Betweenness Centrality Approximations for an Internet Deployed P2P Reputation System

1. Betweenness Centrality Approximations for an Internet Deployed P2P Reputation System Dimitra Gkorou, Johan Pouwelse, and Dick Epema

2. Overview • Tribler • The Bartercast Reputation Mechanism • Betweenness Centrality • Approximations for Betweenness Centrality • Integration of these methods in Bartercast • Conclusion

3. Tribler: main features • based on the BitTorrent P2P file-sharing system • an epidemic protocol for peer and content discovery • social phenomena toimplement distributed control: • content discovery • content recommendation • reputation system • first released on 17 March 2006 • more than 1,000,000 downloads • BarterCast: the reputation system ofTribler against free-riders J.A. Pouwelse, P. Garbacki, J. Wang, A. Bakker, J. Yang, A. Iosup, D.H.J. Epema, M. Reinders, M.R. van Steen, H.J. Sips, "Tribler: A social-based peer-to-peer system," Concurrency and Computation: Practice and Experience Vol. 20, 127-138, 2008.

4. BarterCast 1: Basic Concepts • information exchange: using an epidemic protocol • peers keep the history of their own interactions + the interactions among other peers • each peer i creates a directed, weighted local graph: • vertices: the peers whose activity is known to peer i • weighted edges: the amount of the transferred data between two peers • each peer computes locally the subjective reputations of other peers in the system j wij i wjc c k wki wim wkb m b local subjective graph of peer i

5. Bartercast 2: Information Exchange 8 8 8 w9,8 w9,8 10 12 w10,12 9 10 w9,10 w6,9 w6,9 17 15 14 6 6 w9,10 9 6 w10,4 w2,12 18 2 11 16 7 19 4 local subjective graph of peer 9 8 local subjective graph of peer 10 5 13 1 Bartercast 3 9 10 12 6 2 data transactions 4 21 M. Meulpolder, J.A. Pouwelse, D.H.J. Epema, and H.J. Sips, "BarterCast: A Practical Approach to Prevent Lazy Freeriding in P2P Networks," (HoT-P2P), in conjunction with IPDPS, May 2009.

6. Bartercast 3: Computing Reputation • a peer i willing to interact with a peer g: • considers the amount of transferred data in its local subjective graphas flows • use of the max-flow algorithm to compute fgiand fig • reputation of peer g: the difference of fgiand fig • the computation is restricted to paths of length 2 due to its computational cost a wia Wca\ac i f c Wba\ab wfg g wbi wgc wbj wgk b k wge wjg j local subjective graph of peer i e

7. Bartercast 4: Problem Description • starting the computation from the owner of the subjective graph itself results in bad coverage • starting from the most central node results in better coverage • the most central node is the node with the highest betweenness centrality (BC) a wia i Wca\ac c f wga wfg Wba\ab wbi g wgk b wbj wge wjg k j e local subjective graph of peer i

8. Betweenness Centrality • The BC of a node  is the sum of the ratios of shortest paths between pairs of nodes passing through node : • computation of BC: the all-pair shortest path problem • the fastest algorithm for BC: • explores and counts the shortest paths using Breadth-First Search starting from every node in the network • aggregates efficiently the path counts # shortest paths between nodes s,t passing through node  # shortest paths between nodes s,t

9. Experimental Setup 1: Dataset • growing synthetic and Bartercast graphs • the synthetic graphs grow from 1,000 up to 20,000 nodes • 20 instances, each one containing the previous one + 1,000 new nodes • for the BarterCast graph: • we crawled BarterCast from 24 July to 9 September 2009 • it starts with 1,592 nodes and reaches up to 2,408 nodes

10. Experimental Setup 2: Graph Types • random graph • each new node is connected to every existent node with a constant probability P • power-law graph • each new node is preferentially attached to existent nodes with a probability proportional to their degree. • its degree distribution is expressed as P(k)ck- • only a few nodes are highly connected • graph derived from Bartercast • power-law exponent : 2.2 power-law exponent

11. Approximation 1: Growing Graphs • the most central node in real graphs does not change often due to their structural properties and so, we don’t have to update BC values often. • focus on the stability of the top-n most central nodes • consider the sequences of IDs of the top-n most central nodes in consecutive graph instances • we use two metrics: • the number of common nodes in two consecutive sequences • the minimal number of transpositions needed to get all the common nodes of latter sequence in the order of the previous

12. Approximation 1: Growing Graphs Random Graph Power-law Graph In power-law graphs, the most central nodes remain almost invariant in time and so, BC has not to be recomputed often Number of common nodes Number of common nodes Instance number (t) Instance number (t) Number of transpositions Number of transpositions Instance number (t) Instance number (t)

13. Approximation 2: Large Graphs • Pivot BC (P-BC): random selection of a small subset of nodes (the pivots) to start Breadth-First Search • Overestimation of the BC of nodes close to pivots • Scale BC (S-BC): like P-BC but normalized over the distance of a node from the pivots • k-BC: exploring the paths of length at most equal to k k=2

14. Approximation 2: Large Graphs Power-Law Graph Random Graph • In power-law graphs, the approximations of BC are highly accurate (S-BC achieves the best accuracy) • In random graphs, all the approximations have a lower accuracy (k-BC achieves the best accuracy) Number of correctly identified nodes Number of correctly identified nodes Cost of computation Cost of computation Number of transpositions Number of transpositions Cost of computation Cost of computation

15. Approximation 2: Large Graphs BarterCast Graph In BarterCast graphs, the approximations are accurate enough, with S-BC achieving the best results Number of correctly identified nodes Number of transpositions Cost of computation Cost of computation

16. Integration in BarterCast 1: Setup • we integrate P-BC, S-BC and k-BC in BarterCast evaluating their effect • each peer identifies the most central node in its subjective graph using one of these approximations and then applies max-flow with that node as a start point • two metrics • coverage: the fraction of peers in a subjective graph for which the local reputations turn out to be non-zero • relativeaverage error: the absolute difference of the locally computed reputations of the peers and their actual reputations

17. Integration in BarterCast 2: Results • BC=0: a node with BC equal to 0 • 1/2maxBC: the node with BC equal to 50% of the maximum BC • maxBC: the node with the maximum BC • Using the most central node in the computation of reputation results in better coverage and smaller average error • S-BC and k-BC identify the most central node correctly Coverage Relative Average Error

18. Conclusions & Future Work • power-law graphs: the approximation of BC are efficient and highly accurate • random graphs: it is harder to identify the most central nodes • using the node with the highest BC increases the accuracy and the coverage in Bartercast • k-BC and S-BC identify correctly the most central node in Bartercast • future work: not keeping the complete history of transferred data for the computation of reputation • limited size of memory • computational cost • accuracy

19. Questions? www.pds.ewi.tudelft.nl www.tribler.org contact: d.gkorou@tudelft.nl