Author(s): Abed Zadehgol
In this work, a generalized form of the BGK-Boltzmann equation is proposed, where the velocity, position, and time can be represented by real or complex variables. The real representation leads to the conventional BGK-Boltzmann equation, which can recover the continuity and Navier-Stokes equations. …[Phys. Rev. E] Published Thu Aug 11, 2016
Synchronization over networks depends strongly on the structure of the coupling between the oscillators. When the coupling presents certain regularities, the dynamics can be coarse-grained into clusters by means of External Equitable Partitions of the network graph and their associated quotient graphs. We exploit this graph-theoretical concept to study the phenomenon of cluster synchronization, in which different groups of nodes converge to distinct behaviors. We derive conditions and properties of networks in which such clustered behavior emerges, and show that the ensuing dynamics is the result of the localization of the eigenvectors of the associated graph Laplacians linked to the existence of invariant subspaces. The framework is applied to both linear and non-linear models, first for the standard case of networks with positive edges, before being generalized to the case of signed networks with both positive and negative interactions. We illustrate our results with examples of both signed and unsigned graphs for consensus dynamics and for partial synchronization of oscillator networks under the master stability function as well as Kuramoto oscillators.
We study the Braess paradox in the transport network as originally proposed by Braess with totally asymmetric exclusion processes (TASEPs) on the edges. The Braess paradox describes the counterintuitive situation where adding an additional edge to a road network leads to a user optimum with higher traveltimes for all network users. Traveltimes on the TASEPs are nonlinear in the density and jammed states can occur due to the microscopic exclusion principle. Furthermore the individual edges influence each other. This leads to a much more realistic description of traffic-like transport on the network than in previously studied linear macroscopic mathematical models. Furthermore the stochastic dynamics allows to explore the effects of fluctuations on the network performance. We observe that for low densities the added edge leads to lower traveltimes. For slightly higher densities the Braess paradox in its classical sense occurs in a small density regime. In a large regime of intermediate densities strong fluctuations in the traveltimes dominate the system's behaviour. These fluctuations are due to links that are in a domain wall or coexistence phase. At high densities the added link leads to lower traveltimes. We present a phase diagram predicting in which state the system will be, depending on the global density and crucial length ratios.
Author(s): Petter Holme
We investigate disease spreading on eight empirical data sets of human contacts (mostly proximity networks recording who is close to whom, at what time). We compare three levels of representations of these data sets: temporal networks, static networks, and a fully connected topology. We notice that …
[Phys. Rev. E 94, 022305] Published Mon Aug 15, 2016
Author(s): Lucia Valentina Gambuzza and Mattia Frasca
The position of the coherent and incoherent domain of a chimera state in a ring of nonlocally coupled oscillators is strongly influenced by the initial conditions, making nontrivial the problem of confining them in a specific region of the structure. In this paper we propose the use of spatial pinni…
[Phys. Rev. E 94, 022306] Published Mon Aug 15, 2016
Recently several studies have investigated synchronization in quantum-mechanical limit-cycle oscillators. However, the quantum nature of these systems remained partially hidden, since the dynamics of the oscillator phase was overdamped and therefore incoherent. We show that there exists a regime of underdamped phase motion which would allow for the observation of truly quantum-coherent effects. To this end we study the Van der Pol oscillator, a paradigm for self-oscillating systems, which has recently been used to study synchronization in the quantum regime. We derive an effective quantum model which fully describes the regime of underdamped phase motion. Furthermore, we find a regime of long-lived quantum coherence which opens up new possibilities to study quantum synchronization dynamics. Finally, we identify quantum limit cycles of the phase itself and relate them to recent experimental observations in the classical regime.
