One keyelement in thesemodels is theupdateshedule, whih indiates the order in whih states have to be updated

(1)

networks

J. Araena

a, ∗

E. Fanhon

c

M. Montalva

a

M. Noual

b

a CI ² M A

^and Departamento de Ingeniería Matemátia, Universidad de Conepión,

Av. Esteban Iturra s/n, Casilla 160-C, Conepión, Chile.

b

Laboratoire d'Informatique du Parallélisme de L'ENSLyon,

46,allée d'Italie,69364 Lyon Cedex 07,Frane.

c

TIMC-IMAGLaboratory, TIMBTeam, Faulté de Médeine,

38706 La Tronheedex, Frane.

Abstrat

Disretenetworkshave been usedasmodels ofgeneregulation andother biologial

networks. One keyelement in thesemodels is theupdateshedule, whih indiates

the order in whih states have to be updated. In Araena et al. (2009) was de-

ned equivalene lasses ofdeterministi updateshedules aordingto thelabeled

digraph assoiated to the network (update digraph) and suh that two shedules

in the same lass yield the same dynamial behavior. In this paper we study al-

gorithmial and ombinatorial aspets of update digraphs. We show a polinomial

haraterization of these digraphs,whih enables to haraterize the orresponding

equivalene lasses.We prove thattheupdatedigraphs areexatlytheprojetions,

ontherespetive subgraphs,ofa ompleteupdatedigraph. Finally,theexat num-

ber of omplete update digraphs was determined, whih provides upper and lower

boundson the numberof equivalene lasses.

Key words: Disretenetwork, updateshedule, updatedigraph,feedbakarset.

∗

Correspondingauthor.

Email addresses: jaraenaing-mat.ude.l(J. Araena),

Eri.Fanhonimag.fr(E. Fanhon),mmontalvude.l (M.Montalva),

Mathilde.Noualens-lyon.fr(M.Noual).

(2)

Disretenetworks(DNs)arethemostsimplemodelforgenetiregulatorynet-

works,aswellasforothersimpledistributeddynamialsystems.Despite their

simpliity,theyprovidearealistimodelinwhihdierentphenomenaan be

reprodued and studied, and indeed, many regulatory models published in

the biologialliterature twithin theirframework Kauman(1969);Thomas

(1973);Shmulevih etal.(2003).

A DN isdened by the states set of the nodes of the network, its onnetion

digraph,its loalativation funtions, and the type of update shedule used,

whihmayrangefromtheparallelupdate,themostommonKauman(1969);

Thomas(1991),tothesequentialupdate,passingthroughalltheombinations

of blok-sequentialupdates (whihare sequential over the sets of a partition,

but parallel insideof eah set).

The eet in the dynamial behavior of of a disrete network against per-

turbations in the update shedule has been greatly studied, mainly from a

statistialpoint of view, inrandom Boolean networks(RBN), where the loal

ativationfuntions are probabilistially hosen Chaves etal. (2005).

Some analytial works about perturbations of update shedules have been

made in a partiular lass of disrete dynamial networks, alled sequential

dynamial systems, where the onnetion digraph is symmetri or equiva-

lently an undireted graph and the update shedule is sequential. For this

lass of networks, the team of Barrett, Mortveit and Reidys studied the set

of sequential update shedules preserving the whole dynamial behavior of

the network (2001) and the set of attrators in a ertain lass of Cellular

Automata(2005).

In Araena et al. (2009) was dened equivalene lasses of deterministi up-

date shedules in a partiular kind of disrete networks (Boolean networks)

aording tothe labeled digraphassoiated tothe network (update digraph).

It was proven that two update shedules in the same lasse yield exatlythe

same dynamialbehavior.

In this paper we fous on the study of the update digraphs and the number

and size of equivalene lasses of update shedules assoiated in a disrete

network.

The mainreasonfor our interestinupdate digraphsshedulesistwo-fold.On

one hand, a neessary and good understanding of the objets we are dealing

with.On the other hand,abetterunderstanding of the relationshipsbetween

the arhiteture of onnetion graph and the robustness of disrete network

dynamisthroughthe studyofthe equivalenelassesofdeterminsitiupdate

(3)

2 Denitions

A digraph isan ordered pair of sets

G = ( V, A )

^where

V = {1 , . . . , n}

^is^a ^set

of elementsalled verties(or nodes) and

A

îs â^set ôf ôrdered ^pairs ^(alled

ars) of verties of

V

^. ^The ^vertex ^set ^of

G

^is ^referred ^to ^as

V (G)

^, ^its^ar ^set

as

A ( G )

^.

A walk from a vertex

v ₁

^to ^a ^vertex

v _m

ⁱⁿ ^a ^digraph

G

îs â ^sequene ôf

verties

v ₁ , v ₂ , . . . , v _m

^of

V (G)

^suh ^that

∀k = 1, . . . , m − 1, (v k , v _k+1 ) ∈ A(G)

or

( v _k+1 , v _k ) ∈ A ( G )

^.^The^verties

v ₁

^and

v _m

âre^theînitialând^terminal^vertex

of the walk. A walk is elementary if eah vertex in the walk appears only

onewiththe possibleexeptionthat therstandlastvertex mayoinide. A

walkislosedif itsinitialandterminalvertiesoinide. Airuitisalosed

elementary walk. A walk

v ₁ , v ₂ . . . , v _m

îs â ^path îf

(v k , v _k+1 ) ∈ A(G)

^for ^all

k = 1, . . . , m − 1

^. Â ^yle îs â ^direted îruit, ^that îs â ^losed êlementary

path.

A digraph

G

îs ^said ^to^be ônneted îf^there îs â^walk ^between êvery ^pairôf

itsverties,and stronglyonneted ifthere isapath between every pairof

itsverties.

