Grooming network cohesion and the role of individuals in a captive chimpanzee group.

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Group size, grooming and fission in primates: A modeling approach based
on group structure
Ce´dric Sueur a,b,c,n, Jean-Louis Deneubourg a, Odile Petit b, Iain D. Couzin c
a Unit of Social Ecology, Free University of Brussels, Brussels, Belgium
b Ethologie Evolutive, Department of Ecology, Physiology and Ethology, IPHC CNRS-UDS, Strasbourg, France
c Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544, USA
a r t i c l e i n f o
Article history:
Received 29 June 2010
Received in revised form
15 December 2010
Accepted 21 December 2010
Available online 29 December 2010
Keywords:
Social network
Cohesion
Time allocation
Population structure
ODD protocol
a b s t r a c t
In social animals, fission is a common mode of group proliferation and dispersion and may be affected
by genetic or other social factors. Sociality implies preserving relationships between group members.
An increase in group size and/or in competition for food within the group can result in decrease certain
social interactions between members, and the group may split irreversibly as a consequence. One
individual may try to maintain bonds with a maximum of group members in order to keep group
cohesion, i.e. proximity and stable relationships. However, this strategy needs time and time is often
limited. In addition, previous studies have shown that whatever the group size, an individual interacts
only with certain grooming partners. There, we develop a computational model to assess how dynamics
of group cohesion are related to group size and to the structure of grooming relationships. Groups’ sizes
after simulated fission are compared to observed sizes of 40 groups of primates. Results showed that
the relationship between grooming time and group size is dependent on how each individual attributes
grooming time to its social partners, i.e. grooming a few number of preferred partners or grooming
equally or not all partners. The number of partners seemed to be more important for the group
cohesion than the grooming time itself. This structural constraint has important consequences on group
sociality, as it gives the possibility of competition for grooming partners, attraction for high-ranking
individuals as found in primates’ groups. It could, however, also have implications when considering
the cognitive capacities of primates.
& 2010 Elsevier Ltd. All rights reserved.
1. Introduction
Animals have to balance costs and benefits to be in close
proximity to conspecifics (Krause and Ruxton, 2002). On one
hand, living in groups may offer the advantage of a lower
predation risk and better efficiency when seeking resources. On
the other hand, as group size increases, individuals may experi-
ence more within-group competition for food and have higher
health risks due to the possible spread of contagious diseases
(Krause and Ruxton, 2002). Living in group implies interacting
frequently with other group members in order to maintain group
cohesion (Lehmann et al., 2007). Group cohesion may be defined
using three criteria: stability, coordination and proximity. When
group size or within-group competition for food increases, dis-
advantages may outnumber the advantages of group living,
(Chapman et al., 1995; Janson and Goldsmith, 1995; Ron et al.,
1994). As a consequence, group cohesion decreases and the group
may split either temporarily (Kerth et al., 2006; Popa-Lisseanu
et al., 2008; Wittemyer et al., 2005) or irreversibly (Henzi et al.,
1997a, b; Lehman et al., 2007). In social or pre-social animals,
irreversible fission is a common mode of group proliferation
and dispersion. From ameba to primates, this process may be
affected by genetic or social factors (Chepko-Sade and Sade, 1979;
Gompper et al., 1998; Lehman et al., 2007; Seppa et al., 2008;
Mehdiabadi et al., 2009; Rangel et al., 2009). In primates, irreversible
group fissions rarely occur (about every five/ten years) and separa-
tion of different sub-groups often takes several months to years
(Chepko-Sade and Sade, 1979; Okamoto and Matsumura, 2001; Van
Horn et al., 2007).
In primates, grooming is considered to be the most common
behavior for the maintenance of close social bonds (Schino, 2001).
Previous studies have shown that when an individual regularly
grooms a particular partner, it seems to be more tolerant with this
partner and more likely to support it during a conflict (without
suggesting causality). Likewise, the partner in question typically
reciprocates with the same tolerance and support (Henzi and
Contents lists available at ScienceDirect
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Journal of Theoretical Biology
0022-5193/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jtbi.2010.12.035
n Corresponding author at: Ethologie Evolutive, DEPE, IPHC CNRS-UDS, 23 rue
Becquerel, 67087 Strasbourg Cedex, France. Tel.: +33388107454;
fax: +33388107456.
E-mail addresses: cedric.sueur@c-strasbourg.fr,
csueur@princeton.edu (C. Sueur).
Journal of Theoretical Biology 273 (2011) 156–166

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Barrett, 1999; Dunbar et al., 2009). However, grooming needs time,
and time is a limited resource (Dunbar, 1992b; Lehmann et al.,
2007; Majolo et al., 2008; Pollard and Blumstein, 2008; Dunbar
et al., 2009). In addition to maintaining social relationships, indivi-
duals need to rest, forage and move (Pollard and Blumstein, 2008).
As a consequence, grooming seldom exceeds 15% of day-time
activity for most social species (Dunbar, 1991; Lehmann et al.,
2007). Some authors have investigated how an individual manages
to maintain its social relationships when grooming time is limited
but group size or within-group competition for food has increased
(Chapman et al., 1995; Dunbar, 1992b; Janson and Goldsmith, 1995;
Lehmann et al., 2007; Majolo et al., 2008; Pollard and Blumstein,
2008; Ron et al., 1994). Lehmann et al. (2007) have suggested that
when group size and the number of available partners increase, each
individual will have to spend more time grooming until a certain
group size for which it is impossible to maintain relationships with
all group members (Dunbar, 1992a; Lehmann et al., 2007; Schino
et al., 2009). In this case, group cohesion – social proximity and
stability – decreases and group members permanently split. This
hypothesis implies that an individual tries to develop and maintain
bonds with every group member, or at least, the most of conspe-
cifics. However, other studies have shown that whatever the group
size, an individual mainly interacts only with certain grooming
partners, and does so much more than with other potential partners
(Kudo and Dunbar, 2001; Lehmann and Dunbar, 2009). Individuals
can therefore be considered to have a relatively fixed number of
grooming partners. This is the case, for instance, in blue monkeys
(Cecopithecus mitis, Cords, 2001), savanna baboons (Papio ursinus,
Silk et al., 1999), in chimpanzees (Pan troglodytes, Watts 2000a,b)
and in several macaque species (Macaca sp., Berman et al., 2008; Lin
et al., 2008; Nakamachi and Shizawa, 2003). In this case, an
individual allocates its grooming time to its preferred partners.
