We consider the problem of how to optimize multi-stage campaigning over social networks. The dynamic programming framework is employed to balance the high present reward and large penalty on low future outcome in the presence of extensive uncertainties. In particular, we establish theoretical foundations of optimal campaigning over social networks where the user activities are modeled as a multivariate Hawkes process, and we derive a time dependent linear relation between the intensity of exogenous events and several commonly used objective functions of campaigning. We further develop a convex dynamic programming framework for determining the optimal intervention policy that prescribes the required level of external drive at each stage for the desired campaigning result. Experiments on both synthetic data and the real-world MemeTracker dataset show that our algorithm can steer the user activities for optimal campaigning much more accurately than baselines.
Existing work in stochastic optimal control includes jump diffusion stochastic differential equations (SDE) @cite_34 @cite_18 which focuses on controlling the SDEs with the jump term driven by Poisson processes not for Hawkes processes. Inspired by the opinion dynamics model proposed in @cite_1 , the authors in @cite_32 proposes a multivariate jump diffusion process framework for modeling opinion dynamics over networks and determining the control over such networks.
@cite_18 @cite_34 @cite_1 @cite_32
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Diffusion processes governed by stochastic differential equations (SDEs) are a well-established tool for modelling continuous time data from a wide range of areas. Consequently, techniques have been developed to estimate diffusion parameters from partial and discrete observations. Likelihood-based inference can be problematic as closed form transition densities are rarely available. One widely used solution involves the introduction of latent data points between every pair of observations to allow a Euler-Maruyama approximation of the true transition densities to become accurate. In recent literature, Markov chain Monte Carlo (MCMC) methods have been used to sample the posterior distribution of latent data and model parameters; however, naive schemes suffer from a mixing problem that worsens with the degree of augmentation. A global MCMC scheme that can be applied to a large class of diffusions and whose performance is not adversely affected by the number of latent values is therefore explored. The methodology is illustrated by estimating parameters governing an auto-regulatory gene network, using partial and discrete data that are subject to measurement error. We present a novel approach to inference in conditionally Gaussian continuous time stochastic processes, where the latent process is a Markovian jump process. We first consider the case of jump-diffusion processes, where the drift of a linear stochastic differential equation can jump at arbitrary time points. We derive partial differential equations for exact inference and present a very efficient mean field approximation. By introducing a novel lower bound on the free energy, we then generalise our approach to Gaussian processes with arbitrary covariance, such as the non-Markovian RBF covariance. We present results on both simulated and real data, showing that the approach is very accurate in capturing latent dynamics and can be useful in a number of real data modelling tasks. This paper integrates recent work on Path Integral (PI) and Kullback Leibler (KL) divergence stochastic optimal control theory with earlier work on risk sensitivity and the fundamental dualities between free energy and relative entropy. We derive the path integral optimal control framework and its iterative version based on the aforemetioned dualities. The resulting formulation of iterative path integral control is valid for general feedback policies and in contrast to previous work, it does not rely on pre-specified policy parameterizations. The derivation is based on successive applications of Girsanov's theorem and the use of Radon-Nikodým derivative as applied to diffusion processes due to the change of measure in the stochastic dynamics. We compare the PI control derived based on Dynamic Programming with PI based on the duality between free energy and relative entropy. Moreover we extend our analysis on the applicability of the relationship between free energy and relative entropy to optimal control of markov jump diffusions processes. Furthermore, we present the links between KL stochastic optimal control and the aforementioned dualities and discuss its generalizability. We show that the stochastic differential equation (SDE) model for the merger of two identical two-dimensional vortices proposed by Agullo and Verga [“Exact two vortices solution of Navier–Stokes equation,” Phys. Rev. Lett. 78, 2361 (1997)] is a special case of a more general class of SDE models for N interacting vortex filaments. These toy models include vorticity diffusion via a white noise forcing of the inviscid equations, and thus extend inviscid models to include core dynamics and topology change (e.g., merger in two dimensions and vortex reconnection in three dimensions). We demonstrate that although the N=2 two-dimensional model is qualitatively and quantitatively incorrect, it can be dramatically improved by accounting for self-advection. We then extend the two-dimensional SDE model to three dimensions using the semi-inviscid asymptotic approximation of [“Simplified equations for the interactions of nearly parallel vortex filaments,” J. Fluid Mech. 288, 201 (1995)] for nearly parallel...