# Social contagions with communication channel alternation on multiplex
networks

Wei Wang  Cybersecurity Research Institute, Sichuan University, Chengdu
610065, China  Big Data Research Center, University of Electronic Science and
Technology of China, Chengdu 610054, China  Ming Tang
<EMAIL_ADDRESS>School of Information Science and Technology, East
China Normal University, Shanghai, 200241, China  Shanghai Key Laboratory of
Multidimensional Information Processing, East China Normal University,
Shanghai 200241, China  H. Eugene Stanley  Center for Polymer Studies and
Department of Physics, Boston University, Boston, Massachusetts 02215, USA
Lidia A. Braunstein  Instituto de Investigaciones Físicas de Mar del Plata
(IFIMAR)-Departamento de Física, Facultad de Ciencias Exactas y Naturales,
Universidad Nacional de Mar del Plata-CONICET, Funes 3350, (7600) Mar del
Plata, Argentina  Center for Polymer Studies and Department of Physics, Boston
University, Boston, Massachusetts 02215, USA

###### Abstract

Internet communication channels, e.g., Facebook, Twitter, and email, are
multiplex networks that facilitate interaction and information-sharing among
individuals. During brief time periods users often use a single communication
channel, but then communication channel alteration (CCA) occurs. This means
that we must refine our understanding of the dynamics of social contagions. We
propose a non-Markovian behavior spreading model in multiplex networks that
takes into account the CCA mechanism, and we develop a generalized edge-based
compartmental method to describe the spreading dynamics. Through extensive
numerical simulations and theoretical analyses we find that the time delays
induced by CCA slow the behavior spreading but do not affect the final
adoption size. We also find that the CCA suppresses behavior spreading. On two
coupled random regular networks, the adoption size exhibits hybrid growth,
i.e., it grows first continuously and then discontinuously with the
information transmission probability. CCA in ER-SF multiplex networks in which
two subnetworks are Erdős-Rényi (ER) and scale-free (SF) introduces a
crossover from continuous to hybrid growth in adoption size versus information
transmission probability. Our results extend our understanding of the role of
CCA in spreading dynamics, and may elicit further research.

###### pacs:

89.75.Hc, 87.19.X-, 87.23.Ge

##  I Introduction

Numerous social communication platforms, including Facebook, Twitter, and
email, are a part of the current world-wide information explosion. These
platforms taken together behave as a network of networks (NON) in which each
communication platform functions as a subnetwork Boccaletti et al. (2014); Gao
et al. (2012); D’agostino and Scala (2014); Kivelä et al. (2014). These NONs
can be multilayer, interdependent, or multiplex. In interdependent networks
the functionality of the components in one network depends on the
functionality of nodes in other networks. A multiplex network is a special NON
case in which each agent can be present in more than one layer. Extensive
studies of cascading failure, evolutionary games, synchronization, and
spreading dynamics have found that the dynamics of NONs differ greatly from
those of single networks Kim and Goh (2013); Radicchi (2014); Buldyrev et al.
(2010); Cozzo et al. (2013); Saumell-Mendiola et al. (2012); Zhang et al.
(2015); Xu et al. (2015). For example, For example, Buldyrev et al. found a
first-order percolation phase transition in interdependent networks that is
qualitatively different from the second-order phase transition in single
networks Buldyrev et al. (2010). Baxter et al. found a hybrid phase transition
in the percolation phase transition on multiplex networks Baxter et al.
(2012).

Spreading dynamics in complex networks can be classified as either biological
or social contagions. In biological contagions, such as epidemic spreading,
researchers have found that multilayer networks promote spreading Cozzo et al.
(2013); Saumell-Mendiola et al. (2012), induce the coexistence of mixed phase
transitions Dickison et al. (2012), and produce rare-region phenomena Cozzo
and Moreno (2016). In contrast to biological contagions, social contagions are
strongly affected by social reinforcement Porter and Gleeson (2016). Research
on social contagions has focused primarily on a generalized Watts threshold
model Brummitt et al. (2012); Min and Goh (2014); Yağan and Gligor (2012);
Watts (2002); Lee et al. (2014); Majdandzic et al. (2016). As in biological
contagions, the behavior of NONs promotes social contagions Brummitt et al.
(2012). Majdandzic et al. found multiple tipping points of social contagions
in multilayer networks Majdandzic et al. (2016, 2014). Recently Wang et al.
used a data-driven asymmetric socio-biological coevolutionary model to locate
an optimal information diffusion mechanism for suppressing biological
contagions Wang et al. (2016a) that enables us to understand the effect of
asymmetry in interacting dynamics Wang et al. (2014a); Liu et al. (2016a);
Granell et al. (2013); Funk et al. (2009).

Although there are many different communication channels, we usually select
one to transmit information to friends during short periods of time due to the
inelasticity of such resources as time and energy Holme and Saramäki (2012);
Haerter et al. (2012a); Min et al. (2016). Thus when transmitting information
through a multiplex network we often migrate to other channels De Domenico et
al. (2014); Starnini et al. (2016), and a communication channel alteration
(CCA) occurs. For example, when transmitting information using texting or
email, if we change to another channel a CCA occurs. This event (i) creates
distinct active time periods for individuals in different communication
platforms, and (ii) introduces time delays when obtaining information from
other platforms. Theoretically CCA induces a non-Markovian effect into the
spreading dynamics that further causes strong dynamic correlations among the
states of neighbors.

Because there has been no systematic study of how CCA affects the dynamics of
social contagions on multiplex networks, we propose a non-Markovian behavior
spreading model on multiplex networks. At any given time an individual can be
active in only one communication layer and can transmit behavioral information
to neighbors and obtain behavioral information from neighbors only within the
same subnetwork. Using extensive numerical simulations we find that time
delays induced by CCA slow behavior spreading but do not affect the final
adoption size. We also find that CCA suppresses the final behavior adoption
size. It is significant that CCA changes the growth pattern of the final
adoption size on ER-SF networks, i.e., the growth pattern of the adoption size
versus the behavioral information transmission probability changes from
continuous to hybrid. We develop a generalized edge-based compartmental method
to describe this non-Markovian spreading model and find that the theoretical
predictions agree with the numerical predictions.

##  II Social contagion model on multiplex networks

Communication channels such as Facebook, Twitter, and email facilitate
interaction and information-sharing. Taken together they form a multiplex
network in which each communication channel functions as a subnetwork. During
short periods of time individuals select a single communication platform to
transmit information, and then CCA occurs. To understand how CCA affects the
dynamics of social contagion, we examine the behavior spreading dynamics in
two-layer multiplex networks in which each layer or subnetwork represents a
single communication channel. Figure 1(a) shows a multiplex network. In the
multiplex network model, subnetworks $\mathcal{A}$ and $\mathcal{B}$ have the
same number of nodes, and they randomly match one-to-one, which means that
each individual can communicate with friends through two different
communication channels. To establish the multiplex network we first assign
degrees $k_{\mathcal{A}}$ and $k_{\mathcal{B}}$ to individuals in subnetworks
$\mathcal{A}$ and $\mathcal{B}$, respectively, according to a joint degree
distribution $P(\overrightarrow{K})=P(k_{\mathcal{A}},k_{\mathcal{B}})$. We
then build each subnetwork using an uncorrelated configuration model Catanzaro
et al. (2005). In the thermodynamic limit, i.e., the network size
$N\rightarrow\infty$, there are no intra-layer degree correlations in the
subnetworks.

