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\begin{document}
\section{Introduction}
The purpose of this document is to describe the implementation the
classes used to represent graphs in the \Rpackage{graph} package and
to discuss design issues for future development.
There are many different ways to represent a graph and to deal with
the edges and nodes within that graph. Below we discuss the graph
representations implemented in the \Rpackage{graph} package and define
the set of methods that form the \textit{graph interface} as
determined empiracally by the methods used by packages like
\Rpackage{RBGL} when interacting with \Robject{graph} objects.
A graph is a pair of sets, $G=(V,E)$ where $V$ is the set of nodes and
$E$ is the set of edges, which are determined by relationships that
exist between the nodes. If we let $n = |V|$, be the number of nodes
then, excluding self-loops there are at most $n$ choose 2 edges in
$G$. A \textit{simple graph} is a graph with at most one edge between
any pair of nodes and no self-loops.
\section{The \Rclass{graph} class}
The \Rclass{graph} class and its subclasses support simple graphs as
well as graphs with at most one self-loop on any given node. Not all
graph representations can easily support more general graphs.
Limiting to simple graphs with self-loops allows for reversible
conversions between different graph representations. Furthermore,
this limitation simplifies the interface of edge related methods which
would otherwise have to support ways of identifying one of many edges
between the same pair of nodes.
Arbitrary attributes can be associated with a graph, with a node, or
with an edge. For both nodes and edges if one edge or node has a
particular attribute then all nodes and edges must have that
attribute. Nodes and edges can have more than one attribute
associated with them.
\textit{This raises the question of whether we should use the
\Rclass{AnnotatedDataFrame} class from Biobase here as a way to
implement general node and edge attributes.}
\textit{However, currently AnnotatedDataFrame is based on a
data.frame and cannot easily support arbitrary attributes. Even
having a vector of length greater than one as the value of an
attribute could cause problems.}
The \Rclass{graph} class itself is VIRTUAL and has the following
definition:
<>=
library("graph")
getClass("graph")
@
The \Robject{edgemode} slot indicates whether the graph is
\textit{directed} or \textit{undirected}. Since some graph algorithms
only make sense in a directed graph, the edgemode is a property of the
entire graph, rather than a property of an edge.
The \Robject{graphData} slot was recently added to hold arbitrary
attributes for the graph. Although edgemode is such an attribute, it
isn't clear whether it should move inside the generic container since
edgemode is of such high semantic importance. It probably doesn't
matter as long as methods such as \Rfunction{isDirected} do the right
thing.
The \Robject{edgeData} and \Robject{nodeData} slots store the
attributes for the edges and nodes of the graph, respectively.
There are currently implementations for the \Rclass{graphNEL} class,
where nodes are a vector and edges are a list, each element of the
list correspondes to one node and the values are nodes corresponding
to the out-edges from that node. If the graph is directed then all
edges essentially appear twice.
The \Rclass{graphAM} class, which stores the edge information in an
adjacency matrix. The matrix must be square and the row names must
match the column names. If the graph is undirected then the matrix
must also be symmetric.
There are two specialized classes, \Rclass{distGraph} which takes a
distance matrix directly and has special thresholding capabilities. It
is not clear whether this should be a specialization of the
\Rclass{graphAM} class or not.
The second specialized class is a \Rclass{clusterGraph} which can be
used to represent the output of a clustering algorithm as a graph.
Samples represent nodes and all samples in the same cluster have
edges, while samples in distinct clusters do not. Instances of this
class must have their edgemode as \texttt{undirected}, if the edgemode
is reset then coercion to some other mode of graph is needed.
\subsection{Methods of graphs}
Here are some of the methods that all graph-like objects should support:
\begin{description}
\item[nodes(object)] Return a character vector of the node labels.
The order is not defined.
\item[nodes<-(object)] Return a new graph object with the node labels
set as specified by a character vector. This is slightly fragile
since here order does matter, but the order can only really be
determined by first calling \Rfunction{nodes}. by providing a
character vector of the appropriate length.
\item[addNode(node, object, edges)] Return a new graph object with
additional nodes and (optionally) edges. The methods that have been
implemented expect \Robject{node} to be the node labels of the new
nodes specified as a character vector. Optional edges can be
specified.
\item[removeNode(node, object)] Return a new graph object with nodes
(and their incident edges) removed. Current methods are implemented
for \Robject{node} being a character vector of node labels to remove.
\item[edges(object, which)] Return a list with an element for each
node in the graph. The names of the list are the node labels. Each
element is a character vector giving the node labels of the nodes
which the given element shares an edge with. For undirected graphs,
reciprocal edges should be included. This representation is very
similar to the NEL edgeL structure.
\item[edgeWeights(object, index)]
\item[addEdge(from, to, graph, weights)] Return a new graph object
with additional edges.
