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Dynamical Networks for Smog Pattern Analysis  [PDF]
Linqi Zong,Xinyi Gong,Jia Zhu
Physics , 2015,
Abstract: Smog, as a form of air pollution, poses as a serious problem to the environment, health, and economy of the world[1-4] . Previous studies on smog mostly focused on the components and the effects of smog [5-10]. However, as the smog happens with increased frequency and duration, the smog pattern which is critical for smog forecast and control, is rarely investigated, mainly due to the complexity of the components, the causes, and the spreading processes of smog. Here we report the first analysis on smog pattern applying the model of dynamical networks with spontaneous recovery. We show that many phenomena such as the sudden outbreak and dissipation of smog and the long duration smog can be revealed with the mathematical mechanism under a random walk simulation. We present real-world air quality index data in accord with the predictions of the model. Also we found that compared to external causes such as pollution spreading from nearby, internal causes such as industrial pollution and vehicle emission generated on-site are more critical to the health of the whole network. Most strikingly, we demonstrate that spreading out the pollution sources by moving them to nearby suburban areas, a common practice for smog control, will degrade the overall health of system under certain conditions
Kernel Granger causality and the analysis of dynamical networks  [PDF]
Daniele Marinazzo,Mario Pellicoro,Sebastiano Stramaglia
Quantitative Biology , 2008, DOI: 10.1103/PhysRevE.77.056215
Abstract: We propose a method of analysis of dynamical networks based on a recent measure of Granger causality between time series, based on kernel methods. The generalization of kernel Granger causality to the multivariate case, here presented, shares the following features with the bivariate measures: (i) the nonlinearity of the regression model can be controlled by choosing the kernel function and (ii) the problem of false-causalities, arising as the complexity of the model increases, is addressed by a selection strategy of the eigenvectors of a reduced Gram matrix whose range represents the additional features due to the second time series. Moreover, there is no {\it a priori} assumption that the network must be a directed acyclic graph. We apply the proposed approach to a network of chaotic maps and to a simulated genetic regulatory network: it is shown that the underlying topology of the network can be reconstructed from time series of node's dynamics, provided that a sufficient number of samples is available. Considering a linear dynamical network, built by preferential attachment scheme, we show that for limited data use of bivariate Granger causality is a better choice w.r.t methods using $L1$ minimization. Finally we consider real expression data from HeLa cells, 94 genes and 48 time points. The analysis of static correlations between genes reveals two modules corresponding to well known transcription factors; Granger analysis puts in evidence nineteen causal relationships, all involving genes related to tumor development.
Stability Analysis and Control Synthesis for Dynamical Transportation Networks  [PDF]
Enrico Lovisari,Giacomo Como,Anders Rantzer,Ketan Savla
Mathematics , 2014,
Abstract: We study dynamical transportation networks in a framework that includes extensions of the classical Cell Transmission Model to arbitrary network topologies. The dynamics are modeled as systems of ordinary differential equations describing the traffic flow among a finite number of cells interpreted as links of a directed network. Flows between contiguous cells, in particular at junctions, are determined by merging and splitting rules within constraints imposed by the cells' demand and supply functions as well as by the drivers' turning preferences, while inflows at on-ramps are modeled as exogenous and possibly time-varying. First, we analyze stability properties of dynamical transportation networks. We associate to the dynamics a state-dependent dual graph whose connectivity depends on the signs of the derivatives of the inter-cell flows with respect to the densities. Sufficient conditions for the stability of equilibria and periodic solutions are then provided in terms of the connectivity of such dual graph. Then, we consider synthesis of control policies that use a combination of turning preferences, speed limits, and ramp metering, in order to optimize convex objectives. We first show that, in the general case, the optimal control synthesis problem can be cast as a convex optimization problem, and that the equilibrium of the controlled network is in free-flow. If the control policies are restricted to speed limits and ramp metering, then the resulting synthesis problem is still convex for networks where every node is either a merge or a diverge junction, and where the dynamics is monotone. These results apply both to the optimal selection of equilibria and periodic solutions, as well as to finite-horizon network trajectory optimization. Finally, we illustrate our findings through simulations on a road network inspired by the freeway system in southern Los Angeles.
