Characterization of Fixed Points in Sequential Dynamical Systems

Transcription

1 Characterization of Fixed Points in Sequential Dynamical Systems James M. W. Duvall Virginia Polytechnic Institute and State University Department of Mathematics Abstract Graph dynamical systems are central to the modeling of a wide range of different phenomena on networks. As a particular type of graph dynamical systems, sequential dynamical systems (SDS) have numerous applications ranging from disease dynamics to the mapping of traffic flows. These applications also extend to modeling cellular automata (CA), which have a lot in common, structurally, with SDS. The goal of this paper is to thoroughly examine one of the most commonly studied graph structures, the circle graph, which is denoted by Circ n. Using some function f 3 : {0, 1} 3 {0, 1}, the process of characterizing fixed points over this graph is delineated and the specific outcomes for each result are then discussed. Specifically, this paper will examine a special case, the majority function which is denoted by maj 3, and the outcomes it produces when applied to Circ n. In general, computing these fixed points for an arbitrary graph is hard and computationally intractable. Therefore, the final portion of this paper will discuss the wheel graph, denoted Wheel n, and the impending complications that arise when attempting to compute its local fixed points. 1

2 2 1 Introduction Sequential dynamical systems are important in many areas of modeling and this paper gives a thorough basis into understanding them. The theory learned through the study of SDS can be easily assimilated into many applications which is a large part as to why one would study SDS. One of these relations is to that of cellular automata. CA may be thought of as a collection of colored cells on a regular grid of specified shape that evolves through a number of discrete time steps according to a set of rules based on the states of neighboring cells [7]. Cellular automata share many features with SDS. Each CA has some underlying collection of cells where each cell v has a state x v taken from some finite set and each cell has a function defined over a collection of states associated to nearby cells. An example of the dynamics of a CA is given in the figure below. There is one difference to note though, which is that while SDS always deal with a finite Figure 1. CA dynamics generated in Mathematica with 20 iterations. number of lattice systems, it is not uncommon to consider cellular automata over infinite lattices. There are a number of more specific ways in which CA compare to SDS but for a more specific background into them, refer to either [7] or [5]. The main case this paper deals with is the graph Circ n and how to classify its fixed points. These fixed points represent the long term behavior of a dynamical system which lends them to constant study. However, before they can be classified, a working knowledge of graph theory must be obtained and then the definition of a fixed point must be stated. Interestingly enough, there is a classical problem that relates to the computation of fixed points in the circle graph. Let there be n beads on a necklace where a bead can have one of 2 colors attributed to it. The problem is to find how many different ways the beads can be arranged using a few assumptions. These assumptions are that there can be neither rotations nor reflections and also that none of the beads can be isolated. As it will be seen for the majority function, it will turn out that the solution to this is the same as the number of fixed points in the circle graph. As the reader, try and figure out just how many arrangements can be made from 4 beads with 2 colors (say black and white) before proceeding. The reader may also finish the paper and then show to themselves that this is true by working backwards from the answer. This application can also be solved in another way by the use of Polya theory which provides a systematic approach as the number of beads and colors grows larger. For an introduction to Polya theory, see [1]. 1.1 Graphs A graph is a nonempty, finite set V along with a set E of 2-element subsets of V. The elements of V are called vertices and the elements of E are called edges [3]. A walk (of length k) in a graph Y is a non-empty, alternating sequence v 0 e 0 v 1 e 1...e k 1 v k of vertices and edges in Y such that e i = {v i, v i+1 } for all i < k. The walk is said to be closed if v 0 = v k. A path in a graph Y is a non-empty graph P = (V, E) of Y of the form V = {x 0, x 1,..., x k } E = {x 0 x 1, x 1 x 2,..., x k 1 x k } where the x i s are all distinct [2]. In other words, a path is a walk that never visits the same vertex twice. If P = x 0...x k 1 is a path and k 3, then the graph τ := P + x k 1 x 0 is called a cycle. A non-empty graph Y is called connected if any two of its vertices are linked by a path in Y [2]. Using this, one could also say that