G = (V, A)

^being ^a ^digraph ^and

i ∈ V

ône ôf îts ^verties,

N (i) = {j ∈ V | (j, i) ∈ A}

^denotes ^the ^input neighborhood of

i

^and

d ⁻ (i) = |N (i)|

^is ^the

input degree of

i

^. ^More terminology about digraph an be found in (West, 1996).

Also,inthe sequel,wewillwrite

[[ a, b ]] = {a, . . . , b}

^and

[[ a, b [[= {a, . . . , b − 1}

^,

for any integers

a

^and

b

^.

Denition 1 An update shedule of the verties of a digraph

G = (V, A)

^,

with

|V | = n

^, ^is ^a ^funtion

s : V → {1 , . . . , n}

^suh ^that

s ( V ) = [[1 , m ]]

^for

some

m ≤ n

^. ^If

∀i ∈ V, s(i) = 1

^, ^the ^update ^shedule ^is ^said ^to ^be ^parallel.

Inthis ase,wewillwrite

s = s _p

^. ^If

s

^is^a permutationovertheset

{1, . . . , n}

^,

s

^is ^said ^to ^be sequential. And in all other ases,

s

^is ^said ^to ^be ^blok

sequential.

As mentionned in Demongeot et al. (2008), the number of update shedules

assoiated to a given digraph of order

n

îs êqual ^to ^the ^number ôf ôrdered

partitionsof a set of size

n

^,^that ^is

T _n =

n − 1

X

k=0

n k

!

T _k .

(4)

Let

G = (V, A)

^be ^a ^digraph ^and

s

^an ^update ^shedule. ^We ^denote

s ⁻ ¹ (r) = {i ∈ V | s(i) = r}

^.

Denition 2 Let

G = (V, A)

^be^a^digraph^and

s

^an^update^shedule,^we^dene

the labeling funtion

lab _s : A → { ,

^-

}

⁺ ⁱⁿ ^the ^following ^way ^:

∀( j, i ) ∈ A, lab _s ( j, i ) =







+ ^if

s(j ) ≥ s(i)

- ^if

s(j ) < s(i).

An ar

a ∈ A

^suh ^that

lab _s (a) =

⁺ îs âlle^d â ^positive âr ând ân âr

a ∈ A

^suh ^that

lab _s (a) =

^- îs âlled â ^negative âr. ^Labeling êvery âr

a

^of

A

^by

lab _s ( a )

^, ^we ôbtain â ^labêled ^digraph

( G, lab _s )

^named ^update ^di-

graph. We write

N _s ⁺ (i) = {j ∈ N (i) | lab _s (j, i) = }

⁺ ^and

N _s ⁻ (i) = {j ∈ N (i) | lab _s (j, i) = }

^- ^. ^Thus, ^we ^have

N (i) = N _s ⁺ (i) ∪ N _s ⁻ (i)

^.

2 3 1

+

− +

−

4 −

Fig.1.Adigraph

G = (V, A)

^labeled^by^the^funtion

lab s

^where

∀i ∈ V = {1, . . . , 4}

^,

s(i) = i

^.

Denition 3 A disrete network

N = ( G, F, s )

îs ^dened^by â ^nite^set ôf

variable states

x ∈ Q = [[0, m − 1]]

^, ^a ^digraph

G

^, ^a ^global ^ativation ^funtion

F : Q ⁿ → Q ⁿ

^, ^where

F (x) = (f 1 (x), . . . , f n (x))

^with

f _i : Q ⁿ → Q

^loal ^ati-

vationfuntions suhthat

j ∈ N ( i )

îfândônlyîf

∃ ( x ₁ , . . . , x _n ) ∈ Q ⁿ , a, b ∈ Q

^,

a 6= b,

f _i ( x ₁ , . . . , x _j− ₁ , a, x _j+1 , . . . , x _n ) 6= f _i ( x ₁ , . . . , x _j− ₁ , b, x _j+1 , . . . , x _n ) .

and

s

ân ûpdate ^shedule ôf ^the ^verties ôf

G

^. ^In ^the ^partiular ^ase ^where

Q = {0 , 1}

^the ^network ^is ^said ^to ^be ^Boolean ^network.

The iteration of the disrete network with an update shedule

s

^is ^given^by:

x ^r+1 _i = f _i ( x ^l ₁ ¹ , . . . , x ^l _n ⁿ ) ,

⁽¹⁾

where

l _j = r

^if

s(i) ≤ s(j )

^and

l _j = r + 1

^if

s(i) > s(j)

^.

(5)

Thisis equivalentto applying afuntion

F ^s : Q ⁿ → Q ⁿ

ⁱⁿ ^a^parallel ^way,^with

F ^s (x) = (f ₁ ^s (x), . . . , f _n ^s (x))

^dened ^by:

f _i ^s (x) = f i (g _i,1 ^s (x), . . . , g _i,n ^s (x)),

where the funtion

g _i,j ^s

^is ^dened ^by

g _i,j ^s (x) = x _j

^if

s(i) ≤ s(j)

^and

g ^s _i,j (x) = f _j ^s (x)

^if

s(i) > s(j)

^. ^Thus, ^the ^funtion

F ^s

^orresponds ^to ^the ^dynamial ^be-

havior of the network

N

^. ^We ^will ^say ^that ^two ^networks

N ₁ = ( G, F, s ₁ )

^and

N ₂ = (G, F, s ₂ )

^have ^the ^same ^dynamis ^if

F ^s ¹ = F ^s ²

^.

3 Preliminary results and motivations

Extending a result given in Araena et al. (2009) for Boolean networks, the

following holds:

Theorem 4 Let

N ₁ = (G, F, s ₁ )

^and

N ₂ = (G, F, s ₂ )

^be^two ^disrete ^networks

that dier only in the update shedule. If

(G, lab s 1 ) = (G, lab s 2 )

^, ^then

N ₁

^and

N ₂

^have ^the ^same ^dynamis.

