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Complex Networks
Complex Networks
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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 1/37 WARNING! Network Science is extremely contagious ONCE YOU LEARN IT you. , START seeing Networks everywhere. D Zinoviev.
  2. 2. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 2/37 Outline ● 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 3/37 What is Network Science? Network science is an interdisciplinary academic field which studies complex networks such as:  telecommunication,  transportation,  electrical,  computer,  biological,  cognitive and semantic, and  social.
  4. 4. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 4/37 What 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 5/37 Terms and definitions ● Network = Graph ● Nodes (vertexes, actors, members) represent entities ● Nodes have properties (gender, capacity, political view) ● Edges (arcs, links, ties) represent relationships ● Edges have properties (direction, weight, kind) ● Directed vs undirected ● Multigraph: graph with parallel edges ● Simple graph: undirected, no loops, no parallel edges ● Connected graphs Boston SSAlbany Brunswick Boston NS St Albans Providence Hartford Springfield New Haven New York PS Montreal Rutland
  6. 6. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 6/37 Adjacency Matrix A 7 5 Boston SSAlbany Brunswick Boston NS St Albans Providence Hartford Springfield New Haven New York PS Montreal 6 Rutland 9 12 11 4 8 1 3 2 10 A=  0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 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 7/37 Incidence Matrix B B=  1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0  7 5 Boston SSAlbany Brunswick Boston NS St Albans Providence Hartford Springfield New Haven New York PS Montreal 6 Rutland 9 12 11 4 8 1 3 2 10 A B C D E F G H I J KL Bij =1 if and only if node i is incident to edge j edges nodes A=B2 −2I
  8. 8. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 8/37 PATHS 7 5 Boston SSAlbany Brunswick Boston NS St Albans Providence Hartford Springfield New Haven New York PS Montreal 6 Rutland 9 12 11 4 8 1 3 2 10 A B C D E F G H I J KL  Path = sequence of connected edges (e.g., B – H – I)  Can be simple (no self- intersections)  Can be a loop (ends where it starts)  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 9/37 Small World  We are on average just 4–6 links (“handshakes”) away from any other living person on Earth (Milgram's experiment)— thence, “six degrees of separation”  Not all networks have the “small world” property I Someone I know Boris Berezovsky Vladimir Putin Barak Obama W ait, how do you know Obama?
  10. 10. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 10/37 Centrality ● How “central” is a node in the network? ● Possibly affects influence, resilience, susceptibility, etc. ● Several flavors: degree, closeness, betweenness, eigenvalue, etc.
  11. 11. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 11/37 Degree Centrality[ ] 7 5 Boston 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) 9 12 11 4 8 1 3 2 10 Hartford (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 (N is the total number of nodes, aij ∈A): di=∑j aij di=∑j aij / N −1
  12. 12. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 12/37 Degree Distribution  Degree [centrality] distribution is an important network measure—it relates to the network formation process  Most common distributions in complex networks: binomial (Poisson for n→∞) and power law (a.k.a. Pareto, Zipf, scale free)  Why it is what it is?
  13. 13. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 13/37 Closeness Centrality 7 5 Boston 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) 9 12 11 4 8 1 3 2 10 Hartford (0.5) Albany (0.6)  Calculate average inverse shortest path to all other nodes  Shorter path = closer “friends” = better connectivity  Can be defined in two ways (N is the total number of nodes, pij is a geodesic path from I to j)  Takes care of disconnected networks! ci=∑j 1/ pij ci=∑j 1/ pij/ N −1
  14. 14. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 14/37 Betweenness Centrality 7 5 Boston 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) 9 12 11 4 8 1 3 2 10Hartford (0.06) Albany (0.5)  Calculate how many shortest paths go through the node  Mores paths = better brokerage opportunities (= more vulnerability)  Can be defined in two ways (N is the total number of nodes, pij is a geodesic path from I to j, n is the number of such paths) bwi=∑j≠i≠k n pjik /n p jk  bwi=∑j≠i≠k n p jik /n pjk /N −1 N −2
  15. 15. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 15/37 Eigenvector Centrality 7 5 Boston 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) 9 12 11 4 8 1 3 2 10Hartford (0.33) Albany (0.45)  Recursive definition: A node is as important as its neighbors are ei= 1  ∑j aij e j  A− I  E=0  E ,=eig A
  16. 16. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 16/37 Similarity and Triadic Closure Connectivity between nodes may imply similarity: A is connected to B  A is similar to B (known as homophily in social networks). Two dyads sharing a node become a triad. A B C A B C Alternative interpretation: weak ties become strong ties (Granovetter). A B C A B C
  17. 17. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 17/37 Clustering Coefficient  Clustering coefficient of a node with n neighbors:  Ci =0 — star  Ci =1 — clique (1, 4, 5, 6)  C1 =6/10  Average clustering coefficient: C=(.6+.67+1+1+1+1)/6=.88 Ci=2 ∑j , k aij aik a jk nn−1 “Birds of a feather flock 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 18/37 Modularity and Components  NSSI (self-cutters) online communities in LiveJournal (blogging social Web site) form six components  If these two components are merged, they form a giant component  Modularity Q∈[-1, 1] measures the density of links inside clusters as compared to links between clusters: Q= ∑ij [aij − ∑i aij ∑j aij ∑ij aij ]ij ∑ij aij
  19. 19. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 19/37 Assortativity Assortative networks: nodes connect to nodes with similar degree; high modularity, better community structure Dissassortative networks: nodes connect to nodes with different degree
  20. 20. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 20/37 Network Formation ● Networks are complex systems composed of interconnected parts that as a whole exhibit properties not obvious from the properties of the individual parts. ● Most networks are not an immediate product of intelligent design.
