Biology, Physics, Mathematics, Sociology, Engineering, Computer Science, Etc

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1 Motivation Motifs Algorithms G-Tries Parallelism Complex Networks Networks are ubiquitous! Biology, Physics, Mathematics, Sociology, Engineering, Computer Science, Etc Images: UK Highways Agency, Uriel Law Inc., MKT Portugal, Bigelow Laboratory for Ocean Sciences 1

2 Complex Network Analysis A network is the backbone of a system Network scientists and paleontologists? Extensive list of different metrics for mining interesting characterists Single Node (ex: degree) Global network (ex: average distance) 2

3 Motivation Motifs Algorithms G-Tries Parallelism Building Blocks of Networks Subnetworks, or subgraphs, can characterize the whole network Image: w3.org Some subgraphs can be significant patterns In 2002 the definition of network motifs was coined by Milo et al. (overrepresented subgraphs) 3

4 Motivation Motifs Algorithms G-Tries Parallelism Network Motifs Example Random Networks Motif Original Network Image: Adapted from (Milo et al., 2002) 4

5 Motivation Motifs Algorithms G-Tries Parallelism Goals Discovering Motifs is a computationaly hard problem Motif usage is limited in scale Improving motif discovery has multi-disciplinary impact Image: R. Steeg Main goal: Improve the efficiency of network motifs detection 5

6 Motivation Motifs Algorithms G-Tries Parallelism General Applicability Canon Definition Directed and undirected Colored and uncolored Variations on the concept Frequency concepts Under-represented (anti-motifs) Non-induced subgraphs Constraints for similar random graphs Weighted networks 6

7 Motivation Motifs Algorithms G-Tries Parallelism Network Motifs Discovery How to discover network motifs? Subgraph Census on original network Set of Subgraphs and their frequencies Calculate Significance which is done by Generate ensemble of similar random networks On each generated network Census to determine frequency 7

8 Motivation Motifs Algorithms G-Tries Parallelism Subgraph Census Census is bottleneck (>95% time) General Problem Definition: Input: list of subgraphs S and a bigger graph G Output: frequency count of each subgraph of S in G 8

9 Motivation Motifs Algorithms G-Tries Parallelism Historical Timeline 9

10 Motivation Motifs Algorithms G-Tries Parallelism Algorithms for Census Two main approaches Network-centric: enumerate all subgraphs and compute isomorphisms (ESU, Kavosh) Subgraph-centric: match a single subgraph (Grochow and Kellys, MODA) 10

11 G-Tries Motivation Sequences and prefix trees Can the concept be extended? 11

12 G-Tries Concept Subgraphs have common substructure! Create a tree where each tree node corresponds to a single graph vertex G-Tries (etymology Graph retrieval ) 12

13 GTCanon Example 13

14 Matching Subgraphs Backtracking procedure Candidates for node 1: {0, 1, 2, 3, 4, 5} Try 0: Match = {0}, Neighb. = {1,3,4} Try 1: Match = {0,1}, Neighb. = {2,3,4,5} Try 2: no edge from 2 to 0! FAIL Try 3: no edge from 3 to 1! FAIL Try 4: Match = {0, 1, 4} FOUND! Try 5: no edge from 5 to 1! FAIL 14

15 Search Tree Backtracking produces search tree {} {1} {2} {1,2} {1,2,4} {1,2,5} {1,2,3} 15

16 Global Improvements 30.1X faster, on average, than competing algorithms, for all networks and k sizes considered 37.2X on undirected networks, 23.0X on directed. Considering only last k for each Efficient network, Subgraph average speedup grows to 44.3X. 57.2X on undirected networks, 31.4X on directed. 16

17 Parallelism on Motif Discovery Parallelism on motif discovery has been scarcely used Opportunities for Parallelism Census Parallelization (one census in parallel) Partition (pre-divide network) Tree (recursive tree procedure in parallel) Query (single query in subgraph-centric in parallel) Random Networks Parallelization Significance Parallelization 17

18 General Parallel Approach Motif Discovery as a tree shaped computation (ESU or G-Tries) 18

19 Independent Search Branches Sequential algorithm produces a tree-shaped search space Search tree nodes are independent from each other! {0,1,3} If we know where we are, we can continue from there Tree Nodes -> Work Units 19

20 Parallel Problem Input: set of work units G-Trie: (Network, G-Trie Node, Partial Match) ESU: (Network, Partial Match, Possible Extensions) Goal: efficiently distribute work units among processors Target: distributed memory with message passing Constraints: Tree highly unbalanced Pre-determined static allocation is very hard! Requires dynamic load balancing 20

