CMU SCS : Multimedia Databases and Data Mining CMU SCS Must-read Material CMU SCS Must-read Material (cont d)

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1 15-826: Multimedia Databases and Data Mining Lecture #26: Graph mining - patterns Christos Faloutsos Must-read Material Michalis Faloutsos, Petros Faloutsos and Christos Faloutsos, On Power-Law Relationships of the Internet Topology, SIGCOMM R. Albert, H. Jeong, and A.-L. Barabasi, Diameter of the World Wide Web Nature, 401, (1999). Reka Albert and Albert-Laszlo Barabasi Statistical mechanics of complex networks, Reviews of Modern Physics, 74, 47 (2002). Jure Leskovec, Jon Kleinberg, Christos Faloutsos Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations, KDD 2005, Chicago, IL, USA 2 Must-read Material (cont d) D. Chakrabarti and C. Faloutsos, Graph Mining: Laws, Generators and Algorithms, in ACM Computing Surveys, 3 8(1), 2006 J. Leskovec, D. Chakrabarti, J. Kleinberg, and C. Faloutsos, Realistic, Mathematically Tractable Graph Generation and Evolution, Using Kronecker Multiplication, in PKDD 2005, Porto, Portugal 3 1

Michalis Faloutsos (UCR) George Siganos (UCR) 5 Introduction Internet Map [lumeta.

2 Outline Part 1: Topology, laws and generators Part 2: PageRank, HITS and eigenvalues Part 3: Influence, communities 4 Thanks Deepayan Chakrabarti (CMU) Soumen Chakrabarti (IIT-Bombay) Michalis Faloutsos (UCR) George Siganos (UCR) 5 Introduction Internet Map [lumeta.com] Food Web [Martinez 91] Protein Interactions [genomebiology.com] Graphs are everywhere! Friendship Network [Moody 01] 6 2

3 Graph structures Physical networks Physical Internet Telephone lines Commodity distribution networks 7 Networks derived from "behavior" Telephone call patterns , Blogs, Web, Databases, XML Language processing Web of trust, epinions 8 Outline Topology & laws Generators Discussion and tools Motivating questions: 9 3

4 Motivating questions What do real graphs look like? What properties of nodes, edges are important to model? What local and global properties are important to measure? How to generate realistic graphs? 10 Given a graph: Motivating questions Are there un-natural sub -graphs? (criminals rings or terrorist cells)? How do P2P networks evolve? 11 Why should we care? Internet Map [lumeta.com] Food Web [Martinez 91] Protein Interactions [genomebiology.com] Friendship Network [Moody 01] 12 4

5 Why should we care? Models & patterns help with: A1: extrapolations: how will the Internet /Web look like next year? A2: algorithm design: what is a realistic network topology, to try a new routing protocol? to study virus/rumor propagation, and immunization? 13 Why should we care? (cont d) A3: Sampling: How to get a good sample of a network? A4: Abnormalities: is this sub-graph / sub -community / sub-network normal? (what is normal?) 14 Outline Topology & laws Static graphs Evolving graphs Generators Discussion and tools 15 5

Topology How does the Internet look like? Any rules?

random (Erdos-Renyi) graph 100 nodes, avg degree = 2 before layout

6 Topology How does the Internet look like? Any rules? (Looks random right?) 16 Are real graphs random? random (Erdos-Renyi) graph 100 nodes, avg degree = 2 before layout after layout No obvious patterns (generated with: pajek /pajek/ ) 17 Laws and patterns Real graphs are NOT random!! Diameter in- and out- degree distributions other (surprising) patterns 18 6

7 Laws degree distributions Q: avg degree is ~2 - what is the most probable degree? count?? 2 degree 19 Laws degree distributions Q: avg degree is ~3 - what is the most probable degree? count?? count 2 degree 2 degree 20 S1.Power-law: outdegree O Frequency Exponent = slope O = Nov 97 Outdegree The plot is linear in log-log scale [FFF 99] freq = degree (-2.15) 21 7

