David Sumpters: How to make a reinforced random walk solve transport problems

From a biological perspective, I will present recent work on biological transportation networks, looking at how ants, acellular slime mould and other biological systems build efficient transportation networks. From a mathematical point of view, these systems are examples of 'current' reinforced random walks. In a current reinforced random walk the flow of particles produces a feedback which reduces resistance to further flow. Such a process can then be shown to minimize transportation costs and hence build efficient networks. The form of reinforcement turns out to be crucial in determining whether efficient networks are built (as is the case of current reinforcement) or if particles end up going round in endless self-reinfocing loops (as is the case in, the more commonly studied, 'density' reinforced random walks). I discuss how these models can be directly related to observations of what ants and slime moulds actually do.