INTRODUCTION
Elementary Net System (ENS) known as Conditions/events systems and have been
defined by Thiagarajan (1986), Rozenberg
(1986) and Andre (1989) to describe easily intrinsic
characteristics of realtime systems such as high degree of parallelism and
synchronisation. This model is derived from low level Petri net model.
Although, Petri nets and ENS where been defined several years ago, nothing
was known about their decomposition into subnetworks or the must influential
event or the notion of centrality. The idea of centrality as applied to human
communication was introduced by Bavelas (1948). It is
defined on social network represented as simple graph. We propose to use the
notion of betweenness centrality used for detecting communities in social large
network (Girvan and Newman, 2002) to detect the most
influential event in ENS. To find community (subnetwork) structures in graphs,
the interest in the domain has started with a new divisive approach proposed
by Girvan and Newman (2002), the edges with the largest
betweenness (number of shortest paths passing through an edge) are removed one
by one in order to split hierarchically the graph into communities.
The behavior of Metabolic network is well modelled and analyzed by Petri Nets
(Reddy et al., 1993). The introduction of the
Petri nets to represent the biochemical reactions was carried by many researchers
(Hofestaedt and Telen, 1998; Goss
and Peccoud, 1998; Matsuno et al., 2000;
Chen, 2004) they showed that the graphic structure (static)
of the Petri nets easily (qualitatively) models the interactions in a cell:
the places represent the substances, the transitions represent the reactions,
the arrows the tokens and the weights of the arrows model the quantity of the
substances which is subjected to a reaction to produce a quantity of products
ensuring the expression of temporal evolution of these networks (dynamics).
The execution of Petri nets allows the quantitative analysis (kinetic) (Hofestaedt
and Telen, 1998) of these networks. Structural analysis of large metabolic
network (Schuster et al., 2002) meets with the
problem of combinatorial explosion. A first work to decompose a metabolic network
into subnetwork was proposed by Schuster et al.
(2002). It considers the metabolites degree of network connectivity. Then
others works followed (Holme et al., 2003; Ma
et al., 2004; Ravasz et al., 2002).
All these works represent the metabolic network as a simple graph.
In this study, we investigate an approach to compute betweenness centrality
measure on the model of ENS. As the behavior of metabolic network is well represented
by Petri nets, we propose to use ENS to model it (substances/enzymatic reactions).
This aims to deconstruct the metabolic network into subnetworks based on the
global network structure by using the centrality index. Identifying the modular
organization of metabolic network by certain network decomposition methods can
help us better understand the organization principle of complex metabolic network.
An essential tool for the analysis of social networks are centrality indices
defined on the vertices of the graph (Freeman, 1979;
Newman and Girvan, 2003). They are designed to rank the
actors according to their position in the network and interpreted as the prominence
of actors embedded in a social structure. Many centrality indices are based
on shortest paths linking pairs of actors (Brandes, 2001;
Girvan and Newman, 2002), measuring, e.g., the average
distance from other actors, or the ratio of shortest paths an actor lies on.
Many networkanalytic studies rely at least in part on an evaluation of these
indices. All of them are based on the same idea. If two communities are joined
by only a few intercommunity edges, then all paths through the network from
vertices in one community to vertices in the other must pass along one of those
few edges. Given a suitable set of paths, one can count how many go along each
edge in the graph and this number we then expect to be largest for the intercommunity
edges, thus providing a method for identifying them.
To apply this measure, we first represent ENS as a graph of events in which
events are defined as nodes and conditions as depending connections between
the nodes. Secondly we adapt the definition of shortest path to compute centrality
indices on this graph and we propose to use Brandes (2001)
algorithm.
MATERIALS AND METHODS
Elementary net systems: Elementary net systems are based on low level Petri nets Known as conditions/events systems. Places also called conditions are allowed to contain at most one mark. Consequently, transitions, also called events, may fire not only if input places contain one mark but also if output places are empty. Obviously, such nets are bounded. Note that the firing takes no time, as for reactive systems.
Definition 1: An ENS is a tuple E= (G,M0) where G and M0 are the Petri part:
G = (S,R,Prec,Post) where S is the set of conditions and R is the set of events, the set Prec of arrows represents directed link from preconditions to event and the set Post of arrows represents directed link from event to postconditions.

Fig. 1: 
An elementary net system 
M0 is the initial marking M: S→ {0,1} (number of marks in a condition of set S is zero or one).
Prec(r) = °r where °r represent the set of preconditions needed by an event r, Prec⊆ Rxpartyset (S).
Post(r) = r° where r° represent the set of postconditions of an event r, Post ⊆ Rx partyset (S).
