Enumerating ODE Equivalent Homogeneous Networks

ENUMERA TING ODE EQUIV ALENT HOMOGENEOUS NETW ORKS A. J. WINDSOR Abstract. W e give an alternative criterion for ODE equiv alence in identical edge homogeneous coupled cell netw orks. This allows us to giv e a simple pro of of Theorem 10.3 of Aquiar and Dias, which c haracterizes minimal identical edge homogeneous coupled cell netw orks. Using our criterion we give a form ula for coun ting homogeneous coupled cell netw orks up to ODE equiv alence. Our criterion is purely graph theoretic and makes no explicit use of linear algebra. 1. Introduction Coupled cell net w orks are used to represen t systems of coupled dynamical sys- tems sc hematically . Suc h systems app ear either in v arious biological systems. Net- w orks of eight coupled cells mo deling central pattern generators in quadrup eds can b e used to reco v er the primary animal gaits [3, 7]. One of the imp ortan t conclusions of the theory of coupled cell systems is that the net w ork itself imp oses constrain ts on the possible behaviors of the system ev en when we lac k detailed kno wledge of the b eha vior of the cells within the netw ork. A recent application to head mov emen t that illustrates the imp ortance of this is [6]. F or further applications see [10]. Mathematically coupled cell netw orks are a subclass of v ertex and edge lab eled directed multigraphs with lo ops. V ertices with the same lab el represen t multiple copies of the same dynamical system. Edge lab els represen t the type of coupling. A compatibilit y condition is imposed that requires every vertex with a given lab el to receive the same set of coupling types as inputs. As with any class of graphs there is a natural notion of isomorphic coupled cell netw orks induced b y bijections b et ween the sets of vertices. This representation of coupled cell net w orks follows [8]. An alternative approach is outlined in [5]. F ollowing Stew art and Golubitsky one ma y associate to eac h coupled cell net w ork a class of ordinary differential equations that are compatible with the net w ork structure, the class of coupled cell systems associated to a coupled cell netw ork. As was p oin ted out in [8] it is p ossible for non-isomorphic coupled cell net w orks to ha ve the same class of coupled cell systems. In this case we term the tw o coupled cell net w orks O.D.E. equiv alen t. W e will give a full description of the coupled cell systems asso ciated to a coupled cell net w ork for the simple case of identical edge homogeneous coupled cell netw orks. F or the definition in the general case see [4]. Aguiar and Dias [1] examine the structure of O.D.E. equiv alence classes for suc h coupled cell netw orks. They find a collection of c anonic al normal forms - a collection of netw orks whose num ber of edges is minimal within the equiv alence class. This they term the minimal sub class. 2010 Mathematics Subject Classific ation. 34C15 , 34A34. Key words and phr ases. coupled cell system, coupled cell net works, coupled oscillators. 1 2 A. J. WINDSOR In this pap er w e consider the simplest t ype of coupled cell net w orks, the iden tical edge homogeneous coupled cell net works. These are simply directed multigraphs with lo ops where ev ery vertex has the same indegree. Aldosray and Stew art gav e an enumeration of these netw orks [2] coun ted up to isomorphism. Using a simpler metho d sp ecific to the case of homogeneous net w orks we reco v er Theorem 10.3 of [1] which characterizes the minimal sub class in this case. F urther- more, we are able to give a recursive formula for en umerating the minimal systems with a giv en num b er of v ertices and edges. 