Local cascades induced global contagion: How heterogeneous thresholds, exogenous effects, and unconcerned behaviour govern online adoption spreading

Abstract

Adoption of innovations, products or online services is commonly interpreted as a spreading process driven to large extent by social influence and conditioned by the needs and capacities of individuals. To model this process one usually introduces behavioural threshold mechanisms, which can give rise to the evolution of global cascades if the system satisfies a set of conditions. However, these models do not address temporal aspects of the emerging cascades, which in real systems may evolve through various pathways ranging from slow to rapid patterns. Here we fill this gap through the analysis and modelling of product adoption in the world's largest voice over internet service, the social network of Skype. We provide empirical evidence about the heterogeneous distribution of fractional behavioural thresholds, which appears to be independent of the degree of adopting egos. We show that the structure of real-world adoption clusters is radically different from previous theoretical expectations, since vulnerable adoptions --induced by a single adopting neighbour-- appear to be important only locally, while spontaneous adopters arriving at a constant rate and the involvement of unconcerned individuals govern the global emergence of social spreading.

Full text

Local cascades induced global contagion:
How heterogeneous thresholds, exogenous effects, and
unconcerned behaviour govern online adoption
spreading
M´arton Karsai ∗1, Gerardo I˜niguez2,3, Riivo Kikas4,5, Kimmo Kaski2, and J´anos Kert´esz6,7
1Laboratoire de l’Informatique du Parall´elisme, INRIA-UMR 5668, IXXI, ENS de Lyon, 69364 Lyon,
France
2Department of Computer Science, School of Science, Aalto University, 00076, Finland
3Centro de Investigaci´on y Docencia Econ´omicas, CONACYT, 01210 M´exico D.F., Mexico
4Institute of Computer Science, University of Tartu, 50409 Tartu, Estonia
5Software Technology and Applications Competence Center (STACC), 51003 Tartu, Estonia
6Center for Network Science, Central European University, 1051 Budapest, Hungary
7Institute of Physics, Budapest University of Technology and Economics, 1111 Budapest, Hungary
keywords: cascading behaviour, social spreading phenomena, complex contagion, adoption thresholds
Abstract
Adoption of innovations, products or online services is commonly interpreted as a spreading process
driven to large extent by social influence and conditioned by the needs and capacities of individuals.
To model this process one usually introduces behavioural threshold mechanisms, which can give rise
to the evolution of global cascades if the system satisfies a set of conditions. However, these models
do not address temporal aspects of the emerging cascades, which in real systems may evolve through
various pathways ranging from slow to rapid patterns. Here we fill this gap through the analysis and
modelling of product adoption in the world’s largest voice over internet service, the social network of
Skype. We provide empirical evidence about the heterogeneous distribution of fractional behavioural
thresholds, which appears to be independent of the degree of adopting egos. We show that the structure
of real-world adoption clusters is radically different from previous theoretical expectations, since vul-
nerable adoptions—induced by a single adopting neighbour—appear to be important only locally, while
spontaneous adopters arriving at a constant rate and the involvement of unconcerned individuals govern
the global emergence of social spreading.
Introduction
Spreading of opinions, frauds, behavioural patterns, and product adoptions are all examples of social conta-
gion phenomena where collective patterns emerge due to correlated decisions of a large number of individuals.
Although these choices are personal, they are not independent but potentially driven by several processes
such as social influence [1], homophily [2], and information arriving from external sources like news or mass
media [3]. Social contagion evolves over networks of interconnected individuals, where links associated with
∗Corresponding author: marton.karsai@ens-lyon.fr
1
arXiv:1601.07995v1 [physics.soc-ph] 29 Jan 2016

