Discovering the Most Potential Stars in Social Networks

Discovering the Most Potential Stars in SocialNetworks ZhuoPeng, Chaokun Wang, Lu Han, JingchaoHao and YiyuanBa Proceedings of the Third International Conference on Emerging Databases, Incheon, Korea (August 2011)

Outline • Introduction • Related Work • Preliminary • Algorithm • Experiments • Conclusion

Introduction • Purpose: to find the most potential stars in social networks to be promoted • How to measure the importance • incoming edge and outgoing edge • most potential stars = minimum promotion cost • How to find the most potential stars • Skyline query • promote a non-skyline member into skyline by adding new edges which are directly connected to it • it will take some costs to add a new edge

Problem Definition • member promotion in SNs = to identify the most appropriate non-skyline member(s) which can be promoted to be skyline member(s) by adding edges at minimum cost • To the best of our knowledge, our paper is the first one that raises the member promotion problem in SNs.

Contributions • first one that raises the member promotion problem in SNs and provides the formal definition • propose the general promotion algorithmic framework and bring forward the brute-force method for promotion to solve the problem intuitively • utilize several optimization strategies to improve the efficiency and accordingly propose the IDP algorithm • Extensive experiments were conducted to show the effectiveness and efficiency of the IDP algorithm on both real and synthetic datasets

Outline • Introduction • Related Work • Skyline Query • Skyline Minimum Vector • Preliminary • Algorithm • Experiments • Conclusion

Skyline Query • retrieves a subset of data points that are not dominated by any other points in a set of D-dimensional data points • algorithms • Block Nested Loop (BNL) • Divide-and-Conquer (D&C) • Bitmap method • Nearest Neighbor (NN) • Branched and Bound Skyline(BBS)

Skyline Minimum Vector • studies the query for the points that can be changed to be a skyline point at the minimum cost • The costs are measured by L1 distance of the skyline vectors starting from the original position and pointing to a skyline position. The skyline minimum vector thus indicates minimum L1 distance. • Those non-skyline points which can be changed to be skyline points by the skyline minimum vectors are the solutions to the problem. • Drawbacks • the virtual points which are needed for the computation of the skyline vectors must be provided in advance • the skip region for optimization is not good enough • no theoretical analysis such as time complexity analysis and correctness proof has been provided

Preliminary • An SN is modeled as a directed graph G(V, E, W) • V = the members in the SN • E = the existing directed edges • Each w ∈W : V × V→R+denotes the cost for establishing the edge between any two different members

Authoritativeness and Hubness • Authoritativeness • Given a node v in an SN G(V, E, W), the authoritativeness of v is denoted as the indegree of v, namely din(v) • Shows how much attention v can get • Hubness • Given a node v in an SN G(V, E, W), the hubness of v is denoted as the outdegree of v, namely dout(v) • Shows how the importance of v as a hub

Candidate Set and Dominator Set • Candidate Set • Given an SN G(V, E, W), let the skyline member set be SG, when SG≠ V , the set V-SG, denoted as C*, is the candidate set of G. We say each node c ∈ V-SG is a candidate for member promotion • Dominator Set • Given a member v in an SN G(V, E, W), the dominator set of v, marked as δ(v), is defined as a set of nodes D: {n | n dominates v, n ∈V}.

Promotion • Given an SN G(V, E, W), ∀c ∈C*, p ⊆V × V, a promotion plan against c, denoted as p, is defined as such an edge combination that satisfies: • (1) p ⊆ {e | e = (c, ·) ∨e = (·, c) ∧e ≠(c, c) ∧e ∉E}, • (2) c ∉SG’, where G’ = (V, E + p, W). • In more general cases, the one which only meets (1) is defined as a plan

Promotion Cost • Given an SN G(V, E, W), the cost of any plan p, marked as γ(p), is the sum of the weights corresponding to the edges included in p. As we mark the weight of an edge e as ϵ(e), that is, γ(p) =Σe∈p(ϵ(e)) = Σe∈p (W[e.f rom][e.to]) • in which e.from and e.to represent the source node and the sink node of edge e respectively. Thereby, ∀ c ∈C*, p ∈Pc , the promotion cost of c is the minimum cost among all the promotion plans. We mark it as ζ (c), namely, ζ (c) = minp ∈ Pc (γ(p))

Problem Statement • Member Promotion in SNs • Given an SN G(V, E, W), member promotion in SNs is to find such a member set R which satisfies: (1) R ⊆C*, (2) R = {r | r = argmin(ζ(c))}

Algorithm • A general framework for promotion algorithms • A brute-force method • An index-based dynamic pruning method

General Framework • Offline calculation of the distribution of both measures of all the members • Determine the candidate set by skyline query • Against each candidate, perform promotions by adding edges in the promotion plans and update the minimum promotion cost if necessary • Return the optimal candidate and related optimal promotion plans

Brute-Force Algorithm • Verifies all the possible plans with i edges against all the candidates before we locate the best candidate

IDP : The Index-based Dynamic Pruning Algorithm • A number of “meaningless” promotions will decrease the efficiency, so we should find a way to recognize the skippable plans for pruning • There are some related theorems and lemmas

IDP : Theorem • Given an SN G(V, E, W), if adding an edge e connecting node viand the candidate node c still cannot promote c into the skyline set, all the attempts of adding an edge e′ connecting the node vjand c with the same direction as e are not able to successfully promote c, where vj∈ δ(vi)

IDP : Lemma • Assume a plan p including n edges: e1, e2, …, encannot get its target candidate c promoted. For each edge eiconnecting viand c in p, let li be the list containing all the non-existing edges each of which links one member ∈ δ(vi) and c with the same direction as ei(i = 1, 2,… , n). All the plans with n edges which belong to , the Cartesian product of li, can be skipped in the subsequent verification process against c

Final Verification • Skyline may change after applying a plan, thus the candidate may still be dominated by other members • In the brute-force algorithm is to recalculate the skyline set based on the whole updated network • Theorem • Given a plan p, let M be the set of members relevant to the edges in p except the candidate c. If a member v neither dominates c before the promotion nor belongs to M, v will still not dominate c after p is conducted.

Final Verification • Just need to eliminate the possibility of any member being a dominator of the candidate c to make sure c is successfully promoted • Two cases • the members connected to any edge in the plan may become new dominators of c because at least one of their two measures will increase after the promotion • the members in the skyline member set may still dominate c

Experimental Settings • Implemented using Java with JDK version 1.6.0_10, Inter Core2 Duo CPU T7300 2.00GHz, 1G memory, 120G hard disk, Running Windows XP • Datasets • USAir • Includes 332 nodes and 2126 edges • Power-law set • Used a graph data generator gengraph_win to generate graph datasets

Comparison on Promotion Cost • we verified the effectiveness by comparing the promotion costs between the IDP algorithm and a random promotion algorithm

Comparison on Time Cost • to compare the time cost of the brute-force algorithm and the IDP algorithm on both USAir and Power-law Set respectively

Conclusion • Raised a new interesting problem, namely member promotion in in social networks • Purpose two algorithms • the brute-force algorithm • the IDP algorithm • The future work • Further improve the algorithm • Allows several members to promote concurrently

Discovering the Most Potential Stars in Social Networks