Weak Ties & Their Strength Social and Economic Networks Jafar - - PowerPoint PPT Presentation

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Weak Ties & Their Strength Social and Economic Networks Jafar - - PowerPoint PPT Presentation

Feb 22, 2023 116 likes β€’349 views

Weak Ties & Their Strength Social and Economic Networks Jafar Habibi MohammadAmin Fazli Social and Economic Networks 1 ToC Weak Ties & Their Strength Triadic Closure Local Bridges Strength of Weak ties Neighborhood

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SLIDE 1

Weak Ties & Their Strength

Social and Economic Networks

Jafar Habibi MohammadAmin Fazli

Social and Economic Networks 1

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SLIDE 2

ToC

  • Weak Ties & Their Strength
  • Triadic Closure
  • Local Bridges
  • Strength of Weak ties
  • Neighborhood Overlap
  • Structural Holes
  • Social Capital
  • The Graph Partitioning Problem
  • Partitioning with Betweenness
  • Readings:
  • Chapter 3 from the Jackson book
  • Chapter 3 from the Kleinberg book

Social and Economic Networks 2

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SLIDE 3

Weak Ties are Important

  • How do people find their jobs?
  • Granovetter:
  • Many people find from their social relationships
  • Among them, the majority has found their jobs from their acquaintances (not

their close friends!).

  • Acquaintances have a great impact on our life. We can model their

friendship with weak ties.

Social and Economic Networks 3

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SLIDE 4

Triadic Closure

  • If two people in a social network have a friend in common, then there

is an increased likelihood that they will become friends themselves at some point in the future

Social and Economic Networks 4

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SLIDE 5

Triadic Closure

  • Triadic closure is a procedure for the evolution of links.
  • Clustering coefficient approximately shows how much triadic closure

is used to form the networks.

  • Reasons:
  • the opportunity for B and C to meet: if A spends time with both B and C]
  • the fact that each of B and C is friends with A (provided they are mutually

aware of this) gives them a basis for trusting each other

  • the incentive A may have to bring B and C together: if A is friends with B and

C, then it becomes a source of latent stress in these relationships if B and C are not friends with each other

Social and Economic Networks 5

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SLIDE 6

Bridges & Local Bridges

  • Removing bridges makes the

distance between two nodes A and B, infinite

  • Removing local bridges increase

the distance between two nodes A and B, to a value more than two

  • A and B has not a common

neighbor

  • Span of a local bridge is the

distance between A and B if AB is deleted

Social and Economic Networks 6

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SLIDE 7

Strong Triadic Closure Property

  • We say that a node A

violates the Strong Triadic Closure Property if it has strong ties to two other nodes B and C, and there is no edge at all (either a strong or weak tie) between B and C. We say that a node A satisfies the Strong Triadic Closure Property if it does not violate it.

Social and Economic Networks 7

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SLIDE 8

Strength of Weak ties

  • Weak ties can join you to other clusters of peoples i.e. (local) bridges

are weak

  • Theorem: If a node A in a network satisfies the Strong Triadic Closure

Property and is involved in at least two strong ties, then any local bridge it is involved in must be a weak tie.

  • Proof: black board
  • If we remove edges sequentially according to their strength
  • From the strongest downward
  • From the weakest upward

, which results in smaller giant components

Social and Economic Networks 8

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SLIDE 9

Neighborhood-overlap

  • Neighborhood-overlap for

nodes A and B in graph 𝑕: |𝑂𝐡 𝑕 ∩ 𝑂𝐢(𝑕)| |𝑂𝐡 𝑕 βˆͺ 𝑂𝐢 𝑕 βˆ– {𝐡, 𝐢}|

Social and Economic Networks 9

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SLIDE 10

Strength of Weak Ties & Neighborhood Overlap

  • Neighborhood Overlap is

very correlated with strength

Social and Economic Networks 10

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SLIDE 11

Embeddedness

  • The embeddedness of an edge in

a network to be the number of common neighbors the two endpoints have.

  • (Local) bridges have

embeddedness of zero

  • Linkages with high embeddedness

are more likely to form. Why?

  • What happens if all edges of a

node have significant embeddedness? (See node A)

Social and Economic Networks 11

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SLIDE 12

Structural Holes

  • A structural hole is an empty space

between two sets of nodes that do not otherwise interact closely i.e. absence of connections between two groups

  • There is no formal definition, you

can do by yourselves

  • Filling structural holes leads to

benefits

Social and Economic Networks 12

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SLIDE 13

Structural Holes

  • Advantages of B over A
  • B has early access to information of

different non-interacting groups

  • Can mix different ideas and be more

creative

  • Can be a gate-keeper i.e. a regulator
  • f information between C and D

Social and Economic Networks 13

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SLIDE 14

Social Capital

  • Social capital has not an exact definition
  • Has come to embody a number of different concepts related to how

social relationships lead to individual of aggregate benefits in society.

  • Sometimes it refers to
  • The relationships that an individual has and the potential benefits that those

relationships can bestow.

  • Broader social interactions (norms, sanctions, trust and ...)
  • And….

Social and Economic Networks 14

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SLIDE 15

Social Capital

  • Studies and theories of social capital have resulted in new conceptual

understandings of social networks and their impact on behavior

  • E.g. Coleman emphasizes the role of closure, which roughly corresponds to

high clustering, in the enforcement of social norms. Closure allows agents to coordinate sanctioning of individuals who deviate

  • E.g. Putnam defines bridging capital or bonding capital which shows the

impact of bridging between different tightly-knit groups of people.

Social and Economic Networks 15

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SLIDE 16

An Application: The Graph Partitioning Problem

  • How to break the graph

structure into tightly- knit which are connected with a sparse set of links?

  • How to detect weaker

ties which lies on the boundaries between regions?

Social and Economic Networks 16

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Graph Partitioning General Approaches

  • Divisive Methods:
  • Find spanning links between densely-connected regions
  • Remove these edges sequentially and divide the graph
  • Top-Down approach
  • Agglomerative Methods:
  • Find nodes that may belong to a same densely-connected region
  • Merge them together and repeat the process until some desired criteria meet
  • Bottom-Up approach

Social and Economic Networks 17

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Graph Partitioning General Approaches

  • Both approaches find nested structure of regions

Social and Economic Networks 18

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Using Betweenness

  • Local bridges often connect weakly interacting parts of the network
  • Removing local bridges is a good idea but not strong enough
  • Which local bridge to remove?
  • What to do when there exists no local bridge?

Social and Economic Networks 19

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Using Betweenness

  • Local bridges are important because they are usually parts of shortest

paths between many pairs of nodes. Why?

  • Because they connect different tightly-knit groups.
  • Another intuition about betweenness:
  • Consider a 1-flow between every pair or nodes
  • If there exist k shortest paths between two nodes i and j, divide the 1-flow

into k individual 1/k-flows along every shortest path

  • The total amount of flow an edge or a node carries defines its betweenness
  • A heuristic: Using betweenness for graph partitioning

Social and Economic Networks 20

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Girwan-Newman Algorithm

  • Find the the edge with the highest betweenness and remove it
  • Continue this process until we are satisfied with the found groups

Social and Economic Networks 21

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SLIDE 22

Computing Betweenness Values

  • Perform a BFS
  • Count the number of

shortest paths

Social and Economic Networks 22

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SLIDE 23

Computing Betweenness Values

  • Determine the amount
  • f flows

Social and Economic Networks 23