Latency-Bounded Minimum Influential Node Selection in Social Networks

1. Latency-Bounded Minimum Influential Node Selection in Social Networks Incheol Shin ishin@cise.ufl.edu

2. Introduction • Question • How the influential nodes selection could maximize the speed of the diffusion? • All the existing solutions focus on how to maximize the influence of the initially selected “influential nodes” • Object • Fast Information Propagation Problem • The problem of influential nodes selection regarding to the propagation speed in social network information diffusion. • In a time-essential information diffusion scenario in the social network, (eg. disease prevention in Epidemiology), the faster information diffusion could be, obviously the better.

3. Fast Information Propagation Problem • Social Network Applications • How to utilize social networks to help alleviate social problems in the physical world. • The disease prevention programs • How to choose a subset of individuals to be part of the prevention program so that the positive effect could spread through the whole social network as soon as possible becomes the key problem. • Fast Information Propagation Problem. • Extract the minimum size of nodes to target for initial activation such that it could sent information to all the nodes in the given social network and also the time needed for information propagation is bounded.

4. Fast Information Propagation Problem • Terms • ACTIVE node • adopts the good influence • INACTIVE node • not adopts the good influence • Hop count • to describe the time cost for information propagation • interested in the case that the time requirement for information propagation is exactly 1-hop in this paper. • Fundamental information diffusion model • Linear Threshold (LT) model • If more than half neighbors of an individual are active, then this individual will be influenced to be active • Progressive case • each node could switch from INACTIVE to ACTIVE, but not in the other direction

5. Fast Information Propagation Problem • NP-Hardness Proof • NP-hardness of this problem by using the well known NP-complete decision version of Vertex-Cover problem • Given a graph G = (V,E) and a positive integer k, determine whether G has a vertex cover of size at most k. • construct a polynomial-time many-one reduction from VC to the decision version of this problem: Given a graph G and a positive integer h, determine whether there exists a subset P of size at most h such that for every vertex v ∈ V \ P, at least a half number of its neighbors belongs to P. • Reduction • create |V |+|E| vertices with |V | vertices {v11, ..., v1|V|} representing the vertices in G and |E| vertices {v21, ..., v2|E|} representing the edges in G. • add an edge between vertex v1i and v2j if and only if the vertex v1i represents an endpoint of the edge represented by v2j . • we connect each vertex v2j to one subgraph constructed, called 3−star, and each vertex v1i to d(vi) such subgraphs where d(vi) is the degree of vertex vi in G, represented by v1i.

6. Fast Information Propagation Problem • NP-Hardness Proof • Reduction

7. Fast Information Propagation Problem • NP-Hardness Proof • Reduction • Every vertex not in P has at least a half number of neighbors in P.

8. Latency-Bounded Minimum Influential Node Selection Algorithm • Latency-bounded Minimum Influential Node Selection Algorithm • To solve the 1-hop Fast Information Propagation Problem • Active degree of a node • The number of its ACTIVE neighbors • Inactive degree of a node • The number of its INACTIVE neighbors. • A 1-hop dominating set D of a graph G • A subset of nodes in G such that every vertex not in D has at least one neighbor in D. • The main idea of the algorithm • To select the union of several 1-hop dominating

9. Latency-Bounded Minimum Influential Node Selection Algorithm • Latency-bounded Minimum Influential Node Selection Algorithm • Greedy Algorithm1 • To find the 1-hop dominating set in given a graph G = (V,E). • This algorithm is quite similar to the FASTGREEDY heuristics.

10. Latency-Bounded Minimum Influential Node Selection Algorithm • Latency-bounded Minimum Influential Node Selection Algorithm • Algorithm2 • algorithm 2 outputs a subset P of V , so that every vertex in V \ P has at least d(v)/2 active neighbors belonging to P.

11. Thank you.