Suppose that in a community, people have different opinions on a topic of common interest. Through social interactions, individuals learn about the opinions of others, and as a result may change their own opinion. The goal is to understand and possibly predict how opinions spread in the community. There are numerous mathematical models for such a situation; a very simple deterministic one is the following: the community is modeled as a graph, with edges corresponding to possible interactions between individuals. Opinions spread in rounds, where in each round, each individual adopts the most frequent opinion in its neighborhood. If in case of a tie an individual stays with him/her opinion, it is named majority model, but if in case of a tie s/he always adopts a specific opinion, the process is called biased majority model. These two natural updating rules have various applications, for example in data redundancy , distributed computing , modeling biological interactions , resource allocation for ensuring mutual exclusion , distributed fault-local mending , and modeling diffusion of two competing technologies over a social network .
Scientists from different fields, from Spitzer  to a recent paper by Mitsche , have attempted to study the behavior of these natural updating rules, especially on the two-dimensional torus where the (biased) majority rule can be interpreted as a cellular automaton. Some theoretical and experimental results concerning its behavior have been obtained, which will be discuss in detail in Section 1.2; of particular interest is the consensus time, the time after which the process reaches a periodic sequence of states (which must eventually happen, as the process is deterministic and has finite state space). Also, one would like to understand how these “final” states looks like, depending on the initial distribution of opinions. For example, what are conditions under which some opinion is eventually taken up by all the individuals?
In this paper, we first present some results in general graphs , regarding consensus time, eternal sets (sets of nodes that guarantee the survival of an opinion that they have in common), and robust sets (sets of nodes that will never change an opinion that they have in common). Building on them, our main contribution is for the case where is an torus, with -neighborhoods (Neumann neighborhood), or with -neighborhoods (Moore neighborhood). As mentioned, we study the case of two opinions (modeled by vertex colors blue and red), with an initial coloring that assigns blue to every vertex independently with a probability , and red otherwise (). It is proven that both majority cellular automata and biased majority cellular automata exhibit a threshold behavior with two phase transitions in Neumann neighborhood. In the torus with Neumann neighborhood with high probability, majority model results in a monochromatic generation with red, stable coexistence of both colors, and monochromatic generation with blue for , , and , respectively in number of steps. Furthermore, it is proved that for Neumann neighborhood and the biased majority model, , , and result in final red monochromatic configuration, stable coexistence of both colors, and final complete occupancy by blue, respectively in steps with high probability. We also prove that majority cellular automata show a threshold behavior in the case of Moore neighborhood. Figure 1 summarizes the mentioned threshold behaviors.222 means for two functions and .
These results not only are important in their own right, but also answer two important questions. Firstly, prior empirical research [14, 16, 4, 23] demonstrate that majority cellular automata should show a threshold behavior i.e. if the concentrations of vertices holding the same color is above a certain threshold, the color will survive in all upcoming configurations, otherwise the color might disappear in a few steps. Our results concerning majority cellular automata prove the aforementioned empirical observation and determine the threshold values and the consensus time of the process precisely. Secondly, Schonmann  proved that in biased majority cellular automata with Neumann neighborhood outputs final complete occupancy by blue with high probability. We show that this bound is tight; actually, we prove what exactly happens in . Furthermore, we will present the tight bound of ( is the number of vertices) on the consensus time of both automata which has been a point of interest in the prior works.
Intuitively, if the concentration of blue color increases, the chance of its survival will increase and the chance of survival for red color decreases. In other words, by increasing the initial density of a color from 0 to 1, it will go through different phases: (i) very low concentration results in the disappearance of the color (ii) sufficient initial density for both colors outputs the survival of both of them (iii) very high initial concentration of a color results in the final occupancy by the color. Therefore, intuitively and as also prior experimental results have shown, it is not so surprising if one proves that majority and biased majority automata show a threshold behavior with two phase transitions, but the most surprising part is the substantial change in the value of thresholds by switching form the majority model to the biased majority model. In majority cellular automata with Neumann neighborhood, should be very close to 1 to have a high chance of final complete occupancy by blue, but by just changing the tie breaking rule in favor of blue, the process ends up in a blue monochromatic configuration with high probability even for initial concentration very close to 0. Hence, it not only shows the significant impact of the tie breaking rule, but also demonstrates how small alternations in local behavior can result in considerable changes in global behaviors.
