arXiv:nlin/0310001v1 [nlin.AO] 1 Oct 2003
Universal Mechanisms in the Growth of Voluntary
F L1,2, L´ ı A. N A3and H. E S4
1Department of Sociology, Stockholm University, S-106 91 Stockholm, Sweden
2Swedish Institute for Infectious Disease Control, SE-171 82 Solna, Sweden
3Dept. of Chemical Engineering, Northwestern University, Evanston, IL 60208, USA
4Center for Polymer Studies & Dept. of Physics, Boston University, Boston, MA 02215,USA
PACS. 87.23.Ge – Dynamics of social systems.
PACS. 89.75.-k – Complex systems.
Abstract. – We analyze the growth statistics of Swedish trade unions and find a universal
functional form for the probability distribution of growth rates of union size, and a power law
dependence of the standard deviation of this distribution on the number of members of the
union. We also find that the typical size and the typical number of local chapters scales as a
power law of the union size. Intriguingly, our results are similar to results reported for other
human organizations of a quite different nature. Our findings are consistent with the possibility
that universal mechanisms may exist governing the growth patterns of human organizations.
healthy democracies and healthy market economies. Several studies suggest that the reason
for this failure may be that well-functioning societies  are fostered by the existence of
a dense web of voluntary organizations which facilitate the creation of a “social capital of
trust” among the members of the society. Indeed, many studies support the importance of
institutional settings for the maintenance of healthy societies. A telling example is a study of
the functioning of democratic institutions in the 27 regions of Italy, suggesting a correlation
between a dense web of small voluntary organizations and a dynamic civil society .
Specifically, in regions where people are embedded in a rich environment of decentralized
endeavors of mutual benefit . It would appear then that voluntary organizations promote
the creation of a “social capital of trust” that helps serve the functions of (a) overcoming
the anonymity of life in large societies, which may breed “free-rider” behavior , and (b)
overcoming the difficulties in partitioning the exploitation of public resource , such as use
of public water resources, limitation of air emissions, or determination of fishing quotas.
Because of their societal importance, research on voluntary organizations has been very
active, including many different aspects such as (i) competition between voluntary organiza-
tions and other organizations , (ii) the impact of social networks in membership recruit-
The developments of the last decade and a half in the former Eastern
to overcome “free-rider” behavior . Despite this research activity, one area that has not
Here, we use concepts and methods of statistical physics [9,10] to quantify the growth of
voluntaryorganizations. Specifically,wetestthepossibility (i)thatthestatisticalpropertiesof
fluctuations in the output of a system yield information regarding the underlying processes
responsible for the observed macroscopic behavior [9,10], and (ii) that the precise details
of the interaction between the subunits comprising the system may play virtually no role
in determining the macroscopic behavior of the system . A striking example is the
behavior of response functions in the vicinity of the liquid-gas critical point (the temperature
and density at which liquid and gas become indistinguishable fluids) [9,10]. Close to their
respective critical points, very different liquids—such as water, a polar molecule that forms
hydrogen bonds, and argon, an inert atom—become extremely sensitive to disturbances
yet their responses to those disturbances have identical spatial and temporal scale-invariant
of the growth statistics of a range of different Swedish voluntary organizations—including
trade unions , temperance movements , free churches , and the social democratic
party —during the 50 yr period 1890–1940. We concentrate our study on trade unions
for three reasons. First, there are over 10,000 local chapters (or sections) comprising 60 trade
unions, while there are only 5 free churches, 5 temperance movements and 1 political party
in the database. The larger number of trade unions enables us to make a more significant
statistical analysis of the growth process for the organizations. Second, a number of studies
 indicate that Swedish trade unions played a very important role in the democratization
process in Sweden, making their study particularly relevant. Third, unions are a particularly
We analyze a database  that provides a detailed resource for the study
x = log(Union size)
3.0 3.64.2 4.8 5.4
Fig. 1 – Historical data for total number of union members in Sweden. (a) Time evolution of Swedish
population, workforce, and unionized work forcefor the period1900–1940. In the subsequentanalysis,
we deflate the number of members of a union—its size—by the population growth, to remove the effect
of population growth on the analysis. (b) Probability density function of the size Swedish trade unions
for all years and all unions. The distribution can be well approximated by a log-normal fit (full line in
figure), which suggests, according to Gibrat’s theory , that the growth is a random multiplicative
process. Forsubsequentanalysis, we partition the data into 4bins accordingto union size, as illustrated
by the figure.
