A spectrum of complexity uncovers Dunbars number and other leaps in social structure Mart n Saavedra1 2Jorge Mira3Alberto P Mu nuzuri1 2and Lu s F Seoane4 5

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A spectrum of complexity uncovers Dunbar’s number and other leaps in

social structure

Mart´ın Saavedra,1, 2 Jorge Mira,3Alberto P Mu˜nuzuri,1, 2 and Lu´ıs F Seoane4, 5

1Group of Nonlinear Physics. Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain.

2Galician Center for Mathematical Research and Technology (CITMAga), 15782 Santiago de Compostela, Spain

3Departamento de F´ısica Aplicada and iMATUS, Universidade de Santiago de Compostela, 15782, Santiago de Compostela,

Spain.

4Systems Biology Department, Spanish National Center for Biotechnology (CSIC), C/ Darwin 3, 28049 Madrid, Spain.

5Grupo Interdisciplinar de Sistemas Complejos (GISC), Madrid, Spain.

Social dynamics are shaped by each person’s actions, as well as by collective trends that emerge

when individuals are brought together. These latter kind of inﬂuences escape anyone’s control.

They are, instead, dominated by aggregate societal properties such as size, polarization, cohesion,

or hierarchy. Such features add nuance and complexity to social structure, and might be present,

or not, for societies of diﬀerent sizes. How do societies become more complex? Are there speciﬁc

scales at which they are reorganized into emergent entities? In this paper we introduce the

social complexity spectrum, a methodological tool, inspired by theoretical considerations about

dynamics on complex networks, that addresses these questions empirically. We use as a probe

a sociolinguistic process that has unfolded over decades within the north-western Spanish region

of Galicia, across populations of varied sizes. We estimate how societal complexity increases

monotonously with population size; and how speciﬁc scales stand out, at which complexity would

build up faster. These scales are noted as dips in our spectra, similarly to missing wavelengths in

light spectroscopy. Also, ‘red-’ and ‘blue-shifts’ take place as the general population shifted from

more rural to more urban settings. These shifts help us sharpen our observations. Besides speciﬁc

results around social complexity build-up, our work introduces a powerful tool to be applied in

further study cases.

Keywords: social complexity, Dunbar number, social networks, social dynamics, linguistics

I. INTRODUCTION

If we put together grains of sand, one by one, at some

point they become a mountain. When does that happen?

Daniel Dennett reminds us that such questions might not

have a clear-cut answer, and teaches us how to live with

the ensuing scale of greys [1]. Yet some complex systems

often change radically as their size crosses certain thresh-

olds. At those points, emerging behaviors start shaping

the fate of the whole, overriding the more straightfor-

ward dynamics of the parts. Then, as put beautifully by

William P. Anderson, “More is Diﬀerent” [2].

As humans, we contemplate the duality of our auton-

omy and of belonging to an eusocial species. Tensions

between both levels of Darwinian selection (the organ-

ismal and the societal) largely underlie our conﬂicting

nature [3]. Individually, every modern human has likely

experienced ‘being dragged by the masses’. But, when

does a gathering of people become an emergent entity?

How many such transitions might take place as societies

grow? Are these processes gradual, as with mountains

and grains of sand? Or can we spot relevant scales at

which the dynamics of human behavior is altered by

emerging levels of complexity?

We tackle these questions empirically through a pro-

cess of opinion dynamics that has played out over social

groups of diﬀerent complexity. Speciﬁcally, we tend to

socioliguistic processes of language shift that have oc-

curred in the Spanish Autonomous region of Galicia over

the last ∼100 years [4]. In this process, the vernacular,

Galician, trended to be substituted by Castilian Span-

ish. Both are romance languages with high mutual intel-

ligibility, which might enable long-term bilingualism and

coexistence of both tongues [5]. These dynamics have un-

folded simultaneously, at diﬀerent speeds, in a range of

population centers, from very rural ones to larger cities.

Galicia presents quite unusual demographics: It covers

about 6% of the territory of Spain and houses a similar

percentage of the country’s population. Notwithstand-

ing, Galicia contains roughly a half of all 60 000 Spanish

Singular Population Entities (SPEs, deﬁned as any unit

such as villages, cities, etc.). Among these, as of 2016,

27 000 Galician SPEs had less than 100 inhabitants [6].

This gives us a great sampling of social processes hap-

pening on communities with diﬀerent structures. If we

assume that SPEs of diﬀerent sizes constitute underlying

social groups of varying complexity, we are provided with

a unique opportunity to study at what emerging social

scales human behavior is altered in a noticeable way.

Social exchanges happen within a social substrate, of-

ten modeled as a graph or network. In them, people

are nodes, and edges represent existing interactions—

whether virtual or of physical contact. Networks can

have diﬀerent overall structures—e.g., small world [7],

hierarchical or more horizontal [8], etc. Distinct network

structures enable or hinder the unfolding of diﬀerent phe-

arXiv:2210.15322v1 [physics.soc-ph] 27 Oct 2022

FIG. 1 A complex network structure might slow down social dynamics. In [11] it is observed that certain opinion

dynamics can take longer to reach their absorbing steady state due to a complex structure of the underlying social network.

