We developed a general method to detect hidden nodes in complex networks, using only time series from nodes that are accessible to external observation. Our method is based on compressive sensing and we formulate a general framework encompassing continuous and discrete-time, and evolutionary-game type of dynamical systems as well. For concrete demonstration, we present an example of detecting hidden nodes from an experimental social network. Our paradigm for detecting hidden nodes is expected to find applications in a variety of fields where identifying hidden or black-boxed objects based on limited amount of data is of interest.
R.-Q. Su, W.-X. Wang, and Y.-C. Lai, ``Detecting hidden nodes in complex networks from time series,'' Physical Review E 85, 065201(R) (2012).
(2012) Effect of chaos on relativistic quantum tunneling
We solved the Dirac equation in two spatial dimensions in the setting of resonant tunneling, where the system consists of two symmetric cavities connected by a finite potential barrier. The shape of the cavities can be chosen to yield both regular and chaotic dynamics in the classical limit. We find that certain pointer states about classical periodic orbits can exist, which suppress quantum tunneling, and the effect becomes less severe as the underlying classical dynamics in the cavity is chaotic, leading to regularization of tunneling dynamics even in the relativistic quantum regime. Similar phenomena have been observed in graphene. A physical theory is developed to explain the phenomenon based on the spectrum of complex eigenenergies of the non-Hermitian Hamiltonian describing the effectively open cavity system.
X. Ni, L. Huang, Y.-C. Lai and L. M. Pecora, ``Effect of chaos on relativistic quantum tunneling,'' Europhysics Letters 98, 50007 (2012).
(2012) Controlling complex networks: how much energy is needed?
The outstanding problem of controlling complex networks is relevant to many areas of science and engineering, and has the potential to generate technological breakthroughs as well. We addressed the physically important issue of the energy required for achieving control by deriving and validating scaling laws for the lower and upper energy bounds. These bounds represent a reasonable estimate of the energy cost associated with control, and provide a step forward from the current research on controllability toward ultimate control of complex networked dynamical systems.
G. Yan, J. Ren, Y.-C. Lai, C. H. Lai, and B. Li, ``Controlling complex networks: How much energy is needed?'' Physical Review Letters 108, 218703 (2012).
(2004) A Trick to Remember: Capacity of oscillatory associative-memory networks with error-free retrieval
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 \emph{unstable}, resulting in a near zero capacity. This puts the networks at disadvantage as compared with the classical Hopfield network. We recently proposed 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 Hopfield network. They can thus not only provide insights into the origin of biological memory, but can also be potentially useful for applications in information science and engineering.
T. Nishikawa, Y.-C. Lai, and F. C. Hoppensteadt, ``Capacity of oscillatory associative-memory networks with error-free retrieval,'' Physical Review Letters92, 108101(1-4) (2004). PDF file This research was featured in Physical Review Focus (March 12, 2004).
(2004) Strange nonchaotic attractors in random dynamical systems
Whether strange nonchaotic attractors (SNAs) can occur typically in dynamical systems other than quasiperiodically driven systems has long been an open question. We have recently shown, based on a physical analysis and numerical evidence, that robust SNAs can be induced by small noise in autonomous discrete-time maps and in periodically driven continuous-time systems. These attractors, which are relevant to physical and biological applications, can thus be expected to occur more commonly in dynamical systems than previously thought.
X. Wang, M. Zhan, C.-H. Lai, and Y.-C. Lai, ``Strange nonchaotic attractors in random dynamical systems,'' Physical Review Letters 92, 074102(1-4) (2004). PDF file
(2003) Superpersistent chaotic transients in physical space
Superpersistent chaotic transients are characterized by an exponential-like scaling law for their lifetimes where the exponent in the exponential dependence diverges as a parameter approaches a critical value. So far this type of transient chaos has been illustrated exclusively in the phase space of dynamical systems. Here we report the phenomenon of noise-induced superpersistent transients in the physical space and explain the associated scaling law based on the solutions to a class of stochastic differential equations. The context of our study is advective dynamics of inertial particles in open chaotic flows. Our finding makes direct experimental observation of superpersistent chaotic transients feasible and it also has implications to problems of current concern such as the transport and trapping of chemically or biologically active particles in large scale flows.
