Márcio S. Araújo, Paulo R. Guimarães, Jr., Richard Svanbäck, Aluisio Pinheiro, Paulo Guimarães, Sérgio F. dos Reis, and Daniel I. Bolnick. 2008. Network analysis reveals contrasting effects of intraspecific competition on individual vs. population diets. Ecology 89:1981–1993.

Appendix A. Derivation of the C_ws index of clustering. Click here for a PDF file of the appendix.

The individual niche overlap network

The first step in the development of C_ws is to define a conceptual framework linking patterns of resource use at the population level and complex network theory. We did this by defining the individual niche overlap network as follows. A network is a representation of associations among elements in a system, in which nodes represent elements, and ‘edges’ are lines that connect those nodes that are somehow associated. To describe resource use variation within a population, we can represent individuals as nodes, with each pairwise combination of individuals connected by an edge if they use any resources in common, or disconnected if they do not share resources. A given edge can be weighted to represent the degree of pairwise niche overlap w_ij between two individuals i and j (adapted from Schoener 1968) :

(A.1)

where p_ik is the frequency of category k in individual i’s diet, and p_jk is the frequency of category k in individual j’s diet. The proportion of the k-th resource category in individual i’s diet, p_ik is calculated as:

(A.2)

where  represent the number (or mass) of diet items in individual i’s diet that fall into category k. Numbers of prey consumed may be more appropriate for behavioral ecologists studying prey capture decisions, and lends itself to Monte Carlo resampling schemes to test the null hypothesis of no niche variation. Mass may be more appropriate for studies of energy flux through a community or competition. The pairwise niche overlap ranges from 0 (no overlap) to 1 (total overlap). We therefore use w_ij to weight the edge connecting individuals i and j.

The average network density of connections

Having defined the niche overlap network, we needed to define a measure of the density of connections that can be incorporated into the clustering coefficient. This is because the degree of clustering of a network is directly proportional to the average network density of connections. This means that any clustering coefficient needs to be corrected for the average density of connections before it can be used to compare the degree of clustering among different networks.

First, we define O, the summation of the total pairwise overlap in the individual niche overlap network:

(A.3)

The average density of connections can be calculated by dividing O by , which corresponds to the number of edges of a completely connected network with n individuals.  By doing this we are standardizing the total pairwise overlap O by the number of potential edges in the network, yielding:

,

(A.4)

 which is the average network density of connections.

The index of clustering C_ws

We first take   (Barrat et al. 2004) , which is a combined measure of the number and weight of the edges around individual  and among the nodes directly connected with i defined as:

(A.5)

where  is the sum of the weights (w_i) of all the edges between individual i and the individuals to which it is connected (the neighbors of individual i);  is the number of edges between individual i and its neighbors;  is the weight of the edge between individual i and j;  is the weight of the edge between individual i and h; and , , and  are 1 if an edge is present between each pair ij, ih, and jh respectively, and zero otherwise (Barrat et al. 2004) .  The summation, therefore, quantifies the weights of all edges between individual i and its neighbors (a_ij a_ih) that are also neighbors to each other (a_jh).  We then define , which is the average value of the individual clustering coefficients, , for all nodes in the network.

(A.6)

Now, the degree of clustering is measured relative to , and in a totally random network .  An interesting feature of  is that it can assume both positive and negative values.    will be positive ( ) if the local density of connections is higher than the overall density of connections, indicating that the population is characterized by clusters of individuals sharing common resources. In contrast,  will be negative ( ) when the local density of connections is lower than the overall density of connections, indicating that individuals usually use a very particular combination of resources that differs from that of other individuals (i.e. individuals’ diets are overdispersed).

,

(A.7)

which is positively related to niche variation and ranges from zero when all individuals have identical diets and there is no diet variation, to 1 as diet variation increases. The sampling variance of E is known (see below), allowing parametric comparisons between populations or tests of null hypotheses of no diet variation, making E preferable to previous indices of diet variation (Bolnick et al. 2002) .

(A.8)

i.e.,  is a U-statistic of degree 2 and kernel given by w_ij (A.1).  It is asymptotically normal with mean  and variance  for

(A.9)

(A.10)

 where  and . As long as , the U-statistic is non-degenerate. In our case, given  (the total number of food items of the i-th individual), and taking the distribution of the food items  to be , we can then write

(A.11)

If, and only if, there is one single food category, i.e.  and , so that  almost surely, will the U-statistic be degenerate. This will happen when, and only when, all individuals are specialized on the same single resource. Otherwise, this U-statistic will behave in a reasonable fashion and asymptotic normality is attained. Under asymptotic normality, one can employ the standard deviation for building asymptotic intervals and to perform asymptotically powerful tests. Moreover, the variance of  can be obtained by Jackknifing the U-statistics by the following formula (Sen 1960, Arversen 1969) :

(A.12)

where , for any resample {i₁, i₂, i₃, i₄} from {1, …, n} , c = 0, 1, 2 being the number of coincident indices, and the sum in S_c being taken for all such quadruples. Note that resampling is performed among the individuals and not among food items for a single individual. This is done to preserve the underlying stochastic dependency structure within individual resource distributions and, therefore, produce a more robust estimate, without the need and the associated shortcomings of assuming some specific dependency setup. The variance of the index E in turn is given by

(A.13)

so that the variance of  holds for E.