PageRank (PR), an algorithm originally proposed by Page et al. for ranking web-pages [1] has found manysuccessful applications, including community detection [2,3],linkprediction [4]and recommendersystemdesign[5,6].

Community detection in random graphs or hypergraphs is an interesting fundamental problem in statistics, machine learning and computer vision.

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A.1 CommunityDetection The spectral clustering algorithm for directed networks that we consider in this paper is shown in Algorithm A.1. It can be applied either to the weighted adjacency (count) matrixN or the unweighted adjacency matrixA, where Aij =1{Nij >0} and 1{ } denotes the indicator function of the argument. This algorithm is used for the community detection step in our proposed CHIP estimationprocedure. For undirectednetworks, which we use for the theoreticalanalysisin Section 4, spectral clustering is performed by running k-means clustering on the rows of theeigenvector matrix of N or A, not the rows of the concatenated singular vector matrix. A.2 Estimation of Hawkes process parameters Ozaki (1979) derived the log-likelihood function for Hawkes processes with exponential kernels, which takes the form: logL= µT+ The threeparameters µ,α,β can be estimatedby maximizing (A.1) using standard numerical methods for non-linear optimization (Nocedal & Wright, 2006). We provide closed-form equations for estimating mab =αab/βab and µab in (2).