L, A, S: G V O
−0.6 −0.4 −0.2
0.0 0.2 0.40.6
−6.0−4.0 −2.0 0.0 2.04.06.0
Scaled growth rate
Scaled probability density
Fig. 2 – (a) Probability density function for two different size bins. The annual growth rates of the
unions are “normalized” by subtracting the average growth rate for all trade unions in the specific year.
For clarity, we plot only the results for bins 1 and 4. The figure reveals two interesting points. First, the
width of the distributionof growth rates decreaseswith the size of the union—e.g., the width is smaller
for bin 4 which comprises the largest unions. Second, the distribution of growth rates does not appear
to be Gaussian. This second resultis intriguing because each union comprisesseveral local chapters (or
sections), soone mightexpectthat the central limittheorem applies—leadingto a Gaussian distribution
of growth rates. (b) The non-Gaussian character of the distributions is clearer in this log-linear plot
showing the distributions of union growth rates for all 4 bins scaled by the standard deviations of the
corresponding distributions. The figure also suggests that the functional form of the distribution is
independent of the size of the union.
interesting type of voluntary organization as the decision to join a union is not an easy one:
A prospective new member would ideally balance (i) the benefit of avoiding social pressure
from fellow workers and (ii) hoped-for long-term benefits of membership, against (i) the
investment of time and money into the organization, and (ii) the risk of losing the job or of
being discriminated against.
We start by defining the annual growth rate—that is to say, the size fluctuation—of a
?S(t + 1)
g(t) ≡ log
where S(t) and S(t + 1) are the number of members of a given union in the years t and t + 1,
respectively, deflated by Sweden’s total population. We find that the statistical properties
of the growth rate g depend on S; the magnitude of the fluctuations g will decrease with S
since large organizations have smaller relative fluctuations. We partition the trade unions
into bins according to their number of members—the union size (Fig. 1). Figure 2(a) suggests
that the conditional probability density, p(g|S), has the same functional form, with different
widths, for all S. To test whether p(g|S) has a functional form independent of union size, we
plot the scaled quantities: σ(S)p(g/σ(S)|S) versus g/σ(S). Figure 2(b) shows that the scaled
distributions “collapse” onto a single curve [9,10], consistent with the functional form
We next calculate the standard deviation σ(S) of the distribution of growth rates as a
Standard deviation of growth rates
Scaled growth rate
Fig. 3 – (a) Dependence of the standard deviation of the distribution of growth rates on number of
membersofthe union. The factthat powerlawdependenceofthe standarddeviationonsizeholdsover
three orders of magnitude—from unions with 40 members to unions with 40,000 members—suggests
thatthisfindingisnotspurious. Thestraightlineisapowerlawfittotheregion40 ≤ S ≤ 40,000yielding
an exponent estimate β = 0.19 ± 0.05. (b) Functional form of the distribution of growth rates. We plot
the cumulative distributionof the scaled growthrates fromall bins. The cumulative distribution, which
yields the probability of finding values larger than a certain threshold, is obtained by integrating the
probability distribution function between the threshold value and infinity. The figure confirms that the
distribution is not Gaussian, but may be consistent with either an exponential or stretched exponential
dependence in the tails.
function of S. Figure 3(a) demonstrates that σ(S) decays as a power law
σ(S) ∼ S−β,
with β = 0.19± 0.05.
We next addressthe question of how to interpretour empiricalresults. We first note that a
union is comprisedofseverallocalchapters, spreadaroundthecountry. Areasonablezeroth-
a given trade union will grow independently; so the growth of the size of each union as the
sum of the independent growth of local chapters with different sizes. In a recently-proposed
model [17,18], the subunits comprising the organization grow by an independent, Gaussian-
distributed, random multiplicative process with variance v2. Existing subunits are absorbed
when they become smaller than a “minimum size”, which is a function of the activity they
perform. Subunits can split into two new subunits if they grow by more than the minimum
size for a new subunit to form. The model predicts β = v/[2(v+ w)], where w is the width of
the distribution of minimum sizes .