Synchronization is central to many complex systems in engineering physics (e.g., the power-grid, Josephson junction circuits, and electro-chemical oscillators) and biology (e.g., neuronal, circadian, and cardiac rhythms). Despite these widespread applications---for which proper functionality depends sensitively on the extent of synchronization---there remains a lack of understanding for how systems evolve and adapt to enhance or inhibit synchronization. We study how network modifications affect the synchronization properties of network-coupled dynamical systems that have heterogeneous node dynamics (e.g., phase oscillators with non-identical frequencies), which is often the case for real-world systems. Our approach relies on a synchrony alignment function (SAF) that quantifies the interplay between heterogeneity of the network and of the oscillators and provides an objective measure for a system's ability to synchronize. We conduct a spectral perturbation analysis of the SAF for structural network modifications including the addition and removal of edges, which subsequently ranks the edges according to their importance to synchronization. Based on this analysis, we develop gradient-descent algorithms to efficiently solve optimization problems that aim to maximize phase synchronization via network modifications. We support these and other results with numerical experiments.
In this paper, time delay effect and distributed shear are considered in the Kuramoto model. On the Ott-Antonsen's manifold, through analyzing the associated characteristic equation of the reduced functional differential equation, the stability boundary of the incoherent state is derived in multiple-parameter space. Moreover, very rich dynamical behavior such as stability switches inducing synchronization switches can occur in this equation. With the loss of stability, Hopf bifurcating coherent states arise, and the criticality of Hopf bifurcations is determined by applying the normal form theory and the center manifold theorem. On one hand, theoretical analysis indicates that the width of shear distribution and time delay can both eliminate the synchronization then lead the Kuramoto model to incoherence. On the other, time delay can induce several coexisting coherent states. Finally, some numerical simulations are given to support the obtained results where several bifurcation diagrams are drawn, and the effect of time delay and shear is discussed.
Hysteresis phenomena and multistability play crucial roles in the dynamics of coupled oscillators, which are now interpreted from the point of view of codimension-two bifurcations. On the Ott-Antonsen's manifold, complete bifurcation sets of delay-coupled Kuramoto model are derived regarding coupling strength and delay as bifurcation parameters. It is rigorously proved that the system must undergo Bautin bifurcations for some critical values, thus there always exists saddle-node bifurcation of periodic solutions inducing hysteresis loop. With the aid of center manifold reduction method and the Matlab Package DDE-Biftool, the location of Bautin and double Hopf points and detailed dynamics are theoretically determined. We find that, near these critical points, at most four coherent states (two of which are stable) and a stable incoherent state may coexist, and that the system undergoes Neimark-Sacker bifurcation of periodic solutions. Finally, the clear scenarios about the synchronous transition in delayed Kuramoto model are depicted.
We investigate a model for fatigue crack growth in which damage accumulation is assumed to follow a power law of the local stress amplitude, a form which can be generically justified on the grounds of the approximately self-similar aspect of microcrack distributions. Our aim is to determine the relation between model ingredients and the Paris exponent governing subcritical crack-growth dynamics at the macroscopic scale, starting from a single small notch propagating along a fixed line. By a series of analytical and numerical calculations, we show that, in the absence of disorder, there is a critical damage-accumulation exponent $\gamma$, namely $\gamma_c=2$, separating two distinct regimes of behavior for the Paris exponent $m$. For $\gamma>\gamma_c$, the Paris exponent is shown to assume the value $m=\gamma$, a result which proves robust against the separate introduction of various modifying ingredients. Explicitly, we deal here with (i) the requirement of a minimum stress for damage to occur; (ii) the presence of disorder in local damage thresholds; (iii) the possibility of crack healing. On the other hand, in the regime $\gamma<\gamma_c$ the Paris exponent is seen to be sensitive to the different ingredients added to the model, with rapid healing or a high minimum stress for damage leading to $m=2$ for all $\gamma<\gamma_c$, in contrast with the linear dependence $m=6-2\gamma$ observed for very long characteristic healing times in the absence of a minimum stress for damage. Upon the introduction of disorder on the local fatigue thresholds, which leads to the possible appearance of multiple cracks along the propagation line, the Paris exponent tends to $m\approx 4$ for $\gamma\lesssim 2$, while retaining the behavior $m=\gamma$ for $\gamma\gtrsim 4$.
