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Proceedings of the National Academy of Sciences of the United States of America (20 May 2010)
Abstract
Synchronization, in which individual dynamical units keep in pace with each other in a decentralized fashion, depends both on the dynamical units and on the properties of the interaction network. Yet, the role played by the network has resisted comprehensive characterization within the prevailing paradigm that interactions facilitating pair-wise synchronization also facilitate collective synchronization. Here we challenge this paradigm and show that networks with best complete synchronization, least coupling cost, and maximum dynamical robustness, have arbitrary complexity but quantized total interaction ...
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Complex Sciences (2009), pp. 900-911.
Abstract
To study the effect of parameter mismatch on the stability in a general fashion, we derive variational equations to analyze the stability of synchronization for coupled near-identical oscillators. We define master stability equations and associated master stability functions, which are independent of the network structure. In particular, we present several examples of coupled near-identical Lorenz systems configured in small networks (a ring graph and sequence networks) with a fixed parameter mismatch and a large Barabasi-Albert scale-free network with random parameter mismatch. ...
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SIAM Journal on Applied Dynamical Systems, Vol. 8, No. 1. (2009), pp. 202-221.
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Europhys. Lett., Vol. 85, No. 6. (2009)
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PLoS Comput Biol, Vol. 4, No. 12. (5 December 2008), e1000236.
Abstract
Metabolic reactions of single-cell organisms are routinely observed to become dispensable or even incapable of carrying activity under certain circumstances. Yet, the mechanisms as well as the range of conditions and phenotypes associated with this behavior remain very poorly understood. Here we predict computationally and analytically that any organism evolving to maximize growth rate, ATP production, or any other linear function of metabolic fluxes tends to significantly reduce the number of active metabolic reactions compared to typical nonoptimal states. The reduced ...
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Phys. Rev. E, Vol. 78, No. 2. (2008), pp. 026104-8.
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Physica D, Vol. 224 (2006), pp. 77-89.
Abstract
We consider two optimization problems on synchronization of oscillator networks: maximization of synchronizability and minimization of synchronization cost. We first develop an extension of the well-known master stability framework to the case of non-diagonalizable Laplacian matrices.We then show that the solution sets of the two optimization problems coincide and are simultaneously characterized by a simple condition on the Laplacian eigenvalues. Among the optimal networks, we identify a subclass of hierarchical networks, characterized by the absence of feedback loops and the normalization ...
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Phys. Rev. E, Vol. 73 (2006)
Abstract
We consider maximization of the synchronizability of oscillator networks by assigning weights and directions to the links of a given interaction topology. By extending the master stability formalism to all possible network structures, we show that, unless some oscillator is linked to all the others, maximally synchronizable networks are necessarily nondiagonalizable and can always be obtained by imposing unidirectional information flow with normalized input strengths. The results provide insights into hierarchical structures observed in complex networks in which synchronization is important. ...
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Pramana, Vol. 64, No. 4. (1 April 2005), pp. 483-502.
Abstract
This paper presents a perspective in the study of complex networks by focusing on how dynamics may affect network security under attacks. In particular, we review two related problems: attack-induced cascading breakdown and range-based attacks on links. A cascade in a network means the failure of a substantial fraction of the entire network in a cascading manner, which can be induced by the failure of or attacks on only a few nodes. These have been reported for the internet and for ...
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Physica D: Nonlinear Phenomena, Vol. 197, No. 1-2. (1 October 2004), pp. 134-148.
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Physical Review Letters, Vol. 92, No. 10. (Mar 2004), 108101.
Abstract
Networks of coupled periodic oscillators (similar to the Kuramoto model) have been proposed as models of associative memory. However; error-free retrieval states of such oscillatory networks are typically unstable ; resulting in a near zero capacity. This puts the networks at disadvantage as compared with the classical Hopfield network. Here we propose a simple remedy for this undesirable property and show rigorously that the error-free capacity of our oscillatory; associative-memory networks can be made as high as that of the ...
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Complex Networks (2004), pp. 299-310.
Abstract
This paper reviews two problems in the security of complex networks: cascades of overload failures on nodes and range-based attacks on links. Cascading failures have been reported for numerous networks and refer to the subsequent failure of other parts of the network induced by the failure of or attacks on only a few nodes. We investigate a mechanism leading to cascades of overload failures in complex networks by constructing a simple model incorporating the flow of physical quantities in the network. ...
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Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 14, No. 1. (2004), pp. 72-78.
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Physical Review E, Vol. 68, No. 3. (Sep 2003), 036105.
Abstract
The characterization of large-scale structural organization of social networks is an important interdisciplinary problem. We show; by using scaling analysis and numerical computation; that the following factors are relevant for models of social networks: the correlation between friendship ties among people and the position of their social groups; as well as the correlation between the positions of different social groups to which a person belongs. ...
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Phys. Rev. Lett., Vol. 91 (July 2003)
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SIAM Journal on Applied Mathematics, Vol. 63, No. 5. (2003), pp. 1615-1626.
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Physical Review E, Vol. 66, No. 4. (Oct 2002), 046139.
Abstract
Efficiency in passage times is an important issue in designing networks; such as transportation or computer networks. The small-world networks have structures that yield high efficiency; while keeping the network highly clustered. We show that among all networks with the small-world structure; the most efficient ones have a “single center” node; from which all shortcuts are connected to uniformly distributed nodes over the network. The networks with several centers and a connected subnetwork of shortcuts are shown to be “almost” as ...
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Physical Review E, Vol. 65, No. 2. (Jan 2002), 026216.
Abstract
A small (but finite-size) spherical particle advected by fluid flows obeys equations of motion that are inherently dissipative; due to the Stokes drag. The dynamics of the advected particle can be chaotic even with a flow field that is simply time periodic. Similar to the case of ideal tracers; whose dynamics is Hamiltonian; chemical or biological activity involving such particles can be analyzed using the theory of chaotic dynamics. Using the example of an autocatalytic reaction; A + B →2 ...
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Phys. Rev. E, Vol. 66, No. 6. (Dec 2002)
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Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 12, No. 2. (2002), pp. 408-416.
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Physical Review Letters, Vol. 87, No. 3. (Jun 2001), 038301.
Abstract
We investigate the reaction kinetics of small spherical particles with inertia; obeying coalescence type of reaction; B + B → B ; and being advected by hydrodynamical flows with time-periodic forcing. In contrast to passive tracers; the particle dynamics is governed by the strongly nonlinear Maxey-Riley equations; which typically create chaos in the spatial component of the particle dynamics; appearing as filamental structures in the distribution of the reactants. Defining a stochastic description supported on the natural measure of the ...
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Nonlinearity, Vol. 13, No. 3. (2000), pp. 819-835.
Abstract
We analyse a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and taking modulo one in each component. We show that within this two-parameter family of maps, the set of non-invertible maps is open and dense. For cases where the entries in the matrix are rational we show that the maximal invariant set has positive Lebesgue measure and we give bounds on the measure. For several examples we ...
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Physics Letters A, Vol. 259, No. 5. (23 August 1999), pp. 377-386.
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Physical Review E, Vol. 54, No. 6. (Dec 1996), pp. 6114-6124.
Abstract
Fractalization of a torus and its transition to chaos in a quasiperiodically forced logistic map is reinvestigated in relation to a strange nonchaotic attractor; with the aid of a functional equation for the invariant curve. The existence of a fractal torus in an interval in parameter space is confirmed by the length and the number of extrema of the torus attractor; as well as the Fourier mode analysis. Mechanisms of the onset of a fractal torus and the transition to chaos ...
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