Y. Do and Y.-C. Lai, ``Superpersistent chaotic transients in physical space - advective dynamics of inertial particles in open chaotic flows under noise,'' Physical Review Letters 91, 224101(1-4) (2003). PDF file
(2003) Inability of Lyapunov exponents to predict epileptic seizures
Concepts from nonlinear dynamics such as Lyapunov exponents and fractal dimensions have been increasingly applied to significant biomedical time series for detection, prediction, or control of diseases. In the area of epilepsy, a great deal of effort has been devoted to monitoring the evolution of Lyapunov exponents for early seizure prediction, with claims that seizures can be predicted minutes or even tens of minutes in advance of their clinical onset. Our central concern is that there has been no systematic analysis about the predictive power of Lyapunov exponents from nonstationary time series such as ECoG in the existing literature. Without such analysis, any claim of prediction can be misleading. We have recently investigated the predictive power of the Lyapunov exponents computed from time series. Our analysis and computations indicate that there are two major factors that can prevent the exponents from being effective tools to predict characteristic system changes: statistical fluctuations and noise. The basic message is that for low-dimensional, deterministic chaotic systems, the predictive power of Lyapunov exponents holds only in noiseless or extremely low-noise situations. In reality where an appreciable but reasonable amount of noise is present, especially in a system as high-dimensional and noisy as the brain, it is unlikely that the Lyapunov exponents, or related quantities from nonlinear dynamics, can be used to predict or even to detect epileptic seizures. We feel that this is an important point to keep in mind in light of repeated claims in the literature of seizure prediction with Lyapunov exponents. It is also clear that our result is relevant to many other applications where the temporal evolutions of the Lyapunov exponents estimated from time series are intended for prediction or detection.
Y.-C. Lai, M. A. F. Harrison, M. G. Frei, and I. Osorio, ``Inability of Lyapunov exponents to predict epileptic seizures,'' Physical Review Letters 91, 068102 (2003). PDF file This work was selected by the Virtual Journal of Biological Physics Research for the August 15, 2003 issue See here
(2003) Heterogeneity in oscillator networks: Are smaller worlds easier to synchronize?
Small-world networks are known to be more easily synchronized than regular lattices, which is usually attributed to the smaller network distance between oscillators. Surprisingly, we show that networks with a homogeneous distribution of connectivity are more synchronizable than heterogeneous ones (e.g., scale-free networks), even though the average network distance is larger. Some degree of homogeneity is then expected in naturally evolved structures, such as neural networks, where synchronizability is desirable.
T. Nishikawa, A. E. Motter, Y.-C. Lai, and F. Hoppensteadt, ``Heterogeneity in oscillator networks: Are smaller worlds easier to synchronize?'' Physical Review Letters 91, 014101 (2003). PDF file
(2003) Cascade-based attacks on complex networks
We live in a modern world supported by large, complex networks. Examples range from financial markets to communication and transportation systems. In many realistic situations the flow of physical quantities in the network, as characterized by the loads on nodes, is important. We show that for such networks where loads can redistribute among the nodes, intentional attacks can lead to a cascade of overload failures, which can in turn cause the entire or a substantial part of the network to collapse. This is relevant for real-world networks that possess a highly heterogeneous distribution of {\it loads}, such as the Internet and power grids. We demonstrate that the heterogeneity of these networks makes them particularly vulnerable to attacks in that a large-scale cascade can be triggered by disabling a single key node. This brings obvious concerns on the security of such systems.
A. E. Motter and Y.-C. Lai, ``Cascade-based attacks on complex networks,'' Physical Review E (Rapid Communications) 66, 065102 (2002). PDF file This research was featured in NewsFactor Network (February 6, 2003) PS file
(2002) Can statistical averages of chaotic systems be computed or measured reliably?
Computing or measuring statistical averages is a common practice in science and engineering. A fundamental question concerns the reliability of such computations or measurements. The question, which is of paramount importance for quantitative sciences such as physics and biology, is particularly relevant when the system under investigation exhibits chaos, as characterized by a sensitive dependence on initial conditions. For chaotic systems, numerical trajectories are not always shadowable by true trajectories. However, for statistical averages, it is naturally assumed that they can be computed or measured reliably because, the effect of computational errors or noise would be averaged out due to ergodicity. Is this intuition always true?
