Network Thinking. Complexity: A Guided Tour, Chapters 15-16

Transcription

1 Network Thinking Complexity: A Guided Tour, Chapters 15-16

2 Neural Network (C. Elegans)

3 Food Web s400/food%2bweb.bmp

4 Metabolic Network

5 Genetic Regulatory Network

6 Bank Network From Schweitzer et al., Science, 325, ,

7 Airline Routes

8 US Power Grid

9 Internet

10 World Wide Web (small part) From M. E. J. Newman and M. Girvin, Physical Review Letters E, 69, , 2004.

11 Social Network

12 The Science of Networks

13 The Science of Networks Are there properties common to all complex networks?

14 The Science of Networks Are there properties common to all complex networks? If so, why?

15 The Science of Networks Are there properties common to all complex networks? If so, why? Can we formulate a general theory of the structure, evolution, and dynamics of networks?

16 Small-World Property (Watts and Strogatz, 1998)

17 Small-World Property (Watts and Strogatz, 1998)

18 me Small-World Property (Watts and Strogatz, 1998)

19 Small-World Property (Watts and Strogatz, 1998) me Barack Obama

20 Small-World Property (Watts and Strogatz, 1998) me Barack Obama my mother

21 Small-World Property (Watts and Strogatz, 1998) me Nancy Bekavac Barack Obama my mother

22 Small-World Property (Watts and Strogatz, 1998) me Nancy Bekavac Barack Obama Hillary Clinton my mother

23 Small-World Property (Watts and Strogatz, 1998) me Nancy Bekavac Barack Obama Hillary Clinton my mother

24 Small-World Property (Watts and Strogatz, 1998) me Barack Obama

25 Small-World Property (Watts and Strogatz, 1998) me my cousin Matt Dunne Barack Obama

26 Small-World Property (Watts and Strogatz, 1998) me my cousin Matt Dunne Patrick Leahy Barack Obama

27 Small-World Property (Watts and Strogatz, 1998) me my cousin Matt Dunne Patrick Leahy Barack Obama

28 Stanley Milgram

29 Nebraska farmer Boston stockbroker Stanley Milgram

30 Nebraska farmer Boston stockbroker Stanley Milgram

31 Nebraska farmer Boston stockbroker Stanley Milgram

32 Nebraska farmer Boston stockbroker Stanley Milgram On average: six degrees of separation

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34 The Small-World Property The network has relatively few long-distance links but there are short paths between most pairs of nodes, usually created by hubs.

35 The Small-World Property The network has relatively few long-distance links but there are short paths between most pairs of nodes, usually created by hubs. Most real-world complex networks seem to have the small-world property!

36 The Small-World Property The network has relatively few long-distance links but there are short paths between most pairs of nodes, usually created by hubs. Most real-world complex networks seem to have the small-world property! But why?

37 The Small-World Property And how can the shortest paths actually be found?

38 Six Degrees of Kevin Bacon

39 From Measure the average distance between Kevin Bacon and all other actors. Kevin Bacon No. of movies : 46 No. of actors : 1811 Average separation: 2.79 Is Kevin Bacon the most connected actor? NO! 876 Kevin Bacon

40 From Degree Number of edges connected to a node. In-degree Number of incoming edges. Out-degree Number of outgoing edges.

41 From Network parameters Diameter Maximum distance between any pair of nodes. Path length: number of hops to get from node v 1 to node v 2 Connectivity Number of neighbors of a given node: k := degree. P(k) := Probability of having k neighbors. Clustering Are neighbors of a node also neighbors among them?

42 From Clustering coefficient of a node v C(v) = 4/6 of links between neighbors C(v) = n(n-1)/2 C is the average over all C(v) Clustering: My friends will know each other with high probability! (typical example: social networks)

43 From Duncan J. Watts & Steven H. Strogatz, Nature 393, (1998) Real life networks are clustered, large C, but have small average distance L. L L rand C C rand N WWW Actors Power Grid C. Elegans

44 From Watts-Strogatz model: Generating small world graphs Select a fraction p of edges Reposition on of their endpoints Netlogo: Small Worlds Source: Watts, D.J., Strogatz, S.H.(1998) Collective dynamics of 'small-world' networks. Nature 393:

45 From Watts-Strogatz model: Generating small world graphs Each node has K>=4 nearest neighbors (local) tunable: vary the probability p of rewiring any given edge small p: regular lattice large p: classical random graph

46 From Watts/Strogatz model: What happens in between? Small shortest path means small clustering? Large shortest path means large clustering? Through numerical simulation As we increase p from 0 to 1 Fast decrease of mean distance Slow decrease in clustering

