Fast Graphlet Decomposition: Theory, Algorithms, and Applications
From social science to biology, graphlets have found numerous applications and were used as the building blocks of network analysis. In social science, graphlet analysis (typically known as k-subgraph census) is widely adopted in sociometric studies. Much of the work in this vein focused on analyzing triadic tendencies as important structural features of social networks (e.g., transitivity or triadic closure) as well as analyzing triadic configurations as the basis for various social network theories (e.g., social balance, strength of weak ties, stability of ties, or trust). In biology, graphlets were widely used for protein function prediction, network alignment, and phylogeny to name a few. More recently, there has been an increased interest in exploring the role of graphlet analysis in computer networking (e.g., for web spam detection, analysis of peer-to-peer protocols and Internet AS graphs), chemoinformatics, image segmentation, among others. While graphlet counting and discovery have witnessed a tremendous success and impact in a variety of domains from social science to biology, there has yet to be a fast and efficient approach for computing the frequencies of these patterns. The main contribution of this work is a fast, efficient, and parallel framework and a family of algorithms for counting graphlets of size k-nodes that take only a fraction of the time to compute when compared with the current methods used. The proposed graphlet counting algorithm leverages a number of theoretical combinatorial arguments for different graphlets. For each edge, we count a few graphlets, and with these counts along with the combinatorial arguments, we obtain the exact counts of others in constant time. Furthermore, we show a number of important machine learning tasks that rely on this approach, including graph anomaly detection, as well as using graphlets as features for improving community detection, role discovery, graph classification, and relational learning.
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Fast Graphlet Decomposition: Theory, Algorithms, and Applications
- 4. Network(Motifs:(Simple(Building(Blocks(of(Complex(Networks(– [Milo&et.&al&– Science&2002] The(Structure(and(Function(of(Complex(Networks(– [Newman&– Siam&Review&2003] 2"node' Graphlets 3"node' Graphlets 4"node' Graphlets Connected( Disconnected
- 5. ! Small&k9vertex&induced&subgraphs Network(Motifs:(Simple(Building(Blocks(of(Complex(Networks(– [Milo&et.&al&– Science&2002] The(Structure(and(Function(of(Complex(Networks(– [Newman&– Siam&Review&2003] 2"node' Graphlets 3"node' Graphlets 4"node' Graphlets Connected( Disconnected
- 6. ! Small&k9vertex&induced&subgraphs ! Motifs:&Occur&in&real9world&networks&with&frequencies&significantly( higher than&randomly&generated&networks Network(Motifs:(Simple(Building(Blocks(of(Complex(Networks(– [Milo&et.&al&– Science&2002] The(Structure(and(Function(of(Complex(Networks(– [Newman&– Siam&Review&2003] 2"node' Graphlets 3"node' Graphlets 4"node' Graphlets Connected( Disconnected
- 7. ! Small&k9vertex&induced&subgraphs ! Motifs:&Occur&in&real9world&networks&with&frequencies&significantly( higher than&randomly&generated&networks ! Applied&to&food&web,&genetic,&neural,&web,&and&other&networks • Found&distinct&graphlets in&each&case Network(Motifs:(Simple(Building(Blocks(of(Complex(Networks(– [Milo&et.&al&– Science&2002] The(Structure(and(Function(of(Complex(Networks(– [Newman&– Siam&Review&2003] 2"node' Graphlets 3"node' Graphlets 4"node' Graphlets Connected( Disconnected
- 8. AISTATS'2009
- 10. 9 9 9 9 9 ! Biological&Networks& • network&alignment,&protein&function&prediction ! Social&Networks& • triad&analysis,&community&detection,&Exp.&Random&Models ! Computer&Networks ! Internet&AS ! Cyber&Security& • spam&detection ! Ecology .(.(.
- 12. Ex:(Given(an(input(graph(G 9 How%many%triangles%in%G? 9 How%many%cliques%of%size%49nodes%in%G? 9 How%many%cycles%of%size%49nodes%in%G?
