\section{DYNAMIC INTERACTION NETWORK}
%\section{DYNAMIC INTERACTION NETWORK \\{\bf BASED ON INFLUENCE TABLE}}

The integration of the updating flow in the geographical data base starts the dynamics of the system evolution. Let's recall that updating flow is structured as a set of transactions, each one is a set of canonical operations sequences. So, each canonical operation, applied on an object, activates some falls of updating operations to apply on other objects. Although the starting operation is known, since it is a membership of the input updating flow, the other consequent operations, called influence operations, are not known. Objects to which these operations are applied, called influenced objects, are not either known. \\

Using the previous formalism, we will show in this section (i) how influenced objects are defined, (ii) how influence operations are computed and (iii)  how these objects interact.  We will show that all the computation can be summarized inside a Table of Influences that cross the connection graph to build an interaction network.


% We describe in this section the main processus which leads to implement the updating operations. Using the previous formalism, we will show that all the computation can be summarized inside a Table of Influences that cross the connection graph to build an interaction network.

\subsection{Topological influence area and connection graph}

We consider the target semantic object $O_c$ to be updated. We define the {\em Topological Influence Area} as the following:   
% Etant donné un objet sémantique à mettre à jour et $O_c$ un objet cible, la notion de zone d'influence topologique permet de recupérer les objets candidats qui sont liés topologiquement avec l'objet à mettre à jour et qui appartiennent à l'$O_c$-voisinage. Cette zone sert à délimiter la zone géographique englobant l'objet cible dont les objets pourront \^etre affectés par la mise à jour de l'objet cible.

% \begin{definition}
% Soit $Os$ un objet de la base de données géographique et $O_c$ un objet cible, la {\it zone d'influence topologique}, notée $ \mathcal{Z}^{it}(Os)$, regroupe tous les objets de l'$O_c$-voisinage de $Os$, liés par une relation topologique avec $Os$.
% \end{definition}

\begin{definition}
We consider $O_c$ a target object to be updated from a GDB. Its {\em Topological Influence area} that we note as $ \mathcal{Z}^{it}(Os)$, collects all the GDB objects which are inside a neighbourhood of $O_c$ and which are linked to $0_c$.
\end{definition}

\begin{exemple}
Figure \ref{ZoneInfluence} shows the influence area of the target object ID 9814 which is a camping. The table \ref{TableauInfluence} contains a part of the set of the ID objects which belong to his influence area. For each ID object, its corresponding class and the topological relation with the target object are given.  
\end{exemple}
 
We represent the topological influence area by a connection graph which connects the objects linked by a topological relation.

\begin{exemple}
Figure \ref{connectionGraph} shows some element of the connection graph corresponding to some relation from the target object ID 9814, based on the table \ref{TableauInfluence}.
\end{exemple}
%
\begin{table}[!h]
\begin{center}
\begin{tabular}{|p{1.3cm}|p{3.5cm}|p{2cm}|}
\hline  ID    & CLASS    & RELATION\\
\hline  7130 & BATIMQCQ  &  contains\\
\hline  7135 & BATIMQCQ  &  contains\\
\hline  7134 & BATIMQCQ  &  contains \\
%\hline  7125 & BATIMQCQ  &  contains\\
%\hline  7125 & BATIMQCQ  &  contains\\
%\hline  7548 & BATIMQCQ  &  contains\\
\hline  3640 & ROUTE\_TR  &  borders  \\
\hline  2264 & ROUTE\_TR  &  borders  \\
%\hline  2273 & ROUTE\_TR  &  borders  \\
%\hline  2268 & ROUTE\_TR  &  borders  \\
%\hline  2422 & ROUTE\_TR  &  borders  \\
%\hline  2274 & ROUTE\_TR  &  borders  \\
%\hline  2232 & ROUTE\_TR  &  borders  \\
\hline  452 & CARREFOURNA  & touchs   \\
\hline  453 & CARREFOURNA  & touchs   \\
%\hline  455 & CARREFOURNA  & touchs   \\
%\hline  414 & CARREFOURNA  & touchs   \\
%\hline  436 & CARREFOURNA  & touchs   \\
%\hline  435 & CARREFOURNA  & touchs   \\
%\hline  432 & CARREFOURNA  & touchs   \\
\hline ...& ... & ... \\
\hline
\end{tabular}
\end{center}
\caption{\label{TableauInfluence}\textit{Topological influence area from object ID 9814}}
%Exemple donnant l'ensemble des objets appartenant à la zone d'influence de l'objet %d'identifiant 9814.}}
\end{table}

