Randomized Search of Graphs in Log Space and Probabilistic Computation

Reingold has shown that L = SL, that s-t connectivity in a poly-mixing digraph is complete for promise-RL, and that s-t connectivity for a poly-mixing out-regular digraph with known stationary distribution is in L. Several properties that bound the mixing times of random walks on digraphs have been identified, including the digraph conductance and the digraph spectral expansion. However, rapidly mixing digraphs can still have exponential cover time, thus it is important to specifically identify structural properties of digraphs that effect cover times. We examine the complexity of random walks on a basic parameterized family of unbalanced digraphs called Strong Chains (which model weakly symmetric logspace computations), and a special family of Strong Chains called Harps. We show that the worst case hitting times of Strong Chain families vary smoothly with the number of asymmetric vertices and identify the necessary condition for non-polynomial cover time. This analysis also yields bounds on the cover times of general digraphs. Next we relate random walks on graphs to the random walks that arise in Monte Carlo methods applied to optimization problems. We introduce the notion of the asymmetric states of Markov chains and use this definition to obtain some results about Markov chains. We also obtain some results on the mixing times for Markov Chain Monte Carlo Methods. Finally, we consider the question of whether a single long random walk or many short walks is a better strategy for exploration. These are walks which reset to the start after a fixed number of steps. We exhibit digraph families for which a few short walks are far superior to a single long walk. We introduce an iterative deepening random search. We use this strategy estimate the cover time for poly-mixing subgraphs. Finally we discuss complexity theoretic implications and future work

Information Theoretic Navigability From Biased Random Walks

In this paper, I explore biased random walks between a source and target node on networks embedded in metric spaces. The bias is defined such that walkers tend to go towards neighbors with the smallest distance from the target in the metric space, to imitate the reasoning of people trying to navigate a network. Navigability is then quantified by the amount of information bits required by these random walkers to find a shortest path. From computations on three different random geometric graphs and two different implementations of bias, I find that the distributions of walk lengths are exponential and that the introduction of bias always reduces the rate parameter, resulting in fewer long walks and more short walks. Additionally, I show that different biases have distinct effects on the number of bits needed to find a shortest path.Bachelor of Scienc