Narrow Results

Dramatic advances in the field of complex networks have been witnessed in the past few years. This paper reviews some important results in this direction of rapidly evolving research, with emphasis on the relationship between the dynamics and the topology of complex networks. Basic quantities and typical examples of various complex networks are described; and main network models are introduced, including regular, random, small-world and scale-free models. The robustness of connectivity and the epidemic dynamics in complex networks are also evaluated. To that end, synchronization in various dynamical networks are discussed according to their regular, small-world and scale-free connections.

This paper formulates a new approach to the study of chaos in discrete dynamical systems based on the notions of inverse ill-posed problems, set-valued mappings, generalized and multivalued inverses, graphical convergence of a net of functions in an extended multifunction space [Sengupta & Ray, 2000] and the topological theory of convergence. Order, chaos and complexity are described as distinct components of this unified mathematical structure that can be viewed as an application of the theory of convergence in topological spaces to increasingly nonlinear mappings, with the boundary between order and complexity in the topology of graphical convergence being the region in (Multi(X)) that is susceptible to chaos. The paper uses results from the discretized spectral approximation in neutron transport theory [Sengupta, 1988, 1995] and concludes that the numerically exact results obtained by this approximation of the Case singular eigenfunction solution is due to the graphical convergence of the Poisson and conjugate Poisson kernels to the Dirac delta and the principal value multifunctions respectively. In (Multi(X)), the continuous spectrum is shown to reduce to a point spectrum, and we introduce a notion of latent chaotic states to interpret superposition over generalized eigenfunctions. Along with these latent states, spectral theory of nonlinear operators is used to conclude that nature supports complexity to attain efficiently a multiplicity of states that otherwise would remain unavailable to it.

In this paper, we aim to identify a directed complex network with optimal controllability, for which pinning control is to be applied. Since the controllability of a network can be reflected by the smallest nonzero eigenvalue of a matrix related to its topology, an optimal network design problem is formulated based on the maximization of this eigenvalue. To better consider the practical reality, constraints on node degree sequence are specified. Based on the derived bounds of the eigenvalue, an effective rewiring scheme is designed and solutions close to or equal to the upper bound are obtained. Finally, the relationship between network characteristics and controllability is studied. Through complexity analysis, it is concluded that the network with high controllability should possess two properties, i.e. nodes with high out-degree for pinning and other nodes with uniform degree distribution.

This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie–Poisson bracket of rank two. This Lie–Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.

Characterizing accurately chaotic behaviors is not a trivial problem and must allow to determine the properties that two given chaotic invariant sets share or not. The underlying problem is the classification of chaotic regimes, and their labeling. Addressing these problems corresponds to the development of a dynamical taxonomy, exhibiting the key properties discriminating the variety of chaotic behaviors discussed in the abundant literature. Starting from the hierarchy of chaos initially proposed by one of us, we systematized the description of chaotic regimes observed in three- and four-dimensional spaces, which cover a large variety of known (and less known) examples of chaos. Starting with the spectrum of Lyapunov exponents as the first taxonomic ranks, we extended the description to higher ranks with some concepts inherited from topology (bounding torus, surface of section, first-return map, …).

By treating extensively the Rössler and the Lorenz attractors, we extended the description of branched manifold — the highest known taxonomic rank for classifying chaotic attractor — by a linking matrix (or linker) to multicomponent attractors (bounded by a torus whose genus $g\ge 3$).

We prove that there is a sequence of purely imaginary parameter values ${\lambda }_n$ converging to $0$ such that the Julia set for $z\mapsto {\lambda }_n(z+1/z)$ is homeomorphic to the Sierpiński carpet fractal; however, for any distinct pair of such parameter values, the dynamics of the map restricted to the Julia set are not conjugate.

Based on analyses of specific spatial networks, we compare the accuracy of three models in capturing topologies of two types of spatial networks: electronic circuits and brain networks. The models analyzed are an optimization model trading off multiple-objective constraints, an extended preferential attachment model with spatial constraints, and the generalized random graph model. First, we find that the optimization model and the spatial preferential attachment model can generate similar topological structures under appropriate parameters. Second, our experiments surprisingly show that the simple generalized random graph model outperforms the two proposed models. Third, we find that a series of spatial networks under global optimization of wire length, including the electronic circuits, brain networks, neuronal networks and transportation networks, have high s-metric values close to those of the corresponding generalized random graph models. These s-metric observations explain why the generalized random graph model can match the electronic circuits and the brain networks well from a probabilistic viewpoint, and distinguish their structures from self-organized spatial networks, such as the Internet.

Social conventions are useful self-sustaining protocols for groups to coordinate behavior without a centralized entity enforcing coordination. The emergence of such conventions in different multi agent network topologies has been investigated by several researchers, although exploring only specific cases of the convention emergence process. In this work we will provide multi-dimensional analysis of several factors that we believe determines the process of convention emergence, such as: the size of agents memory, the population size and structure, the learning approach taken by agents, the amount of players in the interactions, or the convention search space dimension. Although we will perform an exhaustive study of different network structures, we are concerned that different topologies will affect the emergence in different ways. Therefore, the main research question in this work is comparing and studying effects of different topologies on the emergence of social conventions. While others have investigated memory for learning algorithms, the effects of memory on the reward have not been investigated thoroughly. We propose a reward metric that is derived directly from the history of the interacting agents. Another research question to be answered is what effect does the history based reward function and the learning approach have on convergence time in different topologies. Experimental results show that all the factors analyzed affect differently the convention emergence process, being such information very useful for policy-makers when designing self-regulated systems.