School of Computer Science Carnegie Mellon Graph Mining, self-similarity and power laws
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1 Graph Mining, self-similarity and power laws Christos Faloutsos University
2 Overview Achievements global patterns and laws (static/dynamic) generators influence propagation communities; graph partitioning local patterns: frequent subgraphs Challenges DoE/DoD, 2007 C. Faloutsos 2
3 Motivating questions: How does the Internet look like? What constitutes a normal social network? network value of a customer? [Domingos+] which gene/species affects the others the most? DoE/DoD, 2007 C. Faloutsos 3
4 Problem #1 - topology How does the Internet look like? Any rules? DoE/DoD, 2007 C. Faloutsos 4
5 Solution#1: Rank exponent R A1: Power law in the degree distribution [SIGCOMM99] internet domains log(degree) ibm.com att.com log(rank) DoE/DoD, 2007 C. Faloutsos 5
6 Power laws In- and out-degree distribution of web sites [Barabasi], [IBM-CLEVER] log(freq) from [Ravi Kumar, Prabhakar Raghavan, Sridhar Rajagopalan, Andrew Tomkins ] log indegree DoE/DoD, 2007 C. Faloutsos 6
7 epinions.com count who-trusts-whom [Richardson + Domingos, KDD 2001] (out) degree DoE/DoD, 2007 C. Faloutsos 7
8 Even more power laws: web hit counts [w/ A. Montgomery] Web Site Traffic log(count) Zipf yahoo.com log(freq) DoE/DoD, 2007 C. Faloutsos 8
![Montgomery] Web Site Traffic](/docs-images/57/6209100/images/page_8.jpg "log(count) Zipf yahoo.")
9 Overview Achievements global patterns and laws (static/dynamic) generators influence propagation communities; graph partitioning local patterns: frequent subgraphs Challenges DoE/DoD, 2007 C. Faloutsos 9
10 Problem#2: evolution Given a graph: how will it look like, next year? [from Lumeta: ISPs 6/1999] DoE/DoD, 2007 C. Faloutsos 10
11 Evolution of diameter? Prior analysis, on power-law-like graphs, hints that diameter ~ O(log(N)) or diameter ~ O( log(log(n))) i.e.., slowly increasing with network size Q: What is happening, in reality? DoE/DoD, 2007 C. Faloutsos 11
12 Evolution of diameter? Prior analysis, on power-law-like graphs, hints that diameter ~ O(log(N)) or diameter ~ O( log(log(n))) i.e.., slowly increasing with network size Q: What is happening, in reality? A: It shrinks(!!), towards a constant value DoE/DoD, 2007 C. Faloutsos 12
13 Shrinking diameter ArXiv physics papers and their citations [Leskovec+05a] DoE/DoD, 2007 C. Faloutsos 13
![citations [Leskovec+05a]](/docs-images/57/6209100/images/page_13.jpg "DoE/DoD, 2007 C.")
