Tomorrow’s speaker will be Ilia Zvedeniouk.

I will give an introduction to random walks on directed graphs and discuss some concepts to do with Markov Processes. Then I will discuss the paper ‘Local Partitioning for Directed Graphs Using PageRank‘ and walk through the proof of the ‘main theorem’ discussed therein.

**Abstract.** A local partitioning algorithm finds a set with small conductance near a specified seed ver-

tex. In this paper, we present a generalization of a local partitioning algorithm for undirected graphs to

strongly connected directed graphs. In particular, we prove that by computing a personalized PageRank

vector in a directed graph, starting from a single seed vertex within a set S that has conductance at

most α, and by performing a sweep over that vector, we can obtain a set of vertices S’ with conduc-

tance Φ_M (S’) = O( sqrt(α log |S|) ). Here, the conductance function Φ_M is defined in terms of the stationary

distribution of a random walk in the directed graph. In addition, we describe how this algorithm may

be applied to the PageRank Markov chain of an arbitrary directed graph, which provides a way to

partition directed graphs that are not strongly connected.

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