The contagion effects of repeated activation in social networks


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Social Networks 54 (2018) 326–335

Contents lists available at ScienceDirect

Social  Networks

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / s o c n e t

he  contagion  effects  of  repeated  activation  in  social  networks

ablo  Piedrahita a,∗,  Javier  Borge-Holthoefer b,  Yamir  Moreno a,c,
andra  González-Bailón d,∗

Institute for Biocomputation and Physics of Complex Systems and Department of Theoretical Physics, University of Zaragoza, Spain
Internet Interdisciplinary Institute (IN3), Open University of Catalonia, Spain
Institute for Scientific Interchange, ISI Foundation, Turin, Italy
Annenberg School for Communication, University of Pennsylvania, USA

 r  t  i c  l e  i  n  f  o

rticle history:
vailable online 1 December 2017

eywords:
nterdependence

a  b  s  t  r  a  c  t

Demonstrations,  protests,  riots,  and  shifts  in  public  opinion  respond  to  the  coordinating  potential  of
communication  networks.  Digital  technologies  have  turned  interpersonal  networks  into  massive,  perva-
sive structures  that constantly  pulsate  with  information.  Here,  we propose  a model  that  aims  to analyze
the  contagion  dynamics  that  emerge  in  networks  when  repeated  activation  is allowed,  that  is, when
actors  can  engage  recurrently  in  a  collective  effort.  We  analyze  how  the structure  of  communication  net-
emporal dynamics
oordination
hresholds
ritical mass
iffusion
ollective action
gent-based simulation

works impacts  on  the ability  to coordinate  actors,  and  we  identify  the conditions  under  which  large-scale
coordination  is more  likely  to  emerge.

© 2017  The  Authors.  Published  by  Elsevier  B.V. This  is  an  open  access  article  under  the  CC  BY-NC-ND
license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
. Introduction

Recent years have seen the emergence of massive events
oordinated through large, decentralized networks. These include
olitical protests and mobilizations like the Occupy movement
f 2011 (Conover et al., 2013; González-Bailón and Wang, 2016),
he Gezi Park demonstrations of 2013 (Barberá et al., 2015), or
he growth of the #BlackLivesMatter campaign during the 2014
rotests in Ferguson (Freelon et al., 2016). These collective events
ffer examples of the coordinating potential of communication net-
orks − which, increasingly, emerge through the use of online

echnologies. This paper pays attention to the coordination dynam-
cs that allow a small movement, a new campaign, or an unknown
ashtag to rise to prominence. We present a formal model that
llows us to answer the following question: How do coordination
ynamics unfold to make individual actions (e.g. using an emerging
ashtag, endorsing a mobilization) converge over time? Our model

ims to disentangle the mechanisms that drive the emergence of
ecentralized, large-scale coordination. The goal is to identify the

∗ Corresponding authors.
E-mail addresses: ppiedrahita@gmail.com (P. Piedrahita),

gonzalezbailon@asc.upenn.edu (S. González-Bailón).

ttps://doi.org/10.1016/j.socnet.2017.11.001
378-8733/© 2017 The Authors. Published by Elsevier B.V. This is an open access article 
/).
conditions under which coordination is more likely to arise from
networks that are constantly pulsating with information.

Threshold models have become the standard for how we think
about interdependence and the collective effects of social influence
(Granovetter, 1978; Granovetter and Soong, 1983; Schelling, 1978).
As originally formulated, the activation of individual thresholds
responds to global information: the group of reference is assumed
to be the same for all actors. In later developments of the basic
model, networks were introduced to add local variance to social
influence: the group of reference was  now determined by connec-
tivity in the network, which changed from actor to actor (Valente,
1996; Watts, 2002). These different variations of the threshold
model share two  important elements: first, activation is modelled
as a step function that goes from 0 to 1 when thresholds are
reached; and second, thresholds can only be reached once, that
is, activation is assumed to be a one-off event. Our model aims to
relax these assumptions and allow actors to repeatedly activate as
a function of the dynamics unfolding in the rest of the network. We
argue that this modification aligns our model of contagion more
closely with what is observed in many empirical networks − in
particular, with the communication dynamics observed in online

networks and the temporal autocorrelation that results from those
dynamics.

Online campaigns are an important manifestation of this type
of repeated activation, and they offer a good example of what we

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P. Piedrahita et al. / Socia

ean by “coordination”: a form of organizational effort to attract
ublic attention or direct mobilization logistics on the ground. The
lack Lives Matter movement, for instance, gained traction when
he hashtag was first adopted in social media in 2013, which fueled
hat has been labeled as “an Internet-driven civil rights move-
ent” (Eligon, 2015, see also Day, 2015). There is agreement that

he movement consolidated with its first peaceful demonstrations
n Ferguson in 2014 (Bosman and Fitzsimmons, 2014); but this

ove “from hashtag to the streets” stands as a new model for
ow “liberations groups in the twenty-first century can organize
n effective freedom rights campaign” (Ruffin, 2015). Online net-
orks were central to the coordination efforts of this campaign;
ere, we aim to illuminate the mechanisms that explain why.

