The Randomized Shortest Path model in a nutshell

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1 Panzacchi M, Van Moorter B, Strand O, Saerens M, Kivimäki I, Cassady St.Clair C., Herfindal I, Boitani L. (2015) Predicting the continuum between corridors and barriers to animal movements using Step Selection Functions and Randomized Shortest Paths. Journal of Animal Ecology Appendix I The Randomized Shortest Path model in a nutshell 1. Introduction This section briefly introduces the Randomized Shortest Path (RSP) model, which was developed in Saerens et al. (2009), Yen et al (2008), and Kivimäki, Shimbo & Saerens (2014), and can be described as follows. The section is based on this material, as well as (Devooght et al 2014) presenting an application of the framework. Suppose an animal has to move from some starting place to a destination on a geographic area. This problem is usually solved in the following way. First, the problem is discretized and a sufficiently dense grid is defined based on the map of the area, providing a graph or network on which the animal can move from node to node by following edges or links. Two important properties are associated to the edges of the graph, (i) a cost quantifying the difficulty of following the link as well as the real distance it represents and (ii) a weight representing the added value of following this edge the affinity of the neighboring node from the point of view of the animal. These two quantities can be set independently or can be related by setting, e.g.,. Second, the animal has to choose a strategy for reaching node from node. Two such common strategies are the shortest path, or geodesic, strategy (choosing the least cost path, see, e.g. Cormen et al 2009) and the pure random walk strategy (walking completely at random until arriving at the destination, see, e.g. Doyle & Snell 1984). However, animals seldom follow such extreme strategies. Instead, they follow suboptimal trajectories, with a balance between exploitation (minimizing total cost along the trajectory) and exploration (exploring 1

2 neighboring nodes looking appealing). This is exactly what the RSP does: it defines a trade-off between following the shortest path and performing a random tour. Technically, the RSP tackles the problem by minimizing an objective function taking into account both the total cost of the path, or trajectory, and the tendency of the animal to examine its neighborhood. Concretely, the model minimizes the expected cost for reaching node (exploitation) while fixing the Kullback-Leibler divergence, also called relative entropy (see, e.g. Cover & Thomas 2006), between the desired strategy and the random walk strategy. The good news is that the model is easily tractable: the main quantities of interest (number of visits to each node before reaching the destination node, expected cost, etc) can be computed in closed form by solving two systems of linear equations. The resulting strategy balancing both goals (exploitation and exploration) actually defines a biased random walk in which the animal is attracted by the destination node. 2. Some notations We assume that the grid defines a weighted directed graph G containing n nodes. Without loss of generality, the starting position A is indexed as node 1 while destination B is node n. We then consider absorbing paths from node 1 to node n, meaning a sequence of adjacent nodes that may contain cycles, except for the destination node. Once the destination node n is reached, the path does not continue. We denote the (usually infinite but countable) set of all absorbing paths starting at 1 and ending in n as. Each individual path is associated with a total cost, given by the sum of the individual costs of each edge along the path. We first introduce and and then define paths and the cost of a path. The cost matrix contains the individual costs as elements. When there is no edge from node to node we consider the cost to be infinite. The graph is also associated to an adjacency matrix containing local affinities between nodes. When there is no link between two nodes,. Matrices and are given by the problem at hand. Also, for the pure random walk strategy, we define the probability that the animal takes the outgoing link in node as. The are gathered in the transition matrix. This 2

3 defines a standard finite Markov chain (see, e.g. Doyle & Snell 1984, Grinstead &. Snell 1997, Kemeny & Snell 1976). Once the animal has reached the destination node, it remains there and the process stops immediately. This implies that and that for each. In this case, we say that node is absorbing. Finally, the likelihood of a path, i.e. the product of the transition probabilities along the path, is denoted by. 3. Probability distribution on paths The RSP model defines a biased random walk on the graph whose moving strategy the probability that the animal chooses a particular path from to consists of minimizing the expected extra-cost of the paths in comparison with the least cost (i.e. favoring shorter paths or exploitation) while in the meantime trying to explore the graph, to a given extent. This tradeoff may be formalized in the following way (Devooght et al 2014; Kivimäki et al. 2014; Saerens et al. 2009): eq. 1 where is the least cost provided by the shortest path (with respect to the costs) between node 1 and node. The first equation expresses the exploitation strategy minimizing the expected extra-cost. The second ensures exploration of the graph by constraining the divergence ( ) from the prior distribution which defines a pure random walk strategy where the animal always chooses his next step according to the unbiased transition probabilities. This tradeoff problem is very common in statistical physics (Jaynes 1957) and has a closed form solution which is provided by the Gibbs-Boltzmann probability density: 3

