`nwkcomponents`

Maximal k-component enumeration

Syntax

nwkcomponents
[netlist]
[,
k(int)
generate(newvarname)
replace
silent]


`k(int)`	Minimum vertex connectivity required; default = 2
`generate(newvarname)`	Required. Name of the Stata variable that stores each node’s largest qualifying k-component size
`replace`	Replace existing variable
`silent`	Suppress display of results

Description

nwkcomponents enumerates every maximal k-component (Kanevsky 1993) in the network(s) in netlist - a subgraph with vertex connectivity of at least k(int), meaning at least k(int) nodes would have to be removed from it to disconnect it (or reduce it to a single node). A k-component is a genuine strengthening of an ordinary connected component (which is just the k(int)=1 case - nwcomponents already implements that, more cheaply, via a simple reachability search rather than a connectivity computation): k(int)=2 excludes single cut-vertices/bridges that would fracture the group, k(int)=3 additionally excludes any 2-node cutset, and so on. This directly formalizes the intuition that a group connected only by a single “weak link” is less cohesive than one where several independent paths connect every pair of members.

k(int) defaults to 2 - the smallest level that is a genuine refinement of nwcomponents’ own plain connectivity - and must be at least 1 (nwcomponents already covers that trivial case directly). Vertex connectivity is computed via the standard node-splitting reduction to max-flow (Even 1979) combined with Menger’s theorem (the minimum vertex set separating any two non-adjacent nodes equals the maximum flow between them in the split graph); the network’s own overall k-components are then found by the standard recursive decomposition also underlying Moody and White’s (2003) cohesive blocking - see Algorithm below for the one respect in which this command deliberately does less than the full Moody-White procedure.

Like cliques/k-plexes/n-cliques/n-clans, k-components can genuinely overlap - the nodes whose removal disconnects a graph (a cutset) remain shared members of every resulting sub-block their removal reveals, not assigned to just one side - so nwkcomponents follows the same output shape as nwclique/nwkplex/nwnclique: a single per-node “largest qualifying k-component size” summary variable (generate(newvarname), required), plus the complete overlapping structure in r(kcomp_matrix) (a k-components-by-nodes 0/1 membership matrix) and r(kcomponents) (count). Unlike those commands there is no minsize() - a k-component’s own minimum possible size is already k(int)+1 (a smaller set cannot reach connectivity k(int) at all, since the maximum possible connectivity of an s-node graph is s-1), so there is no equivalent “trivial small case” to filter out separately.

Examples

. nwwebuse florentine, nwclear
. nwkcomponents flomarriage, generate(_kcompnum)
. nwkcomponents flomarriage, k(3) generate(_kcompnum) replace

Supported network types

Binary: yes. Directed: yes, automatically symmetrized (vertex connectivity in the classical Moody-White sense has no directed generalization here, the same reasoning nwclique/ nwkplex/nwnclique already apply). Weighted: not used - only tie presence/absence affects connectivity. Signed: not checked. Two-mode: not checked. Vertex-connectivity computation via max-flow is polynomial per pair, but nwkcomponents computes it between every non-adjacent pair (a deliberately simple, definitely-correct brute-force rather than the smaller reference-vertex subset a more optimized algorithm would use - see nwclique’s own “Supported network types” section for the same trade-off philosophy applied there), and the overall recursive decomposition can call this repeatedly - fine for the moderate network sizes typical of SNA datasets, not recommended for very large or very dense networks.

Stored results

Scalars

r(kcomponents) number of maximal k-components found

Matrices

r(kcomp_matrix) k-components-by-nodes 0/1 membership matrix, one row per maximal k-component

References

Kanevsky, A. (1993). Finding all minimum-size separating vertex sets in a graph. Networks 23(6), 533-541.

Moody, J., White, D.R. (2003). Structural cohesion and embeddedness: a hierarchical concept of social groups. American Sociological Review 68(1), 103-127.

Even, S. (1979). Graph Algorithms. Computer Science Press. (the vertex-splitting max-flow reduction for vertex connectivity)

Wasserman, S., Faust, K. (1994). Social Network Analysis: Methods and Applications. Cambridge University Press. (k-components and structural cohesion)

nwkcomponents