Emulation and complementarity in onedimensional ... - Semantic Scholar
Emulation and Complementarity in One-Dimensional Alternatives of the Axelrod Model with Binary Features A. ADAMOPOULOS1 AND S. SCARLATOS2∗ 1 Department of Medicine, Medical Physics Laboratory, Democritus University of Thrace, 681 00 Alexandroupolis,
Greece; and 2 Department of Mathematics, Center for Research and Applications of Nonlinear Systems, University of Patras, Patra 265 00, Greece
Received April 16, 2011; revised July 18, 2011; accepted July 19, 2011 We investigate the one-dimensional dynamics of alternatives of the Axelrod model (ξt ) with k binary features and confidence parameter = 0, 1, . . . , k. Simultaneously, the simple Axelrod model is also critically examined. Specifically, for small and large , simulations suggest that the convergent model (ξt ) is emulated by a corresponding attractive model (ηt ) with the same parameters (conditional on bounded confidence). (ηt ) is more mathematically tractable than (ξt ), and the very definitions of the two qualitative behaviors of cyclic particle systems (fluctuation and fixation) are applicable in special cases. Moreover, we observe a complementarity: for not too small k and ≈ k2 , (ηt ) fixates (each site has a final type independent of the possibly infinite size of the lattice), whereas (ξt ) fluctuates (each site changes type at arbitrarily larger times t as the size of the lattice increases). © 2011 Wiley Periodicals, Inc. Complexity 17: 43–49, 2012 Key Words: Axelrod model; cyclic particle systems; voter model; confidence threshold (bounded confidence); random cellular automata
1. INTRODUCTION
C
yclic particle systems, parameterized by the number of colors/types N , are stochastic spatial models with local and dyadic interactions of dominance such that
Correspondence to: S. Scarlatos, Department of Mathematics, Center for Research and Applications of Nonlinear Systems, University of Patras, Patra 265 00, Greece (e-mail: [email protected]) or A. Adamopoulos, Department of Medicine, Medical Physics Laboratory, Democritus University of Thrace, 681 00 Alexandroupolis, Greece (e-mail: [email protected]) ∗ Present address: Kabouridou 44, 552 36 Thessaloniki, Greece.
© 2011 Wiley Periodicals, Inc., Vol. 17, No. 3 DOI 10.1002/cplx.20391 Published online 24 October 2011 in Wiley Online Library (wileyonlinelibrary.com)
the type of a site can change the type of a neighboring site in a cyclic manner [1]. In fact, their interaction mechanism uses an extension of the rock-paper-scissors rule for more than three types. Together with the assumption of cyclical dominance of types/colors, they implicitly incorporate a confidence threshold, = 1, in their formulation. Bramson and Griffeath’s original intention was to study interesting nonlinear systems that apparently exhibit complex behavior. On the one-dimensional integer lattice Z, they fluctuate if N ≤ 4, and they fixate if N ≥ 5. Fluctuation and fixation are defined as follows. Definition 1 If each site has a final type, then the system fixates.
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Definition 2 If each site changes type at arbitrary large times t, then the system fluctuates.
c, provided that (1) holds, together with the assumption of convergence on the hypercube, 0 < |ξ(x) − ξ(y)| = 1 + |c − ξ(y)|.
These two simple definitions make perfect sense for a mathematician, who usually deals with infinite systems [2]. However, for a social scientist, statistical physicist, or mathematician who deals with finite systems, such definitions may seem confusing [3]. Markov chain theory suggests that the currently examined finite models always converge to an absorbing state in finite time, for any range of their parameters. That is, according to the definitions stated earlier, the following systems fixate and never fluctuate on a finite set. To avoid confusion, we will reserve the term “fixation” for the case when a system fixates in the sense that cyclic systems may do so on Z. For instance, recall the dynamics of the lind ear two-type voter model with state space {0, 1}Z , where a site changes its type at a rate proportional to the number of neighboring sites with a different type [2, p. 3]. We would say that the voter model on a finite set fluctuates until consensus. However, it also fixates according to the definition; this is a spurious kind of fixation, caused by fluctuations of finite size. Let (ξt )t≥0 be a spatial version of the underlaid model in the economical and financial articles [4–6] on a social network represented by a finite and connected graph G (with vertex set V and edge set E). The type of a vertex (site) x in V of G, ξ(x), is taken as a k-dimensional vector, ξ(x) = (ξ 0 (x), . . . , ξ k−1 (x)), where its elements are binary features (i.e., features that can take the values 0 or 1). Initially, the 2k types are independently and uniformly distributed among the sites of G. Subsequently, the evolution is Markovian in either discrete or continuous exponentially updated time. At independent times for each x and i, a feature ξ i (x) becomes 1 − ξ i (x) at a rate proportional to the number of neighbors y of x, such that ξ i (x) = ξ i (y) and |ξ(x) − ξ(y)| ≤ ,
(1)
where | · | is the taxicab (L 1 ) norm, and the minus sign is interpreted for the subtraction of vectors. Since the diameter of the k-dimensional hypercube (or k-cube Qk ) is k, it is meaningful for to take values in {1, . . . , k}. The notable difference between (ξt ) and the well-known model for the diffusion of cultures by Axelrod (1997) is the introduction of the parameter , which drops the assumption of homophily, that is, more similar types tend to interact more frequently (cf., [7, p. 208]). In Section 3, we discuss the application of the present study to the one-dimensional Axelrod model with k features and two traits. Under the transition mechanism described above, we realize that ξ(x) converges towards ξ(y) in a minimal sense, for neighbor x, y. This can be clarified by rewriting the interaction mechanism as, at each time, a randomly chosen x picks a random neighbor y, and assumes the (possibly random) type
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(2)