Synergy and redundancy in the Granger causal analysis of dynamical networks  [PDF]
Sebastiano Stramaglia,Jesus M. Cortes,Daniele Marinazzo
Quantitative Biology , 2014, DOI: 10.1088/1367-2630/16/10/105003
Abstract: We analyze by means of Granger causality the effect of synergy and redundancy in the inference (from time series data) of the information flow between subsystems of a complex network. Whilst we show that fully conditioned Granger causality is not affected by synergy, the pairwise analysis fails to put in evidence synergetic effects. In cases when the number of samples is low, thus making the fully conditioned approach unfeasible, we show that partially conditioned Granger causality is an effective approach if the set of conditioning variables is properly chosen. We consider here two different strategies (based either on informational content for the candidate driver or on selecting the variables with highest pairwise influences) for partially conditioned Granger causality and show that depending on the data structure either one or the other might be valid. On the other hand, we observe that fully conditioned approaches do not work well in presence of redundancy, thus suggesting the strategy of separating the pairwise links in two subsets: those corresponding to indirect connections of the fully conditioned Granger causality (which should thus be excluded) and links that can be ascribed to redundancy effects and, together with the results from the fully connected approach, provide a better description of the causality pattern in presence of redundancy. We finally apply these methods to two different real datasets. First, analyzing electrophysiological data from an epileptic brain, we show that synergetic effects are dominant just before seizure occurrences. Second, our analysis applied to gene expression time series from HeLa culture shows that the underlying regulatory networks are characterized by both redundancy and synergy.
Associative networks with diluted patterns: dynamical analysis at low and medium load  [PDF]
Silvia Bartolucci,Alessia Annibale
Physics , 2014, DOI: 10.1088/1751-8113/47/41/415001
Abstract: In this work we solve the dynamics of pattern diluted associative networks, evolving via sequential Glauber update. We derive dynamical equations for the order parameters, that quantify the simultaneous pattern recall of the system, and analyse the nature and stability of the stationary solutions by means of linear stability analysis as well as Monte Carlo simulations. We investigate the parallel retrieval capabilities of the system in different regions of the phase space, in particular in the low and medium storage regimes and for finite and extreme pattern dilution. Results show that in the absence of patterns cross-talk, all patterns are recalled symmetrically for any temperature below criticality, while in the presence of pattern cross-talk, symmetric retrieval becomes unstable as temperature is lowered and a hierarchical retrieval takes over. The shape of the hierarchical retrieval occurring at zero temperature is provided. The parallel retrieval capabilities of the network are seen to degrade gracefully in the regime of strong interference, but they are not destroyed.
Earthquake aftershock networks generated on Euclidean spaces of different fractal geometry  [PDF]
Kamalika Basu Hajra,Parongama Sen
Physics , 2006,
Abstract: According to some recent analysis (M. Baiesi and M. Paczuski, Phys. Rev. E {\bf 69}, 066106, 2004 \cite{maya1}) of earthquake data, aftershock epicenters can be considered to represent the nodes of a network where the linking scheme depends on several factors. In the present paper a model network of earthquake aftershock epicenters is proposed based on this scheme and studied on fractals of different dimensions. The various statistical features of this network, like degree, link length, frequency and correlation distributions are evaluated and compared to the observed data. The results are also found to be independent of the fractal geometry.
