Models of Social Network
Formation
John Navil
Models of Network Formation
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Sociology – Diffusion
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Make friends with individuals who are similar to you
Become similar to your friends
Random Graph Models
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Explains How? of network formation
Erdős-Rényi graphs
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Scale Free networks
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Degree distribution is Poisson
Degree distribution is Power Law
Models from economics - Game theoretic
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Explains the Why? Of network formation
Why do self interested individuals form networks?
Sources
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A Survey of Models of Network Formation Stability and Efficiency (Matthew O. Jackson – 2003)
Models from economics
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Nodes are controlled by self interested individuals
Incorporates costs and benefits (payoffs) to the individuals involved
Trace network formation to incentives to the individuals involved
Questions asked?
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How can we predict which networks are likely to form when individuals have
the discretion to choose their connections?
Do the right networks form in terms of maximizing benefits to society?
Efficiency?
How are these benefits distributed among individuals?
Motivation for economic models
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Example - Obtaining information about job opportunities
Leading source of information about job opportunities is Personal
contacts (before Internet!)
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(Boorman 1975, Granovetter 1974, Montgomery 1991)
Social network structure determines who gets which jobs.
 Is such a system efficient?
 Why is there might be persistent differences in employment
between races?
How to efficiently fill jobs and minimize unemployment?
What are the best policies? Incentives needed?
Value functions
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Measures the overall value generated in a network
V: G => R; V(Ф) =0
Structure affects the value
 V({12, 23, 31}) != V({12, 23})
Value functions ????
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Component Additively
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Allows for externalities, Component additive,
Externalities doesn’t exist across components
Anonymous value functions
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v(gπ)= v(g).
Allocation Rules
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Specifies how value is allocated or distributed among the players
in the society
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Myerson Value
Shapely Value
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Myerson Value
Connections Model
(Jackson and Wolinsky (1996))
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Costs involved in maintaining links.
Direct and indirect benefits
g = {12, 23, 34}
 R(1, 1->2) = δ, δ<1
 R(1, 1->3) = δ2 ..
Payoff for player i,
Value function,
The Co-Author Model
[Jackson and Wolinsky (1996)]
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Each researcher spends time working on some projects
If two researchers are connected, then they are working together on some
projects
Synergy between two players depends on how much time they spend working
together.
Player i’s payoff is given by,
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Total value generated by all researchers is,
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Game Theory
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Extensive Form Games
Strategic Games
Nash Equilibrium
Sub-game perfect Nash Equilibrium
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A sub-game perfect Nash equilibrium is a profile of strategies, one for each
player, that satisfies the requirement that,
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at each point in the game every player’s strategy gives him the highest possible payoff
in the remainder of the game, given the strategies of the other players.
Network Games
[Aumann and Myerson, 1988]
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Described by,
 N -> Set of players, V -> Value function , μ -> Allocation
function
 σ -> External link order
Sequential network formation, players form link one at a time
Links are in some externally specified order
 Pairs of players get opportunities to form links
Players have a global view of the links formed
Links once formed cannot be broken
Result?
 What kinds of networks are supported by a sub-game perfect
Nash Equilibrium?
Network Games
[Aumann and Myerson, 1988]
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N -> {1, 2, 3}
Simplified value function v(g), value only for coalitions
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V = 0 if |g| = 0,1
V = 60 if |g| = 2
V= 72 |g| = 3
Myerson allocation function
U1{Ф}=0
U2{Ф}=0
U3{Ф}=0
U1{(1,2);(3)}=30
U2{(1,2);(3)}=30
U3{(1,2);(3)}=0
U1{(1,2);(2,3)}=14
U2{(1,2);(2,3)}=44
U3{(1,2);(2,3)}=14
U1{(1,2);(2,3);(1,3)}=24
U2{(1,2);(2,3);(1,3)}=24
U3{(1,2);(2,3);(1,3)}=24
Network Games
[Aumann and Myerson, 1988]
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The first link is always formed,
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forming the link increases the payoff for both players involved (0 -> 30)
Both players in isolation would want to form a link with the third player
(30 -> 44)
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but they don’t do that!
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Because, if both players form link with the third player then they get 24
each, (24 < 30)
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So sub game perfect Nash equilibrium predicts that exactly one link is
formed.
Network Games with costs
[Marco Slikkera, Anne van den Nouwelandb 1999]
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What if there are costs for links?
C=22
U1{Ф}=0
U2{Ф}=0
U3{Ф}=0
U1{(1,2);(3)}=30-11=19
U2{(1,2);(3)}=30-11=19
U3{(1,2);(3)}=0
sub game perfect Nash equilibrium
U1{(1,2);(2,3)}=14-11=3
predicts that exactly two links are formed. U2{(1,2);(2,3)}=44-22=22
U3{(1,2);(2,3)}=14-11=3
Costs for links, increases the
U1{(1,2);(2,3);(1,3)}=24-22=2
number of links formed!
U2{(1,2);(2,3);(1,3)}=24-22=2
U3{(1,2);(2,3);(1,3)}=24-22=2
Strategic forms of network formation
(Myerson 1991)
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Single step network formation, Simultaneous move
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Each player has a stategy space which is nothing but the list of other
players that it wants to form ties with.
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Links are formed if both players agree.
Strategic forms of network formation
(Myerson 1991)
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Nash Equilibria
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Strong Nash Equilibria
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A statergy profile that is stable against deviations by a
single player
Many networks can emerge
A statergy profile that is stable against deviations by any
possible coalition of players.
It is possible that no network can emerge
Coalition Proof Nash Equilibria
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Similar to Strong Nash Equilibria, but more limited
More networks can emerge when compared to Strong
Nash Equilibria
Stability
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Pairwise stability
Strong Stability
Network formation by bargaining
Currarini and Morelli (2000)
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Allocation of value among players takes place simultaneously
with link formation.
Players are ordered exogenously
Each player in order announces the set of players with which
they want to link along with some payoff demand.
Links are formed based on the feasibility of the demands of
players.
Problems with game theoretic view
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Assumption of global observation too strong
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(Imperfect information available)
Dynamic models