2. Coupled Cell Systems, Coupled Cell Networks, and O.D.E. Equiv alence. W e will deal exclusively with identical edge homogeneous coupled cell netw orks, hereafter referred to simply as netw orks. Mathematically such a netw ork is a directed multigraph where lo ops are allow ed and where every vertex has the same in-degree. If the constant in-degree is r we will call the netw ork degree r . A directed multigraph consists of a set of vertices V and a multiset of edges E with elemen ts in V × V . A multiset ma y b e thought of as a function E : V × V → N ; we call this function the e dge multiplicity function . The condition that ev ery v ertex has the same in-degree, r , is then v ∈ V P u ∈ V E ( u, v ) = r . Giv en an n cell degree r netw ork G = ( V , E ), a choice of finite dimensional phase space P = R d , and a function F : P × P r → P suc h that F ( x 1 ; y 1 , . . . y r ) is in v ariant under all p erm utations of the v ariables y 1 , · · · , y d , we may pro duce a vector field on P n . The vector field for the v ariable x i asso ciated to cell i is ˙ x i = F ( x i ; x j ( i ) 1 , . . . x j ( i ) r ) where j ( i ) 1 , . . . , j ( i ) r are the source cells for the r arcs that terminate at v ertex i . The complete system is ˙ x 1 = F ( x 1 ; x j (1) 1 , . . . x j (1) r ) . . . ˙ x n = F ( x n ; x j ( n ) 1 , . . . x j ( n ) r ) The set of such vector fields is a subset of the vector fields on P n . A vector field obtained from a coupled cell netw ork G by a choice of phase space and function F is referred to as a c ouple d c el l system , or an admissible ve ctor field , asso ciated to G . W e ma y consider the class of all admissible vector fields for a giv en net work G and phase space P . W e will denote this class of v ector fields b y X P G . Definition: Two coupled cell net works G 1 and G 2 are called O.D.E. e quivalent if there exists a net work G 0 2 isomorphic to G 2 suc h that for all choices of phase space P X P G 1 = X P G 0 2 . More prosaically , an n cell degree r 1 net work G 1 and an n cell degree r 2 net work G 2 are called O.D.E. e quivalent if there exists a netw ork G 0 2 isomorphic to G 2 suc h that ENUMERA TING ODE EQUIV ALENT HOMOGENEOUS NETW ORKS 3 (1) for all choices of phase space P and function F 1 : P × P r 1 → P there exists a function F 2 : P × P r 2 → P such that for all v ertices i F 1 ( x i ; x j ( i ) 1 , . . . x j ( i ) r 1 ) = F 2 ( x i ; x k ( i ) 1 , . . . x k ( i ) r 2 ) where j ( i ) 1 , . . . , j ( i ) r 1 are the source cells for the r 1 arcs that terminate at cell i in net work G 1 and k ( i ) 1 , . . . , k ( i ) r 2 are the source cells for the r 2 arcs that terminate at cell i in netw ork G 0 2 . (2) for all choices of phase space P and function F 2 : P × P r 2 → P there exists a function F 1 : P × P r 1 → P such that for all v ertices i F 1 ( x i ; x j ( i ) 1 , . . . x j ( i ) r 1 ) = F 2 ( x i ; x k ( i ) 1 , . . . x k ( i ) r 2 ) where j ( i ) 1 , . . . , j ( i ) r 1 are the source vertices for the r 1 arcs that terminate at v ertex i in netw ork G 1 and k ( i ) 1 , . . . , k ( i ) r 2 are the source vertices for the r 2 arcs that terminate at v ertex i in netw ork G 0 2 . If w e consider P = R and linear functions F 1 and F 2 then w e obtain the notion of line ar e quivalenc e . It is shown in [4] that linear equiv alence and O.D.E. equiv alence are equiv alen t. 3. Network Opera tions tha t Preser ve O.D.E. equiv alence In this section we introduce tw o op erations that can b e p erformed on a netw ork that preserv e the O.D.E. equiv alence class. Since we are dealing exclusiv ely with homogeneous net works these op erations are a small part of the netw ork op erations considered in [1]. Both Lemma 1 and Lemma 2 can b e deduced from the more general arguments in [1], in particular from Prop osition 7.4. F or completeness we giv e pro ofs of b oth Lemma 1 and Lemma 