Direct Counting Analysis on Network Generated by Discrete Dynamics
aa r X i v : . [ phy s i c s . d a t a - a n ] A p r Direct Counting Analysis on Network Generated by Discrete Dynamics
Shu-Hao Liou ∗ and Chia-Chu Chen † Department of Physics, National Cheng Kung University, Tainan, Taiwan 70101 (Dated: August 28, 2018)A detail study on the In-degree Distribution (ID) of Cellular Automata is obtained by exactenumeration. The results indicate large deviation from multiscaling and classification accordingto ID are discussed. We further augment the transfer matrix as such the distributions for morecomplicated rules are obtained. Dependence of In-degree Distribution on the lattice size have alsobeen found for some rules including R
50 and R PACS numbers:
Discrete dynamical system in general can generate verycomplicated dynamics even though the rule or the equa-tion of evolution for the system can be apparently simple.Examples include the synchronization of Pulse-coupledOscillators, chaos arises from discrete mapping and moreinterestingly the simple Cellular Automata (CA) intro-duced by von Neumann almost half a century ago [1].One of the important problems in discrete dynamical sys-tem is the classification of it’s general behavior. Gener-ally speaking this is a hard question to be answered sincethe classification requests a full understanding of the dy-namics itself. Even for simple system such as CA whichevolves according to simple rules, a complete classifica-tion is still controversial since the work of StevenWolframin the 80’ of last century [2]. Recently, A. Shreim et al .have approached the CA problem by employing conceptsoriginated from network analysis [3]. They classified CAby analyzing the local and global properties of the net-work generated by the dynamics of CA. Their approachcertainly provides new way to address this interestingand difficult problem. In this work we follow up theirapproach and investigate further on this problem. Herewe would only concentrate on the local analysis and willreturn on the global property later.For any discrete dynamical system, the configurationspace can be represented by discrete nodes. The timeevolution of the system in each time step is presented bya connected link starting from one node to its dynamicalsuccessor. As a result, all the trajectories of the dynam-ical system generated a directed network. Since CA aredeterministic systems, each node possess one single out-going link. However, due to the existence of fixed pointsand periodic solutions, the dynamics of CA in general isirreversible and the number of preimages of a state canbe larger than 1. The in-degree (ID) of a node is de-fined as it’s number of preimages and obviously in CAthe in-degree is a local property. In [3] the authors haveperformed analysis on the ID distribution for various onedimensional CA both analytically and numerically. Dueto the simplicity of Rule 4 ( R T ) approach together with combinatory analysis onpreimages, they obtained an approximation solution ofthe probability distribution P ( k ) with k denoting the in-degree. Their results were presented on the log − log plotas curves instead of straight lines and seem to suggest the existence of multiscaling distribution. In our work,instead of using approximation, we re-analyze this prob-lem by performing direct counting of the in-degree forvarious rules with different lattice size L . What we havefound is that the distribution is not as simple as whathave been obtained in [3]. The plan of this report is asfollows. A brief summarization on nomenclatures of CAand network are given and the results on ID of [3] arebriefly discussed in section I. The direct counting resultsof various rules are reported in section II where some fur-ther studies on the ID are also treated. The extension oftransfer matrix for more accurate analysis on other rulesand its implication are treated in Section III. A briefsummary of our findings is provided in the final section. I. INTRODUCTION
A one-dimensional Cellular Automaton is defined ona spatial lattice of L sites where L can either be finiteor infinite. In this work we will keep L finite and thestate of each site is in one of the g states at any time t . Each site follows the same prescribed rules for up-dating. For elementary CA, the number of neighbor-hood of each site is 2. The state of CA starts out witharbitrary initial configuration which is represented by S (0) = { s , s , s , . . . , s L } where s i can be in any oneof the g states. The configuration of system at time t is denoted by S ( t ) = { s t , s t , s t , . . . , s tL } . In this workthe state of each site is restricted to g = 2 such that s i ∈ { , } . Let R be any one of the CA, the value of s i at t + 1 time step is set equal to R ( s ti − s ti s ti +1 ) whichis equal to 0 or 1 according to the rule of interest. Ap-parently for elementary CA there are 256 rules. Follow-ing Wolfram’s notation each rule can be assigned with anumber given by: R (000) + 2 R (001) + 2 R (010) + 2 R (011) + 2 R (100)+2 R (101) + 2 R (110) + 2 R (111) . For example, Rule 237 ( R R (100) =0, R (001) = 0 and all others R ( s s s ) = 1. Basedon large number of numerical studies, Wolfram has sug-gested that CA can be classified into four classes [2]. Cel-lular Automata within each class have the same qualita-tive behavior. Starting from almost all initial conditions,trajectories of CA become concentrated onto attractors,the four classes can then be characterized by their at-tractors. According to Wolfram’s classification, classes I,II and III are roughly corresponding to the limit points,limit cycles and chaotic attractors in continuous dynam-ical systems respectively. More precisely their respectivelong time limits are: (I) spatially homogeneous state,(II) fixed (steady) or periodic structure and (III) chaoticpattern throughout space. The fourth class of CA be-haves in a much more complicated manner and was con-jectured by Wolfram as capable for performing universalcomputation. For finite one-dimensional CA, the latticeis arranged on a circle with periodic boundary conditions.Such Cellular Automata have a finite number of states N = 2 L , and as a result, after sufficient long time evo-lution the system must enter the state which can eitherbe homogenous, steady or periodic state. Therefore, theclass III and IV attractors do not exist in finite CA. How-ever, it is also known that classes III and IV are hard todistinguished in some cases and resulting with disputedclassification. In some sense this is due to the lack of asensible way for defining complexity which is the crite-rion for class IV. In [3] A. Shreim et al . studied networksgenerated by CA, they claimed to have found highly het-erogeneous state space networks for classes III and IV incontrast to the networks generated by classes I and II.One of the characteristic of state space is the in-degreedistribution P ( k ) which is defined as [3] P ( k ) = Σ N . (1)where L is the size of the lattice, N = 2 L is the total num-ber of state and Σ is the number of state with in-degree k .However in their analysis on the local properties, approx-imation has been applied in the analytical calculation ofID which shows nontrivial scaling. For completeness webriefly summarize their method of calculation in this sec-tion. They introduced the transfer matrix T which mapseach pair s i − s i onto the pair s i s i +1 , and such T ( s ) couldbe used to characterize all the preimages of hubstates (Ahubstate is the state which has the maximum in-degree k m .) The rows and columns are order as “00”, “01”, “10”and “11” accordingly. For example the T matrix of R T = . (2)It is easy to check that the maximum in-degree k m isrelated to T as k m = T r ( T L ) . (3)The scaling of k m with the lattice size can be obtained byusing the largest eigenvalue λ m of T , namely, k m ≈ N ν with ν = log λ m . Furthermore, for R n as the number of isolated 1 ′ s , for example two config-urations (010000) and (010100) have n = 1 and 2 re-spectively. Then by assuming a one-one correspondingrelation between k and n , the following expression wasproposed for the in-degree distribution: P ( k ) dk = Ω( n ) dn L . (4)A multiscaling result is then obtained and given by thefollowing expression [3]: y = − − x + log (cid:20) (1 − ǫ ) − ǫ ǫ ǫ (1 − ǫ ) − ǫ (cid:21) . (5)where y = log P ( k ) / log N , x = log k/ log N and ǫ = n/L .The result of Eq. (5) was ploted in [3] and are apparentcurvature appeared on the curve as such a conclusion ofnultiscaling method than finite-size effect was proposed.Similar results of multiscaling in other rules were alsoreported in their article by using the same approxima-tion approach. To compare the results of [3] with exactcalculation we have performed the exact enumeration of R L . For the lack of space, in Table Iwe only list all the in-degree’s for L = 12. The samecharacteristic also exist for other values of L . One no-tices that for the same k there corresponds more thanone value of n . For example, for k = 12, n can either be2 or 3. Therefore the meaning of dn/dk is ambiguous.One might argue that the multi-valueness of n is an arti-fact of small L . However we have checked with L = 10 where different states with the same k were found. Infact, for k = 1, states of different n can be constructedeasily. For example, for k = 1, we have found at least2 states (000100010001 . . . ) and (010101 . . . ) which cor-respond to n = 2500 and 5000 respectively. Since theresults were obtained for small k , there is a possibilitythat dn/dk might be well-defined for large k . To refutethis reasoning, we have performed the large k analysisfor different L . For L = 12, k m = 852 with n = 0 andthe second maximum in-degree k m = 114 with n = 1;even in L = 25, we got k m = 1276941 with n = 0 and k m = 170625 with n = 1. The results indicate that for △ n = 1 the △ k = k m − k m is not a small quantity.In fact △ k rises exponentially as L getting larger. As aresult, dk can never be infinitesimal implying the mean-ingless of dn/dk . Thus the simple multiscaling results in[3] is in doubt.The result of Eq. (4) is presented by the solid line inFigure 1. By substituting the corresponding k m in Eq.(5), one has y = − .