8 S2.Power-law: rank R outdegree Exponent = slope R = R Dec 98 Rank: nodes in decreasing outdegree order The plot is a line in log-log scale 22 S3. Eigenvalues Let A be the adjacency matrix of graph The eigenvalue λ is: A v = λ v, where v some vector Eigenvalues are strongly related to graph topology A B C D 23 S3. Power-law: eigen E Eigenvalue Exponent = slope E = Dec 98 Rank of decreasing eigenvalue Eigenvalues in decreasing order (first 20) [Mihail+, 02]: R = 2 * E 24 8

9 S4. The Node Neighborhood N(h) = # of pairs of nodes within h hops 25 S4. The Node Neighborhood Q: average degree = 3 - how many neighbors should I expect within 1,2, h hops? Potential answer: 1 hop -> 3 neighbors 2 hops -> 3 * 3 h hops -> 3 h 26 S4. The Node Neighborhood Q: average degree = 3 - how many neighbors should I expect within 1,2, h hops? Potential answer: 1 hop -> 3 neighbors 2 hops -> 3 * 3 WE HAVE DUPLICATES! h hops -> 3 h 27 9

S4. The Node Neighborhood Q: average degree = 3 - how many neighbors should I expect within 1,2, h hops? Potential answer: 1 hop -> 3 neighbors 2 hops -> 3 * 3 avg degree: meaningless!

10 S4. The Node Neighborhood Q: average degree = 3 - how many neighbors should I expect within 1,2, h hops? Potential answer: 1 hop -> 3 neighbors 2 hops -> 3 * 3 avg degree: meaningless! h hops -> 3 h 28 S4. Power-law: hopplot H # of Pairs H = 4.86 # of Pairs H = 2.83 Hops Dec 98 Hops Router level 95 Pairs of nodes as a function of hops N(h)= h H 29 Observation Q: Intuition behind hop exponent? A: intrinsic=fractal dimensionality of the network... N(h) ~ h 1 N(h) ~ h

11 But: Q1: How about graphs from other domains? Q2: How about temporal evolution? 31 The Peer-to-Peer Topology [Jovanovic+] Frequency versus degree Number of adjacent peers follows a power-law 32 More Power laws Also hold for other web graphs [Barabasi+, 99], [Kumar+, 99] with additional rules (bi-partite cores follow power laws) 33 11

12 Time Evolution: rank R Domain level #days since Nov. 97 The rank exponent has not changed! [Siganos+, 03] 34 Outline Topology & laws Static graphs Evolving graphs Generators Discussion and tools 35 Any other laws? Yes! 36 12

13 Any other laws? Yes! Small diameter (~ constant!) six degrees of separation / Kevin Bacon small worlds [Watts and Strogatz] 37 Any other laws? Bow-tie, for the web [Kumar+ 99] IN, SCC, OUT, tendrils disconnected components 38 Any other laws? power-laws in communities (bi-partite cores) [Kumar+, 99] Log(count) n:1 n:3 n:2 Log(m) 2:3 core (m:n core) 39 13

14 Any other laws? Jellyfish for Internet [Tauro+ 01] core: ~clique ~5 concentric layers many 1-degree nodes 40 How about triangles? 41 Triangle Laws Real social networks have a lot of triangles 42 14

15 Triangle Laws Real social networks have a lot of triangles Friends of friends are friends Any patterns? 43 S5. Triangle Law: #1 [Tsourakakis ICDM 2008] HEP-TH ASN Epinions X-axis: # of Triangles a node participates in Y-axis: count of such nodes 1-44 S6. Triangle Law: #2 [Tsourakakis ICDM 2008] Reuters SN Epinions X-axis: degree Y-axis: mean # triangles Notice: slope ~ degree exponent (insets) 45 15

16 Triangle Law: Computations [Tsourakakis ICDM 2008] But: triangles are expensive to compute (3-way join; several approx. algos) Q: Can we do that quickly? 46 Triangle Law: Computations [Tsourakakis ICDM 2008] But: triangles are expensive to compute (3-way join; several approx. algos) Q: Can we do that quickly? A: Yes! #triangles = 1/6 Sum ( λ i 3 ) (and, because of skewness, we only need the top few eigenvalues! 47 Triangle Law: Computations [Tsourakakis ICDM 2008] x+ speed-up, (c) 2010 C. Faloutsos high accuracy 48 16

