1 Social and Economic Networks: Lecture 1, Networks? Alper Duman Izmir University Economics, February 26, 2013
2 Conventional economics assume that all agents are either completely connected or totally isolated. All agents are either homogeneous or there exist a representative agent All agents optimize without strategic substitutes or complementarities. Time and topology almost never matters
3 Figure : Input-Output Networks of US Economy,
4 Figure : ISE Firms Network via Interlocking Directors,2012 (Red: Koç group, Blues. Sabancı group)
5 Figure : Ownership Networks of Firms 2010
6 Figure : World Trade Network
7 Figure : Product Space Network
8 Figure : Subway Network
9 Types of Networks Directed Affiliation or Two-Mode Weighted Multigraph Trees
10 Centrality Sabancı, Boyner, Yalçındağ, and Doğan families are all connected Centrality of the actors is very important! Power brokers and bridges; Who and where are they?
11 Figure : Marriage Network among Fortune 100
12 Centrality Which centrality is more important? 1. Degree Centrality 2. Closeness Centrality 3. Betweenness Centrality 4. Random-Walk Centrality 5. Eigenvector Centrality
13 Random Graphs and Networks Seminal model is due Erdös and Renyi (1959) Think about the number of possible networks with just 10 vertices Pick an edge probability 1 > p > 0, and choose any pair of vertices to apply. Do this for every pair. That is a graph G(N, p) with N vertices, and edge is present between any two vertices with a probability p.
14 Figure : Random Network with G(100, 0.01)
15 Figure : Random Network with G(100, 0.02)
16 Figure : Random Network with G(100, 0.03)
17 Characteristics of Random Networks Giant component emerges quickly Average path is very short Clustering is very low Edge formation is independent; very unlikely in social networks, WHY?
18 What is the probability of a fully connected network with N = 3? Three different events, each with a probability of p; so it should be p 3 What about the network with all isolated vertices? In general, any network with n vertices/nodes and m edges/links has a probability p m (1 p) n(n 1) 2 m to form
19 The probability that any given node i has d links is C(n 1; d)p d (1 p) n 1 d Fraction of nodes that have d links in a large network with large n and small p e (n 1)p ((n 1)p) d d! This is an approximation by a Binomial distribution
20 Figure : Frequency Distributions of Random Graphs frequency degree
21 What is the fraction of nodes that have zero degrees, that is the fraction of isolates? For large networks that would be approximated by e (n 1)p, if (n 1)p is sufficiently small If on average we expect only one isolate we have e (n 1)p = 1 n Solve this? How does it relate to the random network we have drawn previously for G(100, 0.02)?
22 Strategic Connection Model Agents form connections strategically There are (1) direct benefits of immediate links and (2) indirect benefits derived from friends of the friends There are costs of forming/keeping direct links Each agent shoud agree on the link; otherwise one is enough to severe the link
23 Figure : Selected 4-Node Networks and Strategic Connection Model
24 Consider agent A in Network 1; the payoff would be δ + δ 2 + δ 3 c where δ is the direct benefit, and c is the cost. What about agent B? Which payoff is higher, A s or B s? Note that if there is no path between two vertices then there can not be any direct or indirect benefit.
25 Come up with your own example of a network Think about the motto of Six degrees of Separation ; why is it a small world? What if our network of connections are random, what would be the diameter in such a world? Why is the Star Network pairwise-stable and efficient network for intermediate levels of cost of link formation?