Thomas

Subway systems span most large cities, and railway networks most countries in the world. These networks are fundamental in the development of countries and their cities, and it is therefore crucial to understand their formation and evolution. However, if the topological properties of these networks are fairly well understood, how they relate to population and socio-economical properties remains an open question. We propose here a general coarse-grained approach, based on a cost-benefit analysis that accounts for the scaling properties of the main quantities characterizing these systems (the number of stations, the total length, and the ridership) with the substrate's population, area and wealth. More precisely, we show that the length, number of stations and ridership of subways and rail networks can be estimated knowing the area, population and wealth of the underlying region. These predictions are in good agreement with data gathered for about $140$ subway systems and more than $50$ railway networks in the world. We also show that train networks and subway systems can be described within the same framework, but with a fundamental difference: while the interstation distance seems to be constant and determined by the typical walking distance for subways, the interstation distance for railways scales with the number of stations.

We investigate the emergence and persistence of communities through a recently proposed mechanism of adaptive rewiring in coevolutionary networks. We characterize the topological structures arising in a coevolutionary network subject to an adaptive rewiring process and a node dynamics given by a simple voter-like rule. We find that, for some values of the parameters describing the adaptive rewiring process, a community structure emerges on a connected network. We show that the emergence of communities is associated to a decrease in the number of active links in the system, i.e. links that connect two nodes in different states. The lifetime of the community structure state scales exponentially with the size of the system. Additionally, we find that a small noise in the node dynamics can sustain a diversity of states and a community structure in time in a finite size system. Thus, large system size and/or local noise can explain the persistence of communities and diversity i...

Dynamic centrality metrics provide a quantitative assessment of the strength of communication between nodes in temporal networks, as well as the overall capacity of the network for the efficient transmission of information. In this article, the behaviours of two variants of the ‘communicability’ metric are examined in simple null models of uncorrelated temporal networks. Analysis of the long-time behaviour of the null models reveals a simple trade-off in the role of the parameters of the metric, suggesting methods to calibrate parameters and to adapt to temporal variations in the network properties. The null models introduced address two main classes of temporal networks (contact sequences and interval graphs), and their predictions are compared and contrasted with results coming from real-world telecommunications data.

This article proposes a network complexity metric based on the concepts of planarity and network communities. This work considers planarity as an analogue for network complexity. It compares the number of network communities that contain non-planar subgraph expansions with the total number of network communities, and considers the networks with less non-planar communities to be less complex. The metric is then used to analyse four networks, three representing real systems and one representing a more abstract system. Finally, the metric itself is analysed according to nine criteria for judging the effectiveness of a complexity metric. Using the network analyses and the effectiveness criteria, this metric is shown to be fast, versatile and effective and to fill a current gap in network tools for assessing the degree of planarity of a system.

We study synchronization of $N$ oscillators indirectly coupled through a medium which is inhomogeneous and has its own dynamics. The system is formalized in terms of a multilayer network, where the top layer is made of disconnected oscillators and the bottom one, modeling the medium, consists of oscillators coupled according to a given topology. The different dynamics of the medium and the top layer is accounted by including a frequency mismatch between them. We show a novel regime of synchronization as intra-layer coherence does not necessarily require inter-layer coherence. This regime appears under mild conditions on the bottom layer: arbitrary topologies may be considered, provided that they support synchronization of the oscillators of the medium. The existence of a density-dependent threshold as in quorum-sensing phenomena is also demonstrated.

Despite the tremendous advancements in the field of network theory, very few studies have taken weights in the interactions into consideration that emerge naturally in all real world systems. Using random matrix analysis of a weighted social network, we demonstrate the profound impact of weights in interactions on emerging structural properties. The analysis reveals that randomness existing in particular time frame affects the decisions of individuals rendering them more freedom of choice in situations of financial security. While the structural organization of networks remain same throughout all datasets, random matrix theory provides insight into interaction pattern of individual of the society in situations of crisis. It has also been contemplated that individual accountability in terms of weighted interactions remains as a key to success unless segregation of tasks comes into play.