\item[removeEdge(from, to, graph)] Return a new graph object with the
specified edges removed.
\item[numNodes(object)] Return a count of the nodes in the graph.
\item[numEdges(object)] Return a count of the edges in the graph.
\item[isDirected(object)] Return TRUE if the graph is directed and
false otherwise.
\item[acc(object, index)] See man page.
\item[adj(object, index)] See man page.
\item[nodeData] Access to node attributes. See man page.
\item[edgeData] Access to edge attributes. See man page.
\end{description}
\subsection{Some Details}
Once both nodes and edges are instances of classes they will be quite
large. In order to reduce the storage requirements (especially for
large graphs) using integer indices may be beneficial.
The minimum amount of storage required is $|V|+|E|$. If we use an
incidence matrix representation then the storage is $|V|^2$.
If we use a node and edge list representation then the storage
requirements are $|V|+2|E|$. When either $|V|$ or $|E|$ are large
these mechanisms will not be especially efficient.
In some cases it may be better to keep the actual node and edge data
stored in hash tables and keep other integer vectors available for
accessing the necessary components.
\subsubsection{Representation of Edges}
\label{sec:edgerep}
We have taken the approach of allowing the representation of the edge
sets to not contain every node. When the graphs are sparse this can
be a fairly large savings in space, but it means that one cannot
determine the nodes in a graph from the edges in the graph.
Also, for the \Rclass{graphNEL} class we do not store the names of the
nodes in the NEL, but rather indexes into a the node vector. This is
important for allowing us to perform permutations on the nodes of a
graph, but causes a number of problems when subsetting graphs, and
means that knowledge of the edges does not provide knowledge of the
nodes.
\section{Multi-graphs}
There are no clear and widely used definitions for multi-graphs, so
here we will make clear a definition that we believe will be useful
for biological computations. We define a multi-graph to consist of
two components, one a set of nodes and the second a list of edge
sets. Each edge set corresponds to a potentially different set of
relationships between the nodes (which are common to all edge
sets). We denote this by $G=(V, E_L)$, where $V$ is the set of nodes
and $E_L = (E_1, \ldots, E_L)$ is a collection of $L$ edge sets. Each
with a potentially distinct set of relationships. The edge sets are
essentially identical to the edge sets for a graph, and hence can have
arbitrary attributes associated with them, the edges can be either
\textit{directed} or \textit{undirected} and self-loops are allowed.
It is not clear whether there should be distinct types of
multigraphs as there are graphs. It will surely be more flexible to
support a list of edge sets, and to allow these to have different
structures.
Current definition does not extend the \Rclass{graph} class. The
definition is:
<>=
getClass("multiGraph")
@
\begin{description}
\item[nodes] A vector of node identifiers.
%% FIXME: if these are node identifiers, then shouldn't we use
%% "character"? Elsewhere, there seems to be an assumption that
%% node labels or identifiers are character.
\item[edgeL] A possibly named list of instances of the \Rclass{edgeSet} class.
\end{description}
The \Rclass{edgeSet} class is a virtual class with several different
extensions. These include a \Rclass{edgeSetNEL} and an
\Rclass{edgeSetAM}, others will be added once the interface
stabilizes.
Edge attributes are in the edgeData slot in the edgeSet class. This
implies that edgeSets in a multiGraph can have completely unrelated
edge attributes. Another approach would be to maintain a list
conforming to the edgeSet list containing edge attributes that would
enforce the same attributes to be defined for all edges in the
multiGraph.
\subsection{Methods}
In some ways it would be most natural to have \Robject{edges} methods
for the \Rclass{edgeSet} class the issues raised in
Section~\ref{sec:edgerep} seem to preclude this and it only seems to
make sense to have \Robject{node} and \Robject{edges} methods for the
\Rclass{multiGraph} class.
It will probably make sense to be able to name the edgeSets within a
multiGraph and to be able to extract graph objects from the multiGraph
representing any of the edgeSets.
There should be methods to produce graph objects based on
intersection, union, and more complex combination algorithms. The
edgeSets may represent interaction data with reliability estimates as
edge weights. The user may want to produce a graph object combining
the available data to obtain most reliable edges.
We may want to consider apply type operations to apply an operation
across all edgeSets in a multiGraph.
\subsection{Use Cases}
An important motivator for the \Rclass{multiGraph} class is the
representation of data from protein interaction experiments. Our goal
is to represent these data in terms of what interactions were tested,
and of those which ones are either positive or negative.
\section{Bipartite Graphs}
A bipartite graph graph is a graph where the nodes can be divided into
two sets, say $V_1$ and $V_2$, such that all edges are between
members of $V_1$ and members of $V_2$ and there are no edges between
any two elements of $V_1$, nor of $V_2$.
\end{document}