Attracting Complex Networks

Abstract

Real phenomena from different areas of Life Sciences can be described by complex networks, whose structure is usually determining their intrinsic dynamics. On the other hand, Dynamical Systems Theory is a powerful tool for the study of evolution processes in real situations. The concept of global attractor is the central one in this theory. In the last decades there has been an intensive research in the geometrical characterization of global attractors. However, there still exists a weak connection between the asymptotic dynamics of a complex network and the structure of associated global attractors. In this paper we show that, in order to analyze the long-time behavior of the dynamics on a complex network, it is the topological and geometrical structure of the attractor the subject to take into account. In fact, given a complex network, a global attractor can be understood as the new attracting complex network which is really describing and determining the forwards dynamics of the phenomena. We illustrate our discussion with models of differential equations related to mutualistic complex networks in Economy and Ecology.

Dynamical informational structures characterize the different human brain states of wakefulness and deep sleep

“…Thus, G A is identified with a directed graph, so that if G A = {Ξ 1 , ..., Ξ n } with Ξ i disjoint invariant subsets (typically, stationary points or periodic orbits 45,[50][51][52] , but it could also contain invariant sets with chaotic dynamics 32,33,53 ), then each Ξ i corresponds to a vertex and a directed connection between Ξ i and Ξ j exists if and only if they are linked by solutions of the DS, in the sense described in the paragraph above. The resulting directed graph is called the Informational Structure (IS), a network describing all the possible future evolution 14,15,18,19 .…”

“…and its characterization is, to some extent, well understood 57,69 . The informational structure of this GA can be computed and, under some conditions, there exists a finite number of equilibria and directed connections among them that create a hierarchical organization with several levels of equilibria sets or energy levels ordered by connections in a gradient-like fashion 18,19 .…”

“…The informational structure of the GA is defined as a directed graph composed of nodes associated with those invariants and links establishing their connections (see Methods and Supplementary Information for a formal rigorous definition). Informational Structure has been used to show the dependence between the topology, the value of the parameters and the state with respect to its level of integration 14 , as such pointing for the small world configuration of the brain 16 (although see recent controversies 17 ).Previous research has applied Informational Structure to population dynamics in complex networks 18,19 . Equally used for modelling mutualistic systems in Theoretical Ecology and Economy 20, 21 , Informational Structure can relate the topology of complex networks and their dynamics.…”

“…Previous research has applied Informational Structure to population dynamics in complex networks 18,19 . Equally used for modelling mutualistic systems in Theoretical Ecology and Economy 20, 21 , Informational Structure can relate the topology of complex networks and their dynamics.…”

Finite-time synchronization of uncertain complex dynamic networks with time-varying delay

“…This system is considered as a complex network, and the units that compose the entire system is regarded as the nodes of the complex network. The introduction of a complex network model provides a huge convenience for people to solve many practical problems and has been extensively investigated [1][2][3]. Synchronization, as a major feature of complex networks, has attracted considerable attention.…”