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Supplementary Material for: Social networks in changing environments
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Behavioral Ecology and Sociobiology
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A.D.M. Wilson*, S. Krause, I.W. Ramnarine, K.K. Borner, R.J.G. Clement, R.H.J.M.
Kurvers & J. Krause
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*Centre for Integrative Ecology, Deakin University, 75 Pigdons Road, Waurn Ponds, Victoria
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3216 Australia
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*Corresponding Author:[email protected]
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Markov chain modelling approach and parameter estimation
Our observations consist of sequences of ‘behavioural states’. In the presence of k potential
neighbours at each time point a focal fish can either be with a nearest neighbour g, 1 ≤ g ≤ k,
denoted by sg or alone (no conspecific within 4 body lengths) denoted by a. A focal fish is
regarded as being social, if it is in state sg for some neighbour g. ‘Being social’ is not an
explicit state in the model, but is implicitly defined by the set of states s1 ... sk. Following
Wilson & Krause et al. (2014) we used the three probabilities
pleave_nn = Prob(staten+1 ≠ sg | staten =sg).
psa = Prob(staten+1 = a | staten  {s1, ..., sk}), and
pas = Prob(staten+1  {s1, ..., sk} | staten = a)
to construct a model that describes the transitions between these states. Here, pleave_nn denotes
the probability of leaving the current nearest neighbour, and psa (pas) the probability that
the focal fish will be alone (social) in the next state when it is currently social (alone). These
probabilities can be estimated based on relative frequencies (Fink 2008). Following Wilson &
Krause et al. 2014 we did not take the specific individual identities into account when
estimating the model probabilities, i.e. we counted the state changes regardless of the
individual identities. Therefore, our model describes the general dynamics common to all
observed individuals. Figure 1 in the main text shows a graphical representation of the
resulting model. Its probabilities define the transition probabilities of a (first-order) Markov
chain without having to introduce any further assumptions or parameters.
From these probabilities the probability pswitch_nn (of switching the nearest neighbours while
staying social) and pretain_nn (of retaining the nearest neighbour) can be derived as follows.
pswitch_nn = pleave_nn  psa.
pretain_nn = 1  pswitch_nn  psa.
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A Markov model can be used to produce state sequences by simply ‘running’ the model. In
our case, this allows predictions regarding the frequency distributions of the lengths of contact
with a specific neighbour, of phases of being social, and of phases of being alone. These
predictions can be used to analyse the goodness of fit of our model, which is described in
more detail in the next section.
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Goodness of fit of the Markov chain model
The time spent in a state of a Markov chain follows a geometric distribution. In our study
system this means, the frequencies of phase lengths of, e.g., being social should decrease
exponentially with increasing phase length. To compare the model predictions with the
observed data we simulated observations of the model’s behaviour where we took into
account the 2 min observation time per focal individual. This is necessary because
incompletely observed phases (that started or ended outside the observation period) will lead
to higher numbers of short phases than theoretically expected. We repeated the simulation 10 4
times and computed the mean frequencies and the 2.5% and 97.5% percentiles for each phase
length. The simulation was based on the estimated probabilities and did not take into account
their confidence intervals. Therefore, the predicted percentile ranges are conservative. Our
results show that the observed data are well approximated by the model predictions (Figure
S1).
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Movement simulation
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In order to construct a movement simulation that can be used as a null model for the
investigation of density changes we tried to introduce as few parameters as possible. We used
a discrete-time simulation where between two successive time points an individual can be
moving or resting. At each time point, an individual can retain its state (moving or resting)
with fixed probabilities pmoving and presting, respectively, or change it. A moving individual
moves by a fixed distance l in a direction given by the individual’s heading h. Additionally, at
each time point an individual can decide to change its heading with probability pchange_heading.
A new heading is computed by adding a randomly chosen value α to the current heading.
We tried out two different probability distributions for α, a uniform distribution on the range
(-π/2, +π/2) and a von Mises distribution, which is a circular analogue of the normal
distribution, on the range (-π, +π) with concentration 1. This choice had little influence on our
results and the trends were exactly the same. Therefore, we decided to only use the von Mises
distribution as it seemed more natural. We also tried out two different ways to define the
simulation ‘world’ in which the individuals move, a circle with fixed radius r and a torus
constructed from a square by pasting the opposite edges together. Our results did not seem to
depend very much on these variants and we decided to use a circular world with a fixed radius
as this seemed more natural to us. However, in this case we need to prevent individuals from
‘leaving’ the simulation world. We did this by letting individuals that are about to leave
choose a new heading (using the von Mises distribution) until a movement in this direction of
length l is within the circle.
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Summarized, our movement simulation consists of the 3 probabilities pmoving, presting, and
pchange_heading, the step length l, and the choices regarding the probability distribution of
heading changes and the shape and size of the simulation world. To be able to draw
comparisons with our real observations we additionally introduced a distance d that defines
the distance within which two individuals are regarded as ‘neighbours’ having social contact.
We examined a wide range of parameter settings for the 3 probabilities, the step length l, and
the distance d and found that regardless of the settings a) the lengths of contact with a
particular neighbour, of being social and of being alone could be described by a Markov chain
model with parameters estimated from the simulated movements and b) the estimated model
probabilities pas (of ending being alone) and pswitch_nn (of switching the current neighbour)
increased with increasing density always following the same pattern (a linear function, see
Figure S2 for an illustration of this for specific parameter settings). This means, although we
do not know whether our movement simulation exactly describes the real movements of our
fish it makes sense to use the movement model to assess the influence of density changes on
the estimated model probabilities pas and pswitch_nn. The same holds for the probability psa
(of ending being social) which equals pleave_nn - pswitch_nn and therefore decreases with
increasing density. The probability pleave_nn (of leaving the current neighbour) does not depend
on the density (however, see the remarks in the next paragraph). By changing the number of
individuals or the radius r of the simulation world while keeping constant all other parameters
of the movement simulation we can investigate the relative magnitude of changes to the
model probabilities caused by increased or decreased density.
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We found that the probability pleave_nn slightly increased with increasing density although
these changes were very minor (Figure S2). One reason for this might be that with higher
density it happens more often that the nearest neighbour of the focal individual changes
because a third individual approaches the focal individual. This leads to shorter contact phases
with the same individual and thus increases pleave_nn.
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References
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Fink GA (2008) Markov Models for Pattern Recognition. Springer-Verlag
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Figures
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Figure S1. Frequency distributions of the lengths of contact with a particular nearest
neighbour, the lengths of social contact, and the lengths of being alone in the observed data
(circles) before manipulations for a) pool 1a, b) pool 1b, c) pool 2, and d) pool 3. Also shown
are the means (x’s) and the 2.5% and 97.5% percentiles as predicted by our Markov chain
models.
(Note that 0 values cannot be displayed in a logarithmic plot and are omitted.)
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Figure S2. Markov chain model probabilities pleave_nn (open circles), psa (filled circles), pas
(squares), and pswitch_nn (triangles) as a function of the density. The R-squared values of all
fitted lines were > 0.98. The size of the simulated world was constant and the number of
individuals was increased to increase the density. Note that the density values do not have an
absolute meaning because the size of the world does not have any unit. The probabilities were
estimated from ‘observations’ of simulated movements. The simulation parameters were set
to the values pmoving = 0.05, presting = 0.2, pchange_heading = 0.05, l = 0.025, and d = 0.1.
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Figure S1
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a)
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Figure S2
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Supplementary Material for: Social networks in changing environments