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\begin{document}
\title{graphBAM and MultiGraph classes.}
\author{N. Gopalakrishnan}
\maketitle

\section{graphBAM class}

\subsection{Introduction}
The \Rclass{graphBAM} class has been created as a more efficient replacement for the 
\Rclass{graphAM} class in the \Rpackage{graph} package. The adjacency matrix in 
the \Rclass{graphBAM} class is represented as a bit array using a \Rcode{raw}
vector. This significantly reduces the memory occupied by graphs having a large 
number of nodes. The bit vector representation also provides advantages in terms
of performing operations such as intersection or union of graphs.

We first load the \Rpackage{graph} package which provides the class definition
and methods for the \Rclass{graphBAM} class.

<<loadGraph>>=
library(graph)
@

One  of the arguments \Rcode{df} to the \Rclass{graphBAM} constructor is a
\Robject{data.frame} containing three columns: "from","to" and "weight", each 
row in the \Robject{data.frame} representing an edge in the graph. The
\Rcode{from} and \Rcode{to} columns can be character vectors or 
factors, while the \Rcode{weight} column must be a numeric vector. The argument
\Rcode{nodes} are calculated from the unique names in the \Rcode{from} and \Rcode{to}
columns of the \Robject{data.frame}. The argument \Rcode{edgeMode} should be a 
character vector, either "directed" or "undirected" indicating whether the graph 
represented should be directed or undirected respectively.

\subsection{ A simple graph represented using graphBAM class}

We proceed to represent a simple graph using the \Rclass{graphBAM} class. Our example 
is a directed graph representing airlines flying between different cities. In this 
example, cities represent the nodes of the graph and each edge represents a flight 
from an originating city (\Rcode{from}) to the destination city (\Rcode{to}). The
weight represents the fare for flying between the \Rcode{from} and \Rcode{to} 
cities.

<<creategraphBAM>>=
df <- data.frame(from = c("SEA", "SFO", "SEA", "LAX", "SEA"),
                   to = c("SFO", "LAX", "LAX", "SEA", "DEN"),
                 weight = c( 90, 96, 124, 115, 259),
                 stringsAsFactors = TRUE)
g <- graphBAM(df, edgemode = "directed")
g 
@

The cities (nodes) included in our \Rclass{graph} object as well as the stored 
fares(\Rcode{weight}) can be obtained using the \Rmethod{nodes} and 
\Rmethod{edgeWeights} methods respectively.

<<nodeAndWeights>>=
nodes(g)
edgeWeights(g, index = c("SEA", "LAX"))
@

Additional nodes or edges can be added to our graph using the \Rmethod{addNode} and 
\Rmethod{addEdge} methods. For our example, we first add a new city "IAH" to 
our graph. We then add a flight connection between "DEN" and "IAH" having a fare
of \$120.

<<addNodeEdge>>=
g <- addNode("IAH", g)
g <- addEdge(from = "DEN", to = "IAH", graph = g, weight = 120)
g
@

Similarly edges and nodes can be removed from the graph using the 
\Rmethod{removeNode} and \Rmethod{removeEdge} methods respectively. We proceed to 
remove the flight connection from "DEN" to "IAH" and subsequently the node
"IAH".

<<removeEdge>>=
g <- removeEdge(from ="DEN", to = "IAH", g)
g <- removeNode(node = "IAH", g)
g
@

We can create a subgraph with only the cities  "DEN", "LAX" and "SEA" 
using the \Rmethod{subGraph} method.

<<subGraph>>=
g <- subGraph(snodes = c("DEN","LAX", "SEA"), g)
g
@

We can extract the \Rcode{from}-\Rcode{to} relationships for our graph
using the \Rmethod{extractFromTo} method.

<<fromTo>>=
extractFromTo(g)
@

\subsection{Mice gene interaction data for brain tissue (SAGE data)}

The C57BL/6J and C3H/HeJ mouse strains exhibit different cardiovascular and 
metabolic phenotypes on the  hyperlipidemic apolipoprotein E (Apoe) null background.
The interaction data for the genes from adipose, brain, liver and muscle tissue 
samples from male and female mice were studied. This interaction data for the 
various genes is included in the \Rpackage{graph} package as a list of 
\Robject{data.frame}s containing information for \Rcode{from-gene}, \Rcode{to-gene} 
and the strength of interaction \Rcode{weight} for each of the tissues studied.

We proceed to load the data for male and female mice. 

<<loadData1>>=
data("esetsFemale")
data("esetsMale")
@

We are interested in studying the interaction data for the genes in the brain 
tissue for male and female mice and hence proceed to represent this data as 
directed graphs using \Rclass{graphBAM} objects for male and female mice.