2 using only what is required for our simpler case. That we may consider only these t wo netw ork operations and not more general op erations is crucial for the results in Section 5. Here we give go t w o simple op erations on netw orks that preserv e the O.D.E. equiv alence class of the netw ork. (1) Adding lo ops: A single loop is added to all v ertices in the netw ork. (2) k -Splitting edges: Each edge in the net work is replaced b y k identical copies of the edge. In tuitively , it should b e clear that these op erations preserve the O.D.E. equiv alence class of the netw ork; how ev er, a formal pro of is surprisingly difficult if one do es not use the notion of linear equiv alence. Lemma 1. If network G 0 is obtaine d fr om network G by either of the two network op er ations ab ove then G and G 0 ar e O.D.E. e quivalent. Pr o of. Using [4] it is enough to prov e that the tw o netw orks are equiv alent when the v ariables x i are taken to be in R and the function F is taken to be linear. In this case we observe that for a degree r net w ork the function F must take the form F ( x ; y 1 , . . . , y r ) = a x + b ( y 1 + · · · + y r ) . Consider a degree r net w ork. Adding a lo op to every v ertex w e obtain a degree r + 1 netw ork. Giv en a function F r : R r → R defined b y F r ( x ; y 1 , . . . , y r ) = a x + b ( y 1 + · · · + y r ) . 4 A. J. WINDSOR w e define a function F r +1 : R r +1 → R by F r +1 ( x ; y 1 , . . . , y r +1 ) = ( a − b ) x + b ( y 1 + · · · + y r +1 ) . Clearly w e hav e F r +1 ( x ; x, y 1 , . . . , y r ) = F r ( x ; y 1 , . . . , y r ) and consequently the lin- ear v ector fields admissible for the degree r net work are a subset of the linear v ector fields admissible for the r + 1 degree netw ork. W e can easily go the other direction. Giv en an y function F r +1 : R r +1 → R of the form F r +1 ( x ; y 1 , . . . , y r +1 ) = a x + b ( y 1 + · · · + y r +1 ) w e ma y define a function F r : R r → R by F r ( x ; y 1 , . . . , y r ) = ( a + b ) x + b ( y 1 + · · · + y r ) . Again w e hav e F r ( x ; y 1 , . . . , y r ) = F r +1 ( x ; x, y 1 , · · · , y r ) and consequen tly we see that the tw o netw orks hav e precisely the same set of admissible linear v ector fields. Consider a degree r net w ork. P erforming the edge splitting op eration w e obtain a degree k × r net w ork. Giv en any function F r : R r → R of the form F r ( x ; y 1 , . . . , y r ) = a x + b ( y 1 + · · · + y r ) . w e ma y define a function F k × r : R k × r → R by F k × r ( x ; y 1 , . . . , y k × r ) = a x + b k ( y 1 + · · · + y k × r ) . Clearly we hav e (1) F k × r ( x ; k − times z }| { y 1 , . . . , y 1 , . . . , k − times z }| { y r , . . . , y r ) = F r ( x ; y 1 , . . . , y r ) and consequen tly the linear vector fields admissible for the degree r netw ork are a subset of the linear v ector fields admissible for the degree k · r net w ork. Giv en an y function F k × r : R k × r → R of the form F k × r ( x ; y 1 , . . . , y k × r ) = a x + b ( y 1 + · · · + y k × r ) w e ma y define a function F r : R r → R by F r ( x ; y 1 , . . . , y r ) = a x + k b ( y 1 + · · · + y r ) . Again equation (1) holds, and consequen tly we see that the tw o net w orks hav e precisely the same set of admissible linear vector fields. In b oth cases w e see that the operation pro duces a new netw ork with precisely the same set of admissible linear vector fields. Th us we hav e that the op erations preserv e the O.D.E. equiv alence class.  