On the Rank and Periodic Rank of Finite Dynamical Systems

A finite dynamical system is a function f : An → An where A is a finite alphabet, used to model a network of interacting entities. The main feature of a finite dynamical system is its interaction graph, which indicates which local functions depend on which variables; the interaction graph is a qualitative representation of the interactions amongst entities on the network. The rank of a finite dynamical system is the cardinality of its image; the periodic rank is the number of its periodic points. In this paper, we determine the maximum rank and the maximum periodic rank of a finite dynamical system with a given interaction graph over any non-Boolean alphabet. The rank and the maximum rank are both computable in polynomial time. We also obtain a similar result for Boolean finite dynamical systems (also known as Boolean networks) whose interaction graphs are contained in a given digraph. We then prove that the average rank is relatively close (as the size of the alphabet is large) to the maximum. The results mentioned above only deal with the parallel update schedule. We finally determine the maximum rank over all block-sequential update schedules and the supremum periodic rank over all complete update schedules. Mathematics Subject Classifications: 05C38, 05C50, 15A03, 06E30

the electronic journal of combinatorics 25(3) (2018), #P3.48 The architecture of an FDS f : [q] n → [q] n can be represented via its interaction graph IG(f ), which indicates which update functions depend on which variables.More formally, IG(f ) has {1, . . ., n} as vertex set and there is an arc from u to v if f v (x) depends on x u .In different contexts, the interaction graph is known-or at least well approximated-, while the actual update functions are not.One main problem of research on FDSs is then to predict their dynamics according to their interaction graphs.However, due to the wide variety of possible local functions, determining properties of an FDS given its interaction graph is in general a difficult problem.
For instance, maximising the number of fixed points of an FDS based on its interaction graph was the subject of a lot of work, e.g. in [1,2,6,13,14].The logarithm of the number of fixed points is notably upper bounded by the transversal number of its interaction graph [2,14].This upper bound is reached for large classes of graphs (e.g.perfect graphs) but is not tight in general [14].Moreover, there is a dramatic change whether we assume that the FDS has an interaction graph equal to a certain digraph or only contained in that digraph (this is the distinction between guessing number and strict guessing number in [5]).
In this paper, we are interested in maximising two other very important dynamical parameters of an FDS given its interaction graph.First, the rank of an FDS f is the number of images of f .In particular, determining the maximum rank also determines whether there exists a bijective FDS with a given interaction graph.This is equivalent to the existence of so-called reversible dynamics, where the whole history of the system can be traced back in time.Second, because there is only a finite number of states, all the asymptotic points of f are periodic.The number of periodic points of f is referred to as its periodic rank.In contrast with the situation for fixed points, we derive a bound on these two quantities which is attained for all interaction graphs and all alphabets.In particular, there exists a bijection with interaction graph contained in D if and only if all the vertices of D can be covered by disjoint cycles.Moreover, we prove that our bound is attained for functions whose interaction graph is equal to a given digraph, and not only contained, for all non-Boolean alphabets.We then show that the average rank is relatively close (as D is fixed and q tends to infinity) to the maximum.
These results can be viewed as the discrete analogue to Poljak's matrix theorem in [11], which proves tat the maximum rank of M p , where M is a real matrix with given support D and p 1, is given by the maximum number of pairwise independent p-walks in D (see the sequel for a precise definition).However, our results extend Poljak's result for the discrete case in three ways (but Poljak's result cannot be viewed as a consequence of our results).Firstly, they hold for all functions, not only linear functions.Secondly, they explicitly determine the maximum periodic rank.Thirdly, the average rank of a real matrix cannot be properly defined, hence our result on the average rank of finite dynamical systems is completely novel.
The results mentioned above hold for the so-called parallel update schedule, where all entities update their local state at the same time, and hence x becomes f (x).We then study complete update schedules, where all entities update their local state at least once, and block-sequential schedules where all entities update their local state exactly once (the parallel schedule being a very particular example of block-sequential schedule).We then prove that the upper bound on the rank in parallel remains valid for any blocksequential schedule but is no longer valid for all complete schedules.We also determine the maximum periodic rank when considering all possible complete schedules.In particular, there exists a function f with interaction graph D and a complete schedule σ such that f σ is a bijection if and only if all the vertices of D belong to a cycle.
