ANALYSIS OF A VOTER MODEL
ALEXANDRU AGAPIE and ROBIN H ¨ONS
We investigate theoretically a voter model – or how opinion changes within a community, subject to neighborhood interaction. Under Markov chain modeling, the stationary probability distribution for certain cases is determined in simple analytical form, then itsentropy is analyzed.
AMS 2000 Subject Classification: 60J10, 68Q80.
Key words: Markov chain, cellular automaton, entropy.
1. INTRODUCTION
The recent research on opinion dynamics within communities has concen- trated on the so-calledvoter models. Formally, these are stochastic dynamical systems working as follows [1]. One is given a lattice or graph whose vertices are called cells or sites. Each sitex has a set of neighbor sites – including the cell itself – that constitute theneighborhoodofx and which may influence the state of the original site, which is initially eitherOnorOff. Uniformly, a site xis chosen at random and its state is updated according to the following rule.
One counts the number ofOnsites in the neighborhood of the chosen site – call this k(x). The site then turns (or stays) Onwith probability f(k(x)), where f is predefined. One then repeats this process by picking another site, updat- ing, and so on. This generates a sequence of configurations whose long-term behavior is of certain interest.
The function f is usually chosen to be increasing so that a site is more and more likely to be On if more and more of its neighboring sites are On.
That would correspond to a majority-biased behavior. According to the mod- eling, the increase of f may be either linear or non-linear (exponential, in particular). Also, the function is chosen so that the system is symmetric un- der color change, meaning that if we invert the state of each cell, apply the update process, and invert the system again, we should get the same result (probabilistically) as if we simply updated the system.
As the above system corresponds not only to the voter model, but also to situations emerging in medicine, physics and biology – see [6] for a recent state of the art – the paradigm is usually referred to as cellular automata.
MATH. REPORTS9(59),2 (2007), 135–145
The sites are called cells, or sometimes spins – especially when referring to the Ising model. Despite the huge interest they generated from the very be- ginning, the analysis of cellular automata was mainly empirical, requiring a huge number of runs, plus an inspired analyst able to derive patterns from the experimental evidence [7, 3]. Yet, this paper demonstrates that in case of the 1-D voter model the stationary distribution of the associated Markov chain can be derived in simple analytical form. The form obtained – the well-known Boltzmann-Gibbs distribution – is extensively used in statistical physics as a theoretical model of Ising spin glasses and ferro-magnetization.
Mathematically, both 1-D (that is, one dimension with respect to spatial structure) and 2-D cellular automaton can be represented as binary strings of fixed length, subject of some probabilistic updating. For the binary values of the cells, one can use both 0/1 and −1/+ 1, depending on the context. As the current state of the automaton is the only responsible for transition to the next state, the process can be modeled by a finite homogeneous Markov chain. A detailed introduction is given in Section 2. In Section 3 we consider a special, basic case: The exponential voter model where the transition depends on the neighborhood, but not on the cell itself. We show that this is the most canonic rule tending towards the Boltzmann distribution and fulfilling thedetailed balance equation. For a very general Boltzmann distribution, an appropriate update rule is derived. In Section 4, we turn to the more difficult case where the transition depends on the state of the cell, and we derive the stationary distribution in general form. For the exponential voter model with neighborhood three, it results in a change of the inverse temperature β. In Section 5 we evaluate the results by investigating the entropy of the stationary distributions.
2. THE VOTER MODEL
A cellular automaton is a vectors= (s1, . . . , sn). In our case each cell can assume only two values which we call{−1,+1}. The values si are also called spins, in agreement with the statistical physics connotation of the model.
Since we consider stochastic automata, there is given a probability dis- tribution for the transition of the automaton. We callp(s|s) the probability of transition from the spin vector s to the vector s. In Markov chain theory, these probabilities form the transition matrix of size 2n×2n. Its entries are denoted asPss:=p(s|s) (note the different ordering).
For asynchronous automata, p(s|s) is zero if the two vectors differ by more than one spin. This leads to a ‘sparse’ transition matrix, each state being connected only to its order-1 neighbours and to itself. A circular connection
among the cells is assumed, such that the right-hand neighbour of celln will be cell 1.
In this article, we consider totalistic automata, in which transition de- pends only on the sum of the states of the neighbouring cells. An example is given in Table 1 for a neighborhood of size three, i.e. the cell itself and its two neighbors. It shows the local probability of transition to state 1 in a cell, given the state sum of its neighboring cells.NLVdenotes the general non-linear voter model with two parameters and α,LIN the linear voter model where the transition probability is proportional to the neighborhood sum, andMAJ is the majority vote where the state of the majority is adopted.EVMdenotes the exponential voter model, where probability depends on the neighborhood sum by a special exponential rule with one parameter, the inverse temperature β. In the case 0< , α <1, the Markov chain is irreducible. That is, for any two statessands, there is a possible chain of transitions, leading fromstos.