To prove that majority cellular automaton has a threshold behavior, we show there exists a blue robust set (a robust set whose all vertices are blue) in starting configuration with high probability for ( in Moore neighborhood) which result in the survival of blue color, but proving (similarly
in Moore) results in a red monochromatic configuration is more difficult. We show in this case with high probability in the initial generation, we can classify the blue cells such that all blue cells in a group behave independently of all other blue cells and the number of blue cells in none of these groups is sufficient for survival. As we will discuss in more detail, this proof technique requires more technical arguments in the case of Moore neighborhood because of switching from 4-neighborhood model to 8-neighborhood model.
In the case of biased majority cellular automata as we mentioned, Schonmann  proved that results in final complete occupancy by blue almost surely. To prove, for in Neumann neighborhood both colors survive, we show there exists at least a blue eternal set (an eternal set whose all vertices are blue) in the initial configuration with high probability, but for proving that red color will never vanish, we need a more complicated argument. For the case of , we intuitively have to overcome the same difficulty that we had in the case of majority model; we have to accurately analyze the behavior of the model in several upcoming configurations to show that blue color in this setting will disappear finally.
We also discuss that majority and biased majority cellular automata reach a configuration of period one or two in steps and these bounds are tight i.e. there are some cases which the process needs steps to stabilize in a configuration of period one or two. We highly depend on the strong results by Poljak an Turzik  concerning our results on consensus time and periodicity.
The layout of the paper is as follows. In the rest of this section, first we introduce majority and biased majority models formally. Then, in Section 1.2 we briefly discuss relevant prior research works. In Section 1.3, the majority model is discussed on cycles (one-dimensional cellular automata) as a simple example before going through two-dimensional cellular automata. In Section 2, we discuss the consensus time and periodicity of the two models on an arbitrary graph . Then, in Sections 3.2 and 3.3, we focus on the threshold behavior of majority cellular automata and biased majority cellular automata, respectively.
1.1 Notation, Preliminaries, and (Biased) Majority Model
Let be a graph that we keep fixed throughout. For a vertex , is the neighborhood of . We also define . Furthermore, for a set , and (similarly and ).
A generation is a function ( and represent blue and red, respectively). If is a constant function, is called a monochromatic generation otherwise it is called a bichromatic generation. is a -community for color and in generation if . For a generation , vertex and color ,
is the set of neighbors of of color in generation . We also define
For a set , we define and similarly .
In addition to for a vertex and , sometimes we also write for a set which means , .
Given an initial generation such that , and independently of all other vertices, and . Let and , equal to the color that occurs most frequently in ’s neighborhood in , and in the case of a tie, conserves its current color. More formally:
The above model is called Majority Model. In the same setting by just changing the tie breaking rule, we have Biased Majority Model. In biased majority model in case of a tie, a vertex always adopts blue color. More formally:
In the present paper, we discuss the behavior of these two deterministic processes with a random initial coloring.
Since both models are deterministic and the number of possible generations for a graph is finite (), they always reach a cycle of generations after finite number of steps. For a graph and (biased) majority model, the number of steps which the process needs to stabilize (reach a cycle of generations) is called Consensus Time. We say a process gets b-monochromatic (similarly r-monochromatic) if finally it reaches a blue (red) monochromatic generation (all vertices blue (red)), otherwise we say it gets bichromatic. Furthermore, for a graph and (biased) majority model, b-monochromatic (r-monochromatic) denotes the probability that the graph gets blue (red) monochromatic, and bichromatic=b-monochromatic-r-monochromatic for a random initial coloring in (biased) majority model.
For a graph , we say an event happens with high probability (or almost surely) if its probability is at least as a function of .333Actually, as we will discuss it is mostly .
1.2 Prior Works
As mentioned, majority-voting rule has been studied in different literatures because of its importance and applications. Therefore, based on different motivations and from a wide spectrum of approaches, various definitions of majority rule have been presented, but in general we may classify them into the three following categories.