The rest of the paper is organised as follows.Section 2 introduces some useful notation and describes our results on the maximum (periodic) rank in parallel.Section 3 then proves our result on the average rank.Finally, the maximum rank and periodic rank under different update schedules are investigated in Section 4.
2 Maximum (periodic) rank in parallel 2.1 Background and notation Let D = (V, E) be a digraph on n vertices; let V = {1, . . ., n} be its set of vertices and E ⊆ V 2 its set of arcs.The digraph may have loops, but no parallel arcs.The adjacency matrix M ∈ {0, 1} n×n has entries m u,v = 1 if and only if (u, v) ∈ E. We denote the in-neighbourhood of a vertex v in D by When there is no confusion, we shall omit the dependence on D. This is extended to sets of vertices: N − (S) = v∈S N − (v).The out-neighbourhood is defined similarly.A source is a vertex with empty in-neighbourhood; a sink is a vertex with empty out-neighbourhood.The in-degree of v is the cardinality of its in-neighbourhood and is denoted by d v .
A walk w = (v 0 , . . ., v p ) is a sequence of (not necessarily distinct) vertices such that (v s , v s+1 ) ∈ E for all 0 s p − 1.A path is a walk where all vertices are distinct.A cycle is a walk where only the first and last vertices are equal.We refer to p as the length of the walk; a p-walk is a walk of length p.We say that two p-walks w = (w 0 , . . ., w p ), w = (w 0 , . . ., w p ) are independent if w s = w s for all 0 s p.We denote the maximum number of pairwise independent p-walks as α p (D).
Edmonds gave a formula for α 1 (D) in [3], based on the König-Ore formula: This was greatly generalised by Poljak, who showed that α p (D) could be computed in polynomial time and who gave a formula for α p (D) for all p 1 in [11].Suppose that C 1 , . . ., C r and P 1 , . . ., P s are vertex-disjoint cycles and paths.The cycle C i = (c 0 , . . ., c l−1 ) produces l independent p-walks of the form W a = (c a , c a+1 , . . ., c a+p−1 ), where indices are computed mod l and 0 a l − 1.The path Poljak's theorem asserts that this is the optimal way of producing pairwise independent p-walks.We denote the number of vertices of a cycle C and of a path P as |C| and |P |, respectively.Theorem 1 ([11]).For every digraph D and a positive integer p, where the maximum is taken over all families of pairwise vertex-disjoint cycles and paths C 1 , . . ., C r and P 1 , . . ., P s .
where the maximum is taken over all families of pairwise vertex-disjoint cycles.
if and only if f v depends essentially on u, i.e. there exist x, y ∈ [q] n which only differ on coordinate u such that f v (x) = f v (y).The set of all functions over an alphabet of size q and whose interaction graph is (contained in) D is denoted as We consider successive iterations of f ; we thus denote f 1 (x) = f (x) and f k+1 (x) = f (f k (x)) for all k 1. Recall that x is an image if there exists y such that x = f (y); x is a periodic point of f if there exists k ∈ N such that f k (x) = x.We are interested in the following quantities: 1. the rank of f is the number of its images: |Ima(f )|; 2. the periodic rank of f is the number of its periodic points: |Per(f )|.
It will be useful to scale these two quantities using the logarithm in base q: Moreover, the maximum (periodic) rank over all functions in F[D, q] is denoted as and ima(D, q) and per(D, q) are defined similarly.We finally note that per(f ) = ima(f p ) for all p q n − 1.Therefore, the main strategy is to maximise the scaled rank of f p for all p; we thus denote ima[D, q, p] := max{ima(f p ) : f ∈ F[D, q]}, ima(D, q, p) := max{ima(f p ) : f ∈ F(D, q)}.
We then have and similarly for ima(D, q) and per(D, q).
The case q = 2 is indeed specific, for there exist graphs D such that max{ima(f p ) : for all p 1. We shall investigate this in the next subsection.
We obtain two immediate consequences of Corollary 4. Firstly, we determine which graphs admit so-called reversible dynamics, i.e. for which graphs D we can find a permutation in F[D, q].Corollary 6 (Reversible dynamics in parallel).For any q 3, there exists f ∈ F[D, q] which is a permutation of [q] n if and only if all the vertices of D can be covered by disjoint cycles.
Secondly, Robert's seminal theorem indicates that if the interaction graph of f is acyclic, then f n is constant (i.e.per(f ) = 0) [16].Since α n (D) = 0 if and only if D is acyclic, we obtain the following result.
