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 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
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
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
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
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
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
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
Neighborhood-overlap • Neighborhood-overlap for nodes A and B in graph : |𝑂 𝐵 ∩ 𝑂 𝐶 ()| |𝑂 𝐵 ∪ 𝑂 𝐶 ∖ {𝐵, 𝐶}| Social and Economic Networks 9
Strength of Weak Ties & Neighborhood Overlap • Neighborhood Overlap is very correlated with strength Social and Economic Networks 10
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
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
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 of information between C and D Social and Economic Networks 13
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
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
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
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
Graph Partitioning General Approaches • Both approaches find nested structure of regions Social and Economic Networks 18
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
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
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
Computing Betweenness Values • Perform a BFS • Count the number of shortest paths Social and Economic Networks 22
Computing Betweenness Values • Determine the amount of flows Social and Economic Networks 23
Recommend
More recommend
