Connectivity and Criticality

Criticality is a phenomenon in which a system exhibits sudden phase changes. Examples include water freezing, crystallization, and epidemic processes. Associated with every critical phenomenon is an order parameter, and the phase change occurs when the order parameter reaches a critical value. For example, water freezes, when its temperature falls to 0 ° C. A wildfire spreads when fuel moisture falls below a critical level (else it dies out).

Changes in the connectivity of a network have important consequences and often underlie critical phenomena. When a network is formed by adding edges at random to a set of N nodes, a connectivity avalanche occurs when the number of edges is approximately N/2. This avalanche is characterized by the formation of a connected subnet, called a unique giant component (UGC), which contains most of the nodes in the full network. The formation of the UGC marks a phase change in which the network shifts rapidly from being disconnected to connected.

Any system that can be identified with nodes and edges forms a network, so the connectivity avalanche occurs in many settings and is the usual mechanism underlying critical phase changes.

The connectivity avalanche has several important implications. For interacting systems, it means that the group behaves either as disconnected individuals, or as a connected whole. Either global properties emerge, or they do not: there is usually very little intermediate behavior. Landscape connectivity provides an important ecological example of critical phase change.

Phase changes in connectivity also underlie criticality in system behavior. The degree of connectivity between states of a system determines the richness of its behavior. Studies based on automata theory show that if connectivity is too low, systems become static or locked in narrow cycles. If connectivity is too high, systems behave chaotically. The transition between these two phases is a critical region, popularly known as the 'edge of chaos'. It has been observed that automata whose state spaces lie in this critical region exhibit the most interesting behavior. This observation led researchers such as James Crutchfield, Christopher Langton, and Stuart Kauffman to suggest that automata need to reside in the critical region to perform universal computation. More speculative is their suggestion that the edge of chaos is an essential requirement for evolvability (the ability to evolve) in complex systems, including living things. Others have proposed that living systems exploit chaos as a source of novelty, and that they evolve to lie near the edge of chaos. These ideas are closely related to self-organized criticality (SOC).

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