Introduction Social and Economic Networks MohammadAmin Fazli Social and Economic Networks 1
Why Study Networks • Social networks permeate our social and economic lives • They play a central role in the transmission of information, … . • Job opportunities • Trade of Goods • Education • … • It is important to understand: • How networks ’ structures affect behavior • Which network structures are likely to emerge in a society • This Course: Studying different network models Social and Economic Networks 2
What is a model? • A tool for better understanding of different phenomena • Think, predict and decide on papers • Compute, analyze and simulate on computers • Building models requires abstraction • Example: Social and Economic Networks 3
This Course • Exercises • Theory + programming • Quizzes • Final Exam Social and Economic Networks 4
ToC • Different aspects of this course • A set of examples • An introduction to network models • Readings: • Chapter 1 from the Jackson book • Chapter 1 from the Kleinberg book Social and Economic Networks 5
Different Aspects of the Course • Information Networks • Graph theory • Game theory • Probabilistic Methods • Strategic Interactions • Network Dynamics • Behavior Aggregation and Institutions Social and Economic Networks 6
Information Networks • The information we deal with online has a fundamental network structure. • Examples: • WWW • Friendship graphs • Links among political blogs • A lot of knowledge can be gained from this data • Visualization Techniques • Data Mining • Business Intelligence • Big Data Social and Economic Networks 7
Graph Theory • Graph theory can say a lot about linked data. • There is a rich repository of measurements, frameworks, proved theories which can support us to study different graphs. • Main focus: Structural properties of graphs and their impact on the behavior • Examples: Florentine Marriage, Add Health dataset Social and Economic Networks 8
Florentine Marriage • In the 15 th century in Florence, Medici family was the “ God Father of Renaissance ” . • Even though other families like Strozzi had more wealth and more seats in the local legislator, Medici family was the commander of the oligarchy. • Why? Social and Economic Networks 9
Florentine Marriage Social and Economic Networks 10
Florentine Marriage • The number of links to other families • Medici: 6, Strozzi: 4, Guadagni: 4 • Good but not effective enough • The betweenness measurement: • Defines the average percentage of shortest paths which passes through k. • Medici: 0.522, Strozzi: 0.103, Guadagni: 0.255 Social and Economic Networks 11
Add Health Social and Economic Networks 12
Add Health • A link denotes a romantic relationship between two students. • As our natural intuition confirms, the graph is nearly bipartite. • The graph has some features of large random graphs: • A giant component • The graph is very treelike i.e. while navigating most of the met nodes are new Social and Economic Networks 13
Add Health • From another view: Homophily • There is a bias in friendship • 52% are white and 85% of whites’ friendships are among themselves. • 38% are black and 85% of blacks’ friendships are among themselves • The students are segregated by race Social and Economic Networks 14
Game theory • To model situations in which a group of people must simultaneously choose to act. • The outcome will depend on the joint decisions made by all of them. • General Model: • A set of n players {p 1 ,p 2 ,…, p n } • A i : the set of p i ’s actions • u i : A 1 ×A 2 …. ×A n ⇾ R is the utility function of p i which maps each action profile (a 1 ,a 2 ,…,a n ) to real numbers • An equilibrium definition for action profiles; e.g. Nash equilibrium ∀ 𝑗 ∀ 𝑏 ′ ∈𝐵 𝑗 : 𝑣 𝑗 𝑏 𝑗 , 𝑏 −𝑗 ≥ 𝑣 𝑗 (𝑏 ′ , 𝑏 −𝑗 ) • Example: Braess’s Paradox Social and Economic Networks 15
Braess’s Paradox • 4000 people wants to travel from A to B. • Choosing ACDB by all people is the unique equilibrium which is bad for all. • What is the social welfare? Social and Economic Networks 16
Probabilistic Methods • Almost nothing is deterministic! • Probabilistic models are important tools to study those non- deterministic phenomena. • These models are used both for simulation and theoretical analysis. • Example: Erdos-Renyi Random Graphs. Social and Economic Networks 17
