graph 1.80.0

The *graph* package provides an implementation of graphs (the kind
with nodes and edges) in R. Software infrastructure is provided by three
different, but related packages,

*graph* Provides the basic class definitions and functionality.

*RBGL* Provides an interface to graph algorithms (such as shortest
path, connectivity etc).

*Rgraphviz* Provides rendering functionality. Different layout
algorithms are provided and node plotting, line type, color etc parameters can
be controlled by the user.

A short description of the R classes and methods is given at the end of this document. But here, we begin by creating some graphs and performing different operations on those graphs.

The reader will benefit greatly from also having the *Rgraphviz*
package available and from using it to render the different graphs as they
proceed through these notes.

We will first create a graph and then spend some time examining some of the different functions that can be applied to the graph. We will create a random graph as the basis for our explorations (but will delay explaining the creation of this graph until Section 3.

First we attach the *graph* package and create a random graph (this
is based on the Erdos-Renyi model for random graphs).

```
library(graph)
set.seed(123)
g1 = randomEGraph(LETTERS[1:15], edges = 100)
g1
```

```
## A graphNEL graph with undirected edges
## Number of Nodes = 15
## Number of Edges = 100
```

We can next list the nodes in our graph, or ask for the degree (since this is an
undirected graph we do not distinguish between in-degree and out-degree). For
any node in `g1`

we can find out which nodes are adjacent to it using the `adj`

function. Or we can find out which nodes are accessible from it using the `acc`

function. Both functions are *vectorized*, that is, the user can supply a vector
of node names, and each returns a named list. The names of the list elements
correspond to the names of the nodes that were supplied. For `acc`

the elements of
the list are named vectors, the names correspond to the nodes that can be
reached and the values correspond to their distance from the starting node.

`nodes(g1)`

`## [1] "A" "B" "C" "D" "E" "F" "G" "H" "I" "J" "K" "L" "M" "N" "O"`

`degree(g1)`

```
## A B C D E F G H I J K L M N O
## 13 13 14 13 14 14 12 14 12 14 12 14 13 14 14
```

`adj(g1, "A")`

```
## $A
## [1] "N" "M" "E" "I" "O" "G" "H" "D" "K" "J" "C" "L" "F"
```

`acc(g1, c("E", "G"))`

```
## $E
## A B C D F G H I J K L M N O
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1
##
## $G
## A B C D E F H I J K L M N O
## 1 1 1 2 1 1 1 2 1 1 1 1 1 1
```

One can obtain subgraphs of a given graph by specifying the set of nodes that
they are interested in. A subgraph is actually a copy of the relevant part of
the original graph. A subgraph is the set of specified nodes plus any edges
between them. We can also compute the `boundary`

of a subgraph. The `boundary`

is the set of all nodes in the original graph that have an edge to the specified
subgraph. The `boundary`

returns a named list with one component for each node
in the subgraph. The elements of this list are vectors which contain all nodes
in the original graph that have an edge to that element of the subgraph.
We also demonstrate two edge related functions in the code chunk below. One
retrieves all edges from a graph and is called `edges`

while the other retrieves
the edge weights and is called `edgeWeights`

.

```
sg1 = subGraph(c("A", "E", "F", "L"), g1)
boundary(sg1, g1)
```

```
## $A
## [1] "C" "D" "G" "H" "I" "J" "K" "M" "N" "O"
##
## $E
## [1] "B" "C" "D" "G" "H" "I" "J" "K" "M" "N" "O"
##
## $F
## [1] "B" "C" "D" "G" "H" "I" "J" "K" "M" "N" "O"
##
## $L
## [1] "B" "C" "D" "G" "H" "I" "J" "K" "M" "N" "O"
```

`edges(sg1)`

```
## $A
## [1] "E" "L" "F"
##
## $E
## [1] "A" "F" "L"
##
## $F
## [1] "E" "A" "L"
##
## $L
## [1] "E" "F" "A"
```

`edgeWeights(sg1)`

```
## $A
## E L F
## 1 1 1
##
## $E
## A F L
## 1 1 1
##
## $F
## E A L
## 1 1 1
##
## $L
## E F A
## 1 1 1
```

The examples here originally came from Chris Volinsky at AT&T, but have been
modified in places as the *graph* package has evolved. In the code
chunk below we demonstrate how to create a graph from scratch. In this code
chunk two graphs are created, `gR`

and `gR2`

, the first is *undirected* while the
second is a *directed graph*.

```
V <- LETTERS[1:4]
edL1 <- vector("list", length = 4)
names(edL1) <- V
for (i in 1:4)
edL1[[i]] <- list(edges = c(2, 1, 4, 3)[i], weights = sqrt(i))
gR <- graphNEL(nodes = V, edgeL = edL1)
edL2 <- vector("list", length = 4)
names(edL2) <- V
for (i in 1:4)
edL2[[i]] <- list(edges = c(2, 1, 2, 1)[i], weights = sqrt(i))
gR2 <- graphNEL(nodes = V,
edgeL = edL2,
edgemode = "directed")
```