17 Outline Topology & laws Static graphs - eigenspokes Evolving graphs Generators Discussion and tools 49 EigenSpokes B. Aditya Prakash, Mukund Seshadri, Ashwin Sridharan, Sridhar Machiraju and Christos Faloutsos: EigenSpokes: Surprising Patterns and Scalable Community Chipping in Large Graphs, PAKDD 2010, Hyderabad, India, June (c) 2010 C. Faloutsos 50 EigenSpokes Eigenvectors of adjacency matrix equivalent to singular vectors (symmetric, undirected graph) (c) 2010 C. Faloutsos 51 17

EigenSpokes Eigenvectors of adjacency matrix equivalent to singular vectors (symmetric, undirected graph) details N N 15-826 (c)

Faloutsos 52 EigenSpokes 2 nd Principal component EE plot: u2 Scatter plot of scores of u1 vs u2 One would expect 15-826 Many

18 EigenSpokes Eigenvectors of adjacency matrix equivalent to singular vectors (symmetric, undirected graph) details N N (c) 2010 C. Faloutsos 52 EigenSpokes 2 nd Principal component EE plot: u2 Scatter plot of scores of u1 vs u2 One would expect Many origin A few scattered ~randomly u1 1 st Principal component (c) 2010 C. Faloutsos 53 EigenSpokes EE plot: Scatter plot of scores of u1 vs u2 One would expect Many origin A few scattered ~randomly u2 90 o u (c) 2010 C. Faloutsos 54 18

19 EigenSpokes - pervasiveness Present in mobile social graph across time and space Patent citation graph (c) 2010 C. Faloutsos 55 EigenSpokes - explanation Near-cliques, or near -bipartite-cores, loosely connected (c) 2010 C. Faloutsos 56 EigenSpokes - explanation Near-cliques, or near -bipartite-cores, loosely connected (c) 2010 C. Faloutsos 57 19

EigenSpokes - explanation Near-cliques, or near -bipartite-cores, loosely

Extract nodes with high scores high connectivity Good communities 15-826 (c)

patents from same inventor(s) cut-and-paste bibliography!

20 EigenSpokes - explanation Near-cliques, or near -bipartite-cores, loosely connected spy plot of top 20 nodes So what? Extract nodes with high scores high connectivity Good communities (c) 2010 C. Faloutsos 58 Bipartite Communities! patents from same inventor(s) cut-and-paste bibliography! magnified bipartite community (c) 2010 C. Faloutsos 59 Outline Topology & laws Static graphs Weighted graphs Evolving graphs Generators Discussion and tools 60 20

21 How about weighted graphs? A: even more laws! 61 Observation W.1: Fortification Q: How do the weights of nodes relate to degree? (c) 2010 C. Faloutsos 62 Observation W.1: Fortification More donors, more $? $10 Reagan $5 $7 Clinton (c) 2010 C. Faloutsos 63 21

22 Observation W.1: fortification: Snapshot Power Law Weight: super-linear on in-degree exponent iw : 1.01 < iw < 1.26 More donors, even more $ $10 In-weights ($) Orgs-Candidates e.g. John Kerry, $10M received, from 1K donors $ Edges (# donors) (c) 2010 C. Faloutsos 64 Fortification: Snapshot Power Law At any time, total incoming weight of a node is proportional to in-degree with PL exponent iw : i.e < iw < 1.26, super-linear More donors, even more $ Orgs-Candidates In-weights ($) e.g. John Kerry, $10M received, from 1K donors Edges (# donors) CIKM Copyright: (c) 2010 Faloutsos, C. Faloutsos Tong (2008) 1-65 Summary of laws static, & weighted graphs Power laws for degree distributions... for eigenvalues, bi-partite cores Small diameter ( 6 degrees ) Bow-tie for web; jelly-fish for internet Power laws for triangles and for weighted graphs 66 22

23 Outline Topology & laws Static graphs Evolving graphs Generators Discussion and tools 67 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? 68 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 69 23

24 T1. Shrinking diameter [Leskovec+05a] diameter Citations among physics papers 2003: 29,555 papers 352,807 citations For each month M, create a graph of all citations up to month M time 70 T1. Shrinking diameter Authors & publications nodes 272 edges ,000 nodes 20,000 authors 38,000 papers 133,000 edges 71 T1. Shrinking diameter Patents & citations ,000 nodes 676,000 edges million nodes 16.5 million edges Each year is a datapoint 72 24

