`nwlambda`

Edge (line) connectivity matrix between all node pairs

Syntax

nwlambda
[netname]
[,
name(newnetname)
xvars
replace
force]


`name(newnetname)`	Name of the new connectivity network; default = lambda
`xvars`	Generate Stata variables for the new network
`replace`	Replace an existing network of the same name
`force`	Run the command although its matrices would not fit in memory (it stops with r(484) above 12 GB otherwise)

Description

nwlambda computes the EDGE (or “line”) connectivity between every pair of nodes - lambda_ij, the maximum number of edge-disjoint paths connecting i and j, equivalently (by Menger’s theorem’s edge version) the size of the smallest set of ties whose removal would disconnect them - and stores the result as a new, valued, undirected network newnetname (default: lambda). This is Borgatti, Everett & Shirey’s (1990) own foundation for lambda sets (also called LS sets): a maximal subset of nodes where every pair inside has HIGHER edge connectivity with each other than either one has with any node outside the set - a classical cohesive-subgroup concept, and the edge-connectivity sibling of nwkcomponents’s own VERTEX-connectivity-based k-components.

Computed via a direct max-flow (Edmonds-Karp) between every pair of nodes on the network’s own symmetrized, binarized adjacency matrix - always undirected and binary, matching every other classical cohesive-subgroup command in this package (nwclique, nwkplex, nwnclique, nwkcomponents all make the identical choice, since none of these concepts has a standard directed/valued generalization in the literature). lambda_ii (self-comparison) is undefined and stored as missing.

Examples

. nwwebuse florentine, nwclear
. nwlambda flomarriage
. nwsummarize lambda, matonly

Supported network types

Binary: yes (only) - similarity is computed from binary edge-connectivity; tie values are ignored. Directed: yes - automatically symmetrized first (edge connectivity has no standard directed generalization in the classical cohesive-subgroup literature, the same choice nwclique, nwkplex, nwnclique, nwkcomponents already make). Weighted: not applicable. Signed: not applicable. Two-mode: same as for the bipartite graph read as a one-mode network (valid; no mode-specific normalisation).

Stored results

Scalars

r(nodes) number of nodes

Macros

r(netname) name of the new connectivity network

References

Borgatti, S.P., Everett, M.G., Shirey, P.R. (1990). LS sets, lambda sets and other cohesive subsets. Social Networks 12(4), 337-357.

Menger, K. (1927). Zur allgemeinen Kurventheorie. Fundamenta Mathematicae 10, 96-115.

The result is an n-by-n network and two matrices are held at once: above about 27,000 nodes (12 GB) the command stops at once with r(484), and force runs it anyway.

nwlambda