Synchronization in networks of coupled oscillators is known to be largely determined by the spectral and symmetry properties of the interaction network. Here, we leverage this relation to study a class of networks for which the threshold coupling strength for global synchronization is the lowest among all networks with the same number of nodes and links. These networks, defined as being uniform, complete, and multi-partite (UCM), appear at each of an infinite sequence of network-complement transitions in a larger class of networks characterized by having near-optimal thresholds for global synchronization. We show that the distinct symmetry structure of the UCM networks, which by design are optimized for global synchronizability, often leads to formation of clusters of synchronous oscillators, and that such states can coexist with the state of global synchronization.
Universal resilience patterns in complex networks
Nature 536, 7615 (2016). doi:10.1038/nature18019
Authors: Jianxi Gao, Baruch Barzel & Albert-László Barabási
Nature530, 307–312 (2016); doi:10.1038/nature16948In the last sentence of page 310 of this Letter, the parameter h should equal 2, rather than 1. In addition, after equation (4), the text should have stated ‘Aij
We study the synchronization of a van der Pol self-oscillator with Kerr anharmonicity to an external drive. We demonstrate that the anharmonic, discrete energy spectrum of the quantum oscillator leads to multiple resonances in both phase locking and frequency entrainment not present in the corresponding classical system. Strong driving close to these resonances leads to nonclassical steady-state Wigner distributions. Experimental realizations of these genuine quantum signatures can be implemented with current technology.
Physics is one of the most successful endeavors in science. Being a prototypic big science it also reflects the growing tendency for scientific collaborations. Utilizing 250,000 papers from ArXiv.org a prepublishing platform prevalent in Physics we construct large coauthorship networks to investigate how individual network positions influence scientific success. In this context, success is seen as getting a paper published in high impact journals of physical subdisciplines as compared to not getting it published at all or in rather peripheral journals only. To control the nested levels of authors and papers, and to consider the time elapsing between working paper and prominent journal publication we employ multilevel eventhistory models with various network measures as covariates. Our results show that the maintenance of even a moderate number of persistent ties is crucial for scientific success. Also, even with low volumes of social capital Physicists who occupy brokerage positions enhance their chances of articles in high impact journals significantly. Surprisingly, inter(sub)disciplinary collaborations decrease the probability of getting a paper published in specialized journals for almost all positions.
We study complex networks of stochastic two-state units. Our aim is to model discrete stochastic excitable dynamics with a rest and an excited state. Both states are assumed to possess different waiting time distributions. The rest state is treated as an activation process with an exponentially distributed life time, whereas the latter in the excited state shall have a constant mean which may originate from any distribution. The activation rate of any single unit is determined by its neighbors according to a random complex network structure. In order to treat this problem in an analytical way, we use a heterogeneous mean-field approximation yielding a set of equations general valid for uncorrelated random networks. Based on this derivation we focus on random binary networks where the network is solely comprised of nodes with either of two degrees. The ratio between the two degrees is shown to be a crucial parameter. Dependent on the composition of the network the steady states show the usual transition from disorder to homogeneous ordered bistability as well as new scenarios that include inhomogeneous ordered and disordered bistability as well as tristability. The various steady states differ in their spiking activity expressed by a state dependent spiking rate. Numerical simulations agree with analytic results of the heterogeneous mean-field approximation.
Author(s): M. E. J. Newman and Gesine Reinert
Community detection, the division of a network into dense subnetworks with only sparse connections between them, has been a topic of vigorous study in recent years. However, while there exist a range of effective methods for dividing a network into a specified number of communities, it is an open qu…
[Phys. Rev. Lett. 117, 078301] Published Thu Aug 11, 2016
Author(s): Menachem Lachiany and Yoram Louzoun
A goal of many epidemic models is to compute the outcome of the epidemics from the observed infected early dynamics. However, often, the total number of infected individuals at the end of the epidemics is much lower than predicted from the early dynamics. This discrepancy is argued to result from hu…
[Phys. Rev. E 94, 022409] Published Thu Aug 11, 2016
Author(s): Tanmoy Banerjee, Partha Sharathi Dutta, Anna Zakharova, and Eckehard Schöll
This paper reports the occurrence of several chimera patterns and the associated transitions among them in a network of coupled oscillators, which are connected by a long range interaction that obeys a distance-dependent power law. This type of interaction is common in physics and biology and consti…[Phys. Rev. E] Published Wed Aug 10, 2016
Vulnerabilities of complex networks have became a trend topic in complex systems recently due to its real world applications. Most real networks tend to be very fragile to high betweenness adaptive attacks. However, recent contributions have shown the importance of interconnected nodes in the integrity of networks and module-based attacks have appeared promising when compared to traditional malicious non-adaptive attacks. In the present work we deeply explore the trade-off associated with attack procedures, introducing a generalized robustness measure and presenting an attack performance index that takes into account both robustness of the network against the attack and the run-time needed to obtained the list of targeted nodes for the attack. Besides, we introduce the concept of deactivation point aimed to mark the point at which the network stops to function properly. We then show empirically that non-adaptive module-based attacks perform better than high degree and betweenness adaptive attacks in networks with well defined community structures and consequent high modularity.