This can be called as subattractive dynamics, and it extends the notion of attractiveness in interacting particle systems ([2], Definition 2.1, p. 134). Whenever |ξ(x) − ξ(y)| ≤ 1 for some x and all neighboring y, interactions are attractive, i.e., x is more likely to assume the type of one of its neighbors if it differs from most of its neighbors than it is if it agrees with most of them. Otherwise, it is distinguished from attractive/exchangeable voter dynamics in a manner that we try to demonstrate in this article. In contrast to disappointing, albeit generic results [8, 9], the introduction of a confidence threshold drastically enhances the dynamics in the particular model of [4], as well as in its spatial version which is presently examined. While investigating the convergent dynamics of (ξt ), it was tempting to do so in comparison to a corresponding process (ηt ) with the same parameters k, , characterized by attractive dynamics (conditional on bounded confidence). According to the new transition mechanism, η(x) becomes of the type c = η(y) at a rate proportional to the number of neighbors y of x, provided that (1) holds. Except for the modification of the convergent assumption (2) to the attractive one η(y) = c, the chance of interaction and the rest of the transition mechanism remains the same as that in (ξt ). Would (ηt ) emulate (ξt ) for all , with respect to their qualitative behaviors at the asymptotic limit, and with a higher rate of convergence to this limit than (ξt )? To answer this question, the present approach is based on the concepts presented in [10], that is, “simulations provide useful evidence for making conjectures that can then be tackled in hopes of proving new theorems,” or the use of synchronously updated cellular automata to model continuous-time Markov processes. On the basis of Ref. 10, we will show that a slight modification to the transition rule of (ξt ) yields the more computationally efficient (ηt ), which exhibits one-dimensional dynamics with identical qualitative behaviors for the largest range of their parameters. In addition, we show that (ηt ) is more mathematically tractable, by applying the definitions of qualitative behaviors and probabilistic methodology described in [1] to special cases. Thus, we jointly formulate a conjecture for both models. In the rest of this article, to compare the present study with the original literature [1] and [7], we assume that G = ⊂ Z is a finite and connected subset of the one-dimensional integer lattice. Conjecture 1 Let (ξt ) and (ηt ) be convergent and attractive alternatives, respectively, of the Axelrod model on ⊂ Z as defined previously, with k binary features and confidence threshold .
© 2011 Wiley Periodicals, Inc. DOI 10.1002/cplx
Starting from the uniform product distribution of 2k types, there are critical fix < flux in {1, . . . , k − 1}, such that:
FIGURE 1
A. If ≤ fix , then (ξt ), (ηt ) fixate (i.e., in the sense that cyclic systems may do so on Z). B. If fix < < flux , then (ηt ) fixates and (ξt ) fluctuates until absorption. C. If ≥ flux , then (ηt ) fluctuates until absorption and (ξt ) fluctuates until consensus. The stated above conjecture can be rephrased as follows. In the range of that the convergent (ξt ) fixates, the attractive (ηt ) supposedly fixates as well (conditional on bounded confidence). That is, for this range of , one process supposedly emulates the other, and vice versa. In the range of that the attractive (ηt ) fluctuates until absorption, the convergent (ξt ) supposedly fluctuates until consensus. In the intermediate range, a complementarity seemingly takes place, in the sense that the convergent process fluctuates, whereas the attractive one fixates (conditional on bounded confidence). The experiments show that this complementarity occurs for k ≥ 4. For k = 3, it seems that the two processes emulate each other with respect to fluctuation and fixation at the asymptotic limit. Note that, when = k, (ηt ) is the multitype voter model, also known as the stepping stone model [11]. In this case, it is well known that (ηt ) fluctuates on the multidimensional lattices Zd ; therefore, it fluctuates on multidimensional boxes in Zd until consensus is reached. (The d-dimensional n-box centered at 0 is defined by the tuples x = (x1 , . . . , xd ) in Zd such that, |xl | ≤ n, for 1 ≤ l ≤ d.) In addition, when = k − 1, (ηt ) fluctuates until absorption on multidimensional boxes in Zd (to either consensus or a bipolar configuration). Two sketches of proofs are presented as follows. One approach is to use an ergodic argument as in [1], Lemma 1, which would also require some graph theoretic notions of the hypercube [12] (Hamilton cycles, and the recursive construction of the hypercube). Another simpler approach, to show fluctuation for = k −1, is by noticing that (ηt ) is a (spatial) political positions process for special graphs of types. Then, one can use the concepts of embedded graphs and processes as in [13], Section 3, to show fluctuation when two vertices of the graph of types are not connected to each other, but are connected to all other vertices (which is, in fact, our case = k − 1). Moreover, the methodology in the original article on cyclic systems [1] applies elegantly to the one-dimensional (ηt ) in an example presented later (fixation for < k3 ). The presented argumentation, together with experiments described in the next section, strongly suggest that in one dimension, fix k2 and flux k2 .
1.1. Two Binary Features In this case, two binary features yield four cultural types, which are represented by a four-vertex cycle. On multidimensional boxes in Zd , this model fluctuates until absorption if
© 2011 Wiley Periodicals, Inc. DOI 10.1002/cplx
(a) Eight cultural types of the Axelrod model with three binary features are represented by a cube. If one removes the grey vertices, a six-vertex cycle is obtained. (b) The cycle with six vertices represents all strong orderings of the three alternatives A, B, and C .
= 1, and it fluctuates until consensus if = 2 (that is, only behavior C. may manifest on Zd ). In spite of its graph of types, which is cyclical, it does not exhibit the interesting local periodicity of basic cyclic systems (see, for instance [14]).
1.2. Three Binary Features In this case, behavior C. (fluctuation for = 2, 3) can be fully proven for (ηt ) on multidimensional boxes in Zd by the argument stated in the previous paragraphs. To prove behavior A. (fixation for = 1) on Z is an interesting problem because it requires the rigorous and unpleasant techniques for N = 5 types/colors described in [1]. Simplification can be achieved by the removal of two cultural types, as showed in Figure 1. It is a meaningful approach if one wants to consider rational agents. This is a typical hypothesis of the transitivity of individual choice followed in social choice theory and welfare economics (for further details and relevant definitions see, for instance [15]). Note that the resulting subgraph after vertex removal is a six vertexcycle, which represents all strong orderings of three alternatives (for the representation of the more general case of all weak orderings of three alternatives, see [16]). Under this cyclical representation of types, the arguments in [1] are directly applicable to the extent that strong evidence of fixation on Z (using block-averaging technique from probability theory) is obtained. There is no need to resort to the tedious methodology described in [1] for N = 5 types/colors. We will briefly refer to these probabilistic techniques in the next example because they are more straightforwardly applicable when k = 4, = 1.
