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).

The spectral clustering algorithm for directed networks that we consider in this paper is shown in Algorithm A.1. Theorem B.1 provides an upper bound to the error rate of spectral clustering on the weighted The effect of this term is negligible as T, so we ignore it. We now present an upper bound on the error rate for communities (analogous to Theorem B.1) estimated from the unweighted adjacency matrix The upper bounds on the error rates in Theorems B.1 and B.2 are not very informative in terms of In Section 4.1, we considered a simplified special case Similarly, we have the following result for spectral clustering using the unweighted adjacency matrix A . 3 Theorem B.3. Hence the unweighted adjacency matrix has a 1 in almost all entries, and the community structure cannot be detected from this matrix. The density of the aggregate adjacency matrix is governed by the parameters of the CHIP model.

In many application settings involving networks, such as messages between users of an on-line social network or transactions between traders in financial markets, the observed data are in the form of relational events with timestamps, which form a continuous-time network. We propose the Community Hawkes Independent Pairs (CHIP) model for community detection on such timestamped relational event data. We demonstrate that applying spectral clustering to adjacency matrices constructed from relational events generated by the CHIP model provides consistent community detection for a growing number of nodes. In particular, we obtain explicit non-asymptotic upper bounds on the misclustering rates based on the separation conditions required on the parameters of the model for consistent community detection. We also develop consistent and computationally efficient estimators for the parameters of the model. We demonstrate that our proposed CHIP model and estimation procedure scales to large networks with tens of thousands of nodes and provides superior fits compared to existing continuous-time network models on several real networks.