Example 1: Figure 1 shows an elementary net system with R1, R2, R3, R4, R5 are events presented by rectangles. s1, s2, s3, s4, s5, s6 are conditions presented by circles. And one mark in s1 (Fig. 1).
Definition 2: An event r is enabled iff all its preconditions are enabled
The firing of an enabled event r from marking M results in marking M’ according to the following:
Events graph: A directed depending events graph is defined as follows:
• 
A directed link from event node A to event node B is added
iff all preconditions of B composed with postcondition of the event A.
We say an event B can occur (to fire) only after A (if B is connected only
with A) or B depend on A 
• 
An event B can occur only after the events A and C iff preconditions
of B composed with postconditions of A and postconditions of C 
• 
An even B can occur only after A1, A2…and An iff preconditions
of B are composed with post conditions of A1 and A2 …An 
Definition 3: Let G = (S, R, Prec, Post) be an ENS. Gr = (R∪T,P) is the directed event graph of G, if the following conditions hold for all z and r in R and all t in T:
In an events graph Gr = (R ∪ T, P), we denote by Pre and Succ the following sets:
Example 2: Figure 2 shows the events graph of Fig. 1.
Shortest path and betweenness Shortest path: A quantity of interest in many social network (modelled with a graph) studies is the betweenness of an actor i, which is defined as the total number of shortest paths between pairs of actors that pass through i. This quantity is an indicator of who the most influential people in the network are, the ones who control the flow of information between most others. We are interesting to do this idea for detecting the most influential event in the events graph of an ENS.
As the events graph is composed with two types of nodes (R or T nodes) an adequate shortest path definition is necessary.
Let w be a weight function on the arrow. We define:
Weights are used to measure, the strength of a link. Define a path from s ∈
R to r ∈ R as an alternating sequence of vertices and edges, beginning
with s (firing the event s) and ending with r (r is enabled), such that each
edge connects its preceding with its succeeding vertex. We use d (s, r) to denote
the distance between vertices s and r, i.e., the minimum length of any path
connecting s and r in Gr.
By definition, d (s, s) = 0 for every s ∈ R and d (s, r) = d (r, s) for s, r ∈ R. We assume familiarity with standard algorithms for shortestpaths problems.
The minimal path of an event t ∈ T to occur after firing an event s is the maximal value of shortest paths of a succeeding events such that preconditions of t are taken on and all events preceding t can be enabled with event s. We consider here the shortestpath as the longest succeeding shortest path of events which makes some preconditions of t taken on. In other words:
Definition 4:
Centrality indices based on shortest paths: Several measures capture variations on the notion of a vertex's importance in a graph. Let σ_{st} denote the number of shortest paths from s ∈ R ∪ T to t ∈ R ∪ T, where σ_{ss} = 1 by convention. Let σ_{st} (v) denote the number of shortest paths from s to t, that some v ∈ R ∪ T lies on.
Given pairwise distances and shortest paths counts, the pairdependency
of a pair s, t ∈ R∪ T on an intermediary v∈ R∪ T, i.e. the ratio of shortest paths between s and t that v lies on, is given by:
To obtain the beetweeness centrality index of a vertex v, we simply have to sum the pairdependencies of all pairs on that vertex,
Therefore, betweenness centrality is determined (Brandes,
2001) in two steps:
• 
Compute the length and number of shortest paths between all
pairs 
• 
sum all pairdependencies 
RESULTS
We adapt the faster Algorithm defined by Brandes according with our definition of shortest path (definition 4). In Brandes algorithm, the betweeness centrality can be computed in O(nm + n2 log n) time and O(n+m) space for weighted graphs. Where n is the number of vertices and m the number of links. For unweighted graphs, running time reduces to O(nm). Brandes count shortest paths using traversal algorithms. Both breadthFirst Search (BFS) for unweighted and Dijkstra's algorithm for weighted graphs start with a specified source vertice and, at each step, add a closest vertex the set of already discovered vertices in order to find shortest paths from the source to all other vertices. He determines the beetweeness centrality index by solving one single source shortestpaths problem for each vertex. At the end of each iteration, the dependencies of the source on each other vertex are added to the centrality score of that vertex. Our adaptation doesn’t modify Brandes’s algorithm complexity. It takes in count tow types of nodes (s ∈ R ∪ T) and applies definition 4 to determine the length of shortest paths between nodes.
Example 3: Figure 3 shows the beetweeness index of Fig. 2.