- 13. Ex:(Given(an(input(graph(G 9 How%many%triangles%in%G? 9 How%many%cliques%of%size%49nodes%in%G? 9 How%many%cycles%of%size%49nodes%in%G? " In%practice,%we%would%like%to%count%all%k9vertex%graphlets
- 15. ! Enumerate&all&possible&graphlets " Exhaustive%enumeration% is%too%expensive%
- 16. ! Enumerate&all&possible&graphlets " Exhaustive%enumeration% is%too%expensive% ! Count&graphlets for&each&node&– and&combine&all&node&counts [Shervashidze et.%al%– AISTAT%2009]%
- 17. ! Enumerate&all&possible&graphlets " Exhaustive%enumeration% is%too%expensive% ! Count&graphlets for&each&node&– and&combine&all&node&counts " Still%expensive%for%relatively%large%k%%%[Shervashidze et.%al%– AISTAT%2009]%
- 18. ! Enumerate&all&possible&graphlets " Exhaustive%enumeration% is%too%expensive% ! Count&graphlets for&each&node&– and&combine&all&node&counts " Still%expensive%for%relatively%large%k% [Shervashidze et.%al%– AISTAT%2009]% ! Other&recent&work&counts&only&connected& graphlets of&size&k=4 [Marcus%&%Shavitt – Computer%Networks%2012]% Not(practical(– scales&only&for&small&graphs&with&few& hundred/thousand&nodes/edges 9 taking%2400%secs for%a%graph%with%26K%nodes
- 19. Most&work&focused&on&graphlets of&k=3&nodes& In&this&work,&we&focus&on&graphlets of&k=3,4&nodes Efficient%Graphlet Counting%for%Large%Networks%%[Ahmed%et%al.,%ICDM%2015] Graphlet Decomposition:% Framework,%Algorithms,%and%Applications [Ahmed%et%al.,%KAIS%Journal%2016%(to%appear)]
- 20. Searching(Edge( Neighborhoods ① For(each(edge(do u v v2 v3v1 v4 v6 v7 edge
- 21. Searching(Edge( Neighborhoods ① For(each(edge(do • Count(All(3<node(graphlets ② Merge(counts(from(all(edges u v v2 v3v1 v4 v6 v7 edge Triangle 2<star 1<edge Independent(
- 22. Searching(Edge( Neighborhoods ① For(each(edge(do • Count(All(3<node(graphlets ② Merge(counts(from(all(edges u v v2 v3v1 V4 v6 v7 edge Triangle 2<star 1<edge Independent( # We(only(need(to( find/count(triangles # Use(equations to(get( counts(of(others(in(o(1) Triangle
- 24. How to count all 4-node graphlets? 4<Clique 4<Cycle4<Chrodal<Cycle Tailed<triangle 4<Path 3<Star 4<node<triangle 4<node<2star 4<node<2edge 4<node<1edge Independent(
- 25. Step(1 Step(2 Step(3 Searching(Edge( Neighborhoods For%each%edge% Find%the%triangles Count(4Gnode(graphlets For%each%edge% Count%49node%cliques% and%49node%cycles only Count(4Gnode(graphlets For%each%edge% Use%combinatorial%%% relationships%to%compute% counts%of%other%graphlets in%constant(time Step(4 Merge(counts(from(all(edges(
- 27. 4<Node(Graphlet Transition(Diagram( ± 1&edge Count(Cliques(&(Cycles(ONLY Use(relationships(&(transitions( to(count(all(other(graphlets in(constant(time 4<Cliques 4<Cycles Maximum&no.&triangles& Incident&to&an&edge Maximum&no.&stars Incident&to&an&edge
- 28. T T Relationship(between(4<cliques(&(4<ChordalCycles 4<Cliques 4<ChordalCycle e T T e No.&49ChordalCycles No.&&49Cliques Proof'in'Lemma'1'" Ahmed'et'al.,'ICDM'2015
