chap. 3 strong and weak ties · chap. 3 strong and weak ties •3.1 triadic closure •3.2 the...

Chap. 3 Strong and weak ties

• 3.1 Triadic closure

• 3.2 The strength of weak ties

• 3.3 Tie Strength and Network Structure in Large-Scale Data

• 3.4 Tie Strength, Social Media, and Passive Engagement

• 3.5 Closure, Structural Holes, and Social Capital

• 3.6 Advanced Material: Betweenness Measures and Graph Partitioning

3.1 Triadic Closure

• Grundidee von Triadic Closure ist:

Wenn 2 Leute einen gemeinsamen Freund haben,

dann sind sie mit größer Wahrscheinlichkeit, dass

sie mit einander befreundet sind.

3.1 Triadic Closure

3.1 Triadic Closure

• Clustering Coefficient :

– The clustering coefficient of a node A is defined as the probability that two randomly selected friends of A are friends with each other.

3.1 Triadic Closure

• Betrachten wir Node A

• Freunde von A : B,C,D,E

• Es gibt 6 Möglichkeiten,

um solche Nodes zu verbinden, gibt aber nur

eine Kante (C,D)

=> Co(A) = 1/6

3.1 Triadic Closure

• Reasons for Triadic Closure:

– Opportunity

• B, C have chances to meet when they both know A

– Basis for Trusting

• When B, C both know A, they can trust each other

better then unconnected people

– Incentive

• A wanted to bring B, C together to avoid relationship’s

problems

3.2 The Strength of Weak Tie

• Bridges:

– An edge (A,B) is a Bridge if deleting it would causeA,B to lie in 2 different components

• Means there is only one route between A,B

• Bridge is extremely rare in real social network


• Local Bridge:– An edge (A,B) is a local Bridge if its endpoints have no

friends in common (if deleting the edge would increase the distance between A and B to a value strictly more than 2.)

• Span:– span of a local bridge is the distance its endpoints

would be from each other if the edge were deleted


• The Strong Triadic Closure Property.

– a node A violates the Strong Triadic ClosureProperty if it has strong ties to two other nodes Band C, and there is no edge at all (either a strongor weak tie) between B and C.

• We say that a node A satisfies the Strong Triadic ClosureProperty if it does not violate it.


• Local Bridges and Weak Ties.

– If a node A in a network satifies 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

3.3 Tie Strength and Network Structure in Large-Scale Data

• Generalizing the Notions of Weak Ties and Local Bridges:

– Neighborhood overlap

– NO(A, B) = 0 if (A,B) is local Bridge

• No common neighbor

3.3 Tie Strength and Network Structure in Large-Scale Data

• Empirical Results on Tie Strength and Neighborhood Overlap.

• Dependence between tie strength of an edge and its neighborhood overlap: – neighborhood overlap

should grow as tie strength grows

• Weak ties servers for keeping giant component intact

3.4 Tie Strength, Social Media, and Passive Engagement

• reciprocal (mutual) communication– user both sent messages to the friend at the other

end of the link, and also received messages from them during the observation period.

• one-way communication– user sent one or more messages to the friend at the

other end of the link

• maintained relationship– user followed information about the friend at the

other end of the link, whether or not actual communication took place (z.B : news von Facebook)

Tie Strength in Twitter

3.5 Closure, Structural Holes, and Social Capital

• Embeddedness:

– embeddedness of an edge in a network is the number of common neighbors the two endpointshave.

– Numerator of neighborhood overlap

• Bridge has embeddedness 0


• Closure and Bridging as Forms of Social Capital.• two important sources of variation

– (1) property of a group, with some groups functioning more effectively than others because of favorable properties of their social structures or networks

– (2) based on whether • social capital is a property that is purely intrinsic to a group

— based only on the social interactions among the group’s members

– or whether • it is also based on the interactions of the group with the

outside world.

3.6 Advanced Material: Betweenness Measures andGraph Partitioning

A. A Method for Graph Partitioning

General Approaches to Graph Partitioning.

1) Identifying and removing the “spanning links” between densely-connected regions. ( which would break network apart)

2) find nodes that are likely to belong to the same region and merge them together. ( which would make each Group larger)


• The Notion of Betweenness.

– betweenness of an edge to be the total amount of flow it carries, counting flow between all pairs of nodes using this edge.

• Using betweenness as unit for traffic

• look for the edges that carry the most of traffic

• if there are k shortest paths from A and B, then 1/k units of flow pass along each one.


• The Girvan-Newman Method: Successively Deleting Edges of High Betweenness.

– (1) Find the edge of highest betweenness — or multiple edges of highest betweenness, if there is a tie and remove these edges from the graph.

– (2) Recalculate all betweennesses, and again remove the edge or edges of highest betweenness.

– Redo (1)


• Es gibt noch eine andere Methode:

– deleting edges of minimum total strength so as to separate two specified nodes (known as the problem of finding a minimum cut)

– Find the same result as Girvan-Newman Method

• Which methode is better is hard to say

– more or less effective on different kinds ofnetworks

– Both work effectively only on small network


• B. Computing Betweenness Values– the definition of betweenness involves reasoning

about the set of all the shortest paths between pairs of nodes.

– how can we efficiently compute betweennesswithout the overhead of actually listing out all such paths?

• We can compute betweennesses efficiently using breadth-first search – Computes total flow from 1 Node to all the other


• (1) Perform a breadth-first search of the graph, starting at a Vetex (A).

• (2) Determine the number of shortest paths from A to each other node.

• (3) Based on these numbers, determine the amount of flow from A to all other nodes that uses each edge.


• Counting Shortest Paths.


• Determining Flow Values.


• Final Observations.

– Methode works well on networks of moderate size (up to a few thousand nodes)

– for larger networks, the need to recomputebetweenness values in every step becomes computationally very expensive.

• Effectiver :

– approximating the betweenness

– divisive and agglomerative methods

chap. 3 strong and weak ties · chap. 3 strong and weak ties •3.1 triadic closure •3.2 the...

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