Computes the trophic incoherence parameter \(q\), which measures how vertically ordered a directed network is (Johnson et al. 2014). For each edge \((u, v)\), the trophic difference is \(x_{uv} = s_v - s_u\) where \(s_i\) is the trophic level of node \(i\). The trophic incoherence parameter is the (population) standard deviation of these differences: $$q = \sqrt{\frac{1}{|E|} \sum_{(u,v) \in E} (x_{uv} - \bar{x})^2}$$
Value
Numeric scalar: the trophic incoherence parameter. NA when
the network has no edges or when some node cannot be reached from a basal
node. An undirected network returns NA with a warning.
Details
Values near 0 indicate a coherent network, in which every edge rises by about one level. High values indicate many level-skipping or downward edges. Johnson et al. (2014) reported that food webs with low \(q\) are more stable.
Trophic levels are computed on the binary adjacency matrix, so edge weights are ignored. The levels are defined only for a directed network in which every node can be reached from a basal node (a node with no incoming edges).
References
Johnson, S., Dominguez-Garcia, V., Donetti, L., & Munoz, M. A. (2014). Trophic coherence determines food-web stability. PNAS, 111(50), 17923-17928.
See also
centrality (the trophic_level measure) for
the per-node levels used in the incoherence calculation.
Examples
strong <- filter_edges(regulation_net, weight > 0.3, keep_isolates = FALSE)
trophic_incoherence(strong)
#> [1] 1.26085