Foundations of Complex Systems

The study of complex networks is a growing area of research with applications in multiple fields ranging from neuroscience through genetics, ecology, social anthropology, informatics, economy and energetics to climate research. A key principle in complex network research is viewing the system at hand as a network of interacting subsystems, with one of the central questions being that of estimating the pattern of mutual or causal interactions of these.

We propose a Cellular Automata (CA) model in which three ubiquitous and relevant processes in nature are present, namely, spatial competition, distinction between dynamically stronger and weaker agents and the existence of an inner resistance to changes in the actual state (=−1,0,+1) of each CA lattice cell n (which we call inertia) [1]. Considering ensembles of initial lattices, we study the average properties of the CA final stationary configuration structures resulting from the system time evolution.

One of the most studied problems in network science is the identification of those nodes that, once activated, maximize the fraction of nodes that are reached by a spreading process of interest. In parallel, scholars have introduced network effective distances as topological metrics to estimate the hitting time of diffusive spreading processes.

The robustness against inclusion of new elements is an essential feature of diverse kinds of real complex systems such as living organisms, ecosystems, social and economic systems, and complex engineering system with bottom-up design principle. A recently proposed simple model has revealed a general mechanism of determining the robustness of such systems against inclusion of elements with totally random interactions [1]. Under this mechanism, the system can grow only if the connection is moderately sparse.

League of Legends is a popular online video game where players battle against one another to climb in skill rank, like chess. Since a single player will encounter only five of approximately a million opponents in any single match, a player must use strategies that are better on average. They must have some understanding of which strategies will beat other strategies, similar to a social aggression hierarchy. Social aggression hierarchies have been well-studied in several species, such as primates, fish, parakeets, and insects [1].

Though interspecific competition profoundly changes community structure and structural stability (species stable coexistence), the current theory does not properly incorporate interspecific competition in mutualistic systems. Inspired by multilayer network theory, here we develop a framework that takes into account interspecific competition derived from and varied with shared mutualists (see Fig. 1 for a minimal model of the proposed framework).

Self-organization is a ubiquitous phenomenon in Nature and it is the expression of the intrinsic complexity of its internal dynamics. Examples range from the skin of some animals or simple bacteria till the more sophisticated dynamics of ecosystems. Among these, maybe the human relationships are one of the most intriguing cases to consider. Several authors have devoted interest to mathematically understand some aspects. In particular, linguistics has benefited from that research and some models have been proposed describing the interrelationships between competing languages.

The concept of emerging behaviours is often quoted within the literature of complex systems, and used as an attribute to assist with the description of these systems. The definition of what constitute an emerging behaviour has been debated in the literature, and we believe that it is, perhaps, the lack of mathematical definition of behaviour in the literature of complex systems what may be contributing to the confusion concerning the concept and even its need.

We consider a network which, under an initial attack, has a fraction of its nodes fail, and the redistribution of the betweenness centrality of the remaining nodes leads to subsequent nodes failing when it exceeds its initial value multiplied by a factor of tolerance . We study the possible disintegration of the network as a function of the size of the initial attack, and the nature of the transition through which the failure takes place, which is determined by the tolerance of the nodes, switching from first order to second when the tolerance increases.

In this work we analyse diffusion processes unfolding on a complex topology. We build an information theoretical framework with the aim of unveiling mesoscopic structures influencing the evolution of the dynamical system, such as core-periphery, community structure, or other types of (block) patterns. In particular, we take advantage of the mutual information between samples of a dynamics aggregated at the level of blocks drawn at every τ time steps, as a measure of dynamical alteration enforced by the projection to the block structure (see Figure A).