Digraph Laplacian and the Degree of Asymmetry

In this paper we extend and generalize the standard random walk theory (or spectral graph theory) on undirected graphs to digraphs. In particular, we introduce and define a (normalized) digraph Laplacian (in short, Diplacian) Γ for digraphs, and prove that 1) its Moore-Penrose pseudo-inverse is the (discrete) Green’s function of the Diplacian matrix (as an operator on digraphs), and 2) it is the normalized fundamental matrix of the Markov chain governing random walks on digraphs. Using these results, we derive new formula for computing hitting and commute times in terms of the MoorePenrose pseudo-inverse of the Diplacian, or equivalently, the singular values and vectors of the Diplacian. Furthermore, we show that the Cheeger constant defined in [7] is intrinsically a quantity associated with undirected graphs. This motivates us to introduce a metric – the largest singular value of the skewed Laplacian ∇ := (Γ − ΓT )/2 – to quantify and measure the degree of asymmetry in a digraph. Using this measure, we establish several new results, such as a tighter bound (than that of Chung’s in [7]) on the Markov chain mixing rate, and a bound on the second smallest singular value of Γ.