New graphs can be constructed from these graphs in many different ways but in
all cases the existing graph itself is not altered, but rather a copy is made
and the changes are carried out on that copy. Nodes and or edges can be added to
the graphs using the functions `addNode`

, `addEdge`

, `removeNode`

and
`removeEdge`

. All functions will take a vector of nodes or edges and add or
remove all of them at one time. One other function in this family is
`combineNodes`

, this function takes a vector of nodes and a graph and combines
those nodes into a single new node (the name of which must be supplied). The
function `clearNode`

removes all edges to the specified nodes.

```
gX = addNode(c("E", "F"), gR)
gX
```

```
## A graphNEL graph with undirected edges
## Number of Nodes = 6
## Number of Edges = 2
```

```
gX2 = addEdge(c("E", "F", "F"), c("A", "D", "E"), gX, c(1, 2, 3))
gX2
```

```
## A graphNEL graph with undirected edges
## Number of Nodes = 6
## Number of Edges = 5
```

```
gR3 = combineNodes(c("A", "B"), gR, "W")
gR3
```

```
## A graphNEL graph with undirected edges
## Number of Nodes = 3
## Number of Edges = 1
```

`clearNode("A", gX)`

```
## A graphNEL graph with undirected edges
## Number of Nodes = 6
## Number of Edges = 1
```

When working with *directed graphs* it is sometimes of interest to find the
*underlying* graph. This is the graph with all edge orientation removed. The
function `ugraph`

provides this functionality.

```
##find the underlying graph
ugraph(gR2)
```

```
## A graphNEL graph with undirected edges
## Number of Nodes = 4
## Number of Edges = 3
```

Other operations that can be carried out on graphs, that are of some interest,
are `unions`

, `intersections`

and `complements.`

We have taken a rather
specialized definition of these operations and it is not one that is widely
used, but it is very useful for the bioinformatics and computational biology
projects that we are working on.

For two or more graphs all with the **same nodes** we define:

`union`

to be the graph with the same set of nodes as the inputs and edges
between any two nodes that were connected in any one graph.

`intersection`

to be the graph with the same set of nodes as the inputs and with
edges between two nodes if there was an edge in all graphs.

`complement`

to be the graph with the same set of nodes as its input and edges
in the complement if there were none in the original graph.

In the code chunk below we generate a random graph and then demonstrate the
concepts of `union`

, `intersection`

and `complement`

.

```
set.seed(123)
gR3 <- randomGraph(LETTERS[1:4], M <- 1:2, p = .5)
x1 <- intersection(gR, gR3)
x1
```

```
## A graphNEL graph with undirected edges
## Number of Nodes = 4
## Number of Edges = 2
```

```
x2 <- union(gR, gR3)
x2
```

```
## A graphNEL graph with undirected edges
## Number of Nodes = 4
## Number of Edges = 4
```

```
x3 <- complement(gR)
x3
```

```
## A graphNEL graph with undirected edges
## Number of Nodes = 4
## Number of Edges = 4
```

Notice that while the graphs `gR`

and `gR2`

have different sets of edge weights
these are lost when the `union`

, `intersection`

and `complement`

are taken. It
is not clear how they should be treated and in the current implementation they
are ignored and replaced by weight 1 in the output.

Three basic strategies for finding random graphs have been implemented:

`randomEGraph`

A random edge graph. In this graph edges are randomly generated
according to a specified probability, or the number of edges can be specified
and they are randomly assigned.

`randomGraph`

For this graph the number of nodes is specified as well as some
latent factor. The user provides both the node labels and a factor with some
fixed number of levels. Each node is randomly assigned levels of the factor and
then edges are created between nodes that share the same levels of the factor.

`randomNodeGraph`

A random graph with a pre-specified node distribution is
generated.

The function `randomEGraph`

will generate graphs using the random edge model. In
the code chunk below we generate a graph, `g1`

on 12 nodes (with labels from the
first 12 letters of the alphabet) and specify that the probability of each edge
existing is 0.1. The graph `g2`

is on the same set of nodes but we specify that it
will contain 20 edges.

```
set.seed(333)
V = letters[1:12]
g1 = randomEGraph(V, .1)
g1
```

```
## A graphNEL graph with undirected edges
## Number of Nodes = 12
## Number of Edges = 7
```

```
g2 = randomEGraph(V, edges = 20)
g2
```

```
## A graphNEL graph with undirected edges
## Number of Nodes = 12
## Number of Edges = 20
```

The function `randomGraph`

generates graphs according to the latent variable
model. In the code chunk below.

```
set.seed(23)
V <- LETTERS[1:20]
M <- 1:4
g1 <- randomGraph(V, M, .2)
```

Our last example involves generating random graphs with a pre-specified node degree distribution. In the example below we require a node degree distribution of 1, 1, 2 and 4. We note that self-loops are allowed (and if someone wants to provide the code to eliminate them, we would be glad to have it).

```
set.seed(123)
c1 <- c(1,1,2,4)
names(c1) <- letters[1:4]
g1 <- randomNodeGraph(c1)
```

In addition to the simple algebraic operations that we have demonstrated in the
preceding sections of this document we also have available implementations of
some more sophisticated graph algorithms. If possible though, one should use the
algorithms provided in the *RBGL*.

