## Graph Theoretic Measures Using Dispersal Data and Spatially Explicit Habitat Data

Graph-theoretic measures of connectivity use spatially explicit habitat data and known biological information about dispersal to estimate potential connectivity. The approach consists of using a habitat graph that summarizes habitat patch arrangement and patch information in a concise way. The graph is then converted to measures of potential connectivity either using fixed dispersal distances to create links if the metric is less than the maximum dispersal distance or this is done probabilistically if a dispersal kernel is available. A dispersal kernel is a function describing the probability of dispersal as a function of distance, and can be combined with structural landscape data by using random draws to decide whether a patch is connected or not depending on the appropriate probability of dispersal for the distance in question. Pairwise metrics such as the maximum distance able to be traveled in a random direction are then scaled up to the entire landscape. Two main approaches are in use, based on percolation theory and the correlation length of spanning clusters, which are both described below.

Percolation is most easily understood for a rasterized grid of habitat and nonhabitat cells and is a measure of the probability that habitat cells are contiguous. Percolation theory comes from mathematics and was developed in physics before being applied to landscape ecology. The main concept is the existence of a percolation threshold, defined in the following way. Suppose p is a parameter that defines the average degree of connectivity between cells (in a grid) of a landscape classified into habitat and nonhabitat. When p = 0, all patches are totally isolated from every other subunit. When p = 1, all habitat cells are connected to (touching) their neighbors. At this point, the landscape is connected from one side to the other, since there are paths that go completely across the system, through spanning clusters. Now suppose, starting at p = 1, habitat cells are randomly removed, so that p, the measure of average connectivity, decreases. The percolation threshold is that value of p, usually denoted pc, at which there is no longer an unbroken path from one side of the system to the other. Measures of structural connectivity based on percolation can be changed into functional connectivity by allowing gaps between habitat cells of certain distances, corresponding to the organism's maximum movement distance across nonhabitat.

The correlation length for a rasterized habitat patch map is based on the average extensiveness of connected cells. The correlation length is the average distance one might traverse across a landscape without leaving what is defined as habitat from a random starting point and moving in a random direction.

An advantage of graph-theoretic approaches is that they can be used to calculate how an individual patch contributes to landscape-scale connectivity. However, such approaches are data intensive, requiring both movement data (maximum dispersal distance or a dispersal kernel) and spatially explicit landscape data representing habitat and nonhabitat areas.

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