Diffusion dynamics of competing information on networks

In the present model, any of the three strategies $\{\mathsf{a},\mathsf{b},\mathsf{ab}\}$ may spread globally, and the shares of each strategy in the stationary state, denoted by $\{\rho^{s}\}$, generally vary depending on the payoff parameters and network structure. This type of spreading process is considered as a multistate dynamical process, for which the approximate master equations (AMEs) method has been used to analytically calculate the dynamical paths and the stationary state [38, 3, 39, 28] (see Appendix B for a description of the AME equations. The Matlab code is based on [40]).

Depending on the inherent attractiveness (i.e., $a$ and $b$), the degree of complementarity $\lambda$ and the mean degree $z$, there are three phases as to which strategy is dominant in equilibrium (Figs. 2a and b, and S1). We observe that the AME solutions (shaded) well predict the corresponding simulation results (lines). It should be noted that the cascade region [1, 2] within which we have $1-\rho^{\mathsf{0}}\gg 0$ is mostly covered by the combined dominant region (Fig. S2), suggesting that a strategy often dominates the others once a global cascade occurs.

While it is natural that the attractiveness parameters $a$ and $b$ explain the differences in popularity between $\mathsf{a}$ and $\mathsf{b}$ (Fig. 2a and b), the following question still remains: what happens when the two memes are equally attractive (i.e., $a=b$) yet mutually exclusive? When $a=b$, we always have $\rho^{\mathsf{a}}=\rho^{\mathsf{b}}$ in the AME solution since there is no intrinsic difference between the two memes (black solid in Fig. 2c).

We find that there are three phases in the AME solutions for the case of $\mathsf{a}=\mathsf{b}$: i) $\rho^{\mathsf{a}},\rho^{\mathsf{b}}>0$ and $\rho^{\mathsf{ab}}=0$, ii) $\rho^{\mathsf{a}},\rho^{\mathsf{b}},\rho^{\mathsf{ab}}>0$, and iii) $\rho^{\mathsf{a}}=\rho^{\mathsf{b}}=0$ and $\rho^{\mathsf{ab}}>0$ (Fig. 2c). It is important to note that while the stationary values of $\rho^{\mathsf{a}}$ and $\rho^{\mathsf{b}}$ are nearly $0.5$ in phase i (i.e., $\lambda<-1$), this does not indicate that each of the strategies $\mathsf{a}$ and $\mathsf{b}$ is adopted by $50\%$ of the population. The average values for simulated $\rho^{\mathsf{a}}$ and $\rho^{\mathsf{b}}$ are nearly $0.5$ because the chance of $\mathsf{a}$ or $\mathsf{b}$ being a dominant strategy (i.e., $\rho^{\mathsf{a}}\approx 1$ or $\rho^{\mathsf{b}}\approx 1$) is close to $0.5$ (Fig. 3a). That is, the popularity of each meme is either 0% or 100% in each simulation (blue circle in the ternary plot in Fig. 3a), although the fractions $\rho^{\mathsf{a}}$ and $\rho^{\mathsf{b}}$ averaged over simulation runs are both $0.5$, which corresponds to the AME value (black cross in Fig. 3a). This indicates that there is no diversity of memes (i.e., the two memes do not coexist) in a stationary state of a spreading process occurring in phase i.

In phase ii (i.e., $-1\lesssim\lambda\lesssim-0.7$), we have a different set of diffused strategies: $\{\mathsf{a},\mathsf{ab}\}$, $\{\mathsf{b},\mathsf{ab}\}$ and $\{\mathsf{ab}\}$ (Fig. 3b). The memes are neither too complementary nor too exclusive, and this is the only phase in which a strategy diversification may be observed. The AME solution indicates that $\rho^{\mathsf{a}}$ and $\rho^{\mathsf{b}}$ are less than $0.5$ (Fig. 2c), but again strategies $\mathsf{a}$ and $\mathsf{b}$ do not coexist in simulation, resulting in a deviation from the theoretical average (Fig. 3b). In phase iii (i.e., $\lambda>-0.7$), the two memes are not strongly mutually exclusive, so that the only strategy adopted in the stationary state is $\mathsf{ab}$ (Fig. 3, right). Since there is only one strategy that prevails in the network, the model is essentially the same as the binary-state cascade model, where the theoretical average is equal to the simulated popularity of $\mathsf{ab}$ in each simulation. These observations suggest that the intrinsic symmetry between the two types of memes leads to a symmetric cascade only in phase iii while symmetry is likely to be broken in the other phases.

