Using graph theory to compare least cost path and circuit theory

Meredith Rainey BIO515 Fall 2009

Using graph theory to compare least cost path and circuit theory connectivity analyses Introduction As spatial habitat data and GIS tools have become increasingly accessible over the past decade, several methods of predicting locations of wildlife movement corridors in complex landscapes have emerged. These methods first quantify resistance to movement across the landscape for a focal species of interest by scoring and combining various habitat layers relevant to the species’ movement (e.g., cover type, slope). This resistance surface is then used to identify likely movement routes between core habitat patches as those with low resistance. While this description summarizes the basic premise of all available methods, each differs in its specific approach and assumptions. These differences may lead to divergent predictions among methods, both in terms of locations of corridors as well as prioritization of patches and linkages for inclusion in reserve networks. In this study, I explore this issue by (1) applying two corridor prediction methods that have received a great deal of attention from researchers and conservation practitioners to the same example focal species and landscape, and (2) using connectivity metrics based on graph theory to compare the predictions made by each approach.

The most commonly used method of predicting wildlife corridors is known as least cost path (or permeability) analysis (e.g., Carroll and Miquelle 2006; Hoctor et al. 2000; Larue and Nielsen 2008; Singleton et al. 2002; Walker and Craighead 1997). This approach compromises between minimum travel distance among habitat patches and minimum exposure to unsuitable habitat (Walker and Craighead 1997). Unlike Euclidian (straight-line) distance, cost-weighted distance accounts for the increased hardship faced by an individual moving through unsuitable habitat. The least cost path is the route that offers the shortest cost-weighted distance between patch pairs. It is considered the optimal route that an individual is most likely to take when moving among patches (Walker and Craighead 1997), assuming that the individual has perfect knowledge of the landscape and that landscape resistance has been quantified appropriately.

Recently, a novel method based on electrical circuit theory has been proposed (McRae 2006; McRae et al. 2008) and is beginning to be evaluated as an alternative to least cost path analysis for a variety of applications (e.g., Schwartz et al. 2009). Circuit theory analysis makes use of the intuitive analogy between movement of individuals through a landscape and movement of charge through an electrical circuit: greater redundancy in travel routes between nodes (patches) enhances flow between them. Circuit theory treats cells in a landscape as electrical nodes connected to neighboring cells by resistors, with resistance values determined by the cells’ landscape resistance values. Consecutive resistors can be linked in series to create a path between two patches; a single route’s total resistance is equivalent to its cost-weighted distance (McRae et al. 2008). Circuit theory is unique in that when all possible paths among patches are treated as resistors connected in parallel, a measure of effective cumulative resistance is obtained that decreases with increasing numbers of paths. Cumulative resistance measures thus account for the positive effects of path redundancy on connectivity (Figure 1). Measures of current flow across the landscape can also be obtained and are useful in that they reflect the probability of movement of individual random walkers through a given cell. Current maps identify important bottlenecks to dispersal, and because current is not weighted by distance, this metric may also help to identify suitable paths that were too long to be considered important routes in least cost path analysis.

A largely separate body of research has focused not on identification of corridor locations, but on quantifying landscape connectivity (e.g., Kadoya 2009; Prugh 2009; Tischendorf and


Fahrig 2000; With et al. 1997). Many of the metrics developed for this purpose only address structural connectivity, or physical connections among patches of a particular habitat type. These indices measure, for example, distance to the nearest forest patch, percentage of forest cover within a given distance of a focal forest patch, or width of linear forested landscape features bridging two forest patches. Graph theory, however, provides a more flexible set of metrics (e.g., Bunn et al. 2000; Minor and Urban 2008; Urban and Keitt 2001). Graphs are networks composed of nodes (representing patches) and edges (representing paths among patches) (Harary 1969). Graph theory metrics quantify, for example, the total length and configuration of edges required to connect all nodes in a graph or the number of edges passing through a given node (indicating the node’s importance to maintaining connectivity of the graph) (e.g., Urban and Keitt 2001). The flexibility of graph theory stems from the fact that edge lengths can be defined in any way deemed relevant to the question of interest, not just by Euclidian distance between patches (Bunn et al. 2000). Graph-based metrics can therefore measure functional connectivity, which accounts for species-specific habitat preferences and movement behaviors. It is these preferences that are captured by the landscape resistance measures used in least cost path and circuit theory analyses. By setting graph edge lengths among patches equal to cost-weighted distances and cumulative resistance values obtained from least cost path and circuit theory analyses, respectively, graph theory metrics effectively capture the differences between these methods in terms of how they represent functional connectivity.

