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Acknowledgements. Ted Klastorin (Tutorials Chair)Param Vir SinghV. Mookerjee, D. Dey, M. Fan, C. Phelps, A. Susarla, A. Jain, G. Zhang, J. Oh, Y. Lee, L. Yan, N. Yu, D. Choi, A. Ozler. Outline. IntroductionEconomic ApproachRandom GraphSocial Network Analysis Social CapitalData CollectionSoftwareSelected Applications.

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2. Acknowledgements Ted Klastorin (Tutorials Chair) Param Vir Singh V. Mookerjee, D. Dey, M. Fan, C. Phelps, A. Susarla, A. Jain, G. Zhang, J. Oh, Y. Lee, L. Yan, N. Yu, D. Choi, A. Ozler

3. Outline Introduction Economic Approach Random Graph Social Network Analysis Social Capital Data Collection Software Selected Applications

4. What Is Social Network A map of relationships (formal or informal) among actors (person, organization, and others) Representations Graph Matrix (Sociomatrix)

5. Friend Network

6. Subscriber Network

7. OSS Collaboration Network

8. Blogs

9. Why Is It Important? Our research focus on economic factors Social embeddedness – Granovetter (1985) Economic actions are “embedded in concrete, ongoing systems of social relations” Networks are central Resource sharing, information dissemination, and knowledge spillover White collar workforce management Wally Hopp et al Successful business models MySpace, FaceBook, YouTube, …

10. It Is Getting Popular! Phelps, Singh and Heidl

11. I. Economic and Social Networks: Stability and Efficiency

12. Introduction Both economic and social interactions involve network relationships The specifics of the network structure are important in determining the outcome The aims are to develop a systematic analysis of how incentives of individuals affect the formation of networks align with social efficiency

13. Definitions A set N = {1,?,n} of individuals are connected in a network relationship. Individuals are the nodes in the graph and links indicate relationships between the individuals Bilateral relationship ? Non-directed Networks Marriage, friendship, alliances, exchange, etc. Both parties should consent to form a link Unilateral relationships ? Directed Networks Advertising or links to web sites etc.

14. Notations ij represents the link {i, j} ij ? g indicates that i and j are linked under network g G = {g ? gN} denotes the set of all possible networks or graphs on N, with gN being the complete network g + ij : network obtained by adding link ij to an existing network g g - ij : network obtained by deleting link ij to an existing network g N(g)={i| ?j s.t. ij ? g} : set of individuals who have at least one link in network g

15. Paths and Components Given a network g ? G, a path in g between i and j is a sequence of individuals i1,i2,…,iK such that ikik+1 ? g for each k ? {1,…, K - 1}, with i1 = i and iK = j.

16. Value Functions Different network configurations lead to different values of overall production or overall utility to a society. These possible valuations are represented via a value function. The set of all possible value functions is denoted V Different networks that connect the same individuals may lead to different values Value function can incorporate costs to links as well as benefits

17. Allocation Rules A value function keeps track of how the total societal value varies across different networks An allocation rule is used to keep track of how that value is distributed among the individuals forming a network is a function Y: G ?V ? RN such that ?iYi(g, v) = v(g) for all v and g depends on both g and v. This allows an allocation rule to take full account of an individual i’s role in the network

18. Pareto Efficiency A network g is Pareto efficient relative to v and Y if there does not exist any g’?G such that Yi(g’,v) ? Yi(g,v) for all i with strict inequality for some i. This definition of efficiency of a network takes Y as fixed, and hence can be thought of as applying to situations where no intervention is possible

19. Efficiency A network g is efficient relative to v if v(g) ? v(g’) for all g’?G. This is a strong notion of efficiency as it takes the perspective that value is fully transferrable Unlimited intervention is possible g is efficient relative to v if g is PE relative to v and Y for all Y

20. Pairwise Stability A network g is pairwise stable with respect to allocation rule Y and value function v if : A network is pairwise stable if it is not defeated by another (necessarily adjacent) network It is a weak notion as it considers only deviations on a single link at a time and only deviations by at most a pair of individuals at a time It is not a sufficient requirement for a network to be stable over time.

21. Existence of Pairwise Stable Networks In some situations, there may not exist any pairwise stable network. Each network is defeated by some adjacent network, and that these “improving paths” form cycles with no undefeated networks existing An improving path is a sequence of networks {g1, g2, …, gK} where each network gk is defeated by the subsequent (adjacent) network gk+1.