8. For R L = 25, the resultsof n − k relation are provided in Figure 1. There is onlyone hubstate with n = 0 in such system, consequentlythe exact P ( k m ) equals to 1 / L [4] which implies y = − L instead of − . TABLE I. The relation of in-degree k and n for R L = 12. number of number of k n states k n states852 0 1 7 3 24114 1 12 4 3 4837 2 12 4 1221 2 12 2 3 2416 2 6 4 314 2 12 1 3 414 2 12 4 5712 2 12 5 363 12 6 2 -2 0 2 4 6 8 10 12 14 160123456789101112 n log k Rule 4 L=25 FIG. 1. The relation of n and in-degree k for Rule 4. Thesolid curve represents Eq. (6) in [3]. II. NETWORK CHARACTERISTICS BYDIRECT ENUMERATION
As suggested in [3], local properties such as the scalingof maximum in-degree with lattice size and the in-degreedistribution might be useful for characterizing networksand hopefully also provide some help to classify CA. How-ever, from previous discussions it is clear that approxi-mation calculations can lead to inadequate conclusionssuch as multiscaling of ID being an example. In order tohave a better understanding on this kind of network, it isunavoidable to address the problem with exact enumera-tion which will be the approach adopted in this work. Wehave studied more rules from different classes suggestedby Wolfram [2] and the in-degree distributions are eval-uated exaclty by direct counting method. The transfermatrix approach proposed in [3] has been used in thiswork extensively.It is important to note at the outset that not all rulescan have a transfer matrix representation. The followingmatrix represents possible candidate of T for elementary TABLE II. The classification for different rules with λ m . λ m ν Rules1.32472 0.4056 110,621.46557 0.551 232,33,61.61803 0.6942 36,18,51.75488 0.811 4,322 1 0,255
CA: T = a a a a a a a a . (6)where a i , i = 1 ∼
8, can take on either 1 or 0. In factthere are totally 256 different matrices represented by T ,however it is also known that not all of them can rep-resent the 256 elementary CA. This is partly due to thefact that not all rules can have a T matrix representa-tion. In particular, by diagonalizing all possible T , wehave obtained only 9 different λ m : 1, 1 . . . . . . . λ m are listed inTable II.In what follows, all ID are evaluated by direct count-ing. To set the stage we present the in-degree distri-butions of various rules with different L in Fig. 2, theresults of R R
22 and R
110 are plotted. The results ofeach rule seem to show the same distribution for threedifferent values of L . Especially for small k the distri-butions are in good agreement for different L with allthe points fall on a simple curve with non-vanishing cur-vature. As k increases, the distribution becomes morespread out but still maintains the same distribution forvarious L . These observation seems to suggest that theID of this kind of network is scale independent. More-over, since these rules belongs to different classes and thescale independent of the distributions seem to be univer-sal for all four classes of CA. Similar results can also beobtained for other rules and will not be shown explicitly.Even though the L independence of ID is very sugges-tive, however a definite answer would require a full scaleanalysis with larger lattice size. Work along this line isnow being studied and will be reported separately. Ona different front, with knowing all the possible values of λ m , it is natural to wonder if rules with the same λ m are correlated. To show whether such correlation existsor not, results of the same ν = log λ m are plotted inFig. 3 and 4. For ν = 0 . R R R R R