25 T1. Shrinking diameter Autonomous systems ,000 nodes 10,000 edges ,000 nodes 26,000 edges One graph per day diameter N 73 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) 74 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) A: over-doubled! 75 25

26 T2. Temporal evolution of graphs A: over-doubled - but obeying: E(t) ~ N(t) a for all t where 1<a<2 76 T2. Densification Power Law ArXiv: Physics papers and their citations E(t) 1.69 N(t) 77 T2. Densification Power Law ArXiv: Physics papers and their citations E(t) tree N(t) 78 26

27 T2. Densification Power Law ArXiv: Physics papers and their citations E(t) clique N(t) 79 T2. Densification Power Law U.S. Patents, citing each other E(t) 1.66 N(t) 80 T2. Densification Power Law Autonomous Systems E(t) 1.18 N(t) 81 27

Faloutsos Weighted Graphs and Disconnected Components: Patterns and a

28 T2. Densification Power Law ArXiv: authors & papers E(t) 1.15 N(t) 82 More on Time-evolving graphs M. McGlohon, L. Akoglu, and C. Faloutsos Weighted Graphs and Disconnected Components: Patterns and a Generator. SIG-KDD T3: Gelling Point Q1: How does the GCC emerge? 84 28

T3: Gelling Point Most real graphs display a gelling point After gelling point,

Diameter t=1914 Time 85 T4: NLCC behavior Q2: How do NLCC s emerge and join with

components) Do they continue to grow in size? or do they shrink? or stabilize?

29 T3: Gelling Point Most real graphs display a gelling point After gelling point, they exhibit typical behavior. This is marked by a spike in IMDB diameter. Diameter t=1914 Time 85 T4: NLCC behavior Q2: How do NLCC s emerge and join with the GCC? (``NLCC = non-largest conn. components) Do they continue to grow in size? or do they shrink? or stabilize? 86 T4: NLCC behavior After the gelling point, the GCC takes off, but NLCC s remain ~constant (actually, oscillate). IMDB CC size 87 29

30 How do new edges appear? [LBKT 08] Microscopic Evolution of Social Networks Jure Leskovec, Lars Backstrom, Ravi Kumar, Andrew Tomkins. (ACM KDD), How do edges appear in time? [LBKT 08] Edge gap δ(d): inter-arrival time between d th and d+1 st edge What is the PDF of δ? Poisson? δ 89 T5: How do edges appear in time? [LBKT 08] LinkedIn Edge gap δ(d): inter-arrival time between d th and d+1 st edge CIKM Copyright: (c) 2010 Faloutsos, C. Faloutsos Tong (2008)

31 # in links T6 : popularity over time lag: days after post Post popularity + lag (c) 2010 C. Faloutsos 91 # in links (log) T6 : popularity over time days after post (log) Post popularity drops-off exponentially? POWER LAW! Exponent? (c) 2010 C. Faloutsos 92 T6 : popularity over time # in links (log) -1.6 Post popularity drops-off exponentially? POWER LAW! Exponent? -1.6 close to -1.5: Barabasi s stack model and like the zero-crossings of a random walk days after post (log) (c) 2010 C. Faloutsos 93 31

32 Summary of laws Power laws for degree distributions... for eigenvalues, bi-partite cores Small & shrinking diameter ( 6 degrees ) Bow-tie for web; jelly-fish for internet ``Densification Power Law, over time Triangles; weights Oscillating NLCC Power-law inter-arrival times 94 Outline Topology & laws Generators Erdos Renyi Degree-based Process-based Recursive generators Discussion and tools 95 Generators How to generate random, realistic graphs? Erdos-Renyi model: beautiful, but unrealistic degree-based generators process-based generators recursive/self-similar generators 96 32

33 random graph 100 nodes, avg degree = 2 Fascinating properties (phase transition) But: unrealistic (Poisson degree distribution!= power law) Erdos-Renyi 97 E-R model & Phase transition Pc vary avg degree D watch Pc = 1 Prob( there is a giant connected component) How do you expect it to be? 0?? D 98 E-R model & Phase transition Pc vary avg degree D watch Pc = 1 Prob( there is a giant connected component) How do you expect it to be? 0 N->infty D 0 N=10^3 D 99 33