To understand the fundamental mechanics behind the observed symmetry breaking, we draw phase diagrams based on a mean-field (MF) approximation using random $z$-regular networks (i.e., the degree distribution $p_{k}=\delta_{kz}$), for which it is assumed that the states of neighbors are independent of each other [7]. In the MF method, the evolution of $\rho^{s}$ for each $s\in S$ is described by the following differential equation [38, 3, 39]:

where ${\bm{\rho}}\equiv(\rho^{\mathsf{0}},\rho^{\mathsf{a}},\rho^{\mathsf{b}},\rho^{\mathsf{ab}})^{\top}$, and $\mathcal{M}_{z}({\bf m},{\bm{\rho}})$ is the multinomial distribution given by

$F_{{\bf m}}(s\to s^{\prime})$ denotes the probability that individuals change their strategy from $s$ to $s^{\prime}$ for a given neighbors’ profile ${\bf m}$: $F_{{\bf m}}(s\to s^{\prime})=1$ if $s^{\prime}=s^{*}({\bf m})$, and $0$ otherwise. The first term in Eq. (16) captures the rate at which a node changes its strategy from $s$ to $s^{\prime}(\neq s)$, and the second term denotes the rate at which a node newly employs strategy $s$. Note that this is a system of four differential equations ($|S|=4$), but it is sufficient to use three of them because there is an obvious constraint $\sum_{s\in S}\rho^{s}=1$.

Fig. 4 presents phase diagrams in the $\rho^{\mathsf{a}}$-$\rho^{\mathsf{b}}$ space for three different values of $\lambda$, representing the phases i–iii defined above. Note that the theoretical equilibrium (indicated by point A) is saddle-path stable in all the three cases, but the diagrams differ in the size of the region in which $\dot{\rho^{\mathsf{ab}}}>0$ (shaded in gray). When the two memes are highly exclusive (Fig. 4a), there is no chance for strategy $\mathsf{ab}$ to gain popularity, so $\dot{\rho^{\mathsf{ab}}}=0$ for any combination of $(\rho^{\mathsf{a}},\rho^{\mathsf{b}})$. In simulations on finite-size networks, the saddle-path equilibrium indicated by the MF/AME method, $(\rho^{\mathsf{a}},\rho^{\mathsf{b}})=(0.5,0.5)$, is not practically reachable; simulated paths of $(\rho^{\mathsf{a}},\rho^{\mathsf{b}})$ converge to $(0,1)$ or $(1,0)$ once they deviate from the stable balanced path: $\rho^{\mathsf{a}}(t)=\rho^{\mathsf{b}}(t)$ for all $t\geq 0$ (red dotted in Fig. 4a, bottom).

In principle, the symmetric MF/AME solution would correspond to the “simulated” equilibrium in the limit of large networks with no structural fluctuations. However, any synthetically generated networks are generally not free from finite-size effects and fluctuations, so it is not guaranteed that $\rho^{\mathsf{a}}(t)=\rho^{\mathsf{b}}(t)$ for all $t\geq 0$ in simulations. Histograms of simulated values of $\rho^{\mathsf{a}}-\rho^{\mathsf{b}}$ at a certain point in time, denoted by $T$, reveal the effect of network size on the likelihood of symmetry breaking (Fig. 5). When $N$ is relatively small, symmetry breaking occurs in the early stage of spreading process, so we often have $\rho^{\mathsf{a}}(T)=1$ or $\rho^{\mathsf{b}}(T)=1$ at $T=100$ (Fig. 5a and b). In contrast, when $N=10^{5}$ or larger, it is much less likely that either of the strategies is adopted by most of the population at $T=100$, indicating that the intrinsic symmetry of the memes is more likely to be maintained for larger networks (Fig. 5c and d).

In phase ii, there arises an area in which $\dot{\rho^{\mathsf{ab}}}>0$ (Fig. 4b). This suggests that the feasible region of $(\rho^{\mathsf{a}},\rho^{\mathsf{b}})$ (i.e., $\{(\rho^{\mathsf{a}},\rho^{\mathsf{b}}):\rho^{\mathsf{a}}\geq 0,\rho^{\mathsf{b}}\geq 0,\rho^{\mathsf{a}}+\rho^{\mathsf{b}}+\rho^{\mathsf{ab}}\leq 1\}$) gradually shrinks as $\rho^{\mathsf{ab}}$ increases as long as the current state of $(\rho^{\mathsf{a}},\rho^{\mathsf{b}})$ is in the gray-shaded area. In this phase, symmetry breaking may occur, but not always (Fig. 4b, bottom). In the latter case, both $\rho^{\mathsf{a}}$ and $\rho^{\mathsf{b}}$ initially increase and then begin to decrease as the feasible region shrinks in accordance with a rise in $\rho^{\mathsf{ab}}$. In phase iii, we always have $\dot{\rho^{\mathsf{ab}}}>0$ (Fig. 4c). This indicates that any path of $(\rho^{\mathsf{a}},\rho^{\mathsf{b}})$ will move toward the origin at some point in time as $\rho^{\mathsf{ab}}$ increases. Therefore, $\mathsf{ab}$ will be the only diffused strategy in equilibrium. The time to reach convergence in simulated cascades follows a heavy-tailed distribution when symmetry breaking always occurs (i.e., in phase i), while the spreading process promptly reaches an equilibrium in the other phases (Fig. S3).

