`nwinduced`

Induced, endogenous and exogenous centrality

Syntax

nwinduced
[netname]
,
measure(string)
[generate(stub)
replace
silent]


`measure(string)`	Underlying centrality measure: one of degree, betweenness, closeness, evcent
`generate(stub)`	Required. Prefix for the three output variables (stub_endog, stub_induced, stub_exog)
`replace`	Replace existing variables
`silent`	Suppress display of results

Description

nwinduced computes Everett and Borgatti’s (2010) three-part decomposition of node centrality, built on top of any of this package’s own existing per-node centrality measures rather than being a single new formula of its own. Given a chosen measure C (e.g. degree, betweenness) and network G:

Endogenous centrality of node v: C_G(v) itself - the node’s ordinary centrality score, exactly as nwdegree/nwbetween/nwcloseness/nwevcent would already report it. Not a new computation; the existing measure, relabeled under this framework.

Induced centrality of node v: sum_i C_G(i) - sum_i C_(G-v)(i) - the total drop in EVERYONE’s summed C-score when v is removed from the network (G-v: the network with v and its own ties deleted). Combines both v’s own direct contribution (its score, which vanishes on removal) and every indirect ripple effect on everyone else’s score.

Exogenous centrality of node v: Induced(v) - Endogenous(v) - the purely indirect part: how much v’s presence props up OTHER nodes’ centrality, isolated from v’s own standing.

A useful free sanity check, not specific to any one network: with measure(degree), induced centrality reduces to EXACTLY twice plain degree, and exogenous centrality reduces to exactly plain degree again, for any undirected network - because removing a node removes exactly its own degree’s worth of edges, each of which contributed 2 to the total degree sum. This holds as an identity, not merely on a hand-picked example (see Examples below).

The real computational cost is the leave-one-out reconstruction: computing induced centrality for every node requires recomputing C on n separate n-1-node subgraphs, one per removed node. For a cheap measure (degree) this is trivial; for an expensive one (betweenness, on a large network) this means n full recomputations of an already-expensive measure - worth being aware of before running this against a large network with measure(betweenness).

Examples

. nwclear
. nwset, mat((0,1,1,1\1,0,0,0\1,0,0,0\1,0,0,0)) name(starnet) undirected labs(A,B,C,D)
. nwinduced starnet, measure(degree) generate(_induced)

A hub (A, degree 3) and three leaves (B/C/D, degree 1) - endogenous exactly reproduces plain
degree; induced is exactly twice that (6 for A, 2 for each leaf); exogenous equals endogenous
again (3 for A, 1 for each leaf) - the free identity described above, verified here on a concrete
example.

. nwwebuse florentine, nwclear
. nwinduced flobusiness, measure(betweenness) generate(bw)

Which Florentine family props up OTHERS’ betweenness the most, independent of its own
standing? - the bw_exog column answers that, distinct from bw_endog (the family’s own
plain betweenness score).

Supported network types

Binary: yes. Directed: measure(betweenness/closeness/evcent) support directed networks directly (each already produces exactly one clean per-node output variable regardless of directedness); measure(degree) does not - nwdegree itself splits into separate _outdegree/_indegree variables on a directed network, so induced degree would need a choice between them with no single obviously-right answer, disclosed as a v1 scope limit (a clear error) rather than guessed at. Weighted/Signed: entirely inherited from whichever measure() is chosen - nwinduced itself does not read tie values directly. Two-mode: not supported - operates on the network’s own one-mode adjacency directly; a bipartite-aware variant is not attempted here.

Stored results

Macros:
r(measure) the underlying measure used
r(name) name of the network
r(endogenous) name of the endogenous-centrality variable
r(induced) name of the induced-centrality variable
r(exogenous) name of the exogenous-centrality variable

References

Everett, M.G., Borgatti, S.P. (2010). Induced, endogenous and exogenous centrality. Social Networks 32(4), 339-344.

nwinduced