P.T. Breznay (USA)

Network Flow, Dynamic Network Flow, Information Diffusion, Innovation and Meme Theory, Memetics.

In this paper we present a modelling paradigm motivated by the physical phenomenon of heat diffusion. The dif fusion equation, or heat equation, in physics describes the dynamics of dissipation of heat in 3-dimensional solid bod ies and many similar phenomena such as the dissipation of agents in solvents. We ﬁnd that the mathematical struc ture governing some important types of network ﬂows is paradigmatically equivalent to diffusion, with conceptual differences imposed by the role of the underlying network, which is a discrete graph as opposed to a continuous mate rial body. In modelling network ﬂows, the 3-dimensional second order partial differential equation that mathemat ically represents heat (and other) diffusion processes de composes to systems of particularly coupled ordinary dif ferential equations. The solutions of these systems of or dinary differential equations describe the evolution of net work ﬂow. The particular ways the equations are coupled are dictated by the graph structure that forms the topology of the network. Of particular interest are the behaviours of scale-free and Granovetter-type networks. In the frame work presented here we can model link strength by a con ductivity factor (or function, if it changes in time), and ob serve a ”clustering in the short term, convergence in the long term” behaviour of Granovetter and scale-free net works. Corresponding results are obtained for important regular and Watts-Strogatz-type ”small world” networks.

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