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Supplementary Material for: Dynamic social networks in guppies (Poecilia
reticulata).
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Ramnarine5, Karoline K. Borner1, Romain J.G. Clement1 & Jens Krause1,6
Alexander D.M. Wilson1, 7, Stefan Krause2,&, Richard James3, Darren P. Croft 4, Indar W.
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and Ecology of Fishes, 12587 Berlin, Germany;
Leibniz Institute of Freshwater Ecology and Inland Fisheries, Department of the Biology
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Applied Sciences, Lübeck, Germany;
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Department of Physics, University of Bath, Bath BA2 7AY, UK;
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Centre for Research in Animal Behaviour, College of Life and Environmental Sciences,
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Washington Singer Labs, University of Exeter, Perry Road, Exeter, EX4 4QG, UK.
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Tobago;
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Humboldt University, LGF, Germany
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Department of Biology, Carleton University, Ottawa, Canada K1S 5B6
Department of Electrical Engineering and Computer Science, Lübeck University of
Department of Life Sciences, University of the West Indies, St Augustine, Trinidad &
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Corresponding author: A.D.M. Wilson. E-mail: alexander.wilson@ymail.com
ADM Wilson and S Krause are shared first authors
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Maximum likelihood estimation of the models’ probabilities
In order to determine the transition probabilities of the Markov chain models, it would in
principle be possible to estimate them based on the observed lengths of contact with a
particular nearest neighbour (q1), of social contact (q2), and of being alone (q3). However,
since our observation periods per focal individual were limited to 1.5 min. (10 data
points) this estimation would be biased, and the number of shorter contact phases would
be overestimated. As an alternative the transition probabilities can be estimated by
computing the relative frequencies of the occurrences of these transitions in the observed
data which constitutes a maximum likelihood estimation (Fink 2008). This is explained in
more detail in the next paragraph.
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In our Markov chain model the next state only depends on the current state. For our
simple model which only distinguishes the states i (“focal fish is social”) and x (“focal
fish is alone”), we need to estimate the conditional probabilities p2 = P(staten+1 = x | staten
= i), and p3 = P(staten+1 = i | staten = x). Here, p2 (p3) denotes the probability that the focal
fish switches to state x (i) when its current state is i (x). The probability p2 can be
estimated by looking at the observed pairs of states and dividing the number of (i,x) pairs
by the number of (i,s) pairs, where s is any state (i or x). In the same way p3 can be
estimated based on the relative frequency of state x being followed by state i.
As an example let us assume that for some focal individual the following state sequence
is observed:
x, x, x, i, i, x, i, x, x, i, i, i, x, i.
This sequence consists of the pairs (x,x), (x,x), (x,i), (i,i), (i,x), (x,i), (i,x), (x,x), (x,i), (i,i),
(i,i), (i,x), and (x,i). The estimate for p2 is 3/6 because 3 out of 6 pairs beginning with i
end with x, and the estimate for p3 is 4/7, because 4 out of 7 pairs beginning with x end
with i. In an analogous way the probability q1 of the detailed model can be estimated.
It is possible to take not only the single preceding state into account when estimating the
probabilities of the next states but the preceding pair, triple or n-tuple of preceding states
(and construct higher-order Markov chains). However, the more preceding states are
taken into account the more data is needed for a robust estimation of the probabilities. In
our case, the behaviour did not seem to depend on more than the current state (Fig. 2).
Therefore, we did not use more than that for the estimation of our model probabilities.
Tests
We used the following types of Monte Carlo tests.
Markov chain Monte Carlo test
Based on a list of groups this test randomises the group compositions while keeping
constant the group sizes and the numbers of occurrences of the group members. Krause et
al. (2009) describe this test in more detail. In our study we used it to analyse the
composition of the groups formed by individuals that were present at the hotspot during
the same session.
Model-based Monte Carlo test
In this test a “randomisation” step consists of running a model to simulate an observation
that has the same number of sessions, the same group compositions per session, and the
same focal individuals as our original observation. In contrast to a pure randomisation
that, e.g., simply permutes individual identities, in the model-based test each data point of
a sequence for some focal individual is generated by the model.
Example:
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If the more detailed model is used and in a session the individuals i1, i2, i3, and i4 were
present, then for a focal individual i0 the following sequence might be generated,
regardless of what the originally observed sequence looked like
x, i2, i1, i2, i2, x, i3, x, x, i1.
For each simulated observation the value of the test statistic was computed. Finally, the
rank position of the originally observed value of the test statistic was determined and the
p-value was computed following the usual definition of Monte Carlo tests (Manly 2007).
Because the simulation does not keep constant the number of contacts and contact phases
we used percentages rather than absolute values in order to compute a test statistic that
measured the association strength of pairs of individuals.
For our tests we used the models described in the methods of the main text. In particular,
we used the more detailed model as a null model for the analysis of differences in
individual-specific behaviour, and the individual-specific models to demonstrate the
goodness of fit of these models regarding certain network measures.
Simulation of disease transmission
The more detailed model and the individual specific models can be used to generate a
behavioural sequence of arbitrary length for some focal individual in the presence of k
potential nearest neighbours, e.g.
x, i2, i1, i2, i2, i2, x, x, x, i3, x, x, i1, i1, i1, i1, x, ...
Under the assumption that it takes m consecutive time steps to transfer a piece of
information or transmit a disease from one individual to another it can easily be
determined how many time steps it takes from the beginning of such a sequence until the
focal individual has been involved in contact phases of length  m with all interesting
(e.g. infected) individuals. By repeating this for multiple sequences the distribution of this
number of time steps can be approximated. In our study we used N=10000 repetitions.
References
Fink GA (2008) Markov Models for Pattern Recognition. Springer-Verlag
Krause S, Mattner L, James R, Guttridge T, Corcoran MJ, Gruber SH, Krause J (2009)
Social network analysis and valid Markov chain Monte Carlo tests of null models.
Behav Ecol Sociobiol 63:1089–1096
Manly BFJ (2007) Randomization, bootstrap, and Monte Carlo methods in biology, 3rd
edn. Chapman and Hall, Boca Raton