Multilayer social reinforcement induces bistability on multiplex networks

Social contagion describes a variety of behavioral imitations caused by social influence and is particularly ubiquitous in the digital age [1, 2, 3]. In order to predict and control the collective contagion phenomena on large-scale social networks, such as the spread of rumors, social norms and online behaviors, scientists have made great efforts on understanding the underlying dynamical mechanisms [4, 5, 6, 7]. Early studies argued that the dynamical processes of social contagion are similar to disease spreading in the sense that a simple contact with a single infected individual can trigger diffusion [8, 9, 10]. Therefore, many epidemic-like models are proposed, which are called simple contagion models. These models provide profound insights into many physical phenomena on social networks, such as the cascading process of information diffusion, the co-contagion dynamics on multiplex networks, and etc [11, 12].

However, simple contagion model can not deal with the spread of more complex social behaviors, especially when the behaviors are risky, costly or polarized, ranging from the spread of public health behaviors (e.g., vaccine, vaping, diet) to social movement [13, 14, 15]. For instance, a well-known complex mechanism of social contagion is called the social reinforcement. It corresponds to the fact that exposure to multiple sources with a same stimuli would give individuals more confidence to participate than multiple exposures to the same source, leading to a significant promotion on social transmission [16, 17]. As McAdam and Paulsen concluded [18], “the fact that we are embedded in many relationships means that any major decision we are contemplating will likely be mediated by a significant subset of those relationships”. Nevertheless, this empirically confirmed phenomenon is fundamentally different from the view of simple contagion model [19, 20]. Hence, more complex mechanisms were further incorporated into dynamical models. These complex contagion models lead to abundant intriguing findings, some of which even conflict with conclusions of simple contagion models [21, 22]. For example, threshold models assumed that social contagion only happens when the number (or fraction) of exposures to multiple sources exceeds a given threshold and suggested the dependence of diffusion results on initial density of adopters [23]. In addition, Centola et al. proved that the clustering network structure suppressed simple contagion while surprisingly facilitated the spread of behaviors that require social reinforcement [24]. Another framework modified epidemic-like models by increasing transmission rate when exposing to multiple sources, and showed the promoting effect of social reinforcement on contagion processes [25, 26]. Another well-known complex mechanism is caused by the interplay between spreading dynamics and network topology. For example, Gross et al. studied contagion process on adaptive networks where the connection between nodes relies on their states, and showed that this dynamics-topology interaction could lead to the emergence of bistable region where healthy and endemic states co-exist [27]. Besides, Iacopini et al. noted that the existence of high-order structures embedded on single-layer networks, such as full triangles, could also induce bistable phenomena [28].

Moreover, complex mechanisms depicting interactions between multiple contagion processes on networked systems, such as the spreading of interacting diseases, were widely investigated [29]. Chen et al. explored cooperative contagion processes where individuals infected by one disease are more susceptible to the other, and showed that large degree of cooperation could lead to the occurrence of abrupt phase transition and bistability [30]. Soriano-Paños et al. described the interaction between cooperative or competitive diseases by increasing or decreasing the susceptibility of individuals infected by one of the diseases, and also found bistability caused by cooperativity between diseases [31]. Sanz et al. proposed a general framework analyzing simultaneous spreading of two interacting diseases and derived complex phase diagrams [32]. Pinotti et al. explored a three-player pathogen system where competition and cooperation coexist, and showed that the presence of cooperative pathogen could lead to some intriguing phenomena including non-monotonic boundaries separating phase diagram and bistability [33]. There were also many works extending the framework from single-layer networks to multilayer networks [34]. For example, Wu et al. studied discrete-time Markov-Chain model depicting the spreading of two diseases in multiplex networks and derived epidemic thresholds [35].

Recently, owing to the development of various social medias, many social behaviors spread in multiple social circles rather than in a single social circle, which are naturally modeled as multilayer networks [36, 37, 38, 39]. Each layer represents a single social circle which could be an online social platform or a network of offline relationships, such as friendships and colleagues. Similar to social reinforcement in a single-layer network, there exists significant difference between exposure to multiple social circles (multilayer reinforcement) and multiple exposures to the same social circle (intra-layer reinforcement)[40]. Specifically, individuals would be more convinced and have higher possibility for diffusion when receiving the same informative stimulus from different social circles. However, previous studies mainly concentrated on simple interactions between nodes and its replicas in different layers [41, 42]. The detailed impacts of multilayer reinforcement mechanism that engineers complex social contagion remain largely unknown.