Theorem 4 allows us to dene equivalene lasses with respet to labeled di-

graphs: if

s

îs ânûpdate ^sheduleôf ^the^verties ôfâ ^digraph

G

^,^we^write

[s] G

the set of update shedules

s ^′

^suh ^that

s ∼ ^G s ^′

^, ^that ^is

[s] G = {s ^′ : (G, lab s ) = (G, lab s ^′ )}.

Thus, anequivalenelass,

[ s ] G

^,îsâ ^set ôfûpdate ^shedules^that âll^yield^the

same labeled digraph,and onsequently, the same dynamison networks.

In this work we study update digraphs and the equivalene lasses of their

update shedules. More preisely, Setion 4 deals with the haraterization

of update digraphs. Setions 5 and 6 fous on the size and the number of

equivalene lassesof update shedules.

4 Charaterization of update digraphs

In this setion,westudy the relation

∼ G

ând ^the ^labelingsôf â^given ^digraph

G

^. ^First, ^we ^give ^a haraterization of the labeling funtions

lab : A(G) → { ,

⁺

}

^- ^thatîndeedôrrespond^to^labeling^funtionsîndued^byûpdate^shed-

ules. Then, we examine update shedules

s

^whih ^satisfy

lab = lab _s

^. ^The

setion ends with some observations that where made to help determine the

number of

[·] G

^lasses. ^First,^let ^us^give^some ^additional^denitions.

(6)

Denition 5 A labeled digraph

(G, lab)

îs ^said ^to ^be ân ûpdate ^digraph

(UD) if there exists an update shedule

s

^suh ^that

lab = lab _s

^, ^that ^is

∀a ∈ A(G)

^,

lab(a) = lab s (a)

^. ^(se^e ^example ⁱⁿ ^gure ⁴⁾

3 2

1 +

− −

3 2

1 −

− −

a) b)

Fig.2.a) Alabeled digraph

(G, lab)

^whih îsûpdate^digraph. ^b)Â ^labeled^digraph

(G, lab ^′ )

^whih îs^not ân ûpdate^digraph.

The goal of this setion is to determine whih labeled digraphs are update

digraphs.

Denition 6 Let

(G, lab)

^be ^a ^labele^d ^digraph ^and

G ^′

^a ^subdigraph ^of

G

^. ^We

denetheprojetionof

(G, lab)

^onto

G ^′

^by^the^labeled^digraph

(G ^′ , lab _G ′ )

^,^where

lab G ^′ (a) = lab(a), ∀a ∈ A(G ^′ )

^.

Denition 7 Let

(G, lab)

^be ^a^labeled^digraph^and

G ^′

^be^a^non^trivial^strongly

onneted subdigraphof

G

^, ^the ^projeted ^lab^eled ^digraph

(G ^′ , lab ^′ )

^is^said ^to^be

a positive strongly onneted omponent of

( G, lab )

^if

∀a ∈ A ( G ^′ ) , lab _G ′ =

⁺

and it is maximal for this property. We will say that

(G, lab)

îs ^redued îf ît

has no positive strongly onnetedomponents.

Nothe that the fatof being

( G, lab )

ânûpdate ^digraphîsindependent ofthe presene or absene of positive strongly onneted omponents, beause the

images of the verties by an update shedule in a positive strongly onnet

omponent are idential. In the sequel and without lossof generality,we will

work onlywith redued labeled digraphs.

Denition 8 Let

(G, lab)

^be ^a ^lab^eled ^digraph. ^We ^dene ^the ^labeled ^reori-

ented digraph assoiated to

(G, lab)

^, ^and ^write

(G R , lab _R )

^, ^to ^refer ^to ^the

labeled digraph in whih all negative ars are inverted:

• V (G R ) = V (G)

^.

• A(G R ) = {(u, v ) ∈ A(G) | lab(u, v) = } ∪ {(v, u)

⁺

| (u, v) ∈ A(G) ∧ lab(u, v) = }

^-

• lab _R ( u, v ) =

^- ^if

( v, u ) ∈ A ( G ) ∧ lab ( v, u ) =

^- ^and

lab _R ( u, v ) =

⁺ ^if

lab(u, v) =

⁺^.

A forbidden yle in

(G R , lab _R )

îs â ^yle ôntaining â ^negative âr.

Let

( G, lab )

^be â ^labeled ^digraph. ^We ân ^determine îf ît îs ^redued ⁱⁿ ^time

O(|A|)

^with ânâlgorithm^that ^searhes ^for^strongly ônnetedômponentsôf

adigraph. We alsoan get

(G R , lab _R )

ⁱⁿ^time

O(|A|)

^.

(7)

Denition 9 Let

(G, lab)

^be^a ^labele^d ^digraph^and

P

^a^path ⁱⁿ

(G R , lab _R )

^, ^we

denote by

l ⁻ (P )

^the ^numbêr ôf ^negative ârs ôf

P

^. ^Thus, ^for ^every

v ∈ V (G)

we dene

L ⁻ (v ) = max{l ⁻ (P v ) | P v

^is ^a ^path ⁱⁿ

(G R , lab R )

^ending ⁱⁿ

v}

^and

L ⁻ (G R , lab _R ) = max

v∈V (G) {L ⁻ (v)},

the number of negative ars of a path with the maximum number of negative

ars in

(G R , lab _R )

^.

Theorem 10 A labeled digraph

( G, lab )

îs ân ûpdate ^digraph îf ând ônly îf

(G R , lab _R )

^does ^not ^ontain ^any ^forbidden ^yle.