social ties transfer influence between peers [4]. Several earlier studies aimed to identify the dominant mech-
anisms at play in social contagion processes [5, 6, 7, 8]. One key element, termed behavioural threshold
by Granovetter [6], is defined as “the number or proportion of others who must make one decision before a
given actor does so”. Following this idea various network models have been introduced [9, 10, 11, 12, 13, 14]
to understand the threshold-driven spreading, commonly known as complex contagion [15]. Although these
models are related to a larger set of collective dynamics, they are particularly different from simple conta-
gion where the exposure of nodes is driven by independent contagion stimuli [16, 17]. In addition, collective
adoption patterns may appear as a consequence of homophilic structural correlations, where connected in-
dividuals adopt due to their similar interests and not due to direct social influence. Distinguishing between
the effects of social influence and homophily at the individual level remains as a challenge [18, 3]. Further-
more, in real social spreading phenomena all these mechanisms are arguably present. However, while in the
case of homophily the adoption behaviour is only seemingly correlated, and for simple contagion only the
number of exposures matters, in complex contagion the fraction of adopting neighbours relative to the total
number of partners determines whether a node adopts or not, capturing the natural mechanisms involved in
individuals’ decision makings [20, 21, 22]. Due to this additional complexity, threshold models are able to
emulate system-wide adoption patterns known as global cascades.
Behavioural cascades are rare but potentially stupendous social spreading phenomena, where collective
patterns of exposure emerge as a consequence of small initial perturbations. Some examples are the rapid
emergence of political and grass-root movements [24, 25, 26], fast spreading of information [27, 28, 29, 12,
30, 31, 32, 33] or behavioural patterns [34], etc. The characterisation [33, 35, 36, 37, 38, 39] and modelling
[9, 40, 41, 42] of such processes have received plenty of attention and provide some basic understanding of
the conditions and structure of empirical and synthetic cascades. However, these studies commonly fail in
addressing the temporal dynamics of the emerging cascades, which may vary considerably between different
cases of social contagion. Moreover, they have not answered why real-world cascades can evolve through
various dynamic pathways ranging from slow to rapid patterning, especially in systems where the threshold
mechanisms play a role and social phenomena spread globally. Besides the case of rapid cascading mentioned
above, an example of the other extreme is the propagation of products in social networks [17], where adoption
evolves gradually even if it is driven by threshold mechanisms and may cover a large fraction of the total
population [22]. This behaviour characterises the adoption of online services such as Facebook, Twitter,
LinkedIn and Skype (Fig.1a), since their yearly maximum relative growth of cumulative adoption [1] (for
definition see Material and Methods (MM)) is lower than in the case of rapid cascades as suggested e.g. by
the Watts threshold (WT) model.
To fill this gap in the modelling of social diffusion, here we will analyse and model real-world examples of
social contagion phenomena. Our aim is to identify the crucial mechanisms necessary to consider in models of
complex contagion to match them better with reality, and define a model that incorporates these mechanisms
and captures the possible dynamics leading to the emergence of real-world global cascades. We follow the
adoption dynamics of the Skype paid service “buy credit” for 89 months since 2004, which evolves over the
social network of one of the largest voice over internet providers in the world. Data includes the time of
first payment of each user, an individual and conscious action that tracks adoption behaviour. In addition
we follow the “subscription” service over 42 months since 2008 (for results see Supplementary Information
[SI]). In contrast to other empirical studies where incomplete knowledge about the underlying social network
leads to unavoidable bias [22], we use here the largest connected component of the aggregated free Skype
service as the underlying structure, where nodes are Skype users and links confirmed contacts between them.
This is a good approximation since it maps all connections in the Skype social network without sampling,
and the paid service is only available for individuals already enrolled in the Skype network. The underlying
structure is an aggregate from September 2003 to November 2011 (i.e. over 99 months) and contains roughly
4.4 billion links and 510 million registered users worldwide [12]. The data is fully anonymised and considers
only confirmed connections between users (for more data details see SI).
In what follows we first provide empirical evidence of the distribution of individual adoption thresholds
and other structural and dynamical features of a worldwide adoption cluster. We incorporate the observed
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structural and threshold heterogeneities into a dynamical threshold model where multiple nodes adopting
spontaneously (i.e. firstly among their neighbours) are allowed [6]. We find that if the fraction of users who
reject to adopt the product is large, the system enters a quenched state where the evolution and structure of
the global adoption cluster is very similar to our empirical observations. Model calculations and the analysis
of the real social contagion process suggest that the evolving structure of an adoption cluster differs radically
from what has been proposed earlier [9], since it is triggered by several spontaneous adoptions arriving at
a constant rate, while stable adopters who are initially resisting exposure, are actually responsible for the
emergence of global social adoption (Fig. 1b and c).
Results
Social contagion phenomena can be modelled as binary-state processes evolving on networks and driven
by threshold mechanisms. In these systems individuals are represented by nodes, each being either in a
susceptible (0) or adopter (1) state and influencing each other by transferring information via social ties [6].
Nodes are connected in a network with degree distribution P(k) and average degree z=hki. In addition,
each node has an individual threshold φ∈[0,1] drawn from a distribution P(φ) with average w=hφi.
This threshold determines the minimum fraction of exposed neighbours that triggers adoption and captures
the resistance of an individual against engaging in spreading behaviour. Once a node reaches its threshold,
it switches state from 0 to 1 and keeps it until the end of the dynamics. In his seminal paper about
threshold dynamics, Watts [9] classified nodes into three categories based on their threshold and degree. He
identified innovator nodes that spontaneously change state to 1, thus starting the process. Such nodes have
a trivial threshold φ= 0. Then there are nodes with threshold 0 < φ ≤1/k, called vulnerable, which need
one adopting neighbour before their own adoption. Finally, there are more resilient nodes with threshold
φ > 1/k, denoted as stable, referring to individuals in need of strong social influence to follow the actions of
their acquaintances.
In the WT model [9], small perturbations (like the spontaneous adoption of a single seed node) can
trigger global cascading patterns. However, their emergence is subject to the so-called cascade condition:
the innovator seed has to be linked to a percolating vulnerable cluster, which adopts immediately afterwards
and further triggers a global cascade (i.e. a set of adopters larger than a fixed fraction of the finite network).
The cascade condition is satisfied if the network is inside a bounded regime in (w, z)-space [9]. This regime
depends on degree and threshold heterogeneities [9] and may change its shape if several innovators start the
process [41].
Empirical observations
Degree and threshold heterogeneities are indeed present in the social network of Skype. The degree distri-
bution P(k) is well approximated by a lognormal function P(k)∝k−1e−(ln k−µD)2/(2σ2
D)(k≥kmin) with
parameters µD= 1.2, σD= 1.39 and kmin = 1 (Fig. 1d), giving an average degree z= 8.56 (for goodness of
fit see SI). Moreover, at the time of adoption we can measure the threshold φ= Φk/k of a user by counting
the number Φkof its neighbours who have adopted the service earlier. We then group users by degree and
calculate the distribution P(Φk) of the integer threshold Φk[37] (Fig. 1e). By using the scaling relation
P(Φk, k) = kP (Φk/k) all distributions collapse to a master curve well approximated by a lognormal function
P(φ)∝φ−1e−(ln φ−µT)2/(2σ2
T), with parameters µT=−2 and σT= 1 as constrained by the average threshold
w= 0.19 (see MM and SI). Note that we observe qualitatively the same scaling and lognormal shape of the
threshold distribution for another service (see SI). These empirical observations, in addition to the broad
degree distribution, provide quantitative evidence about the heterogeneous nature of adoption thresholds.
Since we know the complete structure of the online social network, as well as the first time of service
usage for all adopters, we can follow the temporal evolution of the adoption dynamics. By counting the
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(d)
(c)
(b)
(a)
(e)
(f)
(a)
(c)
(e)
(d)
(b)
Figure 1: Structure and dynamics of online service adoption. (a) Yearly maximum relative growth
rate (RGR) of cumulative adoption (see MM) for several online social-communication services [1], including
three Skype paid services (s1 - ”subscription”, s2 - ”voicemail”, and s3 - ”buy credit”). The red bar
corresponds to a rapid cascade of adoption suggested by the Watts threshold (WT) model, while the green
bar is the model prediction for Skype s3. (b-c) Snowball sample of the Skype social network (gray links)
with nodes and links coloured according to their adoption state: multiple innovators (green nodes), induced
small vulnerable trees (red nodes and links), and the triggered connected stable cluster (blue nodes and
links). Note that some vulnerable and stable clusters seemingly appear without an innovator seed due
to the finite distance used in the snowball sampling method. (d) Degree distribution P(k) of the Skype
network (gray/blue circles for raw/binned data) on double log-scale with arbitrary base n.P(k) is fitted by
a lognormal distribution (see MM and SI) with parameters µD= 1.2 and σD= 1.39, and average z= 8.56
(red line). (e) Distribution P(Φk) of integer thresholds Φkfor several degree groups in Skype s3 (inset). By
using P(Φk, k) = kP (Φk/k), these curves collapse to a master curve approximated by a lognormal function
(dashed line in main panel) with parameters µT=−2 and σT= 1, as constrained by the average threshold
w= 0.19 (see MM and SI). (f) Adoption rate of innovators [Ri(t)], vulnerable nodes [Rv(t)], and stable nodes
[Rs(t)], as well as net service adoption rate [R(t)]. Rates are measured with a 1-month time window, while q
and τare arbitrary constants. The shaded area indicates the regime where innovators adopt approximately
with constant rate.
number of adopting neighbours of an ego, we identify innovators (Φk= 0), and vulnerable (Φk= 1) or stable
(Φk>1) nodes. The adoption rates for these categories behave rather differently from previous suggestions
[9] (Fig. 1f). First, there is not only one seed but an increasing fraction of innovators in the system who, after
an initial period, adopt approximately at a constant rate. Second, vulnerable nodes adopt approximately
with the same rate as innovators suggesting a strong correlation between these types of adoption. Third,
the overall adoption process accelerates due to the increasing rate of stable adoptions induced by social
influence. At the same time a giant adoption cluster grows and percolates through the whole network
(Fig. 3a, main panel). Despite of this expansion dynamics and connected structure of the service adoption
cluster, the service reaches less than 6% of the total number of active Skype users over a period of 7 years
[12]. Therefore we ask whether one can refer to these adoption clusters as cascades. They are not triggered
by a small perturbation but induced by several innovators; their evolution is not instantaneous but ranges
through several years; and although they involve millions of individuals, they reach only a reduced fraction
of the whole network. To answer we incorporate the above mentioned features into a dynamical threshold
model [6] with a growing group of innovators and investigate their effect on the evolution of global social
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adoption. Note that we also perform a null model study to demonstrate, on the system level, that social
influence dominates the contagion process, but not homophily (see section S3 of the SI, together with another
empirical spreading scenario in S7.1).
Model
Our modelling framework is an extension to conventional threshold dynamics on networks studied by Watts,
Gleeson, Singh, and others, where all nodes are initially susceptible and innovators are only introduced as an
initial seed of arbitrary size [9, 42, 41]. Apart from the threshold rule discussed above, our model considers
two additional features: (i) a fraction rof ‘immune’ nodes that never adopt, indicating lack of interest in
the service; (ii) due to external influence, susceptible nodes adopt the innovation spontaneously (i.e. become
innovators) throughout time with constant rate pn, rather than only at the beginning of the dynamics. In
this way, the dynamical evolution of the system is completely defined by the online social network, the
distribution P(φ) and the parameters r,pn. For the sake of simplicity we consider a configuration-model
network, i.e., we ignore correlations in the social network and characterise it solely by its degree distribution
P(k). Furthermore, node degrees and thresholds are considered to be independent [37, 46, 47].
Our threshold model, which has also been introduced in [6], can be studied analytically by extending the
framework of approximate master equations (AMEs) for monotone binary-state dynamics recently developed
by Gleeson [37, 46, 47], where the transition rate between susceptible and adoption states only depends on
the number mof network neighbours that have already adopted. We describe a node by the property vector
k= (k, c), where k=k0, k1,...kM−1is its degree and c= 0,1, . . . , M its type, i.e. c= 0 is the type
of the fraction rof immune nodes, while c6= 0 is the type of all non-immune nodes that have threshold
φc. In this way P(φ) is substituted by the discrete distribution of types P(c) (for c > 0). The integer M
is the maximum number of degrees (or non-zero types) considered in the AME framework, which can be
increased to improve the accuracy of the analytical approximation at the expense of speed in its numerical
computation (see S4.2). Under these conditions, the AME system describing the dynamics of the threshold
model is reduced to the pair of ordinary differential equations (see SI),
˙ρ=h(ν, t)−ρ, (S1a)
˙ν=g(ν, t)−ν, (S1b)
where ρ(t) is the fraction of adopters in the network, ν(t) is the probability that a randomly chosen neighbour
of a susceptible node is an adopter, and the initial conditions are ρ(0) = ν(0) = 0. Here,
h= (1 −r)hft+ (1 −ft)X
k|c6=0
P(k)P(c)X
m≥kφc
Bk,m(ν)i,(S2)
and,
g= (1 −r)hft+ (1 −ft)X
k|c6=0
k
zP(k)P(c)X
m≥kφc
Bk−1,m(ν)i,(S3)
where ft= 1 −(1 −pr)e−prt,pr=pn/(1 −r), and Bk,m(ν) = k
mνm(1 −ν)k−mis the binomial distribution.
The fraction of adopters ρis then obtained by solving Eq. (S27) numerically. Since susceptible nodes adopt
spontaneously with rate pn, the fraction of innovators ρ0(t) in the network is given by (see S4.3),
ρ0=prZt
0
(1 −r−ρ)dt. (S4)
We also implement the threshold model numerically via a Monte Carlo simulation in a network of size
N, with a lognormal degree distribution and a lognormal threshold distribution as observed empirically.
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Figure 2: Threshold model for the adoption of online services. (a-b) Surface plot of the normalised
fraction of adopters ρ/(1 −r) in (w, z)-space, for r= 0.73 and t= 89. Contour lines signal parameter values
for which 20% of non-immune nodes have adopted, for fixed rand varying time (a), and for fixed time
and varying r(b). The continuous contour line and dot indicate parameter values in the last observation
of Skype s3. A regime of maximal adoption (ρ≈1−r) grows as time goes by, and shrinks for larger r.
(c) Time series of the fraction of adopters ρfor fixed pn= 0.00019 and varying r(main), and for fixed
r= 0 and varying pn(inset). These curves are well approximated by the solution of Eq. (S27) for k0= 3,
kM−1= 150 and M= 25 (dashed lines). The dynamics is clearly faster for larger pn. As rincreases, the
system enters a regime where the dynamics is slowed down and adopters are mostly innovators. (d) Final
fraction of innovators ρ0(∞) and time tcwhen 50% of non-immune nodes have adopted as a function of r,
both simulated and theoretical. The crossover to a regime of slow adoption is characterised by a maximal
fraction of innovators and time tc. Unless otherwise stated, pn= 0.00019 and we use N= 104,µD= 1.09,
σD= 1.39, kmin = 1, µT=−2, and σT= 1 to obtain z= 8.56 and w= 0.19 as in Skype s3. The difference
in µDbetween data and model is due to finite-size effects (see Materials and Methods). Numerical results
are averages over 102(a-b) and 103(c-d) realisations.
Thus, we can explore the behaviour of ρand ρ0as a function of z,w,pnand r, both in the numerical
simulation and in the theoretical approximation given by Eqs. (S27) and (S4). For pn>0 some nodes adopt
spontaneously as time passes by, leading to a frozen state characterised by a final fraction ρ(∞) = 1 −r
of adopters. However, the time needed to reach such state depends heavily on the distribution of degrees
and thresholds, as signalled by a region of large adoption (ρ≈1−r) that grows in (w, z)-space with time
(contour lines in Fig. 2a). If we fix a time in the dynamics and vary the fraction of immune nodes instead,
this region shrinks as rincreases (contour lines in Fig. 2b). In other words, the set of networks (defined by
their average degree and threshold) that allow the spread of adoption is larger at later times in the dynamics,
or when the fraction of immune nodes is small. When both tand rare fixed, the normalised fraction of
adopters ρ/(1 −r) gradually decreases for less connected networks with larger thresholds (surface plot in
Fig. 2a and b).
For r≈0 the critical fraction of innovators necessary to trigger a cascade of fast adoption throughout
all susceptible nodes may be identified as the inflection point in the time series of ρ(Fig. 2c, inset). The
adoption cascade appears sooner for larger pn, since this parameter regulates how quickly the critical fraction
of innovators is reached. Yet as we increase rabove a threshold rc, the system enters a regime where rapid
cascades disappear and adoption is slowed down. The crossover between these regimes is gradual, as seen in
the shape of ρfor increasing r(Fig. 2c, main panel). We may identify rcin various ways: by the maximum
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in both the final fraction of innovators ρ0(∞) and the critical time tcwhen ρ= (1 −r)/2 (Fig. 2d), or as
the rvalue where the inflection point in ρdisappears. These measures indicate rc≈0.8 for the chosen
parameters. All global properties of the dynamics (like the functional dependence of ρand ρ0) are very
well approximated by the solution of Eqs. (S27) and (S4) (dashed lines in Fig. 2c and d). Indeed, the AME
framework is able to capture the shape of the ρtime series, the crossover between regimes of fast and slow
adoption, as well as the maximum in ρ0(∞) and tc.
Validation
To better understand how innovation spreads throughout real social networks, we take a closer look at the
internal structure of the service adoption process. By taking into account individual adoption times we con-
struct an evolving adoption network with links between users who have adopted the service before time tand
are connected in the social structure. In order to avoid the effect of instantaneous group adoptions (evidently
not driven by social influence), we only consider links between nodes who are neighbours in the underlying
social network and whose adoption did not happen at the same time. This way links in the adoption graph
indicate ties where social influence among individuals could have existed. The size distribution P(sa) of
connected components in the adoption network shows the emergence of a giant percolating component over
time (Fig. 3a), along with several other small clusters. Moreover, after decomposition we observe that the
giant cluster does not consist of a single innovator seed and percolating vulnerable tree [9], but builds up from
several innovator seeds that induce small vulnerable trees locally (Fig. 3b), each with small depth (Fig. 3d)
[48, 33]. At the same time the stable adoption network (considering connections between all stable adopters
at the time) has a giant connected component, indicating the emergence of a percolating stable cluster with
size comparable to the largest adoption cluster (Fig. 3a, inset). These observations suggest a scenario for
the evolution of the global adoption component different from earlier threshold models [9]. It appears that
here multiple innovators adopt at different times and trigger local vulnerable trees (Fig.1b), which in turn
induce a percolating component of connected stable nodes that holds the global adoption cluster together
(Fig.1c). Consequently, in the structure of the adoption network primary triggering effects are important
only locally, while external and secondary triggering mechanisms seem to be responsible for the emergence
of global-scale adoption.
To model the observed dynamics and explore the effect of immune nodes, we perform extensive numerical
simulations of the threshold model with parameters determined directly from the data (see MM and SI).
We use a network structure with empirical degree and threshold distributions and fix pn= 0.00019 as the
constant rate of innovators, implying that the time scale of a Monte Carlo iteration in the model is 1 month.
We measure the average size of the largest (LC) and second largest (LC2nd) connected components of the
background social network, and of the stable, vulnerable and global adoption networks, as a function of the
fraction of immune nodes r. After T= 89 iterations (matching the length of the real observation period)
we identify three regimes of the dynamics (Fig. 3c): if 0 < r < 0.6 (dark-shaded area) the spreading process
is very rapid and evolves in a global cascade, which reaches most of the nodes of the shrinking susceptible
network in a few iteration steps. About 10% of adopters are connected in a percolating stable cluster, while
vulnerable components remain very small in accordance with empirical observations. In the crossover regime
0.6< r < 0.8 (light-shaded area), the adoption process slows down considerably (Fig. 2d, lower panel), as
stable adoptions become less likely due to the quenching effect of immune nodes. The adoption process
becomes the slowest at rc= 0.8 (Fig. 2d, lower panel) when the percolating stable cluster falls apart, as
demonstrated by a peak in the corresponding LC2nd curve (Fig. 3c, lower panel). Finally, around r= 0.9 the
adoption network becomes fragmented and no global diffusion takes place. We repeat the same calculations
for another service and find qualitatively the same picture, but with the crossover regime shifted towards
larger rvalues due to the different parametrisation of the model process. Note that another possible reason
for the slow adoption could be the time users wait between their threshold has been reached and actual
adoption. We test for the effect of this potential scenario on the empirical curves but find no qualitative
change in the dynamics (see SI).
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(b)
(a)
(c)
(d)
(e)
Figure 3: Empirical cluster statistics and simulation results. (a) Empirical connected-component
size distribution at different times for the adoption [P(sa), main panel] and stable adoption [P(ss), inset]
networks, with saand ssrelative to system size. (b) Empirical connected-component size distribution P(sv)
for the relative size of innovator-induced vulnerable trees at different times. (c) Average size of the largest
(LC) and 2nd largest (LC 2nd ) components of the model network (‘Net’), adoption network (‘Casc’), stable
network (‘Stab’), and induced vulnerable trees (‘Vuln’) as a function of r. Dashed lines show the observed
relative size of the real LC of the adopter network in 2011 [see main panel in (a)] and the predicted rvalue.
(d) Distribution P(d) of depths of induced vulnerable trees in both data and model for several rvalues,
showing a good fit with the data for r= 0.73. The difference in the tail is due to finite-size effects. (e)
Correlation hsvi(k) between innovator degree and average size of vulnerable trees in both data and model
with the same rvalues as in (e). Model calculations for (d) and (e) correspond to networks of size N= 106
and are averaged over 102realisations.
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We can use these calculations to estimate the only unknown parameter (the fraction rof immune nodes
in Skype) by matching the size of the largest component (LCNet ) between real and model adoption networks
at time T. Empirically, this value is the relative size corresponding to the last point on the right-hand side
of the distribution for 2011 (Fig. 3a, main panel). The corresponding value in the model is r= 0.73 (dashed
lines in Fig. 3c; also Fig. 2a and b), suggesting that the real adoption process lies in the crossover regime.
The other analysed service turns out to lie right of the crossover regime, which explains its large innovator
adoption rate and reduced size of stable and vulnerable adoption clusters (see SI).
To test the validity of the prediction of rwe perform three different calculations. First we measure
the maximum relative growth rate of cumulative adoption and find a good match between model and data
(Skype s3 and Model Skype s3 in Fig. 1a). In other words, the model correctly estimates the speed of the
adoption process. Second, we measure the distribution P(d) of depths of induced vulnerable trees (Fig. 3d).
Finally, in order to verify earlier theoretical suggestions [41], we look at the correlation hsvi(k) between the
degree of innovators and the average size of vulnerable trees induced by them (Fig. 3e). We perform the
last two measurements on the real data and in the model process for r= 0.6 and 0.9, as well as for the
predicted value r= 0.73. In the case of hsvi(k), we find a strong positive correlation in the data, explained
partially by degree heterogeneities in the underlying social network, but surprisingly well emulated by the
model. More importantly, although both quantities appear to scale with r, measures for the estimated r
value fit the empirical data remarkably well, confirming our earlier validation based on the matching of
relative component sizes (for further discussion see SI).
Discussion
Although some products and innovations diffusing in society may cover a large fraction of the population,
their spreading tends to follow slow cascading patterns, the dynamics of which have been modelled before
by simple diffusion models like that of Bass [17]. However, this approach neglects threshold mechanisms
that arguably drive the decision making of single individuals. On the other hand, threshold models study
the conditions for cascades in global diffusion but do not address their temporal evolution, which is clearly
a relevant factor in real-world adoption processes. These models are commonly used to predict rapid cas-
cading patterns of adoption, which is a more realistic scenario for the spreading of information, opinions,
or behavioural patterns but are not observed in the case of product or innovation diffusion where adoption
requires additional efforts, e.g., free or paid registration. Here we provide a solution for this conundrum
by analysing and modelling the worldwide spread of an online service in the techno-social communication
network of Skype. Beyond the novel empirical evidence about heterogeneous adoption thresholds and non-
linear dynamics of the adoption process, we identify two additional components necessary to introduce in the
modelling of product adoption, namely: (a) a constant flow of innovators, which may induce rapid adoption
cascades even if the system is initially out of the cascading regime; and (b) a fraction of immune nodes that
forces the system into a quenched state where adoption slows down. These features are responsible for a crit-
ical structure of empirical adoption components that radically differs from previous theoretical expectations.
We incorporate these mechanisms into a threshold model controlled by the rate of innovators and the frac-
tion of immune nodes. The model is able to reproduce several pathways ranging from cascading behaviour
to more realistic dynamics of innovation adoption. By constraining the model with empirically determined
parameters, we provide an estimate for the real fraction of susceptible agents in the social network of Skype,
and validate this prediction through correlated structural features matching empirical observations.
Our aim in this study was to provide empirical observations as well as methods and tools to model the
dynamics of social contagion phenomena with the hope it will foster thoughts for future research. One possible
direction would be the observation of the reported structure and evolution of the global adoption cluster in
other systems similar to the ones studied in [25, 27, 28, 33, 35, 48]. Other promising directions could be the
consideration of homophilic or assortative structural correlations, the evolving nature of the underpinning
9