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We derive rigorous bounds for well-defined community structure in complex networks for a stochastic block model (SBM) benchmark. In particular, we analyze the effect of inter-community "noise" (inter-community edges) on any "community detection" algorithm's ability to correctly group nodes assigned to a planted partition, a problem which has been proven to be NP complete in a standard rendition. Our result does not rely on the use of any one particular algorithm nor on the analysis of the limitations of inference. Rather, we turn the problem on its head and work backwards to examine when, in the first place, well defined structure may exist in SBMs.The method that we introduce here could potentially be applied to other computational problems. The objective of community detection algorithms is to partition a given network into optimally disjoint subgraphs (or communities). Similar to k-SAT and other combinatorial optimization problems, "community detection" exhibits different phases. Networks that lie in the "unsolvable phase" lack well-defined structure and thus have no partition that is meaningful. Solvable systems splinter into two disparate phases: those in the "hard" phase and those in the "easy" phase. As befits its name, within the easy phase, a partition is easy to achieve by known algorithms. When a network lies in the hard phase, it still has an underlying structure yet finding a meaningful partition which can be checked in polynomial time requires an exhaustive computational effort that rapidly increases with the size of the graph. When taken together, (i) the rigorous results that we report here on when graphs have an underlying structure and (ii) recent results concerning the limits of rather general algorithms, suggest bounds on the hard phase.

The entropy of network ensembles characterizes the amount of information encoded in the network structure, and can be used to quantify network complexity, and the relevance of given structural properties observed in real network datasets with respect to a random hypothesis. In many real networks the degrees of individual nodes are not fixed but change in time, while their statistical properties, such as the degree distribution, are preserved. Here we characterize the distribution of entropy of random networks with given degree sequences, where each degree sequence is drawn randomly from a given degree distribution. We show that the leading term of the entropy of scale-free network ensembles depends only on the network size and average degree, and that entropy is self-averaging, meaning that its relative variance vanishes in the thermodynamic limit. We also characterize large fluctuations of entropy that are fully determined by the average degree in the network. Finally, above a certain threshold, large fluctuations of the average degree in the ensemble can lead to condensation, meaning that a single node in a network of size~$N$ can attract $O(N)$ links.

One of the better studied topology metrics of complex networks is the second smallest eigenvalue of the Laplacian matrix of a network's graph, referred to as algebraic connectivity $\mu _{N-1}$. This spectral metric plays a decisive role in synchronization of coupled oscillators, network robustness, consensus problems, belief propagation, graph partitioning and distributed filtering in sensor networks. However, computing the graph spectra is computationally slow and its convergence greatly depends on the topology; thus a number of lower bounds have been proposed over the years in order to find good approximations. To date, the closest bound is the one proposed by Rad et al. (2011. Linear Algebra Appl., 435, 186–192) in 2009. The current paper proposes new approximations for the algebraic connectivity based on three variations of the betweenness centrality, a popular centrality score often used in social studies to characterize the importance of a node or link within a network. Based on numerical and a partly analytic analysis, we show that our approximations provide accurate lower bounds for the algebraic connectivity for a wide range of graphs, including random, power-law, small-world and lattice graphs. In particular, we numerically show that the average weighted Brandes betweenness can be treated as a lower bound for large enough networks, which greatly improves state-of-the-art bounds.

Author(s): Yong Zou, Tiago Pereira, Michael Small, Zonghua Liu, and Jürgen Kurths

Spontaneous explosive emergent behavior takes place in heterogeneous networks when the frequencies of the nodes are positively correlated to the node degree. A central feature of such explosive transitions is a hysteretic behavior at the transition to synchronization. We unravel the underlying mecha...

[Phys. Rev. Lett. 112, 114102] Published Tue Mar 18, 2014

Author(s): Jay M. Newby and Michael A. Schwemmer

The effects of noise on the dynamics of nonlinear systems is known to lead to many counterintuitive behaviors. Using simple planar limit cycle oscillators, we show that the addition of moderate noise leads to qualitatively different dynamics. In particular, the system can appear bistable, rotate in ...

