Information diversity and anomalous scaling in asymmetric social contagion process on low-dimensional static networks

To understand how competition affects the diversity of information, we study the social contagion model introduced by Halvorsen-Pedersen-Sneppen (HPS) [G. S. Halvorsen, B. N. Pedersen, and K. Sneppen, Phys. Rev. E 103, 022303 (2021)] on one-dimensional (1D) and two-dimensional (2D) static networks. By mapping the information value to the height of the interface, we find that the width $W(N,t)$ does not satisfy the well-known Family-Vicsek finite-size scaling ansatz. From the numerical simulations, we find that the dynamic exponent $z$ should be modified for the HPS model. For 1D static networks, the numerical results show that the information landscape is always rough with an anomalously large growth exponent, $\beta$. Based on the analytic derivation of $W(N,t)$, we show that the constant small number of influencers created for unit time and the recruitment of new followers are two processes responsible for the anomalous values of $\beta$ and $z$. Furthermore, we also find that the information landscape on 2D static networks undergoes a roughening transition, and the metastable state emerges only in the vicinity of the transition threshold.

Figure 1

Schematic illustration of the HPS model on a two-dimensional static network. The upper layers represent the influencer-follower (or global asymmetric) networks and the lower layer denotes the two-dimensional static social connection (or local reciprocal interactions). Solid circles are the influencers and the open circles are the followers. Arrows in the upper layers stand for influencer-follower relations. We assume $t_i<t_{i+1}$ ($i=2,3,...$). (a) There are initially two different information clusters. Each cluster is enclosed by a closed path. The node $i$ ($j$) is an influencer with ${\tau }_i=t_1$ (${\tau }_j=t_2$). The dashed lines between the layers denote that the two nodes connected by the dashed line are the same. (b) At $t=t_3$, the node $i$ (marked by the dashed circle) is randomly selected and creates new information (${\tau }_i=t_3$) with a probability $p_i$. (c) The nearest neighbor of $i$ recruits a follower (node $m$) among its nearest neighbors in the static network. Since $t_3>t_2$, $m$ changes its information to ${\tau }_m=t_3$, disconnects the directed link to $j$, and makes a new link to $i$. The thick arrow in the lower layer denotes the direction of the information flow. (d) At $t=t_4$, the node $n$ is randomly selected and becomes a new influencer by creating new information of ${\tau }_n=t_4$ with the probability $p_n$. Thus, $n$ disconnects the directed link to $j$. (e) The new influencer $n$ recruits its nearest neighbor $q$ as its follower. Thus, the information flows form node $n$ to $q$. (f) At $t=t_5$, the node $s$ is randomly selected without the creation of new information and selects $j$ to recruit as a follower of $i$. Let us assume that $t_2<t_3$. Since the node $j$ is an influencer, $j$ only adopts the information of $s$ (${\tau }_j\to t_3$) without the change of the influencer-follower network.

Figure 2

(a) Plot of $W{(N,t)}^{\prime }\mathrm{s}$ for $N=2048\sim 16384$ on 1D static networks with $(N{p}_0,\eta )=(1.0,0.5)$. $t_{\times }$ indicates the crossover time for $N=16384$. The solid line represents $W\left(t\right)\sim t^{1.94}$. The dashed line denotes Eq. (13). Inset: Plot of $W_{\mathrm{sat}}\left(N\right)$ against $N$. The solid line stands for $W_{\mathrm{sat}}\left(N\right)\sim N^{0.5}$. (b) Scaling plot of $W(N,t)$ using Eq. (3). Inset: Plot of $t_{\times }\left(N\right)$ against $N$. The solid line represents the relation $t_{\times }\sim t^{0.5}$.

Figure 3

Scaling collapse of $W(N,t)N^{1/2}$ for (a) $(N{p}_0,\eta )=(0.1,0.0)$, (b) $(N{p}_0,\eta )=(1.0,0.0)$, (c) $(N{p}_0,\eta )=(0.1,1.0)$, and (d) $(N{p}_0,\eta )=(1.0,1.0)$. The solid lines represent the relation $W(N,t)\sim t^{\beta }$ with (a) $\beta \simeq 1.95$, (b) $\beta \simeq 1.85$, (c) $\beta \simeq 1.99$, and (d) $\beta \simeq 1.95$. Insets: Plot of $W_{\mathrm{sat}}\left(N\right)N^{1/2}$ against $N$. The solid lines denote the relation $W_{\mathrm{sat}}\left(N\right)\sim N^{{\alpha }^{\prime }}$ with (a) ${\alpha }^{\prime }\simeq 1.02$, (b) ${\alpha }^{\prime }\simeq 0.99$, (c) ${\alpha }^{\prime }\simeq 1.02$, and (d) ${\alpha }^{\prime }\simeq 0.99$.