In the context of empirical examples like the Black Lives Matter
ovement, activation involves repeatedly using a specific hashtag

o build momentum up to the point when large-scale coordination
s achieved − i.e., the hashtag starts receiving global recognition,

hich can then be used to shape the public agenda or to help
rganize further mobilization efforts. The goal of our model is to
bstract this element of repeated activation and build an analyt-
cal framework around it to answer three interrelated questions:
ow does the structure of interdependence, the variance in indi-
idual propensities to activate, and the strength of social influence
ffect contagion and the emergence of large-scale coordination?
s with many analytical models, ours is a simplification of what is
ssentially a very complex reality. But it offers, we think, important
nsights into the counter-intuitive effects of networks in allowing
oordination to emerge.

The rest of the paper proceeds as follows. First, we consider
rior work analyzing coordination in networks, and the analytical
hoices made when modelling social influence. We introduce our
odel as a continuation of threshold models, well suited to ana-

yze dynamics of adoption (e.g. joining a political movement) but
ot well equipped to analyze the dynamics of coordination that
merge amongst actors that are already part of a movement. We
hen describe our model in detail, highlighting the main differences
ompared to previous approaches and unpacking our assumed
echanisms. In sections five and six we present our findings, which
e organize around two main questions: How do changes in net-
ork topology affect the emergence of coordination under different

ssumptions of social influence? And how does individual hetero-
eneity impact coordination dynamics? We close the paper with

 discussion of our findings, especially as they relate to previous
esearch on contagion in networks.

. Coordination as a two-step selection process

We  can think of coordination dynamics as a two-step selec-
ion process: in the first stage, actors decide if they want to join

 movement; in the second stage, they coordinate their actions
ith those who also opted in. Most threshold models refer to

he first stage, and they focus on the cascading effects of one-
ff activations − the decision, that is, to join a collective effort.
hese models belong to the more general theoretical tradition of
iffusion research, which looks at how ideas or behavior spread

n social systems (Rogers, 2003; Valente, 1995). Diversity in the
otivation to adopt a behavior is modelled as a distribution of

hresholds; what prior research shows is that the shape of this
istribution is one of the key elements explaining the cascading
ffects of individual activations (Watts and Dodds, 2010). Conta-
ion dynamics, however, also depend on the structure of ties and

ow that structure encourages or hinders spreading dynamics. Net-
orks shape coordination dynamics by creating different centrality
istributions, which allow specific individuals to be more or less

nfluential (Freeman, 1979); and by opening more or less structural
orks 54 (2018) 326–335 327

holes (Burt, 1992; also Girvan and Newman, 2002), which constrain
opportunities for chain reactions to the extent that they delimit
the routes that cascades can follow (Watts, 2002). Networks also
delimit the size and the composition of the groups of reference sur-
rounding a given actor and, therefore, the number of social signals
each actor receives (Centola and Macy, 2007; Valente, 1996).

Most threshold models assume that activation happens only
once and that, once activated, the change of state (from inactive
to active) is permanent. This is the reason why threshold models
are appropriate to capture the first stage of coordination dynam-
ics − for instance, the decision to join a movement or start using a
particular hashtag. The model we propose here, on the other hand,
aims to capture dynamics of activation within adoption, that is,
coordination amongst actors who  already opted in and therefore
have an interest in facilitating organizational efforts.

We have theoretical and empirical reasons to allow repeated
activation to be the driving force of contagion dynamics. The empir-
ical reason is that most instances of diffusion do not involve a single
activation but many activations building up momentum in time.
Before a hashtag becomes a trending topic, a period of buzz is first
required; prior to a protest day, calls announcing the mobiliza-
tion are distributed in waves. Actors decide whether they want to
engage in an online conversation or take part in a protest. This is
what threshold models can capture. What threshold models are not
devised to capture is the period of information exchange that fol-
lows the act of joining a collective effort. During this period, social
influence trickles intermittently as a function of the context that
actors inhabit − that is, as a function of activity in the local net-
works to which they are exposed; and this context is not stationary:
it changes, sometimes drastically, over time. Our model aims to
capture this temporal dimension.

We also have a theoretical reason to relax the assumption of sin-
gle activation. The intermittent dripping of information that social
networks facilitate often leads to bursts of activity (Vazquez et al.,
2006), as when news suddenly become trending topics (Lehmann
et al., 2012; Wu  and Huberman, 2007). Coordination dynamics
underlie these bursts of activity: sudden peaks in communication
require the adjustment of individual actions, that is, the align-
ment of many individual decisions so that everybody uses the same
trending hashtag or talks about the same news at the same time.

These dynamics of coordination, and how they lead to collec-
tive outcomes like swift information cascades, trending topics, or
viral hashtags, are overseen if activation is modelled as a perma-
nent change of state − that is, if we only focus on the first stage
of what is, in fact, a two-step process. Our model presumes that,
once in the second stage, individual propensities to activate will be
influenced by the network and the signals it transmits, which in
turn results from how other actors are influenced and react to that
influence over time. These dynamics aim to resemble more closely
the dynamics observed in the context of large-scale mobilizations,
where actors repeatedly engage in activities like spreading calls
for action or increasing the salience of political hashtags (Barberá
et al., 2015; Borge-Holthoefer et al., 2011; Budak and Watts, 2015;
Conover et al., 2013; Jackson and Foucault Welles, 2015). Individual
decisions to contribute to the flow of information, and the deci-
sions of those connected to a focal actor, co-evolve over time; our
analytical approach models that co-evolution explicitly.

