Measures a node's contribution to the base centrality of all other nodes, following Everett and Borgatti (2010): \(E(i)=\sum_{j\ne i}[C_G(j)-C_{G-i}(j)]\). The focal node's own base score is excluded. Three base measures are supported, each calculated without normalization before deletion:
- reverse_closeness
Default. For a graph H with m remaining nodes, \(C_H(j)=\sum_{k\ne j}\max(N-d_H(j,k),0)\), where N is the ORIGINAL input size, including isolates. Unreachable pairs contribute zero. This implements the fixed-size adjustment in section 3.3; it is distinct from ordinary reciprocal-farness closeness.
- betweenness
Raw shortest-path betweenness with endpoints excluded. Each unordered pair counts once on undirected graphs; directed pairs count separately. Exogenous contributions can be negative when removing a node increases the remaining nodes' betweenness.
- degree
Simple degree in the chosen base direction. On an undirected graph the exogenous result equals degree. On a directed graph, outgoing base degree produces incoming exogenous degree, and incoming base degree produces outgoing exogenous degree.
Arguments
- x
Network input accepted by
centrality.- mode
Direction of the base measure: all, out or in. Default all.
- exogenous_base
One of
"reverse_closeness"(default),"betweenness"or"degree". Exact names are required.- ...
Additional arguments to
centrality.
Details
Uses the simple binary topology: parallel connections count once,
self-loops are removed and weights/path inversion are ignored. Mode
"all" projects onto the undirected skeleton; "out" and
"in" use directed paths when the input is directed. For undirected
input all three modes agree. Empty input returns an empty vector;
isolates and singletons score zero. Original size includes other
components, so adding an isolate can change reverse-closeness scores
of connected nodes even though the isolate's own contribution is zero.
normalized = TRUE applies cograph's final division by a positive
maximum, retaining negative values; it does not normalize the base
measure, nor apply the paper's theoretical normalization. If the maximum
is nonpositive, values remain unchanged. Arbitrary normalized base
scores, such as unit-length eigenvectors, are not supported.
Numerical verification uses independent NetworkX base scores, explicit path enumeration and analytical graphs. Some numerical entries in the paper's Florentine tables could not be reproduced from NetworkX's graph plus the Pucci isolate; this implementation follows the stated definition and does not claim complete published-table or UCINET parity.
Betweenness and reverse-closeness require repeated all-pairs distances, with worst-case O(N^4) time using the current dense kernels. The measure is therefore in the costly tier even when the degree base is selected.
References
Everett, M. G., & Borgatti, S. P. (2010). Induced, endogenous and exogenous centrality. Social Networks, 32(4), 339-344. Equations 3 and 8, sections 3.1-3.3. doi:10.1016/j.socnet.2010.06.004 .
Examples
centrality_exogenous(igraph::make_ring(4), exogenous_base = "betweenness")
#> 1 2 3 4
#> 0.5 0.5 0.5 0.5