PROOF. (

⇒

⁾ ^Let ^us ^suppose ^that

( G _R , lab _R )

^ontains ^a ^forbidden ^yle

C : v ₁ , . . . , v _p = v ₁

^suh ^that

(v j , v _j+1 )

îs â ^negative âr. ^Then âny ûpdate

shedule

s

^suh ^that

(G, lab) = (G, lab s )

^must ^satisfy

s(v j ) > s(v j+1 )

^. ^It ^must

also satisfy

s ( v _j ) ≤ s ( v _j+1 )

^sine ^there ^exists ⁱⁿ

( G _R , lab _R )

^a ^path ^from

v _j+1

to

v _j

^. ^Thus, ^weênd ûp ^with âontradition.

(

⇐

⁾ ^Let

L = L ⁻ (G R , lab _R )

^. ^Observe ^rst ^that ^if

P : v ₁ , . . . , v _k

^is ^a ^path ⁱⁿ

G

^suh ^that

l ⁻ ( P ) = L

^with

{( v _i 1 , v _i 2 ) , ( v _i 3 , v _i 4 ) , . . . , ( v _i ₂ _L− ₁ , v _i ₂ _L )}

^the ^set ^of

negative ars of

P

^where

j > k ⇒ i _j > i _k

^, ^and

s

îs ân ûpdate ^shedule ^suh

that

(G, lab) = (G, lab s )

^then

s ( v _i 1 ) > s ( v _i 2 ) > s ( v _i 4 ) > s ( v _i 6 ) > · · · > s ( v _i ₂ _L ) ,

whih implies

max{s(v)| v ∈ V (G)} ≥ L + 1

^. ^Besides,

∀ i = 1 , . . . , k, L ⁻ ( v _i ) = l ⁻ ( v ₁ , . . . , v _i )

^and

L ⁻ ( v ₁ ) = 0 .

Let

s : V (G) → [[1, L + 1]]

^with

s(v) = L − L ⁻ (v) + 1, ∀v ∈ V (G).

Sine obervation mentioned above,

s ( V ( G )) = [[1 , L + 1]]

^, ^that ^is,

s

^is ^an

updatesheduleof

V (G)

^.^T^o^hek^that

s

îsâlsoânûpdate^shedule^satisfying

(G, lab) = (G, lab s )

^, ^we ^must ^show ^that

∀a = (u, v) ∈ A(G R ), s(u) > s(v)

or

L ( v ) > L ( u )

^. ^This ^follows ^from ^the ^fat ^that

( u, v )

^being ân âr ôf

G _R

^, ^it

neessarilyholds that

L(v) ≥ 1 + L(u)

^.

2

We an notie that if

(G, lab)

îs â ^labeled ^digraph, ^the ^forbidden ^yles ôf

( G _R , lab _R )

^orrespond ^to ^what ^we ^will ^refer ^to ^as alternating iruits of

G

^.

That is, they oinide with walks of

G

^,

C = v ₀ , v ₁ , . . . , v _k

^,^where

v ₀ = v _k

^and

either

(v i , v _i+1 ) ∈ A

ⁱⁿ^whih^ase

lab _G (v i , v _i+1 ) =

⁺ ^or

(v i+1 , v _i ) ∈ A

ⁱⁿ^whih

(8)

ase

lab _G (v _i+1 , v _i ) =

^- ^(or vie-versa). Among these alternating iruits,are in partiular iruits suh that

∀i ∈ [[0, k − 1]]

^,

lab(v i , v _i+1 ) =

^- ^as ^well ^as

sub-graphs ontaining two verties

u

^and

v

^, ^a ^walk ^from

u

^to

v

^negatively

labeled and another walk from

u

^to

v

^positively^signed.

Inidently, letusnotie that, asa onsequene of Theorem 10,if

a = (u, v) ∈ A ( G )

îs ân âr ^not ^belonging ^to âny îruit, ^then ^the ^fat ^that

( G, lab )

^is ^an

update digraphor not is independent of

lab(a)

^.

Algorithm1, given below, nds anupdate shedule orresponding to a given

reduedlabeleddigraphasdesribedintheproofofTheorem10.Itisadapted

fromthe famous algorithmvan Leeuwen (1990),givinga topologialorderon

a digraph without yles. For a given redued labeled digraph

( G, lab )

^, ^algo-

rithm 1 works on the labeled reoriented digraph

(G R , lab _R )

^without ^forbid-

den yles. It returns in time

O(|V | + |A|)

^an ^update ^shedule

s

^suh ^that

( G, lab ) = ( G, lab _s )

^and

max{s(v ) | v ∈ V } = min{max{s ^′ (v) | v ∈ V } | s ^′

îs ânûpdate ^shedule ôf

G}.

Figure 4 shows the dierent steps of the algorithm that returns an update

shedule assoiated to an arbitrary possible labeled digraph (not neessarily

redued).

Corollary 11 The following problems an be solved in polynomial time.

(1) Determine whether a labeled digraph

(G, lab)

îsân ûpdate ^digraph,

(2) Given

(G, lab)

â ^labeled ^digraph, ^nd ân ûpdate ^shedule

s

^suh ^that

(G, lab) = (G, lab s )

^.

Indeed, aordingto Theorem10,a labeled digraph

( G, lab )

îsân ûpdateône

if and only if, in

( G _R , lab _R )

^no negative-ar belongs to a strongly onneted omponent. Thus, the rst part of Corollary 11holds sine the strongly on-

neted omponents of a digraph an be identied in polynomial time. The

seond part of Theorem 10 omes from the existene algorithm1 whose run

time isalso polynomial.

5 Sizes of the equivalene lasses

[·] G

In this setion westudy the size of the equivalenelasses

[·] G

^.

