Sensitivity to network perturbations in the Randomized Shortest Paths framework: theory and applications in ecological connectivity

Abstract

Abstract The Randomized Shortest Paths (RSP) framework, developed for network analysis, extends traditional proximity and distance measures between two nodes, such as shortest path distance and commute cost distance (related to resistance distance). Consequently, the RSP framework has gained popularity in studies on landscape connectivity within ecology and conservation, where the behavior of animals is neither random nor optimal.
In this work, we study how local perturbations in a network affect proximity and distance measures derived from the RSP framework. For this sensitivity analysis, we develop computable expressions for derivatives with respect to weights on the edges or nodes of the network. Interestingly, the sensitivity of expected cost to edge or node features provides a new signed network centrality measure, the negative covariance between edge/node visits and path cost, that can be used for pinpointing strong and weak parts of a network. It is also shown that this quantity can be interpreted as minus the endured expected detour (in terms of cost) when constraining the walk to pass through the node or the edge.
Our demonstration of this framework focuses on a migration corridor for wild reindeer (Rangifer rangifer) in Southern Norway. By examining the sensitivity of the expected cost of movement between winter and calving ranges to perturbations in local areas, we have identified priority areas crucial for the conservation of this migration corridor. This innovative approach not only holds great promise for conservation and restoration of migration corridors, but also more generally for connectivity corridors between important areas for biodiversity (e.g. protected areas) and climate adaptation. 
Furthermore, the derivations and computational methods introduced in this work present fundamental features of the RSP framework. These contributions are expected to be of interest to practitioners applying the framework across various disciplines, ranging from ecology, transport and communication networks to machine learning.