`nwconstraint`

Calculate Burt’s constraint

Syntax

nwconstraint
[netname]
[,
nwset_options]
where nwset_options are any options accepted by nwset's mat() form (e.g.
name(), undirected, directed) - they are forwarded unchanged to the internal
nwset, mat() call that stores the result.

Description

Calculates Burt’s (1992) dyadic constraint for a network and stores the result as a new network (not a Stata variable) via nwset, mat() - the constraint matrix becomes the current network afterward. This is a different output shape than most other analysis commands in this package (which generate() a per-node Stata variable): nwconstraint returns the full dyadic c_ij matrix. nwburt computes the standard per-node aggregate constraint (and Burt’s related effective size/efficiency/hierarchy measures) directly as Stata variables - use it instead of this command if the dyadic matrix itself isn’t what you need; see the “Aggregating to the node level” note below for why a plain row sum of this command’s own output is not equivalent to nwburt’s aggregate.

Constraint measures the extent to which a node i’s relationships are concentrated through a single contact or a tightly interconnected group of contacts, rather than spread across independent, unconnected contacts (the latter is Burt’s “structural holes” - low constraint, high brokerage potential). Formally, for each pair i,j:

p_ij = a_ij / sum_k(a_ik) (i’s tie to j, as a proportion of all of i’s outgoing ties)

c_ij = (p_ij + sum_q(p_iq * p_qj))^2 (direct investment in j, plus indirect investment via every other contact q)

The diagonal (self-constraint) is not meaningful and is not part of the returned network.

Aggregating to the node level: the quantity most commonly reported in the literature as “Burt’s constraint” is node i’s aggregate constraint, C_i = sum_j(c_ij) for j in i’s direct contacts only - not summed over every j. This distinction matters because c_ij can be nonzero even when i and j are not directly tied at all (the sum_q(p_iqp_qj)* indirect term alone can make it positive), so naively summing an entire row of this command’s output - sum_j(c_ij) over all j - silently over-counts and gives a different, larger number than the standard aggregate for any node with such indirect-only contributions. nwburt computes the correctly-restricted aggregate directly (as _constraint); this command intentionally does not, since restricting the sum to N(i) requires already knowing which entries of the raw network matrix are direct ties, which is exactly the extra step nwburt takes care of.

Examples

. nwwebuse gang, nwclear
. nwconstraint gang, name(gangconstraint)
. nwtomata gangconstraint, mat(C)
. nwtomata gang, mat(A)
. mata: rowsum(C :* (A :!= 0 :& A :< .))

Supported network types

Binary: yes. Directed: yes, but not symmetrized - p_ij is computed from the raw adjacency matrix exactly as given, so a directed network’s constraint reflects outgoing ties only (a node with no outgoing ties at all has p entirely zero, and therefore zero constraint toward everyone, regardless of its incoming ties). This is a real, asymmetric treatment of directed data - not a symmetrized “Burt” constraint in the traditional (undirected) sense - and is the command’s actual behavior, not silently assumed. Weighted: W1, native - tie weight is used directly as p_ij’s investment proportion (exactly Burt’s own interpretation of tie strength), not as a distance. Signed: not supported - p_ij is a ratio to a sum of outgoing tie weights, which is only mathematically meaningful when all of a node’s tie weights are non-negative; a negative tie included in the same network as positive ties will distort every p_ij for that node (values can fall outside [0,1], and c_ij can exceed 1) rather than being handled as a distinct signed relation. Two-mode: not checked.

References

Ronald S. Burt (1992). Structural Holes: The Social Structure of Competition. Harvard University Press.

nwconstraint