Acknowledgements

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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!