1.3. Four Binary Features and Extension to k Binary Features In this case, on multidimensional boxes in Zd , behavior C. (fluctuation for = 3, 4) can be fully proven on the basis of previous arguments for (ηt ). Moreover, strong evidence can be obtained to show that behavior A. (fixation for = 1) manifests on Z under the
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dynamics of (ξt ) and (ηt ). Specifically, the relatively easily obtainable strong evidence of fixation is that active bonds cannot eliminate all blockades in a sufficiently large interval of Z. Active bonds are the edges connecting each x, x + 1 that satisfies (1) and ξ(x) = ξ(x + 1). Blockades are the edges connecting each x, x + 1 that does not satisfy (1). To understand this argument, it is useful for the reader to construct a graphical representation of the process (details for this technique are provided in particle systems literature; see for instance [2], Chapter III, Section 6, or [1]). If = 1, then a blockade of size |ξ(x)−ξ(x+1)| = 2 is eliminated after a possible collision with an active bond, which jumps to the edge of the blockade after the collision. For blockades of sizes |ξ(x) − ξ(x + 1)| = 3, 4, at least |ξ(x) − ξ(x + 1)| − 2 active bonds are needed to be sacrificed to remove the blockade. The tendency for the elimination of active bonds to remove blockades can be measured by a comparison function φ for each x: φ(x) = 0, if ξ(x) = ξ(x + 1), = |ξ(x) − ξ(x + 1)| − 2, otherwise,
lim P
n→∞
m
φ(x) ≤ 0 for some l ≥ n, m ≥ 0 = 0.
(3)
x=−l
Since |ξ(x) − ξ(x + 1)| are bounded and i.i.d. for each x, the summands of the event in (3) are bounded and i.i.d.. If k = 4, = 1, then E[φ(x)] =
4 6 4 1 1 0+ (−1) + 0+ 1+ 2 > 0. 16 16 16 16 16
Then, a large deviations estimate from probability theory shows that the probability of the event in (3) exponentially tends to 0 as n → ∞. This proves that under the dynamics of (ξt ) and (ηt ), active bonds cannot eliminate all blockades in a sufficiently large interval of Z, when k = 4, = 1. Let us remark that such an argumentation easily extends to k > 4, = 1. Owing to the particular symmetry of the hypercube, the above arguments extend straightforwardly for (ηt ) when k = 3 + 1, > 1. Note that, by recursive definition of the hypercube, the case k = 3 + 1 obeys certain ratios of Q4 with = 1; therefore, fixation is suspected in one dimension, which will be discussed at once. Recall that (ηt ) is taken from (ξt ) by changing convergent interactions to attractive ones (conditional on bounded confidence). The desirable property of (ηt ), which is absent in (ξt ) for > 1 and permits such
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if η(x) = η(x + 1), φ (x) = 0, |η(x) − η(x + 1)| = − 2, otherwise, which yields E[φ (x)] =
1 a b c 1 0 + k (−1) + k 0 + k 1 + k 2 > 0, 2k 2 2 2 2
with a = #{u ∈ Qk : 0 < |u| ≤ }, b = #{u ∈ Qk : < |u| ≤ 2}, and c = #{u ∈ Qk : 2 < |u| ≤ 3}. One can verify that a = c, which proves that E[φ (x)] is strictly positive, and this extends easily to k > 3 + 1. Therefore, by a large deviations argument under the dynamics of (ηt ), active bonds cannot eliminate all blockades in a sufficiently large interval of Z, if < k3 , k ≥ 4.
2. EXPERIMENTS
where ξ = ξ0 is the initial random configuration. The sum of φ over an interval is the number of active bonds necessary to remove all blockades in this interval, minus the number of all active bonds in this interval. Therefore, the relation x∈ φ(x) > 0 implies the lack of sufficient active bonds in to eliminate the blockades in . One can prove that
a straightforward extension, is that active bonds cannot be created. One can use the comparison function
In discrete-time simulations on blocks of the integer lattice Z, the conjectured phase transitions are visible to the eye for small k. In order to demean finite-size effects, so that the phase transition is more clearly visible, periodic boundary conditions were incorporated. Furthermore, instead of updating one site (coordinate) at each time, it is computationally efficient to employ a quasi-synchronous update. One can use a random cellular automaton [17, with both the initial configuration and the transition rule random, p. 2] (see also [18], Chapter 6) with a double clock in the following sense. Simultaneously, at odd times, all odd coordinates are updated, and at even times, all even coordinates are updated from this point on each updated coordinate interacting according to the transition rule employed for (ξt ). That is, the source of randomness in the new transition rule is a collection of ideal coin tosses (Bernoulli trials). Let us denote this new process with (ξ¯t )t=0,1,... defined on an appropriate probability space with a probability law assigning likelihoods to all possible trajectories of the process [17]. (ξ¯t ) is a novel and computationally efficient emulation of (ξt ), and it is different from the partial emulation of (ξt ) by the attractive (ηt ) (conditional on bounded confidence, and in the range of the parameters for which such emulation occurs). According to our experiments, (ηt ) is emulated by an (η¯ t ) with a double clock as described above. Figures 2 and 3 show simulations of (ξ¯t ) on 200 sites up to t = 1000 and t = 104 , respectively. One might think that if (η¯ t ) fixates for k = 3, = 1, then (ηt ) should fixate for all k, < k2 . A syllogism may be construed as follows. Since the hypercube is defined by recursive duplication of itself (i.e., Qk+1 is derived by duplicating Qk and joining their 1-1 corresponding vertices), the model with types in the hypercube with k = 2 + 1, which obeys certain ratios of Q3 with = 1, should fixate in a conjectural sense
© 2011 Wiley Periodicals, Inc. DOI 10.1002/cplx
FIGURE 2
FIGURE 4
Fixation of (ξ¯t ) on Z with three binary features and = 1. Time is increasing downwards on the vertical axis. Since = 1, the figure shows also a simulation of (η¯ t ).
