Michael Small

Superfamily phenomena and motifs of networks induced from time series

We introduce a transformation from time series to complex networks and then study the relative frequency of different subgraphs within that network. The distribution of subgraphs can be used to distinguish between and to characterize different types of continuous dynamics: periodic, chaotic, and periodic with noise. Moreover, although the general types of dynamics generate networks belonging to the same superfamily of networks, specific dynamical systems generate characteristic dynamics. When applied to discrete (map-like) data this technique distinguishes chaotic maps, hyperchaotic maps, and noise data.

Recurrence-based time series analysis by means of complex network methods

Complex networks are an important paradigm of modern complex systems sciences which allows quantitatively assessing the structural properties of systems composed of different interacting entities. During the last years, intensive efforts have been spent on applying network-based concepts also for the analysis of dynamically relevant higher-order statistical properties of time series. Notably, many corresponding approaches are closely related with the concept of recurrence in phase space. In this paper, we review recent methodological advances in time series analysis based on complex networks, with a special emphasis on methods founded on recurrence plots. The potentials and limitations of the individual methods are discussed and illustrated for paradigmatic examples of dynamical systems as well as for real-world time series. Complex network measures are shown to provide information about structural features of dynamical systems that are complementary to those characterized by other methods of time series analysis and, hence, substantially enrich the knowledge gathered from other existing (linear as well as nonlinear) approaches.

Applied nonlinear time series analysis : applications in physics, physiology and finance

# Times Series Embedding and Reconstruction # Dynamics Measures and Topological Invariants # Estimation of Correlation Dimension # The Method of Surrogate Data # Non-Standard and Nonlinear Surrogates # Identifying the Dynamics # Applications

The impact of awareness on epidemic spreading in networks

We explore the impact of awareness on epidemic spreading through a population represented by a scale-free network. Using a network mean-field approach, a mathematical model for epidemic spreading with awareness reactions is proposed and analyzed. We focus on the role of three forms of awareness including local, global, and contact awareness. By theoretical analysis and simulation, we show that the global awareness cannot decrease the likelihood of an epidemic outbreak while both the local awareness and the contact awareness can. Also, the influence degree of the local awareness on disease dynamics is closely related with the contact awareness.

Complex network analysis of time series

Revealing complicated behaviors from time series constitutes a fundamental problem of continuing interest and it has attracted a great deal of attention from a wide variety of fields on account of its significant importance. The past decade has witnessed a rapid development of complex network studies, which allow to characterize many types of systems in nature and technology that contain a large number of components interacting with each other in a complicated manner. Recently, the complex network theory has been incorporated into the analysis of time series and fruitful achievements have been obtained. Complex network analysis of time series opens up new venues to address interdisciplinary challenges in climate dynamics, multiphase flow, brain functions, ECG dynamics, economics and traffic systems.

Pinning synchronization of delayed neural networks

This paper investigates adaptive pinning synchronization of a general weighted neural network with coupling delay. Unlike recent works on pinning synchronization which proposed the possibility that synchronization can be reached by controlling only a small fraction of neurons, this paper aims to answer the following question: Which neurons should be controlled to synchronize a neural network? By using Schur complement and Lyapunov function methods, it is proved that under a mild topology-based condition, some simple adaptive feedback controllers are sufficient to globally synchronize a general delayed neural network. Moreover, for a concrete neurobiological network consisting of identical Hindmarsh-Rose neurons, a specific pinning control technique is introduced and some numerical examples are presented to verify our theoretical results.

Applying the method of surrogate data to cyclic time series

The surrogate data methodology is used to test a given time series for membership of specific classes of dynamical systems. Currently, there are three algorithms that are widely applied in the literature. The most general of these tests the hypothesis of nonlinearly scaled linearly filtered noise. However, these tests and the many extensions of them that have been suggested are inappropriate for data exhibiting strong cyclic components. For such data it is more natural to ask if there exist any long term (of period longer than the data cycle length) determinism. In this paper we discuss existing techniques that attempt to address this hypothesis and introduce a new approach. This new approach generates surrogates that are constrained (i.e., they look like the data) and for cyclic time series tests the null hypothesis of a periodic orbit with uncorrelated noise. We examine various alternative implementations of this algorithm, applying it to a variety of known test systems and experimental time series with unknown dynamics.

SMALL WORLD AND SCALE FREE MODEL OF TRANSMISSION OF SARS

We model transmission of the Severe Acute Respiratory Syndrome (SARS) associated coronavirus (SARS-CoV) in Hong Kong with a complex small world network. Each node in the network is connected to its immediate neighbors and a random number of geographically isolated nodes. Transmission can only occur along these links. We find that this model exhibits dynamics very similar to those observed during the SARS outbreak in 2003. We derive an analytic expression for the rate of infection and confirm this expression with computational simulations. An immediate consequence of this quantity is that the severity of the SARS epidemic in Hong Kong in 2003 was due to ineffectual infection control in hospitals (i.e. nosocomial transmission). If all infectious individuals were isolated as rapidly as they were identified the severity of the outbreak would have been minimal.

Detecting determinism in time series: the method of surrogate data

We review a relatively new statistical test that may be applied to determine whether an observed time series is inconsistent with a specific class of dynamical systems. These surrogate data methods may test an observed time series against the hypotheses of: i) independent and identically distributed noise; ii) linearly filtered noise; and iii) a monotonic nonlinear transformation of linearly filtered noise. A recently suggested fourth algorithm for testing the hypothesis of a periodic orbit with uncorrelated noise is also described. We propose several novel applications of these methods for various engineering problems, including: identifying a deterministic (message) signal in a noisy time series; and separating deterministic and stochastic components. When employed to separate deterministic and noise components, we show that the application of surrogate methods to the residuals of nonlinear models is equivalent to fitting that model subject to an information theoretic model selection criteria.

Optimal embedding parameters: a modelling paradigm

The reconstruction of a dynamical system from a time series requires the selection of two parameters: the embedding dimension de and the embedding lag τ. Many competing criteria to select these parameters exist, and all are heuristic. Within the context of modelling the evolution operator of the underlying dynamical system, we show that one only need be concerned with the product deτ. We introduce an information theoretic criterion for the optimal selection of the embedding window dw = deτ. For infinitely long time series, this method is equivalent to selecting the embedding lag that minimises the nonlinear model prediction error. For short and noisy time series, we find that the results of this new algorithm are data-dependent and are superior to estimation of embedding parameters with the standard techniques.