Complexity, Criticality and Computation

Abstract

What makes a system "complex"? Is it merely the number of components it integrates, a nonlinear nature of the dependencies and feedbacks among its parts, or an unpredictable behavior it exhibits over time? The term "complexity" was initially applied generically to express the lack of predictability, reflecting on the self-organization of a synergistic macroscopic behavior out of interactions between the constituent microscopic parts, and the emergence of global patterns. Without a doubt, by now the concept has acquired a fairly definitive meaning, describing a distinct field of research and education and a new approach to science and engineering. There are abundant examples showing that the enterprise of Complex Systems, having achieved a substantial level of maturity, reaches back into our everyday lives, revealing patterns of complexity that should be considered without employing a reductionist logic [1].Similarly, the idea of criticality was originally motivated by studies of various crises and disruptive events, as well as sensitivities to initial conditions, but over time has developed into a precise field: critical dynamics. Research into critical dynamics is typically focused on the behavior of dynamical spatiotemporal systems during phase transitions where scale invariance prevails and symmetries break. Crucially, such behavior can be understood in terms of the control and order parameters. For instance, a second-order phase transition in a ferromagnetic system, separating two qualitatively different behaviors, can be reached by controlling the temperature parameter: the "disordered" and isotropic (symmetric) high-temperature phase is characterized by the absence of net magnetization, while the "ordered" and anisotropic (less symmetric) low-temperature phase can be described by an order parameter, the net magnetization vector defining a preferred direction in space. Critical phenomena have become associated with the physics of critical points, such as fractal behavior, the divergence of the correlation length, power-law divergences (e.g., the divergence of the magnetic susceptibility in the ferromagnetic phase transition), universality of relevant critical exponents, and so on. Now, these precise theoretical notions begin to reconnect with their motivating applied studies of crisis modeling, forecasting, and response. There is a growing awareness that complexity is strongly related to criticality, and many examples of self-organizing complex systems can be found in applications managing complexity specifically at critical regimes.A similar loop originating in practical studies, maturing to an exact science with precise but narrow definitions, and then reaching back to applied scenarios, can be seen in the realm of distributed computation. These days, complex systems can be viewed as distributed information-processing systems, in the domains ranging from systems biology and artificial life to computational neuroscience, digital circuitry, and transport networks [2]. Consciousness emerging from neuronal ac...