The function `connComp`

returns a list of the connected components of the given
graph. For a *directed graph* or *digraph* the underlying graph is the graph
that results from removing all direction from the edges. This can be achieved
using the function `ugraph`

. A weakly connected component of a digraph is one
that is a connected component of the underlying graph and this is the default
behavior of `connComp`

.

`g1`

```
## A graphNEL graph with undirected edges
## Number of Nodes = 20
## Number of Edges = 58
```

```
g1cc <- connComp(g1)
g1cc
```

```
## [[1]]
## [1] "A"
##
## [[2]]
## [1] "B" "C" "D" "E" "F" "G" "I" "J" "M" "N" "O" "Q" "R" "S" "T"
##
## [[3]]
## [1] "H"
##
## [[4]]
## [1] "K"
##
## [[5]]
## [1] "L"
##
## [[6]]
## [1] "P"
```

```
g1.sub <- subGraph(g1cc[[1]], g1)
g1.sub
```

```
## A graphNEL graph with undirected edges
## Number of Nodes = 1
## Number of Edges = 0
```

Another useful set of graph algorithms are the so-called searching algorithm.
For the *graph* package we have implemented the depth first
searching algorithm as described in Algorithm 4.2.1 of Gross and Yellen (2005). More
efficient and comprehensive algorithms are available through the
*RBGL* package. The returned value is a named vector. The names
correspond to the nodes of the graph and the values correspond to the distance
(often the number of steps) or sum of the edge weights along the path to that
node.

`DFS(gX2, "E")`

```
## A B C D E F
## 1 2 5 4 0 3
```

We have found it useful to define a few special types or classes of graphs for some bioinformatic problems but they likely have broader applicability. All of the functions described above should have methods for these special types of graphs (although we may not yet have implemented all of them, please let the maintainer know if you detect any omissions).

First is the `clusterGraph`

. A cluster graph is a graph where the nodes are
separated into groups or clusters. Within a cluster all nodes are connected (a
complete graph) but between clusters there are no edges. Such graphs are useful
representations of the output of clustering algorithms.

```
cG1 <- new("clusterGraph", clusters = list(a = c(1, 2, 3), b = c(4, 5, 6)))
cG1
```

```
## A graph with undirected edges
## Number of Nodes = 6
## Number of Edges = 6
```

`acc(cG1, c("1", "2"))`

```
## $`1`
## [1] 1 2 3
##
## $`2`
## [1] 1 2 3
```

The other special type of graph that we have implemented is based on distances. This graph is completely connected but the edge weights come from inter-node distances (perhaps computed from an expression experiment).

```
set.seed(123)
x <- rnorm(26)
names(x) <- letters
library(stats)
d1 <- dist(x)
g1 <- new("distGraph", Dist = d1)
g1
```

`## distGraph with 26 nodes`

There are very many different ways to represent graphs. The one chosen for our basic implementation is a node and edge-list representation. However, many others use an adjacency matrix representation. We provide a number of different tools that should help users coerce graphs between the different representations.

Coercion from an adjacency matrix to a `graphNEL`

object requires a numeric
matrix with both row and column names. These are taken to define the nodes of
the graph and the edge weights in the resultant graph are determined by the
values in the array (weights zero are taken to indicate the absence of an edge).

The function `ftM2adjM`

converts a *from-to* matrix into an adjacency matrix.
Conversion to a `graphNEL`

graph can be carried out using the `as`

method for that
class.

An `aM`

is an affiliation matrix which is frequently used in social networks
analysis. The rows of `aM`

represent actors, and the columns represent events. A
one, 1, in the ith row and jth column represents the affiliation of the ith
actor with the jth event. The function `aM2bpG`

coerces a `aM`

into an instance
of the `graphNEL`

where the nodes are both the actors and the events (there is
currently no bipartite graph representation, although one could be added).

The two functions `sparseM2Graph`

and `graph2SparseM`

provide coercion between
`graphNEL`

instances and sparse matrix representations. Currently we rely on the
*SparseM* of Koncker and Ng for the sparse matrix implementation.

We briefly review some of the class structure here and refer the reader to the technical documentation for this package for more details.

The basic class, `graph`

, is a virtual class and all other classes will extend
this class. There are three main implementations available. Which is best will
depend on the particular data set and what the user wants to do with it. The
only slot defined in the virtual class is `edgemode`

which can be either
*directed* or *undirected* indicating whether the edges are directed or not.

The class `graphNEL`

is a node and edge-list representation of a graph. That is
the graph is comprised of two components a list of nodes and a list of the out
edges for each node.

The class `graphAM`

is an adjacency matrix implementation. It will be developed
next and will use the *SparseM* package if it is available.

The class `clusterGraph`

is a special form of graph for clustering. In this
graph each cluster is a completely connected component (a clique) and there are
no between cluster edges.

Gross, J. L., and J. Yellen. 2005. *Graph Theory and Its Applications*. 2nd ed. Textbooks in Mathematics. CRC Press, Taylor & Francis Group. http://www.crcpress.com.