Author(s): Adrian van Kan, Jannes Jegminat, Jonathan F. Donges, and Jürgen Kurths
Transient dynamics are of large interest in many areas of science. Here, a generalization of basin stability (BS) is presented: constrained basin stability (CBS) that is sensitive to various different types of transients arising from finite size perturbations. CBS is applied to the paradigmatic Lore…
[Phys. Rev. E 93, 042205] Published Wed Apr 13, 2016
Author(s): T. L. Carroll and J. M. Byers
Stationary dynamical systems have invariant measures (or densities) that are characteristic of the particular dynamical system. We develop a method to characterize this density by partitioning the attractor into the smallest regions in phase space that contain information about the structure of the …
[Phys. Rev. E 93, 042206] Published Wed Apr 13, 2016
Spatial networks have recently attracted great interest in various fields of research. While the traditional network-theoretic viewpoint is commonly restricted to their topological characteristics (often disregarding existing spatial constraints), this work takes a geometric perspective, which considers vertices and edges as objects in a metric space and quantifies the corresponding spatial distribution and alignment. For this purpose, we introduce the concept of edge anisotropy and define a class of measures characterizing the spatial directedness of connections. Specifically, we demonstrate that the local anisotropy of edges incident to a given vertex provides useful information about the local geometry of geophysical flows based on networks constructed from spatio-temporal data, which is complementary to topological characteristics of the same flow networks. Taken both structural and geometric viewpoints together can thus assist the identification of underlying flow structures from observations of scalar variables.
Article
Flexible information routing underlies the function of many biological and artificial networks. Here, the authors present a theoretical framework that shows how information can be flexibly routed across networks using collective reference dynamics and how local changes may induce remote rerouting.
Nature Communications doi: 10.1038/ncomms11061
Authors: Christoph Kirst, Marc Timme, Demian Battaglia
One of the most celebrated findings in complex systems in the last decade is that different indexes y (e.g., patents) scale nonlinearly with the population~x of the cities in which they appear, i.e., $y\sim x^\beta, \beta \neq 1$. More recently, the generality of this finding has been questioned in studies using new databases and different definitions of city boundaries. In this paper we investigate the existence of nonlinear scaling using a probabilistic framework in which fluctuations are accounted explicitly. In particular, we show that this allows not only to (a) estimate $\beta$ and confidence intervals, but also to (b) quantify the evidence in favor of $\beta \neq 1$ and (c) test the hypothesis that the observations are compatible with the nonlinear scaling. We employ this framework to compare $5$ different models to $15$ different datasets and we find that the answers to points (a)-(c) crucially depend on the fluctuations contained in the data, on how they are modeled, and on the fact that the city sizes are heavy-tailed distributed.
In epidemiological modelling, dynamics on networks, and in particular adaptive and heterogeneous networks have recently received much interest. Here we present a detailed analysis of a previously proposed model that combines heterogeneity in the individuals with adaptive rewiring of the network structure in response to a disease. We show that in this model qualitative changes in the dynamics occur in two phase transitions. In a macroscopic description one of these corresponds to a local bifurcation whereas the other one corresponds to a non-local heteroclinic bifurcation. This model thus provides a rare example of a system where a phase transition is caused by a non-local bifurcation, while both micro- and macro-level dynamics are accessible to mathematical analysis. The bifurcation points mark the onset of a behaviour that we call network inoculation. In the respective parameter region exposure of the system to a pathogen will lead to an outbreak that collapses, but leaves the network in a configuration where the disease cannot reinvade, despite every agent returning to the susceptible class. We argue that this behaviour and the associated phase transitions can be expected to occur in a wide class of models of sufficient complexity.