34 Degree-based Figure out the degree distribution (eg., Zipf ) Assign degrees to nodes Put edges, so that they match the original degree distribution 100 Process-based Barabasi; Barabasi-Albert: Preferential attachment -> power-law tails! rich get richer [Kumar+]: preferential attachment + mimick Create communities 101 Process-based (cont d) [Fabrikant+, 02]: H.O.T.: connect to closest, high connectivity neighbor [Pennock+, 02]: Winner does NOT take all

35 Outline Topology & laws Generators Erdos Renyi Degree-based Process-based Recursive generators Discussion and tools 103 Recursive generators (RMAT [Chakrabarti+, 04]) Kronecker product 104 Wish list for a generator: Power-law-tail in- and out-degrees Power-law-tail scree plots shrinking/constant diameter Densification Power Law communities-within-communities

36 Graph Patterns Power Laws Effective Diameter Count vs Indegree Count vs Outdegree Hop-plot Eigenvalue vs Rank Network values vs Rank Count vs edge-stress 106 Wish list for a generator: Power-law-tail in- and out-degrees Power-law-tail scree plots shrinking/constant diameter Densification Power Law communities-within-communities Q: how to achieve all of them? 107 Wish list for a generator: Power-law-tail in- and out-degrees Power-law-tail scree plots shrinking/constant diameter Densification Power Law communities-within-communities Q: how to achieve all of them? A: Kronecker matrix product [Leskovec+05b]

37 Kronecker product 109 Kronecker product 110 Kronecker product N N*N N**

Kronecker Product Communities within communities 112 Properties of Kronecker graphs: Power-law-tail in- and out-degrees Power-law-tail scree plots constant diameter perfect Densification Power Law

38 Kronecker Product Communities within communities 112 Properties of Kronecker graphs: Power-law-tail in- and out-degrees Power-law-tail scree plots constant diameter perfect Densification Power Law communities-within-communities 113 Properties of Kronecker graphs: Power-law-tail in- and out-degrees Power-law-tail scree plots constant diameter perfect Densification Power Law communities-within-communities and we can prove all of the above (first and only generator that does that)

39 Kronecker - patents Degree Scree Diameter D.P.L. 115 Outline Topology & laws Generators Discussion - tools 116 Power laws Q1: Why so many? A1:

40 Power laws Q1: Why so many? A1: self-similarity; rich-get-richer, etc we saw Newman s paper Can you recall other settings with power laws? 118 log(freq) a A famous power law: Zipf s law the Bible - rank vs frequency (log-log) log(rank) 119 Power laws, cont ed web hit counts [Huberman] Click-stream data [Montgomery+01]

41 u-id s Click-stream data url s Web Site Traffic log(count) Zipf yahoo log(freq) log(count) super-surfer log(freq) 121 Conclusions - Laws Laws and patterns: Power laws for degrees, eigenvalues, communities /cores Small diameter and shrinking diameter Bow-tie; jelly-fish densification power law Triangles; weights Oscillating NLCCs Power-law edge-inter-arrival times 122 Conclusions - Generators Preferential attachment (Barabasi) Variations Recursion Kronecker graphs

42 Conclusions Power laws rank/frequency plots ~ log-log NCDF log-log PDF Self-similarity / recursion / fractals correlation integral = hop-plot 124 Resources Generators: Kronecker (christos@cs.cmu.edu) BRITE INET: Other resources Visualization - graph algo s: Graphviz: pajek: /networks/pajek/ Kevin Bacon web site: /