3 3 a cycle is a connected subgraph where every vertex has precisely two neighbors. Before citing and proving a theorem that will be useful in the latter part of this paper, a little needs to be said of the Transfer-matrix Method. The fundamental problem treated by the transfer-matrix method is the evaluation of A ij (n), the number of walks from i to j of length n, where the first part of the method is to interpret A ij (n) as an entry in a certain matrix. This matrix A is the adjacency matrix which is simply a matrix that is formed by placing a 1 in the ij th entry if there is an edge between i and j and a 0 if there is not. The second part of the method involves the use of linear algebra to analyze the behavior of A ij (n) [6]. The first part of the transfer-matrix method is the one that fits best into this paper with the following theorem being a direct result from this part. Theorem 1. Let χ be a directed graph with adjacency matrix A. The number of walks from u to v in χ with length r is (A r ) uv [4]. Proof. Base Case: Let r = 1, then A r = A 1 = A. When r = 1, it follows that A r = A. Since the matrix A already lists the walks in the directed graph χ of length 1, the case r = 1 is true. Inductive Assumption: Assume that the number of walks from u to v in χ with length k is (A k ) uv and that this is true for k 1. Inductive step: Let the number of walks of length r + 1 from u to v be denoted by Γ uv r+1. If w represents a possible intermediate vertex between u and v, then: Γ uv r+1 = Γ uw r Γ wv 1 wɛv = wɛv (A r ) uw (A) wv (by the inductive hypothesis) = (A r+1 ) uv (by the definition of matrix mulitplication) Therefore by the principle of mathematical induction on r, the number of walks from u to v in χ of length r is (A r ) uv for r What is an SDS? Let Y be an undirected graph of order n with vertex states in a finite set or field K. Let x = (x 1,..., x n ), x[v] be a restricted state denoting the state of v and its neighbors, and also let f v be a vertex function which takes x[v] as its input. Let (f v ) v V be a family of vertex functions, and let π = (π 1,..., π n ) S Y where S Y denotes the set of permutation sequences of V. A sequential dynamical system (SDS) is the triple (Y, (f v ) v V, π). Here, F v : K n K n is the local function and is defined by (1) F vk (x) = (x v1,..., x vk 1, f vk (x[v k ]), x vk+1,..., x vn ), where each of the x vk vertices is fixed and the only thing that gets changed is the state of vertex v k with f vk (x[v k ]) applied to it. The associated SDS-map is F π : K n K n defined by (2) F π = F πn F πn 1... F π1. The graph Y of an SDS is refered to as the base graph. The application of the Y -local map F v is the update of vertex v, and the application of F π is a system update. For certain SDS, the update order is a word w = (w 1,..., w k ) over V, that is, a sequence of Y -vertices. An SDS over a word is defined in almost the exact same way as in Equation 2 except each π is replaced with w. Note that, for future reference, when defined over a permutation sequence, equation (2) is referred to as a permutation-sds and when defined over a word, is referred to as a word-sds [5]. 2 The Circle Graph 2.1 Fixed Points for an SDS over the graph Circ n

4 4 Figure 2. The Circ n graph. For SDS, when the system update F π is applied, system state transitions are generated and these form a directed graph Γ called a phase space which is defined as v[γ] = {x K n }, e[γ] = {(x, F π (x)) x v[γ]}, where v[γ] and e[γ] denote the vertex set of Y and the edge set of Y, respectively. There are three types of points in the phase space that are of interest. First, there is the fixed point. For an SDS, a fixed point is a point x such that F π (x) = x and once a system reaches a fixed point, it remains there. Second is the periodic point. Periodic points are states x such that F k π(x) = x is satisfied. In other words, these points are simply ones that no matter how many update schemes are applied to the graph, they begin at one vertex and come back to this same vertex having a cycle of length greater than 1. Periodic points continue to map to other points and after a certain amount of steps, will form a cycle that repeats itself as the number of update schemes increases. Lastly, there are transient points which are points that are neither fixed points nor periodic points. The main problem that is the focus of the paper is as follows: find the number of fixed points in the graph Circ n using f 3 : {0, 1} 3 {0, 1} and characterize them. The definitions for the phase space and the three types of points it may contain will now be shown through an example. Before this can be done though, an update scheme needs to be defined which is necessary to construct the phase space. A sequential update scheme is the permutation sequence π = (π 1, π 2,..., π n ) specifying the order in which the vertex functions are applied and recall that the SDS map is defined as F π = F πn F πn 1... F π1 from equation (2) above. Consider Circ 4 along with the function maj 3 : {0, 1} 3 {0, 1}. The set of the function, {0,1}, attributes a 0 or 1 to each vertex on Circ n creating a total of 2 n different graph states. Note that since this problem deals with Circ 4, there are a total of 2 4 = 16 different system states. A starting vertex is chosen based on π and one examines this vertex along with its neighbors. Then one checks whether 0 or 1 is the majority and the function output is this majority. Since a sequential update scheme is being used, after finding the output of maj 3 : {0, 1} 3 {0, 1} applied to the starting vertex of Circ 4, one replaces the original state with the value of the function output. Repeat this for the states of the remaining 3 vertices to obtain the transition for that graph state. The 16 transitions are noted as follows in Table 1 below. The sequential phase space Table 1. The graph state transitions for a Majority SDS over Circ n where π = (0, 1, 2, 3) corresponding to these transitions is given in Figure 3 below. From the definitions above, one can use Figure

