Vol.:(0123456789) 1 3 Iranian Journal of Science and Technology, Transactions of Electrical Engineering https://doi.org/10.1007/s40998-019-00178-7 RESEARCH PAPER Integer Linear Programming for Infuence Maximization Farzaneh Ghayour Baghbani 1  · Masoud Asadpour 1  · Heshaam Faili 1 Received: 30 January 2018 / Accepted: 18 January 2019 © Shiraz University 2019 Abstract Infuence Maximization is one of the important research topics in social networks which has many applications, e.g., in marketing, politics and social science. The goal of Infuence Maximization is to select a limited number of vertices (called seed set) in a social graph, so that upon their direct activation, the maximum number of vertices is activated through social interaction of the seed set with the other vertices. Social interaction is modeled by difusion models among which Linear Threshold Model is one of the most popular ones. In Linear Threshold Model, infuence of nodes on each other is quantized by edge weights and nodes have a threshold for activation. If sum of the infuence of activated neighbors of a node reaches a certain threshold, the node is activated. When thresholds are fxed, Infuence Maximization reduces to Target Set Selection Problem. Ackerman et al. solved Target Set Selection Problem by Integer Linear Programming. In this paper, we analyze their work and show that their method cannot properly solve the problem in specifc situations, e.g., when graph has cycle. We fx this problem and propose a new method based on Integer Linear Programming and show in the results that our method can handle graphs with cycles as well. Keywords Infuence Maximization · Linear Threshold Model · Target Set Selection Problem · Integer Linear Programming 1 Introduction Social network analysis has recently attracted many researchers from diferent felds such as social science, com- puter science, economy, politics, biology and physics. Infu- ence Maximization (IM) is one of the important problems in this area which has many applications. An example of the applications of IM is viral marketing (Chen et al. 2013). In viral marketing, some nodes are selected based on some preference criteria to whom free (or discounted) sam- ples of a product are given. If the product is satisfactory enough, it is expected that the receivers of the free samples become advertisers of the product through word of mouth. It is hoped that the people in the network are infuenced indirectly and hopefully by the product. The set of receivers of the free samples are assumed activated and are called seed set, hereafter. The number of activated people at the end is called the spread. The goal of IM is to fnd best seed set that maximizes the spread. Many other applications could have similar scenario, including spread of innovations, informa- tion, news stories, interest, trust, referrals, epidemics, etc. Difusion of the product happens through a social net- work. The social network is modeled here as a directed graph G =(V , E) where V is a fnite set of vertices, E ⊆ V × V is the set of directed edges and E � ⊆ V × V is the set of absent edges. The vertices show social actors or individuals of the network, and the edges show a social relation (e.g., infuence, following) between them. Each edge (u, v) has a weight (or probability) parameter w uv (or p uv ) showing the strength (or probability) of the infuence of u on v. Propaga- tion occurs in discrete time steps t = 0, 1, 2,…. Each node v ∈ V has two possible states, active or inactive. Active nodes in time t are shown by S t ⊆ V and called active set in time t. S 0 is the initial seed and (S 0 ) is the fnal active set when seed set is S 0 . The cardinality of fnal active set (or Electronic supplementary material The online version of this article (https://doi.org/10.1007/s40998-019-00178-7) contains supplementary material, which is available to authorized users. * Masoud Asadpour asadpour@ut.ac.ir Farzaneh Ghayour Baghbani f.ghayour@ut.ac.ir Heshaam Faili hfaili@ut.ac.ir 1 Department of Electrical and Computer Engineering, University of Tehran, Tehran, Iran