47 From Watts/Strogatz model: Change in clustering coefficient and average path length as a function of the proportion of rewired edges C(p)/C(0) l(p)/l(0) 1% of links rewired 10% of links rewired Source: Watts, D.J., Strogatz, S.H.(1998) Collective dynamics of 'small-world' networks. Nature 393:

48 From Structured network high clustering large diameter regular Small-world network high clustering small diameter almost regular Random network small clustering small diameter N = 1000 k =10 D = 100 L = C = 0.67 N =1000 k = 8-13 D = 14 d = 11.1 C = 0.63 N =1000 k = 5-18 D = 5 L = 4.46 C = 0.01

49 Scale-Free Structure (Albert and Barabási, 1998)

50 Scale-Free Structure (Albert and Barabási, 1998) part of WWW Typical structure of a randomly connected network %20network.gif Typical structure of World Wide Web (nodes = web pages, links = links between pages)

51 Concept of Degree Distribution A node with degree 3

52 Concept of Degree Distribution A node with degree 3

53 Concept of Degree Distribution A node with degree 3 Number of Nodes Degree

54 part of WWW Number of nodes Number of nodes Degree Degree

55 part of WWW Number of nodes Number of nodes Degree Degree

56 The Web s approximate Degree Distribution Number of nodes Degree

57 The Web s approximate Degree Distribution Number of nodes Degree

58 The Web s approximate Degree Distribution Number of nodes Degree

59 The Web s approximate Degree Distribution Number of nodes Degree

60 The Web s approximate Degree Distribution Number of nodes Degree

61 The Web s approximate Degree Distribution Scale-free distribution Number of nodes Degree

62 The Web s approximate Degree Distribution Scale-free distribution Number of nodes Number of nodes with degree k 1 k 2 Degree

63 The Web s approximate Degree Distribution Scale-free distribution Number of nodes power law Degree

64 The Web s approximate Degree Distribution Scale-free distribution Number of nodes Scale-free distribution = power law distribution power law Degree

65 Example: Human height follows a normal distribution Frequency Height

66 Example: Population of cities follows a power-law ( scalefree) distribution /09/350px_US_Metro_popultion_graph.png hollandcitypopulation1.png

67 The scale-free structure of the Web helps to explain why Google works so well part of WWW

68 The scale-free structure of the Web helps to explain why Google works so well part of WWW It also explains some of the success of other scalefree networks in nature!

69 Scale-Free Networks are fractal-like

70 Scale-Free Networks have high clustering High Clustering: part of WWW Low Clustering:

71 High-Clustering Helps in Discovering Community Structure in Networks

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73 How are Scale-Free Networks Created?

74 Web pages

75 Web pages

76 Web pages

77 Preferential attachment demo (Netlogo)

78 Robustness of Scale-Free Networks

79 Robustness of Scale-Free Networks Vulnerable to targeted hub failure

80 Robustness of Scale-Free Networks Vulnerable to targeted hub failure Robust to random node failure

81 Robustness of Scale-Free Networks Vulnerable to targeted hub failure Robust to random node failure unless... nodes can cause other nodes to fail Can result in cascading failure

82 August, 2003 electrical blackout in northeast US and Canada 9:29pm 1 day before images/imagerecords/3000/3719/ NE_US_OLS jpg 9:14pm Day of blackout

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84 We see similar patterns of cascading failure in biological systems, ecological systems, computer and communication networks, wars, etc.

85 Normal ( bell-curve) distribution process_simulations_sensitivity_analysis_and_error_analysis_modeling/random_normal_distribution.gif

86 Normal ( bell-curve) distribution Events in tail are highly unlikely process_simulations_sensitivity_analysis_and_error_analysis_modeling/random_normal_distribution.gif

87 Power law ( scale free ) distribution

88 Notion of heavy tail : Events in tail are more likely than in normal distribution Power law ( scale free ) distribution

89 Power law ( scale free ) distribution More normal than normal

90 Few economists saw our current crisis coming, but this predictive failure was the least of the field s problems. More important was the profession s blindness to the very possibility of catastrophic failures in a market economy. -- Paul Krugman, New York Times, September 6, 2009 Power law ( scale free ) distribution More normal than normal

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95 Observed common properties: Small world property Scale-free degree distribution Clustering and community structure Robustness to random node failure Vulnerability to targeted hub attacks Vulnerability to cascading failures

96 Other examples of power-laws in nature Magnitude vs. frequency of earthquakes Magnitude vs. frequency of stock market crashes Income vs. frequency (of people with that income) Populations of cities vs. frequency (of cities with that population) Word rank vs. frequency in English text

97 Binomial distribution demo:

98 Sandpile demo What does a power law distribution look like on a logarithmic plot, and why?

99 Gutenberg-Richter Law By: Bak [1]

100 Regularity of Biological Extinctions By: Bak [1]