Author(s): Reimer Kühn
We describe a method for disentangling giant component and finite cluster contributions to sparse random matrix spectra, using sparse symmetric random matrices defined on Erdős-Rényi graphs as an example and test bed. Our methods apply to sparse matrices defined in terms of arbitrary graphs in the c…
[Phys. Rev. E 93, 042110] Published Tue Apr 12, 2016
Author(s): Marc Wiedermann, Jonathan F. Donges, Jürgen Kurths, and Reik V. Donner
Networks with nodes embedded in a metric space have gained increasing interest in recent years. The effects of spatial embedding on the networks' structural characteristics, however, are rarely taken into account when studying their macroscopic properties. Here, we propose a hierarchy of null models…
[Phys. Rev. E 93, 042308] Published Tue Apr 12, 2016
Author(s): Justus A. Kromer, Lutz Schimansky-Geier, and Alexander B. Neiman
We study the emergence and coherence of stochastic oscillations in star networks of excitable elements in which peripheral nodes receive independent random inputs. A biophysical model of a distal branch of sensory neuron in which peripheral nodes of Ranvier are coupled to a central node by myelinate…
[Phys. Rev. E 93, 042406] Published Fri Apr 08, 2016
Author(s): Deokjae Lee, S. Choi, M. Stippinger, J. Kertész, and B. Kahng
Interdependent networks are more fragile under random attacks than simplex networks, because interlayer dependencies lead to cascading failures and finally to a sudden collapse. This is a hybrid phase transition (HPT), meaning that at the transition point the order parameter has a jump but there are…
[Phys. Rev. E 93, 042109] Published Fri Apr 08, 2016
The study of networks plays a crucial role in investigating the structure, dynamics, and function of a wide variety of complex systems in myriad disciplines. Despite the success of traditional network analysis, standard networks provide a limited representation of complex systems, which often include different types of relationships (i.e., "multiplexity") among their constituent components and/or multiple interacting subsystems. Such structural complexity has a significant effect on both dynamics and function. Throwing away or aggregating available structural information can generate misleading results and be a major obstacle towards attempts to understand complex systems.
The recent "multilayer" approach for modeling networked systems explicitly allows the incorporation of multiplexity and other features of realistic systems. On one hand, it allows one to couple different structural relationships by encoding them in a convenient mathematical object. On the other hand, it also allows one to couple different dynamical processes on top of such interconnected structures. The resulting framework plays a crucial role in helping achieve a thorough, accurate understanding of complex systems. The study of multilayer networks has also revealed new physical phenomena that remain hidden when using ordinary graphs, the traditional network representation. Here we survey progress towards attaining a deeper understanding of spreading processes on multilayer networks, and we highlight some of the physical phenomena related to spreading processes that emerge from multilayer structure.
The recently proposed generalized epidemic modeling framework (GEMF) \cite{sahneh2013generalized} lays the groundwork for systematically constructing a broad spectrum of stochastic spreading processes over complex networks. This article builds an algorithm for exact, continuous-time numerical simulation of GEMF-based processes. Moreover the implementation of this algorithm, GEMFsim, is available in popular scientific programming platforms such as MATLAB, R, Python, and C; GEMFsim facilitates simulating stochastic spreading models that fit in GEMF framework. Using these simulations one can examine the accuracy of mean-field-type approximations that are commonly used for analytical study of spreading processes on complex networks.
We study phase locking in the Kuramoto model of coupled oscillators in the special case where the number of oscillators, $N$, is large but finite, and the oscillators' natural frequencies are evenly spaced on a given interval. In this case, stable phase-locked solutions are known to exist if and only if the frequency interval is narrower than a certain critical width, called the locking threshold. For infinite $N$, the exact value of the locking threshold was calculated 30 years ago; however, the leading corrections to it for finite $N$ have remained unsolved analytically. Here we derive an asymptotic formula for the locking threshold when $N \gg 1$. The leading correction to the infinite-$N$ result scales like either $N^{-3/2}$ or $N^{-1}$, depending on whether the frequencies are evenly spaced according to a midpoint rule or an endpoint rule. These scaling laws agree with numerical results obtained by Paz\'{o} [Phys. Rev. E 72, 046211 (2005)]. Moreover, our analysis yields the exact prefactors in the scaling laws, which also match the numerics.
Uncertain recognition success, unfavorable scaling of connection complexity or dependence on complex external input impair the usefulness of current oscillatory neural networks for pattern recognition or restrict technical realizations to small networks. We propose a new network architecture of coupled oscillators for pattern recognition which shows none of the mentioned aws. Furthermore we illustrate the recognition process with simulation results and analyze the new dynamics analytically: Possible output patterns are isolated attractors of the system. Additionally, simple criteria for recognition success are derived from a lower bound on the basins of attraction.