Table1
Probability of transition to state 1 for the 1-D-3 CA
Nb sum NLV LIN MAJ EV M
−3 0.0000 0 1+e1β
−1 α 0.3333 0 1
1+eβ/3
1 1−α 0.6667 1 1
1+e−β/3
3 1− 1.0000 1 1+e1−β
If this is the case, the Perron-Frobenius theorem applies: There is a unique limit distributionp(s) independent of the initial state or initial distri- bution of states. It is given by the left eigenvector of the transition matrix for the largest eigenvalue λ1 = 1. Unfortunately, the matrix is exponentially large, so generally the classical eigenvalue algorithms are intractable. In the following we shall consider special cases of asynchronous voter models where the stationary distribution can be found analytically.
3. EXPONENTIAL UPDATE RULES
We first investigate the exponential voter model EVM, and regard s = (s1, . . . , sn) as a configuration of spins, each spin taking values in {−1,+1}. The most general form of an asynchronous EVM transition rule is as follows.
Definition 1. The local transition probability from configuration s to a configuration that hassk= 1 and all other spins as in s is given by
(1) q(sk= 1|s) = 1
1 + ef(s,k).
This implies
q(sk=−1|s) = 1−q(sk = 1|s) = 1 1 + e−f(s,k). So ifs is a spin vector, and s−k is the vector
s−k= (s1, . . . , sk−1,−sk, sk+1, . . . , sn)
with the spinsk flipped, the transition probability of flipping the spin sk is
(2) p(s−k|s) = 1
n
1 1 + e−skf(s,k).
In (2) the factor 1/ndenotes the probability that spinsk is chosen within the vector, and the exponential rule gives the probability of flipping the spin (like in Table 1). To avoid confusion, we use the letter p for the global and q for the local transition probability.
As noted in the previous section, these probabilities can be considered as entries of the transition matrix. In fact, they are the non-zero entries in the row belonging to s. The diagonal element in the row of s, corresponding to keeping the current configuration, is
(3) p(s|s) = 1−n
k=1
p(s−k|s).
In this section we consider an inverse problem: We start from a given Boltzmann distribution:
(4) pB(s) = 1
Zeβ
n
i=1si
n
j=1Jijsj
and try to derive an exponential voter model that tends towards that distri- bution. To this end, we derive the functionf in (2).
From (4) we can compute (5) pB(s−k) = 1
Zeβ
isi
jJijsj−2sk
i=ksiJik−2sk
j=kJkjsj
.
By Markov chain theory we have
(6) p(s, t+ 1) =
s
p(s|s)p(s, t).
For the asynchronous voter model, the transition probability is 0 ifs and s differ by more than one spin. Therefore, using (3) we can write
p(s, t+ 1) = n k=1
p(s|s−k)p(s−k, t) +p(s|s)p(s, t)
=p(s, t) + n k=1
(p(s|s−k)p(s−k, t)−p(s−k|s)p(s, t)). (7)
If p is supposed to be a stationary distribution, the sum in (7) must vanish.
There may be many solutions to this equation, but the most trivial one is given by setting all summands to zero. This implies for any k the so-called detailed balance equation
(8) p(s−k)
p(s) = p(s−k|s)
p(s|s−k) = 1 + ef(s|s−k) 1 + ef(s−k|s).
There are also solutions which do not satisfy the detailed balance. In parti- cular, this is the case when the neighborhood is asymmetric, e.g., when the neighbors of a cell are the cell itself and its right-hand neighbor. Inserting (4) and (5) into (8), yields
eβ −2sk
i=ksi(Jik+Jki)
= 1 + ef(s|s−k) 1 + ef(s−k|s). One possible solution to this equation isf(s−k|s) = 2βsk
i=ksi(Jik +Jki), which provides a definition for the local transition rule and leads to the fol- lowing result.
Theorem 1. The Boltzmann distribution pB(s) = 1
Zeβ
n
i=1si
n
j=1Jijsj
is the limit distribution of an automaton with transition probability (9) p(s−k|s) = 1
n
1 1 + e2βsk
i=ksi(Jik+Jki).
Remark 1.1. Note that each cell is not included in its own neighborhood.
In this caseβ used in the local update rule is identical to β of the Boltzmann distribution. Also, the values Jkk do not appear in the transition probability.
Example 1.2. The Boltzmann distribution pB(s) = 1
Zeβ
n
i=1sisi+1
is the limit distribution of an automaton with transition probability
(10) p(s−k|s) = 1
n
1
1 + eβsk(sk−1+sk+1).
