetrically coupled reaction-diffusion systems exhibiting rhythmic spatiotemporal patterns is studied in the framework of the phase reduction theory. The optimal linear filter that maximizes the linear stability of the in-phase synchronized state is derived for the case where the two systems are linearly coupled. The nonlinear optimal interaction function that theoretically gives the largest linear stability of the in-phase synchronized state is also derived. The theory is illustrated by using typical rhythmic patterns in FitzHugh-Nagumo systems as examples.
We consider optimization of linear stability of synchronized states between a pair of weakly coupled limit-cycle oscillators with cross coupling, where different components of state variables of the oscillators are allowed to interact. On the basis of the phase reduction theory, the coupling matrix between different components of the oscillator states that maximizes the linear stability of the synchronized state under given constraints on overall coupling intensity and on stationary phase difference is derived. The improvement in the linear stability is illustrated by using several types of limit-cycle oscillators as examples.
A framework integrating information theory and network science is proposed, giving rise to a potentially new area. By incorporating and integrating concepts such as complexity, coding, topological projections and network dynamics, the proposed network-based framework paves the way not only to extending traditional information science, but also to modeling, characterizing and analyzing a broad class of real-world problems, from language communication to DNA coding. Basically, an original network is supposed to be transmitted, with or without compaction, through a sequence of symbols or time-series obtained by sampling its topology by some network dynamics, such as random walks. We show that the degree of compression is ultimately related to the ability to predict the frequency of symbols based on the topology of the original network and the adopted dynamics. The potential of the proposed approach is illustrated with respect to the efficiency of transmitting several types of topologies by using a variety of random walks. Several interesting results are obtained, including the behavior of the Barab\'asi-Albert model oscillating between high and low performance depending on the considered dynamics, and the distinct performances obtained for two geographical models.
Systems of dynamical elements exhibiting spontaneous rhythms are found in various fields of science and engineering, including physics, chemistry, biology, physiology, and mechanical and electrical engineering. Such dynamical elements are often modeled as nonlinear limit-cycle oscillators. In this article, we briefly review phase reduction theory, which is a simple and powerful method for analyzing the synchronization properties of limit-cycle oscillators exhibiting rhythmic dynamics. Through phase reduction theory, we can systematically simplify the nonlinear multi-dimensional differential equations describing a limit-cycle oscillator to a one-dimensional phase equation, which is much easier to analyze. Classical applications of this theory, i.e., the phase locking of an oscillator to a periodic external forcing and the mutual synchronization of interacting oscillators, are explained. Further, more recent applications of this theory to the synchronization of non-interacting oscillators induced by common noise and the dynamics of coupled oscillators on complex networks are discussed. We also comment on some recent advances in phase reduction theory for noise-driven oscillators and rhythmic spatiotemporal patterns.
The hierarchical product of networks represents a natural tool for building large networks out of two smaller subnetworks: a primary subnetwork and a secondary subnetwork. Here we study the dynamics of diffusion and synchronization processes on hierarchical products. We apply techniques previously used for approximating the eigenvalues of the adjacency matrix to the Laplacian matrix, allowing us to quantify the effects that the primary and secondary subnetworks have on diffusion and synchronization in terms of a coupling parameter that weighs the secondary subnetwork relative to the primary subnetwork. Diffusion processes are separated into two regimes: for small coupling the diffusion rate is determined by the structure of the secondary network, scaling with the coupling parameter, while for large coupling it is determined by the primary network and saturates. Synchronization, on the other hand, is separated into three regimes: for both small and large coupling hierarchical products have poorly synchronization properties, but is optimized at an intermediate value. Moreover, the critical coupling value that optimizes synchronization is shaped by the relative connectivities of the primary and secondary subnetworks, compensating for significant differences between the two subnetworks.
We introduce a practical and computationally not demanding technique for inferring interactions at various microscopic levels between the units of a network from the measurements and the processing of macroscopic signals. Starting from a network model of Kuramoto phase oscillators which evolve adaptively according to homophilic and homeostatic adaptive principles, we give evidence that the increase of synchronization within groups of nodes (and the corresponding formation of synchronous clusters) causes also the defragmentation of the wavelet energy spectrum of the macroscopic signal. Our methodology is then applied for getting a glance to the microscopic interactions occurring in a neurophysiological system, namely, in the thalamo-cortical neural network of an epileptic brain of a rat, where the group electrical activity is registered by means of multichannel EEG. We demonstrate that it is possible to infer the degree of interaction between the interconnected regions of the brain during different types of brain activities, and to estimate the regions' participation in the generation of the different levels of consciousness.