124 and R
54 are given in Fig. 3, all ofwhich show similar distribution. Interestingly these rulesbelong to the same class, namely class four according toWolfram’s scheme. For clarity, in Fig. 4 we have alsoplotted the ID of other rules of class IV. They also sharethe same feature as given in Fig. 3. Similarly, the re-sult for other are given in Fig. 5(a) with ν = 0 . ν is not conclu-sive. Furthermore rules from different classes can havethe same ν . For example, the plot given in Fig. 5(b) forthe in-degree distributions of various rules correspondingto ν = 0 . R
36 and R R
18 and R
126 are classified as class III [5]. To make things worse,the plot for various rules of class II is given in Fig. 5(c),where apparently the patterns get more diverse. Moreimportantly, those rules belong to different ν ’s. As a re-sult, one might conclude that there are good correlationfor class IV which possess ν = 0 . ν can not be used as a indicator forclassification. Thus the local properties such as ν andin-degree distribution are not useful for CA classificationas suggested in [3]. III. AUGMENTATION OF T MATRIX
The T matrix approach for calculating maximum in-degree is quite efficient and there are also cases in whichthe in-degree can also be evaluated exactly. However, asclearly stated in the previous section that 4 × T matrixdo not always exist for elementary CA and therefore it isinteresting to see how modification on the transfer natrixscheme can be extended to other rules. In this section anextension is suggested by augmenting the 4 × × s ti − s ti ontothe pair s ti − s ti , e T will map s ti − s ti − s ti into the triplet s ti − s ti s ti +1 and hence define a 8 × e T matrix which couldbe used to describe the preimage more efficiently. Thebasis states are | i i ∈ { (111) , (110) , (101) , (100) , (011) , (010) , (001) , (000) } . Fig. 6 shows the definition of 8 × e T matrix.There are some merits in this approach. First of all,for elementary CA where all the rules of evolution involvethree adjacent cells, obviously such extension certainlyrespect the evolution dynamics and hence include moreinformation of the dynamical system. Secondly, the aug-mentation of T indeed resolves some of the cases whichcan not be done with 4 × × × × × l og P ( k ) / l og N log k / log N L=15 L=20 L=25 (a) l og P ( k ) / l og N \ log k/ log N L = 15 L = 20 L = 25 (b) l og P ( k ) / l og N log k / log N L = 15 L = 20 L = 25 (c)
FIG. 2. The direct counting results of ID for (a) R
4, (b) R R
110 with different L . (a) R193 R147 R110 l og P ( k ) / l og N log k/log N R137 R124 R54 l og P ( k ) / l og N log k/log N(b) FIG. 3. The In-degree Distributions for (a) R R R
110 and (b) R R
124 and R
54 with L = 25. l og P ( k ) / l og N log k/log N
110 54 73 62(a)
FIG. 4. The In-degree Distributions for R R R
73 and R
62 with L = 25. matrices can be very much different and there is no obvi-ous reason for them to be the same. For all the cases wehave studied the maximum in-degree can be exactly cal-culated by both 4 × × k m with the e T matrices for R . Fig.7 is the evolution rule of R , and the 4 × T matrix l og P ( k ) / l og N log K/log N 33 232 77 6(a) -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-1.0-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2 l og P ( k ) / l og N log k/log N 18 5 126 36(b) R4 R50 R218 l og P ( k ) \ l og N log k/log N(c) FIG. 5. The in-degree distributions for various rules with (a) ν = 0 .
551 and (b) ν = 0 . ν = 0 .
