Integer and combinatorial optimization
Integer and combinatorial optimization
Mining the network value of customers
Proceedings of the seventh ACM SIGKDD international conference on Knowledge discovery and data mining
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Mining knowledge-sharing sites for viral marketing
Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining
Maximizing the spread of influence through a social network
Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining
On the spread of viruses on the internet
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
On the submodularity of influence in social networks
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Maximizing influence in a competitive social network: a follower's perspective
Proceedings of the ninth international conference on Electronic commerce
A note on maximizing the spread of influence in social networks
WINE'07 Proceedings of the 3rd international conference on Internet and network economics
Competitive influence maximization in social networks
WINE'07 Proceedings of the 3rd international conference on Internet and network economics
Competing for customers in a social network: the quasi-linear case
WINE'06 Proceedings of the Second international conference on Internet and Network Economics
Influential nodes in a diffusion model for social networks
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
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Social networks are often represented as directed graphs, 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 Kempe, Kleinberg, and Tardos (KKT) in [Maximizing the spread of influence through a social network, in Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2003, pp. 137-146] and [Influential nodes in a diffusion model for social networks, in Proceedings of the 32nd International Colloquium on Automata, Languages and Programming (ICALP), 2005], where the authors also impose several natural assumptions: the threshold values are random and the activation functions are monotone and submodular. 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 $\sigma(S)$ denote the expected number of active nodes at termination. Here, we prove a conjecture of KKT: we show that the function $\sigma(S)$ 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. Roughly, our results demonstrate that “local” submodularity is preserved “globally” under this diffusion process. This is of natural computational interest, as many optimization problems have good approximation algorithms for submodular functions.