Least cost path and circuit theory analyses seem to often be viewed as interchangeable methods likely to produce comparable results. The purpose of this study is to explore the extent to which this is the case. Using graph theory metrics, results of least cost path and circuit theory analyses will be compared to determine whether these methods predict corridor locations, patterns of landscape connectivity, and patch importance in similar ways. I will conclude by discussing the implications of differences between methods in any of these predictions for reserve network planning.

Figure 1. Patches (white) in H and I have equal area and are separated by equal distances. Least cost path analysis indicates equal cost-weighted distances between each pair, thus equal connectivity. Circuit theory indicates that cumulative resistance decreases with increasing path redundancy, thus pair I has greater connectivity. (McRae et al. 2008)

Methods Study Area This analysis focused on the Vail Pass area approximately 80 miles west of Denver, Colorado. Most of the study area is located within the White River National Forest. Elevation ranges from 1950 to 4400m, and major cover types include conifer and aspen forest, sagebrush steppe, sub-alpine grassland, and mixed shrubland. The area includes several major highways and a number of small towns with population sizes of approximately 5,000 or less. This study utilizes a habitat suitability layer and primary habitat patch delineations for lynx (Lynx canadensis) made available with the FunConn v1 (Theobald et al. 2006) network


analysis package as an example dataset. Habitat suitability was quantified based on cover type (southwest regional gap land cover data) and distance from disturbance sites (roads and development). 48 total primary habitat patches are identified in the dataset; seven of these were selected as focal patches for this analysis to represent a range of distances and conditions between patches. Connectivity Analyses Least cost path analysis was implemented using the ArcGIS 9.3 Spatial Analyst tools ‘Cost Distance’ and ‘Cost Path’. Least cost-weighted distances were calculated for all patch pairs. Circuit theory analysis was conducted using CircuitScape (McRae and Shah 2008). Cumulative resistances among all patch pairs were calculated and a current map was created. Both analyses were conducted on a landscape resistance layer created by subtracting habitat suitability values for each cell from the maximum suitability score and adding one, yielding values ranging from 1 (least resistance) to 101. (Adding a positive value was necessary here to prevent least cost path algorithms from attributing no cost to movement through cells with resistance values of zero.) The resistance layer was then resampled to 90m resolution for use in both analyses due to the memory demands of the CircuitScape program when working with large landscapes (>1 million cells). Graph Theory Analysis Graph-based analyses were conducted using the igraph R package (Csardi and Nepusz 2006). Fully connected graphs (including edges among all node pairs) based on least cost path and circuit analyses were created with edge lengths defined by cost-weighted distances and cumulative resistances, respectively. The minimum spanning tree, defined as the set of edges with minimum total length that connects all graph nodes, was then calculated for each graph. Finally, a suite of patch centrality indices was calculated for each minimum spanning tree to quantify relative importance of each patch to connectivity of the patch network (descriptions of each centrality index are available in the igraph package documentation (Csardi and Nepusz 2006)). Results Connectivity Analyses Pair-wise least cost paths are shown in Figure 2(a). Many of these paths clearly diverge strongly from Euclidian shortest paths, resulting in less predictable patch separation distances and ranks. Cost-weighted distances among all patch pairs are shown in Table 1(a) along with their ranks relative to all other pair-wise distances. The current map output from circuit theory analysis is displayed in Figure 2(b). Areas of high current (high probability of movement) generally coincide with least cost paths but also identify potential alternative paths. Cumulative resistances separating each patch pair are given in Table 1(b) with their ranks. Comparison of these ranks with those produced by least cost path analysis reveals differences between the methods’ predictions of degree of connectivity among patches.


a) b) Figure 2. Study area, Vail Pass, Colorado. a) Least cost paths among all patch pairs. b) Current map output from circuit theory analysis.


