Artificial Life
Sync: The Emerging Science of Spontaneous Order
Sync: The Emerging Science of Spontaneous Order
Small Worlds: The Dynamics of Networks between Order and Randomness
Small Worlds: The Dynamics of Networks between Order and Randomness
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In this paper we present a modelling paradigm motivated by the physical phenomenon of heat diffusion. The diffusion equation, or heat equation, in physics describes the dynamics of dissipation of heat in 3-dimensional solid bodies and many similar phenomena such as the dissipation of agents in solvents. We find that the mathematical structure 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 material body. In modelling network flows, the 3-dimensional second order partial differential equation that mathematically represents heat (and other) diffusion processes decomposes to systems of particularly coupled ordinary differential equations. The solutions of these systems of ordinary differential equations describe the evolution of network 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 framework presented here we can model link strength by a conductivity factor (or function, if it changes in time), and observe a "clustering in the short term, convergence in the long term" behaviour of Granovetter and scale-free networks. Corresponding results are obtained for important regular and Watts-Strogatz-type "small world" networks.