Network Thinking. Complexity: A Guided Tour, Chapters PDF Free Download

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

Neural Network (C. Elegans) http://gephi.org/wp-content/uploads/2008/12/screenshot-celegans.png

Food Web http://1.bp.blogspot.com/_vifbm3t8bou/sbhzqbchiei/aaaaaaaaaxk/rsc-pj45avc/ s400/food%2bweb.bmp

Metabolic Network http://www.funpecrp.com.br/gmr/year2005/vol3-4/wob01_full_text.htm

Genetic Regulatory Network http://expertvoices.nsdl.org/cornell-info204/files/2009/03/figure-3.jpeg

Bank Network From Schweitzer et al., Science, 325, 422-425, 2009 http://www.sciencemag.org/cgi/content/full/325/5939/422

Airline Routes http://virtualskies.arc.nasa.gov/research/tutorial/images/12routemap.gif

US Power Grid http://images.encarta.msn.com/xrefmedia/aencmed/targets/maps/map/000a5302.gif

Internet http://www.visualcomplexity.com/vc/images/270_big01.jpg

World Wide Web (small part) From M. E. J. Newman and M. Girvin, Physical Review Letters E, 69, 026113, 2004.

Social Network http://ucsdnews.ucsd.edu/graphics/images/2007/07-07socialnetworkmaplg.jpg

The Science of Networks

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

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

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?

Small-World Property (Watts and Strogatz, 1998)

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

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

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

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

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

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

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

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

Stanley Milgram

Nebraska farmer Boston stockbroker Stanley Milgram

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

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.

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!

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

Six Degrees of Kevin Bacon http://oracleofbacon.org/

From http://www.dmae.upm.eswebpersonalbartolo/ 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 2.786981 46 1811

From http://www.dmae.upm.eswebpersonalbartolo/ Degree Number of edges connected to a node. In-degree Number of incoming edges. Out-degree Number of outgoing edges.

From http://www.dmae.upm.eswebpersonalbartolo/ 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?

From http://www.dmae.upm.eswebpersonalbartolo/ 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)

From http://www.dmae.upm.eswebpersonalbartolo/ Duncan J. Watts & Steven H. Strogatz, Nature 393, 440-442 (1998) Real life networks are clustered, large C, but have small average distance L. L L rand C C rand N WWW 3.1 3.35 0.11 0.00023 153127 Actors 3.65 2.99 0.79 0.00027 225226 Power Grid 18.7 12.4 0.080 0.005 4914 C. Elegans 2.65 2.25 0.28 0.05 282

From www.cse.unr.edu/~mgunes/cs765/cs790f10/lect18_smallworld.ppt 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 393:440-442.

From www.cse.unr.edu/~mgunes/cs765/cs790f10/lect18_smallworld.ppt 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

From www.cse.unr.edu/~mgunes/cs765/cs790f10/lect18_smallworld.ppt 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

From www.cse.unr.edu/~mgunes/cs765/cs790f10/lect18_smallworld.ppt 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 393:440-442.

From http://www.dmae.upm.eswebpersonalbartolo/ 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 = 49.51 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

Scale-Free Structure (Albert and Barabási, 1998)

Scale-Free Structure (Albert and Barabási, 1998) part of WWW Typical structure of a randomly connected network http://www.dichotomistic.com/images/random %20network.gif Typical structure of World Wide Web (nodes = web pages, links = links between pages)

Concept of Degree Distribution A node with degree 3

Concept of Degree Distribution A node with degree 3 Number of Nodes 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Degree

part of WWW Number of nodes Number of nodes Degree Degree

The Web s approximate Degree Distribution Number of nodes Degree

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

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

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

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

Example: Human height follows a normal distribution Frequency Height http://scienceblogs.com/builtonfacts/2009/02/the_central_limit_theorem_made.php

Example: Population of cities follows a power-law ( scalefree) distribution http://upload.wikimedia.org/wikipedia/commons/4/49/powercitiesrp.png http://www.streetsblog.org/wp-content/uploads 2006/09/350px_US_Metro_popultion_graph.png http://cheapukferries.files.wordpress.com/2010/06/ hollandcitypopulation1.png

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

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!

Scale-Free Networks are fractal-like http://en.wikipedia.org/wiki/file:worldwidewebaroundgoogle.png

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

High-Clustering Helps in Discovering Community Structure in Networks

How are Scale-Free Networks Created?

Web pages

Preferential attachment demo (Netlogo)

Robustness of Scale-Free Networks

Robustness of Scale-Free Networks Vulnerable to targeted hub failure

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

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

August, 2003 electrical blackout in northeast US and Canada 9:29pm 1 day before http://earthobservatory.nasa.gov/ images/imagerecords/3000/3719/ NE_US_OLS2003227.jpg 9:14pm Day of blackout

http://www.geocities.com/wallstreet/exchange/9807/charts/sp500/fdicfail_0907.jpg

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

Normal ( bell-curve) distribution http://resources.esri.com/help/9.3/arcgisdesktop/com/gp_toolref/ process_simulations_sensitivity_analysis_and_error_analysis_modeling/random_normal_distribution.gif

Normal ( bell-curve) distribution Events in tail are highly unlikely http://resources.esri.com/help/9.3/arcgisdesktop/com/gp_toolref/ process_simulations_sensitivity_analysis_and_error_analysis_modeling/random_normal_distribution.gif

Power law ( scale free ) distribution http://www.marketoracle.co.uk/images/mauldin_16_10_07image003.gif

Notion of heavy tail : Events in tail are more likely than in normal distribution Power law ( scale free ) distribution http://www.marketoracle.co.uk/images/mauldin_16_10_07image003.gif

Power law ( scale free ) distribution More normal than normal http://www.marketoracle.co.uk/images/mauldin_16_10_07image003.gif

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 http://www.marketoracle.co.uk/images/mauldin_16_10_07image003.gif

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

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

Binomial distribution demo: http://www.bhsstatistics.com/applets/applets/binomialdemo1.html http://www.jcu.edu/math/isep/quincunx/quincunx.html

Sandpile demo http://www.visualentities.com/sandpile.htm What does a power law distribution look like on a logarithmic plot, and why?

Gutenberg-Richter Law By: Bak [1]

Regularity of Biological Extinctions By: Bak [1]

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