- 29. T T Relationship(between(4<cliques(&(4<ChordalCycles T T No.&49ChordalCycles No.&&49Cliques 4<Cliques 4<ChordalCycle e e Proof'in'Lemma'1'" Ahmed'et'al.,'ICDM'2015
- 31. ! Shared&Memory&Implementation ! Tested&on&graphs&with&over&a&billion&edges ! Largest&systematic&investigation&on&300+&networks • Social,&web,&technological,& biological,&co9authorship,& infrastructure…& • Facebook&100&networks&from&a&variety&of&US&schools • Dense&graphs&from&the&DIMACS&challenge& • Large&collections&of&biological&and&chemical&graphs Details'in'the'paper Data/code'online
- 32. Comparison(to(RAGE([Marcus'&'Shavitt – J.'Computer'Networks'2011]''' Facebook100 Networks from US Schools Ours RAGE Time-in-Seconds
- 33. |V| |E| Ours RAGE Time-in-Seconds Baseline%(RAGE)% did%not%finish%for% most%graphs We'take'~45'mins for'socSorkut (117M'edges) We'take'~40'secs for'caSdblp (15M'edges) Most'graphlet counts'in'orders'of'106'– 1015
- 34. |V| |E| Ours RAGE Time-in-Seconds Baseline%(RAGE)% did%not%finish%for% most%graphs We'take'~4.5'secs for'webSgoogle (4.3M'edges) We'take'~4'secs for'infSroadSusa (29M'edges) Most'graphlet counts'in'orders'of'106'– 1015
- 35. 0 1 2 4 8 16 0 5 10 15 Number of Processing Units Speedup socfb−Texas socfb−OR socfb−UCLA socfb−Berkeley13 socfb−MIT socfb−Penn94 0 1 2 4 8 16 0 5 10 15 Number of Processing Units Speedup 0 1 2 4 8 16 0 2 4 6 8 10 12 14 Number of Processing Units Speedup tech−internet−as tech−WHOIS web−it−2004 web−spam 0 1 2 4 8 16 0 2 4 6 8 10 12 14 Number of Processing Units Speedup Strong'scaling'results Intel%Xeon%3.10%Ghz E592687W%server,%16%cores
- 36. Applications
- 40. ! D&D&– 1178&protein&graphs.&Binary&labeled&as&Enzymes&vs.& Non.&Enzymes ! MUTAG&– 188&mutagenic&compounds. Binary&labeled& (whether&or¬&they&have&a&mutagenic&effect&on&the&Gram9 negative&bacterium) ! 109fold&validation,&Support&Vector&Machine ! Used&2,3,4&node&graphlets as&features
- 41. Previous)work: in)machine)learning)&)biological)networks) Shervashidze et.al [AISTATS'2009] Feature'Extraction'Time: D&D' 2'hours,'45'mins MUTAG 4.73'secs
- 44. ! Unbiased&Estimation&of&Graphlet Counts 10 4 10 5 0.85 0.9 0.95 1 1.05 1.1 1.15 soc−orkut−dir Sample Size 10 4 10 5 0.85 0.9 0.95 1 1.05 1.1 1.15 soc−orkut−dir Sample Size x/y 10 4 10 5 0.9 0.95 1 1.05 1.1 1.15 soc−flickr Sample Size 10 4 10 5 0.9 0.95 1 1.05 1.1 1.15 soc−flickr Sample Size Estimation'of'counts'of'4"vertex'clique
- 45. ! Framework&&&Algorithms& • One&of&the&first¶llel&approaches&for&graphlet counting • On&average&460x&faster&than¤t& methods • Edge9centric&computations&(only&requires&access&to&edge& neighborhood) • Time&and&space9efficient • Sampling/estimation&methods& • Local/global&counting ! Applications • Large<scale graph&comparison,& classification,&and&anomaly&detection • Visual&analytics&and&real<time graphlet mining