Abstract
The influence maximization problem has become one of the fundamental combinatorial optimization problems over the past decade due to its extensive applications in social networks. Although a \(1-1/e\) approximation ratio is easily obtained using a greedy algorithm for the submodular case, how to solve the non-submodular case and enhance the solution quality deserve further study. In this paper, based on the marginal increments, we devise a non-negative decomposition property for monotone function and a non-increasing decomposition property for monotone submodular function (NDP). According to the exchange improvement (EI), a necessary condition for an optimal solution is also proposed. With the help of NDP and EI condition, an exchange improvement algorithm is proposed to improve further the quality of the solution to the non-submodular influence maximization problem. For the influence maximization, we devise effective methods to compute the influence spread and marginal gain in a successive iteration update manner. These methods make it possible to calculate the influence spread directly and accurately. Next, we design a data-dependent approximation algorithm for a non-submodular topology change problem from a marginal increment perspective.
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References
Borgs C, Brautbar M, Chayes J, Lucier B (2014) Maximizing social influence in nearly optimal time. In: SODA, pp 946–957
Chaoji V, Ranu S, Rastogi R, Bhatt R (2012) Recommendations to boost content spread in social networks. In: WWW, pp 529–538
Chen W, Wang C, Wang Y (2010) Scalable influence maximization for prevalent viral marketing in large-scale social networks. In: KDD, pp 1029–1038
Cornuejols G, Fisher M, Nemhauser G (1977) Location of bank accounts to optimize float. Manag Sci 23:789–810
Domingos P, Richardson M (2001) Mining the network value of customers. In: Seventh international conference on knowledge discovery and data mining
Du D-Z, Ko K-I (2000) Theory of computational complexity. Wiley, New York
Du D-Z, Ko K-I, Hu X (2008) Design and analysis of approximation algorithms. Lecture notes
Du D-Z, Graham RL, Pardalos PM, Wan P-J, Wu W, Zhao W (2008) Analysis of greedy approximations with non-submodular potential functions. In: Proceedings of the 19th annual ACM-SIAM symposium on dicrete algorithms (SODA), San Francisco, USA, Jan 20–22, pp 167–175
Feige U, Izsak R (2013) Welfare maximization and the supermodular degree. In: Proceedings of the 4th conference on innovations in theoretical computer science, pp 247–256
Feldman M, Izsak R (2014) Constrained monotone function maximization and the supermodular degree. Eprint Arxiv
Goldenberg J, Libai B, Muller E (2001) Talk of the network: a complex systems look at the underlying process of word-of-mouth. Mark Lett 12:211–223
Granovetter M (1978) Threshold models of collective behavior. Am J Sociol 83:1420–1443
Kempe D, Kleinberg JM, Tardos E (2003) Maximizing the spread of influence through a social network. In: KDD, pp 137–146
Lu W, Chen W, Lakshmanan LV (2015) From competition to complementarity: comparative influence diffusion and maximization. VLDB 9(2):60–71
Murota K (2003) Discrete convex analysis. In: SIMA of discrete mathematics and applications
Nemhauser G, Wolsey L, Fisher M (1978) An analysis of the approximations for maximizing submodular set functions. Math Program 14:265–294
Nguyen HT, Thai MT, Dinh TN (2016) Stop-and-stare: optimal sampling algorithms for viral marketing in billion-scale networks. In: SIGMOD, pp 695–710
Ohsaka N, Akiba T, Yoshida Y, Kawarabayashi K (2014) Fast and accurate influence maximization on large networks with pruned Monte-Carlo simulations. In: AAAI, pp 138–144
Richardson M, Domingos P (2002) Mining knowledge-sharing sites for viral marketing. In: Eighth international conference on knowledge discovery and data mining
Tang Y, Xiao X, Shi Y (2014) Influence maximization: near-optimal time complexity meets practical efficiency. In: SIGMOD, pp 75–86
Tang Y, Shi Y, Xiao X (2015) Influence maximization in near-linear time: a martingale approach. In: SIGMOD, pp 1539–1554
Wang Z, Yang Y, Pei J, Chu L, Chen E (2017) Activity maximization by effective information diffusion in social networks. IEEE Trans Knowl Data Eng 29(11):2374–2387
Zhang H, Dinh TN, Thai MT (2013) Maximizing the spread of positive influence in online social networks. In: IEEE 33rd international conference on distributed computing systems, pp 317–326
Zhu X, Jieun Y, Lee W, Kim D, Shan S, Ding-Zhu D (2010) New dominating sets in social networks. J Glob Optim 48:633–642
Acknowledgements
This work was performed during Wenguo Yang’s visit to the University of Texas at Dallas as a visiting scholar funded by the National Research Foundation of China. It was also supported in part by National Science Foundation under Grant No. 1907472 and by the National Natural Science Foundation of China under Grant No. 11571015. We thank the two anonymous reviewers for their helpful and constructive comments on our work.
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Yang, W., Zhang, Y. & Du, DZ. Influence maximization problem: properties and algorithms. J Comb Optim 40, 907–928 (2020). https://doi.org/10.1007/s10878-020-00638-5
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DOI: https://doi.org/10.1007/s10878-020-00638-5