21 Parallel Program Flow We divide the parallel job in 3 phases: Pre-Processing Phase (before computation) all_in_one, static_partition Work Phase (computation of frequencies) master-worker, distributed_queues, distributed_snapshot Aggregation phase (store frequencies on single processor) naïve, hierarchical, collective 21

22 Distributed Snapshot Receiver-Initiated Strategy 1. While computation not ended If work units available Process work unit Else Someone asked for work?» Stop my computation» Divide work in 2 similar halves» Send half to requester» Return to computation Request work units from other processor 22

23 Running Computation Example Computation G-Trie Node Graph Vertex 23

24 Running Computation Example Computation G-Trie Node Graph Vertex 24

25 Running Computation Example Computation G-Trie Node Graph Vertex 25

26 Running Computation Example Computation G-Trie Node Graph Vertex 26

27 Stopping Computation Example Computation G-Trie Node Graph Vertex Current Work Unit Explored Work Units STOP 27

28 Dividing Computation Example Computation Keep Give to requester Both Diagonal Splitting 28

29 Snapshot Master-Worker and Distributed Queues introduce time overhead Work Units must be added and removed from the queue vs natural recursive calls A space overhead is also introduced Work units share some substructure 1 {1,2,3},{1,2,4},{1,2,5}, {1,3,4},{1,3,5},{1,3,6} 2 3 We create a compact representation of the search state (tree-shaped) Take advantage of common substructure in work units Efficient methods for: stopping, dividing, resuming

30 Work Request When we do not have work, which processor should we contact? No data locality Search trees completely unbalanced Ask a random processor! Random polling ([Sanders 1994]) 30

31 Aggregation Phase Agreement on list of subgraphs G-Tries: given ESU: binary tree Communicate only frequencies (position indicates which graph) Hierarchical: binary tree Collective: MPI collective communication (MPI_Reduce) 31

32 Some Parallel Results Environment used Dedicated cluster, 12 SuperMicro TwinView Server (2 quad Core Xeon 5335, 12GB Ram), 3.8TB space, Max 128 processors Infiniband, OpenMPI Pre-processing phase static_partition better than all_in_one Aggregation collective better then hierarchical and naïve 32

33 Some Parallel Results Compute all k motifs Hybrid approach: ESU + G-Tries Increase k until execution time > 1h Distributed Snapshot always better Ex: dolphins network, 10-motifs, 128 processors, the speedup is: Master-Worker: 92.3 Distributed Queue: Distributed Snapshot:

34 Some Parallel Results Set of 12 representative real networks Nr. Neighbours Network Group Directed V(G) E(G) Average Max dolphins social no circuit physical no neural biological yes 297 2, metabolic biological yes 453 2, links social yes 1,490 19, coauthors social no 1,589 2, ppi biological no 2,361 6, odlis semantic yes 2,909 18, power physical no 4,941 6, company social yes 8,497 6, foldoc Semantic yes 13, , internet Physical no 22,963 48, ,390 34

35 Some Parallel Results Absolute Speedup (distributed snapshots) #CPUs: Speedup Network K dolphins circuit neural metabolic links coauthors ppi odlis power company foldoc internet

36 Some Parallel Results We experimented to find motifs, obtaining similar levels of scalability Varying the number of random networks Using sampling Combine power of parallelism with g-tries Consistently achieve speedups of more than 2000x relative to previous sequential state-of-the-art approaches 36

37 Main Contributions Survey and comparison of previous sequential algorithms Time line, taxonomy, comparison table, pseudo-code, common implementation, empirical evaluation [e-science 2009] G-Trie Data Structure and associated algorithms Novel data structure, new methodology (set-centric), custom canonical implementation, custom symmetry breaking conditions, sampling methodology, implementation, empirical evaluation [ACM-SAC 2010] [WABI 2010] Parallel opportunities in motif discovery Characterization of opportunities, taxonomy, classification of previous approaches [JPDC 2011] General Scalable Parallelization of Subgraph Counting Usage with G-Tries and ESU, efficient methods for stopping, dividing and resuming a computation, implementation, empirical evaluation up to 128 processors [BIOINFORMATICS 2010] [CLUSTER 2010] [JPDC 2011] 37

38 The End G-Tries: an efficient data-structure for subgraph counting, Fernando Silva and Luís Lopes Thank you for listening! Contacts: 38