If a piece of information is released from a set of media sites, can it spread, in 1 month, to a million web pages? Can we efficiently find a small set of media sites among millions that can maximize the spread of the information, in 1 month? The two problems are called influence estimation and maximization problems respectively, which are very challenging since both the time-sensitive nature of the problems and the issue of scalability need to be addressed simultaneously. In this article, we propose two algorithms for influence estimation in continuous-time diffusion networks. The first one uses continuous-time Markov chains to estimate influence  exactly  on networks with exponential, or, more generally, phase-type transmission functions, but does not scale to large-scale networks, and the second one is a highly efficient randomized algorithm, which estimates the influence of every node in a network with general transmission functions, vνv nodes and vev edges to an accuracy of e using  n  e  O (1/e 2 ) randomizations and up to logarithmic factors  O ( n vev+ n vνv computations. We then show that finding the set of most influential source nodes in a continuous time diffusion network is an NP-hard problem and develop an efficient greedy algorithm with provable near-optimal performance. When used as subroutines in the influence maximization algorithm, the exact influence estimation algorithm is guaranteed to find a set of  C  nodes with an influence of at least (1 − 1/ e )OPT and the randomized algorithm is guaranteed to find a set with an influence of at least 1 − 1/ e )OPT − 2 C e, where OPT is the optimal value. Experiments on both synthetic and real-world data show that the proposed algorithms significantly improve over previous state-of-the-art methods in terms of the accuracy of the estimated influence and the quality of the selected nodes to maximize the influence, and the randomized algorithm can easily scale up to networks of millions of nodes.
['Manuel Gomez-Rodriguez', 'Le Song', 'Nan Du', 'Hongyuan Zha', 'Bernhard Schölkopf']
Influence Estimation and Maximization in Continuous-Time Diffusion Networks