Abstract
We consider the problem of maximizing the spread of influence in a social network by choosing a fixed number of initial seeds, formally referred to as the influence maximization problem. It admits a (1−1/e)-factor approximation algorithm if the influence function is submodular. Otherwise, in the worst case, the problem is NP-hard to approximate to within a factor of N1−ε. This article studies whether this worst-case hardness result can be circumvented by making assumptions about either the underlying network topology or the cascade model. All our assumptions are motivated by many real-life social network cascades.
First, we present strong inapproximability results for a very restricted class of networks called the (stochastic) hierarchical blockmodel, a special case of the well-studied (stochastic) blockmodel in which relationships between blocks admit a tree structure. We also provide a dynamic-programming-based polynomial time algorithm, which optimally computes a directed variant of the influence maximization problem on hierarchical blockmodel networks. Our algorithm indicates that the inapproximability result is due to the bidirectionality of influence between agent-blocks.
Second, we present strong inapproximability results for a class of influence functions that are “almost” submodular, called 2-quasi-submodular. Our inapproximability results hold even for any 2-quasi-submodular f fixed in advance. This result also indicates that the “threshold” between submodularity and nonsubmodularity is sharp, regarding the approximability of influence maximization.
- Rico Angell and Grant Schoenebeck. 2017. Don,t be greedy: Leveraging community structure to find high quality seed sets for influence maximization. In Proceedings of the WINE. Springer, 16--29.Google Scholar
- W. B. Arthur. 1989. Competing technologies, increasing returns, and lock-in by historical events. Econ. J. 99, 394 (1989), pp. 116--131. Retrieved from http://www.jstor.org/stable/2234208c.Google Scholar
Cross Ref
- Lars Backstrom, Daniel P. Huttenlocher, Jon M. Kleinberg, and Xiangyang Lan. 2006. Group formation in large social networks: Membership, growth, and evolution. In Proceedings of the ACM SIGKDD. Google Scholar
Digital Library
- Abhijit Banerjee, Arun G Chandrasekhar, Esther Duflo, and Matthew O Jackson. 2013. The diffusion of microfinance. Science 341, 6144 (2013).Google Scholar
- F. M. Bass. 1969. A new product growth for model consumer durables. Manage. Sci. 15, 5 (1969), 215--227. Google Scholar
Digital Library
- Shishir Bharathi, David Kempe, and Mahyar Salek. 2007. In Proceedings of the WINE. Springer, Berlin, 306--311.Google Scholar
- Christian Borgs, Michael Brautbar, Jennifer T. Chayes, and Brendan Lucier. 2012. Influence maximization in social networks: Towards an optimal algorithmic solution. arXiv preprint arXiv:1212.0884 (2012).Google Scholar
- J. J. Brown and P. H. Reingen. 1987. Social ties and word-of-mouth referral behavior. J. Consumer Res. 14 (1987), 350--362.Google Scholar
Cross Ref
- D. Centola. 2010. The spread of behavior in an online social network experiment. Science 329, 5996 (2010), 1194--1197.Google Scholar
- Ning Chen. 2009. On the approximability of influence in social networks. SIAM J. Discrete Math. 23, 3 (2009), 1400--1415. Google Scholar
Digital Library
- Wei Chen, Tian Lin, Zihan Tan, Mingfei Zhao, and Xuren Zhou. 2016. Robust influence maximization. In Proceedings of the ACM SIGKDD. ACM, 795--804. Google Scholar
Digital Library
- Wei Chen, Yajun Wang, and Siyu Yang. 2009. Efficient influence maximization in social networks. In Proceedings of the ACM SIGKDD. ACM, 199--208. Google Scholar
Digital Library
- Wei Chen, Yifei Yuan, and Li Zhang. 2010. Scalable influence maximization in social networks under the linear threshold model. In Proceedings of the ICDM. IEEE, 88--97. Google Scholar
Digital Library
- Aaron Clauset, Cristopher Moore, and Mark EJ Newman. 2008. Hierarchical structure and the prediction of missing links in networks. Nature 453, 7191 (2008), 98--101.Google Scholar
- J. Coleman, E. Katz, and H. Menzel. 1957. The diffusion of an innovation among physicians. Sociometry 20 (1957), 253--270.Google Scholar
Cross Ref
- T. G. Conley and C. R. Udry. 2010. Learning about a new technology: Pineapple in Ghana. Amer. Econ. Rev. 100, 1 (2010), 35--69.Google Scholar
Cross Ref
- Paul DiMaggio. 1986. Structural analysis of organizational fields: A blockmodel approach. Res. Organiz. Behav. 8 (1986), 335--370.Google Scholar
- P. Domingos and M. Richardson. 2001. Mining the network value of customers. In Proceedings of the ACM SIGKDD. Google Scholar
Digital Library
- J. Goldenberg, B. Libai, and E. Muller. 2001. Using complex systems analysis to advance marketing theory development: Modeling heterogeneity effects on new product growth through stochastic cellular automata. Acad. Market. Sci. Rev. 9, 3 (2001), 1--18.Google Scholar
- Sanjeev Goyal and Michael Kearns. 2012. Competitive contagion in networks. In Proceedings of the STOC. 759--774. Google Scholar
Digital Library
- Mark Granovetter. 1978. Threshold models of collective behavior. Amer. J. Sociol. 83, 6 (1978), 1420--1443.Google Scholar
Cross Ref
- Xinran He and David Kempe. 2016. Robust influence maximization. In Proceedings of the ACM SIGKDD. Google Scholar
Digital Library
- Paul W Holland, Kathryn Blackmond Laskey, and Samuel Leinhardt. 1983. Stochastic blockmodels: First steps. Soc. Netw. 5, 2 (1983), 109--137.Google Scholar
Cross Ref
- Matthew O. Jackson. 2008. Social and Economic Networks. Princeton University Press. Google Scholar
Digital Library
- David Kempe, Jon M. Kleinberg, and Éva Tardos. 2003. Maximizing the spread of influence through a social network. In Proceedings of the ACM SIGKDD. 137--146. Google Scholar
Digital Library
- David Kempe, Jon M. Kleinberg, and Éva Tardos. 2005. Influential nodes in a diffusion model for social networks. In Proceedings of the ICALP. 1127--1138. Google Scholar
Digital Library
- K. Lerman and R. Ghosh. 2010. Information contagion: An empirical study of the spread of news on Digg and Twitter social networks. In Proceedings of the ICWSM. 90--97.Google Scholar
- Jure Leskovec, Lada A. Adamic, and Bernardo A. Huberman. 2006. The dynamics of viral marketing. In Proceedings of the EC. 228--237. Google Scholar
Digital Library
- Qiang Li, Wei Chen, Xiaoming Sun, and Jialin Zhang. 2017. Influence maximization with ϵ-almost submodular threshold functions. In Proceedings of the NIPS. 3804--3814. Google Scholar
Digital Library
- Brendan Lucier, Joel Oren, and Yaron Singer. 2015. Influence at scale: Distributed computation of complex contagion in networks. In Proceedings of the ACM SIGKDD. ACM, 735--744. Google Scholar
Digital Library
- Vince Lyzinski, Minh Tang, Avanti Athreya, Youngser Park, and Carey E. Priebe. 2017. Community detection and classification in hierarchical stochastic blockmodels. IEEE Transactions on Network Science and Engineering 4, 1 (2017), 13--26.Google Scholar
Cross Ref
- V. Mahajan, E. Muller, and F. M. Bass. 1990. New product diffusion models in marketing: A review and directions for research. J. Market. 54 (1990), 1--26.Google Scholar
Cross Ref
- S. Morris. 2000. Contagion. Rev. Econ. Studies 67, 1 (2000), 57--78.Google Scholar
Cross Ref
- Elchanan Mossel and Sébastien Roch. 2010. Submodularity of influence in social networks: From local to global. SIAM J. Comput. 39, 6 (2010), 2176--2188.Google Scholar
Digital Library
- George L. Nemhauser, Laurence A. Wolsey, and Marshall L. Fisher. 1978. An analysis of approximations for maximizing submodular set functions. Math. Program. 14, 1 (1978), 265--294. Google Scholar
Digital Library
- M. Richardson and P. Domingos. 2002. Mining knowledge-sharing sites for viral marketing. In Proceedings of the ACM SIGKDD. 61--70. Google Scholar
Digital Library
- Daniel M. Romero, Brendan Meeder, and Jon Kleinberg. 2011. Differences in the mechanics of information diffusion across topics: Idioms, political hashtags, and complex contagion on Twitter. In Proceedings of the WWW. ACM, 695--704. Retrieved from http://dl.acm.org/citation.cfm?id=1963503. Google Scholar
Digital Library
- Lior Seeman and Yaron Singer. 2013. Adaptive seeding in social networks. In Proceedings of the FOCS. IEEE, 459--468. Google Scholar
Digital Library
- Yaron Singer and Thibaut Horel. 2016. Maximization of approximately submodular functions. In Proceedings of the NIPS. Google Scholar
Digital Library
- D. J. Watts. 2002. A simple model of global cascades on random networks. Proc. Natl. Acad. Sci. U.S.A. 99, 9 (2002), 5766--5771. Retrieved from arXiv:http://www.pnas.org/content/99/9/5766.full.pdf+html.Google Scholar
Cross Ref
- Harrison C. White, Scott A. Boorman, and Ronald L. Breiger. 1976. Social structure from multiple networks. I. Blockmodels of roles and positions. Amer. J. Sociol. 81, 4 (1976), 730--780. Retrieved from http://www.jstor.org/stable/2777596.Google Scholar
Cross Ref
Index Terms
Beyond Worst-case (In)approximability of Nonsubmodular Influence Maximization
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