Figure 1: (Color online) Illustration of social contagions on multiplex
networks. Initially, individuals $1$ and $5$ are selected as seeds (adopted),
and the remaining individuals are susceptible. (a) At time $t=1$, individual
$1$ ($5$) successfully transmits the information to his susceptible neighbor
$2$ ($4$) in subnetwork $\mathcal{A}$ ($\mathcal{B}$). The accumulated pieces
of information of $2$ ($4$) in subnetwork network $\mathcal{A}$
($\mathcal{B}$) is $m_{2}^{\mathcal{A}}=1$ ($m_{4}^{\mathcal{B}}=1$). (b) At
time $t=2$, individuals $1$, $3$ and $5$ are in subnetwork $\mathcal{A}$.
Individual $5$ successfully transmits the information to $2$, however,
individual $2$ does not read this information since he belongs to subnetwork
$\mathcal{B}$. (c) At $t=3$, individual $2$ activates in subnetwork
$\mathcal{A}$, he fulfills his adoption threshold $T_{\mathcal{A}}=2$. Thus,
individual $2$ adopts the behavior. Individuals $1$ and $5$ recover with
probability $\gamma=0.6$. (d) At time $t=4$, all adopted individuals recover
and no individual whose received information is greater than his adoption
threshold. The processes terminates. Each individual activates in subnetwork
$\mathcal{A}$ with probability $p=0.6$, and in subnetwork $\mathcal{B}$ with
the complementary probability $1-p=0.4$. Individuals activates in the
subnetwork are marked with red shadow, such as individual $1$ activates in
$\mathcal{A}$.

We propose a generalized non-Markovian susceptible-adopted-recovered (SAR)
model Wang et al. (2015a, 2016b, b) to describe the behavior spreading
dynamics in multiplex networks. Wang et al. Wang et al. (2015a) found a
transition phenomenon in which the dependence of the final adoption size
versus information transmission probability can change from discontinuous to
continuous by decreasing the individual adoption threshold, increasing the
initial seed size, or enhancing the network heterogeneity. In the SAR model,
social reinforcement is triggered by the reception of non-redundant behavioral
information, i.e., a susceptible individual adopts the new behavior only when
the amount of received non-redundant information from neighbors rises above a
given adoption threshold. We only allow the transmission of non-redundant
information. An individual in the susceptible state has not adopted the
behavior. An individual in the adopted state has adopted the behavior and is
willing to transmit the information to neighbors through one communication
channel. An individual in the recovered state has lost interest in the
behavior and no longer transmits the information.

During short periods of time the constraints posed by inelastic resources
limit individuals to a single communication channel Haerter et al. (2012b);
Holme and Saramäki (2012), i.e., at any given time step an individual can only
be active in one subnetwork. Individuals thus use communication channels
alternation (CCA) as the system evolves. To describe CCA, we introduce a
layer-switching parameter $p_{i}$ and assume that individual $i$ is active in
subnetwork $\mathcal{A}$ with a probability $p_{i}$ and active in subnetwork
$\mathcal{B}$ with a probability $1-p_{i}$. For simplicity, we assume that all
individuals have equal values of $p=p_{i}$. When an individual is active in a
subnetwork $\mathcal{X}\in\\{\mathcal{A},\mathcal{B}\\}$, it can transmit the
information to neighbors in subnetwork $\mathcal{X}$ and can read information
from neighbors in subnetwork $\mathcal{X}$ but cannot read information in the
other non-active subnetwork $\mathcal{Y}$. Thus individuals cannot
simultaneously read all information from neighbors in all communication
channels, and this introduces _time delays_ into receiving information.

We use a synchronous updating method Schönfisch and de Roos (1999) to renew
the states of individuals. We first randomly select a fraction $\rho_{0}$ of
individuals to be seeds in the adopted state. The remaining individuals are in
the susceptible state. At each time step, each adopted individual
$v_{\mathcal{X}}$ active in subnetwork $\mathcal{X}$ transmits the information
to each susceptible neighbor $u_{\mathcal{X}}$ with a probability
$\lambda_{\mathcal{X}}$. When individual $v_{\mathcal{X}}$ successfully
transmits information to $u_{\mathcal{X}}$, the information transmission
between them does not occur in subsequent steps, i.e., we only allow
nonredundant information to transmit between two individuals because each
neighbor can only partially guarantee the credibility and legitimacy of the
behavior Centola (2011). Note that if a susceptible individual
$u_{\mathcal{X}}$ is active in subnetwork $\mathcal{X}$, he only reads
information in subnetwork $\mathcal{X}$. Thus the CCA introduces time delays
into receiving information. If an individual $u_{\mathcal{X}}$ in subnetwork
$\mathcal{X}$ receives a new piece of information, he adopts the behavior with
a probability $\pi(m_{u}^{\mathcal{A}},m_{u}^{\mathcal{B}})$, where the
$m_{u}^{\mathcal{X}}$ value is the number of cumulative pieces of nonredundant
information from neighbors in subnetwork $\mathcal{X}$. If individual
$u_{\mathcal{X}}$ adopts the behavior, their counterpart $u_{\mathcal{Y}}$
also adopts the behavior in subnetwork $\mathcal{Y}$. Here we focus on a case
in which an individual in the susceptible state adopts the new behavior when
$m_{u}^{\mathcal{X}}$ exceeds their adoption threshold $T_{u}^{\mathcal{X}}$
in subnetwork $\mathcal{X}$. For simplicity, we assume all individuals have
the same adoption threshold $T_{u}^{\mathcal{X}}=T_{\mathcal{X}}$ in
subnetwork $\mathcal{X}$. The $\pi(m_{u}^{\mathcal{A}},m_{u}^{\mathcal{B}})$
value is

$\pi(m_{u}^{\mathcal{A}},m_{u}^{\mathcal{B}})=\left\\{\begin{array}[]{rcl}1,~{}m_{u}^{\mathcal{A}}\geq
T_{\mathcal{A}}~{}\mathrm{or}~{}m_{u}^{\mathcal{B}}\geq T_{\mathcal{B}},\\\
0,~{}\mathrm{others}.\end{array}\right.$ (1)

To include the social reinforcement effect, both $T_{\mathcal{A}}$ and
$T_{\mathcal{B}}$ values are greater than unity. At each time step we assume
that all individuals in the adopted state lose interest in transmitting the
information and with a probability $\gamma$ they recover. The spreading stops
when all the adopted individuals become recovered, and the received
information of all susceptible individuals does not exceed their threshold in
either subnetwork. Figure 1 shows the behavior spreading dynamics in multiplex
networks.

There are two key features in our proposed spreading dynamics. (1) The memory
effect is induced in our model. Unlike bootstrap percolation Baxter et al.
(2010, 2011); Di Muro et al. (2018) or a threshold model Watts (2002) in which
a susceptible individual becomes active (or adopted) only when its current
number or fraction of adopted neighbors is larger than a given value, in our
model a susceptible individual becomes adopted when their received accumulated
information in either subnetwork is larger than a threshold. (2) CCA is
included in our model. Unlike the models in Refs. Brummitt et al. (2012);
Yağan and Gligor (2012) in which each node can obtain the information from two
subnetworks simultaneously, CCA allows an individual to be active in only one
subnetwork at a time step.

##  III Theoretical method

From the description of the behavior adoption process in Sec. II we know that
there is a _non-Markovian_ characteristic in the dynamics because (i) social
reinforcement occurs when non-redundant behavioral information transmissions
are remembered, and (ii) CCA causes time delays in the reception of
information. This non-Markovian characteristic makes the strong dynamic
correlations among the states of the neighbors difficult to describe. Here we
develop a generalized edge-based compartmental method Wang et al. (2014b);
Miller and Volz (2013); Miller (2011); Wang et al. (2016c) of describing the
spreading of behavior in multiplex networks. In this theoretical method we
assume that the networks are large, the edges sparse, there are no degree-
degree correlations, and the dynamics evolve continuously.

