Network Theory 11/29/2012

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…

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

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

Networks are heterogeneous

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

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

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.

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

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

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.

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

Centrality in Trade

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

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

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

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)

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

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

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

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

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

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

Connection with Matching Theory One-to-one matching Network

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

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

Diffusion of Innovation (Ryan and Gross, ‘43)

Diffusion of Innovation (Griliches, 1957 and 1960)

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%

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

Job Search

Targeted Advertising: Ad Networks

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) www.insidefacebook.com

Risky Transactions (Di Maggio and Louch, ‘98)

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