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It’s a “small world” after all - Social networks and their modeling Michael Li, Ph.D Incubator2.cn [email_address] CBC2006, Hangzhou, PRC October 29, 2006

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<ul><li>What are Social Networks </li></ul><ul><li>History of Social Network Studies </li></ul><ul><li>Modeling Social Networks – Random Net, Small World Net, Scale Free Net </li></ul><ul><li>Possible Future Trends </li></ul>Presentation Contents

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What are Social Networks? “ To speak of social life is to speak of the association between people – their associating in work and in play, in love and in war, to trade or to worship, to help or to hinder. It is in the social relations men establish that their interests find expression and their desires become realized.” Peter M. Blau Exchange and Power in Social Life , 1964

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Society Nodes : individuals Links : social relationship (family/work/friendship/etc.)

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Source: Linton Freeman “See you in the funny pages” Connections , 23, 2000, 32-42.

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Paul Erdös collaboration graph Erdös had 507 direct collaborators (Erdös # of 1), many of whom have other collaborators (Erdös #2). Source: Valdis Krebs

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Bio-Map protein-gene interactions protein-protein interactions PROTEOME GENOME Citrate Cycle METABOLISM Bio-chemical reactions

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<ul><li>Possible earliest mention – in a 1929 short story “Chains” by the famous Hungarian writer Frigyes Karinthy, “5 degrees of separation” </li></ul><ul><li>Rediscovered in 1967 by Stanley Milgram, a Harvard professor of sociology, “6 degrees of separation” between any 2 people in the US </li></ul><ul><li>In 1969, Mark Granovetter, a sociology graduate student at Harvard – how do people network to get a job (Every fresh graduate has to face. Not via a friend, but through just an acquaintance, i.e., a weak tie or link!) </li></ul><ul><li>1998, Watts-Strogatz, applied mathematicians/physicists, proposed the “small world network model” – the significance of clustering, and applications to physical world networks beside that of social networks </li></ul><ul><li>1999, Albert Barabasi and his Notre Dame colleagues – the “scale free network model” , the importance of preferential attachment in a network’s evolution and growth </li></ul>History of Social Network Studies

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The Erdös-Rényi model Albert and Barabasi. REVIEWS OF MODERN PHYSICS, 74 2002 48-97 N nodes, every pair of nodes being connected with probability p Modeling Social Networks – 1. Random Nets

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Watts and Strogatz. Nature (1998) 393 440-442 Starting from a ring lattice with n vertices and k edges per vertex, each edge is rewired at random with probability p. Interpolating between regular and random networks Modeling Social Networks – 2. Small World Nets

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<ul><li>(Watts-Strogatz) - the important thing is that networks are highly clustered such that even when most of the connections are local, any pair of nodes can be linked by a relatively small number of link steps. </li></ul>Modeling Social Networks – 2. Small World Nets

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<ul><li>In a highly clustered, ordered network, a single random connection will create a shortcut that lowers L dramatically </li></ul><ul><li>Watts demonstrates that small world properties can occur in graphs with a surprisingly small number of shortcuts </li></ul>Modeling Social Networks – 2. Small World Nets

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Across a large number of substantive settings, Barab á si points out that the distribution of network involvement ( degree ) is highly and characteristically skewed. Modeling Social Networks – 3. Scale Free Nets

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Many large networks are characterized by a highly skewed distribution of the number of partners (degree)

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<ul><li>Growth : The number of nodes (N) is NOT fixed </li></ul><ul><li>Networks continuously expand by the addition of new nodes </li></ul><ul><li>Examples </li></ul><ul><li>WWW : addition of new documents </li></ul><ul><li>Citation : publication of new papers </li></ul><ul><li>Preferential Attachment : The attachment is NOT uniform </li></ul><ul><li>A node is linked with higher probability to a node that already has a large number of links </li></ul><ul><li>Examples </li></ul><ul><li>WWW : new documents link to well known sites (CNN, YAHOO, NewYork Times, etc) Citation : well cited papers are more likely to be cited again </li></ul>Modeling Social Networks – 3. Scale Free Nets

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Nodes : actors Links : cast jointly N = 212,250 (actors)  k  = 28.78 P(k) ~k^(-  )  =2.3 Days of Thunder (1990) Far and Away (1992) Eyes Wide Shut (1999) Movies Net

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The importance of the connected nodes in the scale-free network: 27% of the nodes are reached by the five most connected nodes, in the scale-free network more than 60% are reached. Comparing Random Vs. Scale-free Networks (both with 130 nodes and 215 links) Modified from Albert et al. Science (2000) 406 378-382 Five nodes with most links First neighbors of red nodes

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A simple model for generating “scale-free” networks <ul><li>Evolution : networks expand continuously by the addition of new vertices, and </li></ul><ul><li>Preferential-attachment (rich get richer) : new vertices attach preferentially to sites that are already well connected. </li></ul>Barabasi and Albert. Science (1999) 286 509-512 Barabasi & Bonabeau Sci. Am. May 2003 60-69

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Examples of real networks with power law degree distributions Modeling Social Networks – 3. Scale Free Nets Network Nodes Links/Edges Attributes World-Wide Web Web pages Hyperlinks Directed Internet Computers and Routers Wires and cables Undirected Actor Collaboration Actors Films Undirected Science Collaboration Authors Papers Undirected Citation Articles Citation Directed Phone-call Telephone Number Phone call Directed Power grid Generators, transformers and substations High voltage transmission lines Directed

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Albert and Barabasi. REVIEWS OF MODERN PHYSICS, 74 2002 48-97 Many real networks are small-world networks

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Future Directions: Network Topology and Dynamics

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Current research on complex networks <ul><li>Network topology characterization: average connectivity, degree distribution, clustering, diameter, network motifs, degree correlations </li></ul><ul><li>Network models: random graphs, small world networks, scale-free networks …. </li></ul><ul><li>Dynamic processes on complex networks: </li></ul><ul><li>Spreading phenomena: epidemics, rumours, computer viruses, information dissemination. </li></ul>Future Directions: Network Topology and Dynamics

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Failure: Removal of a random node. Attack: The selection and removal of a few nodes that play a vital role in maintaining the network’s connectivity. Failure and Attack Albert et al. Science (2000) 406 378-382 a macroscopic snapshot of Internet connectivity by K. C. Claffy Future Directions: Network Topology and Dynamics

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Thanks!