An individual $u$ adopting the new behavior must take into account their
received information in both subnetworks $\mathcal{A}$ and $\mathcal{B}$.
Denoting $u_{\mathcal{X}}$ as an individual $u$ active in subnetwork
$\mathcal{X}\in\\{\mathcal{A},\mathcal{B}\\}$, we quantify the probability
that individual $u$ is in the susceptible state and assume that
$u_{\mathcal{X}}$ is in the cavity state Karrer and Newman (2010), i.e., that
$u_{\mathcal{X}}$ cannot transmit the information to neighbors in subnetwork
$\mathcal{X}$ but can receive information from neighbors in all the
subnetworks. The probability that an individual $v_{\mathcal{X}}$ has not
transmitted the information to a neighbor $u_{\mathcal{X}}$ along a randomly
chosen edge in subnetwork $\mathcal{X}$ by time $t$ is
$\theta_{\mathcal{X}}(t)$. At time $t$, the probability that $u_{\mathcal{X}}$
has received units of information $m_{\mathcal{X}}$ from subnetwork
$\mathcal{X}$ is

$\phi_{m_{\mathcal{X}}}^{\mathcal{X}}(k_{\mathcal{X}},t)={k_{\mathcal{X}}\choose
m_{\mathcal{X}}}[\theta_{\mathcal{X}}(t)]^{k_{\mathcal{X}}-m_{\mathcal{X}}}[1-\theta_{\mathcal{X}}(t)]^{m_{\mathcal{X}}},$
(2)

where $k_{\mathcal{X}}$ is the degree of individual $u$ in subnetwork
$\mathcal{X}$. Here the CCA disallows individual $u_{\mathcal{X}}$ from
reading all information from all subnetworks because he can only focus on the
information in the layer in which he is currently active. At time $t$,
individual $u$ can only read received information from neighbors in subnetwork
$\mathcal{X}$ when he is active in that subnetwork, i.e., when $u$ is active
in $\mathcal{X}$ he cannot read received information in $\mathcal{Y}$.

The CCA between two subnetworks for individual $u$ is a stochastic Poisson
process. The time interval distribution between two successive actions of $u$
in subnetwork $\mathcal{A}$ is $P(\overline{\omega})\sim
e^{-(1-p)\overline{\omega}}$ Mieghem (2014), as shown in Figs. 2(a)–(b) and
(d)–(e) where $p$ is the probability that $u$ is active in subnetwork
$\mathcal{A}$. Note that we use random regular (RR) Newman (2010) networks to
describe the two subnetworks $\mathcal{A}$ and $\mathcal{B}$ in Fig. 2. All
nodes have the same degree in the RR network, i.e.,
$P_{\mathcal{A}}(k_{\mathcal{A}})=P_{\mathcal{B}}(k_{\mathcal{B}})=1$ if
$\langle k_{\mathcal{A}}\rangle=\langle k_{\mathcal{B}}\rangle=10$. Each RR
network can be built using an uncorrelated configuration model Catanzaro et
al. (2005). We find the approximate average time interval of all individuals
active in subnetwork $\mathcal{X}$ to be

$\langle\overline{\omega}_{\mathcal{X}}\rangle=\frac{1}{1-p_{\mathcal{X}}},$
(3)

where $p_{\mathcal{X}}$ is the probability that an individual is active in
subnetwork $\mathcal{X}$. When $\mathcal{X}=\mathcal{A}$, then
$p_{\mathcal{X}}=p$. When $\mathcal{X}=\mathcal{B}$, then
$p_{\mathcal{X}}=1-p$.

If individual $u$ is active in subnetwork $\mathcal{Y}$ at time $t$, on
average his latest active time in subnetwork $\mathcal{X}$ is
$t-\langle\overline{\omega}_{\mathcal{X}}\rangle$. Here he reads all
information in subnetwork $\mathcal{Y}$. The approximate number of information
units he can read in subnetwork $\mathcal{X}$ is the number of information
units in the inbox of subnetwork $\mathcal{X}$ at
$t-\langle\overline{\omega}_{\mathcal{X}}\rangle$. The cumulative number of
information units $n_{\mathcal{X}}$ read by susceptible individual
$u_{\mathcal{X}}$ at time $t$ has a probability

$\chi_{n_{\mathcal{X}}}^{\mathcal{X}}(k_{\mathcal{X}},t)={k_{\mathcal{X}}\choose
n_{\mathcal{X}}}[\theta_{\mathcal{X}}(t-\langle\overline{\omega}_{\mathcal{X}}\rangle)]^{k_{\mathcal{X}}-n_{\mathcal{X}}}[1-\theta_{\mathcal{X}}(t-\langle\overline{\omega}_{\mathcal{X}}\rangle)]^{n_{\mathcal{X}}}.$
(4)

Figure 2: (Color online) Dynamics of social contagions on RR-RR multiplex
networks. Active events in subnetwork $\mathcal{A}$, i.e., individuals are
active in subnetwork $\mathcal{A}$, versus time $\varpi$ with (a) $p=0.8$ and
(c) $p=0.6$, respectively. The time interval distribution $P(\varpi)$ between
two consecutive activates in subnetwork $\mathcal{A}$ with (b) $p=0.8$ and (d)
$p=0.6$, respectively. We set other parameters to be $\rho_{0}=0.1$,
$\gamma=1.0$, $k=10$, $\lambda_{\mathcal{A}}=\lambda_{\mathcal{B}}=0.9$, and
$T_{\mathcal{A}}=T_{\mathcal{A}}=3$.

If individual $u$ with degree
$\overrightarrow{K}=(k_{\mathcal{A}},k_{\mathcal{B}})$ has not adopted the
behavior by time $t$, the cumulative pieces of information individual $u$ has
read in both subnetworks $\mathcal{A}$ and $\mathcal{B}$ will be less than the
adoption thresholds $T_{\mathcal{A}}$ and $T_{\mathcal{B}}$, respectively. At
time $t$, if individual $u$ is active in subnetwork $\mathcal{A}$ he remains
susceptible with a probability

$\begin{split}F_{\mathcal{A}}(\overrightarrow{K},t)&=\sum_{m_{\mathcal{A}}=0}^{k_{\mathcal{A}}}\sum_{n_{\mathcal{B}}=0}^{k_{\mathcal{B}}}\phi_{m_{\mathcal{A}}}^{\mathcal{A}}(k_{\mathcal{A}},t)\chi_{n_{\mathcal{B}}}^{\mathcal{B}}(k_{\mathcal{B}},t)\\\
&\times\prod_{j_{\mathcal{A}}=0}^{m_{\mathcal{A}}}\prod_{j_{\mathcal{B}}=0}^{n_{\mathcal{B}}}[1-\pi(j_{\mathcal{\mathcal{A}}},j_{\mathcal{B}})].\end{split}$
(5)

Similarly, if individual $u$ is active in subnetwork $\mathcal{B}$ at time $t$
he remains susceptible with a probability

$\begin{split}F_{\mathcal{B}}(\overrightarrow{K},t)&=\sum_{n_{\mathcal{A}}=0}^{k_{\mathcal{A}}}\sum_{m_{\mathcal{B}}=0}^{k_{\mathcal{B}}}\chi_{n_{\mathcal{A}}}^{A}(k_{\mathcal{A}},t)\phi_{m_{\mathcal{B}}}^{\mathcal{B}}(k_{\mathcal{B}},t)\\\
&\times\prod_{j_{\mathcal{A}}=0}^{n_{\mathcal{A}}}\prod_{j_{\mathcal{B}}=0}^{m_{\mathcal{B}}}[1-\pi(j_{\mathcal{A}},j_{\mathcal{B}})].\end{split}$
(6)

