Trophic Incoherence Parameter

The trophic incoherence parameter \(q\) is a measure of 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: $$q = \sqrt{\frac{1}{|E|} \sum_{(u,v) \in E} (x_{uv} - \bar{x})^2}$$

Usage

trophic_incoherence(x, cannibalism = TRUE)

Arguments

x

Directed network input.

cannibalism

Logical. If FALSE, self-loops are removed before computing trophic differences. Default TRUE.

Value

A single numeric value (NA_real_ for empty edge sets or undirected input).

Details

Low values (\(q \approx 0\)) indicate a perfectly coherent network (e.g., a pure food web where every edge goes up one level). High values indicate an incoherent network with many level-skipping or downward edges. Johnson et al. 2014 showed that low-\(q\) food webs are dynamically more stable.

Matches networkx.trophic_incoherence_parameter at machine epsilon. Directed-only; requires at least one basal node (node with no incoming edges) for trophic levels to be well-defined.

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

# Small directed 3-node chain: 1 -> 2 -> 3 (perfectly coherent, q = 0)
adj <- matrix(c(0,1,0, 0,0,1, 0,0,0), 3, 3, byrow = TRUE)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
trophic_incoherence(adj)
#> [1] 0