Erdos-Renyi Graphs • Fix a set of n nodes. • Each link is formed with a given probability p • Example: for p=0.02 for 50 nodes Social and Economic Networks 18
Erdos-Renyi Graphs • Probability of a given network with m edges: • The probability that a given node i has degree d: • These graphs are some times called Poisson random networks Social and Economic Networks 19
Erdos-Renyi Graphs Social and Economic Networks 20
Strategic Interactions • Game theoretic interactions on network links • Game theory + Graph theory • Examples: • Network of loans among financial institutions • International trade network • Network Markets which include interactions between buyers and sellers • Strategic network formation models Social and Economic Networks 21
Network of Loans among Financial Instititues • GSCC is a strongly connected component and can be considered as the core of the network. • GIN are senders of the funds • GOUT are the fund receivers Social and Economic Networks 22
International Trade Network Social and Economic Networks 23
Network Markets • Example: A simple Matching Markets • Does there exist a matching? The Philip-Hall theorem: To have a perfect Matching ∀ 𝑇⊆𝑊 𝑂 𝑇 ≥ 𝑇 Social and Economic Networks 24
Strategic Network Formation Models • Example: Symmetric connection model • The relationship between nodes offer benefit in terms of favor, information or … • “Friend of friend”, “friend of friend of friend” and …. Are also beneficial but less than a direct friendship. • The utility function of nodes (each edge costs c ): Social and Economic Networks 25
Strategic Network Formation Models • Example: Symmetric connection model • Each pair of nodes should decide on the link between them. • An efficient network g maximizes 𝑗 𝑣 𝑗 () • For the small value of c ( 𝑑 < 𝜀 − 𝜀 2 ), the complete graph is the unique efficient network (why?) • For the large value of c ( 𝑑 > 𝜀 + 𝑜−2 2 𝜀 2 ) , no edge makes sense. • For the medium values of c , a star is the unique efficient network (exercise). Social and Economic Networks 26
Strategic Network Formation Models • Example: Symmetric connection model • To capture how nodes act, consider a simple definition of equilibrium called Pairwise Stability: • No node benefit from deleting one of his neighboring edges • No two nodes benefit from creating a link between themselves • For 𝑑 < 𝜀 − 𝜀 2 , the complete graph is the only stable and efficient graph. • For 𝜀 > 𝑑 > 𝜀 − 𝜀 2 , a star is a stable and efficient graph, but may not be unique. 𝑜−2 • For 𝜀 < 𝑑 < 𝜀 + 2 𝜀 2 , each node has either no links or else at least two links. Thus, any pairwise stable graph is not efficient. 𝑜−2 2 𝜀 2 , the empty network is the only stable and efficient graph. • For 𝑑 > 𝜀 + Social and Economic Networks 27
Network Dynamics • Recurring patterns by which new ideas, beliefs, opinions, innovations, technologies, social conventions are constantly evolving and emerging. • Social practices can people adopt or not • The way in which new practices spread through a population depends in large part on the fact that people influence each other ’ s behavior Social and Economic Networks 28
Network Dynamics • Population effects: • At a surface level, one could hypothesize that people imitate the decisions of others simply because of an underlying human tendency to conform : we have a fundamental inclination to behave as we see others behaving. • We miss the opportunity to ask why people are influenced? • Example: • Information cascade: Each person make a decision based on his observations from other people. • Consider we have an urn containing 3 marbles colored red or blue. • With probability 50% it contains 2 red and 1 blue (majority-red) and with 50% 1 red and 2 blue (majority-blue) Social and Economic Networks 29
Network Dynamics • Example: • People should sequentially draws a marble from the urn and look at the color and place it back without showing it to others. • Then they should guess whether the urn is majority-blue or majority-red and publicly announce their guess. • The first person: Pr 𝑛𝑏𝑘𝑝𝑠𝑗𝑢𝑧 − 𝑐𝑚𝑣𝑓 𝑐𝑚𝑣𝑓 = 2 3 • The second person: Pr 𝑛𝑏𝑘𝑝𝑠𝑗𝑢𝑧 − 𝑐𝑚𝑣𝑓 𝑐𝑚𝑣𝑓, 𝑐𝑚𝑣𝑓 = 4 5 Social and Economic Networks 30
Social and Economic Networks 31
Network Dynamics • Structural effects: • Network structure is one of the most important key factors influencing on network dynamics. • Example: • Diffusion of Innovation: • v adopts A if 𝑐 𝑞 ≥ 𝑏+𝑐 Social and Economic Networks 32
Recommend
More recommend