social structure as studied in [22], interpersonal influence, or the effects of leader-follower mechanisms on the
social contagion process. Finally, we hope that the reported results may improve efficiency in the strategies
of enhancing the diffusion of products and innovations, by shifting attention from the creation of short-lived
perturbations to the sustenance of external input.
Competing interest statement
The authors have no competing interests.
Authors’ contributions statement
M.K., G.I., R.K., K.K, and J.K designed the research and participated in writing the manuscript. R.K. and
M.K. analysed the empirical data, G.I. made the analytical calculations, and M.K. and G.I. performed the
numerical simulations.
Acknowledgements
The authors gratefully acknowledge the support of M. Dumas, A. Saabas, and A. Dumitras from STACC
and Microsoft/Skype Labs as well as constructive comments by J. Saram¨aki and T. N¨asi.
Funding statement
G.I. acknowledges the Academy of Finland, and J.K. the CIMPLEX FET Open H2020 EU project for
support. This research was partly funded by Microsoft/Skype Labs.
Material and Methods
Data description
We use a static representation of the Skype social network aggregated over 99 months between September
2003 and November 2011. We follow the adoption of the “buy credit” paid service for 89 months starting
from 2004, and the paid service “subscription” for 42 months starting from 2008 (for further details about
the network and service see SI). By considering the online social structure and adoption times, we identify
users as innovator, vulnerable, or stable nodes based on the number Φkof adopting neighbours at the time
of exposure. Thresholds are calculated as φ= Φk/k for users with kcontacts. The adoption network is
constructed by considering confirmed social links between users who adopted the service earlier than t. In
order to avoid the effect of instantaneous group adoptions (evidently not driven by social influence), we only
consider links between nodes who are neighbours in the underlying social network and whose adoption did
not happen at the same time. Note that for the categorisation of nodes we use only the adoption time and
the state of their peers, and thus real categories may differ slightly. For example, an innovator may appear
as a vulnerable or stable node, even if its decision was not driven by social influence but some of its peers
adopted earlier. To consider this bias we measure effective rates of adoption for the model process as well,
just like for the empirical case (Fig.1) and section S3.
10