[Phys. Rev. Lett. 112, 114101] Published Mon Mar 17, 2014

Publication date: 15 June 2014
Source:Physica A: Statistical Mechanics and its Applications, Volume 404
Author(s): J. Martín-Hernández , H. Wang , P. Van Mieghem , G. D’Agostino
The algebraic connectivity μN−1, i.e. the second smallest eigenvalue of the Laplacian matrix, plays a crucial role in dynamic phenomena such as diffusion processes, synchronization stability, and network robustness. In this work we study the algebraic connectivity in the general context of interdependent networks, or network-of-networks (NoN). The present work shows, both analytically and numerically, how the algebraic connectivity of NoNs experiences a transition. The transition is characterized by a saturation of the algebraic connectivity upon the addition of sufficient coupling links (between the two individual networks of a NoN). In practical terms, this shows that NoN topologies require only a fraction of coupling links in order to achieve optimal diffusivity. Furthermore, we observe a footprint of the transition on the properties of Fiedler’s spectral bisection.

We develop a thorough analytical study of the $O(1/N)$ correction to the spectrum of regular random graphs with $N \rightarrow \infty$ nodes. The finite size fluctuations of the resolvent are given in terms of a weighted series over the contributions coming from loops of all possible lengths, from which we obtain the isolated eigenvalue as well as an analytical expression for the $O(1/N)$ correction to the continuous part of the spectrum. The comparison between this analytical formula and direct diagonalization results exhibits an excellent agreement, confirming the correctness of our expression.

Author(s): Federico Battiston, Vincenzo Nicosia, and Vito Latora

Many real-world complex systems consist of a set of elementary units connected by relationships of different kinds. All such systems are better described in terms of multiplex networks, where the links at each layer represent a different type of interaction between the same set of nodes rather than ...

[Phys. Rev. E 89, 032804] Published Wed Mar 12, 2014

Memristors have uses as artificial synapses and perform well in this role in simulations with artificial spiking neurons. Our experiments show that memristor networks natively spike and can exhibit emergent oscillations and bursting spikes. Networks of near-ideal memristors exhibit behaviour similar to a single memristor and combine in circuits like resistors do. Spiking is more likely when filamentary memristors are used or the circuits have a higher degree of compositional complexity (i.e. a larger number of anti-series or anti-parallel interactions). 3-memristor circuits with the same memristor polarity (low compositional complexity) are stabilised and do not show spiking behaviour. 3-memristor circuits with anti-series and/or anti-parallel compositions show richer and more complex dynamics than 2-memristor spiking circuits. We show that the complexity of these dynamics can be quantified by calculating (using partial auto-correlation functions) the minimum order auto-regression function that could fit it. We propose that these oscillations and spikes may be similar phenomena to brainwaves and neural spike trains and suggest that these behaviours can be used to perform neuromorphic computation.

Microscale fluid flows generated by ensembles of beating eukaryotic flagella are crucial to fundamental processes such as development, motility and sensing. Despite significant experimental and theoretical progress, the underlying physical mechanisms behind this striking coordination remain unclear. Here, we present a novel series of experiments in which the flagellar dynamics of two micropipette-held somatic cells of Volvox carteri, with measurably different intrinsic beating frequencies, are studied by high-speed imaging as a function of their mutual separation and orientation. From analysis of beating time series we find that the interflagellar coupling, which is constrained by the lack of chemical and mechanical connections between the cells to be purely hydrodynamical, exhibits a spatial dependence that is consistent with theoretical predictions. At close spacings it produces robust synchrony which can prevail for thousands of flagellar beats, while at increasing separations this synchrony is systematically degraded by stochastic processes. Manipulation of the relative flagellar orientation reveals the existence of both in-phase and antiphase synchronized states, which is consistent with dynamical theories. Through dynamic flagellar tracking with exquisite spatio-temporal precision, we quantify changes in beating waveforms that result from altered coupling configuration and distance of separation. The experimental results of this study prove unequivocally that flagella coupled solely through a fluid medium can achieve robust synchrony despite significant differences in their intrinsic properties.