(9)

Input:

(G = (V, A), lab)

^a ^redued ^labeled^digraph ^suh ^that

(G R , lab _R )

^has

noforbiddenyle

begin

ValMax

←

^table^of^size

|V (G R )|

ⁱⁿ^whih^are^stoked^the^maximal^possible

values of

s(v), v ∈ V (G R )

^.

n ← |V |;

H ← G _R ;

forall

v ∈ V

^do

ValMax

[v] = n;

end

while

∃v ∈ V, d ⁻ ( v ) = 0

ⁱⁿ

H

^do

s ( v ) ←

^ValMax

[ v ];

forall

u ∈ N (v)

^do

if

(u, v)

îs â ^negative âr ^then

ValMax

[u] ← min{

^ValMax

[u], s(v) − 1; }

else

ValMax

[u] ← min{

^ValMax

[u], s(v); }

end

deletethe ar

( u, v )

^from

H

end

s _min ← min{s(v | v ∈ V ); }

forall

v ∈ V

^do

s(v) ← s(v) − s _min ;

end

Letusnowonsiderthe followingquestion:given adigraph

G

ând ânûpdate

shedule

s

^, ^does^thereêxistâny ûdpdate^shedule

s ^′ 6= s

^suh^that

(G, lab s ) = ( G, lab _s ′ )

^? ^Thatîs,^whatônditions^need ^to^be^satisedⁱⁿôrder ^for

|[ s ] G | > 1

tohold?

Corollary 12 Let

(G, lab)

^be ^a ^redued ^update ^digraph ^with

|V (G)| = n

^and

L = L ⁻ ( G _R , lab _R )

^. ^Then,

∀m ∈ [[ L, n − 1]]

^, ^there êxists ân ûpdate ^shedule

s

suh that

max{s(v )| v ∈ V } = m + 1

^and

(G, lab) = (G, lab s )

^.

PROOF. We showthe result by indution on

m

^.

If

m = L

^, ^the ^result ^was ^proved ⁱⁿ ^Theorem^10.

If

L = n − 1

^, ^we ^are ^done. ^Otherwise, ^let

m ∈]]L, n − 1]]

^. ^By ^indution ^hy-

pothesis, there exists an update shedule

s

^suh ^that

(G, lab) = (G, lab s )

^and

(10)

a)

− +

−

+ 4

2

1

3

5

b)

4 2

1

3

5

)

4 2

1

3 5 ^s(5)=5

s(4)=5 s(3)=5

s(2)=4

s(1)=5

d)

4 2

1

3 5 ^s(5)=5

s(3)=5 s(2)=3

s(4)=4 s(1)=5

e)

− +

−

+ 4

2

1

3 5 s(2)=1

s(3)=3 s(4)=2

s(5)=3 s(1)=3

Fig. 3.

a)

^A ^labeled ^digraph

G = ({1, . . . , 5}, A)

^.

b) (G _R , lab _R )

^. ^The ^ars^drawn ⁱⁿ

dotted lines are negative-ars. The others arepositive-ars.

c)

^and

d)

^the^update

shedule omputed by algorithm 1 after the while loop.

e)

^The ^update ^shedule

s

suh that

(G, lab) = (G, lab _s )

^.

max{s ( v )| v ∈ V ( G )} = m

^.^Sine

m < n

^,^there^exists

i ^∗ ∈ [[ L, n − 1]]

^suh^that

|s ⁻ ¹ (i ^∗ )| > 1

^. ^Notie ^that

∀(u, v) ∈ s ⁻ ¹ (i ^∗ ) × s ⁻ ¹ (i ^∗ ) ∩ A(G)

^,

lab _s (u, v) = .

⁺

Besides, beause there are not yles in

(G R , lab _R )

^, ^there ^exists

w ∈ s ⁻ ¹ (i ^∗ )

suhthat

{v ∈ s ⁻ ¹ ( i ^∗ )| ( w, v ) ∈ A ( G _R )} = ∅

^.^Hene,^let^us^dene

s ^′

^as^follows:

s ^′ (v ) =







s ( v ) + 1

^if

s ( v ) ≥ s ( w )

^and

v 6= w

^,

s(v)

^if

s(v) < s(w)

^or

v = w

^.

Hene obviously,

s ^′ (V (G)) = [[1, m + 1]]

^,^i.e.

s ^′

ⁱⁿânûpdate ^sheduleôf

V (G)

^,

and

(G, lab) = (G, lab s ^′ )

^.

2

Corollary 13 Let

(G, lab)

^beâ^reduedûpdate^digraphând

L = L ⁻ (G R , lab _R )

^.

Then,

|[s] G | ≥ |V (G)| − L

^, ^where

s

îsân ûpdate ^shedule ^suh ^that

(G, lab) =

(G, lab s )

^.

(11)

Corollary 14 Let

(G = (V, A), lab s )

^be ^an ^update ^digraph.

|[s] G | > 1

^if ^and

only if

G

^is ^not ^strongly ^onneted, ^neither ^suh ^that

(G R , lab _R )

^is^linear ^and

negative.

PROOF. If

G

^is^not^strongly^onneted,^neither^suh^that

(G R , lab _R )

^is^linear

and negative, then

|[[ L, |V |]]| > 1

^. ^Thus, ^by ^Theorem ^12,

|[ s ] G | > 1

^, ^where

(G, lab) = (G, lab s )

^.

Conversely, there exists two partiular ases inwhih

|[ s ] G | = 1

^:

(1) If

G

^is ^linear ^digraph ^then

L = |V |

^. ^Thus, ⁱⁿ ^the ^ase ^where

∀a ∈ A

^,

lab _G (a) =

^- ^, ônly ône ûpdate ^shedule

s

^satises

(G, lab) = (G, lab s )

^.

On the ontrary, if

∃a ∈ A, lab _G ( a ) =

⁺^, ^there ^are

2 ^k

^suh ^update

shedules, where

k

îs ^the ^number ^positive ârs ôf

(G, lab)

^.