The op erations create a netw ork with a larger degree. How ev er, when a netw ork has the required structure, the inv erse of these op erations ma y be applied to pro duce a netw ork with a smaller degree. Lemma 2. F or any identic al e dge homo gene ous c ouple d c el l network G , ther e exists an O.D.E. e quivalent network G M with the fol lowing pr op erties: (1) At le ast one vertex has no lo ops, and (2) The gr e atest c ommon divisor of the multiplicities of the e dges is 1. We wil l r efer to G M as a reduced network asso ciate d to G . If G is not a r e duc e d network then G M has a lower de gr e e than G . ENUMERA TING ODE EQUIV ALENT HOMOGENEOUS NETW ORKS 5 Pr o of. Let s denote the minim um num b er of lo ops on a vertex in G . Consider the new netw ork G 0 formed by removing exactly s lo ops from every vertex. Clearly G 0 has a v ertex with no lo ops. Since G ma y b e obtained from G 0 b y adding s lo ops we see that G and G 0 are O.D.E. equiv alent. Let d denote the greatest common divisor of the edge multiplicities in G 0 . W e ma y form a new netw ork G M b y dividing all the edge multiplicities by d . Since G 0 had a vertex with no lo ops so does G M . The greatest common divisor of the edge m ultiplicities of G M is 1 b y construction. Since w e may obtain G 0 from G M b y splitting each edge into d edges we see that G 0 and G M are O.D.E. equiv alen t b y Lemma 1. Thus G and G M are O.D.E. equiv alen t and G M has the required prop erties.  The use of G M to denote the reduced netw ork is not accidental. W e will now sho w that G M is indeed the unique minimal netw ork in the O.D.E. equiv alence class of G . Since any netw ork is O.D.E. equiv alent to such a reduced netw ork, it suffices to show that t w o reduced net works that are O.D.E. equiv alen t are isomorphic. Lemma 3. If G 1 and G 2 ar e r e duc e d network,s and G 1 and G 2 ar e O.D.E. e quiv- alent, then G 1 and G 2 ar e isomorphic. Pr o of. Let G 0 2 b e the net w ork isomorphic to G 2 whic h app ears in the definition of O.D.E. equiv alence. W e will sho w that G 1 and G 0 2 are equal. If w e take the phase space P for the cells to b e R and consider linear functions of the form F ( x, y 1 , . . . , y r ) = a x + b ( y 1 + · · · + y r ), then we see that for any choice of a 1 , b 1 there m ust exist a 2 , b 2 , and for any c hoice of a 2 , b 2 there m ust exist a 1 , b 1 , such that ( a 1 Id + b 1 A ) x = ( a 2 Id + b 2 B ) x where A is the adjacency matrix asso ciated to G 1 , B is the adjacency matrix asso ciated to G 0 2 , and x = ( x 1 , . . . , x n ) t ∈ R n . Since this holds for all x ∈ R n w e m ust ha ve (2) a 1 Id + b 1 A = a 2 Id + b 2 B . This matrix condition can b e reduced to a system of linear equations of t wo t yp es: a 1 + b 1 A ii = a 2 + b 2 B ii 1 ≤ i ≤ n (3) b 1 A ij = b 2 B ij 1 ≤ i, j ≤ n, i 6 = j (4) Since b oth A and B hav e a zero on the diagonal they m ust hav e some non-zero off diagonal entries in order to hav e the required row sums. No w by (4) w e see that b 1 and b 2 m ust ha ve the same sign and that A ij 6 = 0 if and only if B ij 6 = 0. Since both A and B ha v e at least one zero entry on the diagonal, either there m ust be an 1 ≤ i ≤ n suc h that A ii = B ii = 0 or there m ust exist i 6 = j such that A ii = 0 but B ii > 0 and A j j > 0 but B j j = 0. If we assume that there exists 1 ≤ i, j ≤ n with i 6 = j such that A ii = 0 but B ii > 0 and A j j > 0 but B j j = 0, then we obtain a 1 = a 2 + b 2 B ii (5) a 1 + b 1 A j j = a 2 (6) from which we immediately get − b 1 A j j = b 2 B ii whic h contradicts our earlier ob- serv ation that b 1 and b 2 m ust ha v e the same sign. Th us there exists an 1 ≤ i ≤ n suc h that A ii = B ii = 0 and w e can obtain from (3) that a 1 = a 2 . Th us w e m ust hav e b 1 A ij = b 2 B ij 6 A. J. WINDSOR for all 1 ≤ i, j ≤ n . No w b 2 divides b 1 A ij for all i, j . Since the greatest common divisor of the en tries of A is 1 w e m ust ha v e b 2 divides b 1 . Similarly b 1 divides b 2 B ij for all i, j . Since the greatest common divisor of the entries of B is 1, we must ha v e b 1 divides b 2 . Since b 1 and b 2 ha ve the same sign, we must hav e b 1 = b 2 . Finally we are able to conclude that A = B so G 1 is equal to G 0 2 as claimed.  