In the model shown above, individuals’ choices are fully reversible where the past strategies do not affect the current strategic choice (Eq. 5). This is a reason why either of the social memes could dominate the other and there is no possibility of polarization: $\rho^{\mathsf{a}}\gg 0,\;\rho^{\mathsf{b}}\gg 0$ and $\rho^{\mathsf{ab}}=0$ [15, 16]. Such a reversible decision making, however, would be practically infeasible when switching costs are high (e.g., switching from Mac to Windows). To investigate irreversible dynamics, we introduce a parameter $q\in[0,1]$ representing the degree of irreversibility; $q=0$ and $1$ respectively correspond to the fully reversible and irreversible cases. When $q=1$, only the following five switching patterns are allowed: $\mathsf{0}\to\mathsf{a}$, $\mathsf{0}\to\mathsf{b}$, $\mathsf{0}\to\mathsf{ab}$, $\mathsf{a}\to\mathsf{ab}$, and $\mathsf{b}\to\mathsf{ab}$. Thus, once a meme is accepted, there is no possibility that the meme will be abandoned (i.e., $\mathsf{a}\nrightarrow\mathsf{0}$, $\mathsf{a}\nrightarrow\mathsf{b}$, $\mathsf{ab}\nrightarrow\mathsf{b}$, etc). The irreversibility parameter $q\in[0,1]$ denotes the rate at which a strategy will not be reverted. The response function with irreversibility constraints, denoted by $\widetilde{F}_{\bf m}(s\to s^{\prime})$, is given in Table 2:

For nodes with $s=\mathsf{0}$, there is no constraint in updating their strategy. For nodes with $s=\mathsf{a}$ (resp. $s=\mathsf{b}$), shifting to $s^{\prime}=\mathsf{b}$ (resp. $s^{\prime}=\mathsf{a}$) or $s^{\prime}=\mathsf{0}$ is restricted, for which the transition probability is multiplied by a factor of $(1-q)$. For nodes with $s=\mathsf{ab}$, any state change is restricted. Note that the unconstrained response function is recovered if $q=0$.

Let $G_{s}$ be a function that represents the right-hand side of the MF equation (16) (i.e., $\dot{\rho}^{s}=G_{s}({\bm{\rho}})$). A stable (resp. unstable) equilibrium is defined as an equilibrium at which $\dot{\rho}^{s}=0$ for all $s\in S$ and the maximum eigenvalue of the Jacobian of vector $\mathbf{G}=(G_{\mathsf{0}},G_{\mathsf{a}},G_{\mathsf{b}},G_{\mathsf{ab}})^{\top}$ is non-positive (resp. positive). We find that introducing a partial irreversibility (i.e., $q<1$) does not qualitatively change the dynamical process; there are still two symmetric unstable equilibria, $(\rho^{\mathsf{a}},\rho^{\mathsf{b}})=(0,0)$ and $(0.5,0.5)$ (red circles in Fig. 6a), and two asymmetric stable equilibria, $(0,1)$ and $(1,0)$ (blue circles in Fig. 6a). Symmetry breaking always occurs in phase i as in the fully reversible model (Fig. 6c). Note, however, that the greater the degree of irreversibility $q$, the longer the time to convergence for $q<1$ (Fig. S4).

In phase i, where $\rho^{\mathsf{ab}}(t)=0$ for all $t$, the saddle equilibrium disappears when the strategies are fully irreversible (i.e., $q=1$). Instead, there arises a continuum of stable equilibria $({\rho}^{\mathsf{a}},{\rho}^{\mathsf{b}})$ such that ${\rho}^{\mathsf{a}}+{\rho}^{\mathsf{b}}=1$ (Fig. 6b). This indicates that equilibrium is indeterminate in irreversible dynamics even in the limit of large networks. Indeed, the simulated equilibria are continuously distributed, at each of which polarization occurs (i.e., $\rho^{\mathsf{a}}\gg 0,\;\rho^{\mathsf{b}}\gg 0$ and $\rho^{\mathsf{ab}}=0$) (Fig. 6d). This is intuitive given that the state-transition process is no longer ergodic when $q=1$. Due to the irreversibility, the time to convergence is minimized at $q=1$ (Fig. S4). We also find that in phases ii and iii, the bilingual strategy $\mathsf{ab}$ promptly becomes the dominant strategy when $q=1$ (Figs. S5 and S6).