Mean size of clusters versus on the one-dimensional torus with 100 sites (k = 20, the ensemble is 1000).
(since fixation is observed in experiments for k = 3, = 1). Then, one may also think that, since (ηt ) supposedly fixates on Z for k = 2 + 1, it should fixate for all k > 2, in analogy to similar cases described in this article (however, the comparison function φ(x) is missing). The emulation by parts of (ξt ) by (ηt ), and vice versa, is intuitive based on the mean size of clusters at absorption. On one-dimensional tori with 100 sites, ensemble-averaged over
1000 simulation runs, the mean size of clusters varies with , as shown in Figure 4. Figures 5–8 show the cumulative distributions of the sizes of clusters in simulations on one-dimensional tori with 100 sites, ensemble-averaged over 100 simulation runs. Fixation is observed for the values of for which the frequencies of cluster sizes decay according to a power law, in analogy to [19]. The critical fix is observed by a power law decay, together with high variance in the frequencies of larger clusters, in
FIGURE 5 FIGURE 3
Fluctuation of (ξ¯t ) on Z with three binary features and = 2. Time is increasing downwards on the vertical axis.
© 2011 Wiley Periodicals, Inc. DOI 10.1002/cplx
Convergent (conditional on bounded confidence) model, 10 binary features, 1x100 torus, ensemble 100.
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FIGURE 6
FIGURE 8
Convergent (conditional on bounded confidence) model, 11 binary features, 1x100 torus, ensemble 100.
Attractive (conditional on bounded confidence) model, 11 binary features, 1 × 100 torus, ensemble 100.
analogy to [4]. Fluctuation is observed by the high frequencies of large clusters, which are comparable to the frequencies of small clusters. Consensus is observed by a single point in the log-log diagram, at the maximum size of a cluster with frequency 1. Simulations were performed in a MATLAB programming environment; the developed MATLAB code is available from the corresponding author.
FIGURE 7
Attractive (conditional on bounded confidence) model, 10 binary features, 1 × 100 torus, ensemble 100.
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3. DISCUSSION We investigated the complex behavior exhibited by variants with a confidence parameter of the binary-featured Axelrod model, by understanding the concepts presented in the first and simplest model, which has been simultaneously studied in depth. Indeed, we show that the definitions of the fundamental behaviors of cyclic particle systems (fluctuation and fixation) can be applied to finite systems with caution, owing to ambiguous situations that may arise because of the finiteness of space. The examined variants of the Axelrod model, conditionally convergent and attractive on bounded confidence, seem to exhibit behavior analogous to one-dimensional cyclic systems with respect to fixation and fluctuation, as expressed by Conjecture 1. The present study applies to the one-dimensional Axelrod model with binary features (two traits) as well, because it is obtained from (ξt ) if = k − 1 after appropriate normalization of the transition rates. In the Axelrod model, the chance of interaction is proportional to the similarity between two neighbors, measured by the proportion of their equal features (homophily), and interacting sites tend to become more similar (convergence). In the dynamics of (ξt ), provided that (1) holds, interacting sites tend to become more similar (conditional convergence on bounded confidence). It is found that (ξt ) with = k − 1 emulates the Axelrod model, i.e., fluctuation until absorption occurs. Therefore, dropping the assumption of homophily does not affect the behavior in one dimension. Through such a critical discussion, we speculated that the examined convergent (ξt ) is emulated for small and large by the attractive (ηt ), conditional on bounded confidence.
© 2011 Wiley Periodicals, Inc. DOI 10.1002/cplx
This partial emulation is achieved with respect to fluctuation and fixation at the asymptotic limit, and with a higher rate of convergence to this limit. To this end, we provide rigorous preliminary results for the attractive model (strong evidence of fixation if < k/3, fluctuation if = k − 1, k). Moreover, it was interesting to observe that the two processes are complementary in the following sense. For not
too small k values, there is a narrow range of around k2 , within which the attractive (ηt ) fixates, whereas the convergent (ξt ) fluctuates (conditional on bounded confidence). Therefore, one can substitute the convergent with attractive/exchangeable interactions, without essentially affecting the behavior in one dimension for the largest range of their parameters.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
Bramson, M.; Griffeath, D. Flux and fixation in cyclic particle systems. Ann Probab 1989, 17, 26–45. Liggett, T. Interacting Particle Systems; Springer-Verlag: New York, 1985. Castellano, C.; Fortunato, S.; Loreto, V. Statistical physics of social dynamics. Rev Mod Phys 2009, 81, 591–646. Banisch, S.; Araújo, T.; Louca, J. Opinion dynamics and communication networks. Adv Complex Sys 2010, 13, 95–111. Araújo, T.; Weisbuch, G. The labour market on the hypercube. Phys A 2008, 387, 1301–1310. Araújo, T.; Mendes, R.V. Innovation success and structural change: An abstract agent based study. Adv Complex Sys 2009, 12, 233–253. Axelrod, R. The dissemination of culture: A model with local convergence and global polarization. J Conflict Resolut 1997, 41, 203–226. Flache, A.; Macy, M.W. Local convergence and global diversity: The robustness of cultural homophily 2007. eprint arxiv:physics/0701333. De Sanctis, L.; Galla, T. Effects of noise and confidence thresholds in nominal and metric Axelrod dynamics of social influence. Phys Rev E 2009, 79, 046108. Fisch, R. The one-dimensional cyclic cellular automaton: A system with deterministic dynamics that emulates an interacting particle system with stochastic dynamics. J Theor Probab 1990, 3, 311–338. Cox, J.T.; Griffeath, D. Recent results for the stepping stone model. In: Percolation Theory and Ergodic Theory of Infinite Particle Systems, IMA Volumes in Mathematics and Its Applications, Vol. 8; Kesten, H., Ed.; Springer: New York, 1986; pp. 73–84. Harary, F. A survey of the theory of hypercube graphs. Comput Math Appl 1998, 15, 277–289. Itoh, Y.; Mallows, C.; Shepp, L. Explicit sufficient invariants for an interacting particle system. J Appl Prob 1998, 35, 633–641. Fisch, R.; Gravner, J.; Griffeath, D. Cyclic cellular automata in two dimensions. In: Spatial Stochastic Processes, A festschrift in honor of the seventienth birthday of T.E. Harris; Alexander, K.S.; Watkins, J.C., Eds.; Birkhäuser Boston: Boston, MA, 1991; pp. 171–185. Arrow, K.J. Social Choice and Individual Values; Wiley & Sons: New York, 1951. Kemeny, J. Mathematics without numbers. Daedalus 1959, 88, 577–591. Griffeath, D. Cyclic random competition: A case history in experimental mathematics. AMS Notices 1988, 35, 1472–1480. Wolfram, S. A New Kind of Science; Wolfram Media: Champaign, IL, 2002. Klemm, K.; Eguíluz, V.M.; Toral, R.; Miguel, M.S. Role of dimensionality in Axelrod’s model for the dissenimation of culture. Physica A 2003, 327, 1–5.