Corollary 7. The graph D is acyclic if and only if f n is constant for all q and all f ∈ F[D, q].The rest of this subsection is devoted to the proof of Theorem 3. We begin with the upper bound on the scaled rank, which follows a form of max-flow min-cut theorem (or at least, the min-cut uper bound).
We now review the communication model based on terms from logic introduced by Riis and Gadouleau in [15].Let {x 1 , . . ., x k } be a set of variables and consider a set of function symbols {f 1 , . . ., f l } with respective arities (numbers of arguments) d 1 , . . ., d l .A term is defined to be an object obtained from applying function symbols to variables recursively.We say that u is a subterm of t if the term u appears in t.Furthermore, u is a direct subterm of t if t = f j (v 1 , . . ., u, . . ., v d j ), and we denote it by u ≺ t.
Let Γ = {t 1 , . . ., t r } be a set of terms built on variables x 1 , . . ., x k and function symbols f 1 , . . ., f l of respective arities d 1 , d 2 , . . ., d l .We denote the set of variables that occur in terms in Γ as Γ var and the collection of subterms of one or more terms in Γ as Γ sub .To the term set Γ we associate the acyclic digraph The set of sources in G Γ is Γ var and the set of sinks is Γ.The min-cut of Γ is the minimum size of a vertex cut of G Γ between Γ var and Γ.
An interpretation for Γ over [q] is an assignment of the function symbols ψ = { f1 , . . ., fl }, where fi : [q] d i → [q] for all 1 i l.We note that fi may not depend essentially on all its d i variables.Once all the function symbols f i are assigned functions fi , then by composition each term t j ∈ Γ is assigned a function tj : [q] k → [q].We shall abuse notations and also denote the induced mapping of the interpretation as ψ : [q] k → [q] r , defined as ψ(a) = t1 (a), . . ., tr (a) .
Intuitively, if S is a vertex cut of G Γ between Γ var and Γ, then the terms in Γ "depend on" the terms in S. As such, the scaled rank of any induced mapping ψ cannot be greater than the size of S. This intuition is given formally as follows.
We illustrate the communication model and Theorem 8 by the following example.Consider the term set The set of variables is Γ var = {x 1 , x 2 , x 3 }, while the set of subterms is The graph G Γ is displayed below.We see that {u, v} forms a vertex cut of G Γ between Γ var and Γ: In fact, the min-cut is indeed 2.
A possible interpretation for Γ over [2] is (all operations mod 2) The corresponding induced mapping is and its scaled rank is log 2 3, which is indeed no more than 2. Lemma 9.For any p 1 and f ∈ F(D, q), ima( f p ) α p (D).
Proof.For all v ∈ V , denoting N − (v; D) = {u 1 , . . ., u k } sorted in increasing order, we have fv (x) = fv (x u 1 , . . ., x u k ).By definition, f p is the induced mapping of an interpretation for Γ p = {t p 1 , . . ., t p n }, where Γ 0 = {t 0 1 = x 1 , . . ., t 0 n = x n } and for all 1 s p, The graph G Γ p = (V Γ p , E Γ p ) is then given by A flow in G Γ p is a set of vertex-disjoint paths from Γ 0 to Γ p .Such a path is of the form t W = (t 0 w 0 , . . ., t p wp ) where w s−1 ∈ N − (w s ; D); it naturally induces a walk in D: W = (w 0 , . . ., w p ).Since the paths t W and t W are vertex-disjoint, the corresponding walks W and W are independent.Therefore, the max-flow of G Γ p is at most α p (D).By the max-flow min-cut theorem and Theorem 8, ima( f p ) α p (D).
Let W 1 , . . ., W α be α := α p (D) independent walks of length p, where we denote W i = (w i,0 , . . ., w i,p ).According to Theorem 1, those arise from families of disjoint cycles and paths.By construction, if w precedes w on one walk and w appears on another walk and has a predecessor there, then w precedes w in the other walk as well.For all 0 s p, we denote W s = {w i,s : We can now construct the finite dynamical systems which attain the upper bound on the scaled rank.The case q = 2 and f ∈ F(D, 2) is easy.We use a finite dynamical system where w i,s+1 simply copies the value x w i,s ; this will transmit the value x w i,0 along the walk W i .
