Social-circles network model
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The generative model of feedback networks studied in mathematics and physics by White, Kejžar, Tsallis, Farmer, and White, or social-circles network model, defines a class of random graphs generated by simple processes that are common to edge formation and feedback loops in social circles. This class is distinct from the small-world network and the scale-free network models but also captures many of the characteristics of real-world networks.[1] [2].
[edit] Rationale
A social-circles network models by a graph-generating process the formation of cohesion and dispersion in connected networks as a function of acitvity and feedback. The process begins with a single node. The next events in the evolution of the network are that: (1) a node is chosen randomly (proportional to the number of current links of each node raised to power alpha) to emit a feedback token, (2) the token attempts to travel by a non-circular path through the current network by choosing its next neighbor randomly, proportional to the number of edges of the neighbor minus the node already traversed, raised to power gamma; (3) the token travels through the network until it reaches a random distance d, proportional to 1 over d raised to a power beta, and failing that, the source node that originated the token gains an edge to a node that is added to the network (dispersion of edges), but if the token succeeds in reaching a target at distance d, a new 'feedback' edge is added between the source and the target.
The three parameters of the model are alpha, beta, gamma. The notion of 'feedback' in this model is very general and may involve intentionality on the part of source nodes as agents looking for a partner or simply nodes in a network emitting signals randomly that contain information that may enable the source and target to lock into a feedback relation, represented by the formation of a new edge. Whether searches or signals have the capacity to locate a target in an existing network is affected by the current size and topology of the network, the distance decay (beta) in traversal, and the intelligence (gamma) exhibited in the method of search. Failure to locate targets results in a generic substitution (new node and an edge connecting to it), like looking in the phone book for a doctor rather that asking for a referral.
[edit] Results
Social-circles network simulation with varying model parameters results in a variety of networks whose topology in terms of network density, clustering, cohesion, and degree distribution varies in ways that resemble the variety of real-world networks. Study of the degree distributions and edge-traversal frequencies shows fit to the statistical distributions of Tsallis entropy, familiar to nonextensive physics. This provides a baseline test of the model to real-world networks, one that contrasts with that of the scale-free network, for which one baseline test is whether the degree distribution follows a power law. The Tsallis entropy distribution asymptotes to a power law for large degree but bends toward the exponential distribution in the lower part of the degree distribution.
[edit] Bibliography
- "Generative Model for Feedback Networks" in Physical Review E, 016119 (2006, Douglas R. White, Nataša Kejžar, Constantino Tsallis, J. Doyne Farmer, Scott D. White). Reviewed 2005 in Europhysicsnews 36(6):218-220 by Stefan Thurner.