We have recently shown that there are common situations in chaotic dynamics where statistical averages of physical quantities depend on the level of computational error or noise. We have derived a universal scaling law governing this noisy dependence obtained experimental evidence with a chaotic electronic circuit. The implication can be quite intriguing. For example, in computational science, if a different precision or a different computer is used, the computed values of statistical averages may be different. In a laboratory experiment, measurements performed under nonidentical circumstances may yield different results.
Y.-C. Lai, Z. Liu, W.-G. Wei, and C.-H. Lai, ``Shadowability of statistical averages in chaotic systems,'' Physical Review Letters 89, 184101 (2002). PDF file
(2002) Topology of the conceptual network of languages
Any language is composed of many thousands of words linked together in an apparently fairly sophisticated way. A language can thus be regarded as a network, in the following sense: (1) the words correspond to nodes of the network, and (2) a link exists between two words if they express similar concepts. Clearly, the underlying network of a language is necessarily sparse in the sense that the average number of links per node is typically much smaller than the total number of nodes. Identifying and understanding the common network topology of languages is of great importance, not only for the study of languages themselves, but also for cognitive science where one of the most fundmental issues concerns associative memory that is intimately related to the network topology.
The network of connections among the many thousands of words that make up a language is important not only for the study of the structure and evolution of languages, but also for cognitive science. We have studied this issue quantitatively by mapping out the conceptual network of the English language, with the connections being defined by the entries in a Thesaurus dictionary. We find that this network presents a small-world structure, with an amazingly small average shortest path, and appears to exhibit an asymptotic scale-free feature with algebraic connectivity distribution.
From the standpoint of retrieval of information in an associative memory, the small-world property of the network represents a maximization of efficiency: on the one hand, similar pieces of information are stored together, due to the high clustering, which makes searching by association possible; on the other hand, even very different pieces of information are never separated by more than a few links, or associations, which guarantees a fast search. We thus speculate that associative memory has arisen partly because of a maximization of efficiency in the retrieval by natural selection. This issue may be related to the fact that the neural network is probably a small-world network as well which is probably necessary for the brain to be able to hold a conceptual network which is needed for associative memory.
A. E. Motter, A. P. S. de Moura, Y.-C. Lai, and P. Dasgupta, ``Language, small-worlds, and cognition,'' Physical Review E (Rapid Communications) 65, 065102(1-4) (2002). PDF file
This research was recently featured in New Scientists (July 13, page 22) PS file AND Nature Science Update See here or PS file AND Wissenschaft in German See here or English Translation AND in newspapers in Europe, Brazil, and China.
This work was also selected by the Virtual Journal of Biological Physics Research for the July 1, 2002 issue See here
(2002) Noise-induced enhancement of chemical reactions in nonlinear flows
The interplay between noise and nonlinear dynamics has long been a topic of tremendous interest in statistical physics. While noise can be detrimental in many situations, it can also be beneficial through, for example, the mechanisms of stochastic and coherence resonances. Recently, a new area of interdisciplinary science emerges: active processes in nonlinear flows. Such processes can be chemical or biological, and are believed to be relevant to a large number of important problems in a variety of areas. Our work focuses on the role of noise in active nonlinear processes. In particular, motivated by the problem of ozone production in atmospheres of urban areas, we investigate how noise influence a general type of chemical reaction, supported on a chaotic flow. To be as realistic as possible, we take into consideration important physical effects such as particle inertia and finite size. Our finding is that noise can enhance the rate of chemical reaction, in a manner similar to that of stochastic resonance. We provide numerical results and also a physical theory, suggesting that at a fundamental level, the resonant behavior is due to the interaction between noise and nonlinearity of the particle (Lagrangian) dynamics.
