Motivated by applications arising on social networks, we study a generalization of the celebrated dominating set problem called the Positive Influence Dominating Set (PIDS). Given a graph G= (V,E), each node i∈ V has a weight bi, and a threshold requirement gi. We seek a minimum weight subset T of V , so that every node i ∈ V \T is adjacent to at least gi members of T . When gi = 1 for all nodes, we obtain the weighted dominating set problem. First, we propose a strong and compact extended formulation for the PIDS problem. We then project the extended formulation onto the space of the natural node-selection variables to obtain an equivalent formulation with an exponential number of valid inequalities. Restricting our attention to trees, we show that the extended formulation is the strongest possible formulation, and its projection (onto the space of the node variables) gives a complete description of the PIDS polytope on trees. We derive the necessary and sufficient facet-defining conditions for the valid inequalities in the projection and discuss their polynomial time separation. We embed this (exponential size) formulation in a branch-and-cut framework and conduct computational experiments using real-world graph instances, with up to approximately 2.5 million nodes and 8 million edges. On a test-bed of 100 real-world graph instances, our approach finds solutions that are on average 0.2% from optimality and solves 51 out of the 100 instances to optimality.

Rapid Influence Maximization on Social Networks: The Positive Influence Dominating Set Problem

Targeted marketing strategies are of significant interest in the smartapp economy. Typically, one seeks to identify individuals to strategically target in a social network so that the network is in...

Influence Maximization with Latency Requirements on Social Networks

We are proposing two greedy and a new linear programming based approximation algorithm for the total positive influence dominating set problem in weighted networks. Applications of this problem in weighted settings include finding: a minimum cost set of nodes to broadcast a message in social networks, such that each node has majority of neighbours broadcasting that message; a maximum trusted set in bitcoin network; an optimal set of hosts when running distributed apps etc. Extensive experiments on different generated and real networks highlight advantages and potential issues for each algorithm.

Total positive influence domination on weighted networks

Targeted marketing strategies are of significant interest in the smartapp economy. Typically, one seeks to identify individuals to strategically target in a social network so that the network is influenced at a minimal cost. In many practical settings, the effects of direct influence predominate, leading to the positive influence dominating set with partial payments (PIDS-PP) problem that we discuss in this paper. The PIDS-PP problem is NP-complete because it generalizes the dominating set problem. We discuss several mixed integer programming formulations for the PIDS-PP problem. First, we describe two compact formulations on the payment space. We then develop a stronger compact extended formulation. We show that when the underlying graph is a tree, this compact extended formulation provides integral solutions for the node selection variables. In conjunction, we describe a polynomial-time dynamic programming algorithm for the PIDS-PP problem on trees. We project the compact extended formulation onto the payment space, providing an equivalently strong formulation that has exponentially many constraints. We present a polynomial time algorithm to solve the associated separation problem. Our computational experience on a test bed of 100 real-world graph instances (with up to approximately 465,000 nodes and 835,000 edges) demonstrates the efficacy of our strongest payment space formulation. It finds solutions that are on average 0.4% from optimality and solves 80 of the 100 instances to optimality. Summary of Contribution: The study of influence propagation is important in a number of applications including marketing, epidemiology, and healthcare. Typically, in these problems, one seeks to identify individuals to strategically target in a social network so that the entire network is influenced at a minimal cost. With the ease of tracking consumers in the smartapp economy, the scope and nature of these problems have become larger. Consequently, there is considerable interest across multiple research communities in computationally solving large-scale influence maximization problems, which thus represent significant opportunities for the development of operations research–based methods and analysis in this interface. This paper introduces the positive influence dominating set with partial payments (PIDS-PP) problem, an influence maximization problem where the effects of direct influence predominate, and it is possible to make partial payments to nodes that are not targeted. The paper focuses on model development to solve large-scale PIDS-PP problems. To this end, starting from an initial base optimization model, it uses several operations research model strengthening techniques to develop two equivalent models that have strong computational performance (and can be theoretically shown to be the best model for trees). Computational experiments on a test bed of 100 real-world graph instances (with up to approximately 465,000 nodes and 835,000 edges) attest to the efficacy of the best model, which finds solutions that are on average 0.4% from optimality and solves 80 of the 100 instances to optimality. 

We are proposing two greedy and a new linear programming based approximation algorithm for the total positive influence dominating set problem in weighted networks. Applications of this problem in weighted settings include finding: a minimum cost set of nodes to broadcast a message in social networks, such that each node has majority of neighbours broadcasting that message; a maximum trusted set in bitcoin network; an optimal set of hosts when running distributed apps etc.. Extensive experiments on different generated and real networks highlight advantages and potential issues for each algorithm.

/pdf/total-positive-influence-domination-on-weighted-networks-2pctdymfq1.pdf

Total Positive Influence Domination on Weighted Networks

Current algorithms in the Positive Influence Dominating Set (PIDS) problem domain are focused on a specific type of PIDS, the Total Positive Influence Dominating Set (TPIDS). We have developed an algorithm specifically targeted towards the non-total type of PIDS. In addition to our new algorithm, we adapted two existing TPIDS algorithms to generate PIDS. We ran simulations for all three algorithms, and our new algorithm consistently generates smaller PIDS than both existing algorithms, with our algorithm generating PIDS approximately 5% smaller than the better of the two existing algorithms.

/pdf/positive-influence-dominating-set-generation-in-social-2a4o3yx09p.pdf

Matthew Rink

Papers

Positive Influence Dominating Set generation in social networks