5 5 Figure 3. The phase space generated from Table 1 3 to identify which points in the graph are fixed points, periodic points or transient points. In Figure 3, the nodes (0000, 1111, 0011, 1100, 1001, 0110) are the fixed points since they map to themselves. The other points in the graph are all transient points. From Figure 3, these points would be (0001, 0100, 0010, 1000, 1101, 1011, 1110, 0111, 0101, 1010). This SDS does not exhibit any periodic points because the phase space does not contain any cycles of length greater than The Fixed Point Graph Now that the concept of a fixed point has been defined, the problem of finding and characterizing fixed points can be solved. To do this, it is first beneficial to construct local fixed points where these are constructed by first examining all of the possible neighborhoods in the graph. The idea of local fixed points is to patch them together to form global fixed points for the full SDS and in order to do this, they need to be compatible wherever they overlap. Going back to the graph Circ n, it is seen that every vertex has two neighors for a total of 3 vertices per neighborhood. Under the function maj 3 : {0, 1} 3 {0, 1}, there are two states per vertex which allows us to construct 2 3 = 8 local fixed point candidates. The idea is to then apply the function to these candidates to find out which ones are local fixed points [5]. To determine this, our local fixed point candidates are set up in the form (x i 1 x i x i+1 )(which is a common form when working with Circ n ), the values in the tuple (either 0 or 1) are examined and as before, whichever value is the majority is the output. Lastly, to determine if the tuple is a local fixed point, simply check the output of the function and if that value is equal to x i, then the point is a local fixed point. The table below (Table 2) labels which of the tuples are local fixed points. As seen in Table 2, there are 6 local fixed points for Circ n under the Table 2. Local fixed points x i 1 x i x i+1 maj 3 local fixed point? Yes Yes No Yes Yes No Yes Yes function maj 3. Using these, a directed graph (call it G) can be constructed in the following manner. Look at the first local fixed point, 000. Take the last two digits of this local fixed point and see which other local fixed

6 6 points begin with those two digits. Then draw a directed edge from 000 to those two points. Specifically in this case for 000, there is a directed edge to 001 and to itself, 000. This process is done for all the local fixed points and a directed graph will result. For maj 3, the directed graph is given in Figure 4. Figure 4. The directed graph G for the maj 3 function. Using Figure 4 above, an adjacency matrix (A) can be constructed to describe G: A = By definition, a cycle of length n in G corresponds to a fixed point of Circ n for a given function f 3 [5]. Using this definition in addition to Theorem 1 that was proved earlier, the total number of fixed points can be calculated in the following manner. Using the running example graph Circ 4, apply Theorem 1 and raise A to the fourth power which will correspond to the number of walks from i to j of length 4. The following matrix results: A 4 = Now it must be that the fixed points occur along the diagonal because a fixed point must begin and end at the same point, making these diagonal entries the ones of interest. In general, the total number of fixed points is the sum along the diagonal of A n or, in other words, Tr A n. Thus, for Circ 4 where n = 4, the total number of fixed points is Tr A 4 = = 6. This value matches the fixed points we observed in the phase space for Circ 4. In the phase space, it was noted that there were 6 fixed points and using this method, the same answer is achieved. 2.3 Recursive Enumeration of Fixed Points