Table 1. Pair-wise distances among focal patches based on a) least cost path analysis (values are cumulative cost-weighted distance between patches) and b) circuit theory analysis (values are cumulative resistances between patches). Ranks of values are shown in parentheses. a) Least cost path Patch 11 22 24 29 36 41

3 1954007 469643 331619 569065 634869 995221 (19) (7) (5) (9) (11) (14)

11 2220029 1880820 1681858 2004219 1930382 (21) (17) (16) (20) (18)

22 309285 511494 421606 1199139 (4) (8) (6) (15)

24 298130 210637 921136 (3) (1) (13)

29 221638 617305 (2) (10)

36 830480 (12)

b) Circuit theory 11 22 24 29 36 41

3 83.5596 28.1263 27.4234 35.2701 43.5879 50.1597 (20) (6) (5) (10) (13) (15)

11 84.3278 73.437 70.2009 80.9291 73.4401 (21) (17) (16) (19) (18)

22 20.3333 31.1382 36.7995 47.3785 (2) (8) (12) (14)

24 18.3889 26.359 36.2306 (1) (4) (11)

29 21.5422 29.2712 (3) (7)

36 32.8426 (9)

Graph Theory Analysis Minimum spanning trees for graphs with edges defined as cost-weighted distances versus cumulative resistances differed only slightly (Figure 3). Both trees indicate that patch 29 is important for connectivity of the network as a whole, but differ in their estimation of the importance of patch 36 as a linkage among other patches.

There was general agreement among patch centrality indices specific to each graph, and together these indices match the conclusions drawn from visual inspection of minimum spanning trees. Ranks of patch centrality based on each index are shown in Table 2. For each graph, the mean rank for each patch is provided as an overall indicator of patch centrality. As expected from the layout of minimum spanning trees, least cost path identifies both patches 29 and 36 as those most important for network connectivity, while circuit theory identifies patch 29 alone as most important. Circuit theory further suggests that patch 36 is of minimal importance, in contrast to least cost path analysis. Both methods agree that patch 3 is least important to connectivity of the landscape.


a) b) Figure 3. Minimum spanning trees based on a) least cost path distances and b) circuit theory resistances. Table 2. Ranked values of patch centrality indices for minimum spanning trees based on a) least cost path analysis and b) circuit theory analysis. a) Least cost path

Patch Close-ness

Kleinberg's hub

Between-ness Mean

3 5 5 4 4.67 11 4 4 4 4 22 3 4 3 3.33 24 2 3 2 2.33 29 2 1 1 1.33 36 1 2 1 1.33 41 4 4 4 4

b) Circuit theory

Patch Close-ness

Kleinberg's hub

Between-ness Mean

3 5 4 4 4.33 11 4 3 4 3.67 22 3 2 3 2.67 24 2 2 2 2 29 1 1 1 1 36 4 3 4 3.67 41 4 3 4 3.67

Discussion This analysis indicates that least cost path and circuit theory analyses are capable of yielding similar predictions of corridor locations as well as landscape connectivity patterns and relative importance of patches. Comparison of the least cost path and current maps in Figure 2 indicates that least cost paths generally follow areas with high probability of movement. As suggested by McRae et al. (2008), outputs of these methods appear to be highly complementary. Least cost paths identify complete optimal routes among patches, which are difficult to identify