  21. 21. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 21/37 exponential 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 22/37 Small 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/Poisson for β→1 (unrealistic)  Small-world—but no clustering! β=0 0<β<1 β=1
  23. 23. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 23/37 Scale Free Networks  A.k.a. Barabási–Albert networks  Start with few nodes  Attach a new node X to m existing nodes Yi with probability proportional to the degrees 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 24/37 Strategic Network formation  Formed on purpose  Start with a fixed set of N nodes  Add links to maximize utility: either globally or pairwise  Topology depends on the costs and benefits  Link cost c  Benefit from direct connection δ  Benefits from indirect connections δ2 , δ3 , δ4 , etc. 3δ-3c 3δ-3c 3δ-3c 3δ-3c δ+2δ2 -c3δ-3c δ+2δ2 -cδ+2δ2 -c 0 0 0 0 δ vs c “cheap” links “expensive”links
  25. 25. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 25/37 Complex Behaviors ● Simple contagion: epidemics, rumor propagation ● Complex contagion: collective action, political views, fashion ● Information diffusion: effect of feedback ● Resilience
  26. 26. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 26/37 Simple Contagion  Susceptible – Infectious – Susceptible (SIS): At each step, a “healthy” (but susceptible) node gets infected by an infected neighbor with probability p, and an infected node recovers with probability r  Susceptible – Infectious – Recovered (SIR): same as in SIS, but a node cannot be reinfected  Spreads fast in power-law networks
  27. 27. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 27/37 Collective Action  A node becomes infected with probability p when either a certain number M or a certain fraction m of its neighbors is infectious ✔ “I will wear red pants if at least 50% of my friends wear red pants.” ✔ “I will use protocol X if at least 10 of my partners support protocol 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 28/37 Information Diffusion  A network of senders and receivers  Each actor has knowledge, credibility, and popularity  Options for sender (speaker):  To send (gain popularity, gain or lose credibility)  Not to send (lose popularity)  Options for receiver (listener):  Listen silently (gain knowledge, lose popularity)  Listen and provide feedback (gain knowledge, gain popularity, gain or lose credibility)  Action based on Nash equilibrium
  29. 29. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 29/37 Resilience Random attacks: Fail random nodes Targeted attacks: Attack selected nodes Exponential random networks No difference: The network gracefully degrades Scale-free networks (robust yet fragile) The giant component survives. The giant component rapidly falls apart.
  30. 30. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 30/37 Tools of the Craft ● Gephi—graph visualization ● Pajek—network algorithms and some visualization ● NetLogo—simple simulation environment (good for small-scale experiments) ● CFinder—community finder ● NodeXL—network visualization plugin for Excel ● networkx—Python library for network algorithms
  31. 31. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 31/37 Gephi  Network Science “Paintbrush”  Analysis and visualization of large networks  Windows, Linux, MacOS  Developed by Gephi consortium  Free and open source
  32. 32. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 32/37 Pajek  “Spider” in Slovene  Analysis and visualization of large networks  Windows (run on Linux in wine)  Developed by Batagelj and Mrvar  Free, but not open source
  33. 33. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 33/37 Unusual applications Reminder: If all you know is Network Science everything looks like a Network.
  34. 34. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 34/37 Unusual 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 35/37 Semantic networks  Two words are similar if they are used by similar people  (But two people are similar if they use similar words!)  Zinoviev, Stefanescu, Swenson, and Fireman, “Semantic Networks of Interests in Online NSSI Communities,” Proc. of Workshop “Words and Networks,” Evanston, IL, June 2012
  36. 36. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 36/37 Textual Networks  Co-occurrence of actors in the New Testament  A node is an actor, an edge is introduced if two actors are mentioned in the same chapter of a book at least once  Bigger nodes—more mentioning  Zinoviev, research in progress, unpublished
  37. 37. April 29, 2013 3rd International Business Complexity and Global Leadership Conference 37/37 Thank you!

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