Computes Lawyer's Expected Force after exactly two transmission events without recovery. For each seed, enumerate ordered sequences of two infected-to-susceptible edge transmissions. Each sequence produces a three-node infected cluster with D outgoing edges to susceptible nodes. Normalize these D values across all sequences and take their Shannon entropy using natural logarithms (Lawyer 2015, equation 1).
Arguments
- x
Network input accepted by
centrality.- ...
Additional arguments to
centrality.normalized = TRUEdivides by the maximum score; all-zero results remain zero.
Details
Different event orders or transmitting parents remain distinct even when they infect the same three nodes. A seed and two adjacent neighbors of an undirected triangle form four sequences, not one. Boundary edges are counted individually even when they reach the same susceptible node. This is not entropy over distinct infected sets or over boundary-degree categories, and is not a probability-weighted epidemic simulation.
Uses the simple unweighted graph, retaining direction. In directed graphs, only outgoing infected-to-susceptible arcs transmit or contribute boundary degree, following the paper's directed extension. Loops and duplicate arcs are removed after generic processing. Weights, mode, inversion and cutoff do not affect the result. The weighted extension and horizons other than two events are outside this implementation.
Zero-degree outcomes use the zero-log-zero entropy limit. If no sequence can perform two transmissions, or every resulting cluster has zero onward force, cograph returns zero. The latter is an explicit extension of the paper's undefined all-zero normalization, not author-code parity. Isolates and components of at most three nodes therefore score zero. A single positive-force outcome also has entropy zero. Empty input returns no scores. The measure is local and does not establish epidemic probability, outbreak size or predictive accuracy on the supplied graph.
Native computation groups three-node clusters by boundary degree while preserving their event multiplicities. Worst-case time is O(n cubed), memory O(n squared), including dense graph preparation. Scores remain independent between components before maximum normalization.
References
Lawyer, G. (2015). Understanding the influence of all nodes in a network. Scientific Reports, 5, 8665. Equations 1 and 2 and the directed extension in the Weighted graphs section. doi:10.1038/srep08665 .
See also
centrality_modified_expected_force for degree
adjustment. centrality_expected computes a different
quantity, the sum of neighbor degrees.
Examples
centrality_expected_force(igraph::make_graph("Zachary"))
#> 1 2 3 4 5 6 7 8
#> 5.665412 4.698245 4.947944 4.210475 3.191167 3.376649 3.376649 3.860885
#> 9 10 11 12 13 14 15 16
#> 4.279408 3.271121 3.191167 2.699524 3.066568 4.258639 3.344324 3.344324
#> 17 18 19 20 21 22 23 24
#> 1.831514 3.191015 3.344324 3.779170 3.344324 3.191015 3.344324 3.953456
#> 25 26 27 28 29 30 31 32
#> 2.639873 2.704936 3.023930 3.710869 3.552187 3.744987 3.904355 4.302340
#> 33 34
#> 5.154976 5.753993