Map Equation Centrality

Map equation centrality (Blocker et al. 2022) is the reduction in codelength, in bits, when a node's codeword is removed from its module codebook and the codebook is redesigned. With $p$ the visit rate of the node and $s$ the use rate of its module codebook, $$MEC_i = -(s - p) \log_2 \frac{s - p}{s}.$$

Usage

centrality_map_equation(
  x,
  membership = NULL,
  map_flow = "unrecorded",
  map_convention = "paper",
  ...
)

Arguments

x: Network input accepted by centrality.
membership: One module label per node. NULL (default) gives one module. A named vector is matched to the node names.
map_flow: "unrecorded" (default) for unrecorded link teleportation or "recorded" for recorded uniform teleportation.
map_convention: "paper" (default) includes module exits in the codebook rate. "infomap" uses node visits only.
...: Further arguments to centrality. The measure uses weighted (use edge weights, default TRUE) and damping (probability of following a link, default 0.85).

Value

A named numeric vector with one score per node, in input node order.

Details

The partition is held fixed and is given by membership, and NULL places every node in one module. With map_convention = "paper" the rate $s$ includes module exits (equations 2 and 9-11), and with "infomap" it includes node visits only, which reproduces Infomap 2.15.1 and Table 1 of the paper. With map_flow = "unrecorded" (Lambiotte and Rosvall 2012) the walk teleports in proportion to out-strength and only link steps are recorded, so on an undirected network the visit rates are proportional to strength for every damping. With "recorded" the walk teleports uniformly and teleportation steps are recorded. Direction is kept, loops are removed and edge weights must be finite and nonnegative. Under unrecorded flow an isolated node scores zero. An invalid membership, map_flow or map_convention, or a damping outside $[0, 1)$, raises an error.

References

Blocker, C., Nieves, J. C. and Rosvall, M. (2022). Map equation centrality: community-aware centrality based on the map equation. Applied Network Science, 7, 56. doi:10.1007/s41109-022-00477-9 .

Lambiotte, R. and Rosvall, M. (2012). Ranking and clustering of nodes in networks with smart teleportation. Physical Review E, 85, 056107. doi:10.1103/PhysRevE.85.056107 .

Examples

centrality_map_equation(regulation_net, membership = rep(1:2, each = 5))
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#> 0.16113220 0.03673693 0.24678669 0.17278681 0.16904742 0.09194596 0.03809941 
#>   Evaluate     Create      Share 
#> 0.10074058 0.18236547 0.12807428