22. Example: Exchange Networks

23. Compatibility of Efficiency and Stability While there are situations where the allocation rule is an object of design, we are also interested in understanding when naturally arising allocation rules lead to pairwise stable networks that are (Pareto) efficient. Example: Coauthor Model: Each individual is a researcher who spends time working on research projects. If two are connected, they are working on a project together. The amount of time researcher i spends on a given project is inversely related to the number of projects, ni. i’s payoff is For n = 4, the complete network is pairwise stable with payoff of 2.5 for each player. For network g = {12,34}, each individual have payoff of 3. So the unique pairwise stable network is Pareto inefficient.

24. Dynamic Model of Network Formation Since network structure affects economic outcomes, it is crucial to know which network configurations will arise Process of network formation in a dynamic framework is analyzed Formation process is found to be path dependent, thus the process often converges to an inefficient network structure

25. Static Model Connection Model (Jackson and Wolinsky) There are n agents, N = {1,2, … ,n}, are able to communicate each other Each agent i ?{1, … ,n} receives a payoff ui(g), from the network g i receives a payoff of 1 > ? > 0 for each direct link he has i pays a cost c > 0 of maintaining each direct link he has t(ij): number of direct links in the shortest path between agent i and j (i ? j). ?t(ij) is the payoff agent i receives from being indirectly connected to agent j

26. Static Model Results For all N, a stable network exists. Further, if c < ? and ? - c > ?2, then gN is stable (unique) if c ? ?, then the empty network is stable (not usually unique) if c < ? and ? - c ? ?2, then a star network is stable (not usually unique) For all N, a unique efficient network exists. Further, if ? - c > ?2, then gN is the efficient network if ? - c > ?2 and , then a star network is efficient if ? - c > ?2 and , then the empty network is efficient

27. Dynamic Model Initially n players are unconnected Players meet over time and have opportunity to form links with each other Time, T, is divided into countable, infinite set, T = {1,2, … ,t, …} gt: network exists at the end of period t ui(gt): payoff of player i at the end of period t In each period, a link ij is randomly identified to be updated with uniform probability If ij ? gt-1, either i or j can decide to sever the link If ij ? gt-1, players i and j can form a link ij and simultaneously sever any of their other links if both agree Each player is myopic If after some time period t, no additional links are formed or broken, then the network formation has reached a stable state

28. Dynamic Network Formation Results If ? - c > ?2 > 0, then every link forms (ASAP) and remains (no links are ever broken). If ? - c < 0, then no links ever form. If player i and j are not directly connected, they will each gain at least (? - c) - ?t(ij) > 0 from forming a direct link. If ? - c > ?2 > 0, connection will take place. If ? - c > ?2 > 0, formation converges to gN (unique efficient and stable network) If ? - c < 0 and such a link forms between i and j, then each agent will receive a payoff ? - c < 0. Since agents are myopic, they will refuse to link If ? - c < 0, the empty network is always stable. It is efficient iff

29. II. Random Graphs

30. Review and Background Network = graph Vertex (node, site) Edge (link, bond) graph (network) is a pair of sets G ={V, E} V is a set of N nodes (vertices) E is a set of edges connecting elements of V Edges do not have length (except in metric spaces)

31. Random graphs Taking n dots and drawing nz/2 lines between random pairs Completely ordered lattice A low dimension regular lattice Watts-Strogatz model (Small-world) A low dimension regular lattice with some degrees of randomness Barabasi-Albert model (Scale-free)

32. Properties Degree of a vertex is a number of edges attached to it (if directed – incoming and outgoing degree) Geodesic path – the shortest path from one node to another (measured in nodes) Diameter of the network – the longest geodesic path between any two vertices (not mean) Average geodesic path length

33. Random Graphs Studied by P. Erdös A. Rényi in 1960s How to build a random graph Take n vertices Connect each pair of vertices with an edge with some probability p There are n(n-1)/2 possible edges The mean number of edges per vertex is

34. Degree Distribution Probability that a vertex of has degree k follows binomial distribution In the limit of n >> kz, Poisson distribution z is the mean

35. Characteristics Small-world effect (Milgram 60s) Diameter (Bollobas) Average vertex-vertex distance Grows slowly (logarithmically with the size) Some inaccuracies describing real-world networks Degree distribution (not Poisson!) Clustering (Network transitivity) If A and B have a common friend C it is more likely that they themselves will be friends. Random graph : z / n social networks, biological networks in nature, artificial networks – power grid, WWW ranging from 0.08 to 0.59

