Parallel Subgraph Listing in a Large-Scale Graph

1. Parallel Subgraph Listing in a Large-Scale Graph Yingxia ShaoBin Cui Lei ChenLin Ma Junjie Yao NingXu   School of EECS, Peking University  Hong Kong University of Science and Technology

2. Outline • Subgraph listing operation • Related work • PSgL framework • Evaluation • Conclusion

3. Motivation Triangle Counting in SN Cascades Counting in RN Motif Detection in Bioinformatics

4. Problem Definition Subgraph Listing Operation • Input:pattern graph, data graph[both are undirected] • Output: all the occurrencesof pattern graph in the data graph. Goal of our work • Efficiently listing subgraph in a large-scale graph Pattern graph Data graph

5. Related Work Centralized algorithms • Enumerate one by one [Chiba ’85, Wernicke ’06, Grochow ’07] Streaming algorithms • Only counting and results are inaccurate [Buriol ’06, Bordino ’08, Zhao ’10] MapReduce based Parallel algorithms • Decompose pattern graph + explicit join operation [Afrati ’13] • Fixed exploration plan + implicit join operation [Plantenga ’13] Other efficient algorithms for specific pattern graph • Triangle [Suri ’11, Chu ’11, Hu ’13]

6. Drawbacks in existing parallel solutions • MapReduce is not friendly to process graphs. • Join operation is expensive. • Do not take care of the balance of data distribution. • Data graph • Intermediate results The novel PSgLframework lists subgraph via graph traversal on in-memory stored native graph.

7. Contributions • We propose an efficient parallel subgraph listing framework, PSgL. • We introduce a cost model for the subgraph listing in PSgL. • We propose a simple but effective workload-aware distribution strategy, which facilitates PSgLto achieve good workload balance. • We design three independent mechanisms to reduce the size of intermediate results.

8. Partial subgraph instance • A data structure that records the mapping between pattern graph and data graph. • Denoted by • Assume the vertices of are numbered from 1 to , we simply state as {map(1), map(2), ..., map()}. {?,?,?,?} {2,3,4,5} {1,5,6,?}

9. Independence Property • Tree • A node is a • The children of a node are derived from expanding one mapped data vertex in the node. • Characteristics • A encodes a set of results. • s are independent from each other except the ones in its generation path. Tree

10. PSgL: Parallel Subgraph Listing Framework • PSgLfollows the popular graph processing paradigm • vertex centric model • BSP model • PSgL iteratively generates in parallel; • Each is expanded by a data vertex.

11. Algorithm of Expanding a - I • Partial Pattern Graph encodes • pattern graph, • , • progress state. • Three types of vertices • BLACK vertex is the one which has been expanded. • GRAY vertex has a mapped data vertex, but it has not been expanded. • WHITE vertex is the one which hasn’t been mapped to any data vertex.

12. Algorithm of Expanding a Gpsi - II • Main logic • Changes one GRAY vertex into BLACK; • Validates the expanding vertex’s GRAY neighbors; • Makes the expanding vertex’s WHITE neighbor become GRAY. • Two observations • In each expansion, at least one pattern vertex is processed. • All GRAYs are the valid candidates for the next expansion. Example: expanding vertex <4, 2>

13. Efficiency of PSgL # of Gpsi processed by worker k # of iterations Total cost cost of processing a Gpsi # of workers • Three metrics: • The number of iterations. • S is bounded by |MVC| ≤ S ≤ |Vp| - 1. • Workload balance. • Required by the max function. • The number of s *Refer to the paper for the details of estimating load().

14. Workload balance - I • Partial subgraph instance distribution problem • There are N s to be processed by K workers, the goal is to find out a distribution strategy to achieve • NP-hard problem! • Naive Solutions • Random distribution strategy • Roulette wheel distribution strategy • has a higher probability to be expanded by a data vertex with smaller degree.

15. Workload balance - II • Workload aware distribution strategy • A general greedy-based heuristic rule. All three strategies have the same worst bound which is K*|OPT|. But in practice, α = 0.5 performs best.

16. Comparison among various approaches Random Roulette =1 =0 =0

17. Partial subgraphinstance reduction - I • Pattern graph automorphism breaking • Using DFS to find the equivalent vertex group • Assign partial order for each equivalent vertex group • Initial pattern vertex selection • Introduce a cost model • General pattern graph • Enumerate all possible selections based on cost model • Cycle and clique • The vertex with lowest rank is the best one. Cost Model Best Initial Pattern Vertex < < < Initial Pattern Vertex Section based on cost model Automorphism Breaking

18. Partial subgraphinstance reduction - II • Online pruning invalid s • Filter by the partial order and degree restriction • Prune with the help of a light weight global edge index • Using bloom filter to index the ends of an edge PG1 PG4 PG5

19. Evaluation - Comparing to MR solutions • Afrati and SGIA-MR are the state-of-art MapReduce solutions. • The ratios exceed 100 times are not visualized. PSgL: 4302s Afrati: 7291s

20. Evaluation - Comparing to GraphLab * using a different traversal order.

21. Conclusion • Subgraph listing is a fundamental operation for massive graph analysis. • We propose an efficient parallel subgraph listing framework, PSgL. • Various distribution strategies • Cost model • Light-weight global edge index • The workload-aware distribution strategy can be extended to other balance problems. • A new execution engine is required for larger pattern graphs.

22. Thanks!

23. Backup Expr. – Scalability of PSgL Performance vs. Worker Number

24. Backup Expr. – Initial pattern vertex selection Random graph Livejournal Influences of the Initial Pattern Vertex on Various Data Graphs