Influence Circle Covering in Large-Scale Social Networks: A Shift Approach

Given a specific propagation speed
$h$
in a social network
$G(V, E)$
, an influence circle(IC) of a node
$s$
in time
$t$
is a node set of its influenced nodes, where the distance between
$s$
and its expected influenced node
$w$
is less than radius
$r=ht$
. Different from the Influence Maximization(IM) problem which finds a set of
$k$
initial seed nodes in a network so that the expected size of cascade is maximized, the aim of the proposed influence circle covering (ICCovering) problem in this work is to find a minimum number of seeds or ICs to cover the whole network. The general approach for this covering problem is greedy strategy, which iteratively selects a seed with the largest influence circle. However, the upper bound of greedy algorithms does not perform very well, and the value will increase further as the network scale expands. In this paper, we propose an
$\alpha $
-approximation partitioning algorithm for large-scale social networks, where
$\alpha $
is the maximum number of outer edges of Voronoi cells appeared in the partition. The algorithm divides the input graph into smaller cells so that each cell can be solved separately, and a feasible solution to the input object can be constructed by combining the solutions of the smaller cells. When solving the smaller cells, we adopt the linear programming method. In order to improve its effectiveness and efficiency, we also propose two optimization algorithms. Extensive experiments on real social networks confirm the superiorities and effectiveness of our solution.