4. Examples First we sho w how Figure 1 and Figure 2 of [1] are related using our netw ork op erations. 4 5 3 2 1 3 1 2 3 12 15 9 6 3 9 1 2 3 2 12 15 9 2 6 5 9 1 2 3 (1) (2) (3) Figure 1. T ransforming Figure 1 to Figure 2 of [1] using netw ork op erations. Edge lab els represen t edge m ultiplicities. Referring to our Figure 1 notice that netw ork (1) satisfies our criterion for b eing a minimal netw ork. If we split eac h edge of netw ork (1) into 3 edges then we obtain net work (2), which is O.D.E. equiv alen t to netw ork (1). If we now adjoin 2 lo ops to eac h vertex of netw ork (2), then we obtain netw ork (3), which is O.D.E. to netw ork (2) and hence O.D.E. equiv alent to net work (1). Next we apply the results of the previous section to the connected 3 cell degree 2 net works examined in [9]. They note that up to permutation there are 38 connected 3 cell degree 2 net works but that 8 of them are O.D.E. equiv alent to the lo w er degree net works. Each of these 8 is obtained from one of the 4 minimal connected 3 cell degree 1 netw orks by either adjoining a lo op to ev ery cell or b y doubling all the edges, see Figure 2. 5. Enumera tion W e begin b y outlining the work of Aldosray and Stew art in en umerating homo- geneous coupled cell netw orks. They use the counting result kno wn as Burnside’s Lemma to en umerate all identical edge homogeneous coupled cell net works with n cells and degree r counted up to isomorphism. T o be explicit let us tak e V = { 1 , . . . , n } . Let us denote the set of all m ultigraphs on V with constan t in-degree r by Ω n,r . The group of bijections on V is the symmetric group on n elemen ts, denoted S n . Each suc h bijection induces a map on Ω n,r . Thus we ha v e a group action of S n on Ω n,r . Two netw orks are related by S n if and only if they are isomorphic netw orks. Since we are counting the netw orks up to isomorphism what we actually wan t to count is the n um b er of distinct S n orbits ENUMERA TING ODE EQUIV ALENT HOMOGENEOUS NETW ORKS 7 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 2 1 3 1 2 3 2 1 3 2 1 3 1 2 3 2 1 3 Figure 2. The minimal connected 3 cell degree 1 netw orks and their asso ciated connected 3 cell degree 2 netw orks in Ω n,r . Burnside’s Lemma is a to ol for counting the n um b er of orbits of a group action, it states | Orb Ω n,r ( S n ) | = 1 | S n | X g ∈ S n | Fix Ω n,r ( g ) | . where Fix Ω n,r ( g ) = { ω ∈ Ω n,r : g · ω = ω } . If g and h are conjugate elements of S n then | Fix Ω n,r ( g ) | = | Fix Ω n,r ( h ) | and consequently w e may sum o v er conjugacy classes rather than individual elemen ts of S n . Supp ose that C 1 . . . , C m are the conjugacy classes in S n . Let g i b e some representativ e of the conjugacy class C i . W e ma y write our sum as (7) | Orb Ω n,r ( S n ) | = 1 | S n | m X i =1 | C i || Fix Ω n,r ( g i ) | . There is a bijection b et ween conjugacy classes of S n and partitions of the in teger n . F ollowing [2] we will denote a partition of n α 1 · 1 + α 2 · 2 + · · · + α n · n = n b y [1 α 1 2 α 2 . . . n α n ]. The multiplicativ e form of this notation is p erhaps unfortunate but should not cause confusion. The strength of this notation b ecomes apparent when w e agree that if α i = 0 then the i α i term in the expression ma y be omitted. Using this notation the 7 partitions of n = 5 ma y b e expressed as follo ws: 5 · 1 [1 5 ] 1 · 1 + 2 · 2 [1 1 2 2 ] 3 · 1 + 1 · 2 [1 3 2 1 ] 1 · 2 + 1 · 3 [2 1 3 1 ] 2 · 1 + 1 · 3 [1 2 3 1 ] 1 · 5 [5 1 ] 1 · 1 + 1 · 4 [1 1 4 1 ] 8 A. J. WINDSOR The set of all partitions of n will b e denoted by Π n . An