36. Clustering If A is connected to B, and B is connected to C, then it is likely that A is connected to C “A friend of your friend is your friend” The average fraction of a node’s neighbor pairs that are also neighbors each other Count up the total number of pairs of vertices on the entire graph that have a common neighbor and the total number of such pairs that are also themselves connected, and divide the one by the other

37. Small-World Model Watts-Strogatz (1998) first introduced small world mode connects regular and random networks Regular Graphs have a high clustering coefficient, but also a high diameter Random Graphs have a low clustering coefficient, but a low diameter Characteristic of the small-world model The length of the shortest chain connecting two vertices grow very slowly, i.e., in general logarithmically, with the size of the network Higher clustering or network transitivity

38. Scale-Free Network A small proportion of the nodes in a scale-free network have high degree of connection Power law distribution A given node has k connections to other nodes with probability as the power law distribution with exponent ? ~ [2, 3] Examples of known scale-free networks: Communication Network - Internet Ecosystems and Cellular Systems Social network responsible for spread of disease

40. Barabasi-Albert Networks Science 286 (1999) Start from a small number of node, add a new node with m links Preferential Attachment Probability of these links to connect to existing nodes is proportional to the node’s degree ‘Rich gets richer’ This creates ‘hubs’: few nodes with very large degrees

41. Analysis

42. Scale-free Networks: Good and Bad Scale-free networks cannot be broken by random node removal ‘Attacks’ can bring them down: hackers’ attacks, major servers (DNS) downed by a computer virus In scale-free networks there is no epidemic threshold: any outbreak should become an epidemic Berger et al, On the spread of viruses on the Internet, Proceedings of the 16th annual ACMSIAM symposium, 2005

43. III. Social Networks Analysis (SNA)

44. 3.1 Centrality and Prestige

50. Eigenvector Centrality Importance of an actor in a network Sociomatrix (Adjacency matrix) Aij = 1, if a link between i and j; 0 otherwise Centrality measure xi In matrix form: Here l is the eigenvalue

62. 3.2 Structural Equivalence Network Position and Role

63. Social Roles and Positions Position A collection of individuals who are similarly embedded in networks of relations (ex. in social activity, ties, or intersections, with regard to actors in other positions) This concept is quite different from the concept of cohesive subgroup (Why? based on the similarity of ties rather than their adjacency, proximity, or reachability.) Example Nurses in different hospitals occupy the position of “nurse” though individual nurses may not know each other, work with the same doctors, or see the same patients

64. Social Roles and Positions Role The patterns of relations which obtain between actors or between positions An associations among relations that link social positions Collections of relations and the associations among relations Example Kinship roles Defined in terms of combinations of the relations of marriage and descent Roles of corporate organization Defined in terms of levels in a chain of command or authority

65. Definition of Structural Equivalence Actor i and j are structurally equivalent if actor i has a tie to k, iff j also has a tie to k, and i has a tie from k iff j also has a tie from k.

66. Structural Equivalence

67. Positional Analysis

68. Position Analysis

69. Position Analysis - Measures Euclidean Distance Single relation Multiple relation

70. Position Analysis - Measures Correlation (Pearson product-moment) Single relation Multiple relation

71. 3.3 Cohesive Subgroups Affiliation Networks

72. Background Cohesive subgroups are subsets of actors among whom there are relatively strong, direct, intense, frequent, or positive ties. Although the literature contains numerous ways to conceptualize the idea of subgroup, there are four general properties: The mutuality of ties The closeness of reachability of subgroup members The frequency of ties among members The relative frequency of ties among subgroup members compared to non-members

73. Subgroups Based on Complete Mutuality “Cliquish” subgroups (Festinger and Luce and Perry) Cohesive subgroups in directional dichotomous relations would be characterized by sets of people among whom all friendship choices were mutual. Definition of a Clique A clique in a graph is a maximal complete subgraph of three or more nodes, all of which are adjacent to each other, and there are no other nodes that are also adjacent to all of the members of the clique.

74. Subgroups Based on Complete Mutuality Example Cliques: {1,2,3},{1,3,5}, and {3,4,5,6}

75. Subgroups Based on Reachability and Diameter n-cliques An n-clique is a maximal subgraph in which the largest geodesic distance between any two nodes is no greater than n. n-clans An n-clan is an n-clique, in which the geodesic distance, d(i,j), between all nodes in the subgraph is no greater than n for paths within the subgraph n-clubs An n-club is defined as a maximal subgraph of diameter n.