It depends only on the sum of the two neighboring spins. The local transition probability to 1 is 12 ifsk−1 =sk+1,= 1+e1β ifsk−1 =sk+1=−1 and 1−if sk−1 =sk+1= +1 .
The derivation leading to Theorem 1 can be extended to more general Boltzmann distributions.
Theorem 2. The Boltzmann distribution pB(s) = 1
Zeβ n
i=1Hisi+ni,j=1Jijsisj
is the limit distribution of an automaton with transition probability p(s−k|s) = 1
n
1
1 + e2βsk(Hk+i=ksi(Jik+Jki)).
4. NEIGHBORHOOD THREE
Next, we analyze the general case of an asynchronous NLV of length n and neighborhood size L = 3. The significant neighbors of cell sk will be {sk−1, sk, sk+1} (addition modn), while the number of Markov chain param- eters is two, and α, as already described in Table 1.
Definition 2. For any configuration s, we denote by nk, k ∈ {0,1,2,3}, the number of cells with k ones in its neighborhood. Furthermore, we split n1 inton1,+, the cells which have value 1 themselves (so they have the neigh- borhood−1,+1,−1) and n1,−, the cells which have value −1 themselves (so they have the neighborhood +1,−1,−1 or −1,−1,+1). In the same manner, we definen2,+ andn2,−.
This induces a partitioning of the space of possible configurations {−1,+1}n into classes (n0, n1,+, n1,−, n2,+, n2,−, n3), where the sum of the six values is n. For a spin vector s, we write s ∈ (n0, n1,+, n1,−, n2,+, n2,−, n3) if the number of cells with the respective neighborhoods comply.
Definition 3. A pair (si, si+1) is called aborderif the two spins are of op- posite signs. We denote byb(s) the number of borders within configurations. Lemma 1. For any configuration s, b(s) is even and, for all k= 0, . . . , [n/2], the number of configurations with 2k borders is 2n
2k
. Proof. The first part is obvious. Next, there are n
2k
possibilities to place 2k borders within a string of length n, ands is completely defined by the position of the borders and the value of s1, which may be either +1 or
−1.
We state now our main result.
Theorem 3. The stationary distribution of P is given by π, where
(11) πs= 1
Z
1−α 12b(s) andZ is a norming constant.
Proof. We have to prove the balance equation
(12) πs=
πtPts for alls∈ {−1,+1}n.
In the asynchronous model, at most one cell can change in a time step.
Given a configuration s in a class (n0, n1,+, n1,−, n2,+, n2,−, n3), the di- agonal term which corresponds to no change in the configuration is computed as in (3), one minus the probability that any cell changes. The formula reads (13) Pss= 1−n0+n3
n −αn1,−+n2,+
n −(1−α)n2,−+n1,+
n .
For the other npositive terms in column Ps, we must consider the pos- sible cases for the previous configuration. There are six possible cases for the switched cell, induced by the six possible neighborhood classes. For each case we deduce the number of borders of the previous configuration and calculate its probability under the assumed distribution π. We just demonstrate here one case.
Case 1. Flip a cell from some t to obtain a cell from n1,+ in s, like in −1,−1,−1 → −1,+1,−1. Then b(t) = b(s)−2. So, under the assumed distribution,t has probability
(14) πt= 1
Z
1−α b(s)−2
2 .
Furthermore, we have Pts =/n, and there are n1,+ transitions of this kind.
So, we can compute the sum over all these configurations
(15)
t
πtPts= nn1,+
1−α
b(s)−2
2 .
Summing up all the six cases and the diagonal term, thus calculating the sum in (12), we can show thatπ is a stationary distribution indeed. For 0< , α <1, the Markov chain is irreducible, so the stationary distribution is unique.
Corollary 3.1. The stationary distribution for = 1−α is the uni- form one.
4.1. THE 1-D EVM WITH THREE NEIGHBORS
We now derive the results for the exponential voter model. From Table 1 we obtain that= 1/(1 + eβ) andα= 1/(1 + eβ/3). Then Theorem 3 becomes
Theorem 4. The limit distribution of the asynchronous EVM with 3- neighborhood including the cell itself is
(16) p(s) = 1
Z
1 + eβ 1 + e−β/3
−b(s)/2
. Sinceb(s) = 12
isi(si−si+1), this turns into a Boltzmann distribution:
Theorem 5. Let
(17) β = ln
1 + eβ 1 + e−β/3
.
Then the limit distribution of the asynchronous EVM with3-neighborhood in- cluding the cell itself is the Boltzmann distribution
(18) p(s) = 1
Z exp
− β 4
i
si(si−si+1)
.