As previous models, our model assumes that exposure to infor-
mation is the driving force underlying contagion. What makes our
model different from previous models is that failure to trigger a
chain reaction depends not only on the distribution of thresholds
or the impact of network structure on activation dynamics; it also

depends on whether the network facilitates coordination, that is,
an alignment of actions in time − which is an important organiza-
tional goal for social movements that want to gain public visibility
in social media or use online networks to manage mobilization



328 P. Piedrahita et al. / Social Networks 54 (2018) 326–335

Fig. 1. Schematic Representation of the Social Influence Model with Recurrent Acti-
vation.
The model, adapted from (Mirollo and Strogatz, 1990) assumes that actors (i.e. the
nodes in the network) reach their activation threshold at different speeds. The speed
of activation is a function of two parameters: ω, which determines how quickly the
actor reaches the threshold zone (i.e. it defines the concavity of the curve that maps
progression towards activation); and ε, or the strength of the signal received from
the neighbors in the network when they activate, a pulse that shifts the state of the
focal actor closer towards the threshold (the timing of which varies over time). The
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Fig. 2. The Impact of the Parameter ω on the Activation Buildup.
When the parameter ω is 0, the progression of actors towards their activation thresh-
old  (x = 1) grows linearly with time; as the parameter ω increases, actors reach their
ower panels in the figure illustrate how actor i advances towards activation. When
 node activates, as node i does in t2, she shifts the state of her neighbors with the

 signal and resets her state back to the beginning of her phase.

ogistics in real time. By focusing on coordination dynamics, our
odel is in a better position to explain why, more often than not,

arge-scale contagion fails to take off. If the network is not con-
ucive to coordination (i.e. if the timing of individual activations
o not align over time), contagion ends up trapped in local activ-

ty clusters and, therefore, fails to synchronize the actions of the
ajority.

. Model and mechanisms

Our model of contagion relaxes the assumption that actors can
nly transition from an inactive to an active state. We  also allow
he effects of each activation to vary over time to the extent that
hey coevolve with the contagion dynamics taking place in the rest
f the network. These modelling choices make sense if we think
bout how online networks facilitate contagion dynamics: users
re constantly exposed to signals that might shift their inclination
o act − for instance, send messages directing attention to spe-
ific issues (e.g. #occupy, #Gezi, #Ferguson, etc). Only when a large
nough number of users converge in their attention to these issues,
heir actions become globally visible − i.e. mass media starts pay-
ng attention. This type of coordination not only affects trending
uzz; it actually has the potential to shape the public agenda in
he same way than more traditional social movements would (e.g.,
etersen-Smith, 2015). The difference is that coordination in social
edia happens spontaneously, from the bottom-up.
To bring these empirical intuitions into a tractable framework,

e follow classic models of synchronized coordination (Mirollo
nd Strogatz, 1990; Piedrahita et al., 2013). These models have
een used extensively to study coordination dynamics in biologi-
al and physical settings (Strogatz, 2003), but they have never been
sed, to the best of our knowledge, to illuminate dynamics relevant
or the study of social mobilization, or to extend classic threshold

odels and their application to sociological questions. Like thresh-
ld models, our model assumes that the motivational structure of
ctors can be defined by a limit that, when reached, triggers activa-
ion; unlike threshold models, we split the motivation to activate
nto two components: a social component, which depends on what

ther actors are doing; and an individual component, which defines
he intrinsic propensity of actors to activate regardless of what oth-
rs are doing. We  model this intrinsic component as a function that
ncreases monotonically over time within the range [0,1] until the
activation zone faster, i.e. a signal received from their neighbors will tip their sate
over the threshold, which means they will send a signal as well (thus helping other
actors to also get closer to their activation zones).

upper bound – which acts as the threshold or activation limit – is
reached. Fig. 1 illustrates the logic of this approach.

Our main assumption is that actors reach their activation zone
at different speeds. The speed of activation is a function of two
parameters: ω, which determines how quickly the actor reaches
the threshold zone (i.e. it defines the concavity of the curve that
maps progression towards activation); and ε, or the strength of the
signal received from other actors − which, in our case, is restricted
to actors one step removed in the network. Every time a neighbor
activates, they send a pulse that shifts the state of the focal actor
closer towards the threshold zone; the parameter ε, in other words,
is the building block we use to introduce social influence in the
model. The lower panels in Fig. 1 illustrate how actor i advances
towards activation, both as a function of her intrinsic propensity ω
and as a response to the activation of the neighbors. When a node
activates, as node i does in t2, she shifts the state of her neighbors
with the ε signal and resets her state back to the beginning of her
phase.

The mathematical expression of these intuitions follows this
functional form:

x = f (t) = 1
ω

ln
(

1 +
[
eω − 1

]
t
)

(1)

The parameter ω determines the shape of this function. It is always
ω > 0 to make the function concave down. We  assume a mono-
tonic increase because it is the most natural choice when modelling
progression towards activation and it follows the same intuition
as threshold models − only that instead of proposing a stepwise
change, it models actors’ propensity to activate as a continuous pro-
gression. As Fig. 2 shows, a larger ω produces a more pronounced
shape, making the function rise very rapidly to then level off.