Maximum relative growth rate
This measure is obtained by taking the maximum of the yearly adoption rate (yearly count of adoptions)
normalised by the final observed adoption number of a given service. It characterises the maximum speed of
adoption a service experienced during its history and takes values between 0 (no cascade) and 1 (instantaneous
cascade). We repeat this measurement for the estimated number of registered users of Facebook, Twitter,
and LinkedIn [1], as well as for the number of active users of Skype and three paid Skype services. Adoption
rates for Facebook, Twitter, and LinkedIn correspond to the period between 2006 and 2012, and for Skype
and its services to the interval from release date until 2011.
Empirical parameter estimation
We use the Skype data to directly determine all model parameters, apart from the fraction rof immune
nodes. To best approximate the degree distribution of the real network, after testing different candidate
functions (see SI) we select a lognormal function P(k) = e−(ln k−µD)2/(2σ2
D)/(kσD√2π) with parameters
µD= 1.2 and σD= 1.39 and minimum degree kmin = 1, leading to the average degree z= 8.56. To account
for finite-size effects in the model results for low N(Fig. 2), we decrease µDslightly to obtain the same value
of zas in the real network.
The threshold distribution of each degree group collapses to a master curve after normalisation by using
the scaling relation P(Φk, k) = kP (Φk/k). This master curve can be well approximated by the lognormal
distribution P(φ) = e−(ln φ−µT)2/(2σ2
T)/(φσT√2π), with parameters µT=−2 and σT= 1 as determined
by the empirical average threshold w= 0.19 and standard deviation 0.233 (for further details see SI). We
estimate a rate of innovators pn= 0.00019 by fitting a constant function to Ri(t) for t > 2τ(Fig. 1f). The
fit to pnalso matches the time scale of a Monte Carlo iteration in the model to 1 month. Model results
(Fig. 3d and e) are calculated with r= 0.73 and pn= 0.00019. Simulation results in Fig. 3c (d and e) are
averaged over 100 configuration-model networks of size N= 105(106) after T= 89 iterations, matching the
length of the observation period in Skype.
Model description
We characterise the static social network by the extended distribution P(k), where P(k) = rP (k) for c= 0
and P(k) = (1 −r)P(k)P(c) for c > 0. Non-immune, susceptible nodes with property vector kadopt
spontaneously at a constant rate pn, else they adopt only if a fraction φcof their kneighbours have adopted
before. These rules are condensed in the probability Fk,m dt that a node will adopt in a small time interval
dt, given that mof its neighbours are already adopters,
Fk,m =(prif m < kφc
1 if m≥kφc
,∀mand k, c 6= 0,(S5)
with F(k,0),m = 0 ∀k, m and F(0,c),0=pr∀c6= 0 (for immune and isolated nodes, respectively). The dynamics
of adoption is well described by an AME for the fraction sk,m(t) of k-nodes that are susceptible at time t
and have m= 0, . . . , k adopting neighbours [7, 46, 47],
˙sk,m =−Fk,m sk,m −βs(k−m)sk,m +βs(k−m+ 1)sk,m−1,(S6)
where βs(t) = PkP(k)Pm(k−m)Fk,msk,m(t)
PkP(k)Pm(k−m)sk,m(t). To reduce the dimensionality of Eq. (S11) we consider the ansatz
sk,m =Bk,m(ν)e−prtfor m < kφc, leading to the condition ˙ν=βs(1 −ν). With ρ= 1 −PkP(k)Pmsk,m
and some algebra, this condition is reduced to Eq. (S27) (see SI).
11