%

\begin{center}
\begin{figure*}[htpb]
\centerline{\epsfig{figure=ZoneTravail9814.eps,width=11cm}}
\caption{Topological influence area from object ID 9814}
%Zone d'influence topologique relative à l'objet 9814 extrait de la BDTopo\textregistered 96 ; zone d'Angers.}
\label{ZoneInfluence}
\end{figure*}
\end{center}
%

\begin{center}
\begin{figure}[htpb]
\centerline{\epsfig{figure=connectionGraph.eps,width=8.5cm}}
\caption{Connection graph (extract from object ID 9814)}
%Zone d'influence topologique relative à l'objet 9814 extrait de la BDTopo\textregistered 96 ; zone d'Angers.}
\label{connectionGraph}
\end{figure}
\end{center}

\subsection{Table of influences} 

Using the canonical decomposition of updating informations, we can define a {\it table of topological influences} which allows to obtain, from an operation applied to an specific object, what are the effects on the other objects in its neighbourhood. Corresponding to the 3 geometric primitives and the 4 canonical operations, we obtain a finite table which can describe all the situations.   

\begin{table}[htbp]
\begin{center}
\begin{tabular}{|p{0.55cm}|p{0.55cm}|p{1.3cm}|p{1.2cm}|p{2.3cm}|}
\hline   $p$     & $p'$    & $Op(p)$     & $R^T(p,p')$   & Influence \\
\hline \hline   Area   & Area & Suppress. & In        & \{ID, Suppress.\} \\
\hline
\end{tabular}
\caption{One line in the table of topological influences}
\label{toti}
\end{center}
\end{table}

In the example corresponding to one line of the table of topological influences shown in the table \ref{toti}, the situation is the suppression of the area object $p$ which constraints another area object $p'$ and that can lead to one of the following operations for this second object, that is to remove it or not. To know what is the resulting operation, we need to explore the semantic influence which is itself managed by another {\it table of semantic influences}. \\


\subsection{From Connection Graph to Interaction Network}  

Because of the canonical decomposition, all the updating processes are summarized in the influence table and the application of each line of this table can be considered as a rule to be applied to the connected graph which describes the objects of the GIS  and the relations between them. \\

Using the influence table, we apply some rules and activate some interactions between GIS objects relatively to the updating operation that we presently manage. In that way, we build an interactive network that is developed dynamically during the processus \cite{barabasi}. The new elements aggregate to this interactive network will generate new processes. The complete algorithmic description is made in the following section.\\



\begin{figure*}[htbp]
\begin{center}
%\rotatebox{-90} 
{\includegraphics[scale=0.8]{graph2network.eps}}
\caption{From cellular automata to interaction networks}
\label{graph2network}
\end{center}
\end{figure*}

% We make some comparisons between the processus used here with the one usually use with cellular automata. These comparisons are summarized in figure \ref{graph2network}. The connection graph which represent the GIS objects in relation can be compared to the cellular automon mesh. The two describe potential connection. In some way, we can consider the mesh and the graph as dual representation. The rules for cellular automata and the influence table applications make activated interactions that are represented in the part (b) of the figure. 

The processus described here and which consists to use an inert support (the connected graph) to build a rules-activated dynamic structures (the interaction network), can be compared with evolutive processes on cellular automata. In this last case, the inert support is the mesh and the rules-activated dynamic structure is the interaction cells network which activates some of them as alive (see figure \ref{graph2network}). This comparaison is here strictly formal and its goal is to explain that, like in cellular automata, some emergent organizations can appear inside the interaction network. The emergent property described in paragraph 6 is one of such example.