In addition to this individual component, the model also takes
into account the activation of other actors in the network, in par-
ticular, those one step removed. If actors i and j are connected,
j’s activation increases i’s propensity to activate by an amount ε

or pushes i directly into activation, whichever is less. This rule of
interdependence is expressed as:

xi = min (1, xi + ε) (2)



P. Piedrahita et al. / Social Networks 54 (2018) 326–335 329

Fig. 3. The Impact of Network Topology on Coordination Dynamics.
T alues 
t  (via ε
t  coord

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his figure summarizes contagion dynamics in toy networks with size N = 10, and v
heir  progression towards activation. As time passes, the impact of social influence
he  network has activated at least once). The tree structure is the least conducive to

ince the parameter ε captures social influence, our model assumes
hat the activation signals sent by neighbors are more consequen-
ial if they are concurrent (as in panel t4 of Fig. 1) than if they are
ot (panel t3). In other words: our model assumes that exposure to
ultiple signals matters not just because it reinforces affirmation

a process that we capture with the sudden increases in the pro-
ression towards the threshold zone specified by Eq. (2)); but also,
nd mostly, because it allows local activity to grow increasingly
orrelated over time. Introducing this temporal correlation is, we
elieve, a necessary ingredient to build realistic models of large-

cale coordination, especially given the available evidence on the
emporal dynamics and bursts of activity characteristic of human
ommunication (Vazquez et al., 2006).
ω = 3 and ε = 0.008. The initial state randomly allocates actors to different points in
) starts aligning nodes to the same timing (a tcycle is complete when every actor in
ination.

Our analytical choice acknowledges the important difference
between having multiple friends participating in, say, #Black-
LivesMatter discussions or encouraging #OccupyCentral actions
at different, uncoordinated times than having them all converge
to the same timing. Convergence in the timing of activations is
more conducive to further activations, which, in turn, reinforces
the feedback mechanism that makes an obscured issue suddenly
jump to the spotlight of media attention. This is what happened
in Ferguson during the first hours of the demonstrations. Journal-
ists learned about the events through their Twitter feeds (where

hashtags created a channel for relevant information to flow in a
coordinated fashion), not from their own  news organizations (Carr,
2014). Large-scale coordination becomes visible only when the tim-
ings of individual activations become highly correlated; and this is



330 P. Piedrahita et al. / Social Networks 54 (2018) 326–335

Fig. 4. Network Topologies Used in the Simulation Experiments.
We  use four topologies to determine how actors influence each other via the ε signal. All networks were generated using the configuration model (Newman, 2010), with
the  exception of the small world network, for which we  used the Watts-Strogatz model 
regular network (we  also run some simulations with p = 0.2, with qualitatively similar re
the  Erdős–Rényi and the scale free networks) and degree (for the regular and small-worl

Fig. 5. Maximum Number of Coordinated Actors as a Function of Time across ω Val-
ues.
The curves track the fraction of actors in the network that activate simultane-
ously. Simulations here run on a small world network with rewiring probability
p  = 0.1 and a fixed ε = 0.01. As expected, high ω values (which here are distributed
homogeneously) lead to faster large-scale coordination. As ω decreases, the time to
f
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ull  coordination increases, down to values for which system-level coordination is
nattainable (i.e. ω = 6, a condition under which only small clusters of coordinated
odes emerge).

n aspect that cannot be captured by models that disregard the
ffects of time on activation dynamics.

To sum up, the motivational structure of actors in our model is
etermined by the parameter ω, which defines how quickly they
each the activation zone; and by the parameter ε, which deter-
ines the strength of social influence. In a world of isolated actors,

 would equal 0; in a world where social influence overrides the
hythms of intrinsic activation, ε would equal 1. Likewise, in a world
f identical actors, ω would be distributed homogeneously; the
ore heterogeneous the distribution, the more unequal actors are

n their propensity to activate. These two parameters open the basic
xperimental space of our model. One additional assumption our
odel makes is that every activation is followed by a reset mech-

nism that brings actors back to the beginning of their activation
ycles. In other words, our model does not incorporate memory or
earning, which we could hypothesize to accelerate activation as
ime goes by − and could be modelled by allowing ω to increase as
earning happens. The results that follow do not allow the intrinsic
ropensity of actors to change; the only thing that changes is the

nformation environment in which they operate, which is defined

y their networks and the activations that take place in those
etworks. Future research, however, should consider the impact
hat allowing actors to change their attitudes (as modelled by the
arameter ω) would have on coordination dynamics.
(Watts and Strogatz, 1998). The small world network rewires 1% of the ties of the
sults). All networks have the same size (N = 104 ) and the same average degree (for
d networks).

4. The dynamics of repeated activation

The combination of values for the two  parameters ω and ε (when
ε > 0) determines the speed at which coordination emerges − that
is, how long it takes for all nodes to start pulsating, or activating,
concurrently (e.g. as when many people simultaneously use a new
hashtag). However, the underlying network determining the path-
ways for influence is also a crucial component of how we  think
about contagion dynamics. The core of our analyses aim, in fact,
to determine the impact that different network structures have on
those dynamics, holding ω and ε constant.