Lemma 10.The function f ∈ F(D, 2) defined as It is easy to show, by induction on s, that for all 0 For q 3 and f ∈ F[D, q], we use a finite dynamical system where w i,s+1 wishes to copy the value x w i,s whenever it can.Each other vertex u ∈ N − (w i,s+1 ) has a red light (the value 2).If all lights are red, then w i,s+1 cannot copy the value x w i,s any more; instead it flips it from 0 to 1 and vice versa.
Lemma 11.For q 3, the function f ∈ F[D, q] defined as )\w i,s = (2, . . ., 2), x w i,s otherwise Proof.The proof is similar, albeit more complex, than the one of Lemma 10.
Proof of Claim 12.We prove the first assertion.First, suppose there exists w i,s ∈ W s where x w i,s 2 and x w i,s = y w i,s , then the electronic journal of combinatorics 25(3) (2018), #P3.48 Second, suppose that for any w i,s ∈ W s such that x w i,s = y w i,s , we have {x w i,s , y w i,s } = {0, 1}.Then For the second assertion, let v ∈ U s+1 , then either v ∈ U or v = w i,t+1 with 0 t = s.If v ∈ U , then f v (x) ∈ {0, 1} for any x.Suppose that v = w i,t+1 such that f w i,t+1 (x) / ∈ {0, 1}.Then x w i,t / ∈ {0, 1}, which implies w i,t ∈ W s , say w i,t = w j,s ; but then, Claim 13.For all 0 s p, |f s W s (X)| = |X| and for any x ∈ X, f s U s (x) ∈ {0, 1} |U s | .Proof of Claim 13.The proof is by induction on s; the statement is clear for s = 0. Suppose it holds for up to s.For any distinct x, y ∈ X, we have

Maximum rank in the Boolean case
We first exhibit a class of digraphs for which the upper bound on the rank is not reached in the Boolean case.Proof.Suppose f ∈ F[D, 2] is a permutation of {0, 1} n , then all the local functions f v must be balanced, i.e. (2).Therefore, f (x) = M x + c, but since every vertex has even in-degree, the sum of all rows in M (in GF(2)) equals zero and M is singular.
For instance, if D is the undirected cycle on n vertices, or the directed cycle on n vertices with a loop on each vertex, then for all p 1, It is unknown whether there exist other such examples.On the other hand, we can easily exhibit a class of digraphs which do reach the bound.For instance, let D = Kn be the clique with a loop on each vertex (alternatively, E = V 2 ).Then the following f ∈ F[ Kn , 2] is a permutation: the electronic journal of combinatorics 25(3) (2018), #P3.48 indeed f is the transposition of (0, . . ., 0) and (1, . . ., 1).Less obviously, the clique also admits a permutation of {0, 1} n .Proposition 15.For any n = 3, ima[K n , 2] = n.
Proof.Firstly, let n be even.Then we claim that f (x) = M x is a permutation, or equivalently that det(M ) = 1.For det(M ) = d(n) mod 2, where d(n) is the number of derangements (fixed point-free permutations) of [n].Enumerating the permutations of [n] according to their number p of fixed points, we have Since n! and n 1 , . . ., n n−1 are all even, it follows that d(n) is odd, thus det(M ) = 1.Secondly, let n 5 be odd.We prove the result by induction on n odd.Let us settle the case where n = 5.We construct f ∈ F[K 5 , 2] as follows: It is easy to check that f is a permutation of [2] 5 .
The inductive case is similar.Suppose that g ∈ F[K n , 2] is a permutation, then construct f ∈ F[K n+2 , 2] as follows: Again, it is easy to check that f is a permutation of [2]  3 Average rank Theorem 17.The average scaled rank in F[D, q] tends to α 1 (D): the electronic journal of combinatorics 25(3) (2018), #P3.48 Proof.The case α 1 (D) = 0 is trivial, thus let a := α 1 (D) 1 and (u 1 , v 1 ), . . ., (u a , v a ) be a collection of pairwise independent arcs.Let q be large enough and f be chosen uniformly at random amongst F[D, q].Let h 0 = (x u 1 , . . ., x ua ) : [q] n → [q] a and for any 1 i a, let Let c i be defined as c 0 = 1 and c i = |Ima(h a )|, all we need is to prove the following claim: with high probability, |Ima(h The proof is by induction on i.The claim clearly holds for i = 0; suppose it holds for i.Let g = (f v 1 , . . ., f v i , x u i+2 , . . ., x ua ) : [q] n → [q] a−1 and consider the set Z of images of g which appear frequently in the image of h i : for otherwise, we would have Now let N be the in-neighbourhood of v i+1 ; note that u i+1 ∈ N .Therefore, for each z ∈ Z, there exist at least 1 2 c i q values of x N such that z = g(x N , y V \N ) for some y V \N ; denote this set of values as X.On X, f v i+1 (x N ) is chosen uniformly at random.