Consequently, grooming time should not be dependent on group
size as it was found in Majolo et al. (2008).
In this study, we wanted to investigate the dynamic of group
cohesion – how group cohesion evolves, from stable groups to
groups having a greater probability to fission – according to group
size and group structure of grooming relationships. By inducing a
variation in the group size and the distribution of grooming time
in a stochastic agent-based model, we make predictions about the
conditions in which a group will irreversibly split. Most of studies
on the link between social structure, grooming and group size
(Kudo and Dunbar, 2001, Lehmann and Dunbar, 2009, Lehman et al.,
2009) followed standard practices in social network analyses and
used a criterion for distinguishing casual from meaningful relation-
ships. Modeling permits, without using this arbitrary criterion for
preferred relationships, the simulation of interactions between group
members (Seyfarth, 1977; Bryson et al., 2007; Meunier et al., 2006;
Sellers et al., 2007; Puga-Gonzalez et al., 2009) and also resulting
sub-grouping patterns (i.e. how individuals are sub-grouped; Ramos-
Fernandez et al., 2006). We attributed a specific grooming time given
by each individual to each other conspecifics. We tested the follow-
ing three different hypotheses for a range of group sizes (from 2 to
200 individuals): (1) an individual divides its grooming time equally
among all other group members; (2) an individual does not divide its
grooming time equally among all other group members (i.e. it
maintains a social bond with all other group members but these
bonds are different depending on the partner concerned) and (3) an
individual divides its grooming time among specific individuals (this
number is fixed to (a) 5 and (b) 10 partners per individual, see
methods for details). This social structure – grooming a specific
number of partners – is suggested by several studies (Berman et al.,
2008; Cords, 2001; Lin et al., 2008; Nakamachi and Shizawa, 2003;
Silk et al., 1999, Watts 2000a,b). Once the social structure was
established in the model, individuals made a decision between two
states (representing the two potential sub-groups). If less than four
individuals split from the main group at the simulation end, we
considered the cohesion maintained (see previous works on fission:
Lefebvre et al., 2003; Ron et al., 1994; Van Horn et al., 2007). Then,
we observed whether, and if so, how the group divided according to
its social structure. According to general rules of cohesion or of
mimetism (the probability to do a behavior depends on the number
of individuals performing this behavior), we expected that if all
individuals are linked together, especially by equivalent grooming
relationships (hypothesis 1), the system would lead to amplification
process – the more individuals join a group, and the more other ones
will join it – and no splitting would be observed (Ame´ et al., 2006;
Dussutour et al., 2005; Nicolis et al., 2003; Meunier et al., 2006).
Then, group cohesion would be influenced by grooming time and
group size when grooming relationships are not equal and especially
when grooming is only given to a small number of partners. The
simulated data were compared to observed data in order to assess
which model most closely corresponds to the observed distribution
of group sizes in primates (Lehmann et al., 2007). This comparison
allowed us to understand which is the best rule affecting group
structure and then population structure. We also tested how
increased within-group competition – leading to grooming time
decrease by a foraging time increase – influences group cohesion,
and whether changes in group cohesion are similar according to
group size. We eventually used a path analysis to investigate the
relationships (direct and/or indirect) between group size, group
structure and group cohesion.
2. Material and methods
2.1. Data
2.1.1. Empirical data
We used published data about group size and grooming time
in order to compare them to our simulation data. Data is taken
from 40 published studies on Old World primate species/popula-
tions (see Lehmann et al., 2007 for details) and are summarized in
Table 1.
2.1.2. Theoretical data
We created theoretical networks using UCINET 6.0 (Borgatti
et al., 2002). Groups contained 2–200 individuals (2, 5, 10, 20, 40,
60 ,y, 200). We set networks as random (Erdos–Renyi random
graph, with a linear distribution of links). The social structure is
different for each generated random network. We did not set
networks as scale-free since recent primate studies showed that
social networks were not scale-free (i.e. with a power distribution
of links) but random (see Flack et al., 2006; McCowan et al., 2008;
Sueur and Petit, 2008; Kasper and Voelkl, 2009; Ramos-Fernandez
et al., 2009 for studies on primate social networks; see Wasserman
and Faust, 1994 for social network theory). We can observe on Fig. 1
that some individuals are only groomed by one partner whilst other
ones are groomed by 7 or 8 partners, even if one individual can only
groom 5 partners. Then, some social characteristics such as dom-
inance of individuals may be taken into account by considering the
network having an Erdos–Renyi random structure. For instance,
individuals groomed by a lot of partners on Fig. 1 might be high-
ranking individuals or matriarchs. Indeed, it was shown that these
individuals are more groomed than other ones (Nakamachi and
Shizawa, 2003; Schino, 2001; Silk et al., 1999).
2.2. Definitions of parameters
We defined a bond (or a link) in a network as the time one
individual groomed another one. Then the relationship is directed
and does not need to be reciprocal The grooming time per
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individual T(G)i was defined as the time an individual spends
grooming, whatever the number of groomed partners n. According
to observed data (see Table 1) in this study, we considered grooming
to represent no more than 15% of total day-time activity.
