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Introduction Social and Economic Networks MohammadAmin Fazli - - PowerPoint PPT Presentation

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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, .

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Introduction

Social and Economic Networks MohammadAmin Fazli

Social and Economic Networks 1

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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

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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:

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This Course

  • Exercises
  • Theory + programming
  • Quizzes
  • Final Exam

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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

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Different Aspects of the Course

  • Information Networks
  • Graph theory
  • Game theory
  • Probabilistic Methods
  • Strategic Interactions
  • Network Dynamics
  • Behavior Aggregation and Institutions

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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

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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

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Florentine Marriage

  • In the 15th 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

  • ligarchy.
  • Why?

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Florentine Marriage

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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

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Add Health

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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

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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

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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 {p1,p2,…,pn}
  • Ai : the set of pi’s actions
  • ui: A1×A2…. ×An⇾ R is the utility function of pi which maps each action profile

(a1,a2,…,an) to real numbers

  • An equilibrium definition for action profiles; e.g. Nash equilibrium

∀𝑗∀𝑏′∈𝐵𝑗: 𝑣𝑗 𝑏𝑗, 𝑏−𝑗 ≥ 𝑣𝑗(𝑏′, 𝑏−𝑗)

  • Example: Braess’s Paradox

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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?

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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.

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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

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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

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Erdos-Renyi Graphs

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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

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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

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International Trade Network

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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

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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):

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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 𝑣𝑗(𝑕)

  • For the small value of c (𝑑 < 𝜀 − 𝜀2), the

complete graph is the unique efficient network (why?)

  • For the large value of c (𝑑 > 𝜀 + 𝑜−2

2 𝜀2),

no edge makes sense.

  • For the medium values of c, a star is the

unique efficient network (exercise).

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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 − 𝜀2, the complete graph is the only stable and efficient graph.
  • For 𝜀 > 𝑑 > 𝜀 − 𝜀2, a star is a stable and efficient graph, but may not be

unique.

  • For 𝜀 < 𝑑 < 𝜀 +

𝑜−2 2 𝜀2, each node has either no links or else at least two

  • links. Thus, any pairwise stable graph is not efficient.
  • For 𝑑 > 𝜀 +

𝑜−2 2 𝜀2, the empty network is the only stable and efficient graph.

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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

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Network Dynamics

  • Population effects:
  • At a surface level, one could hypothesize that people imitate the decisions of
  • thers 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)

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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

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Network Dynamics

  • Structural effects:
  • Network structure is one
  • f the most important key

factors influencing on network dynamics.

  • Example:
  • Diffusion of Innovation:
  • v adopts A if

𝑞 ≥

𝑐 𝑏+𝑐

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Behavior aggregation & Institutions

  • How social-economical designers can use networks and strategic

behaviors to capture which outcome will take place or to enforce certain kind of overall outcomes

  • Based on the rich literature of game theoretical Mechanism Design
  • Reaching truthfulness
  • Examples:
  • Think about the market price as an aggregator of individuals’ beliefs about

the value of an asset.

  • Prediction markets: sell different assets that pay a fixed amount if a certain

event takes place and consider the converged price.

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Behavior aggregation & Institutions

  • The prediction

market to predict Democrats or Republicans win

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Behavior aggregation

  • Examples:
  • Voting is another social institution that aggregate social behavior
  • Each individual has a preference over a set of choices
  • The target is to design a mechanism to aggregate these conflicting

preferences

  • Arrow’s Impossibility Theorem

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