(2) If

G

^is ^strongly ^onneted, ^then

(G, lab)

^has ^only ^positive ^ars. ^Thus,

( G _C , lab _C )

îs ^redued ^to ône âlone ^vertie. Âording ^to ^the ^previous

Lemma, only one shedule satises

(G, lab) = (G, lab s )

^: ^the ^parallel ^up-

date shedule,

s _p

^.

2

Asaonsequene,beause

( G, lab _s _p )

^is^not ^a^negative^linear^digraph,

|[ s _p ] G | >

1

îf ând ônlyîf

G

^is ^not ^strongly ^onneted.

6 Number of update digraphs

Intheprevioussetion,givenalabeleddigraph

(G, lab)

^,^we^were ^interested^by

the existene of update shedules

s

^suh ^that

( G, lab ) = ( G, lab _s )

^. ^And ^when

there did exist suh update shedules, we wanted to know how many there

were.

In the present setion, given a digraph

G

^, ^we ^would ^like ^to ^determine ^how ^it

anbelabeledintoanupdatedigraph,thatis,whiharethelabelingfuntions

lab

^of

G

^suh ^that

(G, lab)

îs îndeed ân ûpdate ^digraph. În ^partiular, ^here,

we fous onthe numberof equivalene lasses

[·] G

^(rather^than ^there ^sizes).

Denition 15 We dene the size of a labeled digraph

(G, lab)

^by^the ^number

of its positive ars.

(12)

DIGRAPH

UPDATE (DU)

problem:



 



 



Input: A digraph

G = (V, A)

ând ân înteger ^k;

Question:

Does there exist a labeling funtion

lab : A → { ,

⁺

}

^- ^suh ^that

(G, lab)

is an update digraph and its size

is at most

k

^?

Theorem 16 DIGRAPH UPDATE is NP-omplete.

PROOF. WearegoingtoproveTheorem16byredutiontotheFASproblem

dened bellow and whih is known to be NP-omplete Garey and Johnson

(1979):

FAS problem:



 



 



Input: A digraph

G = (V, A)

ând ân înteger ^k;

Question:

Does there exist a feedbak ar set

F

^of

G

^suh ^that

|F | ≤ k

^?

where a feedbak ar set (FAS)

F

^of

G

îs â ^set ôf ârs ^suh ^that ^the ^digraph

( V, A\F )

^does^not^have^any^yles.

F

îs^minimalîf^there^does^notêxist

F ^′ ( F

FAS of

G

^.

The redution funtion we use to map an instane of FAS to an instane of

DUissimplytheidentity.Next,foragiveninstane

(G, k)

^we^show^that^there

exists a labeling funtion

lab

^suh ^that

( G, lab )

îs ûpdate ^digraph ôf ^size ât

most

k

îf ând ônlyîf ^there êxists â^FÂS

F

^of

G

^suh^that

|F | ≤ k

^.

(⇒)

^Let

lab

^be ^a ^labeling ^funtion^suh ^that

( G, lab )

îs ânûpdate ^digraphôf

size atmost

k

^and ^let

F = {a ∈ A(G)| lab(a) = }

⁺ ^.^Then,

F

îsâ ^FÂS ôf^size

|F | ≤ k

^.

G ^′ = (V, A \ F )

ânnot ôntain âny ^yle ^sine ôtherwiseît ^would ^be

negative yle of

(G, lab)

^whihîs ^not ^possible ⁱⁿ ânûpdate ^digraph.

(⇐)

^Let

F

^be â ^minimal ^FÂS ôf

G

^suh ^that

|F | ≤ k

^. ^Let

a ∈ F

^. ^If ^every

yleof

G

^ontaining

a

ôntainsâs^wellânotherârôf

F

^then

F \ {a}

îsâ^FÂS

of

G

^smaller ^than

F

^. ^This ^ontradits ^the ^minimality ^of

F

^. ^Thus, ^for ^every

a ∈ F

^, ^there êxists â^yle ôf

G

^ontaining

a

ând ^noôther âr ôf

F

^. ^Now, ^let

usdene the labeling funtion

lab

^as ^follows:

∀a ∈ F, lab(a) =

⁺ ^and

∀a ∈ A \ F, lab(a) = .

^-

Notethatbeausetherearenoylesin

G ^′ = ( V, A\F )

^,^there^are^not^ylesⁱⁿ

(G, lab)

^.^Suppose,^however,^that

(G, lab)

îs^notânûpdate^digraph.În

(G, lab)

^,

there must thus be an alternating iruit (see Theorem 10 and the remarks

(13)

isa forbiddenyle in

(G R , lab _R )

^.^The ^positive ^ars ⁱⁿ^this ^yle ^belong ^to

F

^.

Let

a ∈ A(G)

^be ^suh â ^positiveâr ^belonging ^to ^the ^forbidden ^yle ând ^to

F

^.^F^rom^the ^disussionâbove,^we^derive^that^there êxistsâ^yle

C _a

^of

G

^that

ontains

a

ând ^no ôther âr ôf

F

^. Âll ^the ârs ôf

C _a

^that ^are ^not

a

^are ^thus

negativein

( G, lab )

^.Conatenatingthenegativearsofthe alternatingiruit and of allyles

C _a

⁽

a

^being ân âr ôf

F

ⁱⁿ ^the ^forbidden ^yle) ^we ^obtain ^a

yle in

G ^′ = (V, A(G) \ F )

^(see ^gure ³ ^below) ^whih ^ontradits

F

^being ^a

FAS of

G

^(as^well ^as^the ^fat^that

( G, lab )

^has ^no^negative ^yles).

2 −

+ +

a +

C _a

Fig.4.A forbidden ylein

(G R , lab R )

^with,surrounding it, thenegative yles

C a

mentioned intheproof of Theorem16. Arrows infull linerepresent ars,arrows in

dashedlinesreprensent paths.

