Abstract
We derive formulas for the expected hitting times of general random walks on graphs, in terms of voltages, with very elementary electric means. Under this new light we revise bounds and hitting times for birth-and-death Markov chains and for walks on graphs with cutpoints, and give some exact computations on the necklace graph. We also prove Tetali’s formula for hitting times without making use of the reciprocity principle. In fact this principle follows as a corollary of our argument that also yields as corollaries the triangular inequality for effective resistances and the reversibility of the sum of hitting times around a tour.
Highlights
On a finite connected undirected graph G = (V, E) with |G| = n such that the edge between vertices i and j is given a resistance rij, we can define the random walk on G as the Markov chain Xn, n ≥ 0, that jumps from its current vertex V to the neighboring vertex w with probability pVw = CVw/C(V), where C(V) = ∑w:w∼V CVw and w ∼ V means that w is a neighbor of V
We look at simple random walk (SRW) on graphs with cutpoints
Where TbGb is the hitting time of the random walk restricted to the subgraph Gb
Summary
In this paper we want to depart from these expressions of expected hitting times in terms of effective resistances and give them instead in terms of voltages, using as tools the material in Doyle and Snell’s book, which relies on basic facts of electricity such as Ohm’s law and Kirchhoff ’s law We present this alternative expression for two reasons: on the Journal of Probability one hand, this representation in terms of voltages leads to very simple proofs of many known results, giving new insights into new results and computations which are simpler than those involving effective resistances; on the other hand, using Tetali’s formula means accepting the deep result of the reciprocity theorem for electric networks, which is outside the realm of Doyle and Snell. This principle will follow as a corollary of our argument that yields as corollaries the triangular inequality for effective resistances and the reversibility of the sum of hitting times around a tour
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