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Greece; and 2 Department of Mathematics, Center for Research and Applications of Nonlinear Systems, University of Patras, Patra 265 00, Greece
Received April 16, 2011; revised July 18, 2011; accepted July 19, 2011 We investigate the one-dimensional dynamics of alternatives of the Axelrod model (ξt ) with k binary features and confidence parameter = 0, 1, . . . , k. Simultaneously, the simple Axelrod model is also critically examined. Specifically, for small and large , simulations suggest that the convergent model (ξt ) is emulated by a corresponding attractive model (ηt ) with the same parameters (conditional on bounded confidence). (ηt ) is more mathematically tractable than (ξt ), and the very definitions of the two qualitative behaviors of cyclic particle systems (fluctuation and fixation) are applicable in special cases. Moreover, we observe a complementarity: for not too small k and ≈ k2 , (ηt ) fixates (each site has a final type independent of the possibly infinite size of the lattice), whereas (ξt ) fluctuates (each site changes type at arbitrarily larger times t as the size of the lattice increases). © 2011 Wiley Periodicals, Inc. Complexity 17: 43–49, 2012 Key Words: Axelrod model; cyclic particle systems; voter model; confidence threshold (bounded confidence); random cellular automata
1. INTRODUCTION
C
yclic particle systems, parameterized by the number of colors/types N , are stochastic spatial models with local and dyadic interactions of dominance such that
Correspondence to: S. Scarlatos, Department of Mathematics, Center for Research and Applications of Nonlinear Systems, University of Patras, Patra 265 00, Greece (e-mail: [email protected]) or A. Adamopoulos, Department of Medicine, Medical Physics Laboratory, Democritus University of Thrace, 681 00 Alexandroupolis, Greece (e-mail: [email protected]) ∗ Present address: Kabouridou 44, 552 36 Thessaloniki, Greece.
© 2011 Wiley Periodicals, Inc., Vol. 17, No. 3 DOI 10.1002/cplx.20391 Published online 24 October 2011 in Wiley Online Library (wileyonlinelibrary.com)
the type of a site can change the type of a neighboring site in a cyclic manner [1]. In fact, their interaction mechanism uses an extension of the rock-paper-scissors rule for more than three types. Together with the assumption of cyclical dominance of types/colors, they implicitly incorporate a confidence threshold, = 1, in their formulation. Bramson and Griffeath’s original intention was to study interesting nonlinear systems that apparently exhibit complex behavior. On the one-dimensional integer lattice Z, they fluctuate if N ≤ 4, and they fixate if N ≥ 5. Fluctuation and fixation are defined as follows. Definition 1 If each site has a final type, then the system fixates.
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Definition 2 If each site changes type at arbitrary large times t, then the system fluctuates.
c, provided that (1) holds, together with the assumption of convergence on the hypercube, 0 < |ξ(x) − ξ(y)| = 1 + |c − ξ(y)|.