Corollary 17 Let

(G, lab)

^be ân ûpdate ^digraph. Îf

N _{F AS}

^and

N _{M F AS}

^are,

respetively, the total number of

F AS

^and ^minimal

F AS

^of

G

^. ^Then,

N _{M F AS} ≤ |{[ s ] G | s

îs ûn ûpdate ^shedule ôf

V ( G )}| ≤ N _{F AS}

7 Projetions

Theorem 18 Let

G

^be ^a ^digraph ^and

G ^′

^a ^subdigraph ^of

G

^. ^Then,

(G ^′ , lab ^′ )

isan update digraph if and onlyif there existsa labeling funtion

lab

^of

A(G)

suh that

lab ^′ = lab _G ′

^.

PROOF. Obviously, if

(G, lab)

îs ân ûpdate ^digraph ând

lab ^′ = lab _G ′

^, ^then

(G ^′ , lab ^′ )

îs âlsoânûpdate ^digraph^by ^Theorem ^10.

Conversely, if

( G ^′ , lab ^′ )

îs ân ûpdate ^digraph ^we ^will ^show ^that ^for âll

a = (u, v ) ∈ A(G) \ A(G ^′ )

^,^either

(G ^′ + a, lab ⁺ _a )

^or

(G ^′ + a, lab ⁻ _a )

îsânûpdate

digraph,where

V (G ^′ + a) = V (G ^′ ) ∪ {u, v }

^,

E(G ^′ + a) = E(G ^′ ) ∪ {a}

^and

lab ⁺ _a

(14)

and

lab ⁻ _a

^are ^dened ^by

lab ⁺ _a (e) = lab ⁻ _a (e) = lab(e), ∀e ∈ A(G ^′ )

^,

lab ⁺ _a (a) =

⁺

and

lab ⁻ _a (a) =

^-^.

Let us suppose that there exists

a = ( u, v ) ∈ A ( G ) \ A ( G ^′ )

^suh ^that ^nei-

ther

(G ^′ + a, lab ⁺ _a )

^nor

(G ^′ + a, lab ⁻ _a )

âre ûpdate ^digraphs. ^Then^there êxist â

forbidden yle

C ₁ : x ₁ = u, x ₂ = v, x ₃ , . . . , x _p = u

^with

lab ⁺ _a (x j , x _j+1 ) =

^-

in the reoriented labeled digraph

(( G + a ) R , ( lab ⁺ _a ) R )

^. ^In ^the ^same ^way^,

there exists a forbidden yle

C ₂ : y ₁ = v, y ₂ = u, y ₃ , . . . , y _q = v

ⁱⁿ ^the ^re-

oriented labeled digraph

((G + a) R , (lab ⁻ _a ) R )

^. ^Hene, ^la ^sequene ^of ^nodes

x ₂ = v, . . . , x _j , x _j+1 , . . . , x _p = u = y ₂ , . . . , y _q = v

ⁱⁿ ^the ^reoriented ^labeled

digraph

(G R , lab _R )

ôntains â ^yle înluding ^the âr

(x j , x _j+1 )

^(see ^Fig...),

that is a forbidden yle. Thus

(G, lab)

îs ^not ûn ûpdate ^digraph, ^whih îs â

ontradition.

Therefore, if

A(G) \ A(G ^′ ) = {a ₁ , . . . , a _r }

^, ^then ^by ^indution ^we ^an ^prove

that for all

k

ⁱⁿ

{1, . . . , r}

^there ^exists ^a ^labeling ^funtion

lab ^k

ôf ^the ârs ôf

G ^′ + a ₁ + . . . + a _k

^suh ^that

lab ^k _G ′ = lab ^′

^and

( G ^′ + a ₁ + . . . + a _k , lab _k )

^is ^an

update digraph. In partiular, there exists a labeling funtion

lab

ⁱⁿ

G

^suh

that

(G, lab)

îs ân ûpdate ^digraphând

lab ^′ = lab _G ′

^.

2 C ₂

x j

x _j+1 G R

u v

C ₁

−

Fig.5. Sheme of theforbidden yle in

(G R , lab R )

^mentioned ⁱⁿ ^the^proof ^of^The-

orem 18.

Corollary 19 Let

G

^be â ônneted ^digraph ôf

n > 1

^verties. ^Then,

2 ⁿ⁻¹ ≤ |{(G, lab) : (G, lab)

^is ^update ^digraph

}| ≤ T _n

where

T _n =

n− 1

X

k=0

n k

!

T _k .

PROOF. From Theorem 18for alldigraph

G

^and ^subdigraph

G ^′

^,

|{(G ^′ , lab ^′ ) : (G ^′ , lab ^′ )

^is^UD

}| ≤ |{(G, lab) : (G, lab)

^is^UD

}|.

On other hand, the onneted digraphof

n

^verties ^with ^the ^least ^number^of

ars, equalto

n − 1

^, îs ânôriented ^tree.În ^this âse, âll ^labeling ^funtionsôn

(15)

the digraph yield an update digraph. Thus, there are

2 ⁿ⁻¹

^update ^onneted

digraphs with the least number of ars. In the same way, the onneted di-

graph of

n

^nodes ^with ^the ^greast ^number ôf ârs, êqual ^to

n ²

^(inluding ^the

ars

(u, u)

^), îs ^the ômplete ^digraph. În ^this âse, êah ^labeling ^funtion ôn

a omplete digraphdene a total preorder on the verties. Besides, it is well

known that thenumberoftotal preorders onaset of

n

^elements^is

T _n

^dened

as in the statement of this Theorem. Thus,

T _n

^is ^the ^maximum ^number ^of

update onneted digraphswith

n

^verties.