The probability that individual $u$ with a degree $\overrightarrow{K}$ is
susceptible is

$s(\overrightarrow{K},t)=(1-\rho_{0})[p_{\mathcal{A}}F_{\mathcal{A}}(\overrightarrow{K},t)+p_{\mathcal{B}}F_{\mathcal{B}}(\overrightarrow{K},t)],$
(7)

where the factor $(1-\rho_{0})$ is the probability that individual $u$ is
initially susceptible, and
$p_{\mathcal{X}}=\frac{1}{N}\sum_{u=1}^{N}p_{u}^{\mathcal{X}}$ is the average
probability that individual $u$ is active in subnetwork $\mathcal{X}$. Note
that $p_{\mathcal{A}}=1-p_{\mathcal{B}}$. Examining the degree distribution
$P(\overrightarrow{K})$, the fraction of susceptible individuals at time $t$
is

$S(t)=\sum_{\overrightarrow{K}}P(\overrightarrow{K})s(\overrightarrow{K},t).$
(8)

According to the definition of $\theta_{\mathcal{X}}$, an endpoint individual
$v_{\mathcal{X}}$ of the randomly selected edge of $u_{\mathcal{X}}$ can be in
a susceptible, adopted, or recovered state, thus $\theta_{\mathcal{X}}(t)$ can
be rewritten

$\theta_{\mathcal{X}}(t)=\xi_{S}^{\mathcal{X}}(t)+\xi_{A}^{\mathcal{X}}(t)+\xi_{R}^{\mathcal{X}}(t),$
(9)

where $\xi_{S}^{\mathcal{X}}(t)$ [$\xi_{A}^{\mathcal{X}}(t)$ or
$\xi_{R}^{\mathcal{X}}(t)$] is the probability that individual
$v_{\mathcal{X}}$ is susceptible (adopted or recovered) and has not
transmitted the information to $u_{\mathcal{X}}$ by time $t$.

If individual $u_{\mathcal{X}}$ is initially susceptible, he is in the cavity
state and thus cannot transmit the information to susceptible neighbors
$v_{\mathcal{X}}$ in subnetworks $\mathcal{X}$. When individual
$v_{\mathcal{X}}$ with degree $k_{\mathcal{X}}^{\prime}$ is susceptible and
active in subnetwork $\mathcal{X}$, he can only receive the information from
$k_{\mathcal{X}}^{\prime}-1$ other neighbors in subnetwork $\mathcal{X}$. At
time $t$, if the susceptible individual $v_{\mathcal{X}}$ is active in
subnetwork $\mathcal{X}$ the probability that he can read $m_{\mathcal{X}}$
pieces of information from this subnetwork is

$\tau_{m_{\mathcal{X}}}^{\mathcal{X}}(k_{\mathcal{X}}^{\prime},t)={k_{\mathcal{X}}^{\prime}-1\choose
m_{\mathcal{X}}}\theta_{\mathcal{X}}(t)^{k_{\mathcal{X}}^{\prime}-m_{\mathcal{X}}-1}[1-\theta_{\mathcal{X}}(t)]^{m_{\mathcal{X}}}.$
(10)

In contrast, individual $v$ can only read the information at time
$t-\langle\overline{\omega}_{\mathcal{X}}\rangle$ in subnetwork $\mathcal{X}$
when he is active in subnetwork $\mathcal{Y}$ at time $t$. As in Eq. (4), the
probability that individual $v_{\mathcal{X}}$ reads $n_{\mathcal{X}}$ pieces
of information is

$\begin{split}\varsigma_{n_{\mathcal{X}}}^{\mathcal{X}}(k_{\mathcal{X}}^{\prime},t)&={k_{\mathcal{X}}^{\prime}-1\choose
n_{\mathcal{X}}}\theta_{\mathcal{X}}(t-\langle\overline{\omega}_{\mathcal{X}}\rangle)^{k_{\mathcal{X}}^{\prime}-n_{\mathcal{X}}-1}\\\
&\times[1-\theta_{\mathcal{X}}(t-\langle\overline{\omega}_{\mathcal{X}}\rangle)]^{n_{\mathcal{X}}}.\end{split}$
(11)

Individual $v_{\mathcal{X}}$ remains susceptible when the cumulative pieces of
information read from neighbors in subnetwork $\mathcal{X}$ is lower than
adoption threshold $T_{\mathcal{X}}$ in the absence of individual
$u_{\mathcal{X}}$. When $v$ is active in subnetwork $\mathcal{X}$, he reads
$n_{\mathcal{Y}}$ pieces of information in subnetwork $\mathcal{Y}$ with a
probability $\chi_{n_{\mathcal{Y}}}^{\mathcal{Y}}(k_{\mathcal{Y}}^{\prime},t)$
[see Eq. (4)]. Thus the probability that individual $v$ is susceptible is

$\begin{split}\Psi_{\mathcal{X}}^{\mathcal{X}}(\overrightarrow{K},t)&=\sum_{m_{\mathcal{X}}^{\prime}=0}^{k_{\mathcal{X}}^{\prime}-1}\sum_{n_{\mathcal{Y}}^{\prime}=0}^{k_{\mathcal{Y}}^{\prime}}\tau_{m_{\mathcal{X}}^{\prime}}^{\mathcal{X}}(k_{\mathcal{X}}^{\prime},t)\chi_{n_{\mathcal{Y}}^{\prime}}^{\mathcal{Y}}(k_{\mathcal{Y}}^{\prime},t)\\\
&\times\prod_{j_{\mathcal{X}}^{\prime}=0}^{m_{\mathcal{X}}^{\prime}}\prod_{j_{\mathcal{Y}}^{\prime}=0}^{n_{\mathcal{Y}}^{\prime}}[1-\pi(j_{\mathcal{X}}^{\prime},j_{\mathcal{Y}}^{\prime})].\end{split}$
(12)

Similarly, when individual $v$ is active in subnetwork $\mathcal{Y}$ the
probability that $v$ will read $n_{\mathcal{X}}$ and $m_{\mathcal{Y}}$ pieces
of information from subnetworks $\mathcal{X}$ and $\mathcal{Y}$ is
$\varsigma_{n_{\mathcal{X}}}^{\mathcal{X}}(k_{\mathcal{X}}^{\prime},t)$ and
$\phi_{m_{\mathcal{Y}}}^{\mathcal{Y}}(k_{\mathcal{Y}}^{\prime},t)$,
respectively. The probability that individual $v$ is susceptible is

$\begin{split}\Psi_{\mathcal{X}}^{\mathcal{Y}}(\overrightarrow{K},t)&=\sum_{n_{\mathcal{X}}^{\prime}=0}^{k_{\mathcal{X}}^{\prime}-1}\sum_{m_{\mathcal{Y}}^{\prime}=0}^{k_{\mathcal{Y}}^{\prime}}\varsigma_{n_{\mathcal{X}}^{\prime}}^{\mathcal{X}}(k_{\mathcal{X}}^{\prime},t)\phi_{m_{\mathcal{Y}}^{\prime}}^{\mathcal{Y}}(k_{\mathcal{Y}}^{\prime},t)\\\
&\times\prod_{j_{\mathcal{X}}^{\prime}=0}^{n_{\mathcal{X}}^{\prime}}\prod_{j_{\mathcal{Y}}^{\prime}=0}^{m_{\mathcal{Y}}^{\prime}}[1-\pi(j_{\mathcal{X}}^{\prime},j_{\mathcal{Y}}^{\prime})].\end{split}$
(13)