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The collective dynamics in populations of magnetic spin torque oscillators (STO) is an intensely studied topic in modern magnetism. Here, we show that arrays of STO coupled via dipolar fields can be modeled using a variant of the Kuramoto model, a well-known mathematical model in non-linear dynamics. By investigating the collective dynamics in arrays of STO we find that the synchronization in such systems is a finite size effect and show that the critical coupling-for a complete synchronized state-scales with the number of oscillators. Using realistic values of the dipolar coupling strength between STO we show that this imposes an upper limit for the maximum number of oscillators that can be synchronized. Further, we show that the lack of long range order is associated with the formation of topological defects in the phase field similar to the two-dimensional XY model of ferromagnetism. Our results shed new light on the synchronization of STO, where controlling the mutual synchronization of several oscillators is considered crucial for applications.
Human flourishing is often severely limited by persistent violence. Quantitative conflict research has found common temporal and other statistical patterns in warfare, but very little is understood about its general spatial patterns. While the importance of topology in geostrategy has long been recognised, the role of spatial patterns of cities in determining a region's vulnerability to conflict has gone unexplored. Here, we show that global patterns in war and peace are closely related to the relative position of cities in a global interaction network. We find that regions with betweenness centrality above a certain threshold are often engulfed in entrenched conflict, while a high degree correlates with peace. In fact, betweenness accounts for over 80% of the variance in number of attacks. This metric is also a good predictor of the distance to a conflict zone and can estimate the risk of conflict. We conjecture that a high betweenness identifies areas with fuzzy cultural boundaries, whereas high degree cities are in cores where peace is more easily maintained. This is supported by a simple agent-based model in which cities influence their neighbours, which exhibits the same threshold behaviour with betweenness as seen in conflict data. These findings not only shed new light on the causes of violence, but could be used to estimate the risk associated with actions such as the merging of cities, construction of transportation infrastructure, or interventions in trade or migration patterns.
The study of networks plays a crucial role in investigating the structure, dynamics, and function of a wide variety of complex systems in myriad disciplines. Despite the success of traditional network analysis, standard networks provide a limited representation of complex systems, which often include different types of relationships (i.e., "multiplexity") among their constituent components and/or multiple interacting subsystems. Such structural complexity has a significant effect on both dynamics and function. Throwing away or aggregating available structural information can generate misleading results and be a major obstacle towards attempts to understand complex systems.
The recent "multilayer" approach for modeling networked systems explicitly allows the incorporation of multiplexity and other features of realistic systems. On one hand, it allows one to couple different structural relationships by encoding them in a convenient mathematical object. On the other hand, it also allows one to couple different dynamical processes on top of such interconnected structures. The resulting framework plays a crucial role in helping achieve a thorough, accurate understanding of complex systems. The study of multilayer networks has also revealed new physical phenomena that remain hidden when using ordinary graphs, the traditional network representation. Here we survey progress towards attaining a deeper understanding of spreading processes on multilayer networks, and we highlight some of the physical phenomena related to spreading processes that emerge from multilayer structure.
The study of networks plays a crucial role in investigating the structure, dynamics, and function of a wide variety of complex systems in myriad disciplines. Despite the success of traditional network analysis, standard networks provide a limited representation of these systems, which often includes different types of relationships (i.e., "multiplexity") among their constituent components and/or multiple interacting subsystems. Such structural complexity has a significant effect on both dynamics and function. Throwing away or aggregating available structural information can generate misleading results and provide a major obstacle towards attempts to understand the system under analysis. The recent "multilayer' approach for modeling networked systems explicitly allows the incorporation of multiplexity and other features of realistic networked systems. On one hand, it allows one to couple different structural relationships by encoding them in a convenient mathematical object. On the other hand, it also allows one to couple different dynamical processes on top of such interconnected structures. The resulting framework plays a crucial role in helping to achieve a thorough, accurate understanding of complex systems. The study of multilayer networks has also revealed new physical phenomena that remained hidden when using the traditional network representation of graphs. Here we survey progress towards a deeper understanding of dynamical processes on multilayer networks, and we highlight some of the physical phenomena that emerge from multilayer structure and dynamics.