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Supplementary Information
S1 Detailed data description
This study has been conducted on a dataset of the social network of Skype. The centrepiece of the dataset
is the contact network, where nodes represent users and edges between pairs of users exist if they are in each
other’s contact lists. A user’s contact list is composed of friends. If user uwants to add another user v
to his/her contact list, usends va contact request, and the edge is established at the moment vapproves
the request (or not, if the contact request is rejected). Each edge is labelled with a time stamp indicating
the moment the contact request was approved. As the underpinning social structure we consider the static
representation of the Skype social network, aggregated for 99 months between September 2003 and November
2011. The largest connected component of this structure includes roughly 510 million users and 4.4 billion
edges.
As the chosen service evolving on the Skype network, we follow how users purchase “credits” for calling
phones. For each user, the dataset includes the date when he/she first adopted the paid product “buy credit”
(first credit purchase, for all purposes). We select this service since its lifetime of 89 months is considerably
long (it was introduced in 2004), and it can be adopted by registered Skype users only. This way the
aggregated Skype network provides a complete description of the mediating social structure, which allows us
to calculate the correct degree and adoption threshold for all individuals. To make additional observations
and to further test our model, we perform calculations on a second paid service called “subscription”, which
was introduced in April 2008, lasts for over 42 months, and can also be adopted by registered Skype users
only. Results regarding this service are presented in Section S7.
By considering the online social structure and the adoption times we identify users as innovator, vulner-
able, or stable nodes based on the number Φkof adopting neighbours at the time of exposure. Thresholds
are calculated as φ= Φk/k for users with kcontacts. The adoption network is constructed by considering
confirmed social links between users who adopted the service earlier than the time of observation t. In
order to avoid the effect of instantaneous group adoptions (evidently not driven by social influence), we only
consider links between nodes who are neighbours in the underlying social network and whose adoption did
not happen at the same time. Note that for the categorization of nodes we use only the adoption time and
the state of their peers, and thus ‘real’ categories may differ slightly. For example, an innovator may appear
as a vulnerable or stable node, even if its decision was not driven by social influence but some of its peers
adopted earlier. To consider this bias we also measure ‘effective’ rates of adoption for the model process,
just like for the empirical case (Fig.1, main text) and section S3.
The dataset does not include identity information. All usernames are anonymized and there is no way
of inferring a user’s identity solely from the profile. The dataset does not contain any information about
interpersonal interactions, apart from the contact list.
S2 Empirical determination of model parameters
Parameters in the model are the rate of innovators pn, the degree distribution P(k), the threshold distribution
P(φ), and the fraction of immune nodes r. Other than r, all of them can be estimated from the data as
follows.
15

(a) (b)
Figure 4: Fitted degree distribution. (a) Empirical degree distribution P(k) fitted with a shifted power-law
distribution function [Eq. (S7)] with parameters described in the text. (b) P(k) fitted by a lognormal distribution
function [Eq. (S8)] with parameters determined in the text. Grey symbols are the degree distribution, blue symbols
the log-binned distribution, and solid lines the fitted analytical distribution.
S2.1 Rate of innovators
As discussed in the main text, the rate of spontaneous adoption saturates approximately to a constant value
after an initial transition period, which allows us to determine the rate of innovators by fitting a constant
function on the curve after time 2τ. We estimate this rate to be pn= 0.00019, as demonstrated in Fig. S5a
where the dashed line assigns the fitted constant function.
S2.2 Degree distribution
Degrees in the aggregated Skype network are broadly distributed with a fat tail corresponding to strong
degree heterogeneities. To characterize this distribution analytically we select two candidate distribution
functions. The first is a shifted power-law distribution function of the form,
P(k) = γ−1
C+kmin C+k
C+kmin −γ
for kmin ≤k, (S7)
where kdenotes the degree, γis the power-law exponent scaling the tail of the distribution, and kmin is the
minimum degree (in our case 1). Cis a constant scaling the shift of the distribution, which can be determined
as C=z(γ−2) −kmin(γ−1) since we know the average degree z= 8.56 of the empirical network. This
way our only free parameter during the fit is the degree exponent γ. After fitting this function by using the
non-linear least-square method, we obtain a relatively good match with the empirical distribution (Fig. S4a)
for exponent γ= 3.61.
Our second candidate function is a lognormal distribution function of the form,
P(k) = 1
kσD√2πe−(ln k−µD)2
2σ2
Dfor kmin ≤k, (S8)
where µDand σDare the scaling parameters. After fitting this function by using the non-linear least square
method with two free parameters (µDand σD), we obtain an excellent fit with the empirical distribution for
parameters µD= 1.2 and σD= 1.39.
16

To select the best candidate function, we calculate the corresponding Jensen-Shannon (J S) divergence
values [2] between the empirical and fitted distributions. As a result we find that for the shifted power-law
function the best fit provides JS = 0.0257, while for the lognormal distribution we get JS = 0.0051. Thus
we select the lognormal distribution as the best analytical function describing the degree distribution of the
empirical network.
S2.3 Threshold distribution
The adoption threshold φof a node is defined as φ= Φk/k, i.e. the fraction of adopting neighbours that
trigger the adoption of the central node. Therefore it can only take certain fractional values determined by
the degree k. Although thresholds are defined as a fraction, by considering nodes of the same degree we
can focus on the integer threshold Φk, defined as the number of a node’s neighbours who have adopted the
service earlier.
In our method we first group nodes of the same degree, record their integer thresholds, and then calculate
the threshold distribution for each degree group, as shown in the main text (Fig. 1e, inset). These distri-
butions collapse to a master curve after normalization by using the scaling relation P(Φk, k) = kP (Φk/k)
(Fig. 1e, main panel). Moreover, this master curve can be well approximated by a lognormal distribution of
the form,
P(φ) = 1
φσT√2πe−(ln φ−µT)2
2σ2
T,(S9)
where µT=−2 and σT= 1, as determined by the empirical average threshold w= 0.19 and standard
deviation (STD) 0.233.
These findings indicate that although individual thresholds are strongly determined by degree, their
distribution is degree-invariant, suggesting that the fraction of adopting friends rather than its absolute
number is relevant during the service adoption process. The estimated empirical values of parameters are
summarized in Table 1.
pnhkiµDσDw ST D (φ)µTσT
0.00019 8.56 1.2 1.39 0.19 0.233 −2 1
Table 1: Estimated empirical parameters for service “buy credit”.
S3 Social influence - null model study
Studies of social contagion phenomena assume that social influence is responsible for the correlated adoption
of connected people. However, an alternative explanation for the observed correlated adoption patterns is
homophily: a link creation mechanism by which similar egos get connected in a social structure. In the latter
case, the correlated adoption of a connected group of people would be explained by their similarity and not
necessarily due to social influence. Homophily and influence are two processes that may simultaneously play
a role during the adoption process. Nevertheless, distinguishing between them on the individual level is very
difficult using our or any similar datasets [3]. Fortunately, at the system level one may decide which process
is dominant in the empirical data. To do that first we need to elaborate on the differences between these
two processes.
17

(a) Empirical adoption rates (b) Null model adoption rates
Figure 5: Adoption rates in the original and null model processes. Adoption rates for innovator (green),
vulnerable (red), and stable (blue) nodes as a function of time. (a) Empirical rates where adoptions appear in the
original order. The dashed line assigns a fitted constant function to estimate the innovator adoption rate. (b) Null
model rates where times of adoption are randomly shuffled. Here qand τare arbitrary constant values.
Influence-driven adoption of an ego can happen once one or more of its neighbours have adopted, since
then their actions may influence the decision of the central ego. Consequently, the time ordering of adop-
tions of the ego and its neighbours matters in this case. Homophily-driven adoption is, however, different.
Homophily drives social tie formation such that similar people tend to be connected in the social structure.
In this case connected people may adopt because they have similar interests, but the time ordering of their
adoptions would not matter. Therefore, it is valid to assume that adoption could evolve in clusters due to
homophily, but these adoptions would appear in a random order.
To test our hypothesis we define a null model where we take the adoption times of each adopter and
shuffle them randomly among all adopting egos. This way a randomly selected time is assigned to each
adopter, while the adoption rate and the final set of adopters remain the same. Moreover, this procedure
only destroys correlations between adoption events induced by social influence, but keeps the social network
structure and node degrees unchanged. In this way, during the null model process the same egos appear as
adopters, but the time series of adoption may in principle change (or not), corresponding to social influence
(or homophily) as a dominant factor during the adoption process. If adoption is mostly driven by homophily,
the rates of adoption would not change considerably beyond statistical fluctuations. On the other hand, if
social influence plays a role in the process, rates of adoption in the null model should be very different from
the empirical curves, implying that the time ordering of events matters in the adoption process. In this
case, the rate of innovators should be higher than in the empirical data, since nodes that are in the adoption
cluster originally but not directly connected would have a greater chance to appear as innovators, due to a
random adoption time that is not conditional to the time ordering of the adopting neighbours.
After calculating the adoption rates of different user groups in the shuffled null model sequence we
observe the latter situation: the rate of innovators becomes dominant, while the rate of stable and vulnerable
adoptions drops considerably as they appear only by chance. This suggests that the temporal ordering of
adoption events matters a lot in the evolution of the observed adoption patterns, and thus social influence
may play a strong role here. Of course one cannot decide whether influence is solely driving the process
or homophily has some impact on it; in reality it probably does to some extent. However, we can use this
null model measure to demonstrate the presence and importance of the mechanism of influence during the
adoption process.
18