When an initially susceptible individual $v$ is active in subnetwork
$\mathcal{X}$, the probability that he remains susceptible is

$\Theta_{\mathcal{X}}(\overrightarrow{K},t)=p_{\mathcal{X}}\Psi_{\mathcal{X}}^{\mathcal{X}}(\overrightarrow{K},t)+p_{\mathcal{Y}}\Psi_{\mathcal{X}}^{\mathcal{Y}}(\overrightarrow{K},t).$
(14)

Denoting $H_{\mathcal{X}}(k_{\mathcal{X}}^{\prime}|k_{\mathcal{X}})$ to be the
probability of a node with degree $k_{\mathcal{X}}$ connects to a node with
degree $k_{\mathcal{X}}^{\prime}$ in network $\mathcal{X}$. Thus the
probability that individual $u_{\mathcal{X}}$ connects to a susceptible
individual in subnetwork $\mathcal{X}$ is

$\xi_{S}^{\mathcal{X}}(t)=(1-\rho_{0})\sum_{\overrightarrow{K}}H_{\mathcal{X}}(k_{\mathcal{X}}^{\prime}|k_{\mathcal{X}})\Theta_{\mathcal{X}}(\overrightarrow{K},t).$
(15)

In an uncorrelated network,
$H_{\mathcal{X}}(k_{\mathcal{X}}^{\prime}|k_{\mathcal{X}})=k_{\mathcal{X}}^{\prime}P(\overrightarrow{K})/\langle
k_{\mathcal{X}}\rangle$. We rewrite Eq. (15) to be

$\xi_{S}^{\mathcal{X}}(t)=(1-\rho_{0})\frac{1}{\langle
k_{\mathcal{X}}\rangle}\sum_{\overrightarrow{K}}k_{\mathcal{X}}^{\prime}P(\overrightarrow{K})\Theta_{\mathcal{X}}(\overrightarrow{K},t).$
(16)

When the information transmits through an edge at time $t$ in subnetwork
$\mathcal{X}$, the edge does not fulfill the definition of
$\theta_{\mathcal{X}}(t)$. The decreasing of $\theta_{\mathcal{X}}(t)$ is thus

$\frac{d\theta_{\mathcal{X}}(t)}{dt}=-p_{\mathcal{X}}\lambda_{\mathcal{X}}\xi_{A}^{\mathcal{X}}(t).$
(17)

For $\xi_{R}^{\mathcal{X}}(t)$ to grow, (i) the information cannot be
transmitted through the edge, and (ii) the adopted individual must recover at
time $t$. The evolution of $\xi_{R}^{\mathcal{X}}(t)$ is

$\frac{d\xi_{R}^{\mathcal{X}}(t)}{dt}=\gamma(1-p_{\mathcal{X}}\lambda_{\mathcal{X}})\xi_{A}^{\mathcal{X}}(t).$
(18)

Combining Eqs. (17) and (18) and the initial condition
$\theta_{\mathcal{X}}(0)=1$ and $\xi_{R}^{\mathcal{X}}(0)=0$ gives us

$\xi_{R}^{\mathcal{X}}(t)=\frac{\gamma(1-p_{\mathcal{X}}\lambda_{\mathcal{X}})[1-\theta_{\mathcal{X}}(t)]}{p_{\mathcal{X}}\lambda_{\mathcal{X}}}.$
(19)

We use Eqs. (9), (16), (17), and (19) to obtain the value of
$\theta_{\mathcal{X}}(t)$.

Using the evolution process of the behavior spreading dynamics described in
Sec. II, we derive the evolution equations of the fraction of individuals in
the adopted and recovered states,

$\frac{dA(t)}{dt}=-\frac{dS(t)}{dt}-\gamma A(t),$ (20)

and

$\frac{dR(t)}{dt}=\gamma A(t),$ (21)

respectively. Combining Eqs. (8) and (20)–(21), we obtain the time evolution
of the behavior spreading dynamics in multiplex networks. When
$t\rightarrow\infty$, the final behavior adoption size is denoted $R(\infty)$.

We next examine the growth pattern of the final behavior adoption size
$R(\infty)$ versus the information transmission probability
$\overrightarrow{\lambda}=(\lambda_{\mathcal{A}},\lambda_{\mathcal{B}})$ and
the CCA probability $p$. We first investigate the effects of the time delays
induced by CCA on the dynamics of social contagions by comparing them with a
null model without time delays. Details about the null model are supplied in
the Appendix. Figure 3(a) shows that the time delays induced CCA affect
behavior spreading dynamics, including the time evolutions of susceptible
$S(t)$, adopted $A(t)$, and recovered $R(t)$ individuals. We find that $S(t)$
[$R(t)$] decreases (increases) with $t$, and that $A(t)$ first increases and
then decreases. Note that the time delays induced by CCA slow the behavior
adoption. To quantify the slowing caused by the time delays induced by CCA, we
compute the stabilizing time of the system $t_{\rm max}$, i.e., the average
time needed for the system to reach the final state. When $0\leq p\leq 0.5$,
$t_{\rm max}$ first increases with $p$ and then decreases. When the $p$ value
is small, most individuals are active in subnetwork $\mathcal{B}$. When
susceptible individuals receive information that exceeds their adoption
threshold, and they quickly adopt the behavior. When the $p$ value is
increased, some individuals become active in subnetwork $\mathcal{A}$ but
adopt the behavior only after their received information exceeds their
adoption threshold. Thus $t_{\rm max}$ first increases. When the $p$ value is
large, fewer individuals adopt the behavior (see Fig. 4) and $t_{\rm max}$
thus decreases. The inset of Fig. 3(b) shows the retardation time $\Delta
t_{\rm max}$ for social contagion models with and without time delays. Note
that $\Delta t_{\rm max}$ first increases with $p$ and then decreases. We find
the same phenomena for $0.5\leq p\leq 1.0$ because subnetworks $\mathcal{A}$
and $\mathcal{B}$ are both RR networks.

Figure 3(a) shows that the time delays induced by CCA do not affect the final
behavior adoption size $R(\infty)$. This is because when a susceptible
individual fulfills the behavior adoption conditions the time delays only
affect the behavior adoption time. This also indicates that the critical
points of the system remain the same when there are no time delays in their
behavior adoption, i.e., a susceptible individual adopts the new behavior as
soon as the received pieces of information equal or exceed the adoption
threshold, independent of the subnetwork in which he is currently active. Thus
when examining the final state of the behavior we disregard time delays, i.e.,
Eqs. (4) and (11) are the same as Eqs. (2) and (10), respectively. We perform
numerical simulations and theoretical analyses of the social contagions on ER-
SF multiplex networks, and find that the time delays induced by CCA slow the
spreading dynamics but do not affect the final adoption size.

Figure 3: (Color online) Dynamics of social contagions on RR-RR multiplex
networks. (a) The time evolutions of susceptible $S(t)$, adopted $A(t)$ and
recovered $R(t)$ individuals with (solid symbols) and without (empty symbols)
time delays when $p=0.8$. The symbols are numerical simulation results and
lines are the theoretical predictions. (b) The stabilizing time of the system
$t_{\rm max}$ with (solid symbols) and without (empty symbols) time delays
versus $p$. The inset of (b) shows the retardation time as a function of $p$
for the system with or without time delays. We set other parameters to be
$\rho_{0}=0.1$, $\gamma=1.0$, $k=10$,
$\lambda_{\mathcal{A}}=\lambda_{\mathcal{B}}=0.9$, and
$T_{\mathcal{A}}=T_{\mathcal{A}}=3$.