This ﬁgure illustrates a mechanism behind this phenomenon. In a clique (top), which is a simple kind of graph, all network

individuals interact with each-other. This enables that the average opinion observed by the i-th individual equals the actual

average opinion across the whole group, hxii=hxi. Conﬂicting views are resolved as quickly as possible (a red dashed vertical

line marks when the steady state is reached in each case). In random, yet unstructured networks (middle), which are fairly

simple as well, not everybody is connected with everybody else. However, a node’s neighbors are relatively well distributed

across the network, thus its perceived average opinion comes from a good-enough sampling of the group’s state, hxii' hxi.

Dynamics are resolved only slightly more slowly. In complex, structured networks, individuals cluster locally with few nodes

acting as gate-keepers of larger groups (bottom). Individuals inside each group perceive averaged opinions that might depart

from the network’s consensus, hxii6=hxi. Hence, opinion dynamics are slowed-down as bottlenecks prevent the immediate

invasion of each cluster.

nomena, such as opinion dynamics, epidemic spread, co-

ordination towards a goal, etc. For example a suﬃciently

sparse network of physical contacts can halt the spread

of a virus—as conﬁnement measures during the recent

pandemic have illustrated. Less trivially, Darwinian evo-

lution can be accelerated if it happens over networks with

speciﬁc shapes [9, 10].

A similar interplay between network structure and dy-

namics can apply to other social processes—speciﬁcally,

to opinion dynamics such as the decision to keep or

change your tongue. In a computational study, Toivonen

et al. [11] showed how certain classes of opinion dynamics

have longer relaxation times in more complex networks,

while the same processes converge faster to their steady

states in simpler graphs. In more trivial networks (Fig. 1,

top and middle), each individual samples accurately the

social group’s average opinion, and dynamics can be re-

solved quickly. In more complex graphs (Fig. 1, bottom)

nodes of similar opinion can form clusters guarded by

gatekeepers that hardly ﬂip their opinions. These clus-

ters become diﬃcult to penetrate. Hence, some individ-

uals are cut out of the emerging consensus, and conver-

gence to a homogeneous opinion is hindered. Inspired by

this, it was found empirically that sociolinguistic dynam-

ics in Galicia have unfolded faster in more rural areas,

and slower in more urban ones [4].

This last study assumed that urban communities had

a more complex social structure, and it used the thresh-

old of 5000 inhabitants to consider a SPE as urban. This

choice was based on Spanish legislation [12] that demands

that counties (which usually include several SPEs) over

that size present a series of structures and services (a pub-

lic park and library, a market place, and a waste manage-

ment system), potentially marking a jump in social net-

work complexity. Further demands of urban equipment

do not happen below 5 000 inhabitants, or above until

20 000 and 50 000 inhabitants. However, can we relax

this deﬁnition of urban and come up with a more organic

way to ﬁnd salient leaps in social complexity? Perhaps,

if such changes in complexity aﬀect ongoing opinion dy-

namics (as those simulated by Toivonen et al. [11]), we

can look at empirical data of language shift to spot the

relevant scales at which complexity builds up.

In this paper we study this possibility. We try a

range of scales (based on population sizes), and check

whether each scale is a good separation between simple

versus complex social networks. Following [11], our test is

whether sociolinguistic dynamics tend to play out faster

in ‘simpler’ networks. We measure this through correla-

tions between a region’s purported complexity and the

rate at which sociolinguistic dynamics have unfolded in

that area. This renders a kind of ‘spectrogram’ in which

dips are visible at population scales with non-trivial cor-

relations (much like a spectral line is missing in optical

spectroscopy when light traverses speciﬁc chemical com-

pounds that absorb a speciﬁc wavelength). We ﬁnd two

salient scales that, according to our criterion, would sepa-

rate simpler from more complex social networks. Our re-

sults suggest that at those scales some emergent compo-

nent takes over and alters the pace at which sociolinguis-

tic dynamics play out. One of these scales corresponds to

the threshold used in [4], which was based on urban plan-

ning. The other one, more prominent, happens at much

smaller community sizes (∼200 people). It does not cor-

relate with any feature marked by Spanish law. Rather,

its proximity to Dunbar’s number (an empirical—yet de-

bated [13–15]—cognitive limit to the number of relation-

ships that animals can have [16–23]) suggests an organic

emergence of social complexity.

In Sec. II.A we describe the sociolinguistic dynamics

of interest to us and the empirical data available. In Sec.

II.B we introduce the mathematical model that we ﬁt to

the data. The ﬁtting procedure is explained in Sec. II.C.

We use the resulting model parameters to estimate the

rate at which the dynamics unfold in each geographical

region. In Sec. III.A we introduce our social complex-

ity spectrum. We explain how we deﬁne separations be-

tween potentially simple and potentially complex social

networks, and how we evaluate the goodness of these sep-

arations. Secs. III.A and III.B contain our main results.