These two simple definitions make perfect sense for a mathematician, who usually deals with infinite systems [2]. However, for a social scientist, statistical physicist, or mathematician who deals with finite systems, such definitions may seem confusing [3]. Markov chain theory suggests that the currently examined finite models always converge to an absorbing state in finite time, for any range of their parameters. That is, according to the definitions stated earlier, the following systems fixate and never fluctuate on a finite set. To avoid confusion, we will reserve the term “fixation” for the case when a system fixates in the sense that cyclic systems may do so on Z. For instance, recall the dynamics of the lind ear two-type voter model with state space {0, 1}Z , where a site changes its type at a rate proportional to the number of neighboring sites with a different type [2, p. 3]. We would say that the voter model on a finite set fluctuates until consensus. However, it also fixates according to the definition; this is a spurious kind of fixation, caused by fluctuations of finite size. Let (ξt )t≥0 be a spatial version of the underlaid model in the economical and financial articles [4–6] on a social network represented by a finite and connected graph G (with vertex set V and edge set E). The type of a vertex (site) x in V of G, ξ(x), is taken as a k-dimensional vector, ξ(x) = (ξ 0 (x), . . . , ξ k−1 (x)), where its elements are binary features (i.e., features that can take the values 0 or 1). Initially, the 2k types are independently and uniformly distributed among the sites of G. Subsequently, the evolution is Markovian in either discrete or continuous exponentially updated time. At independent times for each x and i, a feature ξ i (x) becomes 1 − ξ i (x) at a rate proportional to the number of neighbors y of x, such that ξ i (x) = ξ i (y) and |ξ(x) − ξ(y)| ≤ ,
(1)
where | · | is the taxicab (L 1 ) norm, and the minus sign is interpreted for the subtraction of vectors. Since the diameter of the k-dimensional hypercube (or k-cube Qk ) is k, it is meaningful for to take values in {1, . . . , k}. The notable difference between (ξt ) and the well-known model for the diffusion of cultures by Axelrod (1997) is the introduction of the parameter , which drops the assumption of homophily, that is, more similar types tend to interact more frequently (cf., [7, p. 208]). In Section 3, we discuss the application of the present study to the one-dimensional Axelrod model with k features and two traits. Under the transition mechanism described above, we realize that ξ(x) converges towards ξ(y) in a minimal sense, for neighbor x, y. This can be clarified by rewriting the interaction mechanism as, at each time, a randomly chosen x picks a random neighbor y, and assumes the (possibly random) type
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(2)
This can be called as subattractive dynamics, and it extends the notion of attractiveness in interacting particle systems ([2], Definition 2.1, p. 134). Whenever |ξ(x) − ξ(y)| ≤ 1 for some x and all neighboring y, interactions are attractive, i.e., x is more likely to assume the type of one of its neighbors if it differs from most of its neighbors than it is if it agrees with most of them. Otherwise, it is distinguished from attractive/exchangeable voter dynamics in a manner that we try to demonstrate in this article. In contrast to disappointing, albeit generic results [8, 9], the introduction of a confidence threshold drastically enhances the dynamics in the particular model of [4], as well as in its spatial version which is presently examined. While investigating the convergent dynamics of (ξt ), it was tempting to do so in comparison to a corresponding process (ηt ) with the same parameters k, , characterized by attractive dynamics (conditional on bounded confidence). According to the new transition mechanism, η(x) becomes of the type c = η(y) at a rate proportional to the number of neighbors y of x, provided that (1) holds. Except for the modification of the convergent assumption (2) to the attractive one η(y) = c, the chance of interaction and the rest of the transition mechanism remains the same as that in (ξt ). Would (ηt ) emulate (ξt ) for all , with respect to their qualitative behaviors at the asymptotic limit, and with a higher rate of convergence to this limit than (ξt )? To answer this question, the present approach is based on the concepts presented in [10], that is, “simulations provide useful evidence for making conjectures that can then be tackled in hopes of proving new theorems,” or the use of synchronously updated cellular automata to model continuous-time Markov processes. On the basis of Ref. 10, we will show that a slight modification to the transition rule of (ξt ) yields the more computationally efficient (ηt ), which exhibits one-dimensional dynamics with identical qualitative behaviors for the largest range of their parameters. In addition, we show that (ηt ) is more mathematically tractable, by applying the definitions of qualitative behaviors and probabilistic methodology described in [1] to special cases. Thus, we jointly formulate a conjecture for both models. In the rest of this article, to compare the present study with the original literature [1] and [7], we assume that G = ⊂ Z is a finite and connected subset of the one-dimensional integer lattice. Conjecture 1 Let (ξt ) and (ηt ) be convergent and attractive alternatives, respectively, of the Axelrod model on ⊂ Z as defined previously, with k binary features and confidence threshold .
© 2011 Wiley Periodicals, Inc. DOI 10.1002/cplx
Starting from the uniform product distribution of 2k types, there are critical fix < flux in {1, . . . , k − 1}, such that:
FIGURE 1
A. If ≤ fix , then (ξt ), (ηt ) fixate (i.e., in the sense that cyclic systems may do so on Z). B. If fix < < flux , then (ηt ) fixates and (ξt ) fluctuates until absorption. C. If ≥ flux , then (ηt ) fluctuates until absorption and (ξt ) fluctuates until consensus. The stated above conjecture can be rephrased as follows. In the range of that the convergent (ξt ) fixates, the attractive (ηt ) supposedly fixates as well (conditional on bounded confidence). That is, for this range of , one process supposedly emulates the other, and vice versa. In the range of that the attractive (ηt ) fluctuates until absorption, the convergent (ξt ) supposedly fluctuates until consensus. In the intermediate range, a complementarity seemingly takes place, in the sense that the convergent process fluctuates, whereas the attractive one fixates (conditional on bounded confidence). The experiments show that this complementarity occurs for k ≥ 4. For k = 3, it seems that the two processes emulate each other with respect to fluctuation and fixation at the asymptotic limit. Note that, when = k, (ηt ) is the multitype voter model, also known as the stepping stone model [11]. In this case, it is well known that (ηt ) fluctuates on the multidimensional lattices Zd ; therefore, it fluctuates on multidimensional boxes in Zd until consensus is reached. (The d-dimensional n-box centered at 0 is defined by the tuples x = (x1 , . . . , xd ) in Zd such that, |xl | ≤ n, for 1 ≤ l ≤ d.) In addition, when = k − 1, (ηt ) fluctuates until absorption on multidimensional boxes in Zd (to either consensus or a bipolar configuration). Two sketches of proofs are presented as follows. One approach is to use an ergodic argument as in [1], Lemma 1, which would also require some graph theoretic notions of the hypercube [12] (Hamilton cycles, and the recursive construction of the hypercube). Another simpler approach, to show fluctuation for = k −1, is by noticing that (ηt ) is a (spatial) political positions process for special graphs of types. Then, one can use the concepts of embedded graphs and processes as in [13], Section 3, to show fluctuation when two vertices of the graph of types are not connected to each other, but are connected to all other vertices (which is, in fact, our case = k − 1). Moreover, the methodology in the original article on cyclic systems [1] applies elegantly to the one-dimensional (ηt ) in an example presented later (fixation for < k3 ). The presented argumentation, together with experiments described in the next section, strongly suggest that in one dimension, fix k2 and flux k2 .