For hypotheses 1 (an individual equally grooms all group
members) and 2 (an individual non-equally/randomly grooms
all group members), the mean strength of social relationships (i.e.
dyad’s social bonds) was equal to the grooming time divided by
the number of group members. For hypothesis 1, all grooming
time is equal to T(G)i/n. For hypothesis 2, minimum and max-
imum of grooming time are, respectively, 0.4% and 5.4% in a group
of 5 individuals and 0.01% and 0.7% in a group of 200 individuals.
As far as hypothesis 3 is concerned (i.e. an individual grooms a
fixed number of partners), we carried out simulations with two
numbers of groomed partners, n1¼5 and n2¼10. These corre-
spond to the average and to the maximum numbers, respectively,
of groomed partners found in experimental studies regardless of
the group size or the species (Berman et al., 2008; Cords, 2001;
Lin et al., 2008; Nakamachi and Shizawa, 2003; Silk et al., 1999;
Watts 2000a,b; Kudo and Dunbar, 2001; Lehmann and Dunbar,
2009). These studies revealed that the time taken by an individual
to groom a non-preferred group member could be considered
unimportant compared to the time taken to groom its preferred
partners. Previous studies have reported that even if individuals
groomed a specific number of individuals, high-ranking indivi-
duals or matriarchs can be groomed more than others
(Nakamachi and Shizawa, 2003; Schino, 2001; Silk et al., 1999).
We took into account these results to build the theoretical group
Table 1
Data (Genus, species, group size and grooming time) and references of data used in this study.
Genus Species Group size Grooming time (%) Reference(s)
Avahi laniger 2 2 Harcourt (from Dunbar,1991)
Cercocebus galeritus 27 5.5 Homewood (1976)
Cercopithecus ascanius 26.75 3.45 (Struhsaker, 1980; Cords, 1986)
Cercopithecus campbelli 9 2.8 Buzzard (2004)
Cercopithecus diana 28.75 2.48 (Whitesides 1989; Buzzard, 2004)
Cercopithecus mitis 22.65 7.18 Struhsaker and Leland (1979)
(Butynski 1990; Lawes 1991)
Cords (1995, 2002)
Kaplin and Moermond, (2000)
Chlorocebus aethiops 19.7 9.17 (Dunbar, 1974; Lee, 1981)
Baldellou and Adan (1997, 1998)
Colobus angolensis 18 5.25 Bocian (1997)
Colobus guereza 9.04 5.52 Dunbar and Dunbar (1974)
(Oates 1977a,b; Bocian 1997)
Fashing (2001)
Colobus polykomos 12.5 3.49 Dasilva (1989)
Colobus satanas 12 5.51 McKey and Waterman (1982)
Gorilla gorilla 11 0.09 Doran (from Lehmann et al., 2007)
Gorilla G. beringei 6 1 Fossey and Harcourt (1977)
Hylobates agilis 4.4 0 Gittins and Raemakers (1980)
Hylobates klossii 3.8 0 Whiten (1980)
Hylobates lar 3.4 2.1 Ellefson (1974)
Gittins and Raemakers (1980)
Indri indri 4.3 1 Pollock (1977)
Lemur catta 12.2 7.18 Sussmann (1977)
Lemur fulvus 15.33 7.98 Sussmann (1977)
Lophocebus albigena 15 5.8 Struhsaker (1979)
Macaca fascicularis 82.45 7.98 Van Noordwijk (1985) Son (2004)
Macaca fuscata 36.5 10.7 Maruhashi (1981)
Seth and Seth (1986)
Macaca mulatta 32 15 Teas et al. (1980)
Pan paniscus 27.8 5.7 White (1992)
Pan t. Schweinfurthi 59.2 11.67 (Wrangham, 1977; Nishida, 1990)
White and Chapman (1994)
Matsumoto-Oda and Oda (1998)
Fawcett (2000)
Pan t. verus 40.33 8.27 Tutin et al. (1983)
Yamakoshi (1998, 2004)
Boesch and Boesch-Achermann (2000)
Papio anubis 58.8 8.3 (Nagel, 1973; Eley et al., 1989)
Papio ursinus 28.07 12.64 Henzi et al. (1997b)
Papio hamadryas 51 13.5 Nagel (1973)
Piliocolobus badius 42.5 4.5 R. Noe and H. Korstjens (from Lehmann et al., 2007)
Piliocolobus rufomitratus 16.16 0.83 Decker (1994)
Piliocolobus temminckii 26.2 5.4 Starin (1991)
Piliocolobus tephrosceles 51.67 4.99 (Clutton-Brock 1974, 1975; Stanford, 1998)
Struhsaker and Leland (1979)
Chapman and Chapman (2000)
Pongo pygmaeus 1 0 Mackinnon (1974)
Presbytis entellus 33 4.4 Sugiyama (1976)
Presbytis rubicunda 7 0 Davies (1984)
Procolobus verus 3 3.58 R. Noe and H. Korstjens (from Lehmann et al., 2007)
Propithecus verreauxi 5.1 4.7 Howarth et al. (1986)
Trachypithecus leucocephalus 10 11.71 Li and Rogers (2004)
Theropithecus gelada 144.7 17.4 Iwamoto and Dunbar (1983)
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structure. Even if the number of partners an individual grooms is
fixed to (5 and 10), an individual can be groomed by more (or
less) than 5 or 10 grooming partners (see Fig. 1 for an example).
In our model, we decided to simulate an increased within-group
competition by reducing grooming time in steps of 20% (i.e. �20%,
�40%, �60%, �80% and �100%). We attributed the new social
bonds equally for hypothesis 1 and randomly for hypothesis 2. For
hypothesis 3, we deleted one partner at each step (i.e. 4, 3, 2, 1 and
0 partner). Indeed, previous studies reported that decreased groom-
ing time seems to have differing affects on an individual’s social
bonds: in the case of high within-group competition, social bonds
were mainly observed amongst kin, and an attraction was observed
towards the highest-ranking individuals (Berman et al., 2008;
Majolo et al., 2008; Schino, 2001; Watts 2000a,b). We created the
networks varying using UCINET 6.0 (Borgatti et al., 2002; Krause
et al., 2007; Wey et al., 2008).
