Mathematical Programming Models for Influence Maximization on Social Networks

In this dissertation, we apply mathematical programming techniques (i.e., integer programming and polyhedral combinatorics) to develop exact approaches for influence maximization on social networks. We study four combinatorial optimization problems that deal with maximizing influence at minimum cost over a social network. To our knowl- edge, all previous work to date involving influence maximization problems has focused on heuristics and approximation. We start with the following viral marketing problem that has attracted a significant amount of interest from the computer science literature. Given a social network, find a target set of customers to seed with a product. Then, a cascade will be caused by these initial adopters and other people start to adopt this product due to the influence they re- ceive from earlier adopters. The idea is to find the minimum cost that results in the entire network adopting the product. We first study a problem called the Weighted Target Set Selection (WTSS) Prob- lem. In the WTSS problem, the diffusion can take place over as many time periods as needed and a free product is given out to the individuals in the target set. Restricting the number of time periods that the diffusion takes place over to be one, we obtain a problem called the Positive Influence Dominating Set (PIDS) problem. Next, incorporating partial incentives, we consider a problem called the Least Cost Influence Problem (LCIP). The fourth problem studied is the One Time Period Least Cost Influence Problem (1TPLCIP) which is identical to the LCIP except that we restrict the number of time periods that the diffusion takes place over to be one. We apply a common research paradigm to each of these four problems. First, we work on special graphs: trees and cycles. Based on the insights we obtain from special graphs, we develop efficient methods for general graphs. On trees, first, we propose a polynomial time algorithm. More importantly, we present a tight and compact extended formulation. We also project the extended formulation onto the space of the natural vari- ables that gives the polytope on trees. Next, building upon the result for trees---we derive the polytope on cycles for the WTSS problem; as well as a polynomial time algorithm on cycles. This leads to our contribution on general graphs. For the WTSS problem and the LCIP, using the observation that the influence propagation network must be a directed acyclic graph (DAG), the strong formulation for trees can be embedded into a formulation on general graphs. We use this to design and implement a branch-and-cut approach for the WTSS problem and the LCIP. In our computational study, we are able to obtain high quality solutions for random graph instances with up to 10,000 nodes and 20,000 edges (40,000 arcs) within a reasonable amount of time.

Computational Analysis of Intelligent Agents: Social and Strategic Settings

The central motif of this work is prediction and optimization in presence of multiple interacting intelligent agents. We use the phrase `intelligent agents' to imply in some sense, a `bounded rationality', the exact meaning of which varies depending on the setting. Our agents may not be `rational' in the classical game theoretic sense, in that they don't always optimize a global objective. Rather, they rely on heuristics, as is natural for human agents or even software agents operating in the real-world. Within this broad framework we study the problem of influence maximization in social networks where behavior of agents is myopic, but complication stems from the structure of interaction networks. In this setting, we generalize two well-known models and give new algorithms and hardness results for our models. Then we move on to models where the agents reason strategically but are faced with considerable uncertainty. For such games, we give a new solution concept and analyze a real-world game using out techniques. Finally, the richest model we consider is that of Network Cournot Competition which deals with strategic resource allocation in hypergraphs, where agents reason strategically and their interaction is specified indirectly via player's utility functions. For this model, we give the first equilibrium computability results. In all of the above problems, we assume that payoffs for the agents are known. However, for real-world games, getting the payoffs can be quite challenging. To this end, we also study the inverse problem of inferring payoffs, given game history. We propose and evaluate a data analytic framework and we show that it is fast and performant.