4 [ ] [ ] [ ] [ ] eq. 2 with controlling the exploration exploitation tradeoff. The solution can be found using Lagrange multipliers, and the resolution is detailed in, e.g., Mantrach et al. (2010). In statistical physics, the denominator is called the partition function: [ ] eq. 3 We immediately see that implies (pure random walk), and when the probability distribution is concentrated on the path(s) with the lowest cost. In other words, the model interpolates from a random walk strategy to a shortest path strategy. 3.1 Computing the expected number of visits to each node Let us give a quick summary of how to compute the values of interest. First, we define the matrix as [ ] [ ] eq. 4 where is the elementwise (Hadamard) matrix product, and the exponential and logarithm are taken elementwise. In fact, the partition function of Equation 6 (Devooght et al. 2014) can be computed from the matrix in the following way: [ ] [ ] [ ] eq. 5 which enumerates all possible paths and sums up their total costs (see Saerens et al 2009). We defined so that is element of this matrix. From statistical physics fundamentals (e.g. Jaynes 1957; Reichl 1998), it is well-known that many interesting quantities can be computed from the partition function. For instance, using Equations 2 and 3, the expected number of passages through a given edge is given by: 4

5 ( ) [ ] [ ] eq. 6 where is the number of times edge appears on path. Furthermore, the expected number of visits to node is given by its total input. Computing the partial derivative of Equation 6 from Equation 5 provides the expected number of passages (see Saerens et al for details) [ ] eq. 7 and the expected number of visits to node is (Saerens et al. 2009) eq. 8 Interestingly, this model defines a new biased random walk with transition probabilities [ ] eq. 9 which does not depend on the starting node 1 (again, see Saerens et al for details). Consequently, in order to determine the expected number of visits to a node, the only quantities that need to be computed are and, that is, the first row and the last column of matrix. This can be done by solving two systems of linear equations, and. Here, is a basis column vector containing s everywhere, except at position where it contains a. The quantities of interest can then be computed from and. Various other quantities, such as the expected extra-cost according to Equation 4 can be computed using similar techniques as above (Saerens et al. 2009). 5

6 References T. Cormen, C. Leiserson, R. Rivest, and C. Stein. Introduction to algorithms, 3 rd Edition. The MIT Press, T. M. Cover and J. A. Thomas. Elements of Information Theory, 2nd edition. JohnWiley & Sons, R. Devooght, A. Mantrach, I. Kivimaki, H. Bersini, A. Jaimes, and M. Saerens. Random walks based modularity: application to semisupervised learning. Proceedings of the 23rd International World Wide Web Conference (WWW 2014), pages , P. G. Doyle and J. L. Snell. Random walks and electric networks. The Mathematical Association of America, C. Grinstead and J. L. Snell. Introduction to probability, 2nd ed. The Mathematical Association of America, E. T. Jaynes. Information theory and statistical mechanics. Physical Review, 106: , J. G. Kemeny and J. L. Snell. Finite Markov chains. Springer-Verlag, I. Kivimäki, M. Shimbo, and M. Saerens. Developments in the theory of randomized shortest paths with a comparison of graph node distances. Physica A: Statistical Mechanics and its Applications, 393: , A. Mantrach, L. Yen, J. Callut, K. Francoise, M. Shimbo, and M. Saerens. The sum-over-paths covariance kernel: a novel covariance between nodes of a directed graph. IEEE Transactions on Pattern Analysis and Machine Intelligence, 32(6): , L. Reichl. A modern course in statistical physics, 2nd ed. Wiley, M. Saerens, Y. Achbany, F. Fouss, and L. Yen. Randomized shortest-path problems: Two related models. Neural Computation, 21(8): , L. Yen, A. Mantrach, M. Shimbo, and M. Saerens. A family of dissimilarity measures between nodes generalizing both the shortest-path and the commute-time distances. In Proceedings of the 14th SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD 2008), pages ,