811 for R ν = 0 . R
50 and ν = 0 . R L = 25. obtained in Amer Shreim et al. ’s study [3] is: T = . (7)The hubstate of R
18 is (00 . . . s t6 s t5 s t4 s t3 s t2 s t1 s t0 s t+16 s t+15 s t+14 s t+13 s t+12 s t+11 s t+10 FIG. 6. The diagram for 8 × e T matrix. State
Rule FIG. 7. The diagram of R of (00 . . . e T : e T = . (8)The in-degree of hubstate is given by k ( S ) = T r ( T L ),with L = 10 one has: T = . (9)and e T = . (10)It is clear that the traces of these two matrices are thesame and resulting the same value for the maximum in-degree k m .We have also obtained the 8 × e T matrix for morethan 22 rules and also rules related to them by symme-try( by exchanging 0 ↔ R
94, all these rules possess both T and e T which results with the same k m . Further appli-cation of this augmentation procedure is illustrated by TABLE III. The configuration of hubstates in R
43 with dif-ferent L . one of the number of L k m hubstates hubstates10 5 0000110011 4011 8 00100110100 2212 18 001100110100 413 13 0001100110100 2614 13 00001100110100 54 the example of the majority rule with more neighbor-ing cells included in the evolution. The evolution rule is R ( s ti − s ti − s ti s ti +1 s ti +2 ) = s t +1 i where s t +1 i = q ti > q ti < s ti q ti = 0 , (11)and q ti = P µ =2 µ = − s ti + µ . The T matrix for this rule is a32 ×
32 matrix and the maximum in-degree k m is 1414for L = 15. From this example is is also apparent that,in general for the evolution rule involves 2 n neighboringcells, the basic T matrix should be defined by 2 n +1 × n +1 matrix.From the above discussion, one can see that with 8 × e T matrix, more rules can be treated within this approach.As a consequence, such extension of T into larger matrixseems to provide a better description for network anal-ysis. Unfortunately such approach can still not be ableto cover all elementary CA. There exists rules in whichthe configurations of hubstates vary according to L beingeven or odd, such as R
50 and R
77. On the other hand,there are also cases in which T varies with L indefinitely,one of the good examples is R
43. We had analyzed R L and the result is listed in Table III. Thenumber of hubstates will change with L , and the periodof the attractors is large, therefore it is not appropriateto calculate k m by using T matrix. For these situations,there is no scale of k m with arbritrary L .For rules with the above irregularity special treatmentis required for each particular case. In the following wewill discuss R
50 and R
77 for illustration of their com-plication. The maximum in-degree of R
50 and R
77 canbe properly described by 8 × e T matrices only when L is even, since the in-degree distributions will not dependwith lattice length L .Fig. 8 shows the evolution according to R
50, and Ta-ble IV shows the relation between maximum in-degree k m and lattice length L . The hubstate of R
50 is either(0101 . . . . . . L is even.The k m of R
50 can be evaluated by k ( S L = even ) = 1 / T r [ e T L ],the factor 1 / × e T matrixcarries the information of all preimages of both hubstates,namely (0101 . . . . . . α ,all the preimages of L = α + 1 can be obtained by justappending “0” or “1” at the boundary to all the preim-ages of L = α . As a consequence, both even L = α and State
Rule FIG. 8. The diagram of R L in R
50 by direct counting. k m even L ν k m odd L ν
11 10 0.3459 11 11 0.314418 12 0.3474 18 13 0.320729 14 0.3469 29 15 0.323847 16 0.3471 47 17 0.3267 odd L = α + 1 have the same maximum in-degree k m .From these results one can see clearly that for even L k m follows scaling law with ν ≃ . L thereis no scaling behavior for k m . The same conclusion alsoappears in R
77 which is presentated in Table V.Rule 77 is treated in a different way. Fig. 9 is the evo-lution according to R
77. The network configurations arecompletely different for L being odd or even. For illustra-tion it is shown in Fig. 10 the networks for L = 4 and 5.Hence one would expect k m varies with L . The hubstatesof the R
77 are (0101 . . . . . . L is even, but k m of odd L is quite varying, as given inTable V. For R