We perform a thorough study of various characteristics of the asynchronous push-pull protocol for spreading a rumor on Erd\H{o}s-R\'enyi random graphs $G_{n,p}$, for any $p>c\ln(n)/n$ with $c>1$. In particular, we provide a simple strategy for analyzing the asynchronous push-pull protocol on arbitrary graph topologies and apply this strategy to $G_{n,p}$. We prove tight bounds of logarithmic order for the total time that is needed until the information has spread to all nodes. Surprisingly, the time required by the asynchronous push-pull protocol is asymptotically almost unaffected by the average degree of the graph. Similarly tight bounds for Erd\H{o}s-R\'enyi random graphs have previously only been obtained for the synchronous push protocol, where it has been observed that the total running time increases significantly for sparse random graphs. Finally, we quantify the robustness of the protocol with respect to transmission and node failures. Our analysis suggests that the asynchronous protocols are particularly robust with respect to these failures compared to their synchronous counterparts.
We rigorously prove the existence of chaotic dynamics for two nonlinear Cournot triopoly game models with heterogeneous players, for which in the existing literature the presence of complex phenomena and strange attractors has been shown via numerical simulations. In the first model that we analyze, costs are linear but the demand function is isoelastic, while, in the second model, the demand function is linear and production costs are quadratic. As concerns the decisional mechanisms adopted by the firms, in both models one firm adopts a myopic adjustment mechanism, considering the marginal profit of the last period; the second firm maximizes its own expected profit under the assumption that the competitors' production levels will not vary with respect to the previous period; the third firm acts adaptively, changing its output proportionally to the difference between its own output in the previous period and the naive expectation value. The topological method we employ in our analysis is the so-called “Stretching Along the Paths” technique, based on the Poincaré-Miranda Theorem and the properties of the cutting surfaces, which allows to prove the existence of a semi-conjugacy between the system under consideration and the Bernoulli shift, so that the former inherits from the latter several crucial chaotic features, among which a positive topological entropy.
We perform a thorough study of various characteristics of the asynchronous push-pull protocol for spreading a rumor on Erd\H{o}s-R\'enyi random graphs $G_{n,p}$, for any $p>c\ln(n)/n$ with $c>1$. In particular, we provide a simple strategy for analyzing the asynchronous push-pull protocol on arbitrary graph topologies and apply this strategy to $G_{n,p}$. We prove tight bounds of logarithmic order for the total time that is needed until the information has spread to all nodes. Surprisingly, the time required by the asynchronous push-pull protocol is asymptotically almost unaffected by the average degree of the graph. Similarly tight bounds for Erd\H{o}s-R\'enyi random graphs have previously only been obtained for the synchronous push protocol, where it has been observed that the total running time increases significantly for sparse random graphs. Finally, we quantify the robustness of the protocol with respect to transmission and node failures. Our analysis suggests that the asynchronous protocols are particularly robust with respect to these failures compared to their synchronous counterparts.
The concept of city or urban resilience has emerged as one of the key challenges for the next decades. As a consequence, institutions like the United Nations or Rockefeller Foundation have embraced initiatives that increase or improve it. These efforts translate into funded programs both for action on the ground and to develop quantification of resilience, under the for of an index. Ironically, on the academic side there is no clear consensus regarding how resilience should be quantified, or what it exactly refers to in the urban context. Here we attempt to link both extremes providing an example of how to exploit large, publicly available, worldwide urban datasets, to produce objective insight into one of the possible dimensions of urban resilience. We do so via well-established methods in complexity science, such as percolation theory --which has a long tradition at providing valuable information on the vulnerability in complex systems. Our findings uncover large differences among studied cities, both regarding their infrastructural fragility and the imbalances in the distribution of critical services.
