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Network Science workshop

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Prersented at the 3rd International Business Complexity and Global Leadership Conference.

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Network Science workshop

  1. 1. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 1/37WARNING!Network Science is extremelycontagious ONCE YOU LEARN IT you. ,START seeing Networks everywhere.D Zinoviev.
  2. 2. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 2/37Outline●What Is Network Science?●Terms and Definitions●Measures●Formation●Complex Behavior●Tools of the Craft●Unusual Applications of Network Science
  3. 3. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 3/37What is Network Science?Network science is aninterdisciplinary academic fieldwhich studies complex networkssuch as: telecommunication, transportation, electrical, computer, biological, cognitive and semantic, and social.
  4. 4. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 4/37What is it based upon?The field draws on theories and methods including: Graph theory from mathematics (Erdős, Rényi, Strogatz), Game theory from economics (Jackson), Statistical mechanics from physics (Barabási, Newman, Vespignani,Watts), Data mining and information visualization from computer science(Adamic), and Social structure from sociology (Watts).
  5. 5. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 5/37Terms and definitions● Network = Graph● Nodes (vertexes, actors, members)represent entities● Nodes have properties (gender,capacity, political view)● Edges (arcs, links, ties) representrelationships● Edges have properties (direction,weight, kind)● Directed vs undirected● Multigraph: graph with paralleledges● Simple graph: undirected, no loops,no parallel edges● Connected graphsBoston SSAlbanyBrunswickBoston NSSt AlbansProvidenceHartfordSpringfieldNew HavenNew York PSMontrealRutland
  6. 6. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 6/37Adjacency Matrix A75Boston SSAlbanyBrunswickBoston NSSt AlbansProvidenceHartfordSpringfieldNew HavenNew York PSMontreal6 Rutland912114813210A=0 0 1 0 0 0 0 1 0 0 0 00 0 1 0 0 0 0 0 0 0 0 01 1 0 0 0 1 1 0 0 0 0 00 0 0 0 0 0 1 0 0 1 0 00 0 0 0 0 0 1 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 1 1 1 0 0 0 1 0 0 01 0 0 0 0 0 0 0 1 1 0 00 0 0 0 0 0 1 1 0 0 0 00 0 0 1 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 1 0Aij=1 if and only if i and j are connected
  7. 7. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 7/37Incidence Matrix BB=1 0 1 0 0 0 0 0 0 0 0 01 0 0 0 0 0 0 1 0 0 0 00 1 1 0 0 0 0 0 0 0 0 00 0 1 0 0 1 0 0 0 0 0 00 0 1 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 1 0 0 00 0 0 0 0 0 1 0 1 0 0 00 0 0 0 0 0 0 1 0 1 0 00 0 0 1 0 0 0 0 0 1 0 00 0 0 1 0 0 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 10 0 0 0 1 0 1 0 0 0 0 0 75Boston SSAlbanyBrunswickBoston NSSt AlbansProvidenceHartfordSpringfieldNew HavenNew York PSMontreal6 Rutland912114813210ABC DEFGHIJKLBij=1 if and only if node i is incident to edge jedgesnodesA=B2−2I
  8. 8. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 8/37PATHS75Boston SSAlbanyBrunswickBoston NSSt AlbansProvidenceHartfordSpringfieldNew HavenNew York PSMontreal6 Rutland912114813210ABC DEFGHIJKL Path = sequence of connectededges (e.g., B – H – I) Can be simple (no self-intersections) Can be a loop (ends where itstarts) Paths have lengths Geodesic = a shortest path (B– F – G – J is not a geodesic,but B – H – I is) What if edges are weighted?
  9. 9. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 9/37Small World We are on average just 4–6 links(“handshakes”) away from any other livingperson on Earth (Milgrams experiment)—thence, “six degrees of separation” Not all networks have the “small world”propertyISomeoneI knowBorisBerezovskyVladimirPutinBarakObamaWait, howdo you knowObama?
  10. 10. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 10/37Centrality●How “central” is a nodein the network?●Possibly affectsinfluence, resilience,susceptibility, etc.●Several flavors: degree,closeness,betweenness,eigenvalue, etc.