Group fission was considered to have occurred when the group
irreversibly split into two sub-groups (i.e. daughter groups, Ron
et al., 1994) containing more than three individuals each. This
criterion of three individuals was based on group fission studies
(Lefebvre et al., 2003; Ron et al., 1994; Van Horn et al., 2007). It is
suggested that if the number individuals leaving the group is
inferior to three individuals (as dispersing males, females with
juveniles), it is more considered as dispersion than fission.
2.3. The model
We described the model according to the ODD protocol (i.e.
Overview, Design concepts and Details; Grimm et al., 2006).
2.3.1. Purpose
The purpose of the model is to assess how group structure in
terms of group size and distribution of grooming between
individuals (hypotheses 1–3) leads to group fission or allow group
cohesion (see Section 2.2 for definitions). In our model, each
individual has to choose between two sub-groups according to
the social relationships it has with each individual in each sub-
group. Then, the global variable we observed – group fission or
group cohesion – is based on the sum of individual decisions.
2.3.2. State variables and scales
The model is based on rules of mimetism/cohesion (Markov
chain process) described in several studies on collective phenom-
ena (Ame´ et al., 2006; Gautrais et al., 2007; Dussutour et al.,
2005). In this model, the probability of an individual joining a
collective movement in one direction (the future sub-group)
depends on the number but also the strength of relationships it
has with the individuals already in this direction. The number of
individuals, individual identities and the network of affiliative
relationships of each theoretical social network are included in
the model. Then, an individual is only characterized by its affiliative
relationships, based on grooming time he gave and received from its
conspecifics. This model was already used in Sueur et al. (2010). In
this study, authors explained how Tonkean and rhesus macaques
joined a sub-group during short-term fissions. For both species, the
affiliative relationships (i.e. the social network, based on contact
between individuals) explain the sub-grouping patterns. Even if
individuals are all connected together (one individual was at least
observed once with each group member), strength of social relation-
ships leads to fission. In this study, it means that the group fission or
cohesion will not only depend on the group size but also on how
individuals are connected (hypotheses 1–3).
2.3.3. Process overview and scheduling
Each group is characterized by its size (number of individuals
per group) and its structure (how individuals are connected).
Individuals are characterized by their social relationships depend-
ing on the three tested hypotheses and by a state S. At the start of
a simulation all group members are in state s0 (i.e. group 0, initial
group). Then, all individuals will have to choose between state s1
(i.e. sub-group 1) and s2 (i.e. sub-group 2) according to their own
social relationships. This process based on social network will
lead to the group cohesion or the group fission. This is the only
measure we took into consideration at the end of a simulation.
Simulations stop when all group members have changed from
state s0 to states s1 or s2. Groomed partners for each group
member were attributed randomly (see Data for details). The
model was then implemented in Netlogo 3.14 (Wilensky, 1999).
We set the number of simulations to 10,000 for each hypothesis
and for each set of tested parameters.
2.3.4. Design concepts
Emergence: the only phenomenon emerging from individual
decisions in the model is the group fission or cohesion.
Fitness: we did not measure fitness of individuals in this study.
Interaction: individuals are linked to another one by the groom-
ing time they give or they receive (see Section 2.2 for details).
Sensing: to change state s (1 or 2), individuals take into
account an intrinsic probability liS and the relationships they
have with individuals already in the state s.
Stochasticity: the model is stochastic. At each time step, a
number is randomly attributed to each individual and this
number will determine if individual will change of state and for
which state, according to the probabilities to be in each state.
Collectives: collectives are represented as social groups of
primates. Collectives occur as phenomena emerging from indivi-
dual behavior, specifically from the way to choose one sub-group
(i.e. state) or another one according to the relationships an
individual has in each sub-group (i.e. each state). The collective
phenomenon emerging from this choice is the group fission
(or the group cohesion).
Observation: we set the number of simulations to 10,000 for
each hypothesis and for each set of tested parameters (i.e. group
size). Then, for each hypothesis and each group size, we obtained
Fig. 1. Illustration of a social network with 40 individuals (squares). Number
labels indicated the number of group members by which an individual is groomed
(indegree, Wasserman and Faust (1994). We showed from this graph that even if
the number of partners groomed by the same individual was fixed at 5, an
individual can be groomed by more (or less) than 5 grooming partners in the
model. We built this network via Netdraw in UCINET 6.0 (Borgatti et al., 2002).
Distance between individuals represents the strength of associations, and was
calculated using multidimensional scaling (Whitehead 2009; Sueur and Petit,
2008).
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a value of group cohesion (how much time the group staid
cohesive up on the 10,000 simulations).
2.3.5. Initialization
At the start of a simulation all group members are in state s0
(i.e. group 0, initial group). We then induce a change of state in
two randomly chosen individuals: state s1(i.e. sub-group 1) for
one individual and state s2 (i.e. sub-group 2) for the other. These
two individuals are therefore the basis of the formation of the two
sub-groups.
2.3.6. Input
At each time step, a number between 0 and 1 is randomly
attributed to all other individuals i in state s0; when this number is
lower than the theoretical probability cis1 (P1¼[0, cis1 ]) the indivi-
dual changes from state s0 to s1; when this number is comprised
between cis1and cis1 +cis2(P2¼]cis1 ,cis2]), then the individual
changes from state s0 to s2; however, no change of state occurs if
this number is superior to cis1 +cis2(P3¼]cis2, 1]), with P1+P2+P3¼1.
If Sr(k, i)¼0 for an individual i, it changed of state according to
its intrinsic probability l.
