There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation that the above-mentioned common distances in many situations fail to take into account the global structure of the graph. In this article, we develop the theory of one family of graph node distances, known as the randomized shortest path dissimilarity, which has its foundation in statistical physics. We show that the randomized shortest path dissimilarity can be easily computed in closed form for all pairs of nodes of a graph. Moreover, we come up with a new definition of a distance measure that we call the free energy distance. The free energy distance can be seen as an upgrade of the randomized shortest path dissimilarity as it defines a metric, in addition to which it satisfies the graph-geodetic property. The derivation and computation of the free energy distance are also straightforward. We then make a comparison between a set of generalized distances that interpolate between the shortest path distance and the commute time, or resistance distance. This comparison focuses on the applicability of the distances in graph node clustering and classification. The comparison, in general, shows that the parametrized distances perform well in the tasks. In particular, we see that the results obtained with the free energy distance are among the best in all the experiments.

This work develops a generic framework, called the bag-of-paths (BoP), for link and network data analysis. The central idea is to assign a probability distribution on the set of all paths in a network. More precisely, a Gibbs-Boltzmann distribution is defined over a bag of paths in a network, that is, on a representation that considers all paths independently. We show that, under this distribution, the probability of drawing a path connecting two nodes can easily be computed in closed form by simple matrix inversion. This probability captures a notion of relatedness between nodes of the graph: two nodes are considered as highly related when they are connected by many, preferably low-cost, paths. As an application, two families of distances between nodes are derived from the BoP probabilities. Interestingly, the second distance family interpolates between the shortest path distance and the resistance distance. In addition, it extends the Bellman-Ford formula for computing the shortest path distance in order to integrate sub-optimal paths by simply replacing the minimum operator by the soft minimum operator. Experimental results on semi-supervised classification show that both of the new distance families are competitive with other state-ofthe-art approaches. In addition to the distance measures studied in this paper, the bag-of-paths framework enables straightforward computation of many other relevant network measures.

Summary1. The loss, fragmentation and degradation of habitat everywhere on Earth prompts increasing attention to identifying landscape features that support animal movement (corridors) or impedes it (barriers). Most algorithms used to predict corridors assume that animals move through preferred habitat either optimally (e.g. least cost path) or as random walkers (e.g. current models), but neither extreme is realistic. 2. We propose that corridors and barriers are two sides of the same coin and that animals experience landscapes as spatiotemporally dynamic corridor-barrier continua connecting (separating) functional areas where individuals fulfil specific ecological processes. Based on this conceptual framework, we propose a novel methodological approach that uses high-resolution individual-based movement data to predict corridor-barrier continua with increased realism. 3. Our approach consists of two innovations. First, we use step selection functions (SSF) to predict friction maps quantifying corridor-barrier continua for tactical steps between consecutive locations. Secondly, we introduce to movement ecology the randomized shortest path algorithm (RSP) which operates on friction maps to predict the corridor-barrier continuum for strategic movements between functional areas. By modulating the parameter Ѳ, which controls the trade-off between exploration and optimal exploitation of the environment, RSP bridges the gap between algorithms assuming optimal movements (when Ѳ approaches infinity, RSP is equivalent to LCP) or random walk (when Ѳ ? 0, RSP ? current models). 4. Using this approach, we identify migration corridors for GPS-monitored wild reindeer (Rangifer t. tarandus) in Norway. We demonstrate that reindeer movement is best predicted by an intermediate value of Ѳ, indicative of a movement trade-off between optimization and exploration. Model calibration allows identification of a corridor-barrier continuum that closely fits empirical data and demonstrates that RSP outperforms models that assume either optimality or random walk. 5. The proposed approach models the multiscale cognitive maps by which animals likely navigate real landscapes and generalizes the most common algorithms for identifying corridors. Because suboptimal, but non-random, movement strategies are likely widespread, our approach has the potential to predict more realistic corridor-barrier continua for a wide range of species.

This paper introduces two new closely related betweenness centrality measures based on the Randomized Shortest Paths (RSP) framework, which fill a gap between traditional network centrality measures based on shortest paths and more recent methods considering random walks or current flows. The framework defines Boltzmann probability distributions over paths of the network which focus on the shortest paths, but also take into account longer paths depending on an inverse temperature parameter. RSP’s have previously proven to be useful in defining distance measures on networks. In this work we study their utility in quantifying the importance of the nodes of a network. The proposed RSP betweenness centralities combine, in an optimal way, the ideas of using the shortest and purely random paths for analysing the roles of network nodes, avoiding issues involving these two paradigms. We present the derivations of these measures and how they can be computed in an efficient way. In addition, we show with real world examples the potential of the RSP betweenness centralities in identifying interesting nodes of a network that more traditional methods might fail to notice.