14 Shrinking diameter ArXiv: who wrote what DoE/DoD, 2007 C. Faloutsos 14
15 Shrinking diameter U.S. patents citing each other DoE/DoD, 2007 C. Faloutsos 15
16 Shrinking diameter Autonomous systems DoE/DoD, 2007 C. Faloutsos 16
17 Temporal evolution of graphs N(t) nodes; E(t) edges at time t suppose that N(t+1) = 2 * N(t) Q: what is your guess for E(t+1) =? 2 * E(t) DoE/DoD, 2007 C. Faloutsos 17
18 Temporal evolution of graphs N(t) nodes; E(t) edges at time t suppose that N(t+1) = 2 * N(t) Q: what is your guess for E(t+1) =? x2 * E(t) A: over-doubled! DoE/DoD, 2007 C. Faloutsos 18
19 Densification Power Law ArXiv: Physics papers and their citations E(t) 1.69 N(t) DoE/DoD, 2007 C. Faloutsos 19
20 Densification Power Law U.S. Patents, citing each other E(t) 1.66 N(t) DoE/DoD, 2007 C. Faloutsos 20
21 Densification Power Law Autonomous Systems E(t) 1.18 N(t) DoE/DoD, 2007 C. Faloutsos 21
22 Densification Power Law ArXiv: who wrote what E(t) 1.15 N(t) DoE/DoD, 2007 C. Faloutsos 22
23 Summary of laws Static graphs power law degrees; power law eigenvalues communities (within communities) small diameters Dynamic graphs shrinking diameter densification power law DoE/DoD, 2007 C. Faloutsos 23
24 Overview Achievements global patterns and laws (static/dynamic) generators influence propagation communities; graph partitioning local patterns: frequent subgraphs Challenges DoE/DoD, 2007 C. Faloutsos 24
25 Problem#3: Generators Q: what local behavior can generate such graphs? DoE/DoD, 2007 C. Faloutsos 25
26 Problem#3: Generators Q: what local behavior can generate such graphs? A1: Preferential attachment [Barabasi+] A2: copying model [Kleinberg+] A3: forest-fire model [Leskovec+] A4: Kronecker [Leskovec+] A5: Economic reasons [Papadimitriou+] DoE/DoD, 2007 C. Faloutsos 26
27 Problem#3: Generators Q: what local behavior can generate such graphs? A1: Preferential attachment A2: copying model A3: forest-fire model A4: Kronecker A5: Economic reasons power law degree + communities + DPL + easily parallilizable DoE/DoD, 2007 C. Faloutsos 27
28 Overview Achievements global patterns and laws (static/dynamic) generators influence propagation communities; graph partitioning local patterns: frequent subgraphs Challenges DoE/DoD, 2007 C. Faloutsos 28
29 Problem#4: influence propagation how do influence/rumors/viruses propagate? what is the best customer to market to? DoE/DoD, 2007 C. Faloutsos 29
30 Problem#4: influence propagation how do influence/rumors/viruses propagate? tipping point [Kleinberg+] first eigenvalue -> epidemic threshold [Chakrabarti+] what is the best customer to market to? network value of a customer [Domingos+] DoE/DoD, 2007 C. Faloutsos 30
31 Overview Achievements global patterns and laws (static/dynamic) generators influence propagation communities; graph partitioning local patterns: frequent subgraphs Challenges DoE/DoD, 2007 C. Faloutsos 31
32 Problem #5: communities how to find natural communities, quickly? how to find strange /suspicious/valuable edges? DoE/DoD, 2007 C. Faloutsos 32
33 Problem #5: communities how to find natural communities, quickly? network flow [Flake+] node/edge betweeness (~ stress ) cross-associations [Chakrabarti+] 2 nd eigenvalue; METIS [Karypis+] random walks [Newman] etc etc etc DoE/DoD, 2007 C. Faloutsos 33
34 Problem #5: communities connection sub-graphs [Faloutsos+] DoE/DoD, 2007 C. Faloutsos 34
35 Problem #5: communities connection sub-graphs [Faloutsos+] DoE/DoD, 2007 C. Faloutsos 35
36 Problem #5: communities connection sub-graphs [Faloutsos+] BANKS system [Chakrabarti+] ObjectRank [Papakonstantinou+] DoE/DoD, 2007 C. Faloutsos 36
37 Overview Achievements global patterns and laws (static/dynamic) generators influence propagation communities; graph partitioning local patterns: frequent subgraphs Challenges DoE/DoD, 2007 C. Faloutsos 37
38 Problem #6: local patterns Which sub-graphs are common/frequent? H C C N C C C H N C molecule #1 molecule #M DoE/DoD, 2007 C. Faloutsos 38
39 Problem #6: local patterns Which sub-graphs are common/frequent? H C C N C C C H N C molecule #1 molecule #M DoE/DoD, 2007 C. Faloutsos 39
40 Problem #6: local patterns Clever extensions of Association Rules (= frequent itemsets / market basket analysis) [Jiawei Han+] [Jian Pei+] [G. Karypis] [M. Zaki+]... DoE/DoD, 2007 C. Faloutsos 40
41 Overview Achievements Challenges time evolving graphs multi-graphs DoE/DoD, 2007 C. Faloutsos 41
42 Time evolving graphs Q: what will happen next? (eg., on a traffic matrix?) 1/1/2005 DoE/DoD, 2007 C. Faloutsos 42
43 Time evolving graphs Q: what will happen next? (eg., on a traffic matrix?) 1/2/2005 DoE/DoD, 2007 C. Faloutsos 43
44 Time evolving graphs Q: what will happen next? (eg., on a traffic matrix?) 1/3/2005 DoE/DoD, 2007 C. Faloutsos 44
45 Time evolving graphs Q: what will happen next? (eg., on a traffic matrix?) 1/4/2005???? DoE/DoD, 2007 C. Faloutsos 45
46 Overview Achievements Challenges time evolving graphs multi-graphs DoE/DoD, 2007 C. Faloutsos 46
47 Multi-graphs Patterns/outliers? friends co-authors DoE/DoD, 2007 C. Faloutsos 47
48 Multi-graphs Relational Learning link prediction, feature extraction [Jensen+] etc Dis-ambiguation, de-duplication [Getoor+] etc friends co-authors DoE/DoD, 2007 C. Faloutsos 48
49 source person School of Computer Science Promising solution: tensors I x J x K destination person type of contact DoE/DoD, 2007 C. Faloutsos 49
50 Tensors: ~SVD, for >=3 modes C I x J x K Ix R K x R Jx R ¼ A R x R x R B = + + Foil: from Tamara Kolda (Sandia) DoE/DoD, 2007 C. Faloutsos 50
51 Specially Structured Tensors Tucker Tensor Kruskal Tensor Our Notation Our Notation core W K x T K x R W I x J x K Ix R Jx S I x J x K w 1 Ix R w R Jx R = U R x S x T V = = u 1 v 1 U + + V R x R x R u R v R Foil: from Tamara Kolda (Sandia) DoE/DoD, 2007 C. Faloutsos 51
52 Conclusions - achievements Surprising patterns in graphs power laws; communities small/shrinking diameters simple, local behavior can lead to such patterns (eg., preferential attachment, etc) fast algorithms for communities/partitioning fast algorithms for frequent subgraphs DoE/DoD, 2007 C. Faloutsos 52
53 Conclusions - next steps multi-graphs time-evolving graphs scalability (graph sampling) (large graph visualization) DoE/DoD, 2007 C. Faloutsos 53
54 Conclusions - philosophically: deep connections with self-similarity, cellular automata (~agents), and fractals DoE/DoD, 2007 C. Faloutsos 54
55 Resources Manfred Schroeder Chaos, Fractals and Power Laws, 1991 A-L. Barabasi, Linked, 2002 D. Watts, Six Degrees, 2004 DoE/DoD, 2007 C. Faloutsos 55
56 References R. Albert, H. Jeong, and A.-L. Barabási, Diameter of the World Wide Web. Nature 401, (1999) A. Fabrikant, E. Koutsoupias, and C. Papadimitriou. Heuristically Optimized Tradeoffs: A New Paradigm for Power Laws in the Internet. Int. Colloquium on Automata, Languages, and Programming (ICALP), Malaga, Spain, July DoE/DoD, 2007 C. Faloutsos 56
57 References [sigcomm99] Michalis Faloutsos, Petros Faloutsos and Christos Faloutsos, What does the Internet look like? Empirical Laws of the Internet Topology, SIGCOMM 1999 [Leskovec 05] Jure Leskovec, Jon M. Kleinberg, Christos Faloutsos: Graphs over time: densification laws, shrinking diameters and possible explanations. KDD 2005: DoE/DoD, 2007 C. Faloutsos 57
58 References [brite] Alberto Medina, Anukool Lakhina, Ibrahim Matta, and John Byers. BRITE: An Approach to Universal Topology Generation. MASCOTS '01 Xifeng Yan, Xianghong Jasmine Zhou, Jiawei Han: Mining closed relational graphs with connectivity constraints. KDD 2005: DoE/DoD, 2007 C. Faloutsos 58
59 Thank you! Contact info: christos <at> cs.cmu.edu www. cs.cmu.edu /~christos (w/ papers, datasets, code, etc) DoE/DoD, 2007 C. Faloutsos 59
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