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NETWORKS > COHESION > TRANSITIVITY (NEW)

PURPOSE Gives the density of transitive triples in a network., the triples can be ordered or unordered. For valued networks the density of transitive triples defined more generally is given.

DESCRIPTION A path u-v-w of length two from a graph or digraph is transitive if vertex u is connected to vertex w. An unordered triple is transitive if it contains a transitive path. The ordered (unordered) transitivity is the number of paths (triples) which are transitive divided by the number of paths of length 2 (triples contating paths of length 2), i.e. transitivity is the number of triples that are transitive divided by the number of triples which have the potential to be transitive by the addtition of a single edge.

This definition can be extended to valued data.  Ultrametric transitivity occurs only if the final edge is stronger than the two in the original path.  This can be relaxed so that the user can define the minimum value of the final edge (Granovetter transitivity).

PARAMETERS
Input dataset
Name of file containing dataset to be analyzed. Data type: Valued graph.

Output Measures: (Default ='<inputfilename>-transitivity')
Name of UCINET file which will contain tables of values as described in the logfile below.

Treat data as

Directed- treats all data as directed and hence an undirected dataset will be treated as if the edges are reciprocated, doubling some of the counts but the overall transitivity will be the same.

Undirected-treats data as undirected and hence for directed data theunderlying graph will be analyzed.

Auto-detect-automatically checks whether the data is directed or undirected

Model

Ordered
Transitivity is calculated by looking at paths of length two as the base.

Unordered
Transitivity is calculated by taking triples as the base. Hence the triple must contain at least one path of length two to be a base and any base that contains a transitive path will be transitive. It should be noted in this definition the triple may also contain intransitive paths but it will still be counted as transitivie.

Valued data the following values can be user specified, the assumption is that the data is similarity type data.

Valid ties have value greater than (default =0)
Value below which ties are ignored. This is specified as v in the descriptions below.

Strong ties have values greater than
Value g needed in the Granovetter method defined below.

Type of transitivity: (Default = ADJACENCY)
Choices are:

Standard - A triple xik,xij,xjk is transitive if xik is 1 whenever xij and xjk are both 1. Where the network has been dichotomised with a cut-off value of v where v is defined above.

Ultrametric - A triple xik,xij,xjk is transitive if xik ³ min(xij,xjk) provided all are greater than v where v is defined above.

Granovetter - A triple xik,xij,xjk is transitive if whenever min(xij,xjk) ³ v then xik ³ g where v and g are defined above.

LOG FILE For each method selected the following are caculated

Base: number of potentially transitive paths or triples. That is either the number of paths of length two or the number of triples containing paths of length 2. These are called gBase for Granovetter and uBase for ultrametric.

Open: number of base paths or triples that were not transitive, called gOpen for Granovetter and uOpen for ultrametric

Closed: number of base paths or triples that are transitive, note Base= Open+Closed, called gClosed for Granovetter and uClosed for ulrametric.

Transitivity: the density of the transitive paths or triples that is Closed divided by Base. Again called gTransitivity for Granovetter and uTransitivity for ultrametric.

TIMING O(N^3).