Influence Maximization Over Markovian Graphs: A Stochastic Optimization Approach

Abstract

This paper considers the problem of randomized influence maximization over a Markovian graph process: given a fixed set of nodes whose connectivity graph is evolving as a Markov chain, estimate the probability distribution (over this fixed set of nodes) that samples a node which will initiate the largest information cascade (in expectation). Further, it is assumed that the sampling process affects the evolution of the graph i.e. the sampling distribution and the transition probability matrix are functionally dependent. In this setup, recursive stochastic optimization algorithms are presented to estimate the optimal sampling distribution for two cases: 1) transition probabilities of the graph are unknown but, the graph can be observed perfectly 2) transition probabilities of the graph are known but, the graph is observed in noise. These algorithms consist of a neighborhood size estimation algorithm combined with a variance reduction method, a Bayesian filter and a stochastic gradient algorithm. Convergence of the algorithms are established theoretically and, numerical results are provided to illustrate how the algorithms work.

Diffusion in Social Networks: Effects of Monophilic Contagion, Friendship Paradox, and Reactive Networks

“…One major difference between [10] and the third aim of this paper is that the random graph evolves on the same time scale in the current work while it evolves on a slower (compared to the contagion) time scale in [10]. Further, the context for [10] is independent cascade model which is different from the threshold models studied here. provides a simple deterministic approximation of the collective stochastic dynamics of a complex system (an SIS process on a random graph, both evolving on the same time scale).…”

“…3) If the network is a reactive network that randomly evolves depending on the state of the contagion, the collective dynamics of the network and the contagion process can be approximated by an ordinary differential equation (ODE) with an algebraic constraint. From a statistical modeling and machine learning perspective, the importance of this result relies on the fact that it 2 [10], [11] provide further examples of random graph processes that depend on state of diffusions. One major difference between [10] and the third aim of this paper is that the random graph evolves on the same time scale in the current work while it evolves on a slower (compared to the contagion) time scale in [10].…”

“…From a statistical modeling and machine learning perspective, the importance of this result relies on the fact that it 2 [10], [11] provide further examples of random graph processes that depend on state of diffusions. One major difference between [10] and the third aim of this paper is that the random graph evolves on the same time scale in the current work while it evolves on a slower (compared to the contagion) time scale in [10]. Further, the context for [10] is independent cascade model which is different from the threshold models studied here.…”

A Joint Markov Model for Communities, Connectivity and Signals Defined Over Graphs

“…Among others, Markovian models are widely adopted to model complex systems involving several random variables; such models are useful whenever global constraints on single random variables can be expressed exactly, or approximately, in terms of local constraints. Fruitful applications of Markovian models can be found in signal and image processing [5]- [9], and, recently, they have been applied also to graphs [11], [12], especially for graph inference purposes. In particular, SoG are often modeled as Gaussian Markov Random Fields (GMRF), having a precision matrix related to the Laplacian matrix of the underlying graph, see, e.g., [13], [15].…”

Information Diffusion in Social Networks: Friendship Paradox Based Models and Statistical Inference

“…Dynamic models and statistical inference for such information diffusion processes in social networks (such as news, innovations, cultural fads, etc) has witnessed remarkable progress in the last decade due to the proliferation of social media networks such as Facebook, Twitter, Youtube, Snapchat and also online reputation systems such as Yelp and Tripadvisor. Models and inference methods for information diffusion in social networks are useful in a wide range of applications including selecting influential individuals for targeted advertising and marketing [41,67,82], localization of natural disasters [81], forecasting elections [68] and predicting sentiment of investors in financial markets [74,8]. For example, [4] shows that models based on the rate of Tweets for a particular product can outperform market-based prediction methods.…”