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This lecture covers key concepts in network theory, focusing on measures such as cocitation and bibliographic coupling. It discusses the nature of independent paths, including edge-independent and vertex-independent paths, highlighting their significance in understanding vertex connectivity and resilience to failures. Key theorems, such as Menger's theorem, illustrate the relationship between cut sets and independent paths. The lecture also explores structural metrics like clustering coefficients, transitivity, and assortative mixing, elucidating their importance in analyzing networks across various domains.
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Lecture 8 Measures and Metrics
Cocitation and Bibliographic coupling • Cocitation of two vertices i and j is the number of vertices that have outgoing edges to both • Bibliographic coupling is the number of vertices to which both point
Independent paths • Edge independent paths: if they share no common edge • Vertex independent paths: if they share no common vertex except start and end vertices • Vertex-independent => Edge-independent • Also called disjoint paths • These set of paths are not necessarily unique • Connectivity of vertices: the maximal number of independent paths between a pair of vertices • Used to identify bottlenecks and resiliency to failures
Cut Sets and Maximum Flow • A minimum cut set is the smallest cut set that will disconnect a specified pair of vertices • Need not to be unique • Menger’s theorem: If there is no cut set of size less than n between a pair of vertices, then there are at least n independent paths between the same vertices. • Implies that the size of min cut set is equal to maximum number of independent paths • for both edge and vertex independence • Maximum Flow between a pair of vertices is the number of edge independent paths times the edge capacity.
Transitivity • is said to be transitive if a b and b c together imply a c • Perfect transitivity in network → cliques • Partial transitivity • u knows v and v knows w → =
Local Clustering and Redundancy • Redundancy
Reciprocity • How likely is it that the node you point to will point to you as well.
Signed Edges and Structural balance • Friends / Enemies • Friend of friend → • Enemy of my enemy → • Structural balance: only loops of even number of “negative links” • Structurally balanced → partitioned into groups where internal links are positive and between group links are negative
Similarity • Structural Equivalence: share many of the same neighbors • Cosine Similarity: • Pearson Coefficient: Given degree of two nodes, how many common neighbors they have () • Euclidian Distance: • Regular Equivalence: neighbors are the same • Katz Similarity:
Homophily and AssortativeMixing • Assortativity: Tendency to be linked with nodes that are similar in some way • Humans: age, race, nationality, language, income, education level, etc. • Citations: similar fields than others • Web-pages: Language • Disassortativity: Tendency to be linked with nodes that are different in some way • Network providers: End users vs other providers • Assortative mixing can be based on • Enumerative characteristic • Scalar characteristic
Modularity (enumerative) • Extend to which a node is connected to a like in network • + if there are more edges between nodes of the same type than expected value • - otherwise is 1 if ciand cj are of same type, and 0 otherwise err is fraction of edges that join same type of vertices ar is fraction of ends of edges attached to vertices type r
Assortativecoefficient (enumerative) • Modularity is almost always less than 1, hence we can normalize it with the Qmax value
Assortativecoefficient (scalar) • r=1, perfectly assortative • r=-1, perfectly disassortative • r=0, non-assortative • Usually node degree is used as scale
Assortativity Coefficientof Various Networks M.E.J. Newman. Assortative mixing in networks