S4 Threshold model
S4.1 Model description
This model emulates the rise and temporal evolution of system-wide adoption cascades in complex social
networks [4, 5, 6]. Note that this model has been introduced in [6], where its general scaling behaviour has
been explored. In a system of fixed size, a node has social interactions with kother agents and is characterized
by a continuous adoption threshold φ. When faced with the prospect of adopting a given innovation, product,
or fad, susceptible individuals adopt spontaneously with rate pn. Otherwise, the node adopts if at least a
fraction φof its kneighbours have adopted before (the so-called ‘threshold rule’). Moreover, a fraction r
of the system is ‘immune’ to the innovation, in the sense that these agents never adopt regardless of their
values of kand φ. The distributions of degrees and thresholds, P(k) and P(φ) (as well as the values of pn
and r), thus determine the average topological state and dynamical evolution of the system.
The model may be implemented numerically via a Monte Carlo simulation of the rules described above
in a system of size N. Here, the dynamical state of the system is determined by the adoption state (0 or 1)
of all nodes, which change in asynchronous random order in a series of time steps. Once an agent adopts and
its state changes from 0 to 1, it remains so for the rest of the dynamics, thus ensuring a frozen final state
for the finite system where no more adoptions arise. Each time step consists of Nnode updates: In each
node update, a randomly selected node (non-immune and in state 0) adopts spontaneously with probability
pr=pn/(1 −r)1; else it adopts only if the threshold rule is satisfied. The rescaled rate pris necessary if we
wish to obtain a rate pnof innovators in early times of the dynamics, regardless of the value of r. Finally,
we assume that agents with k= 0 receive no social influence (for any value of φ), and therefore can only
adopt spontaneously. We will now explore this dynamics with numerical simulations and a rate equation
formalism.
S4.2 Stochastic binary-state dynamics
Here we extend an approximate master equation (AME) formalism for stochastic binary-state dynamics as
developed recently by Gleeson [7, 8, 9, 10]. In a stochastic binary-state dynamics, each node in the network
can take one of two possible states (susceptible or adopter in the language of innovation adoption) at any
point in time, and state-switching happens randomly with probabilities that only depend on the current state
of the updating agent and on the states of its neighbours. This general definition includes the threshold
model described above as a special case. Such formalism considers configuration-model networks, that is, an
ensemble of networks specified by the degree distribution P(k) but otherwise maximally random [11].
All relevant properties used to describe a node are included in the vector k= (k, c), where k=
k0, k1,...kM−1is the degree of the node and c= 0,1, . . . , M a dummy variable that labels its ‘type’,
i.e. any other property that characterizes the node apart from its degree. In the case of our threshold model,
c= 0 is the type of the fraction rof immune nodes, while c6= 0 labels the type of all non-immune nodes
with given threshold φc. The various values of c6= 0 correspond then to different adoption thresholds φc.
The integer Mis the maximum number of degrees/types considered in the AME framework, which can be
increased to improve the accuracy of the analytical approximation at the expense of speed in its numerical
computation2. Any pair of nodes with identical values of kare considered equivalent in this level of descrip-
tion, forming a node class with the same average dynamics. Moreover, P(k) and P(φ) can be generalized to
1We define pr= 1 for pn>1−r.
2Explicitly, rather than using k0=kmin,kM−1=N−1 and M=N−kmin (i.e, considering all possible degrees in the
empirical/simulated network), we take a small k0> kmin and large kM−1< N −1, M < N −kmin, with the other M−2 degree
values equidistantly distributed between k0and kM−1, thus disregarding some degrees and gaining speed in the computation
of the AMEs. Similarly, the Mthreshold values corresponding to nonzero types are uniformly distributed in the open interval
(0,1).
19

the joint distribution P(k) giving the probability that a randomly selected node has property vector k(i.e.
degree kand type c). Here it is useful to define P(c) as the distribution of all non-zero types, c= 1, . . . , M .
If degrees and thresholds are chosen independently, like in our model, then P(k) = rP (k) for c= 0 and
P(k) = (1 −r)P(k)P(c) for c > 0. The distribution P(c) is, in other words, a discrete, rescaled version of
the continuous threshold distribution P(φ).
In the language of innovation adoption, the dynamics of a node is determined by the number m= 0,1,...k
of its neighbours that have already adopted when the node is deciding whether to adopt or not. During a
small time interval dt, a susceptible node (in state 0) adopts with probability Fk,mdt, while an adopter (in
state 1) becomes susceptible with probability Rk,mdt. The functions Fk,m and Rk,m , known as infection and
recovery rates, respectively, determine the temporal evolution of the node class k. In the particular case of
threshold models, a so-called monotone dynamics, Rk,m = 0 ∀k, m (since no adopters become susceptible
again). As for Fk,m, the rules of spontaneous and threshold adoption imply,
Fk,m =(prif m < kφc
1 if m≥kφc
,∀mand k, c 6= 0,(S10)
that is, a node adopts the innovation either spontaneously with rate pr, or with probability 1 if its number
of adopting neighbours equals or exceeds the integer threshold Φk=dkφce. Immune nodes (c= 0) have an
infection rate of F(k,0),m = 0 ∀k, m, while for isolated nodes (k= 0) F(0,c),0=pr∀c6= 0. In other words,
immune nodes never adopt, and isolated nodes can only adopt spontaneously. We note that Fk,m is written
in terms of pr=pn/(1 −r), not pn, in order to counter the trivial decrease in the rate of spontaneous
adoption for non-zero r.
Let us now turn to the rate equations for our threshold model, called AMEs in the formalism by Gleeson.
We denote by sk,m(t) the fraction of k-class nodes that are susceptible at time tand have madopting
neighbours. Therefore, the fraction of agents with property vector kthat are adopters at time tis ρk(t) =
1−Pk
m=0 sk,m(t), and the fraction of adopters in the system is ρ(t) = PkP(k)ρk(t). Here, the sum over
classes means a sum over all degrees and types, i.e. Pk•=PkPc•. Assuming a monotone dynamics
(Rk,m = 0), the AMEs for sk,m can be written as [7, 8, 10],
dsk,m
dt =−Fk,msk,m −βs(k−m)sk,m +βs(k−m+ 1)sk,m−1,(S11)
where m= 0, . . . , k,sk,−1≡0, Fk,m follows Eq. (S10), and βs(t) (the rate at which edges between pairs of
susceptible nodes transform to edges between a susceptible agent and an adopter) is given by,
βs(t) = PkP(k)Pm(k−m)Fk,msk,m(t)
PkP(k)Pm(k−m)sk,m(t).(S12)
If at time t= 0 we randomly choose a fraction ρ(0) = PkP(k)ρk(0) of nodes as seed for the adoption
process, the initial conditions for Eq. (S11) are sk,m (0) = [1 −ρk(0)]Bk,m[ρ(0)], with ρk(0) the initial
fraction of adopters in class kand Bk,m a binomial factor,
Bk,m(ρ) = k
mρm(1 −ρ)k−m.(S13)
The solution sk,m(t) of the AME system in Eq. (S11) provides a very accurate description of the dynamics
of our model, yet its dimension (i.e. number of equations to solve) grows quadratically with the number of
degrees and linearly with the number of threshold values considered. Fortunately, the AMEs for our model
can be mapped to a reduced-dimension system with a derivation similar to the one used by Gleeson in the
case of the Watts threshold model [4, 5].
20

