The heat kernel as the pagerank of a graph

The concept of pagerank was first started as a way for determining the ranking of Webpages by Web search engines. Based on relations in interconnected networks, pagerank has become a major tool for addressing fundamental problems arising in general graphs, especially for large information networks with hundreds of thousands of nodes. A notable notion of pagerank, introduced by Brin and Page and denoted by PageRank, is based on random walks as a geometric sum. In this paper we consider a new notion of pagerank which is based on the (discrete) heat kernel and can be expressed as an exponential sum of random walks. The heat kernel satisfies the heat equation and can be used to analyze many useful properties of random walks in a graph. A local Cheeger inequality is established which implies that by focusing on cuts determined by linear orderings of vertices using the heat kernel pageranks, the resulting partition is within a quadratic factor of the optimum. This is true, even if we restrict the volume of the small part separated by the cut to be close to some specified target value. This leads to a graph partitioning algorithm for which the running time is proportional to the size of the targeted volume (instead of the size of the whole graph). Introduction In the development of quantitative ranking for Webpages, many mathematical methods have come into play. The Hub-and-Authority algorithm by Kleinberg [11] uses eigenvectors. The PageRank introduced by Brin and Page [3] basically uses random walks. These pagerank algorithms mainly rely on the network structure of the Web. The viewpoint is to regard the Web as a graph, with vertices to be Webpages and edges as links between pairs of Webpages. Various notions of pagerank are computed using the Webgraph which are then used for numerous applications, such as identifying communities or finding hot spots in various information networks. Another example is to use PageRank to derive a local graph partitioning algorithm [1], which can be computed very efficiently in the sense that the cost of computing is proportional to the size of the small part of the partition, in contrast with the generic partitioning algorithm having cost depending on the size of the whole graph. In this paper, we introduce a new notion of pagerank by using the heat kernel of a graph. Similar to PageRank, the heat kernel pagerank is based on random walks but having the extra benefit of satisfying the heat equation. Originally rooted in spectral geometry [15], the heat equation for graphs involves a parameter t, the heat, which allows additional control of the rate of diffusion (see detailed definitions later). Using the heat equation, the heat kernel pagerank is amenable to various mathematical analyses of the graph. A key isoperimetric invariant of a graph is the Cheeger constant which provides an evaluation of how good a cut can be found. The classical Cheeger inequality concerns the relationship between the Cheeger constant and eigenvalues of the (normalized) Laplacian of a graph. (A graph can be viewed as a discrete version of a manifold where the original Cheeger inequality applies [4].) Here we will prove several variations of the Cheeger inequality, establishing relationships between the Cheeger constant and the heat kernel pagerank. One of the consequences of the local Cheeger inequality is that, for a given value s, the minimum Cheeger ratio of subsets of ∗Research supported in part by NSF Grants DMS 0457215 and ITR 0426858