  11. 11. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 11/37Degree Centrality[ ]75Boston SS (2)Albany (4)Brunswick(1)Boston NS (1)St Albans (1)Providence (2)Springfield (4)New Haven (3)New York PS (2)Montreal (1)6 Rutland (1)912114813210Hartford (2) Just count the neighbors! More neighbors = more“friends” = more importance Distinguish in-degree, out-degree, and [total] degree Can be defined in two ways (Nis the total number of nodes,aij∈A):di=∑jaijdi=∑jaij / N −1
  12. 12. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 12/37Degree Distribution Degree [centrality]distribution is animportant networkmeasure—it relatesto the networkformation process Most commondistributions incomplex networks:binomial (Poissonfor n→∞) andpower law (a.k.a.Pareto, Zipf, scalefree) Why it is what itis?
  13. 13. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 13/37Closeness Centrality75Boston SS (0.5)Brunswick(1)Boston NS (1)St Albans (0.4)Providence (0.4)Springfield (0.6)New Haven (0.5)New York PS (0.5)Montreal (0.4)6 Rutland (0.4)912114813210Hartford (0.5)Albany (0.6) Calculate average inverseshortest path to all other nodes Shorter path = closer “friends”= better connectivity Can be defined in two ways (Nis the total number of nodes, pijis a geodesic path from I to j) Takes care of disconnectednetworks!ci=∑j1/ pijci=∑j1/ pij/ N −1
  14. 14. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 14/37Betweenness Centrality75Boston SS (0.1)Brunswick(0)Boston NS (0)St Albans (0)Providence (0.04)Springfield (0.5)New Haven (0.14)New York PS (0.13)Montreal (0)6 Rutland (0)912114813210Hartford (0.06)Albany (0.5) Calculate how many shortestpaths go through the node Mores paths = better brokerageopportunities (= morevulnerability) Can be defined in two ways (Nis the total number of nodes, pijis a geodesic path from I to j, nis the number of such paths)bwi=∑j≠i≠kn pjik /n p jk bwi=∑j≠i≠kn p jik /n pjk /N −1 N −2
  15. 15. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 15/37Eigenvector Centrality75Boston SS (0.29)Brunswick(0)Boston NS (0)St Albans (0.19)Providence (0.25)Springfield (0.49)New Haven (0.34)New York PS (0.31)Montreal (0.17)6 Rutland (0.17)912114813210Hartford (0.33)Albany (0.45) Recursive definition: A node isas important as its neighborsareei=1 ∑jaij e j A− I  E=0 E ,=eig A
  16. 16. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 16/37Similarity and Triadic ClosureConnectivity between nodes may implysimilarity: A is connected to B  A issimilar to B (known as homophily insocial networks). Two dyads sharing anode become a triad.ABCABCAlternative interpretation: weak tiesbecome strong ties (Granovetter).ABC ABC
  17. 17. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 17/37Clustering Coefficient Clustering coefficient of anode with n neighbors: Ci=0 — star Ci=1 — clique (1, 4, 5, 6) C1=6/10 Average clusteringcoefficient:C=(.6+.67+1+1+1+1)/6=.88Ci=2∑j , kaij aik a jknn−1“Birds of a featherflock together...”(William Turner)1 (.6)2 (.67)3 (1.)4 (1.)5 (1.)6 (1.)
  18. 18. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 18/37Modularity and Components NSSI (self-cutters) onlinecommunities in LiveJournal (bloggingsocial Web site) form sixcomponents If these two components are merged,they form a giant component Modularity Q∈[-1, 1] measures thedensity of links inside clusters ascompared to links between clusters:Q=∑ij[aij −∑iaij ∑jaij∑ijaij ]ij∑ijaij
  19. 19. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 19/37AssortativityAssortative networks: nodes connect tonodes with similar degree; highmodularity, better community structureDissassortative networks: nodesconnect to nodes with different degree
  20. 20. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 20/37Network Formation●Networks are complexsystems composed ofinterconnected parts thatas a whole exhibitproperties not obviousfrom the properties of theindividual parts.●Most networks are not animmediate product ofintelligent design.
  21. 21. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 21/37exponential Networks A.k.a. Erdős–Rényi networks Start with a fixed set of N nodes Randomly connect them with probability p Average degree λ=pN Binomial / Poisson degree distribution(decays exponentially after max) No small-world property!