<<dataFrames>>=
dfMale <- esetsMale[["brain"]]
dfFemale <- esetsFemale[["brain"]]
head(dfMale)
@

<<creategraphBAMs>>=
male <- graphBAM(dfMale, edgemode = "directed")
female <- graphBAM(dfFemale, edgemode = "directed")
@

We are interested in pathways that are common to both male and female graphs for 
the brain tissue and hence proceed to perform a graph intersection operation using 
the \Rmethod{graphIntersect} method. Since edges can have different values of the 
weight attribute, we would like the result to have the sum of the weight attribute
in the male and female graphs. We pass in \Rcode{sum} as the function for handling 
weights to the \Rcode{edgeFun} argument. The \Rcode{edgeFun} argument should be 
passed a list of named functions corresponding to the edge attributes to be 
handled during the intersection process.

<<bamIntersect>>=
intrsct <- graphIntersect(male, female, edgeFun=list(weight =  sum))
intrsct
@

If node attributes were present in the \Robject{graphBAM} objects, a list of
named function could be passed as input to the \Rcode{graphIntersect} method for 
handling them during the intersection process.

We proceed to remove edges from the \Robject{graphBAM} result we just calculated
with a weight attribute less than a numeric value of 0.8 using the
\Rmethod{removeEdgesByWeight} method.

<<removeEdges>>=
resWt <- removeEdgesByWeight(intrsct, lessThan = 1.5)
@

Once we have narrowed down to the edges that we are interested in, we would like
to change the color attribute for these edges in our original \Robject{graphBAM}
objects for the male and female graphs to "red". Before an attribute can be
added, we have to set its default value using the \Rfunction{edgedataDefaults}
method. For our example, we set the default value for the color attribute to
white.

We first obtain the from - to relationship for the \Rcode{resWt}
graph using the \Rmethod{extractFromTo} method and then make use of the 
\Rmethod{edgeData} method to update the "color" edge attribute.

<<updateColor>>=
ftSub <- extractFromTo(resWt)
edgeDataDefaults(male, attr = "color") <- "white"
edgeDataDefaults(female, attr = "color") <- "white"
edgeData(male, from = as.character(ftSub[,"from"]), 
        to = as.character(ftSub[,"to"]), attr = "color") <- "red"

edgeData(female, from = as.character(ftSub[,"from"]), 
        to = as.character(ftSub[,"to"]), attr = "color") <- "red"

@

\section{MultiGraphs}

\subsection{Introduction}

The \Rclass{MultiGraph} class can be used to represent graphs that share a single
node set and have have one or more edge sets, each edge set representing
 a different type of interaction between the nodes. An \Robject{edgeSet} object
can be described as representing the relationship between a set of from-nodes and 
to-nodes which can either be directed or undirected. A numeric value (weight) 
indicates the strength of interaction between the connected edges. 

Self loops are permitted in the \Rclass{MultiGraph} class representation (i.e. the
from-node is the same as the to-node). The \Rclass{MultiGraph} class supports 
the handling of arbitrary node and edge attributes. These attributes are stored 
separately from the edge weights to facilitate efficient edge weight computation.

We shall load the \Rpackage{graph} and \Rpackage{RBGL} packages that we will be
using. We will then create a \Rclass{MultiGraph} object and then spend some time 
examining some of the different functions that can be applied to \Rclass{MultiGraph}
objects.

<<loadRBGL>>=
library(graph)
library(RBGL)
@

\subsection{ A simple MultiGraph example}

We proceed to construct a \Rclass{MultiGraph} object with directed \Robject{edgeSets}
to represent the flight connections of airlines Alaska, Delta, United and American
that fly between the cities Baltimore, Denver, Houston, Los Angeles, Seattle and 
San Francisco. For our example, the cities represent the nodes of the
\Rclass{MultiGraph} and we have one \Robject{edgeSet} each for the airlines. 
Each \Robject{edgeSet} represents the flight connections from an originating 
city(\Rcode{from}) to the destination city(\Rcode{to}). The weight represents 
the fare for flying between the \Rcode{from} and \Rcode{to} cities.

For each airline, we proceed to create a \Rclass{data.frame} containing the 
originating city, the destination city and the fare. 

<<createDataFrames>>=
ft1 <- data.frame(
        from = c("SEA", "SFO", "SEA", "LAX", "SEA"),
          to = c("SFO", "LAX", "LAX", "SEA", "DEN"),
      weight = c( 90, 96, 124, 115, 259),
      stringsAsFactors = TRUE)

ft2 <- data.frame(
        from = c("SEA", "SFO", "SEA", "LAX", "SEA", "DEN", "SEA", "IAH", "DEN"),
          to = c("SFO", "LAX", "LAX", "SEA", "DEN", "IAH", "IAH", "DEN", "BWI"),
       weight= c(169, 65, 110, 110, 269, 256, 304, 256, 271),
       stringsAsFactors = TRUE)

ft3 <- data.frame( 
   from = c("SEA", "SFO", "SEA", "LAX", "SEA", "DEN", "SEA", "IAH", "DEN", "BWI"),
    to  = c("SFO", "LAX", "LAX", "SEA", "DEN", "IAH", "IAH", "DEN", "BWI", "SFO"),
 weight = c(237, 65, 156, 139, 281, 161, 282, 265, 298, 244),
 stringsAsFactors = TRUE)

ft4 <- data.frame( 
    from = c("SEA", "SFO", "SEA", "SEA", "DEN", "SEA", "BWI"),
     to  = c("SFO", "LAX", "LAX", "DEN", "IAH", "IAH", "SFO"),
  weight = c(237, 60, 125, 259, 265, 349, 191),
  stringsAsFactors = TRUE)
@

These data frames are then passed to \Rclass{MultiGraph} class constructor as a 
named \Robject{list}, each member of the list being a \Robject{data.frame} for
an airline. A logical vector passed to the \Rcode{directed} argument 
of the \Rclass{MultiGraph} constructor indicates whether the \Robject{MultiGraph} 
to be created should have directed or undirected edge sets.