The Kuramoto model is a system of nonlinear differential equations that models networks of coupled oscillators and is often used to study synchronization among the oscillators. In this paper we study steady state solutions of the Kuramoto model by assigning to each steady state a tuple of integers which records how the state twists around the cycles in the network. We then use this new classification of steady states to obtain a "Weyl" type of asymptotic estimate for the number of steady states as the number of oscillators becomes arbitrarily large while preserving the cycle structure. We further show how this asymptotic estimate can be maximized, and as a special case we obtain an asymptotic lower bound for the number of stable steady states of the model.
Author(s): Antonio Politi
A powerful approach is proposed for the characterization of chaotic signals. It is based on the combined use of two classes of indicators: (i) the probability of suitable symbolic sequences (obtained from the ordinal patterns of the corresponding time series); (ii) the width of the corresponding cyl…
[Phys. Rev. Lett. 118, 144101] Published Fri Apr 07, 2017
Author(s): Antonio Politi
A powerful approach is proposed for the characterization of chaotic signals. It is based on the combined use of two classes of indicators: (i) the probability of suitable symbolic sequences (obtained from the ordinal patterns of the corresponding time series); (ii) the width of the corresponding cyl...
[Phys. Rev. Lett. 118, 144101] Published Fri Apr 07, 2017
The dynamical behavior of networked complex systems is shaped not only by the direct links among the units, but also by the long-range interactions occurring through the many existing paths connecting the network nodes. In this work, we study how synchronization dynamics is influenced by these long-range interactions, formulating a model of coupled oscillators that incorporates this type of interactions through the use of $d-$path Laplacian matrices. We study synchronizability of these networks by the analysis of the Laplacian spectra, both theoretically and numerically, for real-world networks and artificial models. Our analysis reveal that in all networks long-range interactions improve network synchronizability with an impact that depends on the original structure, for instance it is greater for graphs having a larger diameter. We also investigate the effects of edge removal in graphs with long-range interactions and, as a major result, find that the removal process becomes more critical, since also the long-range influence of the removed link disappears.
The dynamical behavior of networked complex systems is shaped not only by the direct links among the units, but also by the long-range interactions occurring through the many existing paths connecting the network nodes. In this work, we study how synchronization dynamics is influenced by these long-range interactions, formulating a model of coupled oscillators that incorporates this type of interactions through the use of $d-$path Laplacian matrices. We study synchronizability of these networks by the analysis of the Laplacian spectra, both theoretically and numerically, for real-world networks and artificial models. Our analysis reveal that in all networks long-range interactions improve network synchronizability with an impact that depends on the original structure, for instance it is greater for graphs having a larger diameter. We also investigate the effects of edge removal in graphs with long-range interactions and, as a major result, find that the removal process becomes more critical, since also the long-range influence of the removed link disappears.
For smooth vector fields the classical method of characteristics provides a link between the ordinary differential equation and the corresponding continuity equation (or transport equation). We study an analog of this connection for merely bounded Borel vector fields. In particular we show that, given a non-negative Borel measure $\bar \mu$ on $\mathbb{R}^d$, existence of $\bar \mu$-measurable flow of a bounded Borel vector field is equivalent to existence of a measure-valued solution to the corresponding continuity equation with the initial data $\bar \mu$.
Many real-world multilayer systems such as critical infrastructure are interdependent and embedded in space with links of a characteristic length. They are also vulnerable to localized attacks or failures, such as terrorist attacks or natural catastrophes, which affect all nodes within a given radius. Here we study the effects of localized attacks on spatial multiplex networks of two layers. We find a metastable region where a localized attack larger than a critical size induces a nucleation transition as a cascade of failures spreads throughout the system, leading to its collapse. We develop a theory to predict the critical attack size and find that it exhibits novel scaling behavior. We further find that localized attacks in these multiplex systems can induce a previously unobserved combination of random and spatial cascades. Our results demonstrate important vulnerabilities in real-world interdependent networks and show new theoretical features of spatial networks.
