Dynamic Opinion Model and Invasion Percolation

We propose a “nonconsensus” opinion model that allows for stable coexistence of two opinions by forming clusters of agents holding the same opinion. We study this nonconsensus model on lattices, several model complex networks, and a real-life social network. We find that the model displays a phase transition behavior characterized by a large spanning cluster of nodes holding the same opinion appearing when the concentration of nodes holding the same opinion (even minority) is above a certain threshold. Because of the clustering (community support) of agents holding the same opinion, these clusters cannot be invaded by the other opinion (similar to incompressible fluids). Our extensive simulations show that the nonconsensus opinion model appears to belong to the same universality class as invasion percolation.

Figure 1

Dynamics of the NCO model showing the approach to a stable state on a network with $N=9$ nodes. For simplicity, we assume $p=0$. (a) At $t=0$, five nodes are randomly assigned to be ${\sigma }_{+}$ (filled circle). The remaining four nodes are assigned ${\sigma }_{-}$ (open circle). In the set comprising of node $A$ and its 4 neighboring nodes (dashed box), node $A$ is in a local minority opinion (2 ${\sigma }_{+}$ nodes and 3 ${\sigma }_{-}$ nodes), while the remaining nodes are not. At the end of simulation step $t=0$, node $A$ is converted into ${\sigma }_{-}$. (b) At $t=1$, in the set of nodes comprising node $B$ and its 6 neighboring nodes (dotted box), node $B$ becomes in a local minority opinion (3 ${\sigma }_{+}$ nodes and 4 ${\sigma }_{-}$ nodes), while the remaining nodes are not. Node $B$ is converted into ${\sigma }_{-}$ at the end of simulation step $t=1$. (c) At $t=2$, the nine-nodes system reaches a stable state.

Figure 2

Plot of the normalized size of the largest cluster $s_1$ (dotted line), the second largest cluster $s_2$ (full line), and the fraction of ${\sigma }_{-}$ nodes $F$ (dashed line) in the stable state as a function of $f$ for (a) a SF network with $\lambda =2.5$ and $N={10}^5$, (b) an ER network with $⟨k⟩=4$ and $N={10}^5$, (c) a HEP network with $\lambda \approx 2.9$, and (d) a HX lattice of size $1000\times 1000$ [20]. Each curve represents an average over 100 realizations. The sharp increase of $s_1$ and the peak of $s_2$ at $f^{*}$ indicate a second-order phase transition. The insets in (a) and (d) show $g(S_1,N)$ [Eq. (1)] as a function of $N$ at the critical threshold $f^{*}$ (●), $f^{*}+\Delta$ (◇), and $f^{*}-\Delta$ (□), where $\Delta =0.01$ for SF and $\Delta =0.005$ for HX. At $f^{*}$, $g(S_1,N)$ approaches a constant, indicating the size of the largest cluster $S_1$ proportional to $N^{\zeta }$ at $f^{*}$ [Eq. (1)], which is another characteristic of a second-order phase transition.

Figure 3

(a) The cumulative distribution function of cluster sizes $P(S^{\prime }>S)$ at $f^{*}$ for a NCO model on a SQ lattice with $N=9\times {10}^6$ ($3000\times 3000$) and a SF network with $\lambda =2.5$ and $N={10}^5$. $P(S^{\prime }>S)\sim S^{1-\tau }$ for both SQ and SF, where $\tau \approx 1.89$ (SQ) and $\tau \approx 2.5$ (SF). $P(S^{\prime }>S)$ of the sizes of pores of TIP on SQ with $N=9\times {10}^6$ is also shown, which also takes the form $P(S^{\prime }>S)\sim S^{1-\tau }$ with $\tau \approx 1.90$. Averages over 100 realizations are shown for all curves. (b) For the NCO model, at criticality, $R_g/S^{1/d_f}$ as functions of $S$ for SQ and TR with $N=9\times {10}^6$. For the trial value $d_f=1.84$, $R_g/S^{1/d_f}$ approaches a constant. For $d_f=1.896$ (regular percolation fractal dimension), $R_g/S^{1/d_f}$ is an increasing function of $S$. (c) For the NCO model, at criticality, $\ell /S^{1/d_{\ell }}$ as a function of $S$ for SQ and TR with $N=9\times {10}^6$. For the trial value $d_{\ell }=1.54$, $\ell /S^{1/d_{\ell }}$ approaches a constant. For $d_{\ell }=1.678$ (regular percolation fractal dimension), $\ell /S^{1/d_{\ell }}$ is an increasing function of $S$. In (b) and (c), each curve is averaged over 1000 realizations.

Figure 4

(a) The average degree $⟨k(f)⟩$ of ${\sigma }_{-}$ nodes in the stable state as a function of $f$ for a SF network with $\lambda =2.5$ and $N={10}^5$ and the HEP network. It is seen that, for $f<0.5$, $⟨k(f)⟩$ is significantly smaller than that for $f>0.5$, since the high degree nodes join the majority opinion [18]. (b) The cumulative degree distribution $P_f(k^{\prime }>k)$ of ${\sigma }_{-}$ nodes in the stable state for a SF network with $\lambda =2.5$ and $N={10}^5$. $P_f(k^{\prime }>k)$ for $f=0.450$ and 0.5 are plotted. Notice that $f^{*}\approx 0.450$. The absence of high degree ${\sigma }_{-}$ nodes in a stable state for $f<0.5$ is again confirmed.