76. Subgroups Based on Reachability and Diameter Example 2-cliques: {1,2,3,4,5} {2,3,4,5,6} 2-clans: {2,3,4,5,6} 2-clubs: {1,2,3,4} {1,2,3,5} {2,3,4,5,6}

77. Measures of Subgroup Cohesion A measure of degree to which strong ties are within rather than outside is given by the ratio:

78. Affiliation Networks Affiliation networks are two-mode networks Affiliation networks consist of subsets of actors, rather than simply pairs of actors Connections among members of one of the modes are based on linkages established through the second mode Affiliation networks allow one to study the dual perspectives of the actors and the events

79. Collaboration Network

80. IV. SOCIAL CAPITAL

81. Why Is Social Capital Important?

82. Network Relationships and Knowledge Benefits

83. Network Elements – Direct Ties

84. Network Elements – Indirect Ties

85. Cohesiveness means that ties are redundant To the degree that they lead back to the same actors Such redundancy increases the information transmission capacity in a group of developers having cohesive ties It promotes sharing and makes information exchange Speedy Reliable Effective Information between two developers in a cohesive group can flow through multiple pathways; this increases the speed as well as reliability of information transfer.

86. Cohesiveness in the group gives rise to trust, reciprocity norms, and a shared identity leads to a high level of cooperation facilitate collaboration by providing self-enforcing informal governance mechanisms Cohesive ties enable richer and greater amounts of information and knowledge to be reliably exchanged The groups also provide meaningful context for information and resource sharing The trust among the members in the group affords them to be creative This creativity helps in coming up with alternative interpretation of current problems, or novel approaches to solve these problems

87. Cohesion Lead to norms of adhering to established standards and conventions Potentially stifle innovation The standards, conventions and knowledge stocks vary across groups. Structural holes are the gaps in the information flow between these groups. Developers who connect different groups are said to fill these structural holes. Teams composed of developers who span different groups may have several advantages.

88. Illustrative Partial Developer Network

89. Teams composed of developers who span different groups may have several advantages. Technical or organizational problems and difficulties of a developer in one group can be easily and reliably relayed to developers in other groups. The solutions for these problems may be obvious to someone and would be quickly and reliably relayed back. Each group has its own best practices (organizational or technical perspectives) which may have value for other groups. The developers who connect these groups can see how resources or practices in one may create value for other and synthesize, translate as well as transfer them across groups. Resource pooling across groups provide developers opportunities to work on different but related problem domains, which may help them in developing a better understanding of their own problems.

90. Structural Hole vs Network Closure

91. Measures Direct Ties Indirect Ties (Burt 1992) n: total number of developers in the network wij: number of developers that lie at a path length of j from i zij: decay associated with the information that is received from developers at path length j fij: number of developers that i can reach within and including path length j Ni: total number of developers that i can reach in the network

92. Measures (2) Network Closure or Structural Hole (Burt 1992) Network Cohesion is indirect structural constraint Computed as: Where Mi ? number of direct ties for developer i piq ? proportion of i’s relations invested in the relationship with j

93. Social Capital and OSS Success Singh, Tan, and Mookerjee (2007) Data 5191 projects and 10973 developers

94. Data Collection The challenge is determining network boundary Two approaches Whole network Not easy Ego-centric Problematic Snowballing

95. Software UCINET Software for Social Network Analysis (Borgatti, Everett, and Freeman) Pajek R SNA module StOCNET SIENA module Longitudinal data (Snijders 2004) Dynamic network MCMC

96. SELECTED APPLICATIONS

97. Strategy and Organization Firm alliance (R&D) network Board of director network Venture capital network Team social network

98. Marketing Social contagion Estimating customer value Gupta et al (2006) “Social Network and Marketing” Ven den Bulte and Wuyts (2007)

99. Information Systems Open Source Software Development Collaboration Network Singh (2007) Singh, Tan, and Mookerjee (2007) Online Communities Productivity, Information Aral and Van Alstyne (2007) Aral, Brynjolfsson and Van Alstyne (2007)

100. Finance Financial Networks Leinter (2005) Venture Capital Networks and Investment Performance Hochberg et al (2007)

101. Thank You!

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