The collective dynamics in populations of magnetic spin torque oscillators (STO) is an intensely studied topic in modern magnetism. Here, we show that arrays of STO coupled via dipolar fields can be modeled using a variant of the Kuramoto model, a well-known mathematical model in non-linear dynamics. By investigating the collective dynamics in arrays of STO we find that the synchronization in such systems is a finite size effect and show that the critical coupling-for a complete synchronized state-scales with the number of oscillators. Using realistic values of the dipolar coupling strength between STO we show that this imposes an upper limit for the maximum number of oscillators that can be synchronized. Further, we show that the lack of long range order is associated with the formation of topological defects in the phase field similar to the two-dimensional XY model of ferromagnetism. Our results shed new light on the synchronization of STO, where controlling the mutual synchronization of several oscillators is considered crucial for applications.
Author(s): P. Van Mieghem
A general two-layer network consists of two networks G1 and G2, whose interconnection pattern is specified by the interconnectivity matrix B. We deduce desirable properties of B from a dynamic process point of view. Many dynamic processes are described by the Laplacian matrix Q. A regular topologica…
[Phys. Rev. E 93, 042305] Published Thu Apr 07, 2016
Author(s): Kay Jörg Wiese
In renormalized field theories there are in general one or few fixed points that are accessible by the renormalization-group flow. They can be identified from the fixed-point equations. Exceptionally, an infinite family of fixed points exists, parameterized by a scaling exponent ζ, itself a function…
[Phys. Rev. E 93, 042105] Published Wed Apr 06, 2016
Network models with latent geometry have been used successfully in many applications in network science and other disciplines, yet it is usually impossible to tell if a given real network is geometric, meaning if it is a typical element in an ensemble of random geometric graphs. Here we identify structural properties of networks that guarantee that random graphs having these properties are geometric. Specifically we show that random graphs in which expected degree and clustering of every node are fixed to some constants are equivalent to random geometric graphs on the real line, if clustering is sufficiently strong. Large numbers of triangles, homogeneously distributed across all nodes as in real networks, are thus a consequence of network geometricity. The methods we use to prove this are quite general and applicable to other network ensembles, geometric or not, and to certain problems in quantum gravity.
In stochastic systems with weak noise, the logarithm of the stationary distribution becomes proportional to a large deviation rate function called the quasi-potential. The quasi-potential, and its characterization through a variational problem, lies at the core of the Freidlin–Wentzell large deviations theory (Freidlin and Wentzell, Random perturbations of dynamical systems, 2012). In many interacting particle systems, the particle density is described by fluctuating hydrodynamics governed by Macroscopic Fluctuation Theory (Bertini et al., arXiv:1404.6466, 2014), which formally fits within Freidlin–Wentzell’s framework with a weak noise proportional to \(1/\sqrt{N}\) , where N is the number of particles. The quasi-potential then appears as a natural generalization of the equilibrium free energy to non-equilibrium particle systems. A key physical and practical issue is to actually compute quasi-potentials from their variational characterization for non-equilibrium systems for which detailed balance does not hold. We discuss how to perform such a computation perturbatively in an external parameter \(\lambda \) , starting from a known quasi-potential for \(\lambda =0\) . In a general setup, explicit iterative formulae for all terms of the power-series expansion of the quasi-potential are given for the first time. The key point is a proof of solvability conditions that assure the existence of the perturbation expansion to all orders. We apply the perturbative approach to diffusive particles interacting through a mean-field potential. For such systems, the variational characterization of the quasi-potential was proven by Dawson and Gartner (Stochastics 20:247–308, 1987; Stochastic differential systems, vol 96, pp 1–10, 1987). Our perturbative analysis provides new explicit results about the quasi-potential and about fluctuations of one-particle observables in a simple example of mean field diffusions: the Shinomoto–Kuramoto model of coupled rotators (Prog Theoret Phys 75:1105–1110, [74]). This is one of few systems for which non-equilibrium free energies can be computed and analyzed in an effective way, at least perturbatively.