Author(s): Gordon Robb and Antonio Politi
Thorough numerical studies reveal that spatially extended dissipative systems with long-range interactions may give rise to a large-scale dynamics. This phenomenon, which generalizes mean-field chaos, can be interpreted as a form of subtle pattern formation, where a chaotic microscopic dynamics coex…
[Phys. Rev. E 95, 040201(R)] Published Mon Apr 03, 2017
Reconstructing the states of the nodes of a dynamical network is a problem of fundamental importance in the study of neuronal and genetic networks. An underlying related problem is that of observability, i.e., identifying the conditions under which such a reconstruction is possible. In this paper we study observability of complex dynamical networks, where we consider the effects of network symmetries on observability. We present an efficient algorithm that returns a minimal set of necessary sensor nodes for observability in the presence of symmetries.
I will report below on a few examples of raving and insane (or maybe utterly genial) sentences that can be found in famous and otherwise admirable books of physics, because I genuinely believe it is amusing.
The largest eigenvalue of a network's adjacency matrix and its associated principal eigenvector are key elements for determining the topological structure and the properties of dynamical processes mediated by it. We present a physically grounded expression relating the value of the largest eigenvalue of any network to the largest eigenvalue of two network subgraphs, considered as isolated: The hub with its immediate neighbors and the densely connected set of nodes with maximum $K$-core index. We validate this formula showing that it predicts with good accuracy the largest eigenvalue of a large set of synthetic and real-world topologies, with no exception. We also present evidence of the consequences of these findings for broad classes of dynamics taking place on the networks. As a byproduct, we reveal that the spectral properties of heterogeneous networks built according to the linear preferential attachment model are qualitatively different from those of their static counterparts.
Author(s): Yoji Kawamura
We formulate a theory for the collective phase reduction of globally coupled noisy dynamical elements exhibiting macroscopic rhythms. We first transform the Langevin-type equation that represents a group of globally coupled noisy dynamical elements into the corresponding nonlinear Fokker-Planck equa…
[Phys. Rev. E 95, 032225] Published Fri Mar 31, 2017
Author(s): Vincenzo Nicosia, Per Sebastian Skardal, Alex Arenas, and Vito Latora
We introduce a framework to intertwine dynamical processes of different nature, each with its own distinct network topology, using a multilayer network approach. As an example of collective phenomena emerging from the interactions of multiple dynamical processes, we study a model where neural dynami…
[Phys. Rev. Lett. 118, 138302] Published Fri Mar 31, 2017
Author(s): Demian Levis, Ignacio Pagonabarraga, and Albert Díaz-Guilera
[Phys. Rev. X 7, 019904] Published Thu Mar 30, 2017
Reconstructing the states of the nodes of a dynamical network is a problem of fundamental importance in the study of neuronal and genetic networks. An underlying related problem is that of observability, i.e., identifying the conditions under which such a reconstruction is possible. In this paper we study observability of complex dynamical networks, where we consider the effects of network symmetries on observability. We present an efficient algorithm that returns a minimal set of necessary sensor nodes for observability in the presence of symmetries.
The dynamics of an oscillator driven by both low- and high- frequency external signals is studied. It is shown that both two- and three-frequency resonances arise due to a nonlinear interaction of these harmonic forces. Conditions which must be met for oscillator synchronization under these resonances are estimated analytically by considering the Van der Pol oscillator with modulated natural frequency as mathematical model. It is demonstrated that due to the low-frequency modulation, additional synchronization regions arise in the control parameter space. Feasibility of the theoretical findings is confirmed in experiments with a Hartley-type oscillator.
Author(s): Alexander Schmidt, Theodoros Kasimatis, Johanne Hizanidis, Astero Provata, and Philipp Hövel
We discuss synchronization patterns in networks of FitzHugh-Nagumo and leaky integrate-and-fire oscillators coupled in a two-dimensional toroidal geometry. A common feature between the two models is the presence of fast and slow dynamics, a typical characteristic of neurons. Earlier studies have dem…
[Phys. Rev. E 95, 032224] Published Tue Mar 28, 2017