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The authors have declared that no competing interests exist.

Conceived and designed the experiments: MTG NM GP MD. Performed the experiments: MTG NM. Analyzed the data: MTG NM GP. Wrote the paper: MTG NM GP MD.

Statistical physicists have become interested in models of collective social behavior such as opinion formation, where individuals change their inherently preferred opinion if their friends disagree. Real preferences often depend on regional cultural differences, which we model here as a spatial gradient

Disagreement between neighbors costs energy, in human societies as well as in ferromagnetic spin interactions. Because of this similarity, statistical physicists have recently shown great interest in models of opinion formation (e.g.

Many opinion formation models embedded in two-dimensional space have only one stable solution, namely complete consensus

In this article we tackle the open question: can opinion dynamics, with or without a stochastic element, fundamentally alter percolation properties such as the clusters' fractal dimensions or the cluster size distribution? We show that in many cases we retrieve the scaling laws of independent percolation. Moreover, we also give one example where a slight change of the dynamic rules leads to a radically different scaling behavior.

We focus on models where the nodes are placed on a square lattice with edges linking them to their four nearest neighbors. Each node holds one of two possible opinions: “black” or “white”. Initially, the probability to be black is independent at all sites and given by_{c}

Including a non-zero gradient in the numerical simulations also has advantages for studying percolation properties

In the present work we consider opinion formation according to the following local rules.

Majority vote (MV): the node follows the majority opinion of its four nearest neighbors. If both opinions are equally represented, no opinion change occurs.

Unanimity rule (UR): the node changes its current opinion if and only if all of its nearest neighbors hold the opposite opinion

Independent percolation (IP): the node keeps its current opinion irrespective of the surrounding opinions.

When a node is updated, it follows the local rule with probability

All nodes simultaneously update their opinion at each time step, but other choices such as random sequential updates do not change our findings noticeably. The latter may have the more immediate social interpretation as an ongoing opinion formation with agents re-considering choices with a fixed rate, but simultaneous updates are, surprisingly, slightly more accessible analytically. For a fixed value of _{q}_{q}_{c}

Once the model reaches the steady state, we study the geometric properties of the clusters formed. On the left of

We show typical steady-state opinion distributions for ^{−3} and (a) MV_{1}, (b) UR_{1}, (c) MV_{0.8}. The two opposing opinions are shown as black and white squares. The sites marked by gray squares form the spanning cluster's hull. (d) Illustration how the hull can be parameterized by a left-turning walk

Our numerical and analytical findings are summarized in

Model | Exponents | Universality Class | |

Independent Percolation (IP) | IP (by definition) | ||

Deterministic Majority Vote Model (MV_{1}) |
1 | IP | |

Deterministic Unanimity Rule (UR_{1}) |
1 | IP | |

Stochastic Majority Vote Model (MV_{0.8}) |
0.8 | Edwards-Wilkinson | |

Stochastic Unanimity Rule (UR_{0.8}) |
0.8 | IP |

If _{1} is identical to the non-consensus opinion model of Ref. _{c}_{1} to have reached its steady state. The convergence is quick: a non-periodic node freezes after a mean of only 0.8 time steps. In UR_{1}, oscillatory opinions can occur only if the initial opinions form a perfect checkerboard pattern. Because the gradient pins the left (right) edge to be entirely white (black), a checkerboard pattern is impossible. Hence, every node reaches a stationary opinion, on average after just 0.06 updates at _{c}_{c}

If _{q}_{q}_{1} and UR_{1} on the one hand and MV_{0.8} on the other hand. In the latter case, the spanning cluster appears significantly more compact and the hull, which is centered at _{c}_{0.8} anneals rather than roughens the surface compared to MV_{1} and UR_{1}.

We can quantify this observation by computing the hull's width

As the numerical results in _{0.8} with _{0.8} so that we must look elsewhere for an explanation.

Insets: slope in doubly-logarithmic scales (i.e.

We will briefly summarize why _{q}_{0.8}, as opposed to UR_{q}_{0.8}. Although we have here derived the scaling law only for the MV model, numerical evidence suggests that

The scaling laws for _{0.8} is not in the same universality class as IP. In Ref. _{1} is in yet another class, namely invasion percolation with trapping (IPT). Although _{1} belongs to the IP class after all, thus supporting the arguments of Ref. _{max} of the largest cluster in a lattice whose linear size is _{c}_{max} is expected to satisfy the ansatz

Here _{1}, we obtain a data collapse with the same IP exponents (_{1} is in the same universality class as IPT. Changing the exponent 4/7 on _{0.8} with

For the correct exponents _{1}, _{0.8} cluster, we obtain a data collapse if

The cluster size distribution provides further support for this classification. We count all non-spanning clusters with at least one site in the stripe _{1} the exponent _{1} and IP share the following critical exponents: the hull width and length exponents _{1} is in the IP universality class. As shown in the _{1} and UR_{0.8}.

(a) The rescaled distribution _{1} the data collapse is much better (b) for the IP exponent _{0.8} distribution does not follow the same asymptotic power law as IP.

The situation is different in MV_{0.8} where the cluster size distribution appears to drop more sharply with a cutoff that varies much less with the gradient. We want to assess the lack of scaling quantitatively and distinguish it from a power law with large exponent

We plot the moment ratios of IP, UR_{1}, MV_{1}, UR_{0.8} and MV_{0.8} for _{0.8}, all of these cases are in excellent agreement with the prediction of _{0.8}, by contrast, does not diverge as a power law for _{0.8} has the IP value

The moment ratios _{1}, MV_{1}, and UR_{0.8} scale in the same manner as in IP, namely _{0.8} appear to reach an asymptotic limit for

We have studied in total five opinion dynamics models on a gradient, as summarized in

One model, MV_{0.8}, differs from all of the above. At

MV_{0.8} differs from the other models in two important points. First, its stochastic nature helps anneal boundaries between opposite opinions. The second difference is that the majority rule makes small clusters more prone to invasion by the opposing opinion. The combination of these two features results in what appears to be a first order transition. Nevertheless, the opinion interface displays scaling, found to be in the Edwards-Wilkinson universality class, which differs significantly from independent percolation.

The birth-death model of Ref. _{0.8} transition. It would also be insightful to investigate more complex network topologies that are based on real social interactions rather than a regular square lattice. We emphasize that, in the light of previous work on explosive percolation _{0.8} is an example of a dynamic rule that leads to percolation outside the IP universality class.

From a sociological perspective, our study shows that small variations in the innate propensity towards one or another opinion may turn into a spatial discontinuity in the opinions. Interestingly, the sharpest division occurs when agents do not follow the local majority all the time. Hence, processes that may be perceived as having the effect of making the interface between different opinions more blurred, such as the majority rule with stochasticity involved, have the opposite effect. They anneal that interface and contribute to the collapse of minority clusters, which are sustained in the presence of stricter rules, such as the deterministic unanimity rule.

_{1} and UR_{0.8} are in the IP universality class.

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