Figure 4

Schematic diagram for the early-time evolution of clusters on the 1D static network at $t={\tau }_2$ (${\tau }_2>{\tau }_1$). (a) The (red) rectangle denotes the cluster of nodes following the same influencer whose information value is ${\tau }_1$. The solid arrows on the top of nodes denote the follower network. The (white) open circles are nodes of $\tau =0$ whose state is $\varphi =F$, but they do not follow any influencer. (b) A node (green circle) creates new information at $t={\tau }_2$ with a probability $p_0$ and becomes an influencer. The (blue) dashed circles are the nodes at the boundary of each cluster. Each boundary node randomly selects one of its neighbors on a static network [represented by (black) dotted arrows]. (c) The node at the right boundary of the cluster of ${\tau }_1$ recruits a new follower and $s_{{\tau }_1}=5$, while the node at the left boundary cannot recruit a new node because it selects the node in the same cluster. At the same time, the new influencer recruits one of its neighbors and $s_{{\tau }_2}\left({\tau }_2\right)=2$.

Figure 5

(a) Plot of $s_{\tau }{\left(t\right)}^{\prime }\mathrm{s}$ against $\tau$ for $N{p}_0=1.0$, $\eta =1.0$, and $N=16384$ when ${\tau }_{\max }$ reaches ${\tau }_{\max }=20$ and ${\tau }_{\max }=30$. The solid lines represent the relation $s_{\tau }\left(t\right)\sim -\tau$. (b) Plot of $K\left(t\right)$ against $t$. The solid line denotes the relation $K\left(t\right)\sim t^{\delta }$ with $\delta =1.8$. The dashed line represents Eq. (15).

Figure 6

Plot of $W{(N,t)}^{\prime }\mathrm{s}$ on 2D static networks for (a) $(N{p}_0,\eta )=(0.1,0.0)$ and (b) $(N{p}_0,\eta )=(1.0,1.0)$. The solid lines represent the relations (a) $W(N,t)\sim t^{2.5}$ and (b) $W(N,t)\sim t^{2.6}$. Insets: Plots of $W_{\mathrm{sat}}{\left(N\right)}^{\prime }\mathrm{s}$. The solid lines denote (a) $W_{\mathrm{sat}}\left(N\right)\sim N^{0.32}$ and (b) $W_{\mathrm{sat}}\left(N\right)\sim N^{-0.5}$.

Figure 7

(a) Phase diagram of the HPS model. Dark green (gray) represents the rough phase and light green (gray) corresponds to the flat phase. Inset: Plot of ${\eta }^{*}$ against $N{p}_0$. The solid line represents the relation ${\eta }^{*}\sim {\left(N{p}_0\right)}^{-1/2}$. Black circles in panels (b) and (c) denote the index of the influencer node belonging to the largest cluster at $t$ for (b) $(N{p}_0,\eta )=(1.0,0.15)$ and (c) $(N{p}_0,\eta )=(1.0,1.0)$. Blue (gray) lines represent the evolution of $s_{\max }/N$ for $N=400\times 400$.

Figure 8

Scaling collapse of $W(N,t)N^{1/2}$ with $\beta =2.5$, ${\alpha }^{\prime }=0.82$, and $z^{\prime }=0.32$ for (a) $(N{p}_0,\eta )=(0.1,0.0)$ and (b) $(N{p}_0,\eta )=(0.5,0.1)$.

Figure 9

(a) Plot of $N_{\mathrm{cls}}\left(N\right)$ against $N$ for $(N{p}_0,\eta )=(1.0,1.0)$ (open squares) and $(N{p}_0,\eta )=(1.0,0.1)$ (open circles) in the steady state. The dashed line represents $N_{\mathrm{cls}}\sim \mathrm{const}.$ and the solid line stands for $N_{\mathrm{cls}}\sim N^{0.4}$. Inset: Plot of $s_{\max }$ against $N$ for $(N{p}_0,\eta )=(1.0,0.1)$. The solid line denotes $s_{\max }\sim N^{0.72}$. (b) Plot of $P\left(s\right)$ for $(N{p}_0,\eta )=(1.0,0.1)$. The solid line represents $P\left(s\right)\sim s^{0.69}$. Inset: Plot of $\tau \left(s\right)$ against $s$ for $N=400\times 400$.