The emergence of self-organization in complex systems-Preface

The interest towards self-organized and cooperative systems has rapidly increased in recent years. This is largely due to the recent availability of large data-sets where the links among data can be easily described in terms of a network with complex link topology and structure. Important examples of complex networks are social networks (e.g., the web) and the -omics emerging in several research fields such as biology (proteomics, genomics, metabolic networks), neurophysiology (connectomics) and physics (condensed matter). This has triggered an increasing interest towards the dynamics of complex networks. This hot research field is sometimes denoted as complexity science. Actually, a definition of a complex system that is universally accepted by the scientific community does not yet exist, but there are some basic features that are recognized to characterize complex systems. Firstly, a complex system is multi-component, i.e., it is composed of many degrees of freedom: many individuals, particles, units or, in general, many sub-systems that are embedded in a network of strong nonlinear interactions. A complex system is often described as a network with a given set of nodes that interact by means of a set of links with complex topology. This aspect explains why statistical physics and network science have been the first research fields with a rapidly increasing interest towards complexity. It is worth noting that the presence of many degrees of freedom and nonlinearity is not sufficient to get a complex behavior. In a complex system the nonlinear dynamics must be cooperative, thus giving rise to the so-called emergent properties, and this seems to be the most peculiar and crucial feature characterizing complexity. Emergent properties are associated with the emergence of self-organized or coherent structures. These structures are states of the system whose temporal and spatial scale spectra (or, equivalently, long-range time-space correlations) cannot be represented as a simple function of external forcing or derived by the microscopic dynamics through some coarse graining procedure. In other words, it is not possible to link the large temporal, spatial and/or topological scales of self-organized structures with the correspondent small scales of the micro-dynamics. In summary, a multi-component system can be considered “complex” when its dynamics trigger the emergence of self-organized structures, so that the typical approach to complexity is focused on identifying self-organized structures, on their analysis and description, and on the modeling of their dynamical evolution.