Claim 18. With probability exponentially small
.
Therefore, with high probability, |f v i+1 (X)| > 1 2 |X| for all z ∈ Z, and hence the electronic journal of combinatorics 25(3) (2018), #P3.48 Conversely, let us remove all the arcs connecting strong components of D and all the chords of any cycle in D. We obtain a new graph D which is the disjoint union of strong chordless graphs; the trivial components T 1 , . . ., T t of D are exactly those of D. Let C 1 , . . ., C k be a collection of cycles of D which cover all the vertices that do not belong to a trivial component and σ x u mod q, where an empty sum is equal to zero and the neighbourhood is according to D .It is easy to check that f is a permutation for all 1 i k and hence that {x ∈ [q] n : x T 1 = . . .x Tt = 0} is a set of q n−T (D) periodic points of f σ .
Next, by a similar argument we prove that per(D, q) actually approaches n − T (D).
Theorem 23.For all D, sup Proof.Let C 1 , . . ., C k be a collection of cycles which cover all vertices belonging to a cycle, W denote the set of remaining vertices and let σ = (W, C 1 , . . ., C k ).Let q − 1 = 2 m be large enough (m 2 n 2 +1 ) and let α be a primitive element of GF(q − 1).Denote the arcs in D as e 1 , . . ., e l .Let A ∈ GF(q − 1) n×n such that a u,v = α 2 i if (u, v) = e i and a u,v = 0 if (u, v) / ∈ E and let g(x) = Ax.Now f ∈ F[D, q] is given as follows: view [q] = GF(q − 1) ∪ {q − 1} and f w (x) = 0 if x u ∈ GF(q − 1) for all u ∈ N − (w), q − 1 otherwise, ∀w ∈ W f v (x) = g v (x) if x u ∈ GF(q − 1) for all u ∈ N − (v), q − 1 otherwise, ∀v / ∈ W.
Then f acts like g on the set of states X = {x ∈ GF(q − 1) n : x W = (0, . . ., 0)}; in particular, we have f (X) ⊆ X.We can then remove W and consider h ∈ F[D \ W, q − 1] such that h v (x V \W ) = g v (x V \W , 0 W ) for all v / ∈ W instead.All we need to prove is that h (C 1 ,...,C k ) is a permutation of GF(q − 1) n−T (D) .
Denote the square submatrix of A induced by the vertices of C j as A j .Then we remark that det(A j ) = 0 for any 1 j k.Indeed, let K 1 , . . ., K l denote all the hamiltonian cycles in the subgraph induced by the vertices of C j (and without loss, K 1 = C j ).For any 1 a l, let S(a) = e i ∈K l 2 i .We note that S(1), . . ., S(l) are all distinct, hence α S(1) , . . ., α S(l) are all linearly independent (when viewed as vectors over GF(2)) and det(A j ) = Now h (C j ) (x) = A j x, where A j = A j B j 0 I , where (A j |B j ) are the rows of A corresponding to C j and I is the identity matrix of order n − T (D) − |C j |.Since A j is nonsingular, so is A j .Hence h (C j ) is a permutation of GF(q − 1) n−T (D) , and by composition, so is h (C 1 ,...,C k ) .
If W is empty, then we can simplify the proof of Theorem 23 and work with GF(q) n instead of GF(q − 1) n−T (D) (this time q = 2 p ), hence we obtain a permutation.This yields the following corollary on the presence of reversible dynamics.
Corollary 24.There exist q, σ and f ∈ F[D, q] such that f σ is a permutation of [q] n if and only if all the vertices of D belong to a cycle.
The theorem brings the following natural question.
Problem 25.Is there an analogue of Theorem 23 for the rank?

Proposition 14 .
Let D be a digraph such that α 1 (D) = n and d v = 2 for all vertices v ∈ V .Then ima(f p ) < α p (D) for all f ∈ F[D, 2] and all p 1.

n . Problem 16 .
Find a good lower bound on the maximum rank or maximum periodic rank in F[D, 2].