{"id":{"repo_id":"wfu","oai_identifier":"oai:wakespace.lib.wfu.edu:10339/39015"},"canonical_url":"https://search.dev.ndltd.org/etd/wfu/oai:wakespace.lib.wfu.edu:10339/39015","repository":{"repo_id":"wfu","name":"Wake Forest University","base_url":"https://wakespace.lib.wfu.edu/oai/request"},"display":{"title":"Random Walks with Pheromone","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Wei, Shuowen"],"institution":"Wake Forest University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013","date_published":"2013","updated_at":"2026-07-27T22:01:39Z","subjects":["generating function"],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10339/39015","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Wei, Shuowen"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2013-08-23T08:35:15Z"]},{"key":"dc:date.issued","label":"Date","values":["2013"]},{"key":"dc:publisher","label":"Institution","values":["Wake Forest University"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["generating function"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10339/39015"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:title","label":"Title","values":["Random Walks with Pheromone"]}]}],"canonical_facts":{"dc:creator":["Wei, Shuowen"],"dc:date.accessioned":["2013-08-23T08:35:15Z"],"dc:date.issued":["2013"],"dc:description.abstract":["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."],"dc:identifier.uri":["http://hdl.handle.net/10339/39015"],"dc:language.iso":["en"],"dc:publisher":["Wake Forest University"],"dc:subject":["generating function"],"dc:title":["Random Walks with Pheromone"],"dc:type":["Thesis"]},"updated_at":"2026-07-27T22:01:39Z"}