Distribution dynamics of complex systems

Abstract

A complex system is defined as a system with many interdependent parts having emergent self-organization; analyzing and designing such complex systems is a new challenge. A common observable structure of many complex systems is the network, which is connections among nodes, and thus inherently difficult to describe. The goal of this research is to introduce an effective methodology to describe complex systems, and thus we will construct a population balance (distribution kinetics) model based on the association-dissociation process to describe the evolution of complex systems. Networks are commonly observed structures in complex systems with interdependent parts that self-organize. How networks come into existence and how they change with time are fundamental issues in numerous networked systems. Based on the nodal-linkage distribution, we propose a unified population dynamics approach for the network evolution. Size-independent rate coefficients yield an exponential network without preferential attachment, and size-dependent rate coefficients produce a power law network with preferential attachment. For nonlinearly growing networks, when the total number of connections increases faster than the total number of nodes, the network is said to accelerate. We propose a systematic model, a population dynamics model, for the dynamics of growing networks represented by distribution kinetics equations, and perform the moment calculations to describe the dynamics of such networks. Power law distributions have been observed in numerous physical and social systems; for example, the size distributions of particles and cities are often power laws. Each system is an ensemble of clusters, comprising units that combine with or dissociate from the cluster. To describe the growth of clusters, we hypothesize that a distribution obeys a governing population dynamics equation based on reversible association-dissociation processes. The rate coefficients considered to depend on the cluster size as power expressions provide an explanation for the asymptotic evolution of power law distributions. To mathematically represent human-initiated phenomena, which recently recognized as power law distributions, we apply the framework of cluster kinetics to the study of waiting-time distributions of human activities. The model yields both exponential and power law distributed systems, depending on the expressions for the rate coefficients in a Fokker-Planck equation.