To illustrate why networks matter in the context of our model,
Fig. 3 summarizes contagion dynamics in toy networks with size
N = 10, and values ω = 3 and ε = 0.008. The initial state randomly
allocates actors to different points in their progression towards
activation, which means that they pulsate at different times, as
matrices tcycle = 1 show (a tcycle is complete when every actor in the
network has activated at least once). As time progresses, however,
the impact of social influence (via ε) starts aligning nodes to the
same timing. This is particularly clear in the case of the directed
cycle. The undirected version of the cycle requires more time for
actors to coordinate their activations; in fact, there is still an actor
that activates with its own timing at tcycle = 70. The tree structure,
also undirected, is the least conducive to coordination: the pres-
ence of hubs, and their greater influence over the peripheral nodes
that are only connected through them, hampers the spontaneous
emergence of coordination.

5. Social context as communication networks

The topology on which interactions take place is, therefore, a
crucial element in the dynamics we  want to model. We  run exper-
iments on four network topologies, summarized in Fig. 4. These
networks determine how actors influence each other via the ε
signal, and they capture different hypothetical scenarios where
interactions might unfold empirically.

In the Erdős–Rényi network, for instance, ties connecting the
actors are formed at random. Although we know that social net-
works are never formed at random, this topology could account for
a scenario where actors are connected through their online search
patterns, i.e. by looking at what others are posting on websites or
blogs beyond social media platforms. This network also offers a

standard benchmark with which to assess the performance of the
other three topologies.

The regular network offers a way  of mapping interdependence
when it is highly structured by logistical or space constraints. Dur-



P. Piedrahita et al. / Social Networks 54 (2018) 326–335 331

Fig. 6. The Impact of Social Influence on Large-Scale Coordination across � Values.
The  panels summarize coordination dynamics for different values of ω (the intrinsic motivation parameter) and � (social influence strength) across the four network topologies.
Every dot in the plots corresponds to a combination of parameters ε and ω; the distribution of ω and ε is homogenous across nodes and edges, respectively. The color scheme
indicates how long it takes, for each combination, to reach large-scale coordination (which here we define as at least 75% of the nodes activating simultaneously); time
is  averaged over 100 realizations of the simulation. Lighter colors indicate earlier coordination, darker colors indicate later coordination; black signals that no large-scale
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oordination was possible. The findings suggest that random, homogenous network
y  the presence of hubs (i.e. scale free networks) do not allow large-scale coordi
estrictive in the emergence of coordination than regular networks, in spite of the g

ng the 2014 Umbrella Revolution in Hong Kong, for instance, social
edia and other Internet-based modes of communication were

ensored by the Chinese government, so protesters used the Blue-
ooth technologies in their cell phones to create mesh networks
nd coordinate their actions while on the streets (Knibbs, 2014;
arker, 2014; Rutkin and Aron, 2014). These networks do not rely
n online servers (and are, therefore, more difficult to monitor by
hird parties); but they require physical proximity: they are only
easible when there is a large number of people concentrated in
estricted spaces (like concert halls, stadiums or, as in this case, a
ew streets within the same city district). Regular networks offer
n approximation to that sort of empirical scenario.

The small world and scale free networks are the topologies we
se to approximate most observed networks. There is ample evi-
ence that social networks exhibit the small world property (Watts,
003) and they also tend to have a very skewed degree distribu-
ion, especially those that emerge online (Barabási, 2009). Twitter,
or instance, has a long tail in the allocation of connections, with

 minority of accounts being disproportionately better connected
han the vast majority (Kwak et al., 2010). Similar properties have
een found in other social media platforms like Facebook or the
hinese Sina Weibo (Backstrom et al., 2012; Ugander et al., 2011;
hengbiao et al., 2011). We  reproduce these structural features in
ur simulation experiments because social media networks have
een shown to play an important part in the emergence of large-

cale coordination, from agreeing on which hashtags to use to
rganizing massive demonstrations (Barberá et al., 2015; Conover
t al., 2013; González-Bailón et al., 2011; Romero et al., 2011;
teinert-Threlkeld et al., 2015; González-Bailón and Wang, 2016).
ore conducive to large-scale coordination. Heterogeneous networks characterized
 when the social influence signal weakens. Small world networks are also more
shortcuts created by random rewiring (or because of them).

To recover the example introduced above, the growth of the #Black-
LivesMatter movement relied heavily on the coordinating potential
of social media.

6. The effects of network topology

Our analyses aim to identify the conditions that need to be in
place for large-scale coordination to emerge. Given that the time
to full coordination depends on the specific combination of ω and
ε, but also on the underlying network, we  measure time in terms
of tcycles, which were illustrated in Fig. 3. This definition allows us
to normalize time across conditions and directly compare coordi-
nation dynamics across networks and parametric settings. From an
empirical point of view, every step in the evolution of our model
(every tcycle) can be interpreted as a different time window, e.g.
hourly, daily, weekly, or monthly activity. Finding the appropriate
temporal resolution to empirically analyze evolving dynamics in
networks is not a trivial issue (Holme and Saramäki, 2012; Moody,
2002). Our model does not make any specific assumptions about the
right resolution to aggregate observed activation data; the time it
takes for a cycle to complete can correspond to different empirical
windows − and, in fact, the appropriate width for that window is
likely to change as periods of bursts in activity unfold in chrono-
logical time (Borge-Holthoefer et al., 2016).