Namely, that we identify two outstanding scales at which

leaps, or buildups of social complexity would happen ac-

cording to our criterion. Sec. III.C further explores the

spectrum of social complexity as a methodological tool.

Similarly to physics and optical spectroscopy, we observe

‘red-’ and ‘blue-shifts’ as the overall demographics has

changed over two decades. We illustrate how this can

help us reﬁne the separation of simple and complex so-

cial networks. We wrap up the paper with a discussion of

our ﬁndings in Sec. IV. We further argue that the “com-

plexity spectrum” might be a powerful tool to uncover

scales of social relevance when applied to similar dynam-

ics.

II. METHODS

A. Sociolinguistic dynamics and data

We investigate the emergence of singular scales of so-

cial complexity by looking at how certain sociolinguis-

tic dynamics have unfolded at diﬀerent speeds in regions

that, potentially, contain distinctly complex social net-

works. The dynamics that we use as a proxy is the co-

existence of Castillian Spanish and Galician. Both these

tongues are romance languages that coexist in the Au-

tonomous Community of Galicia, in north-western Spain.

Mutual intelligibility between them is large, allowing

broad bilingual communities and, potentially, a sustained

coexistence between the two of them. While Galician is

the vernacular, a shift towards Castillian Spanish has

been underway for centuries, and has been especially ac-

centuated during the 20th century.

The Galician Statistical Oﬃce (Instituto Galego de Es-

tat´ıstica, IGE) has tracked language use in diﬀerent Gali-

cian regions and across demographic groups. In peri-

odic polls, informants would self-assess their language use

as ‘only Galician’, ‘mostly Galician’, ‘same use of both

tongues’, ‘mostly Spanish’, and ‘only Spanish’. We took

informants at either extreme of this scale as monolingual

individuals of the corresponding language, and grouped

the central categories as bilinguals. We are interested in

the fractions of speakers in these groups.

IGE polls are stratiﬁed by age, which allows us to build

a time-series of fractions of speakers by projecting age

groups in apparent time [24–28] (meaning that the frac-

tion of speakers of people of a certain age become es-

timators of the fraction of speakers when those people

were born). Additionally, data is split into 20 indepen-

dent Galician subregions, each of which is made up of

a collection of smaller counties—but data for individual

counties is not available. Hence, our empirical dataset of

sociolinguistic dynamics consists of 20 time series with

the fractions of monolingual Galician speakers, bilingual

speakers, and monolingual Spanish speakers (Fig. 2a-b).

We base our work on the IGE poll that allowed us of to

estimate the fractions of those born in 2001 [29].

B. Mathematical model

Beginning in the early 90s [30, 31], a growing commu-

nity of mathematicians, physicists, ecologists, and com-

plexity researchers started using systems of diﬀerential

equations to model possible trajectories of speakers of co-

existing languages over time [32]. A turning point was the

work by Abrams and Strogatz [33], who ﬁtted their equa-

tions to data from dozens of cohabiting tongues. This

inspired a wave of new models whose stability and dy-

namical classes were thoroughly analyzed [5, 34–44], and

which could in some occasions be tested against empirical

data [4, 28, 44–52].

Diﬀerent authors would emphasize distinct ingredients

that might aﬀect language coexistence, such as their spa-

tial distribution, or bilingualism (elements that the origi-

nal model by Abrams and Strogatz did not contemplate).

We use one such variation that includes bilingualism [34],

whose stability and dynamics have been studied in detail

[5, 39, 42, 43], and that has been ﬁtted to data of diﬀer-

ent cohabiting languages, including the Galician-Spanish

case [4, 28, 52]. Contrary to some other models with

bilingualism, the one that we use is compatible with ei-

ther the stable coexistence of both tongues, or that one

language takes over and drives the other to extinction.

Thus, the model is agnostic regarding the stability of the

coexisting couple, and the empirical data can constrain

model parameters towards either outcome.

The model consists of a system of coupled diﬀerential

equations that tracks the time-evolution of the fraction,

x, of monolinguals of language X (here, Galician); of the

fraction, b, of bilinguals; and of the fraction, y, of mono-

linguals of language Y(here, Spanish). Population is

normalized such that x+y+b= 1, hence two equations

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AspectrumofcomplexityuncoversDunbar'snumberandotherleapsinsocialstructureMartnSaavedra,1,2JorgeMira,3AlbertoPMu~nuzuri,1,2andLusFSeoane4,51GroupofNonlinearPhysics.UniversidadedeSantiagodeCompostela,15782,SantiagodeCompostela,Spain.2GalicianCenterforMathematicalResearchandTechnology(CITMAga),1578...

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A spectrum of complexity uncovers Dunbars number and other leaps in social structure Mart n Saavedra1 2Jorge Mira3Alberto P Mu nuzuri1 2and Lu s F Seoane4 5

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