  22. 22. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 22/37Small World Networks A.k.a. Watts–Strogatz networks Start with a fixed set of N nodes Connect each node to its m neighbors Rewire the connections with probability β Degree distribution: δ-function for β→0, binomial/Poissonfor β→1 (unrealistic) Small-world—but no clustering!β=00<β<1β=1
  23. 23. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 23/37Scale Free Networks A.k.a. Barabási–Albert networks Start with few nodes Attach a new node X to m existing nodesYiwith probability proportional to thedegrees of Yi(preferential attachment) Power law degree distribution Small-world, community structure No meaningful average degree (scale-free) Fat tail
  24. 24. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 24/37Strategic Network formation Formed on purpose Start with a fixed set of N nodes Add links to maximize utility: eitherglobally or pairwise Topology depends on the costs andbenefits Link cost c Benefit from directconnection δ Benefits from indirectconnections δ2, δ3, δ4,etc.3δ-3c3δ-3c3δ-3c3δ-3cδ+2δ2-c3δ-3cδ+2δ2-cδ+2δ2-c0000δ vs c“cheap” links“expensive”links
  25. 25. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 25/37Complex Behaviors●Simple contagion: epidemics, rumorpropagation●Complex contagion: collective action,political views, fashion●Information diffusion: effect offeedback●Resilience
  26. 26. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 26/37Simple Contagion Susceptible – Infectious – Susceptible (SIS): At each step, a “healthy” (butsusceptible) node gets infected by an infected neighbor with probability p, and aninfected node recovers with probability r Susceptible – Infectious – Recovered (SIR): same as in SIS, but a node cannotbe reinfected Spreads fast in power-law networks
  27. 27. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 27/37Collective Action A node becomes infected with probability p when either a certainnumber M or a certain fraction m of its neighbors is infectious✔ “I will wear red pants if at least 50% of my friends wear redpants.”✔ “I will use protocol X if at least 10 of my partners supportprotocol X.”✔ “I will go to protest tax hikes if all my friends go with me.”✔ “I will feel happy if people around me are happy.” Supported by community structure:✔ Structural trapping (few external links)✔ Social reinforcement (many internal links)✔ Homophily (“connected” means “similar”) Success depends on the point of origin
  28. 28. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 28/37Information Diffusion A network of senders and receivers Each actor has knowledge, credibility,and popularity Options for sender (speaker): To send (gain popularity, gain or losecredibility) Not to send (lose popularity) Options for receiver (listener): Listen silently (gain knowledge, losepopularity) Listen and provide feedback (gainknowledge, gain popularity, gain orlose credibility) Action based on Nash equilibrium
  29. 29. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 29/37ResilienceRandomattacks: FailrandomnodesTargetedattacks:AttackselectednodesExponentialrandomnetworksNo difference: The networkgracefully degradesScale-freenetworks(robust yetfragile)The giantcomponentsurvives.The giantcomponentrapidly fallsapart.
  30. 30. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 30/37Tools of the Craft●Gephi—graph visualization●Pajek—network algorithms and somevisualization●NetLogo—simple simulation environment (goodfor small-scale experiments)●CFinder—community finder●NodeXL—network visualization plugin for Excel●networkx—Python library for networkalgorithms
  31. 31. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 31/37Gephi NetworkScience“Paintbrush” Analysis andvisualizationof largenetworks Windows,Linux, MacOS Developed byGephiconsortium Free and opensource
  32. 32. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 32/37Pajek “Spider” inSlovene Analysis andvisualization oflarge networks Windows (runon Linux inwine) Developed byBatagelj andMrvar Free, but notopen source
  33. 33. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 33/37Unusual applicationsReminder:If all you know is Network Scienceeverything looks like a Network.
  34. 34. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 34/37Unusual networks●Networks of recipes and cooking ingredients(Adamic)●Product space networks (Hidalgo)●Human disease networks (Barabási)●Flavor networks (Ahn)●Soccer player networks (Onody / de Castro)●And more!..
  35. 35. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 35/37Semantic networks Two words are similar if theyare used by similar people (But two people are similar ifthey use similar words!) Zinoviev, Stefanescu,Swenson, and Fireman,“Semantic Networks ofInterests in Online NSSICommunities,” Proc. ofWorkshop “Words andNetworks,” Evanston, IL,June 2012
  36. 36. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 36/37Textual Networks Co-occurrence of actors inthe New Testament A node is an actor, anedge is introduced if twoactors are mentioned in thesame chapter of a book atleast once Bigger nodes—morementioning Zinoviev, research inprogress, unpublished
  37. 37. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 37/37Thank you!

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