The probability cis for an individual i in state 0 to turn into
state s (1 or 2) was
cis ¼ liþ
XN�1
k ¼ 1
rðk,iÞs
!p
where li was the intrinsic probability to change state. We
considered that all group members had the same intrinsic prob-
ability. li¼0.0001.
p determined the degree of non-linearity in the response
shown by individual i. The higher the value of p was, the higher
was the resulting discrimination between the both directions
(i.e. the higher the individual probability C to go into state s),
suggesting a deterministic response in this study (Ame´ et al.,
2006; Dussutour et al., 2005; Nicolis et al., 2003). p¼5.
li and p were chosen according to previous studies using
similar models in primates (Meunier et al., 2006; Petit et al., 2009;
Sueur et al., 2009, 2010; Jacobs et al., In press).
r(k, i)s was the social bond of i towards k when individual k was
already in state s. If individual k was not yet in state s, then r(k, i)s¼0.
Sr(k, i)s represented the sum of social bonds for individual i in
state s.
2.3.7. Submodels
Hypothesis 1. (an individual grooms all group members equally)
rðk,iÞs ¼ok,i�TðGÞi
whereok,i is the weight (strength of social bonds) between individual
i and individual k and T(G)i is the grooming time per individual.
For this hypothesis,
8i, ok:i ¼ 1=ðN�1Þ ðall rðk,iÞs were equalÞ:
N is the number of individuals in the group.
TðGÞi ¼ 0:15
Hypothesis 2. (an individual does not groom all group members
equally)
rðk,iÞs ¼ok,i�TðGÞi
with all ok,i were attributed randomly according to a normal
distribution in order that ok,i40 and Sok,i¼1 for each individual
of each group.
TðGÞi ¼ 0:15
For this second hypothesis, an individual groomed all other
group members but almost all the r(k, i)s were seen to be different
(see Data for details).
Hypothesis 3. (an individual grooms a fixed number n of
partners)
8kAGi, ok,i ¼ 1=n
8k=2Gi, ok,i ¼ 0
where n is the number of groomed partners (see Definitions of
parameters for explanation) and Gi the group of G partners
groomed by the individual i.
2.4. Group structure analyses
Group structure was analyzed using the social networking
approach (Krause et al., 2007; Wasserman and Faust, 1994; Wey
et al., 2008; Whitehead 2009). We used two indexes to determine
a group structure.
Group density: number of observed bonds divided by the
number of possible bonds in the group.
Group mean path length: an average number of all paths
(shortest number of connections between two individuals)
between all pairs of individuals in the group. For the same density
(i.e. same number of social bonds in a group), group mean path
length can be different. This depends on how individuals are
connected (see Wasserman and Faust, 1994 for details on social
network theory; see Flack et al., 2006; McCowan et al., 2008;
Sueur and Petit, 2008; Kasper and Voelkl, 2009; Ramos-Fernandez
et al., 2009; for studies on primate social networks).
We calculated these indices using UCINET 6.0 (Borgatti et al.,
2002).
2.5. Statistical analyses
The relationship between group size and group structure i.e.
group mean path length and density, was determined for the
three hypotheses. Curve estimation tests were used to assess
whether the density and the group mean path length of a given
group depends on group size, and to establish the relationship
between these variables (linear, logarithmic and exponential)
(Newman et al., 2006). The same curve estimation tests were
used to assess the dynamics of group cohesion for each hypoth-
esis according to group size. In order to understand the relations
between group cohesion and group size, we finally verified for
each hypothesis how group cohesion is affected by group density
and group mean path length.
Distributions of group sizes after simulations were compared
to observed distribution (data from 40 studied groups, see Data
for details) using the Kolmogorov–Smirnov test with Monte–Carlo
significance estimation (the number of simulations for this test
was set at 10,000). This allows to assess if simulated dynamics of
group cohesion/fission fits with global patterns observed in the
wild. The theoretical values were obtained by dividing the
number of cohesive groups (i.e. that have not split) in each group
size by the total number of cohesive groups.
A Mann–Whitney test was used to assess how increased
within-group competition – a decrease in the grooming time –
influences group cohesion within our model. The initial condition
– 15% of grooming time – was compared to each other condition
(12%, 9%, 6%, 3% and 0%). Analyses were performed using SPSS 10
(SPSS Inc., Chicago, USA).
Path analysis was used to assess direct and indirect effects
between group size, group structure and group cohesion. Path
analyses and diagrams were carried out using AMOS5 software
(AMOS Development Corporation, Spring House, PA, USA.) with
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maximum likelihood estimations (for non-parametric analyses).
This analysis was the most suitable for obtaining the best model
of possible relationships and causal effects between variables
(identified using AIC – Akaike Information Criterion – values, a
measure a goodness of fit of a model).
a was set at 0.05. Means were Standard Error (7SE). Tests are
two-tailed.
3. Results
3.1. Dynamics of group cohesion
3.1.1. Relationship between group structure and group size
Group mean path length. For both hypotheses 1 (an individual
grooms all other group members equally) and 2 (an individual
does not groom all other group members equally), the group
mean path length was constant (1.00) whatever the group size
(Fig. 2a). For hypothesis 3 (an individual grooms a fixed number
of partners), the best-fit equation between the group mean path
length and the group size was logarithmic, whatever the number
of groomed partners (5 or 10) (N¼13 tests group sizes,
F1,11Z195.945, R2Z0.947, Po0.000001; Fig. 2a), showing that
path length and, thus connectivity between individuals highly
vary for small group size (until about 40 individuals per group)
whilst it is more constant for large group sizes.
Group density. For hypotheses 1 and 2, the density was
constant (1.00) whatever the group size (Fig. 2b). For hypothesis
3, the best fit equation between the group mean path length
and the group size was logarithmic, whatever the number of
groomed partners (5 or 10), (N¼13, F1,11Z96.611, R2Z0.898,
Po0.000001; Fig. 2b).
