Techniques for Graph Analytics on Big Data

1. Techniques for Graph Analytics on Big Data UsmanNisar, ArashFard, John A. Miller Computer Science Department University of Georgia, Athens, GA

2. Outline • Introduction • Subgraph Pattern Matching • Types of Subgraph Pattern Matching • Models of Computation • Distributed Algorithms • Performance Evaluation • Graph Partitioning • Results Analysis • Conclusion

3. Introduction • Directed graph G(V, E, l) • V is a set ofvertices • E ⊆ V x V is a set of directed edges • l: V →𝓝 is a function mapping vertices to labels • Versatility and expressivity • Social networks, web search engines, genome sequencing, etc. • Data sizes are growing rapidly • Facebook: 1 billion + users, average degree of 140 • Twitter: 200 million + users, 400 million tweets each day • Bigger datasets mean bigger graphs

4. Subgraph Pattern Matching • A graphG’(V’, E’, l’) is a subgraph of G(V, E, l) if: • V’ ⊆ V • E’ ⊆ V’ x V’ ⊆ E • ∀v ∈ V’ , l’(v) = l(v) • A pattern (or query) is a graph that we want to find in a bigger graph • Subgraph pattern matching: Given a query graph, find all subgraphs of another graph (the datagraph) that are similar to the query based on certain criteria

5. Types of Subgraph Pattern Matching • Exact • SubgraphIsomorphism • Heuristic • Graph Simulation • Dual Simulation • Strong Simulation

6. Subgraph Isomorphism Exact matching from the pattern to data graph Labels must be the same Ullmann’salgorithm, VF2 NP-hard problem in general case Current solutions not practical for large graphs

7. Example: SubgraphIsomorphism Matches: {4, 8, 6, 7}, {5, 8, 6, 7}

8. Graph Simulation [1]  M. R. Henzinger, T. A. Henzinger, and P. W. Kopke, “Computing simulations on finite and infinite graphs,” in Foundations of Computer Science, 1995. Proceedings., 36th Annual Symposium on. IEEE, 1995, pp. 453–462. • A vertex in a data graph G matches a vertex in query graph Q via graph simulation iff • both have the same label • a subset of its children match all the children of its corresponding vertex in the query graph • HHK algorithm[1]

9. Graph Simulation

10. Dual Simulation • Adds a duality condition to graph simulation • A vertex in a data graph becomes a match with a vertex in the query graph iff • Both have the same label • A subset of its children matches all the children of its corresponding vertex in the data graph • A subset of its parents matches all the parents of its corresponding vertex in the data graph • O((|Vq| + |Eq|) (|V| + |E|)) • More restrictive than graph simulation

11. Example: Graph  Dual Simulation

12. Example: Graph  Dual Simulation

13. Strong Simulation • Extends dual simulation with a locality condition • A ball G[v, r]is a subset of a graph G containing: • all vertices VBwithin an undirected distance r of the vertex v • all the edges between the vertices in VB

14. Strong Simulation • Extends dual simulation with a locality condition • A ball G[v, r]is a subset of a graph G containing: • all vertices VBwithin an undirected distance r of the vertex v • all the edges between the vertices in VB

15. Strong Simulation • Extends dual simulation with a locality condition • A ball G[v, r]is a subset of a graph G containing: • all vertices VBwithin an undirected distance r of the vertex v • all the edges between the vertices in VB • O(|V|(|V| + (|Vq| + |Eq|) (|V| + |E|)) • Query graph Q(Vq,Eq,l) matches data graph G(V, E, l) via strong simulation if there exists a vertex v ∈ Vs.t. • Q matches G[v, dq] via dual simulation with match relation Rb • v is contained in Rb

16. Example: Dual  Strong Simulation

17. Example: Dual  Strong Simulation

18. Example: Dual  Strong Simulation

19. Example: Dual  Strong Simulation

20. Models of Computation

21. Distributed Graph Simulation

22. Distributed Graph Simulation

23. Other Distributed Graph Algorithms • Dual simulation • Similar to graph simulation • In addition to storing the match sets of its children, a vertex also stores the match sets of its parents • During match set evaluation, a vertex takes both of these into account • Strong simulation (optimized) • First run dual simulation to obtain match relation R • For each matching vertex v in R: • Create a ball centered at v with radius dqcontaining only vertices in R • Perform dual simulation on the ball

24. Performance Evaluation • Experimental setup • Cluster with 12 machines • Each with two 2Ghz Intel Xeon E5-2620 CPUs, each with six cores • Ethernet: 1 Gb/s • Set up HDFS/GPS on all of the machines

25. Runtime Evaluation Synthesized, |V|= 108

26. Runtime Evaluation uk-2005, |V|= 3.9 x 107

27. Runtime Evaluation enwiki-2013, |V|= 4.2 x 106

28. Speedup Evaluation Synthesized, |V|= 108

29. Speedup Evaluation uk-2005, |V|= 3.9 x 107

30. Speedup Evaluation enwiki-2013, |V|= 4.2 x 106

31. Conclusions • The three algorithms exhibit excellent scalable behavior in terms of speedup and efficiency as we increase the number of workers • Distributed implementations (GPS): • Graph simulation • Dual simulation • Strong simulation • Ongoing work: • Distributed implementations with message passing in Akka • Our own sequential/distributed isomorphism algorithm

32. Questions?

33. Thanks

34. Depth-First Ball Algorithm works in a depth-first fashion. Message is generated at the center which is then propagated through the system for ballSizesupersteps. The approach results in an exponential number of messages that slows down the whole system and renders the approach impractical

35. Breadth-First Ball Works on a simple ping-reply model. Center vertex starts of by sending a ping message to all of its adjacent nodes in the first superstep. In the second superstep, all the recipient nodes reply back with their label and the ids of their children and parents. Center vertex upon receiving this information in the third superstep, saves the returned labels and then ping the boundary nodes. This process is repeated till we have a ball of size dq.

36. Breadth-First Ball

37. Efficiency • Calculated as Efficiencyk = Speedupk/k Synthesized, |V|=108 uk-2005, |V|=3.9x107 enwiki-2013, |V|=4.2x106

38. Graph Partitioning 1 http://glaros.dtc.umn.edu/gkhome/metis/metis/overview • By default, the data graph is partitioned in a round-robin fashion among workers • The goals of min-cut partitioning are two-fold: • to create well-balanced partitions • to reduce the inter-partition edges • Number of other algorithms have been shown to use min-cut graph partitioning successfully for speed-ups • METIS1 – graph partitioning tool • Written in C • Takes an undirected graph and outputs the partitions

39. Performance Evaluation • Experimental Setup • Cluster with 5 machines • Each with two 2Ghz Intel Xeon E5-2620 CPU, each with six cores • Ethernet: 1 Gb/sec • Setup HDFS/GPS on all the machines • Datasets: • Labels(l) = 200

40. Results - Runtime Graph Simulation Dual Simulation Strict Simulation Synthesized Dataset, |V|=107, = 1.2

41. Results - Runtime Graph Simulation Dual Simulation Strict Simulation uk-2002, |V|=1.8x107

42. Results – Network I/O Graph Simulation Dual Simulation Strict Simulation Synthesized Dataset, |V|=107,

43. Results – Network I/O Graph Simulation Dual Simulation Strict Simulation uk-2002, |V|=1.8x107

44. Planned Pattern Matching Implementations