A Note on the PageRank of Undirected Graphs

The PageRank is a widely used scoring function of networks in general and of the World Wide Web graph in particular. The PageRank is defined for directed graphs, but in some special cases applications for undirected graphs occur. In the literature it is widely noted that the PageRank for undirected graphs are proportional to the degrees of the vertices of the graph. We prove that statement for a particular personalization vector in the definition of the PageRank, and we also show that in general, the PageRank of an undirected graph is not exactly proportional to the degree distribution of the graph: our main theorem gives an upper and a lower bound to the L1 norm of the difference of the PageRank and the degree distribution vectors. Introduction In this short note we are examining the PageRank [3] of the undirected graphs. While the PageRank is usually applied for directed graphs (e.g., for the World Wide Web), in the literature it is sometimes mentioned in connection with undirected graphs [6], [9], [2], [10], [1]. In the literature it is frequently noted, that the PageRank of the vertices of an undirected graph is simply proportional to their degree. We intend to refine this statement here. Let G(V,E) denote an n-vertex undirected graph with vertex-set V and edge-set E. Let B = {bij} denote the n× n adjacency matrix of graph G(V,E): that is, if the vertices are V = {v1, v2, . . . , vn}, then bij = { 1 if vi is connected to vj , 0 otherwise. Matrix A is derived from matrix B by dividing every entry of row i by the degree of vertex vi; therefore the sum of every row of A equals to 1 (i.e., A is a row-stochastic matrix). Note, that while matrix B is symmetric, matrix A is usually not. Suppose that G(V,E) is a connected, undirected, non-bipartite graph. It is well known [8] that the limit probability distribution of the random walk on the vertices of graph G, corresponding to the transition matrix A, exists and unique, and it is given by the column-vector