element of S 5 can b e asso ciated to eac h ρ ∈ Π n as follows: [1 5 ] (1)(2)(3)(4)(5) [1 1 2 2 ] (1)(2 3)(4 5) [1 3 2 1 ] (1)(2)(3)(4 5) [2 1 3 1 ] (1 2)(3 4 5) [1 2 3 1 ] (1)(2)(3 4 5) [5 1 ] (1 2 3 4 5) [1 1 4 1 ] (1)(2 3 4 5) Ev ery p ermutation in S 5 is conjugate to one of the p ermutations that correspond to a partition of 5. Ev ery permutation σ ∈ S n ma y be written as a pro duct of disjoint cycles in a fashion that is unique up to the order to the cycles. The lengths of these cycles form a partition on n called the cycle typ e of the p ermutate σ . The p erm utation corresp onding to a given cycle type is called the normal form of the cycle t yp e. Ev ery permutation is conjugate to the normal form of its cycle type. Lo oking at the formula (7) we see that it w ould b e adv antageous to kno w the size of the conjugacy class asso ciated to a given partition of n . The size of the conjugacy class corresp onding to [1 α 1 2 α 2 · · · n α n ] is (8) n ! 1 α 1 2 α 2 · · · n α n α 1 ! α 2 ! . . . α n ! . If we consider the partition determines the pattern of paren theses [1 2 2 2 3 1 ] ( )( )( )( )( ) then n ! is the n um b er of w a ys of writing 1 , . . . , n in the blanks.Observing that w e can p ermute each cycle cyclically , that is (123) (231) (312) are all the same 3-cycle, w e must factor out the 1 α 1 2 α 2 · · · n α n p ossible wa ys of expressing all the cycles. Finally we observe that we may p ermute cycles of the same length freely , so w e must factor out the a 1 ! α 2 ! . . . α n ! p ossible orderings of the cycles. The main difficult y in enumerating the orbits of S n lies in determining the size of the fixed p oint set Fix Ω n,r ( g i ). W e will give the form ula for this here and refer the reader to the details in [2]. Definition : Giv en ρ ∈ Π n and s ∈ { 1 , . . . , n } w e may define Φ s,ρ ( z ) = n Y k =1 (1 − z k h ) − α ρ k h where h = gcd( s, k ). Clearly Φ s,ρ ( z ) is analytic about 0 and hence we may write Φ s,ρ ( z ) = ∞ X r =1 φ r ( s, ρ ) z r . Theorem 1 (Theorem 8.3 [2]) . L et n, r ∈ N \ { 0 } . L et H n,r denote the numb er of n c el l de gr e e r networks c ounte d up to isomorphism. H n,r is given by H n,r = 1 n ! X ρ ∈ Π n n ! 1 α 1 2 α 2 · · · n α n α 1 ! α 2 ! . . . α n ! n Y k =1 φ r ( k , ρ ) α ρ k . ENUMERA TING ODE EQUIV ALENT HOMOGENEOUS NETW ORKS 9 r 1 2 3 4 5 6 1 1 1 1 1 1 1 2 3 6 10 15 21 28 n 3 7 44 180 590 1582 3724 4 19 475 6915 63420 412230 2080827 5 47 6874 444722 14072268 265076184 3405665412 6 130 126750 43242604 5569677210 355906501686 13508534834704 T able 1. The num b er of n cell degree r netw orks counted up to isomorphism, H n,r . W e use this theorem to generate T able 1. This coun t how ever includes disconnected coupled cell net w orks. F rom the p er- sp ectiv e of dynamical systems w e are in terested only in the connected iden tical edge coupled cell netw orks. A disconnected system can b e decomp osed into a num ber of connected systems. Th us a disconnected n cell net w ork corresponds to a partition of n with α n = 0 i.e. any partition of n except [ n 1 ]. If we denote the n um b er of connected n cell degree r netw orks by K n,r then we ma y en umerate the num ber of disconnected coupled cell net w orks as follows X ρ ∈ Π n α ρ n =0 n − 1 Y m =1  K m,r + α ρ m − 1 α ρ m  where  K m,r + α ρ m − 1 α ρ m  is the n umber of wa ys of choosing α ρ m net works from the K m,r distinct connected m cell netw orks with replacemen t and where order does not matter. F rom this we obtain Theorem 2 (Theorem 10.1 [2]) . L et n, r ∈ N \ { 0 } . L et K n,r denote the numb er of minimal c onne cte d n c el l de gr e e r networks. We have K 1 ,r = H 1 ,r = 1 and for n ≥ 2 K n,r = H n,r − X ρ ∈ Π n α π n =0 n − 1 Y m =1  K m,r + α ρ m − 1 α ρ m  . W e use this theorem to generate T able 2. No w w e will use the w ork of Section 3 to giv e a recursiv e form ula for en umerating the connected minimal coupled n cell degree r netw orks. Theorem 3. L et M n,r denote the numb er of minimal c onne cte d n c el l de gr e e r networks. F or n ≥ 2 we have M n, 1 = K n, 1 and M n,r = K n,r − r − 1 X s =1  r s  M n,s . F or n = 1 note that M n,r = 0 . 