We investigate the influence that adding a new coupling has on the linear stability of the synchronous state in coupled oscillators networks. Using a simple model we show that, depending on its location, the new coupling can lead to enhanced or reduced stability. We extend these results to electric power grids where a new line can lead to four different scenarios corresponding to enhanced or reduced grid stability as well as increased or decreased power flows. Our analysis shows that the Braess paradox may occur in any complex coupled system, where the synchronous state may be weakened and sometimes even destroyed by additional couplings.
The present work introduces an efficient Monte Carlo algorithm for continuum percolation composed of randomly-oriented rectangles. By conducting extensive simulations, we report high precision percolation thresholds for a variety of homogeneous systems with different rectangle aspect ratios. This work verifies and extends the excluded area theory. It is confirmed that percolation thresholds are dominated by the average excluded areas for both homogeneous and heterogeneous rectangle systems (except for some special heterogeneous systems where the rectangle lengths differ too much from one another). In terms of the excluded areas, generalized formulae are proposed to effectively predict precise percolation thresholds for all these rectangle systems. This work is therefore helpful for both practical applications and theoretical studies concerning relevant systems.
The Erratum addresses errors in our earlier paper "Percolation thresholds of two-dimensional continuum systems of rectangles" published in [Phys. Rev. E 88, 012101 (2013)].
In this work we extend previous analyses of linguistic networks by adopting a multi-layer network framework for modelling the human mental lexicon, i.e. an abstract mental repository where words and concepts are stored together with their linguistic patterns. Across a three-layer linguistic multiplex, we model English words as nodes and connect them according to (i) phonological similarities, (ii) synonym relationships and (iii) free word associations. Our main aim is to exploit this multi-layered structure to explore the influence of phonological and semantic relationships on lexicon assembly over time. We propose a model of lexicon growth which is driven by the phonological layer: words are suggested according to different orderings of insertion (e.g. shorter word length, highest frequency, semantic multiplex features) and accepted or rejected subject to constraints. We then measure times of network assembly and compare these to empirical data about the age of acquisition of words. In agreement with empirical studies in psycholinguistics, our results provide quantitative evidence for the hypothesis that word acquisition is driven by features at multiple levels of organisation within language.
A problem closely related to epidemiology, where a subgraph of 'infected' links is defined inside a larger network, is investigated. This subgraph is generated from the underlying network by a random variable, which decides whether a link is able to propagate a disease/information. The relaxation timescale of this random variable is examined in both annealed and quenched limits, and the effectiveness of propagation of disease/information is analyzed. The dynamics of the model is governed by a master equation and two types of underlying network are considered: one is scale-free and the other has exponential degree distribution. We have shown that the relaxation timescale of the contagion variable has a major influence on the topology of the subgraph of infected links, which determines the efficiency of spreading of disease/information over the network.
We investigate the influence that adding a new coupling has on the linear stability of the synchronous state in coupled oscillators networks. Using a simple model we show that, depending on its location, the new coupling can lead to enhanced or reduced stability. We extend these results to electric power grids where a new line can lead to four different scenarios corresponding to enhanced or reduced grid stability as well as increased or decreased power flows. Our analysis shows that the Braess paradox may occur in any complex coupled system, where the synchronous state may be weakened and sometimes even destroyed by additional couplings.
Discovering communities in complex networks helps to understand the behaviour of the network. Some works in this promising research area exist, but communities uncovering in time-dependent and/or multiplex networks has not deeply investigated yet. In this paper, we propose a communities detection approach for multislice networks based on modularity optimization. We first present a method to reduce the network size that still preserves modularity. Then we introduce an algorithm that approximates modularity optimization (as usually adopted) for multislice networks, thus finding communities. The network size reduction allows us to maintain acceptable performances without affecting the effectiveness of the proposed approach.
We investigate critical behaviors of a social contagion model on weighted networks. An edge-weight compartmental approach is applied to analyze the weighted social contagion on strongly heterogenous networks with skewed degree and weight distributions. We find that degree heterogeneity can not only alter the nature of contagion transition from discontinuous to continuous but also can enhance or hamper the size of adoption, depending on the unit transmission probability. We also show that, the heterogeneity of weight distribution always hinder social contagions, and does not alter the transition type.