At the end of every tcycle, i.e., once every node has activated at

least once, we count the number of nodes that activated simultane-
ously − i.e. the size of the clusters in the matrices of Fig. 3. Our model
allows large-scale coordination to arise when small local islands of
coordinated nodes start merging together through the cascading



3 l Networks 54 (2018) 326–335

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(
s
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t
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m
t
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Fig. 7. Heterogeneous Distribution of the Speed-to-Activation Parameter ω.
32 P. Piedrahita et al. / Socia

ffects of social influence, as captured by the parameter ε and as
hanneled by the network. Fig. 5 shows what happens with the
evels of coordination as the system evolves with a fixed ε = 0.01 on

 small world graph with size N = 104. The curves track the fraction
f actors that activate simultaneously. As expected, high ω values
which, in this example, is the same for all actors) lead to faster
arge-scale coordination. As ω decreases, the time to full coordina-
ion increases. At low values (i.e. ω = 6), system-level coordination
s unattainable: this is a condition under which only small clusters
f coordinated nodes emerge.

The question we are interested in is: How do contagion dynam-
cs differ when ω and ε are held constant but the underlying
etworks change? Fig. 6 shows a first set of results to answer this
uestion. Every dot in the heatmaps corresponds to a combination
ε, ω). On the left of the horizontal axis we have systems where
ocial influence is very strong; as we move to the right, the impact
f neighbor activations on the focal actor starts diminishing. At the
ottom of the vertical axis, we have actors that progress slowly
owards the activation zone; at the top, we have those that get
ery quickly into a tipping-point state. In this set of simulations, the
ropensity to activate (the ω value) is distributed homogeneously
cross all actors in the network; what changes is the structure of
he underlying network.

The color scheme indicates the time it takes under each para-
etric combination to reach large-scale coordination, measured as

cycles. We  define large-scale coordination as having at least 75% of
he nodes activating simultaneously. For each point, time is aver-
ged over 100 realizations of the simulation, with different initial
onditions. In this scheme, lighter colors indicate earlier coordina-
ion; as the colors get darker, coordination takes longer to emerge.
lack signals that no coordination was possible within the limit of
00 tcycles, when the simulations stopped.

These results suggest that all networks are capable of generating
oordination in scenarios with strong to moderate social influence
0.5 < � < 1), regardless of the actors’ propensity to activate (regard-
ess of the ω value). As ε starts getting smaller (i.e. as the strength
f social influence diminishes), actors need to have steeper incli-
ations to reach the tipping point for coordination to emerge. A
etwork where ties channel little impact takes more time, and
equires more motivated actors, to generate the same level of coor-
ination than a network with stronger ties. After some critical
oint, no amount of actor predisposition can overcome the lack of
ubstantive social influence. This critical point, however, changes
cross networks: in the random, Erdős–Rényi network, large-scale
oordination emerges for most social influence conditions when ω
s high, even when the impact of each neighbor activation is really
ow. This is not the case for the regular, the small world, and the
cale free networks, which are way more restrictive in their support
o spontaneous coordination. The scale-free network is particularly
imiting: it either allows coordination to emerge fast (white region)
r it prevents it very abruptly (black region). The existence of hubs,
o characteristic in the structure of these networks, explains why
uch an abrupt transition takes place: because hubs are so much
etter connected than the other nodes, they have a wide impact
hen they activate; but hubs, which are surrounded by many struc-

ural holes (Burt 1992), also restrict the pathways for contagion, and
or the alignment of local dynamics.

Given that most social media networks are well represented
y the scale free structure, our simulations suggest two possibili-
ies: either online ties channel stronger influence than traditionally
cknowledged (e.g., Gladwell, 2010); or users are so ready to acti-
ate that coordination is possible even with weak social influence

but not too weak). This is indeed what seems to happen during
he emergence of campaign hashtags. Social media users tend to be
roactive in their behavior to facilitate coordination; in fact, the use
f hashtags in Twitter emerged itself as a user-driven convention
In  a second set of experiments we introduced actor heterogeneity by drawing the
parameter ω from different normal distributions, centered around mean ω = 50 and
with a standard deviation in the range � = [1,10].

(Parker, 2011). This high predisposition, especially amongst those
who opted into a movement or mobilization (as our model pre-
sumes), compensates for the constraints imposed by the network
to spreading dynamics.

7. The effects of actor heterogeneity

The findings above are interesting because they cast light on
the importance that network topology has to delimit the possibility
space for large-scale coordination. However, it is a big simplifica-
tion to assume that all actors have the same propensity to reach
their activation zone. In a second set of simulations, we introduced
heterogeneity in the distribution of the ω parameter, as illustrated
in Fig. 7. We  randomly drew N = 104 values from a normal dis-
tribution centered around ω = 50 and a standard deviation in the
interval � = [1,10], with 0.1 increases. A condition where actors
differ slightly in their predispositions to act corresponds to sce-
narios where exogenous events instill a sense of urgency in the
need to act, as it happened in Ferguson. A condition where actors
are very heterogeneous, on the other hand, corresponds to situ-
ations where the level of commitment to a cause varies amongst
those willing to participate. For instance, in the Hong Kong protests
students triggered a movement that soon escalated to involve a
larger group of participants, partly thanks to the aid of social media
(Parker, 2014). Students had the ability to camp on the streets and
the time to generate the messages, photos, and videos that others
(including mainstream media) picked up soon after. Other demo-
graphic groups (e.g., parents, middle class professionals) might
have wanted to join the protests but they were unable to do so
with similar dedication because of their job schedules or other time
constraints. Sociological factors like these could be a source of het-
erogeneity in the ability to activate for actors that are, otherwise,
equally interested in a political cause.

The results of this second set of simulations are shown in
Fig. 8. In general, the simulations reveal that heterogeneity reduces
opportunities for large-scale coordination across all networks. This
supports the intuition that, for a cause to grow large, actors need
to share predispositions, that is, they need to be as similar as possi-
ble in their willingness to act. The scale-free network is, again, the
most restrictive structure − but as long as ties channel some influ-
ence, coordination arises fast, which is important for time-sensitive
mobilizations (for instance, during the first hours of the 2013 Gezi
Park protests, when mainstream media were censorig news of the
events on the ground, Barberá et al., 2015). Given that large-scale
coordination emerges repeatedly (and swiftly) in social media sites,

our simulation results provide further evidence that online ties
weave relevant interdependence, that is, they act as a significant
source of social influence. This is consistent with experimental evi-
dence on the mobilizing potential of online networks (Bond et al.,



P. Piedrahita et al. / Social Networks 54 (2018) 326–335 333

Fig. 8. The Impact of Actor Heterogeneity on Large-Scale Coordination across ε Values.
The panels summarize coordination dynamics for different distributions of ω and ε values (social influence strength). The distribution of ω depends on the standard deviation
(vertical axis); ε is homogenous across edges. The color scheme indicates, again, the time it takes to reach large-scale coordination (i.e. at least 75% of the nodes activating
s  more,
b ity inc
i

2
m
T
r
t
a

8

e
g
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t
a
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t
d
(
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f
w

imultaneously); time is averaged over 100 realizations. The results show that, once
enchmark provided by the Erdős–Rényi topology. Overall, low to mild heterogene

t.

012), which shows that exposure to information through social
edia has a positive and significant impact on political behavior.

his positive impact is what we capture with the ε parameter. Our
esults show that as long as the impact of social influence is not
oo low, it can drive the network towards coordination even when
ctor heterogeneity is high.

. Discussion

Our results show that network topology has counter-intuitive
ffects on coordination when repeated activation is allowed. Homo-
eneous networks, that is, networks where the degree distribution
s not significantly skewed, are more conducive to coordina-
ion: the parametric combinations (ω, ε) leading to coordination
re wider for more egalitarian networks, following this order:
rdős–Rényi > small world > regular networks. This ranking applies
o conditions where ω is fixed but also where it is distributed ran-
omly. Networks characterized by a skewed degree distribution
that is, by the presence of a small group of nodes exerting more
nfluence over other nodes) are clearly less favorable to coordi-

ation: they require stronger influence and actors that are more
imilar in their propensity to activate. We  refer to this as the “scale-
ree paradox”: on the one hand, scale-free networks clearly create
orse conditions for contagion dynamics to spread under repeated
 all networks are less efficient in allowing large-scale coordination than the random
reases the probability of global coordination, whereas high heterogeneity hinders

activation; on the other hand, an increasing body of observational
evidence shows that these networks are also very good at help-
ing coordinate the actions of many (Barberá et al., 2015; Conover
et al., 2013; González-Bailón et al., 2011; Romero et al., 2011;
Steinert-Threlkeld et al., 2015). This empirical evidence suggests
that heterogeneous networks are indeed behind many observed
episodes of mass mobilization, regardless of the topological restric-
tions uncovered by our simulation results.

Related to this, our findings also suggest that online networks
must channel enough social influence to allow individual actions
to align over time. Our results show that there is a critical ε for
all topologies, that is, for a given network and ω there is always
a value for the social influence parameter below which actors do
not achieve coordination. This critical value changes across topolo-
gies, and it is particularly stringent for the scale-free networks.
Since most online networks are skewed in their degree distribu-
tion, we  contend that those networks must channel moderate to
strong social influence – otherwise, it is unlikely that large-scale
coordination would emerge so often though online channels.

Time-varying dynamics in networks have so far been largely
disregarded by analytical approaches to collective action – and

yet these dynamics are crucial, as our model suggests, to under-
stand the feedback mechanisms that activate cascading reactions
and the consolidation of a critical mass. Prior research has shown



3 l Netw

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34 P. Piedrahita et al. / Socia

hat attaining this critical mass depends on the network topology,
n particular the density and the centralization of ties (Marwell
nd Prahl, 1988). That work suggests that centralization always
as a positive effect on collective action because it increases the
robability that involved actors will be tied to a large number of
ontributors, allowing for more efficient coordination. Our model
uggests that highly centralized networks (in the form of scale-free
tructures) can indeed be very efficient in coordinating efforts but
nly when certain conditions are met. The strength of social influ-
nce and the distribution of propensities to activate need both to be
onducive to the critical mass. Compared to other network topolo-
ies, however, centralized structures perform significantly worse,
ll else equal.

By allowing activation to re-occur, we shift attention from the
iffusion of activations (the focus of traditional threshold mod-
ls) to their coordination (which happens during the second stage
f activity within adoption). What we find is that for a range of
arametric combinations (ω, ε), the four network topologies we
nalyze are equally successful at generating coordination. What
akes them differ is the impact that social influence has on col-

ective dynamics. As networks grow more heterogeneous in their
onnectivity, and as they open more structural holes, the space for
he emergence of large-scale coordination diminishes. This differ-
nce across networks results from how the underlying structure of
ommunication activates feedback mechanisms of reinforcement
hat align, with more or less success, individual decisions to acti-
ate.

There are two aspects of our modelling approach that deserve
uture consideration: the distribution of ε (which we  keep con-
tant across ties) and the way in which ω values are distributed
randomly, when heterogeneous). There are a number of reasons
hy these two choices could be modified. We  know that in social
etworks ties vary in their strength: the actions of relatives, friends
nd acquaintances, for instance, do not have the same effect on an
ctor’s behavior. Our model assumes that all ties channel the same
mount of influence. Although some ties activate more often than
thers (and are de facto more influential), their impact on activa-
ion responds to changing local events in the network, not to an
ttribute of the tie itself. Future work should consider coordination
ynamics under different distributional assumptions of ε. Likewise,
uture research should also analyze scenarios where the propensity
o activate (ω) is not distributed randomly but as a function of the
etwork topology itself. For instance, there is ample empirical evi-
ence to suggest that the values of ω might be more similar within
lusters in a network – if we assume that this is another dimension
n which homophily operates (McPherson et al., 2001). Students,
or example, are more likely to share the same predispositions and
e better connected to each other compared to other demographic
roups. At the same time, observational and analytical evidence
uggests that the importance of social influence can be overrated if
omophily is not properly taken into account by studies of conta-
ion (Aral et al. 2009, 2013; Aral and Walker, 2012). Future research
hould consider whether critical mass dynamics and the timing of
oordination differ substantially if we constrain the distribution of

 to the position of nodes in the network and, in particular, to their
lustering.

Another important question for future research is how much the
esults would vary if actors were equipped with memory, that is, if
hey did not reset their progression towards activation to 0 at the
nd of every tcycle. Equipping actors with memory would open the
oor to more explicit theorization on the impact that mechanisms

ike social learning have on activation dynamics. As it currently

tands, our model is aseptic about the specific mechanisms that
ive shape to the function expressed in Eq. (1). Our main explana-
ory variables are the networks assumed to underlie coordination
ynamics; we treat ω as a black box that determines the timing
orks 54 (2018) 326–335

of individual activations. When we  allow ω to differ from actor to
actor, the assumed heterogeneity can relate to different empirical
possibilities: more or less interest in a political cause, more or less
time to devote to the cause, etc. In any case, adding memory to
how our actors behave would require a solid empirical justification
of how memory operates in the context of coordination through
decentralized networks.

Finally, another important question that we  do not consider
directly relates to finding a temporal scale that is the most appro-
priate to empirically analyze coordination dynamics. As with most
analytical models, ours is developed on a level of abstraction that
allows generalizing across possible scenarios but does not give
precise guidelines as to how to aggregate empirical data. Digital
technologies are providing richer sources of data that could help
test empirically models like ours (Golder and Macy, 2014; Lazer
et al., 2009; Watts, 2007). Our model, in particular, requires a sys-
tematic approach to the analysis of time-evolving networks and
time-dependent activations (Holme and Saramäki, 2012; Moody,
2002). In data tracking social media activity, the temporal scale
can be expressed in terms of days, hours, or minutes – and the
most informative temporal scale might not even remain constant
during the observation window (Borge-Holthoefer et al., 2016).
Bringing closer the results of simulation models with the patterns
observed in empirical data requires solving first the temporal res-
olution problem. More research is necessary in this area, which
would help calibrate our model with the most likely parameters, as
inferred from observational data.

9. Conclusion

The model presented here casts light on how contagion dynam-
ics emerge when actors are allowed to activate repeatedly and
contribute intermittently to activity around a collective cause.
Theories of a critical mass and threshold models emphasize the
importance of interdependence, and highlight that collective action
is not about obtaining unanimous participation but about mobiliz-
ing enough people to make the effort self-sustaining. Our model
contributes to this broad line of research by focusing on the second
stage of coordination within adoption, that is, on the exchange of
information among actors who are already part of a political cause.
We emphasize the importance of temporal correlations in network
activity, so far largely disregarded in previous modelling efforts
but characteristic of many recent examples of observed large-scale
coordination. Our model shows that many contagion conditions
are not conducive to coordination. In particular, networks that are
more homogenous in their degree distribution facilitate coordina-
tion under a wider range of actor predisposition and social influence
conditions; as inequality in the degree distribution increases, how-
ever, so does the time required to achieve coordination – time that,
from an empirical point of view, might not always be available. Our
model also shows that when social influence has a moderate to
strong impact, large-scale coordination emerges regardless of the
underlying structure of communication, and regardless of actor’s
predisposition to act. To the extent that digital technologies are
inserting networks in every aspect of social life, our results sug-
gest that we  should expect to see more instances of large-scale
coordination cascading from the bottom-up.

Acknowledgements
P.P. acknowledges partial support from MINECO through grant
FIS2012- 38266; Y. M.  acknowledges partial support from the Gov-
ernment of Aragón, Spain through a grant to the group FENOL and
by MINECO and FEDER funds (grant FIS2014-55867-P).



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	The contagion effects of repeated activation in social networks
	1 Introduction
	2 Coordination as a two-step selection process
	3 Model and mechanisms
	4 The dynamics of repeated activation
	5 Social context as communication networks
	6 The effects of network topology
	7 The effects of actor heterogeneity
	8 Discussion
	9 Conclusion
	Acknowledgements
	References