The first class is the -monotone model in which, at each step a vertex becomes blue if at least of its neighbors are blue, and once blue no cell ever becomes red (in the literature this is also known as bootstrap percolation). For example in , the authors discussed the case of on hypercubes (such that is the degree of vertex ). Flocchini et al.  also studied the minimum number of blue vertices (in the initial generation) which can finally result in a completely blue generation and the necessary time for this transition to happen, especially on planar graphs, rings, and butterflies. Moreover, Mitsche and et al.  considered ; they proved for any integer , there exists a family of regular graphs such that with high probability all vertices become blue at the end. Recently, Koch and Lengler  mathematically analyzed the role of geometry on bootstrap percolation for geometric scale-free networks.
The second one is the -threshold model in which, at each step, a vertex becomes blue if at least of its neighbors are blue, otherwise it becomes red. For instance, Schonmann  considered the state of (tie is in the favor of blue), and he showed that for any initial density of blue vertices in a torus444For a formal definition of torus and grid see Definitions 3.1 and 3.2, the probability of final complete occupancy by blue converges to 1 as the torus grows. Fazli et al.  also discussed the same model while it seems they were not aware that this model was presented (probably for the first time) in . They presented some thresholds regarding the minimum-cardinality of an initial set of blue nodes which would eventually converge to the steady state where all nodes are blue. In addition, Moore  surprisingly showed that in -dimensional grid for , this model can simulate boolean circuits of and gates.
Another model is the random -threshold model such that a vertex takes the value that the -threshold model would give with probability and its complement with probability . For instance, Balister et al.  considered the case of on 2-dimensional grids. They showed that if is sufficiently small, then the process spends almost half of its time in each of two generations, all vertices blue or all red.
The (Biased) Majority model is a subcategory of -threshold model, which was introduced by Gray . Majority and biased majority models are often called the(discrete time synchronous) majority-vote model in the literature. Most of prior research regarding (biased) majority model [14, 16, 4, 23] are by physicists, and they mostly do computer simulations (i.e., Monte-Carlo methods). Specifically, their computer simulations show majority model allows stable coexistence of two colors by forming clusters of vertices holding the same color in a 2-dimensional torus. Actually, their experimental results show a phase transition behavior characterized by a large connected component of vertices holding the same color appearing when the concentrations of vertices holding the same color is above a certain threshold. However, there are some rigorous mathematical results; Poljak and Turzik  presented an upper bound on the consensus time of (biased) majority model on a graph . For instance, their results imply that for a 2-dimensional torus , number of steps is sufficient to stabilize. Furthermore, Frischknecht, Keller, and Wattenhofer proved there exist graphs which need steps to stabilize for some initial colorings in majority model. Recently, Gärtner and Zehmakan  studied the behavior of the majority model on the random -regular graph . It is shown that in by starting from the initial density for , the process reaches red monochromatic configuration in steps with high probability, provided that for a suitable constant .
Different versions of majority updating rule have been discussed in different literatures and from various aspects for diverse aims, but as mentioned, there are just few concrete mathematical results regarding the most natural versions (majority and biased majority models). In the present paper, we study some essential and interesting aspects of the behavior of these two models and present some strong results regarding their stability, periodicity, and consensus time. As a main result we prove both majority and biased majority cellular automata exhibit a threshold behavior with two phase transitions. More precisely, we prove for a two-dimensional torus , there are two thresholds such that , , and result in r-monochromatic configuration, stable coexistence of both colors, and b-monochromatic configuration, respectively in number of steps.
1.3 One-dimensional Majority Cellular Automata
Before going through (biased) majority model in 2-dimensional torus, we discuss the case of one-dimensional majority cellular automaton (a cycle in majority model) which might help the reader to have a better primary intuition of the techniques and the results that are presented in the rest of the paper.
In majority model and a cycle , if , with high probability the process reaches a bichromatic configuration at the end, but results in a red monochromatic generation in at most steps.
Proof: Consider a maximum matching which divides into pairs for (for odd, one vertex remains). We say a pair is red (blue) in generation , if both of its vertices are red (blue). It is easy to see that a blue (red) pair stays blue (red) in all next generations.
Let Bernoulli random variablefor be if and only if pair is blue in initial generation . Therefore, . If we assume , then by considering and for :
Therefore, with high probability there exists a blue pair and also a red pair in (with a similar argument).
At each step, a red (blue) path (with more than one vertex) extends its size at least by two unless it is adjacent to a red (or blue) path of size more than one. Actually, these red and blue paths grow constantly until they meet each other. Therefore, after at most steps, majority model reaches a stable generation including red and blue paths.
