Integrating species and interactions into similarity metrics: a graph theory-based approach to understanding community similarity

Biological data

We analyzed a dataset (https://www.nceas.ucsb.edu/interactionweb/html/thomps_towns.html) of consumer-resource interactions obtained from the National Center for Ecological Analysis and Synthesis (NCEAS). The dataset includes 16 communities (Akatore A, Berwick, Blackrock, Broad, Canton, Catlins, DempSp, German, Kyeburn, Narrowdale, NorthCol, Powder, Stony, SuttonSp, and Venlaw), covering ca. 200 km of the Taieri River tributaries in Otago, New Zealand. These sites include pine forest, broadleaf forest, pasture grassland, and tussock grassland with recorded taxonomic identities of aquatic insects, algae, and fish species and their trophic interactions (Jaarsma et al., 1998; Thompson & Townsend, 2005; Thompson & Townsend, 2003; Thompson & Townsend, 2000; Thompson & Townsend, 1999). The size of these networks varies from 48 to 113 taxonomic identities and from 110 to 832 consumer-resource interactions. We assign an identification code to taxonomic identities to facilitate comparison among different networks. We made some modifications to the network dataset prior to the analyses (see Data S1), checking scientific names and correcting typographical errors. Finally, similarity measures were calculated for paired combinations of sites in Taieri River (120 paired comparisons for each scenario).

Graph edit distance (GED): concept and application

We used the graph edit distance (GED) as a metric that includes both compositional and interaction similarities. GED is a widely used graph transformation method in which each transformation (edition) necessary to pass from one network to another has a cost, so that a greater number of changes implies a higher cost, and this mirrors higher dissimilarity between the compared communities (Bunke, 1983; Emmert-Streib, Dehmer & Shi, 2016). This feature of GED represents an advantage over the composition similarity analysis, because it allows the inclusion of interaction similarity in the metrics and to assign different degrees of importance to species or interactions in the network through differential costs for each type of edition (Bunke & Allermann, 1983; Emmert-Streib, Dehmer & Shi, 2016). Despite the fact that our study only deals with trophic interactions, GED could be applied to networks containing different kinds of links, including other ecological interactions like facilitation, competition, and parasitism.

Let us consider two networks represented by the graphs g₁ = (V₁, E₁) and g₂ = (V₂, E₂), where V is a set of nodes, E a set of links (u, v), where u ∈ V is the source node and v ∈ V the target node of a directed link (Dehmer, 2010; Riesen, 2015). The idea behind GED includes transforming one graph into another using edit operations (e_i) such as deletions, substitutions, and insertions of nodes and links (Bunke & Allermann, 1983; Dehmer, 2010; Emmert-Streib, Dehmer & Shi, 2016; Riesen, 2015). A given transformation is represented by an edit path (λ), which is a set of edit operations that transform g₁ to g₂. The set of all possible λ is called υ (g₁, g₂) (Riesen, 2015). Assuming each edit operation e_i has an associated cost c(e_i), we can assign a relative cost to the k^th edit path λ_k: ${\displaystyle C\left({\lambda }_{\mathrm{k}}\right)=\sum c\left(e_i\right),\forall e_i\in {\lambda }_{\mathrm{k}}}$ With this information we can define GED as the edit path with the minimum cost (Dehmer, 2010; Ibragimov et al., 2013), ${\displaystyle GED=min\left[C\left({\lambda }_k\right)\right],\forall {\lambda }_k\in Y\left(g1,g2\right).}$ The application of GED to the study of trophic networks allows us to evaluate differences in species compositions and those due to the absence of interactions despite species co-occurrences.

Cost operations

Given the nature of food webs, we considered that the lowest cost was that of deleting or adding an interaction, followed by deleting or adding a species, and finally the most costly edition was that of flipping an interaction, this is a change to the hierarchy of the consumer-resource interactions (Table 1, scenarios 30, 35, 36). We considered 49 scenarios of editing costs (8,820 paired comparisons of our networks), which differed from each other in the relative magnitude of the costs of each edition operation c(e_i), as shown on Table 1. We include two cost scenarios of flipping an interaction (0.25 and 0.75) represent two contrasting scenarios (low and high cost), and 5 levels of costs for deleting/adding an interaction or node. Also, eight scenarios were included specifically to assess how much GED changed by minimizing the importance of trophic interactions (reducing the costs to deleting or adding interactions, Table 1).