77, where L = 10 we observed that thereexist only two hubstates where 22 hubstates are foundfor L = 11. Concentrating on one of L = 10 hubstates, itis clear that the hubstate of L = 11 is just adding 1 or 0to the end of L = 10’s hubstate. Furthermore, there aretotally 46 preimages for L = 10 and only 28 for L = 11.The 18 italic configurations with L = 10 do not associatewith the preimages in L = 11. From Fig. 11 it is knownthat only those preimages of L = 10 without (111) at theboundary for can associate to the preimages of L = 11.To evaluate the k m with odd L we follow the approachsuggested in [3] in obtaining arbitrary in-degree of R e T matrix itself an auxiliary matrixis constructed to resolve the difficulty. Therefore, by thisapproach we propose the following formula: k ( S L = odd ) = T r [ e T L − × e T multiplied ] . (12)This is due to the fact that the hubstates of odd L can beviewed as just adding “0” or “1” in the hubstates of even L . According to Fig. 10, we could assign the e T multiplied ,which just corresponds to adding “0” in the hubstateobtained by referring to Fig. 11, since the mapping atthe boundary of preimages with L = 11 has only two State
Rule FIG. 9. The diagram of R Pajek
Pajek
FIG. 10. The networks of R
77 with (a) L = 4 and (b) L = 5which are drawn by PAJEK.TABLE V. Maximum in-degree corresponding to latticelength L in R
77 by direct counting. k m even L ν k m odd L ν
46 10 0.552356 28 11 0.43703298 12 0.551226 60 13 0.454376211 14 0.551507 129 15 0.467415453 16 0.55146 277 17 0.477279
Preimages of (0101010101) with L=10
Preimages of (10101010100) with L=11
FIG. 11. The lists of pre-images of particular hubstates in R
77 with different L . TABLE VI. Summary for maximum in-degree calculation for R
50 and R R k ( S L = even ) = T r [ e T L ] k ( S L = odd ) = T r [ e T L − ] R k ( S L = odd ) = T r [ e T L ] k ( S L = odd ) = T r [ e T L − × e T multiplied ] possibilities: e T multiplied | > = | > . (13a) e T multiplied | > = | > . (13b)Therefore we have e T multiplied as e T multiplied = . (14)Then we can get the exact maximum in-degree of hub-state with any lattice length L . This analytical resultsagree with direct counting for L ≤
25. A summary of theformulas for the maximum in-degree of R
50 and R
77 aregiven in Table VI.In this section we have found more rules which couldnot be treated with 4 × T matrix can now be dealt withby 8 × IV. CONCLUSION
A detail discussion of the local characteristic of net-work generated by elementary CA is presented in thiswork. First we have located the possible source of er-rors in claiming the multiscaling of such network in [3].This is due to a misused of derivative on large difference.Secondly the in-degree distributions of many rules werecalculated by direct counting method and the multiscal-ing characteristic does not appear. From the studies upto L = 25, it suggests that the in-degree distribution ofCA is L independent regardless of its classes. Further-more all the possible largest eigenvalues λ m for 4 × T are obtained and analysis on the cor-relation between ν = log λ m and classification of CA isdiscussed. We have found that there might be more rulessharing the same in-degree distribution but correspond-ing to different classification according to S. Wolfram. Ithas been found that there are good correlation for classIV which possess ν = 0 . ν can not be used as an indicator forclassification. Therefore the in-degree distribution is nota proper characteristic to classification which was alsopointed out in [3]. In section III, the transfer matrix wasextended to 8 × e T matrix. This augmentation in somesense respects the evolution of CA rule and is also morenatural to reflect the characters of hubstates’ pre-images.As a result the 8 × × T matrix. The32 × T matrix has also applied to CA beyond neighborevolution. The L dependence of the hubstates of someparticular rules such as R
50 and R
77 are discussed indetail. The calculation of their k m are also treated inthis work. ACKNOWLEDGMENTS
This work was supported by National Science Councilof Taiwan, NSC-97-2112-M-006-003-MY2. ∗ [email protected] † [email protected][1] J. von Neuman, University of Illinois, Urbana, IL (1966).[2] S. Wolfram, Physica (Amsterdam) , 1 (1984). [3] A. Shreim et al. , Phy. Rev. Lett. , 198701 (2007).[4] M. E. J. Newman, SIAM Review (2003).[5] Y. Kayama et al. , Physics Letters A198