On the other hand, if , one can prove that with high probability there is no blue pair in , but there is at least a red pair. Since there is no blue pair, this red pair extends its size at least by two at each step. Actually, each red path (with more than one vertex) stays red forever and extends its size at least by two at each step, and in an alternating path (both end points are blue and it contains no two consecutive blue (red) vertices) all internal vertices switch from blue to red (or from red to blue) and two end points get red in each step. Therefore after at most steps, it gets monochromatic by red because red (alternating) paths grow (shrink) constantly and no blue pair is created.
2 (Biased) Majority Model
In this section, we first introduce two basic concepts of robust set and eternal set. Then, by using these concepts, we present sufficient conditions which guarantee the survival of a color in all upcoming generations in (biased) majority model on a graph, depending on the graph structure and the concentration of the color in the initial configuration. Specifically, we will exploit these results in Sections 3.2 and 3.3 to prove the threshold behavior of (biased) majority cellular automata. Furthermore, in Section 2.2 we discuss the number of steps which a graph needs to stabilize, consensus time.
2.1 Eternal and Robust Sets
Recall that, if generation is a constant function, is called a monochromatic generation, and a set is a -community in generation for color if . We are interested in sets of vertices that guarantee the survival of a color forever if they create a -community. More specifically, we are also interested in sets of vertices which will keep a common color forever when they create a community, regardless of the colors of the other vertices.
Let in a graph . is called -eternal for color in (biased) majority model whenever the following holds: if forms a -community in some generation for , then for all generations where , such that .
In a graph , a set is -robust for color in (biased) majority model whenever the following holds: if forms a -community in some generation for , then for all generations for .
It follows that once a -robust set forms a -community, it will remain a -community forever and if a -eternal set creates a -community once, color will survive forever. Therefore, in a graph a -robust set is also a -eternal set, but a -eternal set is not necessarily a -robust set. Furthermore, in majority model a set is a b-robust (b-eternal) set if and only if it is an r-robust (r-eternal) set; however, in the biased majority model an r-robust (eternal) set is a b-robust (eternal) set, but not necessarily the other way around. As a simple example, the reader might check that in a star graph , i.e. a tree with one internal node and leaves, the internal node is the smallest eternal set (both b-eternal and r-eternal) in the majority model. What is the size of the smallest robust set in this case?
A blue (red) robust set in a generation is a b-robust (r-robust) set which is a b-community (r-community) in .
Similarly, A blue (red) eternal set in a generation is a b-eternal (r-eternal) set which is a b-community (r-community) in .
Now, we discuss two theorems which present sufficient conditions, on the structure of a graph and the initial generation, which guarantee the survival of a color forever in (biased) majority model (without loss of generality we consider blue as color ).
Let for a graph , for be b-eternal sets; then, we define and such that . Theorem 2.4 says if there are disjoint b-eternal sets in a graph , then with high probability in (biased) majority model color will survive forever. On the other hand, Theorem 2.5 explains if there are (not necessarily disjoint) b-eternal sets in graph and is a constant, then color survives with high probability.
For a graph and (biased) majority model, if for are disjoint b-eternal sets, then r-monochromatic.
Proof: We define random variables such that for :
where for . Therefore, if , then (biased) majority model does not get r-monochromatic because there is at least a blue eternal set. We show which implies r-monochromatic.
is the summation of independent Bernoulli random variables, then:
In a graph and (biased) majority model, if are b-eternal sets (not necessarily disjoint), then r-monochromatic.
Proof: Let random variable denote the number of sets for which create a b-community in i.e. . If we prove , then r-monochromatic.
If we are given discrete probability spaces , then their product is defined to be the probability space over the base set … with the probability function
where . Now, we have random variable so that is the product of discrete probability spaces which correspond to independent random coloring of all vertices in .
We say that the effect of the -th coordinate is at most if for all which differ in the -h coordinate we have . is actually equal to the number of b-eternal sets which contain vertex i.e .
Now, by utilizing Azuma’s Inequality , we have:
We have , therefore:
If and is constant, then r-monochromatic.