Ecosystem functioning depends on multiple successful interactions, many supported by individual movements. The degree to which the landscape allows these interactions to take place has been referred to as 'effective connectivity' (EC). Many of the cumulative impacts of anthropogenic activities on ecosystem functioning arise from changes in EC. Therefore, a coherent framework to quantify EC is urgently needed. Recent theoretical developments propose that studying EC requires the simultaneous consideration of spatial, environmental and species intrinsic characteristics (SEI framework).In this paper we further expand the SEI framework by integrating advances in geographic information science, ecological niche modelling, movement ecology, island biogeography and network sciences to develop a comprehensive three-step methodological approach for quantifying EC. First, using niche modelling and movement ecology, we quantify the species movement probabilities with respect to local environmental conditions. Second, we quantify ecological distances between non-adjacent locations by integrating species movement responses to the local environment with the spatial configuration of the landscape using the expected cost obtained from the randomized shortest paths (RSP) framework. This expected cost generalizes the two most frequently used ecological distance metrics, i.e. least-cost distance and resistance distance. Moreover the 'absorbing random walk' properties of RSP allow the integration of new developments in connectivity research, i.e. spatial absorbing Markov chains, to account for movement-related mortality. Third, drawing from island biogeography and metapopulation ecology, we scale ecological distances by relevant speciesand area-specific parameters to estimate EC for the ecological process of interest, e.g. migration, dispersal or gene flow.The integrative and highly interdisciplinary approach we propose can lead to increasingly more realistic measures of EC at different organizational levels. Moreover efficient computation allows its application to large-scale high-resolution landscapes for theoretical studies, conservation planning and sustainable management of real landscapes.

The recently developed bag-of-paths (BoP) framework consists in setting a Gibbs–Boltzmann distribution on all feasible paths of a graph. This probability distribution favors short paths over long ones, with a free parameter (the temperature T) controlling the entropic level of the distribution. This formalism enables the computation of new distances or dissimilarities, interpolating between the shortest-path and the resistance distance, which have been shown to perform well in clustering and classification tasks. In this work, the bag-of-paths formalism is extended by adding two independent equality constraints fixing starting and ending nodes distributions of paths (margins).When the temperature is low, this formalism is shown to be equivalent to a relaxation of the optimal transport problem on a network where paths carry a flow between two discrete distributions on nodes. The randomization is achieved by considering free energy minimization instead of traditional cost minimization. Algorithms computing the optimal free energy solution are developed for two types of paths: hitting (or absorbing) paths and non-hitting, regular, paths and require the inversion of an n × n matrix with n being the number of nodes. Interestingly, for regular paths on an undirected graph, the resulting optimal policy interpolates between the deterministic optimal transport policy (T → 0+) and the solution to the corresponding electrical circuit (T → ∞). Two distance measures between nodes and a dissimilarity between groups of nodes, both integrating weights on nodes, are derived from this framework.

1. Increasingly precise spatial data (e.g. high-resolution imagery from remote sensing) allow for improved representations of the landscape network for assessing the combined effects of habitat loss and connectivity declines on biodiversity. However, evaluating large landscape networks presents a major computational challenge both in terms of working memory and computation time. We present the ConScape (i.e. "connected landscapes") software library implemented in the high-performance open-source Julia language to compute metrics for connected habitat and movement flow on high-resolution landscapes.2. The combination of Julia's 'just-in-time' compiler, efficient algorithms and 'landmarks' to reduce the computational load allows ConScape to compute landscape ecological metrics-originally developed in metapopulation ecology (such as 'metapopulation capacity' and 'probability of connectivity')-for large landscapes. An additional major innovation in ConScape is the adoption of the randomized shortest paths framework to represent connectivity along the continuum from optimal to random movements, instead of only those extremes. We demonstrateConScape's potential for using large datasets in sustainable land planning by modelling landscape connectivity based on remote-sensing data paired with GPS tracking of wild reindeer in Norway. To guide users, we discuss other applications, and provide a series of worked examples to showcase all ConScape's functionalities in Supplementary Material.4. Built by a team of ecologists, network scientists and software developers, ConScape is able to efficiently compute landscape metrics for highresolution landscape representations to leverage the availability of large data for sustainable land use and biodiversity conservation. As a Julia implementation, ConScape combines computational efficiency with a transparent code base, which facilitates continued innovation through contributions

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