3.1.2. Relationship between group cohesion and group size
For hypothesis 1, group cohesion (defined as the ratio of the
number of cases where ‘‘one of the sub-groups containing three
individuals or less’’ divided by the total number of simulations)
was a constant (1.00; Fig. 2c). For hypothesis 2, the best fit
equation between group cohesion and group size was logarithmic
(N¼13, F1,11¼24.255, R2¼0.68, P¼0.0004; Fig. 2c). The decrease
in group cohesion was only seen to be 0.08570.004% in groups of
10–200 members. For hypothesis 3, whatever the number of
groomed partners (5 or 10), the best fit equation between the
group cohesion and the group size was exponential (N¼13,
F1,11Z327.615, R2Z0.967, Po0.000001; Fig. 2c).
These results suggest that when an individual has social bonds
with all other group members, whatever the quality of these
social bonds, group size has little influence on group cohesion and
consequently on group fission probability.
3.1.3. Relationship between group cohesion and group structure
Group mean path length. For hypotheses 1 and 2, group cohesion
was constant (1.00) whatever the group mean path length. For
hypothesis 3, the best fit equation between mean path length
and group cohesion is negatively linear, whatever the number of
groomed partners (5 or 10) (N¼13, F1,11Z219.38, R2Z0.952,
Po0.000001).
Group density. For hypotheses 1 and 2, the group cohesion was
constant (1.00) whatever the density. For hypothesis 3, the best fit
equation between density and group cohesion is logarithmic,
whatever the number of groomed partners (5 or 10) (N¼13,
F1,11Z471.13, R2Z0.977, Po0.000001).
The equation best describing the relationship between density
(and the group mean path length) and group cohesion for 5 partners
was similar to that calculated for 10 partners (Table 2). These results
suggest that group size seems to not directly influence group
cohesion, but it seems to indirectly do it through group structure,
and only in the case of hypothesis 3. However, this hypothesis needs
to be statistically tested a test allowing to describe direct and
indirect effects of variables (see. Section 3.2 in Results).
3.1.4. Comparison of observed group distribution and simulated
group distributions
For the observed data, and for each hypothesis, we calculated
the relative distribution of group sizes. The best fit equation for
the observed distribution was exponential (N¼6, F1,4¼21.29,
R2¼0.852, P¼0.008; Fig. 3). For hypotheses 1 and 2, there was no
relationship between relative frequency and group size (N¼11,
F1,9r1.74, R2r0.16, PZ0.215; Fig. 3). Their distributions signifi-
cantly differed from the observed distribution (Nobserved¼6,
Nsimulated¼11, z¼1.314, Pr0.021). For hypothesis 3, whatever the
number of groomed partners (5 or 10), the best fit was exponential
(N¼11, F1,9Z720.55, R2Z0.98, Po0.000001; Fig. 3). For 5 and 10
partners, the simulated distribution did not significantly differ from
the observed distribution (Nobserved¼6, Nsimulated¼11, z¼0.806,
P¼0.356 for 5 partners; Nobserved¼6, Nsimulated¼11, z¼1.134,
P¼0.086 for 10 partners; Fig. 3). However, P-values suggested that
there was less difference between the observed distribution and the
1
1.5
2
2.5
3
3.5
4
0 50 100 150 200
M
e
a
n
p
a
th
le
n
g
th

0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
D
e
n
si
ty
hypothesis 1
hypothesis 2
hypothesis 3 (5 partners)
hypothesis 3 (10 partners)
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
C
o
h
e
si
o
n
Group size
Fig. 2. Influence of group size on group structure ((a) on groupmean path length and
(b) on density) and on group cohesion (c) for each hypothesis. For hypotheses 1 and 2,
where an individual groomed all other group members, there is no relationship
between group size and group structure, nor between group size and group cohesion,
contrary to hypothesis 3 (where an individual groomed a fixed number of partners).
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Author's personal copy
distribution for 5 partners than for 10 partners. The ‘‘5 partners’’
condition seemed the best model to explain how an individual
attributed grooming time to group-mates.
3.2. How does an increase of within-group competition affect group
cohesion?
When within-group food competition increased, group members
had to spend more time foraging (and therefore to move from one
patch to another). As a consequence, grooming time decreased
(Berman et al., 2008; Lehmann et al., 2007; Sterck et al., 1997). For
hypotheses 1 and 2, a decrease in grooming time did not affect
group cohesion, whatever the group size (Mann–Whitney test:
UZ68.5, PZ0.418; see Table 3 for means), except when grooming
time is null (Mann–Whitney test: Uo0.001, Po0.00001, Table 3).
For hypothesis 3, a decrease in grooming time (represented by a
decrease in the number of partners) from 40% (3 partners) to 100%
(0 partner), influenced group cohesion (Mann–Whitney test: Ur46,
Pr0.048). This effect followed an exponential law (N¼13,
F1,11Z31, R2Z0.74, Pr0.0001). We can therefore conclude that
according to group size, group cohesion was affected non-linearly by
decreased grooming time and a decreased number of partners.
3.3. Causal relations between group size, grooming time, group
structure and group cohesion
The most likely causal relations among the different variables
previously tested were assessed using path analyses. For hypotheses
1 and 2, the most parsimonious causal model showed that grooming
time and group size did not affect group cohesion either directly or
indirectly (AIC¼18; df¼2; P¼1; Fig. 4a). These results confirmed
what we expected below. In the case of hypothesis 3, grooming time
and group size affected group cohesion but did so indirectly through
Table 2
Equations of relationship between mean path length and group cohesion and between density and group cohesion, for 5 and 10 partners. Tests showed that the equations
are similar for 5 and 10 partners. We used a comparison test for two linear regressions (transformation from logarithmic to linear for the relations between density and
number of partners). We first used a Snedecor test to compare variances of each distribution (df1¼df2¼11, fr0.75). Then, we tested if variable (b15 partners�b110 partners,
df¼22, Tr0.27) and variable (constant5 partners�constant10 partners, df¼22, Tr0.21) followed a Student law (df¼22).
N¼13 Group cohesion
5 partners 10 partners Similar functions
Function Constant b1 Function Constant b1
Group mean path length Linear 1.51 �0.45 Linear 1.48 �0.42 Yes Po0.05
Density Logarithmic 1.03 0.23 Logarithmic 1.05 0.26 Yes Po0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 50 100 150 200
re
la
tiv
e
f
re
q
u
e
n
cy
group size
observed data
hypothesis 1
hypothesis 2
hypothesis 3 (5 partners)
hypothesis 3 (10 partners)
Expon. (observed data)
Fig. 3. Relative distribution of group size for observed data and under each
hypothesis. The curve line is the best-fit model explaining the distribution of
observed data and corresponds to an exponential law.
Table 3
Influence of grooming time on mean group cohesion for each hypothesis.
Grooming time Initial time
Mean group cohesion
(% per day time) Decrease (%) Hypothesis 1 Hypothesis 2 Hypothesis 3
15 (initial time) �0 1.00 0.93 0.44
12 �20 1.00 0.93 0.33
8 �40 1.00 0.92 0.27
6 �60 1.00 0.93 0.22
3 �80 1.00 0.92 0.17
0 �100 0.00 0.00 0.00
Group size Grooming timeNumber of partners
Group density Mean path length
Cohesion
- 0.65
0.43
- 0.58
1
- 0.47
0.20 - 0.84
Group size Grooming time
Mean strength
of social relationships
Group density Mean path length
Cohesion
0.33
- 0.63
1
a. Hypotheses 1 and 2
a. Hypothesis 3
Fig. 4. Path diagram indicating causal relations between group size, grooming time
and group structure, i.e. mean strength of social relationships (a), number of partners
(b), density and group mean path length, and group cohesion. Rectangles indicated
observed variables. Arrows indicated presumed causal relations: solid arrows repre-
sented the best model based on AIC value, using AMOS5; dotted arrows represented
the relations we included in the analysis but not selected in the best model.
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Author's personal copy
the group structure and especially through the group mean path
length (AIC¼38; df¼2; P¼0.021; see Fig. 4b for details about
different influences).
4. Discussion
Time is a finite resource and grooming time, therefore, has to be
limited if an individual also wishes to forage, rest or move (Dunbar,
1992b; Lehmann et al., 2007; Majolo et al., 2008; Pollard and
Blumstein, 2008). Nevertheless, grooming may be the most impor-
tant behavior used by primates for maintaining social rela-
tionships (Henzi and Barrett, 1999; Schino, 2001). As a consequence,
the time an individual attributes to grooming will likely influence the
quality and/or the quantity of its social bonds (Dunbar, 1991, 1992b;
Lehmann et al., 2007). If the grooming time of an individual
decreases, the quality and/or the quantity of its social bonds should
also decrease. Moreover, the structure of social networks – the
number of partners per individual and the strength of these connec-
tions – may influence group fission probability and therefore group
size if the group splits (Koyama, 2003; Van Horn et al., 2007). As a
consequence, grooming time should logically influence group cohe-
sion and also thus group size. Our study suggests that the relation-
ship between grooming time, group size and group cohesion
depends on the way an individual distributes its grooming time to
its specific social partners. If an individual grooms all its conspecifics,
in an equal way or not, group size does not affect group structure and
then group cohesion. If grooming time is, however, distributed to a
specific number of partners, group structure and then group cohe-
sion are affected.
It has previously been shown that the time primates invest in
grooming increases with group size (Dunbar, 1991). Nevertheless,
Lehmann et al. (2007) suggested that when group size increases,
each individual should spend more time grooming until a certain
group size (about 40 individuals) for which individuals cannot
devote more time to maintain relationships with all group members
(Dunbar, 1992a; Lehmann et al., 2007; Schino et al., 2009). As a
consequence, group cohesion decreases and the group splits (Henzi
et al., 1997a, b). This hypothesis implies that individuals should be
expected to try to maintain social bonds with (i.e. to groom) all, or at
least most of, group members. Our results, by contrast, suggest that
if this assumption is applied (hypotheses 1 and 2 in our study),
group cohesion is not influenced by grooming time, even if time for
this activity decreases when group size increases. In the same way,
within-group competition is not predicted to affect group cohesion
when an individual grooms all its partners, but does have an effect
when an individual has a specific number of partners which it
grooms. This result may be explained by the fact that, as all indi-
viduals were linked to each other in one way or another (whatever
the quality/strength of these social bonds), group cohesion was high
and therefore the probability of the group splitting was low. This
result is not dependent on our model but directly influenced by
social network and sociality of individuals. For instance, Ame´ et al.
(2006) showed that when several shelters are proposed, cockroaches
(Blatta germanica) always aggregate together in only one shelter if
this one is able to host all individuals. Even if fission is possible, the
group cohesion of individuals does not lead them to split. Sueur
et al. (2010) also showed that in macaques, the highest the group
cohesion is (more contacts between individuals, less clustered),
the lowest the probability to split is. An alternative to grooming all
other group members is to groom a specific number of partners
(hypothesis 3). Several authors have shown that individuals do not
groom all available partners but rather a fixed number of individuals,
whatever the group size (Silk et al., 1999; Cords, 2001; Nakamachi
and Shizawa, 2003; Berman et al., 2008; Lin et al., 2008; Watts,
2000a,v). Under this hypothesis and as found in Majolo et al. (2008),
the grooming time an individual devotes to another is not directly
dependent on group size. However, the study of Majolo et al. (2008)
was the only one to propose this indirect link before our study and
this result needs to be checked. Our model suggests that group size,
grooming time and group cohesion are linked when an individual
grooms only a specific number of partners. In this scenario, group
cohesion decreases when group size increases and/or grooming time
decreases. Kudo and Dunbar (2001) also showed that structure of
small and large groups differ. Indeed, large groups seem to be more
sub-structured. This sub-grouping might be because animals delib-
erately invest their grooming in core coalition partners (Kudo and
Dunbar, 2001; Lehmann and Dunbar, 2009). Indeed, a theoretical
study on social network graphs showed a similar decrease of connec-
tivity according to nodes’ number (Wu, 2005). Moreover, we showed
that this relationship between group cohesion and group size is non-
linear: when group size exceeded 40 individuals, group cohesion was
almost null. This link was however indirect: as (1) group size and
grooming time directly influenced group structure, and (2) group
structure directly influenced group cohesion (in our study the inverse
probability that a group splits in two sub-groups), therefore (3) group
size and grooming time influenced group cohesion. To confirm this
result and to understand how the group structure evolved according
to both group size and the number of groomed partners, it would be
interesting to study groups at the same size but with a different
number of partners per individual.
Moreover, Lehmann et al. (2007) showed for instance that
female dispersion and sex ratio influenced grooming: species
with female philopatry spend more time grooming than species
with female dispersal. Even if social characteristics such as philopa-
tric sex or the sex ratio, that represent the variability of a social
structure, are already included in the different random networks we
tested with our model, we did not identify them. However, it would
be interesting to assess how these characteristics influence the
social network and then the group cohesion.
The distribution of the number of groups staying cohesive
according to group size was similar, in our model with a fixed
number of 5 partners, to that based on observed data. Previous
studies have confirmed that, on average, an individual preferentially
grooms 5 partners (Berman et al., 2008; Cords, 2001; Lin et al., 2008;
Nakamachi and Shizawa, 2003; Silk et al., 1999; Watts 2000a,b),
giving further support for our hypotheses. Even if an individual
grooms 5 preferred partners, it can still be groomed by more or less
individuals than 5 congeners itself, since grooming is not necessarily
reciprocal (Nakamachi and Shizawa, 2003; Schino, 2001; Silk et al.,
1999). This pattern has important consequences on group sociality:
it allows competition for grooming partners, attraction to high-
ranking individuals (Schino, 2001) and may allow the emergence of
phenomena such as the biological market (i.e., exchanges of com-
modities according to supply and demand; Fruteau et al., 2009). On
the other hand, if an individual grooms all its partners, this kind of
competition for partners could not emerge. Kinship may also
constrain the relationships of individuals: they will groom their
relatives. In these conditions, the two new sub-groups will be more
composed on kin related individuals (Chepko-Sade and Sade, 1979;
Van Horn et al., 2007). This influence did not affect, however, our
results, since groups having an influence of kinship can also be
represented using random networks as the ones we used in this
study (Flack et al., 2006; McCowan et al., 2008; Sueur and Petit,
2008; Kasper and Voelkl, 2009; Ramos-Fernandez et al., 2009).
Grooming a specific and low number of partners could also be of
interest when managing time (individuals do not need to change
their grooming time when group size increases) but could also have
implications when considering cognitive capacities. Several authors
(Dunbar 1992a, 1996; Lehmann et al., 2007; Stevens et al., 2005)
have suggested that there is a relationship between the cognitive
capacities (measured as the neocortex ratio) and the number of
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Author's personal copy
relationships an individual can keep track of. Remembering the
grooming relationships for 5 partners would be the easiest solution
for an individual, and would be a more parsimonious pro-
cess than having to remember its ties with all group members.
Moreover, grooming 5 partners rather than all group members may
favor sub-grouping patterns and reduce within-group conflict by
regrouping individuals with similar social/physiological affinities
(Aureli and Schaffner 2007; Couzin and Laidre, 2009; Ramos-
Fernandez et al., 2006).
We believe that a relationship may exist between group size and
grooming time but that this relationship would probably be indirect:
an increase in group size could result in more food competition
between group members (Majolo et al., 2008). As a consequence, an
individual would have to increase its foraging time and therefore
decrease its grooming time. The consequence is not due to a higher
number of partners, but to lower food availability. We did not
directly test the relation between food competition and group size in
our model, but several studies have already supported this hypoth-
esis (Berman et al., 2008; Dittus, 1988). Moreover, group size and
grooming time, elements that can be influenced by ecological factors
such as food competition or predation (Lehmann et al., 2007; Majolo
et al., 2008; Pollard and Blumstein, 2008), did not directly influence
the probability that a group would split. Social relationships directly
influenced group fission probability. And these social relationships
are resultants of the combined influence of group size and food
competition, but also of other factors as internal or structural
constraints (Thierry et al., 2004).
This study was based on grooming interactions but we suggested
that similar results may be obtained for any positive interaction such
as proximities, contacts or frequency of lips-making for instance.
Even if our model did not test all factors having a potential influence
on the structure of social relationships, it does show that the key
characteristic of group cohesion and stability is group structure (Wey
et al., 2008). It is interesting to note that the probability to find large
group sizes decreases exponentially with group size in the context of
stable groups after irreversible fission but also in fission–fusion
populations (Couzin and Laidre, 2009). We do not think that a group
splits irreversibly in one event as it does in our model. This irre-
versible fission might be long, from several months to several years
(Chepko-Sade and Sade, 1979; Okamoto and Matsumura, 2001; Van
Horn et al., 2007). We still lack of data about dynamic of fission. It
should be interesting to conduct more studies about dynamical
processes allowing a group to split. General principles seem to
underlie rules of group cohesion at different time scales. All factors,
whether social or ecological, seem to influence group cohesion
through its structure. In the end, group fission probability may not
depend on the sum of individual decision-making based on physio-
logical states and/or on their complex interactions (Schino, 2001;
Tomasello and Call, 1997), but may simply depend on the properties
of the social structure, as observed in several self-organized systems
(Camazine et al., 2001; Couzin and Krause, 2003).
Acknowledgements
We are grateful to J. Munro for the language editing and
N. Poulain, biostatistician at the DEPE, IPHC, for his help on analyses.
This work was supported by the Wallonia Brussels International, the
Belgian National Funds for Scientific Research, the Franco-American
Commission, the Alsace Region and the Fyssen Foundation.
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