2.2 Periodicity and Consensus Time
For a graph and (biased) majority model, the number of possible generations is , and (biased) majority model is a deterministic process; therefore, the process always reaches a cycle of generations after a finite number of steps and stays there forever, but there are two natural questions which arise. What is the length of the cycle and how long does it take to reach it?
Goles and Olivos  and independently Poljak and Sura  proved that a large class of majority-based models, including (biased) majority model, always reach a cycle of period one or two. More precisely, they consider a set of individuals such that every has an initial color from set for i.e. there is a function . Furthermore, the function for measure the influence of on and it is symmetric which means . Now, consider a system which evolves over time so that for every , the function which maps a member to its color at time is defined as follows for ( adopts the most frequent color in its neighborhood and in case of a tie, it chooses the largest one.):
 A system with symmetric following
always reaches a cycle of period one or two.
In (biased) majority model, an arbitrary graph (from any initial coloring) always reaches a cycle of generations of length one or two.
Proof: Let and two colors (red) and (blue). The initial generation corresponds to and we consider symmetric function as follow
One can easily see that in this case the model is equivalent to the majority model. Therefore, the proposition is true in this case. For the case of biased majority model is sufficient to change the state of from to .
As mentioned, we are also interested in the consensus time of the process, and it is trivial that it is at most . However, Poljak and Turzik  proved a very strong proposition (see Theorem 2.9), in the literature of cyclically monotonous mappings and symmetric matrices which provides a tight upper bound on the consensus time of (biased) majority model. More precisely, they present an upper bound on the pre-period of mappings of form where is a linear mapping given by a symmetric matrix of size
and the vector(for more details see Theorem 2.9). Furthermore, pre-period means the maximal such that all , , , are distinct which is equivalent to the consensus time of the process in our terminology.
 Let be defined by where if and if . Let be a symmetric matrix with integral entities, then the pre-period of is at most where .
In a graph for majority model and biased majority model, the consensus time is at most and , respectively.
Proof: Set . Let be an arbitrary ordering of vertices in and let denote the initial coloring of the vertices. The entries of the matrix of size are defined as follows for :
and for as follows:
is a symmetric matrix since if and only if .
Set if and if . For a vertex with odd, there is always a color that occurs most frequently in the neighborhood since the number of neighbors is odd. Note that if the majority of the neighbors of is red and otherwise. For a vertex with even, the rules of majority model state that in the following generation takes the color that occurs most frequently in its neighborhood and in case of a tie, it conserves its color. In other words, takes the color that occurs most frequently in . Note that if the majority of the vertices in is red and otherwise. This shows that applying the mapping to the vector corresponds to one step in majority model on the graph .
Furthermore, since every row of contains an odd number of ’s and every other entry is by construction. Theorem 2.9 implies that the consensus time is at most . For with even, it holds that and for with odd, it holds that . Therefore, the consensus time is at most which is equal to because (where and are equal to the number of vertices with even degree and odd degree, respectively) which is obviously smaller than which is equal to .
The case of biased majority model also can be proved very similarly by defining for :
3 (Biased) Majority Model in Grid and Torus
In this section, first some primary definitions concerning torus and grid are presented. Then, it is proved that majority cellular automata and biased majority cellular automata show a threshold behavior respectively in Section 3.2 and Section 3.3. Furthermore, we also discuss the consensus time and the periodicity of both automata by exploiting the aforementioned results in Sections 2.
The grid is the graph such that and .
The torus is the graph such that and .
A torus is a wrap-around version of the grid which can be visualized as taping the left and right edges of the rectangle to form a tube, then taping the top and bottom edges of the tube to form a torus (See Figure 2 (b)).
The aforementioned definitions of grid and torus follow a neighborhood model which is called Neumann neighborhood or 4-neighborhoods (see Figure 2 (a)). On the other hand, there is another common neighborhood model which is called Moore model (see Figure 2 (a)), and in a torus or grid (by skipping the borders), each cell instead of four neighbors has eight neighbors. More accurately, the grid with Moore neighborhood is the graph such that and .
We sometimes use the term of cell instead of vertex in grids and tori.
, column and row .
