We study the spread of influence in a social network based on the Linear Threshold model. We derive an analytical expression for evaluating the expected size of the eventual influenced set for a given initial set, using the probability of activation for each node in the social network. We then provide an equivalent interpretation for the influence spread, in terms of acyclic path probabilities in the Markov chain obtained by reversing the edges in the social network influence graph. We use some properties of such acyclic path probabilities to provide an alternate proof for the submodularity of the influence function. We illustrate the usefulness of the analytical expression in estimating the most influential set, in special cases such as the UILT(Uniform Influence Linear Threshold), USLT(Uniform Susceptance Linear Threshold) and node-degree based influence models. We show that the PageRank heuristic is either provably optimal or performs very well in the above models, and explore its limitations in more general cases. Finally, based on the insights obtained from the analytical expressions, we provide an efficient algorithm which approximates the greedy algorithm for the influence maximization problem.
A social network models a set of entities (such as individuals or organizations) that are tied by one or more types of interdependency (such as friendship, collaboration or coauthorship). Typically each individual is a node in the social network, and there is an edge between two nodes, if there exists some form of interaction between them. Real world social networks such as scientific collaboration networks, have been observed [1] to exhibit several properties of complex networks, such as scale-free degree distribution and the small-world phenomenon. Given a social network, there are several well established node-selection heuristics such as degree centrality and distance centrality whose effectiveness have been analysed in [2]. In this paper we analyze and derive new insights on the spread of influence under the Linear Threshold model studied by Kempe et al. [8].
Related Literature: Social networks play a fundamental role as a medium for the spread of information, ideas and influence among its members. Network diffusion processes have been investigated extensively in the past, with focus on spread of epidemics, diffusion of innovation and decision models. The concept of using threshold models to explain collective behaviour was first put forward by Granovetter in [4], where he discusses the spread of binary decisions, among a group of rational agents, for instance in voting models. Similar behaviours can also be observed in cases of innovation adoption, rumour and disease spreading. Newman [5] studied the spread of disease on networks under the susceptible-infected-removed (SIR) model and showed how concepts from percolation theory can be used to study these models on a wide variety of networks. Domingos and Richardson [6,7] were the first to study information diffusion under the viral marketing perspective, and they proposed the concept of a customer’s network value, apart from his intrinsic value. They were also the first to pose the combinatorial optimization problem of choosing the initial set of customers to maximize the net profits, and showed that choosing the right set of users for the marketing campaign could make a large difference. Kempe et al. [8] studied the problem of choosing the most influential initial set using two different models of information propagation, namely the Linear Threshold model (LT model) and the Independent Cascade model (IC model), and showed that the problem is NP-hard and the objective function is submodular. They proposed a greedy approximation algorithm that was shown to achieve an approximation factor of (1 -1/e). They also provided generalizations of the two models, and showed how the two generalized models can be made equivalent.
Web page ranking algorithms such as Google’s PageRank [11] can also be extended as a heuristic to the social network context, for ranking nodes in order of influence. Kimura et al. [13] develop upon the Independent Cascade model introduced in [8] and suggest two special cases of the IC model, which are computationally more efficient, and are good approximations to the IC model when the propagation probabilities are small. Kimura et al. [14] have also used the concept of bond percolation, to easily evaluate the expected influence of a given set of nodes, and hence proposed a faster version of the greedy algorithm. In [16] the authors propose a general framework for cost effective outbreak detection, of which the influence maximization problem is a special case, and, by exploiting the submodularity of the influence function, propose the CELF algorithm which achieves close to greedy algorithm performance. Wei Chen et al. [17] study the IC model and propose an improved version of the greedy algorithm and also the degree discount heuristic which are found to perform on par with the greedy algorithm.
Our Contributions:We develop upon the Linear Threshold model studied by Kempe et al. [8]. Our major contributions are as follows:
• We derive recursive expressions for the expected influence of a given initial set (in Section 3), provide an interpretation via Acyclic Path Probabilities in Markov chains,and provide an alternate proof of submodularity of the objective function (in Sections 4 and 5).
• We provide some sample cases where the PageRank algorithm is provably optimal or performs very well (in Section 6) and subsequently we discuss the limitations of PageRank in more general cases.
• We also propose the G1-Sieving algorithm to find the most influential set, based on the insights derived from the recursive expression(in Section 7) and find that G1-sieving performs almost on par with the Greedy algorithm and is also very efficient in terms of computation.
N -weighted directed graph of the entire social network N \A -graph obtained by removing nodes in A ⊆ N and all links to or from these nodes W -influence matrix with wi,j as entries, gives the edge weights of N Θj -U [0, 1] random threshold chosen by node j bj (A) = i∈A wi,j , total in
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