Fig. 3: 
The betweeness indices 
Application to metabolic network: The metabolic networks are represented by the biochemists as complex graphs whose nodes constitute the metabolites and the arrows represent the reactions which are catalysed by specific molecules called enzymes. The structural (topological) analysis of metabolic pathways only uses information about stochiometric structure and the reversibility or irreversibility of enzymatic reactions. It consists in computing a minimal enzymes set that describe the conical steady state solution space for flux distribution in a metabolic network.
In the modelling of metabolic systems, usually a distinction is made between external and internal metabolites. External metabolites are the sources and sinks of the network; their concentrations are assumed to be buffered. Internal metabolites (intermediates) have to be balanced with respect to production and consumption at steady state. In many cases, there are specific biochemical reasons to treat a metabolite as internal (for example, when it is known to be merely an intermediate) or as external (e.g., nutrient or excreted product).
Classification of metabolites is not clear: an external metabolite is considered
to be present in large excess so that its concentration is virtually unaffected
by the reactions under study. An internal metabolite, however, must fulfil a
balance equation implying that production is equals consumption. A first work
to classify metabolites was proposed by Schuster et al.
(2002). It considers the metabolite degree of network connectivity. Above
a given connectivity threshold, some metabolites are considered as external.
Then others works followed to decompose metabolic networks (Holme
et al., 2003; Ma et al., 2004;
Ravasz et al., 2002). Identifying the modular organization of metabolic
network by certain network decomposition methods can help us better understand
the organization principle of complex systems. Here we represent a metabolic
network as an ENS and than represent it as a reaction depending graph by definition
3. The connectivity in this graph describes the sequences and the dependency
of enzymatic reactions and not only connectivity.
Definition 5: A metabolic network is represented as ENS = (S, R, Prec, Post) where S is the set of metabolites and R is the set of enzymatic reactions, the set Prec of arrows represents directed link from metabolite substrates to reactions and the set Post of arrows represents directed link from reactions to products.
R = Rirev ∪ Rrev (irreversibles and reversibles enzymatics reactions)
Prec(r)=°r where °r represent the set of substances needed by a reaction r, Prec⊆Rx party set(S).
Post(r) = r° where r° represent the set of products of a reaction
r, Post⊆Rxpartyset(S)
Example 4: The pentosephosphatepathway (PPP) (Fig. 4,
5). In this example we represent the PPP by ENS and give the
corresponding reaction depending graph.

Fig. 4: 
Metabolic network with ENS 

Fig. 5: 
The reaction depending graph of metabolic network of Fig.
4 
S={ G6P, NADP, 6PL, NADPH, H, H2O, 6GP, CO2, Ru5P, Ru5P, R5P, Xu5P, R5P,GAP,
S7P, F6P, E4P}
R={ g6pdh, 6pgl, 6pgd, rpi, rpe, tkt, tal}
g6pdh: G6P + NADP _ 6PL + NADPH + H
6pgl: 6PL + H2O _6GP + H
6pgd: 6GP + NADP _ NADPH + CO2 + Ru5P
rpi: Ru5P __ R5P rpi is reversible reaction we can represent it as rpi and rpi_rev
as follow:
rpi: Ru5P _ R5P
rpi_rev: R5P _ Ru5P
tkt: Xu5P + R5P _ GAP + S7P
tkt_rev: GAP + S7P _ Xu5P + R5P
tal: GAP + S7P _ F6P + E4P
tal_rev: F6P + E4P _ GAP + S7P
tkt2: Xu5P + E4P _ GAP + F6P
tkt2_rev: GAP + F6P _ Xu5P + E4P
DISCUSSION
This investigation of modelling the metabolic network as a reaction depending
graph and the application of the betweenness centrality algorithm helps to determine
the most influential reaction and eventually to decompose the metabolic network
into subnetworks. The strategy of representing the metabolic network as a reaction
depending graph (event graph definition 3) gives a graph which has only reactions
types of nodes (reactions without metabolites). This is important to apply the
Brandes algorithm. What makes the difference with the Holme
et al. (2003) work. We preferred to compute the betweenness centrality
on a graph which is primarily composed with only reaction nodes. Secondly, the
connectivity in this graph describes the sequences and the dependency of enzymatic
reactions (definition 3).
CONCLUSION
In this study we have investigated a centrality structure in ENS. This betweenness centrality is based on compute of the length and number of shortest paths between all pairs events and sum all pairdependencies. Our approach permits firstly to describe an ENS network as an event depending graph, secondly, to give a new definition of shortest path on graph which defined with tow reaction types of nodes and we have adapt the betweenness algorithm for our definitions. The use of ENS to model metabolic network permits to represent the metabolic network as a reaction (event) depending graph.