Recently, there has been considerable interest in the study of spontaneous synchronization, particularly within the framework of the Kuramoto model. The model comprises oscillators with distributed natural frequencies interacting through a mean-field coupling, and serves as a paradigm to study synchronization. In this review, we put forward a general framework in which we discuss in a unified way known results with more recent developments obtained for a generalized Kuramoto model that includes inertial effects and noise. We describe the model from a different perspective, highlighting the long-range nature of the interaction between the oscillators, and emphasizing the equilibrium and out-of-equilibrium aspects of its dynamics from a statistical physics point of view. In the first main section, we introduce the model and discuss for the noiseless and noisy dynamics and unimodal frequency distributions the synchronization transition that occurs in the stationary state. In the second section, we introduce the generalized model, and discuss its synchronization phase diagram for unimodal frequency distributions. In the third section, we describe deviations from the mean-field setting of the Kuramoto model by considering the generalized dynamics on a one-dimensional periodic lattice on the sites of which the oscillators reside and interact with a coupling that decays as an inverse power-law of their separation. For two specific cases, namely, in the absence of noise and inertia, and in the case when the natural frequencies are the same for all the oscillators, we discuss how the long-time transition to synchrony is governed by the dynamics of the mean-field mode (zero Fourier mode) of the spatial distribution of the oscillator phases.

Author(s): L. Timms and L. Q. English

We explore both analytically and numerically an ensemble of coupled phase oscillators governed by a Kuramoto-type system of differential equations. However, we have included the effects of time delay (due to finite signal-propagation speeds) and network plasticity (via dynamic coupling constants) in...

[Phys. Rev. E 89, 032906] Published Mon Mar 10, 2014

Author(s): Anders Nordenfelt, Alexandre Wagemakers, and Miguel A. F. Sanjuán

We study the synchronization and frequency distribution in networks of time-delayed Kuramoto oscillators with identical natural frequency. It is found that a pronounced frequency dispersion occurs in networks with nonidentical degree distributions. The deviation of the average network frequency from...

[Phys. Rev. E 89, 032905] Published Mon Mar 10, 2014

Birds in a flock move in a correlated way, resulting in large polarization of velocities. A good understanding of this collective behavior exists for linear motion of the flock. Yet observing actual birds, the center of mass of the group often turns giving rise to more complicated dynamics, still keeping strong polarization of the flock. Here we propose novel dynamical equations for the collective motion of polarized animal groups that account for correlated turning including solely social forces. We exploit rotational symmetries and conservation laws of the problem to formulate a theory in terms of generalized coordinates of motion for the velocity directions akin to a Hamiltonian formulation for rotations. We explicitly derive the correspondence between this formulation and the dynamics of the individual velocities, thus obtaining a new model of collective motion. In the appropriate overdamped limit we recover the well-known Vicsek model, which dissipates rotational information and does not allow for polarized turns. Although the new model has its most vivid success in describing turning groups, its dynamics is intrinsically different from previous ones in a wide dynamical regime, while reducing to the hydrodynamic description of Toner and Tu at very large length-scales. The derived framework is therefore general and it may describe the collective motion of any strongly polarized active matter system.

The analysis of the activity of neuronal cultures is considered to be a good proxy of the functional connectivity of in vivo neuronal tissues. Thus, the functional complex network inferred from activity patterns is a promising way to unravel the interplay between structure and functionality of neuronal systems. Here, we monitor the spontaneous self-sustained dynamics in neuronal clusters formed by interconnected aggregates of neurons. The analysis of the time delays between ignition sequences of the clusters allows the reconstruction of the directed functional connectivity of the network. We propose a method to statistically infer this connectivity and analyze the resulting properties of the associated complex networks. Surprisingly enough, in contrast to what has been reported for many biological networks, the clustered neuronal cultures present assortative mixing connectivity values, meaning that there is a preference for clusters to link to other clusters that share similar functional connectivity. These results point out that the grouping of neurons and the assortative connectivity between clusters are intrinsic survival mechanisms of the culture.

