An Implied Latent Position Process for Doubly Stochastic Messaging Activities

This paper studies the problem of identifying an inhomogeneous interaction structure amongst social agents. We construct the social network by a random graph and model the messaging activities via a multi-channel self-exciting point process. We design a methodology that divides the agents into two disjoint groups so that members within each group are considered to be of similar attributes. Our methodology and algorithm are useful for investigating and detecting abnormal activities within a network. We provide numerical illustrations based on a large email dataset from Enron. KeywordsSocial network; Multiple self-exciting point processes; Hypothesis testing; Risk mitigation. In this paper, we propose a model to estimate and analyze the structure of messaging activities in a social network. This is motivated by the recent proliferation of mobile technology, along with spread of blogs, social networking site, and media-sharing technology. For classification, detection, tracking and other practical purposes, robust statistical analysis as well as a good understanding of the data structure are essential. In this paper, we consider a collection of messaging data, made public by the Federal Energy Regulatory Commission in 2003, that contains highly accurate information about the time at which each message was exchanged. We introduce a meaningful way to reduce messaging data to a random graph and explore its possible application to a community detection problem. A simple and popular existing method to achieve this is to “pairwise threshold”, where for each pair of agents, an edge between vertex i and vertex j is formed if the number of messaging events between them exceeds a certain threshold. Such graphs are often thought to reveal a structure of an underlying social dynamic, motivating several successful models for a social network with sub-communities, and many tools for detecting a community with a particular graph theoretic and statistical properties have been proposed (Goldenberg et al. 2010, Kolaczyk 2009). On the other hand, some recent research (De Choudhury et al. 2010), has documented that changing the thresholds in the reduction procedure can produce dramatically different graphs, resulting in vastly dissimilar communities. This issue has motivated the work such as Heard et al. (2010) and Perry and Wolfe (2010). In both studies, as a remedy, the messaging events are modeled by way of point processes. In Heard et al. (2010), a piecewise-constant interaction rate is considered, while in Perry and Wolfe (2010) a Cox multiplicative intensity model is used with covariates that depend on the history of the process. Our approach is to model the dynamic network via n lowdimensional latent processes. One challenge to overcome is to estimate the intensity processes. Given an estimate of the intensity processes, we propose a novel embedding methodology to facilitate inference. Based on our methodology, we can produce a random graph with a particular probabilistic structure that could be very useful for the purpose of community detection. I. MESSAGING EVENTS LABELED WITH LINEARLY-ORDERED RISK LEVELS We consider a network of n vertices, and denote V = {1, . . . , n}. For ` ∈ N, we denote by τ`, {i`, j`} and k`, respectively, the occurrence time, the messaging pair and the risk level of the `-th message. Collectively, d` = (τ`, {i`, j`}, k`) represents the `-th messaging event. We require that τ` k2. For each (undirected) pair ij of the vertices and t ∈ [0, T ], we denote by Nij,k(t) the number of (undirected) messaging events on the topic k. between vertex i and vertex j during [0, t]. For each t ∈ [0, T ], let Dk(t) be the collection of all communication messaging events by time t on topic label k, i.e., Dk(t) = { (τk,`, {ik,`, jk,`}, k) : ` = 1, . . . , ∑