# On minimizing budget and time in influence propagation over social networks

- 1k Downloads
- 47 Citations

## Abstract

In recent years, study of influence propagation in social networks has gained tremendous attention. In this context, we can identify three orthogonal dimensions—the number of *seed* nodes activated at the beginning (known as *budget*), the expected number of activated nodes at the end of the propagation (known as *expected spread* or *coverage*), and the *time* taken for the propagation. We can constrain one or two of these and try to optimize the third. In their seminal paper, Kempe et al. constrained the budget, left time unconstrained, and maximized the coverage: this problem is known as *Influence Maximization* (or MAXINF for short). In this paper, we study alternative optimization problems which are naturally motivated by resource and time constraints on viral marketing campaigns. In the first problem, termed *minimum target set selection* (or MINTSS for short), a coverage threshold η is given and the task is to find the *minimum size seed set* such that by activating it, at least η nodes are eventually activated in the expected sense. This naturally captures the problem of deploying a viral campaign on a budget. In the second problem, termed MINTIME, the goal is to minimize the time in which a predefined coverage is achieved. More precisely, in MINTIME, a coverage threshold η and a budget threshold *k* are given, and the task is to find a seed set of size at most *k* such that by activating it, at least η nodes are activated in the expected sense, *in the minimum possible time*. This problem addresses the issue of *timing* when deploying viral campaigns. Both these problems are **NP**-hard, which motivates our interest in their approximation. For MINTSS, we develop a simple greedy algorithm and show that it provides a bicriteria approximation. We also establish a generic hardness result suggesting that improving this bicriteria approximation is likely to be hard. For MINTIME, we show that even bicriteria and tricriteria approximations are hard under several conditions. We show, however, that if we allow the budget for number of seeds *k* to be boosted by a logarithmic factor and allow the coverage to fall short, then the problem can be solved *exactly* in PTIME, i.e., we can achieve the required coverage within the time achieved by the optimal solution to MINTIME with budget *k* and coverage threshold η. Finally, we establish the value of the approximation algorithms, by conducting an experimental evaluation, comparing their quality against that achieved by various heuristics.

## Keywords

Social networks Social influence Influence propagation Viral marketing Approximation analysis MINTSS MINTIME## References

- Agarwal N, Liu H, Tang L, Yu P (2011) Modeling blogger influence in a community. Social Netw Anal Min 1–24. doi: 10.1007/s13278-011-0039-3
- Bakshy E, Hofman JM, Mason WA, Watts DJ (2011) Everyone’s an influencer: quantifying influence on twitter. In: Proceedings of the fourth ACM international conference on Web search and data mining, ACM, WSDM ’11, pp 65–74Google Scholar
- Bar-Ilan J, Kortsarz G, Peleg D (2001) Generalized submodular cover problems and applications. Theor Comput Sci 250(1–2):179–200MathSciNetzbMATHCrossRefGoogle Scholar
- Ben-Zwi O, Hermelin D, Lokshtanov D, Newman I (2009) An exact almost optimal algorithm for target set selection in social networks. In: EC ’09: Proceedings of the tenth ACM conference on electronic commerce, ACM, New York, NY, USA, pp 355–362Google Scholar
- Bhagat S, Goyal A, Lakshmanan LVS (2012) Maximizing product adoption in social networks. In: Web search and data mining, WSDMGoogle Scholar
- Bross J, Richly K, Kohnen M, Meinel C (2011) Identifying the top-dogs of the blogosphere. Social Netw Anal Min 1–15. doi: 10.1007/s13278-011-0027-7
- Cha M, Trez JP, Haddadi H (2011) The spread of media content through blogs. Social Netw Anal Min 1–16. doi: 10.1007/s13278-011-0040-x
- Chen N (2008) On the approximability of influence in social networks. In: SODA ’08: Proceedings of the nineteenth annual ACM–SIAM symposium on discrete algorithms, pp 1029–1037Google Scholar
- Chen W, Wang Y, Yang S (2009) Efficient influence maximization in social networks. In: Proceedings of the 15th ACM SIGKDD international conference on knowledge discovery and data mining (KDD’09)Google Scholar
- Chen W, Wang C, Wang Y (2010a) Scalable influence maximization for prevalent viral marketing in large-scale social networks. In: Proceedings of the 16th ACM SIGKDD international conference on knowledge discovery and data mining (KDD’10)Google Scholar
- Chen W, Yuan Y, Zhang L (2010b) Scalable influence maximization in social networks under the linear threshold model. In: Proceedings of the 10th IEEE international conference on data mining (ICDM’2010)Google Scholar
- Domingos P, Richardson M (2001) Mining the network value of customers. In: Proceedings of the seventh ACM SIGKDD international conference on knowledge discovery and data mining, ACM, New York, NY, USA, KDD ’01, pp 57–66Google Scholar
- Feige U (1998) A threshold of XXX for approximating set cover. J ACM 45(4):634–652MathSciNetzbMATHCrossRefGoogle Scholar
- Fujito T (1999) On approximation of the submodular set cover problem. Oper Res Lett 25(4):169–174MathSciNetzbMATHCrossRefGoogle Scholar
- Fujito T (2000) Approximation algorithms for submodular set cover with applications. IEICE Trans Inf Syst 83Google Scholar
- Goyal A, Bonchi F, Lakshmanan LVS (2008) Discovering leaders from community actions. In: Proceeding of the 17th ACM conference on information and knowledge management, ACM, New York, NY, USA, CIKM ’08, pp 499–508Google Scholar
- Goyal A, Bonchi F, Lakshmanan LVS (2010) Learning influence probabilities in social networks. In: Proceedings of the third ACM international conference on web search and data mining, ACM, New York, NY, USA, WSDM ’10, pp 241–250Google Scholar
- Goyal A, Bonchi F, Lakshmanan LVS (2011) A data-based approach to social influence maximization. PVLDB 5(1)Google Scholar
- Kempe D, Kleinberg JM, Tardos É (2003) Maximizing the spread of influence through a social network. In: Proceedings of the ninth ACM SIGKDD international conference on knowledge discovery and data mining (KDD’03)Google Scholar
- Kempe D, Kleinberg J, Tardos É (2005) Influential nodes in a diffusion model for social networks. In: ICALP, Springer, Berlin, pp 1127–1138Google Scholar
- Khuller S, Moss A, Naor JS (1999) The budgeted maximum coverage problem. Inf Process Lett 70(1):39–45MathSciNetzbMATHCrossRefGoogle Scholar
- Kimura M, Saito K (2006) Tractable models for information diffusion in social networks. In: Proceedings of PKDD 2006, Lecture notes in computer science, vol 4213Google Scholar
- Leskovec J, Krause A, Guestrin C, Faloutsos C, VanBriesen J, Glance NS (2007) Cost-effective outbreak detection in networks. In: Proceedings of the 13th ACM SIGKDD international conference on knowledge discovery and data mining (KDD’07)Google Scholar
- Li Gørtz I, Wirth A (2006) Asymmetry in
*k*-center variants. Theor Comput Sci 361(2):188–199CrossRefGoogle Scholar - Nemhauser GL, Wolsey LA, Fisher ML (1978) An analysis of approximations for maximizing submodular set functions-I. Math Program 14(1):265–294MathSciNetzbMATHCrossRefGoogle Scholar
- Panigrahy R, Vishwanathan S (1998) An
*O*(log^{*}*n*) approximation algorithm for the asymmetric*p*-center problem. J Algorithms 27(2):259–268MathSciNetzbMATHCrossRefGoogle Scholar - Richardson M, Domingos P (2002) Mining knowledge-sharing sites for viral marketing. In: Proceedings of the eighth ACM SIGKDD international conference on knowledge discovery and data mining, ACM, New York, NY, USA, KDD ’02, pp 61–70Google Scholar
- Slaví k P (1997) Improved performance of the greedy algorithm for partial cover. Inform Process Lett 64(5):251–254MathSciNetCrossRefGoogle Scholar
- Sviridenko M (2004) A note on maximizing a submodular set function subject to a knapsack constraint. Oper Res Lett 32(1):41–43MathSciNetzbMATHCrossRefGoogle Scholar
- Weng J, Lim EP, Jiang J, He Q (2010) Twitterrank: finding topic-sensitive influential twitterers. In: Proceedings of the third ACM international conference on web search and data mining, ACM, New York, NY, USA, WSDM ’10, pp 261–270Google Scholar
- Wolsey LA (1982) An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica 2(4):385–393MathSciNetzbMATHCrossRefGoogle Scholar