## Introduction

Key goals of the emerging field of landscape genetics are to gain an understanding of how processes such as migration, genetic drift, and the distribution and connectivity of populations affect genetic structure (Manel et al. 2003; Storfer et al. 2007). Less attention has been paid to identifying general system-level features that arise from patterns of connectivity. In many complex systems, patterns of connectivity give rise to system-level properties that are not apparent from analysis of pairwise relationships between components. The ability to characterize these system-level properties, along with the local properties of individual landscapes, could improve resource management in complex, natural ecosystems. Valid inference at this level of analysis requires genetic samples from multiple populations and an analytical framework within which the influence of landscape variables on genetic variation can be determined.

We were interested in investigating the link between landscape quality and genetic connectivity among fishers (*Martes pennanti*) sampled from 34 landscapes in the Great Lakes Region of ON, Canada (Carr et al. 2007a). We used graph theory to model gene flow within the resulting network of genetic connectivity among fishers in order to relate system- and node-level biological characteristics to landscape quality.

Graph theory (see Table 1 for a glossary of terms) has provided a powerful framework for characterizing processes that take place in complex interconnected systems in such diverse disciplines as physics, mathematics, and sociology (Newman 2003), as well as in biology where protein–protein interactions, social structure, and food webs have been modeled (Proulx et al. 2005; May 2006). The distribution of genetic variation can also be intuitively conceptualized as a network of genetically interconnected nodes representing individuals from sampled sites connected by gene flow (Dyer and Nason 2004; Dyer 2007). We employed a recently developed technique to construct our network based solely upon genotypes of individuals sampled from multiple landscapes, avoiding the need for *a priori* assignment of barriers to gene flow (Dyer and Nason 2004). We considered the fishers sampled within each landscape as a node in the network.

Betweenness: the number of shortest paths that a particular node or edge lies on. Assuming that interactions take place through the shortest path, then betweenness is a measure of the importance of a node or edge in terms of the bottleneck it creates. |

Centrality: a measure of the relative position of a node or an edge in terms of connectivity or facilitation of node interaction (e.g., betweenness, degree, eigenvector centrality). |

Characteristic path length: the mean of all pairwise graph distances connecting nodes. It can be used as a ‘fitness’ measure describing the ease of node communication. |

Clustering coefficient: a measure of the probability that two nodes connected to a particular other node are themselves connected. |

Degree: the number of edges connected to a node. If the edges are weighted, then edge weights are summed and this measure is generally termed ‘strength’. |

Degree distribution: the distribution of node degree values of a network. The degree distribution is a particularly important measure of network topology and together with other metrics is diagnostic of certain classes of networks and some general properties of network topology. |

Eigenvector centrality: a similar in concept to ‘degree’ but accounts for the fact that not all connections are equally connected. Here connections to well-connected nodes will likely be more influential than connections to less well-connected nodes and are weighted as such. |

Graph theory: a branch of mathematics that deals with describing and understanding the properties of networks. |

Modularity: a measure of community structure within a network. |

Network: a set of entities (represented as nodes) that interact (represented as edges). Interactions can be represented as simple binary connections, can have direction, or weighted values representing the strength of interactions. |

Graph distance: the sum of the shortest number of distinct edges (or edge weights) connecting a pair of nodes. |

There are three particularly well-described classes of networks that may be of general interest to landscape geneticists: small-world, scale-free, and random (described in Table 2; Barabasi and Albert 1999; Erdos and Renyi 1959; Watts and Strogatz 1998). These classes are of interest because they each imply characteristic dynamic features that can be interpreted in the context of dispersal, gene flow, resilience to extirpations, and genetic structure (Table 2). Generally, small-world networks are characterized by highly clustered nodes, suggesting efficient transfer of information (in our case alleles) and a decentralized network structure. Scale-free networks are characterized by a few highly connected nodes that have disproportionate importance in maintaining network connectivity. These connected ‘hubs’ are also points of vulnerability for the network if removed. Finally, random networks are useful ‘straw-men’. When constructed with similar node and edge properties as empirical networks, random networks can aid in determining whether observed properties of a network are a consequence of some nonrandom process or simply a byproduct of random linkages among nodes.

