Hierarchical Directed Spectral Graph Partitioning

In this report, we examine the generalization of the Laplacian of a graph due to Fan Chung. We show that Fan Chung’s generalization reduces to examining one particular symmetrization of the adjacency matrix for a directed graph. From this result, the directed Cheeger bounds trivially follow. Additionally, we implement and examine the benefits of directed hierarchical spectral clustering empirically on a dataset from Wikipedia. Finally, we examine a set of competing heuristic methods on the same dataset. 1 Clustering for Directed Graphs Clustering problems often arise when looking at graphs and networks. At the highest level, the problem of clustering is to partition a set of objects such that each partition contains similar objects. The choice of how to define the similarity between objects is the key component to a clustering algorithm. If we restrict ourselves to clustering graphs, then a natural way to define a clustering is a graph partition. That is, each of the objects is a vertex in our graph and we want to partition the vertices of the graph to optimize a function on the edges and vertices. For example, one common function is the normalized cut objective function. Given a set of vertices S, ncut(S) = vol ∂S ( 1 volS + 1 vol S̄ ) , where S̄ = V − S, vol ∂S = ∑ (u,v)∈E|u∈S,v∈S̄ wi,j , and volS = ∑