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oai:null:10339/390152026-09-02T08:40:46Zcom_10339_14934col_10339_38132

Wei, Shuowen 2013-08-23T08:35:15Z 2013 https://wakespace.lib.wfu.edu/handle/10339/39015 In this thesis we are interested in random walks on graphs where transition probabilities from each vertex depend on values of a function, f, at neighboring nodes. The work is motivated by applications that arise in bio-inspired models wherein questions of dynamics are effected by pheromone trails. In Chapter 2, we discuss when the graph is the n-cycle, and the set of non-zero function values (the trail) is generated by visits of a simple random walk. We show that an optimal trail length is, in some sense, approximately one third of the cycle length. In Chapter 3, we consider non-contiguous subsets of the n-cycle. Here, a first random walker leaves maps (indicating a shortest path to a point s) in a possibly non-contiguous subset, S. We are then interested in minimizing the expected time required for a second uniformly random located second walker to reach s (utilizing these maps when found). We show that the expected time is minimized when the maps are in a sense, evenly distributed on the cycle. The thesis concludes with some results (and a conjecture) regarding when f values are determined by a large number of random walks departing from a point s in Z and leaving accumulating pheromone. en Wake Forest University generating function pheromone random walks toroidal graph Random Walks with Pheromone Thesis Berenhaut, Kenneth S Rouse, Jeremy Erhardt, Robert Mathematics forever 10000-01-01 restricted