Z. Liu, Y.-C. Lai, and J. M. Lopez, ``Noise-induced enhancement of chemical reactions in chaotic flows,'' Chaos 12, 417-425 (2002). PDF file
This research recently appeared in Physics News Updates of American Institute of Physics See here
(2002/2003) Transition to chaos in random dynamical systems
The problem of noise-induced chaos is fundamental to understanding the interplay between stochasticity and nonlinearity, which is important for a variety of fields. We consider situations where in a continuous-time dynamical system, a nonchaotic attractor coexists with a nonattracting chaotic saddle, as in a periodic window. Under the influence of noise, chaos can arise. We investigate the fundamental dynamical mechanism responsible for the transition and obtain a general scaling law for the largest Lyapunov exponent. A striking finding is that the topology of the flow is fundamentally disturbed after the onset of noisy chaos, and we point out that such a disturbance is due to changes in the number of unstable eigendirections along a continuous trajectory under the influence of noise.
Z. Liu, Y.-C. Lai, L. Billings, and I. B. Schwartz, ``Transition to chaos in continuous-time random dynamical systems,'' Physical Review Letters 88, 124101 (2002). PDF file
B. Xu, Y.-C. Lai, L. Zhu, and Y. Do, ``Experimental characterization of transition to chaos in random dynamical systems,'' Physical Review Letters 90, 164101 (2003). PDF file
(2002) Tunneling and nonhyperbolicity in quantum dots
Electronic transport in semiconductor nanostructures is a frontier problem in condensed matter physics and nonlinear science. On sub-micron scales, quantum interference gives rise to phenomena as conductance fluctuations and the Aharonov-Bohm effect. We have recently reported our finding concerning the fundamental role played by dynamical tunneling in conductance fluctuations in realistic quantum dots. We present strong evidence that dynamical tunneling through regular phase-space structures, such as Kolmogorov-Arnold-Moser (KAM) islands, fundamentally determines the characteristics of conductance fluctuations in typical quantum dots. Theoretical analysis based on the tunneling mechanism gives quantitative predictions (the average frequency of the fluctuations) in excellent agreement with experimental measurements. To our knowledge, this is the first time that the major characteristics of experimentally observable conductance fluctuations in quantum dots are explained in a quantitative way, indicating the fundamental importance of dynamical tunneling in the transport dynamics of electrons in semiconductor nanostructures.
A. P. S. de Moura, Y.-C. Lai, R. Akis, J. Bird, and D. K. Ferry, ``Tunneling and nonhyperbolicity in quantum dots,'' Physical Review Letters 88, 236804(1-4) (2002). PDF file
(2002) Tracking epileptic seizures
An outstanding problem in biomedical sciences is to devise techniques to understand and, more importantly, to predict in advance of clinical onset, epileptic seizures which affect about $1\%$ of the population in industrialized countries. Epileptic seizures are characterized electrographically by sudden simultaneous changes in power spectral density and increases in wave rhythmicity. These changes in brain activity, whether local or global, can be monitored via electrodes on the scalp (EEG), or intracranially (ECoG). These recordings provide a window, perhaps the only practically accessible window at present, through which the dynamics of epilepsy can be investigated. Analysis of EEG/ECoG has thus become the subject of renewed interest in this field.
We focus on an anomalous scaling region in correlation-integral analysis of electrocorticogram (ECoG) in epilepsy patients. We find that epileptic seizures typically are accompanied by wide fluctuations in the slope of this scaling region. An explanation, based on analyzing the interplay between the autocorrelation and $C(\epsilon)$, is provided for these fluctuations. This anomalous slope appears to be a sensitive measure for tracking (but not predicting) seizures. Our study suggests that the underlying dynamical process contains a significant stochastic component. Though epileptic seizures are characterized by fluctuations in the value of the anomalous slope, the fluctuations correspond to those of the autocorrelation, a more computationally efficient measure for seizure tracking. Thus, it is questionable whether any correlation-integral based techniques can be more effective at predicting seizures than traditional signal processing methods. This is apparently in sharp contrast to the recent claims that such techniques are powerful for prediction of seizures.
Y.-C. Lai, I. Oserio, M. A. Harrison, and M. Frei, ``Correlation-dimension and autocorrelation fluctuations in seizure dynamics,'' Physical Review E 65, 031921(1-5) (2002). PDF file
This work was selected by the Virtual Journal of Biological Physics Research for the March 15, 2002 issue See here
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