This is the second in a series of blog entries exploring the metrics used for assessing dominance hierarchies: see the introductory post for the rationale behind doing this, with other metrics visible through the index page.
Clutton-Brock et al. (1979) were interested in giving a metric to fighting success in red deer stags, where individuals were studied over long periods of time. Studying any network system over longer periods of time is going to cause a problem, as the status of individuals may change during that period (see Rands 2014 for some discussion of this problem), and the authors of this paper were aware that a male’s dominance could change within a mating period as his energy levels flagged or he became injured. Simply counting the number of fights won and lost will not give a very accurate reflection of how an individual is placed within the herd, as his success is also going to depend upon the idntities of the individuals he beats: a male who consistently fights and wins against weak opponents is not necessarily going to be of similar quality to a male who consistently fights and wins against strong opponents. So, Clutton-Brock and colleagues designed a simple metric that takes account of the quality of opponents individuals win and lose against.
I’ll illustrate how this is calculated with by considering the fighting ability metric of two individuals (labelled black and blue) within the following group structure:
To gauge an individual animal’s fighting success, you need to calculate B, the number of other animals that the focal individual has won against, and note the identities of all the losers. For each of these marked losers, you also need to calculate the number of individuals that they in turn have beaten, and sum these to give Σb. Because we define one individual in an interacting pair as a winner, and the other a loser, this means that none of the summed interactions contributing to Σb are against the focal individual.
As well as assessing wins, you also need to calculate L, the number of other individuals that the focal loses against. These winning animals are marked and the summed number of animals that they themselves lose against is calculated, giving Σl.
Having collated these numbers, the fighting success of a focal individual (which I will refer to as DCB) is calculated as
DCB = (B + Σb + 1)/(L + Σl + 1),
where the “+1” term on both the top and bottom of the equation allows a meaningful metric to be calculated for individuals that are either never seen to win or lose.
Using the group interactions given in Figure 1, we calculate DCB for the individual coloured black using the following reasoning:
Following Figure 2, we see that B = 8, Σb = 2 + 2 + 2 + 1 + 1 + 0 + 0 + 0 = 8, L = 3, and Σl = 2 + 2 + 0 = 4. So, DCB = (8 + 8 + 1)/(3 + 4 + 1) = 2.125 for the black individual. Similarly, using the reasoning given in Figure 3, DCB =1.167 for the blue individual.
A larger value of DCB will notify a greater fighting ability, and the maximum size of the statistic within an observed group is going to depend on both the size of the group and the maximum number of other animals that each individual in the group interacts with. In their original paper, Clutton-Brock and his colleagues found DCB for red deer ranged between 0 and a little over 3.
This is a simple statistic to compute, but I would caution that it should really only be used for comparing individuals within a group, given that it is dependent upon both group size and number of interactions recorded. The metric is also dependent upon observed relationships being fixed: an individual that wins an interaction will always win future interactions with the same opponent. This suggests that caution should be used if this metric were to be transferred to observed interactions where the dynamic between a dyad could change over time.
Clutton-Brock TH, Albon SD, Gibson RM & Guinness FE (1979). The logical stag: adaptive aspects of fighting in red deer (Cervus elaphus L.). Animal Behaviour 27: 211-225 | pdf
Rands SA (2014). We must consider dynamic changes in behavior in social networks, and conduct manipulations: comment on Pinter-Wollman et al. Behavioral Ecology 25: 259-260 | full text | pdf
Technical Note: The network diagrams were drawn on a Mac with Dia Diagram Editor (open source freeware), and coerced into nice smooth images with GIMP (GNU Image Manipulation Program: open source freeware).