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
3.2 The Strength of Weak Tie
• 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
3.2 The Strength of Weak Tie
3.2 The Strength of Weak Tie
• 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.
3.2 The Strength of Weak Tie
• 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
3.5 Closure, Structural Holes, and Social Capital
3.5 Closure, Structural Holes, and Social Capital
• 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)
3.6 Advanced Material: Betweenness Measures andGraph Partitioning
• 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.
3.6 Advanced Material: Betweenness Measures andGraph Partitioning
3.6 Advanced Material: Betweenness Measures andGraph Partitioning
3.6 Advanced Material: Betweenness Measures andGraph Partitioning
3.6 Advanced Material: Betweenness Measures andGraph Partitioning
3.6 Advanced Material: Betweenness Measures andGraph Partitioning
3.6 Advanced Material: Betweenness Measures andGraph Partitioning
• 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)
3.6 Advanced Material: Betweenness Measures andGraph Partitioning
• 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
3.6 Advanced Material: Betweenness Measures andGraph Partitioning
• 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
3.6 Advanced Material: Betweenness Measures andGraph Partitioning
• (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.
3.6 Advanced Material: Betweenness Measures andGraph Partitioning
3.6 Advanced Material: Betweenness Measures andGraph Partitioning
• Counting Shortest Paths.
3.6 Advanced Material: Betweenness Measures andGraph Partitioning
• Determining Flow Values.
3.6 Advanced Material: Betweenness Measures andGraph Partitioning
• 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