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# Scalable Influence Maximization in Social Networks under the Linear Threshold Model

ICDM, pp.88-97, (2010)

EI WOS SCOPUS

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Abstract

Influence maximization is the problem of finding a small set of most influential nodes in a social network so that their aggregated influence in the network is maximized. In this paper, we study influence maximization in the linear threshold model, one of the important models formalizing the behavior of influence propagation in social net...More

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Introduction

- Influence maximization is the problem of finding a small set of most influential nodes in a social network so that their aggregated influence in the network is maximized.
- The greedy algorithm relies on the computation of influence spread given a seed set, the exact solution of which is left as an open problem in [1] for both models.

Highlights

- Influence maximization is the problem of finding a small set of most influential nodes in a social network so that their aggregated influence in the network is maximized
- Influence maximization has the obvious application in viral marketing through social networks, where companies try to promote their products and services through the word-of-mouth propagations among friends in the social networks
- [1] Kempe et al proposed two basic stochastic influence cascade models, the independent cascade (IC) model and the linear threshold (LT) model, which are extracted from earlier work on social network analysis, interactive particle systems, and marketing
- Our reduction uses the interpolation technique, and is more involved than the simple reduction used in [7] to show the #P-hardness in the independent cascade model. This hardness result closes the open problem left in [1] and further indicates that the greedy algorithm may have intrinsic difficulty to be made more efficient. To constrast with this hardness result, we show that computing influence spread in directed acyclic graphs (DAGs) can be done in linear time, which relies on an important linear relationship in activation probabilities between a node and its in-neighbors in directed acyclic graphs
- We show that in directed acyclic graphs (DAGs), the computation instead can be done in time linear to the size of the graph

Results

- To constrast with this hardness result, the authors show that computing influence spread in directed acyclic graphs (DAGs) can be done in linear time, which relies on an important linear relationship in activation probabilities between a node and its in-neighbors in DAGs. based on the fast influence computation for DAGs the authors propose the first scalable heuristic algorithm tailored for influence maximization in the LT model, which the authors refer to as the LDAG algorithm (Section IV).
- The influence maximization problem under the linear threshold model is, when given the influence graph G and an integer k, finding a seed set S of size k such that its influence spread σL(S) is the maximum.
- Finding the optimal seed set in the LDAG influence model is NP-hard, computing σD(S) given S is in polynomial-time because all computations are on DAGs. It is easy to see that σD(S) is still monotone and submodular
- To circumvent the NP-hardness result, the authors use an efficient greedy heuristic algorithm shown in Algorithm 3 to compute a local DAG LDAG(v, θ) for each node v given a threshold θ.
- After selecting the LDAGs rooted at all nodes, the authors may follow the greedy Algorithm 1 to select the k seeds, and use Algorithm 2 to compute influence spread.
- Consider a DAG D = (V, E, w) and a seed set S ⊆ V , and for all u ∈ V , let ap(u) denote the activation probability of u as computed by Algorithm 2.
- For all nodes u reachable from s in LDAG(v, θ), they need to update apv(u) but not αv(u), and the update of apv(u) follows Algorithm 2 by computing the changes ∆apv(u), with the initial condition ∆apv(s) = 1 − apv(s) and for all seeds u ∈ S ∆apv(u) = 0.

Conclusion

- The authors believe that the LDAG algorithm is suitable as the scalable solution to the influence maximization problem in the LT model.
- One may further pursue the theoretical problems related to influence maximization, for example, finding efficient approximation algorithms for computing influence in the IC or LT model, constructing LDAGs with approximation ratio guarantees, etc.

- Table1: STATISTICS OF FOUR REAL-WORLD NETWORKS
- Table2: AVERAGE NUMBER OF NODES AND EDGES IN LDAGS IN THE

Reference

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