10 A. J. WINDSOR r 1 2 3 4 5 6 1 1 1 1 1 1 1 2 2 5 9 14 20 27 n 3 4 38 170 575 1561 3696 4 9 416 6690 62725 410438 2076725 5 20 6209 436277 14000798 264632734 3403484793 6 51 117020 42722972 5554560632 355631996061 13505066262007 T able 2. The n umber of connected n cell degree r netw orks coun ted up to isomorphism, K n,r Pro of: If a connected n cell degree r netw ork is not minimal then it is O.D.E. equiv alent to a minimal n cell s degree net w ork where s < n . Given a minimal n cell degree s net w ork G the question thus b ecomes how many non-isomorphic n cell degree r netw orks can b e obtained that are O.D.E. equiv alent to G . W e ha ve seen that any netw ork G 0 O.D.E. equiv alent to a minimal netw ork G may b e obtained from G b y a combination of adjoining loops and splitting edges (and an isomorphism which w e ma y ignore). Let A b e the op eration of adjoining a ro ot and T k the operation of k -splitting the edge. Clearly we hav e T k ◦ T l = T kl . There is a comm utation relation b et ween T k and A , T k ◦ A = A k ◦ T k . Using this commutation relation we see that an y com bination of adjoining lo ops and edge splitting can b e reduced to a single k -splitting for some k ≥ 1 follow ed b y adjoining some num b er of lo ops. Given that G has degree s and G 0 has degree r the p ossible c hoices of k are constrained b y k s ≤ r . Thus there are b r /s c p ossible v alues of k . W e then adjoin sufficiently many lo ops to bring the degree to r . The num b er of connected minimal n cell degree r netw orks is th us given by M n,r = K n,r − r − 1 X s =1  r s  M n,s with the initial condition that M n, 1 = K n, 1 for n ≥ 2.  Using this theorem we generate T able 3. r 1 2 3 4 5 6 1 0 0 0 0 0 0 2 2 1 2 2 4 2 n 3 4 30 128 371 982 1973 4 9 398 6265 55628 347704 1659615 5 20 6169 430048 13558332 250631916 3138415822 6 51 116918 42605901 5511720691 350077435378 13149391543076 T able 3. The num ber of minimal connected n cell degree r net- w orks coun ted up to isomorphism, M n,r . It is interesting to note that the num ber of connected minimal 2 cell degree r net works for r ≥ 2 is given by φ ( r ) where φ is the Euler totien t function. The app earance of the Euler T otient is explained by the following netw ork diagram: ENUMERA TING ODE EQUIV ALENT HOMOGENEOUS NETW ORKS 11 r - k r k 1 2 Figure 3. A connected 2 cell degree r net wor k. Edge labels represen t edge m ultiplicities. In order for a 2 cell net w ork to b e minimal at least one vertex m ust hav e no lo ops. Without losing generalit y we may supp ose that vertex 2 has no lo ops. Thus v ertex 2 must receive r inputs from vertex 1. If we let k , with k ≤ r , denote the n umber of edges from v ertex 2 to v ertex 1 then w e see that v ertex 1 m ust ha ve r − k lo ops. If this net w ork is to b e minimal then the three edge multiplicities, r , k , and r − k , must b e relatively prime. This o ccurs if and only if r and k are relatively prime. F or a fixed r the n um b er of 1 ≤ k ≤ r for which r and k are relativ ely prime is φ ( r ). Provided that r ≥ 2 w e may exclude k = 0 since then r − k = r and all edge multiplicities hav e divisor r and hence the netw ork is not minimal. If r = 1 then there are in fact tw o minimal 2 cell degree 1 netw orks. 1 2 2 1 Figure 4. The t wo minimal 2 cell degree 1 netw orks. References [1] Manuela A. D. 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