S4.3 Reduced-dimension AMEs
To reduce the dimension of Eq. (S11), we need to consider system-wide quantities that are more aggre-
gated than sk,m. One of them is the probability that a randomly chosen node is an adopter, ρ(t) =
1−PkP(k)Pmsk,m(t), i.e. the fraction of adopters in the network. The other one is the probability that a
randomly chosen neighbour of a susceptible node is an adopter, ν(t) = PkP(k)Pmmsk,m(t)/Pmksk,m(t).
We start by proposing an exact solution for the AME system in terms of the following ansatz,
sk,m(t) = [1 −ρk(0)]Bk,m[ν(t)]e−prtfor m < kφcand c6= 0,(S14)
and s(k,0),m =Bk,m(ν) for c= 0, where Bk,m follows Eq. (S13). The meaning of the ansatz in Eq. (S14)
is quite intuitive and considers two processes. First, a susceptible agent with degree kand madopting
neighbours is not selected as part of the initial adoption seed with probability 1 −ρk(0) and is connected to
madopters with the binomially distributed probability Bk,m(ν). Second, for m < kφca susceptible node
does not fulfill the threshold rule and can only adopt spontaneously with probability e−prt, since the system
is progressively been filled by adopters. Considering these two processes as independent we end up with
the product in Eq. (S14). Finally, since immune nodes do not adopt and are distributed randomly over the
network, s(k,0),m is determined only by a binomial factor.
The next step is to insert the ansatz (S14) into the AME system (S11) and derive a set of differential
equations for the aggregated quantities ρand ν. Taking the time derivative ˙sk,m of Eq. (S14) (i.e. the
left-hand side of Eq. (S11)) we get,
˙sk,m =m
ν−k−m
1−ν˙ν−prsk,m .(S15)
Then, we use the threshold rule (S10) for m < kφc, the ansatz (S14) and the binomial identity,
Bk,m−1(ν) = 1−ν
ν
m
k−m+ 1Bk,m (ν),(S16)
in the right-hand side of Eq. (S11) to obtain,
−Fk,msk,m −βs(k−m)sk,m +βs(k−m+ 1)sk,m−1=−pr+βsm−k+1−ν
νmsk,m.(S17)
Equating Eqs. (S15) and (S17) as in the AME system (S11) leads to,
˙ν=βs(1 −ν),(S18)
a condition on νso that the ansatz (S14) is a solution of Eq. (S11). This differential equation has the
initial condition ν(0) = ρ(0), obtained by evaluating Eq. (S14) at t= 0 and comparing with the expression
[1 −ρk(0)]Bk,m[ρ(0)], which corresponds to a random distribution of initial adopters among kclasses. Fur-
thermore, by assuming a (yet to be determined) function g(ν, t) such that ˙ν=g(ν, t)−ν, Eq. (S18) reduces
to,
βs=g(ν, t)−ν
1−ν.(S19)
Now, we consider the following general result derived by Gleeson in [8] (Eqs. (F6)–(F10) therein),
X
k
P(k)X
m
(k−m)sk,m =z(1 −ν)2,(S20)
21

with z=PkkP (k) the average degree in the network. Eq. (S20) is valid for functions sk,m and νthat satisfy
Eqs. (S11) and (S18) respectively, for any Fk,m and random initial conditions on sk,m and ν, and is thus
applicable in our case. Our goal here is to use Eq. (S20) to find an expression for g(ν) and therefore write
the differential equation (S18) explicitly. Noting that the left-hand side of Eq. (S20) is the denominator in
the definition (S12) of βsand that F(k,0),m = 0 (i.e. immune nodes do not adopt), Eq. (S12) gives,
βs=1−r
z(1 −ν)2
prX
k|c6=0
P(k)P(c)X
m<kφc
(k−m)sk,m +X
k|c6=0
P(k)P(c)X
m≥kφc
(k−m)sk,m

=1
z(1 −ν)2"X
k
P(k)X
m
(k−m)sk,m −rX
k
P(k)X
m
(k−m)s(k,0),m
−(1 −r)(1 −pr)X
k|c6=0
P(k)P(c)X
m<kφc
(k−m)sk,m
,(S21)
where we have written P(k) explicitly as P(k) = rP (k) for c= 0 and P(k) = (1 −r)P(k)P(c) for c > 0, in
order to notice the dependence on r. Then, we insert the ansatz (S14) (with its special case s(k,0),m =Bk,m (ν)
for immune nodes), as well as the identities (k−m)Bk,m(ν) = k(1 −ν)Bk−1,m(ν) and Pm<kφcBk−1,m(ν) =
1−Pm≥kφcBk−1,m(ν) to obtain,
βs=1
1−ν (1 −r)"1−(1 −pr)e−prt
+(1 −pr)e−prtX
k|c6=0
k
zP(k)P(c)
ρk(0) + [1 −ρk(0)] X
m≥kφc
Bk−1,m(ν)

−ν
.(S22)
A comparison of Eqs. (S19) and (S22) gives us the following expression for g(ν, t),
g(ν, t) = (1 −r)
ft+ (1 −ft)X
k|c6=0
k
zP(k)P(c)
ρk(0) + [1 −ρk(0)] X
m≥kφc
Bk−1,m(ν)

,(S23)
where we define ftas ft= 1 −(1 −pr)e−prt. Thus, the AME system (S11) gets reduced to the differential
equation ˙ν=g(ν, t)−ν, with g(ν, t) given explicitly by Eq. (S23).
Even though the equation ˙ν=g(ν, t)−νis closed and in this sense equivalent to Eq. (S11), we can
also derive the corresponding equation for ρ, since we are mainly interested in the temporal evolution of the
fraction of adopters in the network. From the definition of ρand Eq. (S11) we have,
˙ρ=−X
k
P(k)X
m
˙sk,m =X
k
P(k)X
m
Fk,msk,m
+βsX
k
P(k)X
m(k−m)sk,m −(k−m+ 1)sk,m−1,(S24)
where the second term in the right-hand side telescopes to zero. Then, we use an algebraic manipulation
22

similar to that of Eqs. (S21) and (S22) to obtain,
X
k
P(k)X
m
Fk,msk,m = (1 −r)
prX
k|c6=0
P(k)P(c)X
m<kφc
sk,m +X
k|c6=0
P(k)P(c)X
m≥kφc
sk,m

= (1 −r)
1−(1 −r)(1 −pr)X
k|c6=0
P(k)P(c)X
m<kφc
sk,m
−ρ
= (1 −r)
ft+ (1 −ft)X
k|c6=0
P(k)P(c)
ρk(0) + [1 −ρk(0)] X
m≥kφc
Bk,m(ν)

−ρ. (S25)
In this way, Eqs. (S24) and (S25) can be rewritten as ˙ρ=h(ν, t)−ρ, where,
h(ν, t) = (1 −r)
ft+ (1 −ft)X
k|c6=0
P(k)P(c)
ρk(0) + [1 −ρk(0)] X
m≥kφc
Bk,m(ν)

.(S26)
Joining all of these results, the AME system (S11) gets reduced to the system of two ordinary differential
equations,
˙ρ=h(ν, t)−ρ, (S27a)
˙ν=g(ν, t)−ν, (S27b)
with the quantities g(ν) and h(ν) given explicitly by Eqs. (S23) and (S26).
The system (S27) can be solved numerically to obtain ρ(t) and thus characterize the temporal evolution
of the adoption process. Let us further separate the fraction of adopters as ρ(t) = ρ0(t) + ρ1(t), where ρ0
and ρ1are the fractions of innovators and induced adopters (i.e. vulnerable and stable nodes), respectively.
Now consider the identity,
1−ρ=X
k
P(k)X
m
sk,m =r+ (1 −r)X
k|c6=0
P(k)P(c)X
m
sk,m =r+ρs,(S28)
where ρs(t) is the fraction of non-immune, susceptible nodes that can eventually adopt, either spontaneously
or not. Since such susceptible nodes adopt spontaneously at a rate pr, the rate equation for innovators is
˙ρ0=prρs. Then, with Eq. (S28) we obtain,
ρ0(t) = prZt
0
[1 −r−ρ(t)]dt, (S29)
which can be calculated explicitly with the numerical solution of Eq. (S27).
S5 Waiting time of adoption
Another reason behind the non-rapid evolution of the adoption process could be the time users wait after
their personal adoption threshold is reached and before adopting the service. This lag in adoption can be
due to individual characteristics, or come from the fact that social influence does not spread instantaneously
(as commonly assumed in threshold models, including ours). However, the waiting time τwcan be estimated
by measuring the time difference between the last adoption in a user’s egocentric network and the time of
23