from circuit theory current maps. Current maps, however, help to identify potentially suitable alternative routes as well as clarifying which portions of least cost paths are embedded within a broad swath of suitable habitat and which are potential bottlenecks meriting closer attention. Based on this analysis, it appears that least cost path and circuit theory analyses should not be considered alternative methods for corridor delineation, but should perhaps instead be applied in tandem, as together they may be more informative than either is alone. The strong overlap in corridor location predictions from least cost path and circuit theory analyses suggests that their representation of functional connectivity of the landscape is also likely to be similar, which is confirmed by graph theory analysis. Minimum spanning trees based on cost-weighted distance and cumulative resistance measures differ only in the role of patch 36 in connecting the rest of the patch network. Examination of the current map (Figure 2b) provides a likely explanation for this difference. Patches 24 and 29, which are linked by circuit theory’s minimum spanning tree but not by that of least cost path, are separated by a continuous swath of suitable habitat with high movement probabilities. Patches 24 and 36, however, which are linked by least cost path’s minimum spanning tree but not by that of circuit theory, have several narrower suitable routes weaving through areas with very low movement probability. The differences between these areas in availability of suitable alternative paths are captured by circuit theory, but not by least cost path analysis. Circuit theory therefore assigns the link between patches 24 and 29 a lower resistance value relative to the link between patches 24 and 36 than does least cost path analysis. This difference is also carried through to the patch centrality indices for each method, which disagree as to the importance of patch 36. The general agreement between these approaches is encouraging for conservation practitioners faced with choosing a method by which to predict potential corridor locations and to prioritize patches and linkages for inclusion in reserve networks. This analysis suggests that use of least cost path and circuit theory analyses together, rather than choosing a single method, may be the most informative approach for identifying corridors. Both methods are straightforward to implement, making the gain in information from using both worth the small additional effort of applying two analyses. However, the memory requirements of the CircuitScape program may be an impediment to its use on large landscapes if relatively fine-grain analysis is needed.

Application of both methods may also be beneficial when using graph theory metrics to prioritize acquisition of patches and linkages for protection. Exploration of any differences in predictions between methods may improve understanding of the role of each patch in maintaining connectivity of the network and of the importance of maintaining broad areas of quality habitat between key patches. This comparison may also help to identify alternative reserve network designs if protection of particular linkages is not feasible. For example, based on this analysis one might wish to focus protection on the area between patch 24 and 29, but if this area consisted of unavailable private land or was too costly to maintain as a whole, protection of a corridor between patches 24 and 36 may provide a viable alternative.

If a single method were to be implemented to predict patch and linkage importance, circuit theory appears likely to be more informative than least cost path analysis by quantifying resistance of the landscape as a whole rather than that of a single optimal path. However, choice of a method may depend on practical constraints faced by a practitioner. If resources will only be available to acquire and protect single, relatively narrow corridors among patches, least cost path is likely to be more useful for identifying optimal sites because circuit theory measures of resistance are dependent on the entire landscape separating two patches remaining intact.


This analysis has utilized only a handful of the many graph theory metrics that are likely to be informative for reserve network planning. Two general classes of graph operations – node removal and edge removal – are relevant to connectivity and provide more advanced indicators of patch and linkage importance. These operations measure the effects of removal of each individual graph component from the minimum spanning tree on the rest of the graph structure. For example, threshold effects of loss of edges or nodes can be measured by removing edges longer than a given length or nodes smaller than a given patch size and measuring the size of the largest remaining cluster in the graph or the number of separate clusters produced (Bunn et al. 2000; Urban and Keitt 2001). These methods would not be particularly informative when applied to the very small, simple graph used in this study, but are likely to provide greater insight into large, complex networks than the simple patch importance metrics demonstrated here.

Future work comparing least cost path, circuit theory, and other corridor identification methods such as individual-based models is needed to more fully understand the effects of differences among these methods’ assumptions and approaches on their predictions. Extensions of this study applying these alternative methods to a broader variety of more complex landscapes and using more advanced graph theory operations to compare their predictions of functional connectivity patterns is expected to be informative. As landscapes and metrics become more complex, however, it may be increasingly difficult to decipher the sources of differences between each method’s predictions and to determine which is likely to be most relevant for the goals of conservation practitioners. Movement frequencies of focal species individuals among patches provide a true measure of functional connectivity. Therefore empirical movement data from GPS collars, marked individuals, or genetic distances (depending on the species and scale of interest) are expected to be crucial for identifying methods that most accurately represent patterns of functional connectivity and that are most likely to be useful for sound reserve network planning. References Bunn A.G., Urban D.L. and Keitt T.H. 2000. Landscape connectivity: A conservation application

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Using graph theory to compare least cost path and circuit theory