GEDs were calculated using the software Cytoscape (Shannon et al., 2003) with the GEDEVO plugin (Ibragimov et al., 2013), which implements an evolutionary algorithm for GED calculation. Due to GEDEVO implementing a evolutionary algorithm for estimating GED, there is no straightforward rule for stopping the iterative process. In this case we used 1,000 iterations without improvements as termination criterion for the minimization of GED. In the case of GEDEVO, GED scores ranged from 0 to 1, where 0 represented maximum similarity and 1 represented perfect dissimilarity. According to Malek (2015), the method implemented in GEDEVO is that the GED of a given edit path is the result of the sum of the GED estimated for each node. The GED forming each node is transformed into a score based on the highest possible GED for a single node in the whole network.

Compositional similarity and comparison with GED scores

In order to compare the GED results with those from a traditional compositional approach, we first computed the Jaccard index from the presence/absence species data of each network. Jaccard dissimilarity index was computed as (a + b)∕(a + b + c), where a is the number of species only present in the first network, b is the number of species only present in the second network, and c is the number of species shared between both networks.

In our results, the GED and Jaccard scores were expressed in terms of similarity (i.e., 1-GED and 1-Jaccard, respectively). We used the Adjusted Mutual Information metric (AMI) to assess the amount of information shared between GED and Jaccard.

Mutual Information (MI) is a measure from information theory and based on entropy estimations. MI quantifies the mutual dependence between the variables or, in other words, how much information about one variable is possible to obtain by knowing the second variable. MI can theoretically range from 0 to ∞. So, to standardize the values and make them comparable we used the method from Vinh, Epps & Bailey (2010). In this case AMI is: ${\displaystyle AM{I}_{max}=I\left(U,V\right)-E\left\{I\left(U,V\right)\right\}∕max\left\{H\left(U\right),H\left(V\right)\right\}-E\left\{I\left(U,V\right)\right\}}$ where, U and V are two datasets, I (U, V) represents mutual information, E {I (U,V)} the expected mutual information, and finally H (U, V) joint entropy, thus: ${\displaystyle I\left(\mathrm{U},\mathrm{V}\right)=\sum _{u\in U}\sum _{v\in V}p\left(\mathrm{u},\mathrm{v}\right)logp\left(\mathrm{u},\mathrm{v}\right)∕p\left(u\right)p\left(v\right),}$ ${\displaystyle H\left(\mathrm{U},\mathrm{V}\right)=-\sum _{u\in U}\sum _{v\in V}p\left(\mathrm{u},\mathrm{v}\right)logp\left(\mathrm{u},\mathrm{v}\right),\mathrm{where the entropy of a random variable is defined by}:}$ ${\displaystyle H\left(U\right)=-\sum p\left(u\right)logp\left(u\right)\mathrm{or}}$ ${\displaystyle H\left(V\right)=-\sum p\left(v\right)logp\left(v\right),}$ with probability functions p (u) and p (v).

The advantage of AMI is that it allows us to quantify linear and non-linear relationship between variables. AMI scores ranged from 0 to 1, where 0 represented perfect independence (no information is shared between indices) and 1 means both variables contain exactly the same information (Vinh, Epps & Bailey, 2010). Jaccard index and AMI calculations were performed in the R environment (R Development Core Team, 2016, Data S2). Jaccard index values were calculated using vegan package for R (Oksanen et al., 2013) and AMI values were calculated using the function discretize from the R package infotheo (Meyer, 2014).

We used the same approach to compare dissimilarity of interactions (B_WN) and the dissimilarity of interactions due to species turnover (B_ST) as there are defined in Poisot et al. (2012) with the estimated GED values in the 49 scenarios (Table 2). B_WN and B_STvalues were calculated using betalink package for R (Poisot et al., 2012). Finally, we used the AMI to compared the amount of information shared between GED and B_WN, and between GED and B_ST. The objective of this comparison is to evaluate the relationship of GED with another recent methodological approach to the problem, and also to evaluate the relationship of GED under different cost scenarios with the result obtained by partitioning network similarity into species and interaction β-diversity as is proposed by Canard (2011) and Poisot et al. (2012).