The Asymptotic Synchronization Analysis for Two Kinds of Complex Dynamical Networks  [PDF]
Ze Tang,Jianwen Feng
Advances in Mathematical Physics , 2012, DOI: 10.1155/2012/309289
Abstract: We consider a class of complex networks with both delayed and nondelayed coupling. In particular, we consider the situation for both time delay-independent and time delay-dependent complex dynamical networks and obtain sufficient conditions for their asymptotic synchronization by using the Lyapunov-Krasovskii stability theorem and the linear matrix inequality (LMI). We also present some simulation results to support the validity of the theories. 1. Introduction A complex dynamical network is a large set of interconnected nodes that represent the individual elements of the system and their mutual relationships. Owing to their immense potential for applications to various fields, complex networks have been intensively investigated in the past decade in areas as diverse as mathematics, physics, biology, engineering, and even the social sciences [1–3]. The synchronization problem for complex networks was first posed by Saber and Murray [4, 5] who also introduced a theoretical framework for their investigation by viewing them as the adjustments of the rhythms of their interaction states [6] and many different kinds of synchronization phenomena and models have also been discovered such as complete synchronization, phase synchronization, lag synchronization, antisynchronization, impulsive synchronization, and projective synchronization. Time delays are an important consideration for complex networks although these were usually ignored in early investigations of synchronization and control problems [6–11]. To make up for this deficiency, uniformly distributed time delays have recently been incorporated into network models [12–25] and Wang et al. [18] even considered networks with both delayed and nondelayed couplings and obtained sufficient conditions for asymptotic stability. Similarly, Wu and Lu [19] investigated the exponential synchronization of general weighted delay and nondelay coupled complex dynamical networks with different topological structures. There remains, however, much room for improvement in both the scope of the systems considered by Wang and Xu as well as in their methods of proofs. The main contributions of this paper are two-fold. Firstly, we present a more general model for networks with both delayed and nondelayed couplings and derive criteria for their asymptotical synchronization. Secondly, we apply the Lyapunov-Krasovskii theorem and the LMIs to ensure the inevitable attainment of the required synchronization. The rest of the paper is organized as follows. In Section 2, we present the general complex dynamical network model under
Stability Analysis and Hadamard Synergic Control for a Class of Dynamical Networks  [PDF]
Xinjin Liu, Yun Zou
Intelligent Control and Automation (ICA) , 2010, DOI: 10.4236/ica.2010.11005
Abstract: Hadamard synergic control is a new kind of control problem which is achieved via a composite strategy of the state feedback control and the direct regulation of the part of connection coefficients of system state variables. Such a control is actually used very often in the practical areas. In this paper, we discuss Hadamard synergic stabilization problem for a class of dynamical networks. We analyze three cases: 1) Synergic stabilization problem for the general twonodenetwork. 2) Synergic stabilization problem for a special kind of networks. 3) Synergic stabilization problem for special kind of networks with communication timedelays. The mechanism of the synergic action between two control strategies: feedback control and the connection coefficients regulations are presented.
Topology Identification of Directed Dynamical Networks via Power Spectral Analysis  [PDF]
Shahin Shahrampour,Victor M. Preciado
Mathematics , 2013,
Abstract: We address the problem of identifying the topology of an unknown weighted, directed network of LTI systems stimulated by wide-sense stationary noises of unknown power spectral densities. We propose several reconstruction algorithms based on the cross-power spectral densities of the network's response to the input noises. Our first algorithm reconstructs the Boolean structure (i.e., existence and directions of links) of a directed network from a series of dynamical responses. Moreover, we propose a second algorithm to recover the exact structure of the network (including edge weights), as well as the power spectral density of the input noises, when an eigenvalue-eigenvector pair of the connectivity matrix is known (for example, Laplacian connectivity matrices). Finally, for the particular cases of nonreciprocal networks (i.e., networks with no directed edges pointing in opposite directions) and undirected networks, we propose specialized algorithms that result in a lower computational cost.
Signature stability analysis for networks of coupled dynamical systems with Hermitian Jacobian  [PDF]
Anne-Ly Do,Stefano Boccaletti,Jeremias Epperlein,Stefan Siegmund,Thilo Gross
Physics , 2012,
Abstract: The central theme of complex systems research is understanding the emergent macroscopic properties of a system from the interplay of its microscopic constituents. Here, we ask what conditions a complex network of microscopic dynamical units has to meet to permit stationary macroscopic dynamics, such as stable equilibria or phase-locked states. We present an analytical approach which is based on a graphical notation that allows rewriting Jacobi's signature criterion in an interpretable form. The derived conditions pertain to topological structures on all scales, ranging from individual nodes to the interaction network as a whole. Our approach can be applied to many systems of symmetrically coupled units. For the purpose of illustration, we consider the example of synchronization, specifically the (heterogeneous) Kuramoto model and an adaptive variant. Moreover, we discuss how the graphical notation can be employed to study isospectrality in Hermitian matrices. The results complete and extend the previous analysis of Do et al. [Phys. Rev. Lett. 108, 194102 (2012)].
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