Evolutionary games model a common type of interactions in a variety of complex, networked, natural systems and social systems. Given such a system, uncovering the interacting structure of the underlying network is key to understanding its collective dynamics. Based on compressive sensing, we develop an efficient approach to reconstructing complex networks under game-based interactions from small amounts of data. The method is validated by using a variety of model networks and by conducting an actual experiment to reconstruct a social network. While most existing methods in this area assume oscillator networks that generate continuous-time data, our work successfully demonstrates that the extremely challenging problem of reverse engineering of complex networks can also be addressed even when the underlying dynamical processes are governed by realistic, evolutionary-game type of interactions in discrete time. In many fields of science and engineering, one encounters the situation where the system of interest is composed of networked elements, called nodes, but the pattern of the node-to-node interaction or the network topology is totally unknown. It is desirable and of significant interest to uncover the network topology based on time series of certain observable quantities extracted from experiments or observations. Examples of potential applications abound: reconstruction of gene-regulatory networks based on expression data in systems biology [1][2][3][4], extraction of various functional networks in the human brain from activation data in neuroscience [5][6][7][8], and uncovering organizational networks based on discrete data or information in social science and homeland defense. In the past few years, the problem of network reconstruction has received growing attention [9][10][11][12][13][14][15][16]. Most existing works were based, however, on networks of oscillators whose dynamics are mathematically described by coupled, continuous differential equations. In particular, either some knowledge about the dynamical evolution of the underlying networked system is needed [9][10][11] or long, oscillatory signals in continuous time are required [12][13][14][15][16]. The advantage of availing oneself of continuous-time data is lost for networks in social, economic, and even biological sciences where node-to-node interactions are governed by evolutionary-game types of dynamics [17][18][19][20][21]. In addition to being discrete, the available data may be sporadic and the amount may be small. To our knowledge, the problem of reconstructing the full topology of a network based on discrete and ''rare'' data remains outstanding [22].In this paper, we articulate a general method of addressing the problem of how to uncover network topology using evolutionary-game data based on compressive sensing, a recently developed paradigm for sparse-signal reconstruction [23][24][25][26][27][28] with broad applications ranging from image compression/reconstruction to the analysis of large-scale sensor-network data. Although convex opti...
Complex dynamical networks consisting of a large number of interacting units are ubiquitous in nature and society. There are situations where the interactions in a network of interest are unknown and one wishes to reconstruct the full topology of the network through measured time series. We present a general method based on compressive sensing. In particular, by using power series expansions to arbitrary order, we demonstrate that the network-reconstruction problem can be casted into the form X = G • a, where the vector X and matrix G are determined by the time series and a is a sparse vector to be estimated that contains all nonzero power series coefficients in the mathematical functions of all existing couplings among the nodes. Since a is sparse, it can be solved by the standard L1-norm technique in compressive sensing. The main advantages of our approach include sparse data requirement and broad applicability to a variety of complex networked dynamical systems, and these are illustrated by concrete examples of model and real-world complex networks.
Strange nonchaotic attractors are attractors that are geometrically strange, but have nonpositive Lyapunov exponents. We show that for dynamical systems with an invariant subspace in which there is a quasiperiodic torus, the loss of the transverse stability of the torus can lead to the birth of a strange nonchaotic attractor. A physical phenomenon accompanying this route to strange nonchaotic attractors is an extreme type of intermittency. [S0031-9007(96)01861-3]
When a chaotic attractor lies in an invariant subspace, as in systems with symmetry, riddling can occur. Riddling refers to the situation where the basin of a chaotic attractor is riddled with holes that belong to the basin of another attractor. We establish properties of the riddling bifurcation that occurs when an unstable periodic orbit embedded in the chaotic attractor, usually of low period, becomes transversely unstable. An immediate physical consequence of the riddling bifurcation is that an extraordinarily low fraction of the trajectories in the invariant subspace diverge when there is a symmetry breaking. [S0031-9007(96)00503-0]
Recent work has considered the possibility of utilizing symbolic representations of controlled chaotic orbits for digital communication. We argue that dynamically a coding scheme usually leads to trajectories that live on a nonattracting but noise-resisting chaotic saddle embedded in the chaotic attractor. We present analyses and numerical computation which indicate that the channel capacity of the chaotic saddle has a devil-staircase-like behavior as a function of the noise-resisting strength. The implication is that nonlinear digital communication using chaos can yield a substantial channel capacity even in a noisy environment. [S0031-9007(97)04462-1]
The interplay between individual behaviors and epidemic dynamics in complex networks is a topic of recent interest. In particular, individuals can obtain different types of information about the disease and respond by altering their behaviors, and this can affect the spreading dynamics, possibly in a significant way. We propose a model where individuals' behavioral response is based on a generic type of local information, i.e., the number of neighbors that has been infected with the disease. Mathematically, the response can be characterized by a reduction in the transmission rate by a factor that depends on the number of infected neighbors. Utilizing the standard susceptible-infected-susceptible and susceptible-infected-recovery dynamical models for epidemic spreading, we derive a theoretical formula for the epidemic threshold and provide numerical verification. Our analysis lays on a solid quantitative footing the intuition that individual behavioral response can in general suppress epidemic spreading. Furthermore, we find that the hub nodes play the role of “double-edged sword” in that they can either suppress or promote outbreak, depending on their responses to the epidemic, providing additional support for the idea that these nodes are key to controlling epidemic spreading in complex networks.
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