In the final state, i.e., when $t\rightarrow\infty$, there are no nodes in the
adopted state, and no information is transmitted through edges. Thus we have
$d\theta_{\mathcal{X}}(t)/dt=0$ and
$\theta_{\mathcal{X}}(t)=\theta_{\mathcal{X}}(t-\langle\overline{\omega}_{\mathcal{X}}\rangle)=\theta_{\mathcal{X}}^{\ast}$.
Combining Eqs. (9), (16)–(17), and (19) we obtain

$\begin{split}\theta_{\mathcal{X}}^{\ast}&=(1-\rho_{0})\frac{\sum_{\overrightarrow{K}}k_{\mathcal{X}}P(\overrightarrow{K})\Theta_{\mathcal{X}}(\overrightarrow{K},\infty)}{\langle
k_{\mathcal{X}}\rangle}\\\
&+\frac{\gamma(1-p_{\mathcal{X}}\lambda_{\mathcal{X}})[1-\theta_{\mathcal{X}}^{\ast}]}{p_{\mathcal{X}}\lambda_{\mathcal{X}}}\\\
&=f_{\mathcal{X}}(\theta_{\mathcal{A}}^{\ast},\theta_{\mathcal{B}}^{\ast}).\end{split}$
(22)

When $\rho_{0}\rightarrow 0$, then $\theta_{\mathcal{X}}^{*}=1$ is a trivial
solution of Eq. (22), but this vanishingly small fraction of seeds cannot
trigger global behavior adoption because $T_{\mathcal{X}}>1$ Wang et al.
(2015a). To stimulate global behavior adoption, we must have a finite fraction
of seed individuals. Here $\theta_{\mathcal{X}}^{*}=1$ is no longer the
solution of Eq. (22), which now has either one or three fixed points
(including multiplicity). If Eq. (22) has only one solution at all values of
$\lambda_{\mathcal{X}}$, then $\theta_{\mathcal{X}}^{*}$ decreases
continuously with $\lambda_{\mathcal{X}}$, and this leads to a continuous
growth pattern in the final behavior adoption $R(\infty)$. If the number of
the solutions of Eq. (22) varies with $\lambda_{\mathcal{X}}$, the situation
is different. For a given $\lambda_{\mathcal{X}}$, if there is only one fixed
point of Eq. (22), it is the physically meaningful solution. If there are
three fixed points, which are stable, unstable, and saddle points, only the
maximum solution is physically meaningful in our irreversible behavior
spreading dynamics when we randomly select a relatively small fraction of
seeds, since the individuals in the adopted state persistently transmit the
information to their neighbors, and $\theta_{\mathcal{X}}^{*}$ decreases from
unity. Thus a saddle-node bifurcation occurs Strogatz (2014); Gleeson and
Cahalane (2007). Through a bifurcation analysis of Eq. (22), we find that the
system undergoes a cusp catastrophe: varying $\lambda_{\mathcal{X}}$ the
physically meaningful stable solution of $\theta_{\mathcal{X}}^{*}$ suddenly
produces a different outcome. Therefore, the growth patten of $R(\infty)$ will
be discontinuous because a meaningful solution decreases abruptly at such
critical conditions as the critical information transmission probability
$\overrightarrow{\lambda}_{c}$ and the critical CCA probability $p_{c}$.

To determine $p_{c}$, we first rewrite Eq. (22) to be

$F_{\mathcal{A}}(\theta_{\mathcal{A}}^{\ast},\theta_{\mathcal{B}}^{\ast})=\theta_{\mathcal{A}}^{\ast}-f_{\mathcal{A}}(\theta_{\mathcal{A}}^{\ast},\theta_{\mathcal{B}}^{\ast})=0,$
(23)

and

$F_{\mathcal{B}}(\theta_{\mathcal{A}}^{\ast},\theta_{\mathcal{B}}^{\ast})=\theta_{\mathcal{B}}^{\ast}-f_{\mathcal{B}}(\theta_{\mathcal{A}}^{\ast},\theta_{\mathcal{B}}^{\ast})=0.$
(24)

At the discontinuous critical point, the curves of
$f_{\mathcal{A}}(\theta_{\mathcal{A}}^{\ast},\theta_{\mathcal{B}}^{\ast})$ and
$f_{\mathcal{B}}(\theta_{\mathcal{A}}^{\ast},\theta_{\mathcal{B}}^{\ast})$ are
tangent to each other at the discontinuous critical point Yuan et al. (2017);
Parshani et al. (2010). Thus we find that the critical point for the
discontinuous growth pattern is given by
$\frac{d\theta_{\mathcal{A}}^{\ast}(\theta_{\mathcal{B}}^{\ast})}{d\theta_{\mathcal{B}}^{\ast}}\frac{d\theta_{\mathcal{B}}^{\ast}(\theta_{\mathcal{A}}^{\ast})}{d\theta_{\mathcal{A}}^{\ast}}=1$.
Combining Eqs. (23) and (24), we further obtain the discontinuous critical
points of $\overrightarrow{\lambda}_{c}$ and $p_{c}$ by solving both Eq. (22)
and

$\frac{\partial
f_{\mathcal{A}}(\theta_{\mathcal{A}}^{\ast},\theta_{\mathcal{B}}^{\ast})}{\partial\theta_{\mathcal{B}}^{\ast}}\frac{\partial
f_{\mathcal{B}}(\theta_{\mathcal{A}}^{\ast},\theta_{\mathcal{B}}^{\ast})}{\partial\theta_{\mathcal{A}}^{\ast}}=1.$
(25)

##  IV Numerical simulations

Here we perform extensive simulations on multiplex networks, including RR-RR
networks [i.e., in which each subnetwork is a random regular (RR) network] and
ER-SF networks [i.e., in which subnetworks $\mathcal{A}$ and $\mathcal{B}$ are
Erdős-Rényi (ER) Erdös and Rényi (1959) and scale-free (SF) Catanzaro et al.
(2005) networks, respectively]. In each case we set the network size, average
degree, and recovery probability to be $N=10^{4}$, $\rho_{0}=0.1$, $\langle
k_{\mathcal{A}}\rangle=\langle k_{\mathcal{B}}\rangle=10$, and $\gamma=1.0$,
respectively, unless stated otherwise.

###  IV.1 RR-RR multiplex networks

Figure 4: (Color online) Behavior spreading on RR-RR networks. (a) Final
behavior adoption size $R(\infty)$, (b) the fraction of individuals in the
subcritical state $g(\infty)$, and (c) NOI versus information transmission
probability $\lambda$ for CCA probability $p=0.0$, $0.2$ and $0.4$. (d)
$R(\infty)$, (e) $g(\infty)$, and (f) NOI versus $p$ for $\lambda=0.7$, $0.8$
and $0.9$. The inset of (d) shows $R(\infty)$ versus $p$ for $\lambda=0.6$.
The empty (full) symbols are the simulated values of $R(\infty)$ at network
size $N=10^{4}$ ($N=10^{6}$). The lines in (a), (b), (d) and (e) are the
theoretical values, and are numerical simulation results in (c) and (f). Other
parameters are $\rho_{0}=0.1$, $\gamma=1.0$, $\langle k\rangle=10$, and
$T_{\mathcal{A}}=T_{\mathcal{A}}=3$, respectively.

