Random Walks on Colored Graphs

13 years 8 months ago

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This thesis introduces a model of a random walk on a colored undirected graph. Such a graph has a single vertex set and distinct sets of edges, each of which has a color. A particle begins at a designated starting vertex and an inﬁnite color sequence ¡ is speciﬁed. At time ¢ the particle traverses an edge chosen uniformly at random from those edges of color ¡¤£ incident to the current vertex. The ﬁrst part of this thesis addresses the extent to which an adversary, by choosing the color sequence, can affect the behavior of the random walk. In particular, we consider graphs that are covered with probability one on all inﬁnite sequences, and study their expected cover time in the worst case over all color sequences and starting vertices. We prove tight doubly exponential upper and lower bounds for graphs with three or more colors, and exponential bounds for the special case of two-colored graphs. We obtain stronger bounds in several interesting special cases, including ra...

Anne Condon, Diane Hernek

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