An “Estimate & Score Algorithm” for simultaneous parameter estimation and reconstruction of incomplete data on social networks

Abstract

Dynamic activity involving social networks often has distinctive temporal patterns that can be exploited in situations involving incomplete information. Gang rivalry networks, in particular, display a high degree of temporal clustering of activity associated with retaliatory behavior. A recent study of a Los Angeles gang network shows that known gang activity between rivals can be modeled as a self-exciting point process on an edge of the rivalry network. In real-life situations, data is incomplete and law-enforcement agencies may not know which gang is involved. However, even when gang activity is highly stochastic, localized excitations in parts of the known dataset can help identify gangs responsible for unsolved crimes. Previous work successfully incorporated the observed clustering in time of the data to identify gangs responsible for unsolved crimes. However, the authors assumed that the parameters of the model are known, when in reality they have to be estimated from the data itself. We propose an iterative method that simultaneously estimates the parameters in the underlying point process and assigns weights to the unknown events with a directly calculable score function. The results of the estimation, weights, error propagation, convergence and runtime are presented.

Spatio-temporal patterns of IED usage by the Provisional Irish Republican Army

“…The model chosen to explore this question is a Hawkes self-exciting point process. It makes use of a response function (or kernel) which holds information pertaining to the long-term influence of previous events and has been shown to well represent a number of past dependent processes including gang related violence [18,24,52], email exchanges to infer organisation leadership [21], burglary [34,50] and violent deaths in conflicts as previously mentioned [29]. It also provides an opportunity to examine each of the five phases of PIRA activity separately, as is done in Model 1 and all subsequent models, offering direct quantitative insights into how the group behaved and reacted through the stages of the Republican campaign.…”

Dyadic event attribution in social networks with mixtures of hawkes processes

“…We conducted experiments on synthetic and real-world data sets, and compared the performance of our MHP/AMHP algorithm with state-of-the-art DEAP methods including: Parameter-Fixed Hawkes Process (PFHP) proposed in [16], Estimate & Score Algorithm (ESA) proposed in [8] and Latent Point Process Model (LPPM) proposed in [3]. …”

“…Gang activities are successfully modeled using Hawkes processes [16, 8]. However, these existing works employed substantial approximation schemes and failed to optimize the data likelihood.…”

Point-process models of social network interactions: Parameter estimation and missing data recovery

“…We take a maximum-likelihood approach, using an expectation-maximisation numerical method to combat the problem's ill conditioning [36]. Finally, we give several examples for different choices of the triggering function g. It is most common in the literature to assume an exponential form for g [5,11,15,22,34], though other forms are also in use, including power law [6,27] and the exponential multiplied by a polynomial [28]. Our comparison of exponential and power-law forms suggests that it does not matter which is used, validating the frequent use of the exponential form.…”

“…The Hawkes process appears in the seismology literature as a model for the timing of earthquakes and their aftershocks [26]. As interest in and availability of large data sets of human activities have grown, Hawkes processes have been used to model electronic Point-process models of social network interactions communications [5], gang crimes [10,15,34], and even terrorist and insurgent activity [19,25].…”