We first study social contagions on RR-RR multiplex networks in which each
node in each subnetwork has a degree $k=10$. Figure 4(a) shows $R(\infty)$ as
a function of $\lambda$ under different CCA probabilities $p$. We find that
$R(\infty)$ first grows continuously for small values of $\lambda$, and then
increases discontinuously at $\lambda_{c}$, i.e., exhibiting a hybrid growth,
regardless of $p$. We can understand the discontinuous increasing of
$R(\infty)$ by studying the fraction of individuals $g(\infty)$ in the
subcritical state. When an individual is in the subcritical state he has
received the information but have not adopted the behavior, and the number of
information units from distinct neighbors is one less than the adoption
threshold in subnetwork $\mathcal{A}$ or $\mathcal{B}$ Wang et al. (2015a).
Slightly increasing the value of $\lambda$ can increase the number of
subcritical state individuals with information units equal to or greater than
their threshold [see Fig. 4(b)] and lead to a discontinuous jump in
$R(\infty)$. In addition, $R(\infty)$ decreases with $p$, since an increasing
number of susceptible individuals are unable to fulfill the behavior adoption
condition. Because near the critical point the system exhibits a critical
slowing, we locate the numerical critical point $\lambda_{c}^{\rm I}$ by
examining the number of iterations (NOI), which is widely used in the
bootstrap percolation [41,42] and cascading failures Baxter et al. (2010,
2011); Di Muro et al. (2018) and cascading failures Gao et al. (2011); Liu et
al. (2016b), required to reach the final state, and we take into account only
iterations in which at least one individual adopts the behavior. Figure 4(c)
shows that at the critical point the NOI exhibit a peak. Note that our
theoretical method accurately predicts $R(\infty)$ and the growth pattern of
$R(\infty)$. The deviations around the critical points are caused by finite-
size network effects [see Fig. 4(a)].

Figures 4(d)–4(f) show the effect of $p$ on social contagions. Unlike when
$T_{\mathcal{A}}=T_{\mathcal{B}}=1$ in Fig. 7 of Appendix B, Fig. 4(d) shows
that the final behavior adoption size $R(\infty)$ versus $p$ is non-monotonic.
Specifically, for relatively small values of $\lambda=0.6$ few susceptible
individuals adopt the behavior, and $R(\infty)$ first decreases continuously
with $p$ and then increases continuously [see the inset in Fig. 4(d)]. For
large values of $\lambda=0.7,0.8,0.9$, $R(\infty)$ first decreases
_discontinuously_ with $p$ and then increases _discontinuously_ [see Fig.
4(d)]. We can similarly understand the growth pattern of $R(\infty)$ by
studying $g(\infty)$. For a small value of $p$, e.g., $p=0.1$ when
$\lambda=0.9$, most individuals in subnetwork $\mathcal{A}$ are active, many
individuals adopt the behavior, and few individuals remain in the subcritical
state [see Fig. 4(e)]. With an increase of $p$, fewer individuals are active
in subnetwork $\mathcal{B}$, many individuals in subnetwork $\mathcal{A}$ are
in the subcritical state [see Fig. 4(e)], and there is a sharp decrease in
$R(\infty)$. By further increasing $p$, many individuals “jump” between
subnetworks $\mathcal{A}$ and $\mathcal{B}$, and fewer individuals receive one
fewer information units than the adoption threshold in the two subnetworks.
Thus $g(\infty)$ decreases [see Fig. 4(e)]. Similarly $g(\infty)$ first
increases and then decreases discontinuously when the value of $p$ is large.
Note that NOI versus $p$ exhibits two peaks at $p_{c}^{1}=1-p_{c}^{2}$ because
the two subnetworks are RR networks [see Fig. 4(f)].

Figure 5: (Color online) Dynamics of social contagions on uncorrelated ER-SF
multiplex networks. (a) The final behavior adoption size $R(\infty)$ versus
information transmission probability $\lambda$ under different communication
channels alternation probability $p$. (b) $R(\infty)$ as a function of $p$ for
different $\lambda$. Lines are the theoretical predictions from Eqs. (7) and
(17)-(19). The error bars indicate the standard deviations. Other parameters
are set to be $\langle k_{\mathcal{A}}\rangle=10$, $\langle
k_{\mathcal{B}}\rangle=10$, $\upsilon_{\mathcal{B}}=3.0$, $\rho_{0}=0.1$, and
$T_{\mathcal{A}}=T_{\mathcal{A}}=3$.

###  IV.2 ER-SF multiplex networks

When studying social contagions on ER-SF multiplex networks, we assume that
there are no degree-degree correlations in the intralayers and interlayers. We
generate the SF networks using the same method as that used in uncorrelated
configuration networks that have a power-low degree distribution
$P_{\mathcal{B}}(k_{\mathcal{B}})=\frac{1}{\sum_{k_{\mathcal{B}}}k_{\mathcal{B}}^{-v_{\mathcal{B}}}}k_{\mathcal{B}}^{-v_{\mathcal{B}}}$.
In network $\mathcal{B}$ without degree-degree correlations, the maximum
degree follows a structural cut-off Boguná et al. (2004), i.e., $k_{\rm
max}\sim\sqrt{N}$. The SF network is built using the uncorrelated
configuration method in Ref. Catanzaro et al. (2005). Figure 5 shows the
social contagions on ER-SF networks. Figure 5(a) shows that CCA changes the
growth pattern of $R(\infty)$ versus $\lambda$. When $p=0.0$ ($1.0$), the
number of individuals only active in subnetwork $\mathcal{B}$ ($\mathcal{A}$)
and $R(\infty)$ grows continuously (discontinuously) versus $\lambda$ (see
Ref. Wang et al. (2015a)). Increasing $p$ increases the number of individuals
active in the homogeneous subnetwork $\mathcal{A}$, and there are more
individuals in the subcritical state who are likely to simultaneously adopt
the behavior. Thus we see a hybrid growth in $R(\infty)$, i.e., $R(\infty)$
first grows continuously for small values of $\lambda$, and then grows
discontinuously at $\lambda_{c}$.

Figure 6: (Color online) Final behavior adoption size $R(\infty)$ versus
information transmission probability $\lambda$ and communication channels
alternation probability $p$ on uncorrelated ER-SF networks. Color-coded values
of $R(\infty)$ from numerical simulations (a) and theoretical solutions (b) in
the parameter plane $(\lambda,p)$. The white horizontal line $p_{c}$ separates
the plane into regions I and II. In region I (II), $R(\infty)$ exhibits a
hybrid (continuous) growth with $\lambda$. The white curves denote the
theoretical discontinuous critical points $\lambda_{c}^{\rm I}$. We set other
parameters as $\langle k_{\mathcal{A}}\rangle=10$, $\langle
k_{\mathcal{B}}\rangle=10$, $\upsilon_{\mathcal{B}}=3.0$, $\rho_{0}=0.1$, and
$T_{\mathcal{A}}=T_{\mathcal{A}}=3$.