Remark 5.1. If the spin is included in its neighborhood, the local update rule and the global Boltzmann distribution have a different inverse tempera- tureβ. The difference is shown in Figure 1. β is always less thanβ given by the local update rule. The maximal difference isβ−β = ln34 at β= ln 8.
-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0
0 5 10 15 20
beta’ - beta
beta
Fig. 1. The difference betweenβ andβ, the inverse temperatures of the stationary distribution and of the transition rule.
5. ENTROPY ANALYSIS
A common measure for characterizing the probability distributions over the space of configurations is the entropy. In statistical mechanics and spin
glass models, a normalized entropy per cell is defined as
(19) H(p) =−1
n
s
p(s) lnp(s).
In the case of the 1-D general voter model with neighbourhood three, Lemma 1 and Theorem 3 lead to
(20) H(π) =−1 n
1 Z
[n2]
k=0
2 n
2k
1−α
k
kln
1−α −ZlnZ
,
where the norming constant is
(21) Z =
[n2]
k=0
2 n
2k
1−α
k .
In Figure 2 the entropy (20) versus α and is plotted. The dependency of the (normalized) entropy on the size of the cellular automaton is very small – the cases n= 10, n= 100 and n= 1000 plot actually to the same picture, having a maximum of ln 2 at the line= 1−α. This was no surprise, as the stationary distribution in that case corresponds to the uniform distribution (see Corollary 3.1), known to provide the maximum of entropy. For a fixed value of , the entropy increases from α = 0 to α = 1−, then it decreases again. The closer to zero is, the steeper is the final descent for 1− < α <1.
0 0.2
0.4 0.6
0.8 1 alpha
0 0.2 0.4 0.6 0.8 1
epsilon 0.3
0.4 0.5 0.6 0.7
entropy
Fig. 2. Entropy per cell for the NLV forn= 100.
Lemma 2. In the interior of the parameter domainD={0≤, α≤1}, the following symmetries of the entropy function hold:
1. H(, α) =H(1−α,1−), if nis even;
2. H(, α) =H(1−,1−α), if a) = 1−α, or b)=α andn is even.
Proof. The proof for cases 1, respectively 2.b, relies on the mirror symme- try of the normalized stationary distribution with respect ton/2, forn= even.
As for case 2.a, it corresponds to the uniform stationary distribution, whose entropy doesnotdepend on orα.
It is easy to see that, for large values ofN, the effect of the disjunction odd/even vanishes, so the cases 1, respectively 2.b will provide an entropy which is almost symmetric.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 5 10 15 20
entropy
beta
(ln2)/N
Fig. 3. Entropy per cell for the exponential voter model.
The exponential voter model behaves smoothly with respect to β – see Figure 3. The entropy decreases continuously with rising β. For β = 0 the entropy is maximal (again, corresponding to the uniform distribution), while for β → ∞ the entropy per cell tends to ln 2/n. This corresponds to a sta- tionary distribution equally concentrated on only two configurations, namely (+1,+1, . . . ,+1) and (−1,−1, . . . ,−1).
Acknowledgements. This work was done while the first author was with Fraunhofer Institute for Autonomous Intelligent Systems, Sankt Augustin, Germany.
REFERENCES
[1] R.C. Butler,http://www.math.utah.edu/∼rbutler/votermodel/index.html (2002).
[2] R. Durret,http://www.math.cornell.edu/∼durret/survey/survhome.html(2001).
[3] A. Ilachinski, Cellular Automata. A Discrete Universe. World Scientific, Singapore, 2002.
[4] J. Molofsky, R. Durret, J. Dushoff, D. Griffeath and S. Levin,Local frequency dependence and global coexistence.Theoret. Population Biology55(1999), 270–282.
[5] N.A. Oomes,Emerging markets and persistent inequality in a non-linear voter model.In:
D. Griffeath and C. Moore (Eds.),New Constructions in Cellular Automata, pp. 207–
229. Oxford University Press, 2002.
[6] F. Schweitzer (Ed.),Applications of Cellular Automata in Complex Systems.Advances in Complex Systems5(2002),2, 101–337.
[7] S. Wolfram, Cellular Automata and Complexity: Collected Papers. Addison-Wesley, Reading, 1994.
Received 15 October 2005 Romanian Academy
Institute of Mathematical Statistics and Applied Mathematics Casa Academiei Romane Calea 13 Septembrie nr. 13 050711 Bucharest 5, Romania
agapie@rdslink.ro and
Fraunhofer Institute AIS
Faculty of Mathematics and Computer Science D-53754 Sankt Augustin, Germany
robin.hoens@ais.fhg.de