43 References [Aiello+, '00] William Aiello, Fan R. K. Chung, Linyuan Lu: A random graph model for massive graphs. STOC 2000: [Albert+] Reka Albert, Hawoong Jeong, and Albert -Laszlo Barabasi: Diameter of the World Wide Web, Nature (1999) [Barabasi, '03] Albert-Laszlo Barabasi Linked: How Everything Is Connected to Everything Else and What It Means (Plume, 2003) 127 References, cont d [Barabasi+, '99] Albert-Laszlo Barabasi and Reka Albert. Emergence of scaling in random networks. Science, 286: , 1999 [Broder+, '00] Andrei Broder, Ravi Kumar, Farzin Maghoul, Prabhakar Raghavan, Sridhar Rajagopalan, Raymie Stata, Andrew Tomkins, and Janet Wiener. Graph structure in the web, WWW, References, cont d [Chakrabarti+, 04] RMAT: A recursive graph generator, D. Chakrabarti, Y. Zhan, C. Faloutsos, SIAM-DM 2004 [Dill+, '01] Stephen Dill, Ravi Kumar, Kevin S. McCurley, Sridhar Rajagopalan, D. Sivakumar, Andrew Tomkins: Self-similarity in the Web. VLDB 2001:

44 References, cont d [Fabrikant+, '02] A. Fabrikant, E. Koutsoupias, and C.H. Papadimitriou. Heuristically Optimized Trade-offs: A New Paradigm for Power Laws in the Internet. ICALP, Malaga, Spain, July 2002 [FFF, 99] M. Faloutsos, P. Faloutsos, and C. Faloutsos, "On power-law relationships of the Internet topology," in SIGCOMM, References, cont d [Jovanovic+, '01] M. Jovanovic, F.S. Annexstein, and K.A. Berman. Modeling Peer-to-Peer Network Topologies through "Small-World" Models and Power Laws. In TELFOR, Belgrade, Yugoslavia, November, 2001 [Kumar+ '99] Ravi Kumar, Prabhakar Raghavan, Sridhar Rajagopalan, Andrew Tomkins: Extracting Large-Scale Knowledge Bases from the Web. VLDB 1999: References, cont d [Leland+, '94] W. E. Leland, M.S. Taqqu, W. Willinger, D.V. Wilson, On the Self-Similar Nature of Ethernet Traffic, IEEE Transactions on Networking, 2, 1, pp 1-15, Feb [Leskovec+05a] Jure Leskovec, Jon Kleinberg, Christos Faloutsos, Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations (KDD 2005)

45 References, cont d [Leskovec+05b] Jure Leskovec, Deepayan Chakrabarti, Jon Kleinberg, Christos Faloutsos Realistic, Mathematically Tractable Graph Generation and Evolution, Using Kronecker Multiplication (ECML/PKDD 2005), Porto, Portugal, References, cont d [Leskovec+07] Jure Leskovec and Christos Faloutsos, Scalable Modeling of Real Graphs using Kronecker Multiplication, ICML [Mihail+, '02] Milena Mihail, Christos H. Papadimitriou: On the Eigenvalue Power Law. RANDOM 2002: References, cont d [Milgram '67] Stanley Milgram: The Small World Problem, Psychology Today 1(1), (1967) [Montgomery+, 01] Alan L. Montgomery, Christos Faloutsos: Identifying Web Browsing Trends and Patterns. IEEE Computer 34(7): (2001)

46 References, cont d [Palmer+, 01] Chris Palmer, Georgos Siganos, Michalis Faloutsos, Christos Faloutsos and Phil Gibbons The connectivity and fault-tolerance of the Internet topology (NRDM 2001), Santa Barbara, CA, May 25, 2001 [Pennock+, '02] David M. Pennock, Gary William Flake, Steve Lawrence, Eric J. Glover, C. Lee Giles: Winners don't take all: Characterizing the competition for links on the web Proc. Natl. Acad. Sci. USA 99(8): (2002) 136 References, cont d [Schroeder, 91] Manfred Schroeder Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise W H Freeman & Co., References, cont d [Siganos+, '03] G. Siganos, M. Faloutsos, P. Faloutsos, C. Faloutsos Power-Laws and the AS -level Internet Topology, Transactions on Networking, August [Watts+ Strogatz, '98] D. J. Watts and S. H. Strogatz Collective dynamics of 'small-world' networks, Nature, 393: (1998) [Watts, '03] Duncan J. Watts Six Degrees: The Science of a Connected Age W.W. Norton & Company; (February 2003)

Finding patterns in large, real networks

Thanks to Finding patterns in large, real networks Christos Faloutsos CMU www.cs.cmu.edu/~christos/talks/ucla-05 Deepayan Chakrabarti (CMU) Michalis Faloutsos (UCR) George Siganos (UCR) UCLA-IPAM 05 (c)