To fill this gap, here we propose a theoretical framework that incorporates multilayer reinforcement into ignorant-spreader-ignorant (SIS) model to study the spreading dynamics on multiplex networks. We find this complex social contagion mechanism not only expands dissemination, but also results in the emergence of bistable region, within which extinction and outbreak states coexist. Within bistable region, the final state depends on the initial density of adopters. We also detailedly discuss the hysteresis loop and the unstable equilibrium manifolds occurred in the bistable region. Through observing phase diagrams, we highlight two conditions for the outbreak of social contagion: (i) to possess large transmissibility; (ii) to possess large initial density of adopters with strong multilayer reinforcement. As the second condition cannot be obtained by simple contagion model and is less intuitive, our results stress the unneglectable role of complex social contagion and are in line with the findings of previous experimental studies that critical masses are required for establishing collective behaviors such as social changes [43]. In addition, similar phenomena are also observed on finite-size heterogeneous multiplex networks. Our findings provide valuable insights toward dynamical evolutions of complex social contagion on multiplex networks, which are of vital importance for understanding collective online behaviors in the era of social media. In particular, the emerging bistable phenomenon on multiplex networks indicates that the information (truth, ideas, advertisements, etc.) can stride across the trap of extinction and become widespread by selecting a large proportion of initial spreaders on multiple platforms, which implies effective strategies for controlling rumors, promoting innovations and marketing. [44, 45, 46].

Consider an undirected multiplex network with two layers, denoted as layer 1 and layer 2. Each layer stands for a social circle, which is an online social platform or an offline relationship network composed of interacting individuals such as friends, families and colleges. Both layers have the same nodes and interlayer edges only connect entities with their replicas, as shown in figure 1(a).

Here we stress that our main purpose is to explore the influence of complex social contagion aroused by multiple social circles, i.e., multilayer reinforcement mechanism. Therefore, we adopt ignorant-spreader-ignorant (SIS) model rather than complex contagion model to characterize the intralayer spreading process, which provides better analytical insights while at expense of being less realistic. In this situation, spreader ($S$) represents individuals who adopt norms (information, cognition, attitudes, behaviors and so on) and are willing to spread, while ignorant ($I$) stands for nodes who do not adopt norms or have no motivation to spread, corresponding to infected and susceptible state in epidemiology, respectively. In each layer $k$ $(k=1,2)$, spreader has a probability $\lambda_{k}$ to spread norms to its ignorant neighbors and becomes ignorant with probability $\mu_{k}$, as shown in figure 1(b).

Beyond intralayer contagion processes, norms also diffuse across layers, i.e., the interactions between layers. First, complying with previous studies, we utilize interlayer contagion processes where ignorant has a probability $p$ to become a spreader once its counterpart is in $S$ state, which depicts the interaction between agents and their replicas, as shown in figure 1(c) [47]. Then, we introduce multilayer reinforcement mechanism to mimic the fact that exposure to multiple social circles is more convincing than multiple exposures to a single social circle. At each time step, if an individual can receive norms from both social circles, i.e. he/she has at least one spreader neighbor in each layer, then with probability $\gamma$ the individual would randomly choose one layer to spread the norms (figure 1(d)). The parameter $\gamma$ reflects the strength of multilayer reinforcement.

In summary, our model is composed of three processes: intralayer contagion, interlayer contagion and multilayer reinforcement, which is as follows:

1. 1.

   Intralayer contagion. At each time step, in layer $k$, $S$ makes its neighbors in $I$ state become $S$ with probability $\lambda_{k}$. Meanwhile, $S$ turns into $I$ with probability $\mu_{k}$.

2. 2.

   Interlayer contagion. At each time step, $I$ becomes $S$ with probability $p$ if its replica is a spreader.

3. 3.

   Multilayer reinforcement. At each time step, a node has a probability $\gamma$ to randomly choose a layer and become a spreader if there exists at least one spreader in its neighborhood of each layer.

In this section, we explore dynamical equations of our model on homogeneous multiplex systems, composed of two Erdös-Rényi (ER) networks with average degree $\langle k_{1}\rangle$ and $\langle k_{2}\rangle$. The major challenge is the dynamical correlation between two layers caused by interlayer contagion and multilayer reinforcement. Specifically, the counterpart of a spreader is more likely to be a spreader than the counterpart of an ignorant. Classic Mean-field approximation does not consider the dynamical correlation, which leads to the deviation of theoretical predictions from Monte Carlo simulations (see A). To overcome the difficulty, we conduct theoretical analysis by combining pairwise method and mean-field theory.