In a new type of percolation phase transition, which was observed in a set of non-equilibrium models, each new connection between vertices is chosen from a number of possibilities by an Achlioptas-like algorithm. This causes preferential merging of small components and delays the emergence of the percolation cluster. First simulations led to a conclusion that a percolation cluster in this irreversible process is born discontinuously, by a discontinuous phase transition, which results in the term "explosive percolation transition". We have shown that this transition is actually continuous (second-order) though with anomalously small critical exponent of the percolation cluster. Here we propose an efficient numerical method enabling us to find the critical exponents and other characteristics of this second order transition for a representative set of explosive percolation models with different number of choices. The method is based on sewing together the numerical solutions of evolution equations for the cluster size distribution and power-law asymptotics. For each of the models, with high precision, we obtain critical exponents and the critical point.

Driven by various kinds of noise, ensembles of limit cycle oscillators can synchronize. In this letter, we propose a general formulation of synchronization of the oscillator ensembles driven by common colored noise with an arbitrary power spectrum. To explore statistical properties of such colored noise-induced synchronization, we derive the stationary distribution of the phase difference between two oscillators in the ensemble. This analytical result theoretically predicts various synchronized and clustered states induced by colored noise and also clarifies that these phenomena have a different synchronization mechanism from the case of white noise.

For millimeter-scale aquatic crustaceans such as copepods, ensuring reproductive success is a challenge as potential mates are often separated by hundreds of body lengths in a 3D environment. At the evolutionary scale, this led to the development of remote sensing abilities and behavioral strategies to locate, to track, and to...

The most important driver of climate variability is the El Niño Southern Oscillation, which can trigger disasters in various parts of the globe. Despite its importance, conventional forecasting is still limited to 6 mo ahead. Recently, we developed an approach based on network analysis, which allows projection of an El...

Complex networks have been mostly characterized from the point of view of the degree distribution of their nodes and a few other motifs (or modules), with a special attention to triangles and cliques. The most exotic phenomena have been observed when the exponent γ of the associated power-law degree distribution is sufficiently small. In particular, a zero percolation threshold takes place for ##IMG## [http://ej.iop.org/images/0295-5075/105/2/28005/epl16046ieqn1.gif] {$\gamma

We propose a bare-bones stochastic model that takes into account both the geographical distribution of people within a country and their complex network of connections. The model, which is designed to give rise to a scale-free network of social connections and to visually resemble the geographical spread seen in satellite pictures of the Earth at night, gives rise to a power-law distribution for the ranking of cities by population size (but for the largest cities) and reflects the notion that highly connected individuals tend to live in highly populated areas. It also yields some interesting insights regarding Gibrat's law for the rates of city growth (by population size), in partial support of the findings in a recent analysis of real data [Rozenfeld et al., Proc. Natl. Acad. Sci. U.S.A. 105, 18702 (2008)]. The model produces a nontrivial relation between city population and city population density and a superlinear relationship between social connectivity and city population, both of which seem quite in line with real data.

We describe the emergence of the giant mutually connected component in networks of networks in which each node has a single replica node in any layer and can be interdependent only on its replica nodes in the interdependent layers. We prove that if in these networks, all the nodes of one network (layer) are interdependent on the nodes of the same other interconnected layer, then, remarkably, the mutually connected component does not depend on the topology of the network of networks. This component coincides with the mutual component of the fully connected network of networks constructed from the same set of layers, i.e., a multiplex network.

Recently much attention has been paid to the study of the robustness of interdependent and multiplex networks and, in particular, networks of networks. The robustness of interdependent networks can be evaluated by the size of a mutually connected component when a fraction of nodes have been removed from these networks. Here we characterize the emergence of the mutually connected component in a network of networks in which every node of a network (layer) $\alpha$ is connected with $q_{\alpha}$ randomly chosen replicas in some other networks and is interdependent of these nodes with probability $r$. We find that when the superdegrees $q_{\alpha}$ of different layers in the network of networks are distributed heterogeneously, multiple percolation phase transition can occur, and depending on the value of $r$ these transition are continuous or discontinuous.

Author(s): James P. Gleeson, Jonathan A. Ward, Kevin P. O’Sullivan, and William T. Lee

Heavy-tailed distributions of meme popularity occur naturally in a model of meme diffusion on social networks. Competition between multiple memes for the limited resource of user attention is identified as the mechanism that poises the system at criticality. The popularity growth of each meme is des...

[Phys. Rev. Lett. 112, 048701] Published Thu Jan 30, 2014