On the submodularity of influence in social networks

ABSTRACT

We prove and extend a conjecture of Kempe, Kleinberg, and Tardos (KKT) on the spread of influence in social networks.

A social network can be represented by a directed graph where the nodes are individuals and the edges indicate a form of social relationship. A simple way to model the diffusion of ideas, innovative behavior, or "word-of-mouth" effects on such a graph is to consider an increasing process of "infected" (or active) nodes: each node becomes infected once an activation function of the set of its infected neighbors crosses a certain threshold value. Such a model was introduced by KKT in [7,8] where the authors also impose several natural assumptions: the threshold values are (uniformly) random to account for our lack of knowledge of the true values; and the activation functions are monotone and submodular, i.e. have "diminishing returns." The monotonicity condition indicates that a node is more likely to become active if more of its neighbors are active, while the submodularity condition, indicates that the marginal effect of each neighbor is decreasing when the set of active neighbors increases.

For an initial set of active nodes s, let σ(S) denote the expected number of active nodes at termination. Here we prove a conjecture of KKT: we show that the function σ(S) is submodular under the assumptions above. We prove the same result for the expected value of any monotone, submodular function of the set of active nodes at termination.

In other words, our results demonstrate that "local" submodularity is preserved "globally" under diffusion processes. This is of natural computational interest, as many optimization problems have good approximation algorithms for submodular functions. In particular, our results coupled with an argument in [7] imply that a greedy algorithm gives an (1-1/e-ε)-approximation algorithm for maximizing σ(S) among all sets s of a given size. This result has important practical implications for many social network analysis problems, notably viral marketing.

REFERENCES

• 1: Noam Berger , Christian Borgs , Jennifer T. Chayes , Amin Saberi, On the spread of viruses on the internet, Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, January 23-25, 2005, Vancouver, British Columbia
• 2: Pedro Domingos , Matt Richardson, Mining the network value of customers, Proceedings of the seventh ACM SIGKDD international conference on Knowledge discovery and data mining, p.57-66, August 26-29, 2001, San Francisco, California [doi>10.1145/502512.502525]
• 3: Matthew Richardson , Pedro Domingos, Mining knowledge-sharing sites for viral marketing, Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining, July 23-26, 2002, Edmonton, Alberta, Canada [doi>10.1145/775047.775057]
• 4: Rick Durrett. Lecture notes on particle systems and percolation. 1988.
• 5: Rick Durrett and Paul Jung. Two phase transitions for the contact process on small worlds. Submitted, 2006.
• 6: M. Granovetter. Threshold models of collective behavior. American Journal of Sociology, 83(6):1420--1443, 1978.
• 7: David Kempe , Jon Kleinberg , Éva Tardos, Maximizing the spread of influence through a social network, Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, August 24-27, 2003, Washington, D.C. [doi>10.1145/956750.956769]
• 8: D. Kempe, J. Kleinberg, and E. Tardos. Influential nodes in a diffusion model for social networks. In Proc. 32nd International Colloquium on Automata, Languages and Programming (ICALP), 2005.
• 9: Thomas M. Liggett. Interacting particle systems, volume 276 of Grundlehren der Mathematischen Wissenschaften {Fundamental Principles of Mathematical Sciences}. Springer-Verlag, New York, 1985.
• 10: Thomas M. Liggett. Stochastic interacting systems: contact, voter and exclusion processes, volume 324 of Grundlehren der Mathematischen Wissenschaften {Fundamental Principles of Mathematical Sciences}. Springer-Verlag, Berlin, 1999.
• 11: T. Lindvall. Lectures on the Coupling Method. Wiley, New York, 1992.
• 12: M. Morris, editor. Network Epidemiology. Oxford University Press, 2004.
• 13: George L. Nemhauser , Laurence A. Wolsey, Integer and combinatorial optimization, Wiley-Interscience, New York, NY, 1988
• 14: S. Wasserman and K. Faust. Social Network Analysis. Cambridge University Press, 1994.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.3 PROBABILITY AND STATISTICS
      Subjects: Stochastic processes

General Terms:
Algorithms, Economics, Theory

Keywords:
coupling, social networks, submodularity, viral marketing