1.1. Two Binary Features In this case, two binary features yield four cultural types, which are represented by a four-vertex cycle. On multidimensional boxes in Zd , this model fluctuates until absorption if
© 2011 Wiley Periodicals, Inc. DOI 10.1002/cplx
(a) Eight cultural types of the Axelrod model with three binary features are represented by a cube. If one removes the grey vertices, a six-vertex cycle is obtained. (b) The cycle with six vertices represents all strong orderings of the three alternatives A, B, and C .
= 1, and it fluctuates until consensus if = 2 (that is, only behavior C. may manifest on Zd ). In spite of its graph of types, which is cyclical, it does not exhibit the interesting local periodicity of basic cyclic systems (see, for instance [14]).
1.2. Three Binary Features In this case, behavior C. (fluctuation for = 2, 3) can be fully proven for (ηt ) on multidimensional boxes in Zd by the argument stated in the previous paragraphs. To prove behavior A. (fixation for = 1) on Z is an interesting problem because it requires the rigorous and unpleasant techniques for N = 5 types/colors described in [1]. Simplification can be achieved by the removal of two cultural types, as showed in Figure 1. It is a meaningful approach if one wants to consider rational agents. This is a typical hypothesis of the transitivity of individual choice followed in social choice theory and welfare economics (for further details and relevant definitions see, for instance [15]). Note that the resulting subgraph after vertex removal is a six vertexcycle, which represents all strong orderings of three alternatives (for the representation of the more general case of all weak orderings of three alternatives, see [16]). Under this cyclical representation of types, the arguments in [1] are directly applicable to the extent that strong evidence of fixation on Z (using block-averaging technique from probability theory) is obtained. There is no need to resort to the tedious methodology described in [1] for N = 5 types/colors. We will briefly refer to these probabilistic techniques in the next example because they are more straightforwardly applicable when k = 4, = 1.
1.3. Four Binary Features and Extension to k Binary Features In this case, on multidimensional boxes in Zd , behavior C. (fluctuation for = 3, 4) can be fully proven on the basis of previous arguments for (ηt ). Moreover, strong evidence can be obtained to show that behavior A. (fixation for = 1) manifests on Z under the
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dynamics of (ξt ) and (ηt ). Specifically, the relatively easily obtainable strong evidence of fixation is that active bonds cannot eliminate all blockades in a sufficiently large interval of Z. Active bonds are the edges connecting each x, x + 1 that satisfies (1) and ξ(x) = ξ(x + 1). Blockades are the edges connecting each x, x + 1 that does not satisfy (1). To understand this argument, it is useful for the reader to construct a graphical representation of the process (details for this technique are provided in particle systems literature; see for instance [2], Chapter III, Section 6, or [1]). If = 1, then a blockade of size |ξ(x)−ξ(x+1)| = 2 is eliminated after a possible collision with an active bond, which jumps to the edge of the blockade after the collision. For blockades of sizes |ξ(x) − ξ(x + 1)| = 3, 4, at least |ξ(x) − ξ(x + 1)| − 2 active bonds are needed to be sacrificed to remove the blockade. The tendency for the elimination of active bonds to remove blockades can be measured by a comparison function φ for each x: φ(x) = 0, if ξ(x) = ξ(x + 1), = |ξ(x) − ξ(x + 1)| − 2, otherwise,
lim P
n→∞
m
φ(x) ≤ 0 for some l ≥ n, m ≥ 0 = 0.
(3)
x=−l
Since |ξ(x) − ξ(x + 1)| are bounded and i.i.d. for each x, the summands of the event in (3) are bounded and i.i.d.. If k = 4, = 1, then E[φ(x)] =
4 6 4 1 1 0+ (−1) + 0+ 1+ 2 > 0. 16 16 16 16 16
Then, a large deviations estimate from probability theory shows that the probability of the event in (3) exponentially tends to 0 as n → ∞. This proves that under the dynamics of (ξt ) and (ηt ), active bonds cannot eliminate all blockades in a sufficiently large interval of Z, when k = 4, = 1. Let us remark that such an argumentation easily extends to k > 4, = 1. Owing to the particular symmetry of the hypercube, the above arguments extend straightforwardly for (ηt ) when k = 3 + 1, > 1. Note that, by recursive definition of the hypercube, the case k = 3 + 1 obeys certain ratios of Q4 with = 1; therefore, fixation is suspected in one dimension, which will be discussed at once. Recall that (ηt ) is taken from (ξt ) by changing convergent interactions to attractive ones (conditional on bounded confidence). The desirable property of (ηt ), which is absent in (ξt ) for > 1 and permits such
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if η(x) = η(x + 1), φ (x) = 0, |η(x) − η(x + 1)| = − 2, otherwise, which yields E[φ (x)] =
1 a b c 1 0 + k (−1) + k 0 + k 1 + k 2 > 0, 2k 2 2 2 2
with a = #{u ∈ Qk : 0 < |u| ≤ }, b = #{u ∈ Qk : < |u| ≤ 2}, and c = #{u ∈ Qk : 2 < |u| ≤ 3}. One can verify that a = c, which proves that E[φ (x)] is strictly positive, and this extends easily to k > 3 + 1. Therefore, by a large deviations argument under the dynamics of (ηt ), active bonds cannot eliminate all blockades in a sufficiently large interval of Z, if < k3 , k ≥ 4.