Internal organization of the unions. –
is natural to inquire what are the statistical properties of the set of local chapters comprising
a given union [17,19,20]. To this end, we quantify how the internal structure of a trade union
depends on its size. Specifically, we calculate the conditional probability density ρ(ξ|S) to
find a local chapterwith ξ members in a union with S members; Fig. 4(a). The model predicts
that ρ(ξ|S) obeys the scaling form 
As trade unions have a complex internal structure, it
L, A, S: G V O
Scaled local size
Scaled probability density
Fig. 4 – (a) Probability density function of number of members of a local chapter, conditional on the
size of the union it belongs to. We plot our results for two bins of union size. The figure reveals two
interesting points: (i) the typical size of the local chapters increases with union size, (ii) the functional
formof the distributionappears to beindependentof unionmembership. (b)Scaled probabilitydensity
function of scaled local chapter size conditional on union size (see text immediately following Eq. (4)
for details). The data for the four bins collapse onto a single universal curve, suggesting that the the
structure of different unions is independent of union size except for a scale factor.
where ξt(S) ∼ Sαis the typical size of a local chapter in a trade union of size S, and F1(u)
appears to decay as a stretched exponential or a power law for u ≫ 1. As a test of the scaling
hypothesis (4), we plot the scaled quantities Sαρ(ξ|S) versus ξ/Sαand obtain a good data
collapse, that is, all data points fall onto a single universal curve; Fig. 4(b). To estimate α, we
use the fact that Eq. (4) implies that the typical number of local chapters in a trade union with
S members increases proportionally to S1−αwith α = 0.32 ± 0.05, while the typical number
of members of these local chapters is proportional to Sαwith the independent estimate
α = 0.30± 0.05; Figs. 5(a),(b).
creation of social capital is a subject of debate in the literature . One may reasonably
hypothesize that the size of the subunits will be negatively correlated with their capacity for
creating social capital, since members in large subunits will likely (i) not be able to create
strong links among one another, and (ii) not be able to participatein the governing process as
fully as the members of small subunits. Our results support this hypothesis and suggest that
large organizations—because they typically consist of larger subunits—will be less effective
in creatingsocial capitalthan small organizations. This resultmay find support in the current
trendin high-tech firms to organizeprojectsaroundsmall teamsthat split, when they become
too large, in order to facilitate cooperation.
Our findings are also of note for other reasons: First, our approach differs from the sta-
tistical methods traditionally used in macrosociology, which typically assumes that systems
are linear and in an equilibrium state . It also stands in contrast to the view that sociolog-
ical explanations ideally would only make reference to individual agents and their actions
(“methodological individualism” ). We show that techniques successfully used in statisti-
cal physics can be applied to a central sociological topic—voluntary organizations—to reveal
nontrivial patterns and relationships.
Second, an intriguing aspect of our findings is that they provide evidence for growth
Which characteristics of a voluntary organization are important for the
ln (Number of locals)
ln (Local size)
Fig. 5 – (a) Number n of local union chapters comprising a given trade union as a function of union
size. The thick continuous line is a linear least squares fit between lnn and lnS. The gray areas defines
the 95% confidence interval. The data surprisingly fall along a straight line in a log-log plot, indicating
a simple power law dependence on the union size of the number of local chapters comprising a union,
with an exponent α = 1 − 0.68 ± 0.05 = 0.32 ± 0.05. (b) Size of local union chapters comprising a
given trade union as a function of union size. The thick continuous line is a linear least squares fit
between lnξ and lnS. The gray areas defines the 95% confidence interval. We find a similar power
law dependence of the typical size of the local chapters comprising a union with a given size with an
exponent α = 0.3 ± 0.05. Note that the two independent estimates of α are within error bars.
dynamics similar to those found for other organizations, such as business firms . This
similarity is rather surprising as the reasons for the growth of a voluntary organization are
quite different from those for a business firm. In particular, the profit motive—perhaps
the most important factor in the growth of business firms—is not evident for voluntary or-
ganizations. The similarity between the empirical laws describing the growth of voluntary
organizationsandbusiness firms,andthefactthesetwo typesoforganizationsareappar-
ently so different, raises an intriguing analogy between the growth of human organizations
comprised of many animate interacting units and the physics of natural systems comprised of
many inanimate interacting units. Our findings areconsistent with the possibility that univer-
salmechanismsgoverning thegrowth ofhumanorganizations—suchasthecomplexinternal
structure of units, stochastic growth, and a broad range of scales—are more important than
the idiosyncratic characteristics of the system that are customarily believed to determine the
∗ ∗ ∗
We thank S. V. Buldyrev, C. Edling, P. Gopikrishnan, P. Hedstr¨ om, M. Macy, and V. Plerou
for stimulating discussions and A. L. Stinchcombe for numerous comments and suggestions.
FL thanks STINT (97/1837) and HSFR (F0688/97). The CPS is supported by NSF and NIH.
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