Systems of mobile physical entities exchanging information with their neighborhood can be found in many different situations. The understanding of their emergent cooperative behaviour has become an important issue across disciplines, requiring a general conceptual framework in order to harvest the potential of these systems. We study the synchronization of coupled oscillators in time-evolving networks defined by the positions of self-propelled agents interacting in real space. In order to understand the impact of mobility in the synchronization process on general grounds, we introduce a simple model of self-propelled hard disks performing persistent random walks in 2$d$ space and carrying an internal Kuramoto phase oscillator. For non-interacting particles, self-propulsion accelerates synchronization. The competition between agent mobility and excluded volume interactions gives rise to a richer scenario, leading to an optimal self-propulsion speed. We identify two extreme dynamic regimes where synchronization can be understood from theoretical considerations. A systematic analysis of our model quantifies the departure from the latter ideal situations and characterizes the different mechanisms leading the evolution of the system. We show that the synchronization of locally coupled mobile oscillators generically proceeds through coarsening verifying dynamic scaling and sharing strong similarities with the phase ordering dynamics of the 2$d$ XY model following a quench. Our results shed light into the generic mechanisms leading the synchronization of mobile agents, providing a efficient way to understand more complex or specific situations involving time-dependent networks where synchronization, mobility and excluded volume are at play.
A general scheme is proposed and tested to control the symmetry breaking instability of a homogeneous solution of a spatially extended multispecies model, defined on a network. The inherent discreteness of the space makes it possible to act on the topology of the inter-nodes contacts to achieve the desired degree of stabilization, without altering the dynamical parameters of the model. Both symmetric and asymmetric couplings are considered. In this latter setting the web of contacts is assumed to be balanced, for the homogeneous equilibrium to exist. The performance of the proposed method are assessed, assuming the Complex Ginzburg-Landau equation as a reference model. In this case, the implemented control allows one to stabilize the synchronous limit cycle, hence time-dependent, uniform solution. A system of coupled real Ginzburg-Landau equations is also investigated to obtain the topological stabilization of a homogeneous and constant fixed point.
We study the dynamics of a Josephson junction connected to a dc current supply via a distributed parameter capacitor, which serves as a resonator. We reveal multistability in the current-voltage characteristic of the system; this multistability is related to resonances between the generated frequency and the resonator. The resonant pattern requires detailed consideration, in particular, because its basic features may resemble those of patterns reported in experiments with arrays of Josephson junctions demonstrating coherent stimulated emission. From the viewpoint of nonlinear dynamics, the resonances between a Josephson junction and a resonator are of interest because of specificity of the former; its oscillation frequency is directly governed by control parameters of the system and can vary in a wide range. Our analytical results are in good agreement with the results of numerical simulations.
Author(s): Shunsuke Watanabe and Yoshiyuki Kabashima
In this study, we investigate the resilience of duplex networked layers (\alpha and \beta) coupled with antagonistic interlinks, each layer of which inhibits its counterpart at the microscopic level, changing the following factors: whether the influence of the initial failures in \alpha remains (que…[Phys. Rev. E] Published Thu Aug 04, 2016
Author(s): Li Chen, Cristián Huepe, and Thilo Gross
We consider a class of adaptive network models where links can only be created or deleted between nodes in different states. These models provide an approximate description of a set of systems where nodes represent agents moving in physical or abstract space, the state of each node represents the ag…[Phys. Rev. E] Published Thu Aug 04, 2016
Author(s): Lucas Illing
It is known that amplitude death can occur in networks of coupled identical oscillators if they interact via diffusive time-delayed coupling links. Here, we consider networks of oscillators that interact via direct time-delayed coupling links. It is shown analytically that amplitude death is impossi…[Phys. Rev. E] Published Thu Aug 04, 2016