1. Network Theory
11/29/2012

2. Network Effects in Platforms
In previous classes, we learned that a consumer’s value for a platform depends on
Intrinsic value: b
Price charged by platform: p
Number of other users: f(n)
Consumer’s value: b-p+f(n)
A consumer can derive
different benefits from
different users
Social Media: Facebook,
Twitter, Pinterest…

3. What is a Network?
“Real Networks represent populations of individual components that are actually doing something” (Watts, 2002)
Some examples:
Social Groups: sharing of information, experiences, risk
Cities: trade of goods and services, transport connections
Diffusion of viruses and diseases

4. Networks are heterogeneous
Banerjee, Chandrasekhar, Duflo, Jackson (2012)

5. Networks are heterogeneous

6. Networks share a common structure
Facebook network of friends
Worldwide air transportation network

7. Outline Basics of network structure Network formation and dynamics
Definitions: nodes, links, paths
Centrality measures: degree, closeness, betweenness
Network formation and dynamics
Symmetric connections model (Jackson and Wolinsky, 1996)
Efficient and stable networks
Connection with matching theory
Relevance for market design
Networks as conduits of information and trust
Examples:
Diffusion of innovation
Job search
Targeted advertising
Risky transactions

8. Network Structure
Florentine Marriages (Padgett and Ansell, ‘93)
Node: an agent involved in a network of relationships (people, firms, countries, web pages)
Link: relationship between any two nodes i and j
Path between nodes
i and j: sequence of links
connecting i and j such that
each node in the sequence
is distinct.

9. Measures of Centrality
Degree: how connected a node i is.
where di is the number of links involving node i, and N is
the total number of nodes.
Degree
Strozzi
0.29
Guadagni
Medici
0.43

10. Measures of Centrality
Closeness: how easily a node can reach other nodes
where l(i,j) is the number of links in the shortest path
between i and j.
Closeness
Strozzi
0.42
Guadagni
0.47
Medici
0.56

11. Measures of Centrality
Betweenness: how important a node is in terms of connecting other nodes
where P(kj) is the number of shortest paths between k
and j, and Pi(kj) is the number of shortest paths between
k and j that involve i.

12. Centrality in Florence
Betweenness
Strozzi
0.10
Guadagni
0.25
Medici
0.52
Degree
Closeness
Betweenness
Strozzi
0.29
0.42
0.10
Guadagni
0.47
0.25
Medici
0.43
0.56
0.52

13. Centrality in Trade

14. How do networks form? Modeling:
Strategic, game theoretic approach
Random graph approach
Hybrids
Strategic network formation: analysis of networks that form when links are chosen by the agents in the network

15. Symmetric Connections Model (Jackson and Wolinsky, ‘96)
A node is an agent
Links represent social relationships (friendships, marriages)
Direct relationships offer benefits (favors, information, risk sharing) and costs
Players benefit from indirect relationships (“friends of friends”, “friends of friends of friends”,…)
Benefits decrease with the distance of the relationship

16. Symmetric Connections Model
Benefit of a direct link: δ between 0 and 1
Cost of a direct link: c
Benefit from an indirect relationship of length t: δt
δ+δ2+δ3-c
2δ+δ2-2c
2δ+δ2-2c
δ+δ2+δ3-c

17. Efficient Networks
A network is efficient if it maximizes the total utility to all players in the society
c < δ-δ2: it is efficient to include all links in the network
4(δ-c)
4(δ-c)
4(δ-c)
4(δ-c)

18. Efficient Networks
δ-δ2 < c < δ+δ2(n-2)/2 : a star network is efficient
c > δ+δ2(n-2)/2 : the empty network is efficient
δ+2δ2-c
δ+2δ2-c
3(δ-c)
δ+2δ2-c

19. Which networks are formed by the agents?
A network is pairwise stable if:
No agent gains from severing a link
No two agents both gain from adding a link 3 3 3
3.25
3.25 3 3 3 2 2 2 2 2
2.2
2.5
2.5 2 2
2.2 2
2.5
2.5

20. Which networks are formed by the agents?
Stable vs. Efficient Networks
Efficient 3 3 3
3.25
3.25 3 3 3 2 2
Pairwise Stable 2 2 2
2.2
2.5
2.5 2 2
2.2 2
2.5
2.5

21. Which networks are formed by the agents?
Stable vs. Pareto Efficient Networks
Pareto Efficient
Pareto Efficient 3 3 3
3.25
3.25 3 3 3 2 2
Pairwise Stable 2 2 2
2.2
2.5
2.5 2 2
2.2 2
2.5
2.5

22. Pairwise Stability in the Symmetric Connections Model
c < δ-δ2: complete network is pairwise stable
δ-δ2 < c < δ:
A star is pairwise stable
Other networks are also pairwise stable
4(δ-c)
δ+2δ2-c
3(δ-c)
3(δ-c)
2(δ-c)+δ2
2(δ-c)+δ2
δ+δ2+δ3-c
Since
δ2 >δ-c
Since
δ2 >δ-c If
δ3 >δ-c

23. Pairwise Stability in the Symmetric Connections Model
δ < c < δ+δ2(n-2)/2
A star is not pairwise stable
Empty network is pairwise stable
Other networks are also pairwise stable
c >δ+δ2(n-2)/2: empty network is pairwise stable
3(δ-c)
Since δ<c

24. Connection with Matching Theory
One-to-one matching
Network

25. Connection with Matching Theory
Some hybrids:
Path: limits each user’s social network to 150 friends
FamilyLeaf: limits social network to family members
Pair: limits connections to just 1

26. Relevance for Market Design
Role of social networks as:
Conduits of information
Reputation mechanisms: ensuring trust
People learn from one another  Important implications:
Which technologies they adopt
How they find employment
Which products they purchase

27. Diffusion of Innovation (Ryan and Gross, ‘43)

28. Diffusion of Innovation (Griliches, 1957 and 1960)

29. Diffusion of Innovation (Coleman, Katz, Menzel, ’66)
Named as friend
Percent of doctors adopting by
By no other doctors
By one or two other doctors
By three or more other doctors
6 months
31%
52%
70%
8 months
42%
66%
91%
10 months
47%
94%
17 months
83%
84%
97%

30. Diffusion of Microfinance (Banerjee, Chandrasekhar, Duflo, Jackson, ‘12)

31. Job Search

32. Targeted Advertising: Ad Networks

33. Targeted Advertising Facebook “Sponsored Stories”:
“The whole premise of the site is that everything is more valuable when you have context about what your friends are doing. That’s true for ads as well. An advertiser can produce the best creative ad in the world, but knowing your friends really love drinking Coke is the best endorsement for Coke you can possibly get.” (Mark Zuckerberg)

34. Risky Transactions (Di Maggio and Louch, ‘98)

35. Summary Brief overview of:
Network heterogeneity
Network common structures
How networks form: agents choose their relationships
Network theory offers an alternative approach in market design
Accounts for diffusion of information and technology
Affects preferences and adoption of new products
Ensures trust in case of risky transactions