Non-equilibrium dynamics in complex networks [HBNI Th141]

Abstract

Complex systems comprising a large number of strongly interacting components that possess non-trivial dynamics are ubiquitous in nature, examples ranging from the immune system of organisms to ecological food. In the economic domain also we observe a variety of complex systems, such as financial markets and the network of international trade between nations. A common framework often used for describing the structure of such systems is that of networks, defined as a set of nodes (or vertices) that are connected with one another through links (or edges). For example, in the context of ecology, nodes are the different species and links are the possible predator-prey, competitive or mutualistic interactions between them. While the topological arrangement of the connections between the constituent parts of a complex system can often reveal fascinating insights, such an exclusively structural approach cannot often explain its dynamical behavior. Complex systems in nature are often subject to various types of environmental stimuli and perturbations that keep them far from equilibrium. To describe the behavior of such non-equilibrium systems we need to take recourse to a dynamical perspective. In this thesis, we look at the relation between the structural properties of a complex system and the features arising from the collective dynamics of its elements. We focus on understanding how robustness of a complex system (either in terms of survival of activity in its components or stability of the statistical properties of its dynamics) can arise as a result of the interaction between its different parts. We show that a set of concepts from statistical physics can provide a common toolbox for explaining features of widely different out-of-equilibrium complex systems, e.g., random walks for explaining movement of prices in markets as well as extinction of species in ecological communities. Similarly the concept of waiting time is useful when considering phenomena involving the intervals between successive extinctions in an ecological system or the duration between two successive transactions in a financial market.

Recent studies of percolation of failure processes in a system of two or more connected networks have suggested that interdependence makes the entire system fragile. However, a proper appraisal of the role of interdependence on the stability of complex systems necessarily needs to take into account the dynamical processes occurring on them. Using this novel approach we find that an optimally strong interdependence between networks can increase the robustness of the system in terms of its dynamical stability. We show that the system has a much higher likelihood of survival for an optimal interdependence, with both networks facing almost certain catastrophic collapse in isolation. Thus, interdependence need not always have negative repercussions. Instead its impact may depend strongly on the context, e.g., the nature of coupling and the type of dynamics being considered. The thesis also considers the stability-diversity debate, viz., the question of whether higher complexity (e.g., as a result of increasing the number of nodes, connection density or the range of interactions strengths in a network) makes a system more vulnerable to disturbances arising from small perturbations in the state variables. We view this question in the perspective of long-term dynamical evolution of many coupled dynamical elements where activity in each of the nodes may cease (corresponding to extinction) as a result of interaction with other nodes. We focus on the case when the average interaction strength of a node is inversely proportional to its in-degree (i.e., the total number of incoming connections from other nodes of the network). Such a reciprocal relation between strength and degree is motivated by empirical observations. We show that, the more connected a system, the longer is the duration of the transient period characterized by the network possessing a large fraction of active nodes. Thus, observations made in short timescales may well conclude that systems having more nodes (i.e., more diversity) and higher link density (i.e., more connected) will be more stable but at extremely long times, the results will show that more complex systems would tend to have lower number of active nodes surviving. Thus, we provide a novel resolution to the long-standing debate over the relation between complexity and stability by showing that the answer depends on the time-scale of observation. We also look at financial markets as examples of complex systems. Using high-frequency (HF) trade data from different intervals we show that the gross statistical properties of the market as a whole are in general stationary even though those of its constituents, i.e., individual stocks, are not. In particular we see that the distribution of transaction sizes (i.e., units of stock involved in a single transaction) when trades carried out in the entire market are considered does not change its nature over time. However, for individual stocks, this distribution can differ significantly between one period and another.