2

8 Aknowledgments

ThisworkwaspartiallysupportedbyFONDECYTprojet1061008(J.A.),by

FONDAP andBASALprojetsCMM,UniversidaddeChile,byCentrode In-

vestigaión en IngenieríaMatemátia (

CI ² M A

^), Universidadde Conepión, and by CONICYT fellowship (M.M.).

Referenes

J. Araena, E. Goles, A. Moreira, L. Salinas, On the robustness of update

shedulesin Boolean networks, Biosystems 97 (2009)18.

S. Kauman,Metabolistability and epigenesis in randomlyonneted nets,

Journalof Theoretial Biology22 (1969)43767.

R.Thomas, Booleanformalizationof genetiontroliruits, Journalof The-

oretialBiology 42(1973)563585.

I. Shmulevih, H. Lähdesmäki, E. Dougherty, J. Astola, W. Zhang, The role

ofertainPost lasses inBooleannetworkmodels ofgenetinetworks,Pro-

eedingsoftheNationalAademyodSienesUSA100(2003)1073410739.

R. Thomas, Regulatory networks seen as asynhronous automata: A logial

desription,Journal of TheoretialBiology 153 (1991) 123.

M. Chaves, R. Albert, E.Sontag, Robustness and fragilityof Booleanmodels

for geneti regulatorynetworks, Journal of Theoretial Biology 235 (2005)

431449.

H. Mortveit, C. Reidys, Disrete, sequential dynamial systems, Disrete

Mathematis226 (2001)281295.

A.Hansson,H. Mortveit, C. Reidys,On asynhronous ellularautomata,Ad-

vanes inComplex Systems 8(4) (2005)521538.

J.Demongeot,A.Elena,S.Sen,RobustnessinRegulatoryNetworks:aMulti-

DisiplinaryApproah,Ata Biotheoretia 56 (1-2)(2008) 2749.

J. vanLeeuwen, Handbook of theoretialomputer siene, Elsevier, 1990.

(16)

NP-ompleteness, W.H. Freeman, 1979.

One keyelement in thesemodels is theupdateshedule, whih indiates the order in whih states have to be updated

a, ∗

c

a

b

a CI 2 M A

b

c

∗

G = ( V, A )

V = {1 , . . . , n}

A

V

G

V (G)

A ( G )

v 1

v m

G

v 1 , v 2 , . . . , v m

V (G)

∀k = 1, . . . , m − 1, (v k , v k+1 ) ∈ A(G)

( v k+1 , v k ) ∈ A ( G )

v 1

v m

v 1 , v 2 . . . , v m

(v k , v k+1 ) ∈ A(G)

k = 1, . . . , m − 1

G

G = (V, A)

i ∈ V

N (i) = {j ∈ V | (j, i) ∈ A}

i

d − (i) = |N (i)|

i

[[ a, b ]] = {a, . . . , b}

[[ a, b [[= {a, . . . , b − 1}

a

b

G = (V, A)

|V | = n

s : V → {1 , . . . , n}

s ( V ) = [[1 , m ]]

m ≤ n

∀i ∈ V, s(i) = 1

s = s p

s

{1, . . . , n}

s

s

n

n

T n =

n − 1

X

k=0

n k

!

T k .

G = (V, A)

s

s − 1 (r) = {i ∈ V | s(i) = r}

G = (V, A)

s

lab s : A → { ,

}

∀( j, i ) ∈ A, lab s ( j, i ) =







s(j ) ≥ s(i)

s(j ) < s(i).

a ∈ A

lab s (a) =

a ∈ A

lab s (a) =

a

A

lab s ( a )

( G, lab s )

a CI ² M A

v ₁

v _m

v ₁ , v ₂ , . . . , v _m

∀k = 1, . . . , m − 1, (v k , v _k+1 ) ∈ A(G)

( v _k+1 , v _k ) ∈ A ( G )

v ₁

v _m

v ₁ , v ₂ . . . , v _m

(v k , v _k+1 ) ∈ A(G)

d ⁻ (i) = |N (i)|

s = s _p

T _n =

T _k .

s ⁻ ¹ (r) = {i ∈ V | s(i) = r}

lab _s : A → { ,

∀( j, i ) ∈ A, lab _s ( j, i ) =

lab _s (a) =

lab _s (a) =

lab _s ( a )

( G, lab _s )

N _s ⁺ (i) = {j ∈ N (i) | lab _s (j, i) = }

N _s ⁻ (i) = {j ∈ N (i) | lab _s (j, i) = }

N (i) = N _s ⁺ (i) ∪ N _s ⁻ (i)

F : Q ⁿ → Q ⁿ

f _i : Q ⁿ → Q

∃ ( x ₁ , . . . , x _n ) ∈ Q ⁿ , a, b ∈ Q

f _i ( x ₁ , . . . , x _j− ₁ , a, x _j+1 , . . . , x _n ) 6= f _i ( x ₁ , . . . , x _j− ₁ , b, x _j+1 , . . . , x _n ) .

x ^r+1 _i = f _i ( x ^l ₁ ¹ , . . . , x ^l _n ⁿ ) ,

l _j = r

l _j = r + 1

F ^s : Q ⁿ → Q ⁿ

F ^s (x) = (f ₁ ^s (x), . . . , f _n ^s (x))

f _i ^s (x) = f i (g _i,1 ^s (x), . . . , g _i,n ^s (x)),

g _i,j ^s

g _i,j ^s (x) = x _j

g ^s _i,j (x) = f _j ^s (x)

F ^s

N ₁ = ( G, F, s ₁ )

N ₂ = (G, F, s ₂ )

F ^s ¹ = F ^s ²

N ₁ = (G, F, s ₁ )

N ₂ = (G, F, s ₂ )

N ₁

N ₂

s ^′

s ∼ ^G s ^′

[s] G = {s ^′ : (G, lab s ) = (G, lab s ^′ )}.

lab = lab _s

lab = lab _s