On the Approximability of Influence in Social Networks Yilin Shen - PowerPoint PPT Presentation

Contents Problem Definition and Threshold Model Inapproximability Results On the Approximability of Influence in Social Networks Yilin Shen January 27, 2010 Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Contents 1 Problem Definition and Threshold Model Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Contents 1 Problem Definition and Threshold Model 2 Inapproximability Results Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Contents 1 Problem Definition and Threshold Model 2 Inapproximability Results Inapproximability Result on General Threshold Model 1 Addition Inapproximability Result on Different Threshold 2 Models Majority Thresholds 1 Small Thresholds 2 Unanimous Thresholds 3 Tree Structure 4 Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Problem Definition and Threshold Model Definition (Threshold Model) Given a connected undirected graph G = ( V , E ), let d ( v ) be the degree of v ∈ V . For each v ∈ V , there is a threshold value t ( v ) ∈ N , where 1 ≤ t ( v ) ≤ d ( v ). Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Problem Definition and Threshold Model Definition (Threshold Model) Given a connected undirected graph G = ( V , E ), let d ( v ) be the degree of v ∈ V . For each v ∈ V , there is a threshold value t ( v ) ∈ N , where 1 ≤ t ( v ) ≤ d ( v ). Definition (Target Set Selection Problem) Given a threshold model, initially the states of all vertices are inactive. The Target Set Selection problem is to pick the minimum subset of vertices, the target set, and set their state to be active. After that, in each discrete time step, the states of vertices are updated according to following rule: An inactive vertex v becomes active if at least t ( v ) of its neighbors are active. The process runs until either all vertices are active or no additional vertices can update states from inactive to active. Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Inapproximability Result on General Threshold Model Theorem (2.1) The Target Set Selection problem can not be approximated within � 2 log 1 − ǫ n � the ratio of O , for any fixed constant ǫ > 0 , unless NP ⊆ DTIME ( n poly log( n ) ) . Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Inapproximability Result on General Threshold Model Theorem (2.1) The Target Set Selection problem can not be approximated within � 2 log 1 − ǫ n � the ratio of O , for any fixed constant ǫ > 0 , unless NP ⊆ DTIME ( n poly log( n ) ) . Proof. We will prove the theorem by a reduction from the Minimum Representative (MinRep) problem. Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Minimum Representative (MinRep) problem Definition Given a bipartite graph G = ( A , B ; E ), where A and B are disjoint sets of vertices, there are explicit partitions of A and B into i =1 A i and B = ∪ β equal-sized subsets. That is, A = ∪ α j =1 B j , where all sets A i have the same size | A | /α and all sets B j have the same size | B | /β . The partition of G induces a super-graph H as follows: There are α + β super-vertices, corresponding to each A i and B j respectively, and there is a super-edge between A i and B j if there exist some a ∈ A i and b ∈ B j that are adjacent in G . Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Minimum Representative (MinRep) problem Definition Given a bipartite graph G = ( A , B ; E ), where A and B are disjoint sets of vertices, there are explicit partitions of A and B into i =1 A i and B = ∪ β equal-sized subsets. That is, A = ∪ α j =1 B j , where all sets A i have the same size | A | /α and all sets B j have the same size | B | /β . The partition of G induces a super-graph H as follows: There are α + β super-vertices, corresponding to each A i and B j respectively, and there is a super-edge between A i and B j if there exist some a ∈ A i and b ∈ B j that are adjacent in G . The goal of the MinRep problem is to select the minimum number of representatives from each set A i and B j such that all super-edges are covered. That is, we wish to find subsets A ′ ⊆ A and B ′ ⊆ B with the minimum total size A ′ + B ′ such that, for every super-edge ( A i , B j ), there exist representatives a ∈ A ′ ∩ A i and a ∈ B ′ ∩ B j that are adjacent in G . Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Minimum Representative (MinRep) problem (Cont.) A 1 A i A α u v B 1 B j B β Figure: An instance of the MinRep problem Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Minimum Representative (MinRep) problem (Cont.) Theorem (2.2. R. Raz) For any fixed ǫ > 0 , the MinRep problem can not be approximated � 2 log 1 − ǫ n � , unless NP ⊆ DTIME ( n poly log( n ) ) . within the ratio of O Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Proof of Theorem 2.1 Definition (Basic Gadget Γ l ) denoted by Γ ℓ v 1 v 2 v ℓ Figure: The basic gadget Γ l Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Proof of Theorem 2.1 (Cont.) The Construction of Graph G ′ for the Target Set Selection Problem For any given MinRep instance G = ( A , B ; E ), let M be the number of super-edges and N be the total input size. Basically, G ′ consists of four different groups of vertices V 1 , V 2 , V 3 , V 4 , where the vertices between two groups are connected by the basic gadgets described above. Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Proof of Theorem 2.1 (Cont.) The Construction of Graph G ′ for the Target Set Selection Problem For any given MinRep instance G = ( A , B ; E ), let M be the number of super-edges and N be the total input size. Basically, G ′ consists of four different groups of vertices V 1 , V 2 , V 3 , V 4 , where the vertices between two groups are connected by the basic gadgets described above. • V 1 = { a | a ∈ A } ∪ { b | b ∈ B } and each vertex has threshold N 2 . Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Proof of Theorem 2.1 (Cont.) The Construction of Graph G ′ for the Target Set Selection Problem For any given MinRep instance G = ( A , B ; E ), let M be the number of super-edges and N be the total input size. Basically, G ′ consists of four different groups of vertices V 1 , V 2 , V 3 , V 4 , where the vertices between two groups are connected by the basic gadgets described above. • V 1 = { a | a ∈ A } ∪ { b | b ∈ B } and each vertex has threshold N 2 . • V 2 = { u a , b | ( a , b ) ∈ E } and each vertex has threshold 2 N 5 . Vertex u a , b ∈ V 2 is connected to each of a , b ∈ V 1 by a basic gadget Γ N 5 . Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Proof of Theorem 2.1 (Cont.) The Construction of Graph G ′ for the Target Set Selection Problem (Cont.) • V 3 = { v i , j | A i , B j is connected by a super-edge } and each vertex has threshold N 4 . Vertex u a , b ∈ V 2 is connected to v i , j ∈ V 3 by a basic gadget Γ N 4 if a ∈ A i and b ∈ B j . Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Proof of Theorem 2.1 (Cont.) The Construction of Graph G ′ for the Target Set Selection Problem (Cont.) • V 3 = { v i , j | A i , B j is connected by a super-edge } and each vertex has threshold N 4 . Vertex u a , b ∈ V 2 is connected to v i , j ∈ V 3 by a basic gadget Γ N 4 if a ∈ A i and b ∈ B j . • V 4 = { w 1 , . . . , w N } and each vertex has threshold M · N 2 . Each vertex v i , j ∈ V 3 is connected to each w k ∈ V 4 by a basic gadget Γ N 2 , and each vertex a , b ∈ V 1 is connected to each w k ∈ V 4 by a basic gadget Γ N . Yilin Shen On the Approximability of Influence in Social Networks

Contents Problem Definition and Threshold Model Inapproximability Results Proof of Theorem 2.1 (Cont.) a b V 1 threshold = N 2 Γ N 5 Γ N 5 u a,b V 2 threshold = 2 N 5 Γ N Γ N Γ N Γ N 4 Γ N 4 Γ N 4 v i,j V 3 threshold = N 4 Γ N 2 Γ N 2 Γ N 2 w 1 w k w N V 4 threshold = M · N 2 Figure: The structure of graph G ′ Yilin Shen On the Approximability of Influence in Social Networks