3.2 Majority Cellular Automata
In this section, we prove that is a threshold for getting monochromatic or bichromatic in majority cellular automata with Neumann neighborhood. More precisely, results in a cycle of bichromatic generations, but outputs a red monochromatic generation with high probability. For proving the first part, we exploit the concept of robustness and show that if , there exists a blue robust and a red robust set in with high probability which guarantee reaching a cycle of bichromatic generations. On the other hand, for proving the second part we consider a constant number of initial generations instead of just considering initial generation , and interestingly, this technique helps us to prove that results in a red monochromatic generation with high probability. Furthermore, it is shown the aforementioned threshold property works also for Moore neighborhood by considering instead of .
As we know, in the case of majority model, there is no difference between red and blue in the sense of updating rule; then, in this section we simply utilize the term of robust set instead of b-robust set and r-robust set and without loss of generality we also assume .
In the torus with Neumann neighborhood (Moore neighborhood) there is a robust set of size () as shown in Figure 3 (assume is large enough, say () in the case of Neumann (Moore) neighborhood). Actually, in the proof of Theorem 3.9 (Theorem 3.10), we show that in the torus with Neumann (Moore) neighborhood if in a generation , there are at most () blue vertices, the process reaches an r-monochromatic generation in constant number of steps; therefore, the size of the smallest robust set in the case of Neumann and Moore neighborhood is and , respectively. As we will discuss, the size of the smallest robust set plays a critical role in the threshold behavior of majority cellular automaton.
For the torus with Neumann neighborhood, a cluster is a connected subgraph by considering the same vertex set but Moore neighborhood instead of Neumann neighborhood. Therefore, in a torus with Neumann neighborhood, a cluster is not necessarily a connected component.
For the torus with Neumann neighborhood, () is the size of the largest blue (red) cluster in generation for .
In majority model, the torus with Neumann neighborhood and for a generation , if , then .
Proof: Let and . First we prove if , then . To prove, assume can result in i.e there exists a cell such that . We show contradicts . because a blue cell for staying blue in needs at least two blue cells in its neighborhood in which contradicts . implies that , and results in the existence of at least a blue cluster of size in ’s neighborhood in which contradicts .
Now, it is proved that if or , then i.e. blue clusters of size or smaller cannot create a blue cluster of size larger than . It is enough to prove implies . Assume there exists a blue cluster of size in generation . can have two different structures which are shown in Figure 4 (notice a torus is symmetric).
For the first structure (see Figure 4 (a)), we have three states:
(i) if , then for staying blue in needs at least a blue cell among in and needs at least a blue cell among which result in .
(ii) if and (or similarly and ), then for getting blue in needs at least two blue cells among in , and needs at least one blue cell among which imply .
(iii) if , then implies which means .
For the second structure (see Figure 4), also there are three possibilities:
(i) if , then , for staying blue in , needs at least two blue cells among in which implies .
(ii) if and (or similarly and ), then , for getting blue in , needs at least three blue cells among its neighbors () in which implies again.
(iii) if , then implies that three cells in and three cells in are blue in which mean that .
Therefore, implies which means results in . Furthermore, we proved outputs . Then, provides .
In the torus with Neumann neighborhoods and a generation , if , then generation is red monochromatic.
In the majority model and the torus with Neumann neighborhood, results in a red monochromatic generation in at most steps almost surely, but outputs a cycle of bichromatic generations of size one or two in steps with high probability.
Proof: First consider the case of in torus . Let random variable be the number of blue clusters of size in . We claim , then by Markov’s Inequality  with the probability of , . Based on Corollary 3.8, i.e. is red monochromatic. Then, it is enough to show that . Since the number of clusters of size in is (every vertex could be contained in at most a constant number of clusters of size ):
Now, we discuss the state of . Consider (see Figure 3 (b)) as disjoint robust sets. Based on Theorem 2.4, majority model in this state with high probability reaches a cycle of bichromatic generations. Actually, by utilizing Theorem 2.8 and Corollary 2.10, we can say it reaches a cycle of bichromatic generations of size one or two in steps.
In majority model and the torus with Moore neighborhood, results in an r-monochromatic generation in constant number of steps, but outputs a cycle of bichromatic generations of size one or two in steps with high probability.
Proof: First, we prove outputs a red monochromatic generation in constant number of steps with high probability. Let for arbitrary and , set be a rectangle of size in torus . Let random variable denote the number of squares of size (squares for ) in the torus which include more than blue vertices in . Therefore: