Discrete Hierarchical Organization of Social Group Sizes

Abstract

The 'social brain hypothesis' for the evolution of large brains in primates has led to evidence for the coevolution of neocortical size and social group sizes, suggesting that there is a cognitive constraint on group size that depends, in some way, on the volume of neural material available for processing and synthesizing information on social relationships. More recently, work on both human and non-human primates has suggested that social groups are often hierarchically structured. We combine data on human grouping patterns in a comprehensive and systematic study. Using fractal analysis, we identify, with high statistical confidence, a discrete hierarchy of group sizes with a preferred scaling ratio close to three: rather than a single or a continuous spectrum of group sizes, humans spontaneously form groups of preferred sizes organized in a geometrical series approximating 3-5, 9-15, 30-45, etc. Such discrete scale invariance could be related to that identified in signatures of herding behaviour in financial markets and might reflect a hierarchical processing of social nearness by human brains.

Full text

doi: 10.1098/rspb.2004.2970
, 439-444272 2005 Proc. R. Soc. B
W.-X. Zhou, D. Sornette, R. A. Hill and R. I. M. Dunbar
Discrete hierarchical organization of social group sizes
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Proc. R. Soc. B (2005) 272, 439–444
doi:10.1098/rspb.2004.2970
Published online 17 February 2005
Discrete hierarchical organization of social group
sizes
W.-X. Zhou
1,2
, D. Sornette
2,3,4
, R. A. Hill
5
and R. I. M. Dunbar
6

1
State Key Laboratory of Chemical Reaction Engineering, East China University of Science and Technology, Shanghai 200237,
China
2,3
Institute of Geophysics and Planetary Physics, and Department of Earth and Space Sciences, University of California,
Los Angeles, CA 90095, USA
4
Laboratoire de Physique de la Matie
`
re Condense
´
e, CNRS UMR 6622 and Universite
´
de Nice-Sophia Antipolis,
06108 Nice Cedex 2, France
5
Evolutionary Anthropology Research Group, Department of Anthropology, University of Durham, 43 Old Elvet,
Durham DH1 3HN, UK
6
British Academy Centenary Project, School of Biological Sciences, University of Liverpool, Crown Street,
Liverpool L69 7ZB, UK
The ‘social brain hypothesis’ for the evolution of large brains in primates has led to evidence for the coevolu-
tion of neocortical size and social group sizes, suggesting that there is a cognitive constraint on group size
that depends, in some way, on the volume of neural material available for processing and synthesizing infor-
mation on social relationships. More recently, work on both human and non-human primates has suggested
that social groups are often hierarchically structured. We combine data on human grouping patterns in a
comprehensive and systematic study. Using fractal analysis, we identify, with high statistical confidence, a
discrete hierarchy of group sizes with a preferred scaling ratio close to three: rather than a single or a continu-
ous spectrum of group sizes, humans spontaneously form groups of preferred sizes organized in a geometri-
cal series approximating 3–5, 9–15, 30–45, etc. Such discrete scale invariance could be related to that
identified in signatures of herding behaviour in financial markets and might reflect a hierarchical processing
of social nearness by human brains.
Keywords: social brain hypothesis; social group size; log-periodicity; fractal analysis
1. INTRODUCTION
Attempts to understand the grouping patterns of humans
have a long history in both sociology (Coleman 1964) and
social anthropology (Kottak 1991; Scupin 1992). While
these approaches have been largely sociological in focus,
attempts to understand grouping patterns in non-human
primates have had a largely ecological focus (see Dunbar
1988). However, there has been recent interest in the
extent to which group size and grouping patterns in
primates might be constrained by cognitive factors
(Dunbar 1992, 1998). The latter interests arise out of what
has become known as the ‘social brain hypothesis’.
The social brain hypothesis (Byrne & Whiten 1988;
Barton & Dunbar 1997) argues that the evolution of
primate brains was driven by the need to coordinate and
manage increasingly large social groups. Since the stability
of these groupings is based on intimate knowledge of other
individuals and the ability to use this knowledge to manage
social relationships effectively, the computational capacity
of the brain (presumed to be broadly a function of its size)
is assumed to impose a species-specific limit on group size.
Attempts to increase group size beyond this threshold must
inevitably result in reduced social stability and, ultimately,
group fission. Dunbar (1992, 1998; Joffe & Dunbar 1997;
also Sawaguchi & Kudo 1990) showed that there is a log-
linear relationship between social group size and relative
neocortex volume in primates, and argued that this
relationship reflected the computational capacity that any
given species could bring to bear on its social relationships.
Extrapolating these findings to humans led to the predic-
tion that humans had a cognitive limit of approximately
150 on the average number of individuals with whom
coherent personal relationships could be maintained (Dun-
bar 1993). Evidence to support this prediction has come
from a number of ethnographic and sociological sources
(Dunbar 1993). The fact that these relationships are not
simply a matter of memory for individuals but, rather, of
integrating and managing information about the constantly
changing relationships between individuals within a group,
is indicated by the fact that relative neocortex size corre-
lates with a number of core aspects of social behaviour and
socialization in primates (Byrne 1995; Pawlowski et al.
1998; Joffe 1997; Lewis 2000; Byrne & Corp 2004).
It has, however, always been recognized that both
human and non-human primate groups are internally
highly structured (e.g. Dunbar 1988). Further analyses
(Kudo & Dunbar 2001) have indicated that at least one
level of structuring (the grooming clique) also correlates
with neocortex size. While the significance of these tiered
groupings is not always apparent, there is strong prima facie
evidence to suggest that human social groups (like those of
other primates) consist of a series of sub-groupings

Author for correspondence (rimd@liverpool.ac.uk).
Received 29 April 2004
Accepted 29 September 2004
439
#
2005 The Royal Society
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arranged in a hierarchically inclusive sequence (Hill &
Dunbar 2003).
In this sequence, the core social grouping is the support
clique, defined as the set of individuals from whom the
respondent would seek personal advice or help in times of
severe emotional and financial distress; its mean size is
typically 3–5 individuals (Dunbar & Spoor 1995). Above
this may be discerned a grouping of 12–20 individuals
(often referred to as a sympathy group) that characteristi-
cally consists of all the individuals with whom one has
special ties; these individuals are typically contacted at least
once per month (Dunbar & Spoor 1995; Hill & Dunbar
2003). The ethnographic data on hunter-gatherer societies
(summarized in Dunbar 1993) point to a grouping of
30–50 individuals as the typical size of overnight camps
(sometimes referred to as bands); these groupings are often
unstable, but their membership is always drawn from the
same set of individuals, who typically number ca. 150 indi-
viduals. This last grouping is often identified in small-scale
traditional societies as the clan or regional group. Beyond
these, at least two larger-scale groupings have been
identified in the ethnographic literature: the megaband of
ca. 500 individuals and the tribe (a linguistic unit,
commonly of 1000–2000 individuals) (Dunbar 1993).
In this paper, we provide the first systematic analysis of
human grouping patterns, using data collated from the
literature. Using spectral analysis, we show that there is a
consistent pattern in the size of these groupings and, more
importantly, that successive groupings in the hierarchy
have a constant ratio.
2. MATERIAL AND METHODS
There is no universally accepted procedure for analysing human
social groups, and all methods attempting to identify group sizes
suffer from at least some sources of bias (small sample size, large
inter-individual variability or differences in the criteria used to
include individuals). Our strategy is to include all reasonable data
and attempt to extract useful signals above the noise level by a
careful analysis of the global dataset. We therefore searched the
sociological and other literatures for quantitative data on social
group and social network sizes in humans. For these purposes, we
sought studies that provided quantitative data on the size of indivi-
duals’ social networks, irrespective of how the social network itself
was defined.
Most such studies focus on a particular kind of network (among
those defined above in x 1). In addition to the data listed in
Dunbar (1993), Dunbar & Spoor (1995) and Kudo & Dunbar
(2001), we add the following data. The USA 1998 General Social
Survey reports a mean size of 3.3 for support cliques in the USA
(Marsden 2003). The mean sizes of sympathy groups are reported
by Buys (1992) to be 14.0 in Egypt, 15.1 in Malaysia, 13.5 in
Mexico, 13.8 in South Africa and 10.2 in the USA (Latkin et al.
1995). In separate samples in The Netherlands, they were repor-
ted to be 15.0 in 1995 (Kef 1997; Kef et al. 2000), 15.0 in 1992,
14.3 in 1992–1993, 14.8 in 1995–1996 and 14.2 in 1998–1999
(van Tilburg & van Groenou 2002), finally, Adams et al. (2002)
reported them to be 14.4 in Mali (West Africa). Although a num-
ber of these studies have been carried out in the same country, we
have considered each study to be an independent sample since
they involve different datasets; nevertheless, averaging
The Netherlands samples and treating them as a single data point
does not alter the conclusions drawn.
Only one study sought to estimate the size of successive social
groupings for individual subjects (Hill & Dunbar 2003). These
data were obtained from an analysis of Christmas card
distribution lists, in which 42 UK-domiciled subjects logged the
identities of all individuals in the households to which cards were
sent and their relationships to these individuals. Participants were
asked both to list everyone in the household to which they were
sending a card and to state the quality of their relationship with
each individual (using two metrics: how often they contacted the
individual, and the emotional intensity of the relationship scored
on a 0–10 Likert-type scale: for details, see Hill & Dunbar
(2003)). Because this study uniquely provides data on the differ-
ent grouping levels of which any one individual is a member, we
treat these data separately from the census data obtained from the
literature search.
3. RESULTS
We begin by analysing the data on groupings reported in
the social networks literature. (The Christmas card distri-
bution data will be dealt with separately: see below.)
Figure 1 plots the sizes of the different grouping levels
identified in the various studies.
We begin with a qualitative analysis of the data in figure
1, using the groupings that have conventionally been
defined (see x 1). First, we denote S
1
as the mean support
clique size, S
2
the mean sympathy group size, S
3
the mean
band size, S
4
the mean community group size, and S
5
and
S
6
the mean sizes of mega-bands and large tribes, respect-
ively. Averaging across these grouping levels, the data give
mean values of S
0
¼ 1 (individual or ego), S
1
¼ 4:6,
S
2
¼ 14:3, S
3
¼ 42:6, S
4
¼ 132:5, S
5
¼ 566:6 and S
6
¼
1728. To determine the possible existence of a
discrete hierarchy, we construct the series of ratios S
i
/S
i1
10
0
10
1
10
2
10
3
10
4
0
5
10
15
20
25
30
network sizes
references
Figure 1. Presentation of our dataset of 61 group sizes. The
ordinate is an arbitrary ordering of data sources and the
abscissa gives the group sizes reported in each source. The
symbols refer to the classification used in each of the studies:
circles (support cliques), triangles (sympathy groups),
diamonds (bands), stars (cognitive groups), and squares
(small and large tribes). This classification is not used in our
systematic analysis summarized in the other figures, to avoid
any bias.
440 W.-X. Zhou and others Social group size organization
Proc. R. Soc. B (2005)
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of successive mean sizes:
S
i
=S
i 1
¼ 4:58, 3:12, 2:98, 3:11, 4:28, 3:05,
for i ¼ 1, ...,6: ð3:1Þ
This suggests that humans form groups according to a
discrete hierarchy with a preferred scaling ratio between 3
and 4 (the mean of S
i
/S
i1
is 3.52).
To avoid any biases that might be present in
the published census data, we next undertake a more
systematic analysis that uses all the available data rather
than just their means. The sample in figure 1 has 61 group-
ing clusters (including the ego) with estimates of mean size
s
i
available for i ¼ 1,2, ...,61 clusters. We consider this
sample to be a realization of a distribution whose sample
estimation can be written as:
fsðÞ¼
X
61
i ¼1
d s  s
i
ðÞ, ð3:2Þ
where d is Dirac’s delta function. Figure 2 shows the prob-
ability density function f(s) obtained by applying a Gaus-
sian kernel estimation approach (Silverman 1986).
Our challenge is to extract a possible periodicity in this
function in the ln(s) variable, if any. If the grouping clusters
form a series of harmonics, the harmonics will have a con-
stant ratio, and we would expect a periodic oscillation of
f(s) expressed in the variable ln(s) (known as its ‘log-period-
icity’; Sornette 1998).
Standard spectral analysis applied to f(s) is dominated
by the trend seen in figure 2, with a peak at a very low log-
frequency corresponding to the whole range of the group
sizes. We thus turn to generalized q-analysis or (H, q )-
analysis (Zhou & Sornette 2002a), which has been shown
to be very sensitive and efficient for such tasks. The q-
analysis is a natural tool to describe discrete scale invar-
iance (DSI) in fractals and multifractals (Erzan 1997;
Erzan & Eckmann 1997). The (H, q )-analysis consists in
constructing the (H, q )-derivative
D
H
q
fs
ðÞ
¼
fsðÞfqsðÞ
1  qðÞs½
H
: ð3:3Þ
Introducing an exponent H different from 1 allows us to
detrend f(s) in an adaptive way (that is, detrend it with dif-
ferent values of [(1  q )s]
H
at different s values). Note that
the limit H ¼ 1 and q ! 1 retrieves the standard definition
of the derivative of f. A value of q strictly less than 1 makes it
possible to enhance possible discrete scale structures in the
data. To keep a good resolution, we work with
0:65 6 q 6 0:95, because smaller values of q require more
data for small values of s. To put more weight on the small
group sizes (which are probably more reliable since they are
obtained by conducting general surveys in larger represen-
tative populations), we use 0:5 6 H 6 0:9. A typical (H,
q)-derivative with H ¼ 0:5 and q ¼ 0:8 is illustrated in a
semi-log plot in figure 3.
We then use a Lomb periodogram analysis (Press et al.
1996) to extract the log-periodicity in f(s). Figure 4
presents the normalized Lomb periodograms of D
H
q
fsðÞfor
different pairs of (H, q ) with 0:5 6 H 6 0:9 and
0:65 6 q 6 0:95. This figure illustrates the robustness of
our result. For the specific values H ¼ 0:5 and q ¼ 0:8
shown in figure 4, the highest peak is at x
1
¼ 5:40 with
height P
N
¼ 8:67. The preferred scaling ratio is thus
k ¼ exp 2p
=
x
1
ðÞ3:2. The confidence level is 0.993
under the null hypothesis of white noise (Press et al. 1996).
If the underlying noise decorating the log-periodic struc-
ture is correlated with a Hurst index of 0.6, the confidence
level decreases to 0.99; if the Hurst index is 0.7 (which cor-
responds to an unreasonably large noise correlation), the
confidence level falls to 0.85 (Zhou & Sornette 2002b).
The Lomb periodograms also exhibit a second peak at
x
2
¼ 9:80 with height P
N
¼ 5:48. This can be interpreted
as the second harmonic component x
2
 2x
1
of the funda-
mental component at x
1
¼ 5:40. The amplitude ratio of
the fundamental and the harmonic is 1.26. The coexistence
of the two peaks at x
1
and x
2
 2x
1
strengthens the stat-
istical significance of a log-periodic structure. To see this,
we constructed 10
4
synthetic sets of 61 values uniformly
distributed in the variable ln(s) within the interval [0,
ln(2000)]. By construction, these 10
4
sets, which are
exactly of the same size as our data and span the same
interval, do not have log-periodicity and thus have no
characteristic sizes. We then applied the same procedure as
10
–1
10
0
10
1
10
2
10
3
10
4
0
0.1
0.2
0.3
0.4
0.5
0.6
s
f (s)
Figure 2. Probability density function f(s) of size s estimated
with a Gaussian kernel estimator in the variable ln(s) with a
bandwidth h ¼ 0:14. Varying h by 100% does not change f(s)
significantly.
10
0
10
1
10
2
10
3
10
4
–0.20
–0.15
–0.10
–0.05
0
0.05
0.10
0.15
0.20
s
D
q
f (s)
H
Figure 3. Typical (H, q)-derivative D
H
q
f(s) of the probability
density f(s) as a function of size s with H ¼ 0:5 and q ¼ 0:8.
Social group size organization W.-X. Zhou and others 441
Proc. R. Soc. B (2005)
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for the real dataset to these synthetic datasets and obtained
10
4
corresponding Lomb periodograms. Finally, we per-
formed the following tests on these Lomb periodograms:
find the highest Lomb peak (x, P
N
). If P
N
> 8:5, check if
there is at least another peak at 2x
^
1 with its P
N
larger
than 5.5. A total of 238 sets among the 10
4
passed the test,
suggesting a probability that our signal results from chance
equal to 0.024. The probability that there are at least two
peaks (one in 4:9 < x < 5:9 with P
N
> 8:5 and the other in
9:5 < x < 11:5 with P
N
> 5:5) is found equal to 77/10
4
,
giving another estimation of 0.992 for the statistical confi-
dence of our results.
Another metric consists in quantifying the area below the
significant peaks found in the Lomb periodogram of our
data and comparing them with those in the synthetic sets.
We count the area of the main peak of the Lomb period-
ogram at x and add to it the areas of its harmonics whose
local maxima fall in the intervals [(k ð1=5ÞÞx,
ðk þð1=5ÞÞx] for k ¼ 2,3, ..., around all its harmonics.
The area associated with a peak is defined as the region
around a local maximum delimited by the two closest local
minima bracketing it. The fraction of synthetic sets which
give an area thus defined larger than the value found for the
real data is 6–7%, depending on the specific values H and q
used in the analysis.
We applied the same analysis to individual social
networks based upon the exchange of Christmas cards
(Hill & Dunbar 2003). This study indicated that contem-
porary social networks might be differentiated based on the
frequency of contact between individuals, but that both
‘passive’ and ‘active’ factors may determine contact
frequency. Controlling for the passive factors (distance
apart, and whether the contact was overseas or a work
colleague) allowed the hierarchical network structure to be
examined based on the residual (active) contact frequency.
Starting from the residual contact frequencies, we
constructed their (H, q )-derivative with respect to the num-
ber of people contacted for each individual, obtained the
Lomb spectrum of the (H, q )-derivative and then averaged
them over the 42 individuals in the sample (figure 5). The
very strong peak at x ¼ 5:2 is consistent with the previous
results with a preferred scaling ratio from the expression k
¼ exp 2p=x
1
ðÞ3:3 (Sornette 1998) for the smaller
grouping levels in this study (i.e. group sizes below 150).
In summary, all these tests suggest that the evidence in
support of our hypothesis is significantly unlikely to result
from chance, but rather reflects the fact that human group
sizes are naturally structured into a discrete hierarchy with
a preferred scaling ratio close to 3.
4. DISCUSSION
Collating a variety of measures collected under a wide
range of conditions and in different countries, we have
documented a coherent set of characteristic group sizes
organized according to a geometric series with a preferred
scaling ratio close to three. The fact that the signature of
this scaling ratio comes through so strongly despite the fact
that the data derive from a variety of different small- and
large-scale societies suggests that it is very much a universal
feature. Were it the case that scaling ratios differed between
societies, pooling data would have tended to obscure any
relationships that might have been present.
Indeed, it turns out that similar hierarchies can be found
in other types of human organizations, although the
consistency of the patterning has not previously attracted
comment. Of these, the military probably provides the best
examples. In the land armies of many countries, one
typically finds sections (or squads) of ca. 10–15 soldiers,
platoons (of three sections, ca. 35), companies (3–4
platoons, ca. 120–150), battalions (usually 3–4 companies
plus support units, ca. 550–800), regiments (or brigades)
(usually three battalions, plus support; 2500þ), divisions
(usually three regiments) and corps (2–3 divisions). This
gives a series with a multiplying factor from one level to the
next close to three. Could it be that the army’s structures
have evolved to mimic the natural hierarchical groupings of
everyday social structures, thereby optimizing the cognitive
processing of within-group interactions?
0 5 10 15 20 25 30 35 40
0
2
4
6
8
10
12
ω
P
N
( )
ω
Figure 4. Normalized Lomb periodograms P
N
(x)asa
function of angular log-frequency x of the (H, q )-derivative
D
H
q
f(s) for different pairs of (H, q ) with 0:5 6 H 6 0:9 and
0:65 6 q 6 0:95.
0 10 20 30 40 50
0.5
10
1.5
20
2.5
30
3.5
40
4.5
ω
P
N
( )
ω
Figure 5. Average Lomb periodogram P
N
(x) of the (H, q )-
derivative D
H
q
f(s) with respect to the number of receivers of
the residual contact frequency for each individual in the
Christmas card experiment, as a function of the angular log-
frequency x of the (H, q )-derivative, over the 42 individuals
and different pairs of (H, q ) with 1 6 H 6 1 and
0:80 6 q 6 0:95.
442 W.-X. Zhou and others Social group size organization
Proc. R. Soc. B (2005)
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Stock market behaviour provides another example of the
same kind of phenomenon (and one that we happen to
have investigated). The existence of a discrete hierarchy of
group sizes may provide a key ingredient in rationalizing
the reported existence of DSI in financial time series in so-
called ‘bubble’ regimes characterized by strong herding
behaviours between investors (Sornette 2003). Johansen et
al. (1999, 2000) have proposed a model to explain the
observed DSI in stock market prices as resulting from a dis-
crete hierarchy in the interactions between investors.
Recent analyses of DSI in market regimes with a strong
herding component have also identified the presence of a
strong harmonic at 2x, similar to the findings reported here
( Johansen & Sornette 1999; Sornette & Zhou 2002).
Strong herding behaviour occurs only when groups of
investors coordinate their buy and sell orders; the coordi-
nated buy and sell orders that occur during a strong herd-
ing market phase thus expresses, better than at any other
time, the natural inner structure of the community of tra-
ders. By contrast, herding is absent when investors disagree
on what will be the next market move; as a result, the aggre-
gate market orders do not express the inner hierarchical
structure of the community.
The fact that DSI is found only during stock market
regimes associated with a strong herding behaviour sug-
gests that it may reflect the fact that a discrete hierarchy of
naturally occurring group sizes characterizes human inter-
actions whether they be hunter-gatherers or traders. The
findings reported here suggest that this discrete hierarchy
may have its origins in the fundamental organization of any
social structure and be deeply rooted within the cognitive
processing abilities of human brains.
When dealing with discrete hierarchies, it may be impor-
tant to distinguish between the specific group sizes and
their successive ratios. It may be that the absolute values of
the group sizes are less important than the ratios between
successive group sizes. If the ratio of group sizes is inter-
preted as a fractal dimension (specifically, the ratio is
related to the imaginary part of a fractal dimension: see
Sornette (1998) and references therein), this would imply
that, depending on the social context, the minimum
‘nucleation’ size (in the range 3–5 in previous examples)
may vary, but the ratio (close to three) might be universal.
The fundamental question, then, is to determine the origin
of this discrete hierarchy. At present, there is no obvious
reason why a ratio of three should be important.
Equally, however, we have little real understanding of
what mechanisms might limit the nucleation point to a
particular value. We do not even know, for example,
whether the constraint is a cognitive one (e.g. memory for
individual identities versus capacity to manage information
about relationships); or a time budgeting one (how much
time has to be invested in interaction with an individual to
create a bond of a particular strength, and then how many
such bonds can be fitted into a given time-scale). Nor do
we know much about how larger-scale groupings are built
up out of smaller ones. A hierarchical structure could, for
example, be built up by each individual interacting with,
say, three new individuals in an expanding network, or it
might be the result of rather discrete small subgroups held
together through a subset of individuals who act as ‘weak
links’ in the small-worlds sense—although there is some
evidence for the latter in respect of both primate social
groups (Kudo & Dunbar 2001) and at least some aspects of
human behaviour (Stiller et al. 2004). Considerable
additional work will need to be done on both these compo-
nents if we are to understand why these constraints on
human grouping patterns exist and exactly what their sig-
nificance might be.
Research by R.A.H. and R.I.M.D. was funded by the ESRC’s
Research Centre in Economic Learning and Social Evolution
(ELSE). R.I.M.D.’s research is supported by the British Acad-
emy Centenary Project and by a British Academy Research
Professorship.
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