dPC decomposition

dPC (delta PC) measures the importance of each individual habitat patch to the landscape's overall connectivity. It answers the question: how much would connectivity decline if this patch were lost?

1. What dPC measures

$$ \text{dPC}_k = \frac{\text{PC} - \text{PC}^{-k}}{\text{PC}} \times 100 $$

where $\text{PC}^{-k}$ is the Probability of Connectivity computed after removing patch $k$ and all its edges from the network.

dPC is expressed as a percentage loss of EC(PC). A patch with dPC = 3.5 is one whose removal would reduce the landscape's equivalent connected area by 3.5%.

The removal-based definition has an important implication: dPC measures the value of a patch given the rest of the landscape as it exists. A patch that sits at a critical bridge position — connecting two otherwise-isolated clusters — will have much higher dPC than a patch of identical size in a well-connected landscape, because its removal severs connectivity that has no alternative route.

2. The three components

dPC can be decomposed into three non-negative, additive terms (Saura & Rubio 2010):

$$ \text{dPC}_k = \text{dPC}^{\text{intra}}_k + \text{dPC}^{\text{flux}}_k + \text{dPC}^{\text{connector}}_k $$

dPCintra

$$ \text{dPC}^{\text{intra}}_k = \frac{a_k^2}{A_L^2 \cdot \text{PC}} \times 100 $$

The contribution from the patch's own area — the self-term in the PC double sum. dPCintra is proportional to $a_k^2$ and depends on nothing else in the network. It represents the irreplaceable value of the patch as habitat in its own right, independent of its connections.

Ecological meaning: a patch with high dPCintra is large and important even if isolated. Losing it reduces the landscape's total habitat and, through the squared-area term, reduces connectivity more than proportionally to area.

dPCflux

The contribution from dispersal flows to and from the patch as an endpoint — weighted connections to every other patch in the network:

$$ \text{dPC}^{\text{flux}}_k \propto \sum_{j \neq k} a_k \cdot a_j \cdot p^*_{kj} $$

dPCflux depends on both the patch's own area and its connectivity to the rest of the network. A large, centrally-located patch scores high on dPCflux; a small peripheral patch scores low even if well-connected.

Ecological meaning: a patch with high dPCflux is a well-connected hub — a population source or sink that sustains dispersal flows across the network. Losing it reduces the quantity of dispersal the landscape supports.

dPCconnector

The contribution from the patch as a stepping stone — its role in improving the maximum-probability path between other patch pairs that do not involve it as an endpoint:

$$ \text{dPC}^{\text{connector}}_k \propto \sum_{i \neq k} \sum_{j \neq k} a_i \cdot a_j \cdot (p^*_{ij} - p^*_{ij,-k}) $$

where $p^*_{ij,-k}$ is the best path from $i$ to $j$ without routing through $k$. For a patch that lies on no shortest path, this term is zero. For a patch that is the only route between two clusters, this term can be very large.

Ecological meaning: a patch with high dPCconnector is a strategic bottleneck. It may not be large or particularly species-rich, but removing it would disconnect parts of the network that currently exchange individuals. These patches are most critical for maintaining metapopulation viability at the landscape scale.

3. Reading the decomposition in practice

The decomposition is particularly useful when deciding between two patches of similar overall dPC: a high-dPCconnector patch may be ecologically more urgent to protect than a high-dPCflux patch of equal total importance, because its loss cannot be compensated by any existing alternative route.

4. dBC_PC — betweenness centrality

dBC_PC is an alternative patch importance metric based on betweenness centrality: the fraction of all maximum-probability paths between patch pairs that pass through the focal patch. It is purely topological — patch area does not enter the formula.

dBC_PC is useful as a cross-check:

• A patch with both high dPC and high dBC_PC is important under two independent definitions — its importance is robust.
• A patch with high dPC but low dBC_PC is important mainly because of its size (high dPCintra), not because of its structural position in the network.
• A patch with low dPC but high dBC_PC is a stepping stone that connects many patch pairs but is too small to contribute much area-weighted flux.

5. Interpretation across dispersal distances

dPC rankings are scenario-specific. A patch that ranks first in dPC at $d = 500$ m — where it connects many small patches in a local cluster — may rank much lower at $d = 3\,000$ m, where direct long-distance connections bypass it entirely.

This is not a limitation but an ecological signal: the patch is important for short-range species (insects, small mammals) and less critical for wide-ranging ones. Conservation decisions informed by multiple distances can identify patches that are broadly important vs those that matter primarily for specific taxa.

6. Normalisation: ekokrati.graph vs Conefor

ekokrati.graph computes dPC using $\text{PC}_\text{full}$ (computed without any minimum-path-probability threshold) as the denominator, even when a PC heuristic is used in the numerator $\text{PC}^{-k}$. Conefor uses its thresholded PC value as the denominator.

The consequence is that ekokrati.graph dPC values are generally smaller than Conefor dPC values on sparse landscapes where the heuristic prunes many path pairs. The relative rankings of patches within a single ekokrati.graph run are unaffected.

See PC normalisation: ekokrati.graph vs Conefor for the full derivation and a worked comparison.

Key references

• Saura, S. & Rubio, L. (2010). A common currency for the different ways in which patches and links can contribute to habitat availability and connectivity in the landscape. Ecography, 33(3), 523–537.
• Bodin, Ö. & Saura, S. (2010). Ranking individual habitat patches as connectivity providers: integrating network analysis and patch removal experiments. Ecological Modelling, 221(19), 2393–2405.