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 cause A,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 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. 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 endpoints have. – 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 and Graph 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 and Graph 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 and Graph Partitioning 3.6 Advanced Material: Betweenness Measures and Graph Partitioning 3.6 Advanced Material: Betweenness Measures and Graph Partitioning 3.6 Advanced Material: Betweenness Measures and Graph Partitioning 3.6 Advanced Material: Betweenness Measures and Graph Partitioning 3.6 Advanced Material: Betweenness Measures and Graph 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 and Graph 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 of networks – Both work effectively only on small network 3.6 Advanced Material: Betweenness Measures and Graph 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 betweenness without 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 and Graph 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 and Graph Partitioning 3.6 Advanced Material: Betweenness Measures and Graph Partitioning • Counting Shortest Paths. 3.6 Advanced Material: Betweenness Measures and Graph Partitioning • Determining Flow Values. 3.6 Advanced Material: Betweenness Measures and Graph Partitioning • Final Observations. – Methode works well on networks of moderate size (up to a few thousand nodes) – for larger networks, the need to recompute betweenness values in every step becomes computationally very expensive. • Effectiver : – approximating the betweenness – divisive and agglomerative methods.