Random networks: A class of networks characterized by a short characteristic path length, binomial degree distribution, and a small average clustering coefficient. Because each node is approximately equally well connected, the characteristic path length increases monotonically after random or targeted node removal. If the genetic connectivity among populations displays random graph properties, this would suggest that dispersal among populations was entirely random and unstructured and that populations are separated by short paths (direct or through intermediate populations). Extirpations of populations would steadily decrease the ease through which genes were exchanged among populations. |

Small-world networks: A class of networks characterized by a short characteristic path length, binomial degree distribution, and a large mean clustering coefficient. Small-world networks are similar to random networks in that each node has approximately the same influence on the characteristic path length if removed, however the added feature of clustering might create alternate paths between nodes such that impact of node removal could be less than on random networks. If genetic connectivity has these characteristics genes can be efficiently exchanged among populations ‘locally’ and ‘globally’. Given that there will likely be increasing fitness costs of dispersal with increasing geographic distances and greater robustness to losses of populations, we might predict the small-world network characteristics to be common to well connected populations. |

Scale-free networks: A class of networks characterized by a short characteristic path length and a power-law degree distribution. The average clustering coefficient can vary. Most nodes have relatively few connections while a few nodes are highly connected hubs Because most nodes are not particularly well connected, the random removal of even a high proportion of nodes tends to have little impact on the network characteristic path length. However, the targeted removal of the most connected nodes leads to a rapid increase in the characteristic path length and network fragmentation. From a biological perspective, the random removal of population nodes could be considered analogous to stochastic extirpation perhaps due to severe weather events, whereas removal of the most connected nodes might occur, for example, due to over harvest of populations in high quality habitats. In this case ‘hub’ populations would warrant considerable concern within management and conservation strategies. |

Previous research has demonstrated that fishers are territorial and relatively philopatric, exhibiting short dispersal distances for a carnivore of their size (Arthur et al. 1993; Kyle et al. 2001; Koen et al. 2007). We hypothesized that this would lead to a highly clustered network of genetic connectivity, with either small-world or scale-free properties. A tendency for philopatry could lead to the clustered nodes of a small-world network. A few important source populations in high quality habitat, however, could act as the hubs of a scale-free network. We also simulated the effect of local extirpation (i.e., node removal) on network structure. A small-world network should be more resilient than a scale-free network to node removal.

Aside from system-level properties of the network, we were also interested in how the topological positions of individual nodes on the network characterized their influence on the network’s dynamic processes. There are numerous metrics of node and edge position (Costa et al. 2007). Three particularly relevant measures of node centrality are degree, eigenvector centrality, and betweenness (Table 1). The degree of a node is a measure of its connectivity (number of connections); eigenvector centrality incorporates both direct and indirect connectivity (how connected a node’s immediate connections are); and betweenness is a measure of the bottleneck a particular node forms in the network. Assessing the effects of targeted removal of nodes with high values for these measures on the characteristic path length should help identify nodes that are particularly important for maintaining genetic connectivity.

Metrics of node position may also be valuable for understanding the ecological properties of the network. To investigate this, we related indices of node centrality (degree and eigenvector centrality; Table 1) to the proportion of genetically identified immigrants in each node and to other measures of habitat suitability so as to identify biologically meaningful traits of the topological position of nodes. The fisher population that we studied was increasing (Bowman et al. 2006) and fishers appear to exhibit density-dependent dispersal such that, during a population increase, the proportion of immigrants into each landscape is negatively related to habitat suitability (Carr et al. 2007b). In this ecological context, central nodes should be productive and well connected. Thus, we hypothesized that node centrality should be inversely related to the proportion of immigrants to each node, and therefore directly related to habitat suitability.

Finally, a network approach to the analysis of gene flow among nodes may provide an informative method for identifying genetic structure. Many other naturally occurring networks including biological networks of social interactions (Lusseau et al. 2006) and metabolic pathways (Guimera and Amaral 2005) display structure within networks. These networks are characterized by communities of nodes with dense node connectivity within groups and relatively more sparse connections among groups. Thus, existing graph-theoretic methods for clustering based upon network topology may also be useful for clustering genetically well-connected nodes and identifying cryptic population structure. An effective method for detecting structure within networks is ‘modularity’ optimization (Danon et al. 2005; Gustafsson et al. 2006) as characterized by eigenvectors of the network matrix (Newman 2006). This method searches for natural divisions such that there are more connections within clusters or fewer connections among clusters than expected at random.

In our analyses we have compared the network’s characteristic path length and clustering to that of similarly structured random networks to determine whether network clustering was a feature of the number of nodes and edges or a result of some other potentially non-random process (i.e., we tested whether the network had small-world properties). We then assessed the distribution of node connectivity to determine whether the network was characterized by a few hubs of particularly well-connected nodes (was the network scale-free?). We examined the effects of the sequential removal of the most central nodes on the network’s characteristic path length to gain an understanding of possible effects of node extirpation on gene flow. Measures of node centrality were related to the proportion of immigrants in each node, as well as the show depth and proportion of coniferous forest cover within the area encompassed by the node. Finally, we assessed the ability of a network clustering technique, modularity optimization, to identify population genetic structure.