2. EXPERIMENTS
where ξ = ξ0 is the initial random configuration. The sum of φ over an interval is the number of active bonds necessary to remove all blockades in this interval, minus the number of all active bonds in this interval. Therefore, the relation x∈ φ(x) > 0 implies the lack of sufficient active bonds in to eliminate the blockades in . One can prove that
a straightforward extension, is that active bonds cannot be created. One can use the comparison function
In discrete-time simulations on blocks of the integer lattice Z, the conjectured phase transitions are visible to the eye for small k. In order to demean finite-size effects, so that the phase transition is more clearly visible, periodic boundary conditions were incorporated. Furthermore, instead of updating one site (coordinate) at each time, it is computationally efficient to employ a quasi-synchronous update. One can use a random cellular automaton [17, with both the initial configuration and the transition rule random, p. 2] (see also [18], Chapter 6) with a double clock in the following sense. Simultaneously, at odd times, all odd coordinates are updated, and at even times, all even coordinates are updated from this point on each updated coordinate interacting according to the transition rule employed for (ξt ). That is, the source of randomness in the new transition rule is a collection of ideal coin tosses (Bernoulli trials). Let us denote this new process with (ξ¯t )t=0,1,... defined on an appropriate probability space with a probability law assigning likelihoods to all possible trajectories of the process [17]. (ξ¯t ) is a novel and computationally efficient emulation of (ξt ), and it is different from the partial emulation of (ξt ) by the attractive (ηt ) (conditional on bounded confidence, and in the range of the parameters for which such emulation occurs). According to our experiments, (ηt ) is emulated by an (η¯ t ) with a double clock as described above. Figures 2 and 3 show simulations of (ξ¯t ) on 200 sites up to t = 1000 and t = 104 , respectively. One might think that if (η¯ t ) fixates for k = 3, = 1, then (ηt ) should fixate for all k, < k2 . A syllogism may be construed as follows. Since the hypercube is defined by recursive duplication of itself (i.e., Qk+1 is derived by duplicating Qk and joining their 1-1 corresponding vertices), the model with types in the hypercube with k = 2 + 1, which obeys certain ratios of Q3 with = 1, should fixate in a conjectural sense
© 2011 Wiley Periodicals, Inc. DOI 10.1002/cplx
FIGURE 2
FIGURE 4
Fixation of (ξ¯t ) on Z with three binary features and = 1. Time is increasing downwards on the vertical axis. Since = 1, the figure shows also a simulation of (η¯ t ).
Mean size of clusters versus on the one-dimensional torus with 100 sites (k = 20, the ensemble is 1000).
(since fixation is observed in experiments for k = 3, = 1). Then, one may also think that, since (ηt ) supposedly fixates on Z for k = 2 + 1, it should fixate for all k > 2, in analogy to similar cases described in this article (however, the comparison function φ(x) is missing). The emulation by parts of (ξt ) by (ηt ), and vice versa, is intuitive based on the mean size of clusters at absorption. On one-dimensional tori with 100 sites, ensemble-averaged over
1000 simulation runs, the mean size of clusters varies with , as shown in Figure 4. Figures 5–8 show the cumulative distributions of the sizes of clusters in simulations on one-dimensional tori with 100 sites, ensemble-averaged over 100 simulation runs. Fixation is observed for the values of for which the frequencies of cluster sizes decay according to a power law, in analogy to [19]. The critical fix is observed by a power law decay, together with high variance in the frequencies of larger clusters, in
FIGURE 5 FIGURE 3
Fluctuation of (ξ¯t ) on Z with three binary features and = 2. Time is increasing downwards on the vertical axis.
© 2011 Wiley Periodicals, Inc. DOI 10.1002/cplx
Convergent (conditional on bounded confidence) model, 10 binary features, 1x100 torus, ensemble 100.
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FIGURE 6
FIGURE 8
Convergent (conditional on bounded confidence) model, 11 binary features, 1x100 torus, ensemble 100.
Attractive (conditional on bounded confidence) model, 11 binary features, 1 × 100 torus, ensemble 100.
analogy to [4]. Fluctuation is observed by the high frequencies of large clusters, which are comparable to the frequencies of small clusters. Consensus is observed by a single point in the log-log diagram, at the maximum size of a cluster with frequency 1. Simulations were performed in a MATLAB programming environment; the developed MATLAB code is available from the corresponding author.
FIGURE 7
Attractive (conditional on bounded confidence) model, 10 binary features, 1 × 100 torus, ensemble 100.
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3. DISCUSSION We investigated the complex behavior exhibited by variants with a confidence parameter of the binary-featured Axelrod model, by understanding the concepts presented in the first and simplest model, which has been simultaneously studied in depth. Indeed, we show that the definitions of the fundamental behaviors of cyclic particle systems (fluctuation and fixation) can be applied to finite systems with caution, owing to ambiguous situations that may arise because of the finiteness of space. The examined variants of the Axelrod model, conditionally convergent and attractive on bounded confidence, seem to exhibit behavior analogous to one-dimensional cyclic systems with respect to fixation and fluctuation, as expressed by Conjecture 1. The present study applies to the one-dimensional Axelrod model with binary features (two traits) as well, because it is obtained from (ξt ) if = k − 1 after appropriate normalization of the transition rates. In the Axelrod model, the chance of interaction is proportional to the similarity between two neighbors, measured by the proportion of their equal features (homophily), and interacting sites tend to become more similar (convergence). In the dynamics of (ξt ), provided that (1) holds, interacting sites tend to become more similar (conditional convergence on bounded confidence). It is found that (ξt ) with = k − 1 emulates the Axelrod model, i.e., fluctuation until absorption occurs. Therefore, dropping the assumption of homophily does not affect the behavior in one dimension. Through such a critical discussion, we speculated that the examined convergent (ξt ) is emulated for small and large by the attractive (ηt ), conditional on bounded confidence.
© 2011 Wiley Periodicals, Inc. DOI 10.1002/cplx
This partial emulation is achieved with respect to fluctuation and fixation at the asymptotic limit, and with a higher rate of convergence to this limit. To this end, we provide rigorous preliminary results for the attractive model (strong evidence of fixation if < k/3, fluctuation if = k − 1, k). Moreover, it was interesting to observe that the two processes are complementary in the following sense. For not
too small k values, there is a narrow range of around k2 , within which the attractive (ηt ) fixates, whereas the convergent (ξt ) fluctuates (conditional on bounded confidence). Therefore, one can substitute the convergent with attractive/exchangeable interactions, without essentially affecting the behavior in one dimension for the largest range of their parameters.
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