(a) (b)
Figure 6: Waiting time distribution and its effect on the adoption process (a) Distribution P(τw) of times
between the last adoption in the egocentric network of an individual and his/her own adoption. (b) Cumulative
adoption rates after waiting time removal [CR(t) and C Rτw(t), respectively]. nand τare arbitrary constant values.
adoption. We define τw= 0 for innovators, but τwcan take any positive value for vulnerable and stable
adopters up to the length of the observation period.
Waiting times are broadly distributed for adopters (Fig. S6a), meaning that many users adopt the service
shortly after their personal threshold is reached, but a considerable fraction waits long before adopting the
service. The heterogeneous nature of waiting times may be a key element behind the observed adoption
dynamics. One way to figure out the effect of waiting times on the speed of cascade evolution is by removing
them. We can extract waiting times from adoption times and thus calculate rescaled adoption times. The
rescaled adoption time of a user is the last time when his/her fraction of adopting neighbours changed and
the adoption threshold was (hypothetically) reached. After this procedure we can calculate a new adoption
rate function by using rescaled adoption times and compare it to the original. From Fig. S6b we can conclude
that although adoption becomes faster, the rescaled adoption dynamics is still not rapid. On the contrary, it
suggests that the rescaled adoption dynamics is still very slow and quite similar to the original. Consequently,
waiting times cannot explain the observed dynamics of adoption.
Note that long waiting times can have a further effect on the measured dynamics. After the ‘real’
threshold of a user is reached and he/she waits to adopt, some neighbours may adopt the product. Hence all
observed measures are in this sense ‘effective’: observed thresholds are larger or equal than real thresholds;
the innovator rate is smaller or equal; the vulnerable and stable rates will be larger or equal; and waiting
times will be shorter or equal than the real values. Consequently the process may be actually faster than
that we observe in Fig. S6b after removing effective waiting times. However, this bias becomes important
only after the majority of individuals in the social network has adopted the service and the spontaneous
emergence of adopting neighbours becomes more frequent. As the fraction of adopters in our dataset is
always less than 6% [12], we expect minor effects of this observational bias on measurements.
S6 Empirical and model cluster statistics
As described in the main text, we perform extensive model calculations using empirically determined pa-
rameters to estimate the only unknown parameter, the fraction of immune nodes r. We match the relative
size of the largest connected component of the real adoption network with its corresponding measure in the
model at the end of the observation period, and estimate the fraction of immune nodes in the real system
24

(a) (b)
Figure 7: Empirical and model cluster statistics (a) Distribution P(d) of the depth of induced vulnerable trees
in the empirical and model systems. (b) Correlation hsvi(k) between the degree of innovators and the average size
of vulnerable trees they induce. Empty symbols denote model calculations for r= 0.6 (blue), 0.73 (black), and 0.9
(green), and full red symbols the empirical measurements. Model calculations correspond to networks of size N= 106
and are averaged over 100 independent realizations.
as r= 0.73. To support our estimation we also measure the distribution P(d) of the depth of induced
vulnerable trees and the correlation hsvi(k) between the degree of innovator nodes and the average size of
induced vulnerable trees in the model, and match them with the equivalent empirical measures. To provide
further support for the estimated rvalue we show the dependence of these quantities of different rvalues.
We measure P(d) and hsvi(k) for r= 0.6 and 0.9, as well as for the predicted value r= 0.73 (Fig. S7). It is
clear that both quantities scale with r. For smaller rmore nodes are susceptible for adoption, allowing deeper
and larger vulnerable trees, while for larger rno large induced cluster can emerge as the system is forced into
a quenched state. Moreover, measures for the estimated rvalue fit the empirical data considerably well. This
collapse is remarkable, since we neglect any higher-order structural and temporal correlations in the model
(like assortative mixing, community structure, bursty adoption patterns, periodic activity fluctuations, etc.),
which are present in the empirical system. Differences in the tails of the measures are due to finite-size effects
since the modelled network is two orders of magnitude smaller than the empirical social structure. Note that
although we can look for an rfraction that produces a better fit between model and data in terms of P(d)
and hsvi(k), the collapse in Fig. S7 demonstrates the quality of an independent procedure of estimating r
(i.e. by matching the relative size of components). Therefore, these results are intended for validation only
and not as a method to estimate the correct value of r.
S7 Calculations for additional service
S7.1 Empirical observations
In order to support our empirical observations and modelling of the social spreading of Skype, we examine
the adoption dynamics of an additional paid service called “subscription”, introduced in April 2008 and with
adoption data for over 42 months until the end of the observation period. This service is only available for
registered Skype users, and we can therefore use the accumulated static Skype network as background social
structure. In order to investigate the adoption of this service we repeat all calculations described previously.
First we measure the decoupled rate of innovator, vulnerable, and stable adopters (Fig. S8a). We see that
after a short initial period innovators adopt approximately with a constant rate, setting the model parameter
25

(a) (b)
Figure 8: Adoption rates and threshold distributions for service “subscription”.(a) Net adoption rate
(black), as well as rates for innovator (green), vulnerable (red), and stable (blue) nodes as function of time. The
dashed line is a fitted constant function to estimate the innovator adoption rate as pn= 0.00012. (b) Distribution
of integer thresholds Φkfor several degree groups (inset). By using P(Φk, k ) = kP (Φk/k) these curves collapse to a
master curve well approximated by a lognormal function (dashed line) with average w= 0.063 and STD 0.153 (for
further details see Section S2.3).
to pn= 0.00012. Moreover, here innovators dominate social spreading since the rate of vulnerable and stable
adoptions is relatively low.
We also measure the integer threshold distribution for different degree groups (Fig. S8b, inset) just
as described in Section S2.3. These distributions scale together after normalization with the scaling rela-
tion P(Φk, k) = kP (Φk/k) (Fig. S8b, main panel) and are well approximated by a lognormal distribution
[Eq. (S9)] with parameters µT=−3.73 and σT= 1.39, as determined by the average threshold w= 0.063
and STD 0.153. Note that since the adoption dynamics of this service is dominated by innovators, the
average threshold wis smaller than in the case of the “buy credit” service. All parameters are summarized
in Table 2. Since the background network is the same for both services, network parameters are those of
Table.1.
Although the adoption process is dominated by innovators, a giant connected component evolves in the
adoption network (Fig. S8a, main panel). On the other hand, its relative size is considerable smaller than
for the “buy credit” service. The stable adoption network is also dominated by a giant component, but
its relative size is even smaller when compared to the adoption network (Fig. S8a, inset). Moreover, the
largest vulnerable trees are only two orders of magnitude smaller than the stable giant cluster (Fig. S8b).
For comparison, this difference is five order of magnitude for the “buy credit” service.
pnw ST D(φ)µTσT
0.00012 0.063 0.153 −3.73 1.39
Table 2: Estimated empirical parameters for service “subscription”.
S7.2 Model and validation
We repeat all model calculations with the parameters of the “subscription” service to see whether we can
recover its adoption dynamics by using the dynamical threshold model introduced in the main text and
26

(a) (b)
Figure 9: Empirical cluster statistics. (a) Relative connected-component size distribution P(s) at different
times for the empirical adoption network (main panel) and the stable adoption network (inset), with sizes saand ss,
respectively. (b) Relative connected-component size distribution P(sv) of the empirical innovator-induced vulnerable
trees at different times.
Figure 10: Modeled adoption process of the service “subscription” Average size of the largest (LC) and 2nd
largest (LC2nd ) components of the model network (‘Net’), model adoption network (‘Casc’), model stable network
(‘Stab’), and induced vulnerable trees (‘Vuln’) as a function of r. Dashed lines show the observed relative size of the
real LC of the adopter network in 2011 (Fig. S9, main panel) and the predicted rvalue. The lower panel depicts
the time t50% when the adoption process has reached 50% of the susceptible network as a function of r. We use 100
realizations of configuration-model networks with size N= 105and lognormal degree distribution parametrized as
described in Section S2.2. Model calculations correspond to the parameters of Table 2 for 42 iteration steps (matching
the length of the observation period).
in Section S4. We check the dependence on rof the average size of the largest connected component of
the network (LC) of susceptible nodes available for the adoption process, the adoption network, the stable
27

adoption network, and of vulnerable trees (Fig. S10, upper panel). In addition we record the average size
LC2nd of the second largest connected component (Fig. S10, middle panel). Finally we show the time when
the adoption process has reached the 50% of available susceptible nodes in the adoption network (Fig. S10,
lower panel).
The rdependence of the adoption process appears to be qualitatively similar to our earlier calculations
on the “buy credit” service, but there are remarkable differences. Firstly, the crossover regime (depicted by
the light grey area in Fig. S10) is shifted towards larger rvalues due to the different threshold distribution
and innovator adoption rate. Secondly, after matching the relative size of the largest connected component
of the empirical adoption network (last point on the right-hand side of Fig. S9, main panel), the predicted
r= 0.928 is out of the crossover regime. At this point the background social network is still not fragmented
(as evidenced by the black line in Fig. S10, which has not reached its maximum yet) and it allows for
the emergence of large connected adoption clusters. It is very sparse, however, which explains: (a) the
dominating innovator adoption rate observed empirically; (b) the reduced size of the giant component of the
adoption and stable adoption networks; and (c) the relatively large innovator trees as compared to the stable
adoption network components. We observe that the largest vulnerable trees are smaller than the largest
stable clusters in the empirical data, while the opposite is true for the model. A possible explanation of
this difference is the assumption in the model that the network is degree-uncorrelated. This is a necessary
approximation in order to treat the model analytically, but it might not hold for the empirical network. All in
all, this picture suggests that the “subscription” service is out of the rapid and even the crossover cascading
regimes, and that its dynamics is mostly driven by independent innovators rather than social influence, on
a network of which a large majority is not susceptible to innovation.
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