Here, we define $\rho_{X,Y}(t)$ as the probability that the individual is $X$ state in layer 1 and $Y$ state in layer 2. Clearly, each individual has four possible states and we have $\rho_{S,S}(t)+\rho_{S,I}(t)+\rho_{I,S}(t)+\rho_{I,I}(t)=1$. To begin, we explore intralayer contagion process. Let $\theta_{k}(t)$ represent the probability that a node in layer $k$ is not convinced by its neighbors of the same layer. Utilizing the mean-field theory, $\theta_{k}(t)$ can be approximated by

Then, we examine the influence of multilayer reinforcement. Denote $\delta_{k}(t)$ as the probability that the multilayer reinforcement does not make ignorant turn into a spreader in layer $k$. We have $\delta_{1}(t)=\delta_{2}(t)=\delta(t)$, which can be approximated by

where $(1-(1-\rho_{S,S}(t)-\rho_{S,I}(t))^{\langle k_{1}\rangle})$ and $(1-(1-\rho_{S,S}(t)-\rho_{I,S}(t))^{\langle k_{2}\rangle})$ account for the probabilities that there exists at least one spreader neighbor in layer 1 and in layer 2, respectively.

Furthermore, let $g^{I}_{k}(t)$ and $g_{k}^{S}(t)$ represent the probabilities that ignorant in layer $k$ is not convinced if its counterpart is in ignorant and spreader state, respectively. The probabilities describe the integrated influence of intralayer contagion, interlayer contagion and multilayer reinforcement. Because the three processes are approximately independent in our model, the probabilities read

Finally, the temporal evolutionary equations of $\rho_{S,S}(t)$ can be written as

where the first term represents the outflow from the $S-S$ class and the last three terms stand for transition from the other states to $S-S$ state. Similarly, we can derive the dynamical evolutions of $\rho_{S,I}(t)$, $\rho_{I,S}(t)$ and $\rho_{I,I}(t)$. For the sake of theoretical analysis and readability, $\rho_{I,I}(t)$ is automatically substituted for $(1-\rho_{S,S}(t)-\rho_{S,I}(t)-\rho_{I,S}(t))$. Thus, the evolutionary equations of our model read as follows:

Equation (5) can be simply written as

where $\boldsymbol{\rho}(t)=(\rho_{S,S}(t),\rho_{S,I}(t),\rho_{I,S}(t))$. This indicates that the dynamical system is 3-dimension autonomous. Thus, the stability of the fixed points directly determines evolutionary results of the system. We define $\boldsymbol{\rho}_{f}$ as the fixed point of equation (5), which satisfies

In particular, $\boldsymbol{\rho}_{f}$ represents the final state of the system if and only if the fixed point is stable, i.e.,

where $\Lambda_{max}(J)$ is the largest eigenvalue of matrix $J$.

We notice that there is at least one fixed point ($\boldsymbol{\rho}_{f}=\boldsymbol{0}$), which represents that all individuals are in $I-I$ state. When the initial density of spreaders is small, norms go extinct if and only if $\boldsymbol{\rho}_{f}=\boldsymbol{0}$ is stable, i.e.,

Complex social contagion describes complicated behavioral evolutions that cannot be characterized by simple contact-contagion models, such as public opinion formation on controversial events, the spread of conspiracies and the establishment of new social norms [54, 55]. These risky or polarized collective behaviors are ubiquitous on various social networks and are largely determined by the high-order interactions regarding to the enhancement effects of multiple exposures, i.e., the social reinforcement [56]. However, it remains unclear how multilayer reinforcement which mimics complex contagion process among multiple social circles influences the final diffusion results.

In this paper, we propose a modified SIS model which incorporates multilayer reinforcement to describe spreading dynamics on multiplex networks. Firstly, we examine how our model behaves on homogeneous multiplex networks. In particular, we stress our efforts on the detailed impact of multilayer reinforcement. A theoretical framework combining pairwise method and mean-field theory is proposed and is verified by large-scale simulations. Interestingly, we find that the multilayer reinforcement induces the emergence of bistability, where extinction and outbreak states coexist. Furthermore, we show that the final state of bistable region is determined by the initial density of adopters whose threshold decreases as intralayer transmissibility, interlayer transmissibility or multilayer reinforcement increases. We also illustrate the hysteresis phenomena where two evolutionary routes occur, known as an ascending path and a descending path, accounting for the phenomenon that rumors are hard to dispel among online social platforms. The detailed conditions of phase transition for bistability are further derived analytically and are shown in phase diagrams. Within the bistable region, we show the existence of unstable manifolds. Results highlight that the chance for the outbreak of social contagion is either to possess large contagion transmissibility or to own large initial density of adopters with strong multilayer reinforcement. Our results are also valid on finite-size heterogeneous multiplex networks.

Our basic yet powerful multilayer reinforcement mechanism reveals the dramatic promoting impacts of complex social contagion on multiplex networks, which indicates the possibility of facilitating spread outbreaks via controlling initial spreaders. These insights are in line with the empirical studies that only large initial density of adopters can successfully initiate social changes [43]. While current results are performed on uncorrelated multiplex networks, further study may extend the framework to partially overlapped multiple networks where edges in different layers are correlated.