Figure 5(b) shows that $R(\infty)$ versus $p$ exhibits differing patterns
under different values of $\lambda$. For a small value $\lambda=0.5$,
$R(\infty)$ first decreases continuously with $p$. For a relatively large
value, e.g., $\lambda=0.6$, $R(\infty)$ first decreases continuously with $p$
and then increases discontinuously. Note that when $\lambda=0.7$, $R(\infty)$
first decreases continuously with $p$, then increases to a peak at some $p$,
then decreases discontinuously, and finally increases discontinuously. For a
very large value $\lambda=0.9$, $R(\infty)$ first decreases continuously and
then increases continuously. For intermediate values of $0<p<1$, the CCA
emerges and constrains user ability to receive enough information to exceed
the adoption threshold of a subnetwork, and thus there is a non-monotonous
varying of $R(\infty)$. We can understand the different growth patterns by
studying high degree nodes (hubs), which (i) are more likely to adopt the
behavior than those in homogeneous network networks for small values of
$\lambda$, and (ii) lead to individuals adopting the behavior gradually with
fewer individuals in the subcritical state simultaneously adopting the
behavior Wang et al. (2015a). The fraction of hubs in the live subnetwork
varies with $p$. We define a live subnetwork to be a user’s active subnetwork
at different time steps. For small $\lambda=0.5$, individuals active in
heterogeneous (homogeneous) networks can easily (with difficulty) adopt the
behavior, adding the second role of hubs, and $R(\infty)$ thus decreases with
$p$. For the larger $\lambda=0.6$, $R(\infty)$ first decreases continuously
because some individuals are active in heterogeneous networks when $p$ is
relatively small (e.g., $p<0.5$), and $R(\infty)$ then increases
discontinuously because many individuals are active in homogeneous networks
and in the subcritical state for large $p$. The strange phenomena when
$\lambda=0.7$ is also induced by the live network. When $p\approx 0.4$, many
individuals are active in homogeneous subnetwork $\mathcal{A}$, and
$R(\infty)$ thus increases with $p$. A further increase of $p$ is a slight
perturbation that moves many individuals into the subcritical state, and
$R(\infty)$ decreases discontinuously. When $\lambda=0.8$, most individuals
adopt the behavior, few individuals are in the subcritical state, and there is
only a non-monotonous varying of $R(\infty)$ versus $p$. These phenomena do
not occur when $T_{\mathcal{A}}=T_{\mathcal{B}}=1$, as shown in Fig. 7. We can
predict these phenomena using our theoretical method. Note that the deviations
near the discontinuous points can be eliminated by enlarging the network size
$N$.

Figure 6 shows $R(\infty)$ versus the $\lambda-p$ plane. Using the growth
pattern of $R(\infty)$ versus $\lambda$, we divide the plane into regions I
and II according to a critical CCA probability $p_{c}$. In region I,
$R(\infty)$ first grows continuously with $\lambda$ then increases
discontinuously at $\lambda_{c}$, and in region II, $R(\infty)$ grows
continuously with $\lambda$. There is thus a crossover phenomenon in the
growth pattern: When $p>p_{c}$, the growth pattern of $R(\infty)$ is
discontinuous; otherwise it is continuous. We can explain the growth pattern
of $R(\infty)$ using bifurcation theory. The discontinuous critical points
$\lambda_{c}^{\rm I}$ exhibit a non-monotonic change with $p$ because of the
CCA. When $p>p_{c}$, individuals are more likely to be active in subnetwork
$\mathcal{A}$, and thus there are some hubs and many low-degree individuals in
the live subnetwork. These hubs promote behavior adoption. Increasing $p$
decreases the number of hubs and low-degree individuals in the live network,
and $p_{c}$ thus first increases then decreases. The results from our
theoretical method agree with those from the numerical simulations.

##  V Conclusions

We have investigated how communication channel alteration (CCA) affects the
dynamics of social contagions. We first propose a non-Markovian behavior
spreading model for multiplex networks in which each individual can only
transmit and obtain the information from neighbors in their own subnetwork. To
include CCA, we assume that an individual can be active in only one
communication layer and at any given time can only transmit behavioral
information to neighbors and read behavioral information from neighbors within
the same subnetwork. The CCA slows a user’s ability to receive information
from both subnetworks. Thus time delays in obtaining the information from both
subnetworks are introduced. We then perform numerical simulations of
artificial multiplex networks and find that the time delays caused by CCA slow
the behavior adoption process but do not affect the final behavior adoption
size. In addition, CCA suppresses the final behavior adoption size $R(\infty)$
but does not change the growth pattern of $R(\infty)$ on RR-RR networks. We
find in ER-SF networks that the growth pattern of $R(\infty)$ can be changed
from hybrid to continuous by decreasing the layer-switching probability. To
quantify the non-Markovian spreading dynamics, we develop an edge-based
compartmental method that produces results that agree with the numerical
simulation results.

We have examined how CCA—an important inter-layer switching mechanism—affects
social contagions. We first construct the connections between human dynamics
Barabasi (2005); Vazquez (2005); Brockmann et al. (2006) and social contagions
on multiplex networks. Individuals accomplishing different tasks using
different communication channels do so in patterns that exhibits memory and
burst characteristics. Our results here are the first to investigate the
effects of human dynamics on social contagions in multiplex networks, and they
expand our understanding of phase transition phenomena. The hybrid growth in
the final adoption size is similar to the hybrid phase transition observed in
other dynamics Dorogovtsev et al. (2008); Di Muro et al. (2017); Wu et al.
(2014), and the critical phenomena of our proposed social contagions near the
critical point need further investigation Our theoretical method allows us to
understand how CCA shapes spreading dynamics and to analyze different dynamic
processes on multiplex networks. Our work may stimulate further research on
social contagions that takes into account both realistic spreading mechanisms
and network topologies and provides theoretical insights into how to control
the spread of epidemics. In addition, social contagions with a heterogeneous
layer-switching probability is an intriguing subject for examination Starnini
et al. (2016). For example, a layer-switching probability follows a power-law
distribution, and the layer-switching probability of each individual is
dependent on their inherent characteristics.

## Acknowledgements

This work was partially supported by the National Natural Science Foundation
of China (Grant Nos. 11575041, and 61672238), China Postdoctoral Science
Foundation (CPSF) (Grant No. 2018M631073), the Fundamental Research Funds for
the Central Universities, and Natural Science Foundation of Shanghai (Grant
No. 18ZR1412200). L.A.B. acknowledges the support of UNMdP, PICT 0429/13, and
DTRA Grant HDTRA1-14-1-0017. The Boston University Center for Polymer Studies
is supported by NSF Grants No. PHY-1505000, No. CMMI-1125290, and No. 624
CHE-1213217, and by DOE Contract No. DE-AC07-05Id14517.

## Appendix A: Null model for social contagions with CCA

We use the layer-switching parameter $p$ to describe CCA in the null model.
The only difference between this null model and the model described in Sec. 2
is that in this null model we assume a susceptible individual becomes adopted
when the accumulated units of received information is equal to or larger than
the adoption threshold in any subnetwork, regardless of whether the individual
is active in the subnetwork. This difference allows the susceptible individual
to obtain the behavioral information without time delays. The null model
includes CCA but not time delays.

## Appendix B: $T_{\mathcal{A}}=T_{\mathcal{B}}=1$

Figure 7 shows the contagions on multiplex networks with
$T_{\mathcal{A}}=T_{\mathcal{B}}=1$, i.e., the contagions on both subnetworks
return to the simple contagion. We find that $R(\infty)$ does not change with
$p$ on RR-RR multiplex networks and increases with $p$ on ER-SF networks.

Figure 7: (Color online) Final behavior adoption size $R(\infty)$ versus
communication channels alternation probability $p$ on multiplex networks when
$T_{\mathcal{A}}=T_{\mathcal{B}}=1$. The symbols are the numerical simulation
results, and lines are the theoretical predictions. We set other parameters as
$\langle k_{\mathcal{A}}\rangle=10$, $\rho_{0}=0.1$, $\langle
k_{\mathcal{B}}\rangle=10$ and $\upsilon_{\mathcal{B}}=3.0$ for SF networks.

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