# Universal Mechanisms in the Growth of Voluntary Organizations

**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.

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**ABSTRACT:**We investigate the timing of messages sent in two online communities with respect to growth fluctuations and long-term correlations. We find that the timing of sending and receiving messages comprises pronounced long-term persistence. Considering the activity of the community members as growing entities, i.e. the cumulative number of messages sent (or received) by the individuals, we identify non-trivial scaling in the growth fluctuations which we relate to the long-term correlations. We find a connection between the scaling exponents of the growth and the long-term correlations which is supported by numerical simulations based on peaks over threshold. In addition, we find that the activity on directed links between pairs of members exhibits long-term correlations, indicating that communication activity with the most liked partners may be responsible for the long-term persistence in the timing of messages. Finally, we show that the number of messages, $M$, and the number of communication partners, $K$, of the individual members are correlated following a power-law, $K\sim M^\lambda$, with exponent $\lambda\approx 3/4$.Physics of Condensed Matter 02/2010; 84(1). · 1.46 Impact Factor

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arXiv:nlin/0310001v1 [nlin.AO] 1 Oct 2003

Universal Mechanisms in the Growth of Voluntary

Organizations

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.

Introduction. –

Blockcountriesshowthatdemocratizationandmarketreformsdonotautomaticallygenerate

healthy democracies and healthy market economies. Several studies suggest that the reason

for this failure may be that well-functioning societies [1] 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 [1].

Specifically, in regions where people are embedded in a rich environment of decentralized

civicnetworks,thereisanincreasinglikelihoodthattheindividualswillbeabletocooperatein

endeavors of mutual benefit [2]. 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 [3], and (b)

overcoming the difficulties in partitioning the exploitation of public resource [4], 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 [5], (ii) the impact of social networks in membership recruit-

ment[6],(iii)socialbackgroundofmembers[7],and(iv)theabilityofvoluntaryorganizations

The developments of the last decade and a half in the former Eastern

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to overcome “free-rider” behavior [8]. Despite this research activity, one area that has not

beenpursuedconcernsthequantitativecharacterizationofthegrowthdynamicsofvoluntary

organizations.

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 [10]. 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

properties.

Results. –

of the growth statistics of a range of different Swedish voluntary organizations—including

trade unions [12], temperance movements [13], free churches [14], and the social democratic

party [15]—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

[16] 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 [11] that provides a detailed resource for the study

1900 19101920

Year

19301940

10

4

10

5

10

6

10

7

Total number

Population

Work force

Unionized workers

a

1.21.8 2.4

x = log(Union size)

3.03.64.2 4.8 5.4

10

−2

10

−1

10

0

Probability density

b

Bin 1

Bin 2

Bin 3

Bin 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 [18], that the growth is a random multiplicative

process. Forsubsequentanalysis, we partition the data into 4bins accordingto union size, as illustrated

by the figure.

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L, A, S: G V O

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−0.6−0.4 −0.2

Growth rate

0.00.2 0.40.6

10

−2

10

−1

10

0

10

1

Probability density

Bin 1

Bin 4

a

−6.0−4.0 −2.00.02.0 4.0 6.0

Scaled growth rate

10

−2

10

−1

10

0

Scaled probability density

Bin 1

Bin 2

Bin 3

Bin 4

b

Gaussian

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

union,

?S(t + 1)

g(t) ≡ log

S(t)

?

,

(1)

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

p(g|S) ∼

1

σ(S)F

?

g

σ(S)

?

.

(2)

We next calculate the standard deviation σ(S) of the distribution of growth rates as a

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4

10

1

10

2

10

3

10

4

10

5

Union size

10

−2

10

−1

Standard deviation of growth rates

a

−0.19

0.0 2.04.06.0 8.0

Scaled growth rate

10

−3

10

−2

10

−1

10

0

Cumulative probability

Positive tail

Negative tail

b

Gaussian

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,

(3)

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-

orderapproximationisthatthenumberofmembersofthedifferentlocalchapterscomprising

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 [17].

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 [17]

As trade unions have a complex internal structure, it

ρ(ξ|S) ∼

1

ξt(S)F1

?

ξ

ξt(S)

?

,

(4)

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5

10

0

10

1

10

2

10

3

10

4

Local size

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

Probability density

Bin 1

Bin 4

a

10

−2

10

−1

Scaled local size

10

0

10

1

10

2

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

10

1

Scaled probability density

Bin 1

Bin 2

Bin 3

Bin 4

b

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).

Discussion. –

creation of social capital is a subject of debate in the literature [21]. 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 [22]. 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” [1]). 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

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102

103

104

105

Union size

1.0

2.0

3.0

4.0

5.0

6.0

7.0

ln (Number of locals)

a

102

103

104

105

Union size

2.0

3.0

4.0

5.0

6.0

ln (Local size)

b

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 [23]. 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[23],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

system’s dynamics.

∗ ∗ ∗

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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[16] B. Rothstein, in R. D. Putnam, Democracies in Flux (Oxford University Press, Oxford, 2002).

[17] L.A. N. Amaral et al., Phys. Lev. Lett. 80, 1385 (1998).

[18] J. Sutton, J. Econ. Lit. 35, 40 (1997); Ibid, Physica A (in press);

[19] V. Plerou et al., Nature 400, 433 (1999).

[20] T. Keitt et al., Phil. Trans. Roy. Soc. B 357, 627 (2002).

[21] M. Levi, Politics and Society 24, 45 (1996).

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Social Theory (Cambridge U. Press, Cambridge, 1998).

[23] M. H. R. Stanley et al., Nature 379, 804 (1996).

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