7 7 From the last section, it was found that over Circ n, the total number of fixed points, L n, induced by the function, f 3 : {0, 1} 3 {0, 1}, was: (3) L n = Tr A n, where A was the adjacency matrix of the graph G. The following theorem is a consequence of this result. Theorem 2. Let χ A (x) = a i x k i = 0 be the characteristic polynomial of A. The number of fixed points L n satisfies the recursion relation (4) a i L n i = 0, where the a i s are the coefficients to the characteristic polynomial [5]. Proof. Using Equation 3, the last part of it can be written in the following manner: (5) L n = Tr A n = [A n ] ii = e i A n e T i, i=1 where e i is the ith unit vector. Through the use of Equation 5 and the result of the Hamilton-Cayley theorem that states that χ A (A) = 0, Equation 4 is shown to be true from the following: a i L n i = a i ( e j A n 1 e T j ) = = = = 0 j=1 j=1 ( e j a i A n 1 e T j ) i=1 e j (a 0 A n + a 1 A n a k A n k )e T j j=1 e j χ A (A)A n k e T j j=1 The total number of fixed points for Circ n can now be defined recursively. The characteristic polynomial of A is found to be x 6 2x 5 + x 4 x 2. Applying Theorem 2, the recursion relation becomes L n 2L n 1 + L n 2 L n 4 = 0. Bringing the last three terms to the right side, the recursion relation is found to be: (6) L n = 2L n 1 L n 2 + L n 4, with initial values L 3 = 2, L 4 = 6, L 5 = 12, and L 6 = Problems that Arise with other SDS Graphs One might try to see if these same techniques that have been developed can be applied to other graph classes to obtain a nice, closed form solution for the enumeration of fixed points. For a few special graph classes, the same techniques can be applied to generate such a solution. However, for most graph classes, a closed-form solution can not be obtained and to demonstrate some of the issues that arise when trying to do so, the wheel graph (Wheel n ) is used. Structurally, Wheel n has a few things in common with Circ n but it is the differences that matter. Most notably, Circ n is a regular graph while Wheel n is not and note that a

8 8 Figure 5. The Wheel n graph. regular graph is defined as a graph such that every vertex has the same degree (number of incident edges) [5]. Every vertex of Circ n has degree 2 while in Wheel n, the vertices along the circle all have degree 3 while the center vertex has degree n. This forces the state of the center vertex to depend on every other vertex while the states along the circle depend on the states of the neighbors on the circle (as is the case in Circ n ) and also the center vertex itself. This is going to yield two neighborhoods of different sizes, the neighborhood about the center vertex which contains n + 1 vertices and the neighborhood about all the other points which have 4 vertices. Relating this to the case of the circle graph, the local fixed point candidates for the wheel graph can be set up in the following manner: (x 0 x i 1 x i x i+1 ). Note that in this setup, the right side is the same as how the local fixed points were determined in Circ n. Thus, if x 0 is the center vertex and it is taken to be a dependent variable which the right half is based off of, then the result will give two different local fixed point graphs. One of them is the same as the one obtained for Circ 4 while the other is much different. As it turns out, this other graph only yields 1 fixed point, the one where every vertex has the same state. The fixed point graph that is the same as the circle case is evaluated in almost the same way as the original except for the fact that after the graph is obtained, each vertex has to match up with the center vertex. In other words, there is an algebraic constraint placed on the system over the wheel graph. This constraint is that f 0 (x) = x 0 which says to take the function at vertex zero, apply it, and see if it equals the state at vertex zero. If this is satisfied, it will give a local fixed point at vertex zero which is necessary for the entire system to have global fixed points. This creates a problem in finding a way to determine the fixed points and it turns out that the fixed points can only be found by cranking them out one by one. As a result, the total number of fixed points will be the union of the fixed points of the two separate graphs while being subjected to an algebraic constraint. References [1] Daniel I. A. Cohen. Basic Techniques of Combinatorial Theory. John Wiley and Sons, Inc., New York, [2] Reinhard Diestel. Graph Theory. Springer Science+Business Media, Berlin, third edition, [3] John A. Dossey, Albert D. Otto, Lawrence E. Spence, and Charles Vanden Eynden. Discrete Mathematics. Pearson Education, Inc., Massachusettes, fifth edition, [4] Chris Godsil and Gordon Royle. Algebraic Graph Theory. Springer-Verlag New York, Inc., New York, [5] Henning S. Mortveit and Christian M. Reidys. An Introduction to Sequential Dynamical Systems. Springer Science+Business Media LLC, New York, [6] Richard P. Stanley. Enumerative Combinatorics, volume 1. Wadsworth and Brooks/Cole Advanced Books and Software, California, [7] Eric W. Weisstein. Cellular automaton.