<<createMG>>= 
esets <- list(Alaska = ft1, United = ft2, Delta = ft3, American = ft4)
mg <- MultiGraph(esets, directed = TRUE)
mg
@
    
The nodes (cities) of the \Rclass{MultiGraph} object can be obtained by using the 
\Rmethod{nodes} method.
<<cities>>=
nodes(mg)
@
   
To find the fares for all the flights that originate from SEA for the Delta 
airline, we can use the \Rmethod{mgEdgeData} method.

<<DeltafromSeattle>>=
mgEdgeData(mg, "Delta", from = "SEA", attr = "weight")
@

We proceed to add some node attributes to the \Robject{MultiGraph} using the 
\Rfunction{nodeData} method. Before node attributes can be added, we have to set
a default value for each node attribute using the \Rfunction{nodeDataDefuault}
method. For our example, we would like to set a default value of square for the
node attribute shape.

We would like to set the node attribute "shape" for Seattle to the value \Rcode{"triangle"}
and that for the cities that connect with Seattle to the value \Rcode{"circle"}.

<<nodeData>>=
nodeDataDefaults(mg, attr="shape") <- "square"
nodeData(mg, n = c("SEA", "DEN", "IAH", "LAX", "SFO"), attr = "shape")  <- 
    c("triangle", "circle", "circle", "circle", "circle")
@

The node attribute shape for cities we have not specifically assigned a value 
(such as BWI) gets assigned the default value of "square".

<<nodeDataVal>>=
nodeData(mg,  attr = "shape")  
@

We then update the edge attribute \Rcode{color} for the Delta airline flights
that connect with Seattle to "green". For the remaining Delta flights that
connect to other destination in the MultiGraph, we would like to assign a
default color of "red". 

Before edge attributes can be added to the MultiGraph, 
their default values must be set using the \Rfunction{mgEdgeDataDefaults} method.
Subsequently, the \Rfunction{megEdgeData<-} method can be used to update
specific edge attributes.

<<edgeDataVal>>=
mgEdgeDataDefaults(mg, "Delta", attr = "color")  <- "red"
mgEdgeData(mg, "Delta", from = c("SEA", "SEA", "SEA", "SEA"), 
        to = c("DEN", "IAH", "LAX", "SFO"), attr = "color") <- "green"
@

<<mgEdgeDataVal>>=
mgEdgeData(mg, "Delta", attr = "color")
@

We are only interested in studying the fares for the airlines  Alaska, United and 
Delta and hence would like to create a smaller \Rclass{MultiGraph} object 
containing edge sets for only these airlines. This can be achieved using the 
\Rmethod{subsetEdgeSets} method.

<<subsetMG>>=
g <- subsetEdgeSets(mg, edgeSets = c("Alaska", "United", "Delta"))
@

We proceed to find out the lowest fares for Alaska, United and Delta along the 
routes common to them. To do this, we make use of the \Rmethod{edgeSetIntersect0}
method which computes the intersection of all the edgesets in a MultiGraph. While 
computing the intersection of edge sets, we are interesting in retaining the 
lowest fares in cases where different airlines flying along a route have different
fares. To do this, we pass in a named list containing the \Rmethod{weight}
function that calculates the minimum of the fares as the input to the 
\Rmethod{edgeSetIntersect0} method. (The user has the option of specifying any
function for appropriate handling of edge attributes ). 

<<intersecmg>>=
edgeFun <- list( weight = min)
gInt <- edgeSetIntersect0(g, edgeFun = edgeFun)
gInt
@

The edge set by the \Rmethod{edgeSetIntersect0} operation is named by 
concatenating the names of the edgeSets passed as input to the function.
  
<<intersectWeights>>=
mgEdgeData(gInt, "Alaska_United_Delta", attr= "weight")
@

\subsection{MultiGraph representation of mice gene interaction data. (SAGE)}

The C57BL/6J and C3H/HeJ mouse strains exhibit different cardiovascular and 
metabolic phenotypes on the  hyperlipidemic apolipoprotein E (Apoe) null background.
The interaction data for the genes from adipose, brain, liver and muscle tissue 
samples from male and female mice were studied. This interaction data for the 
various genes is included in the \Rpackage{graph} package as a list of 
\Robject{data.frame}s containing information for \Rcode{from-gene}, \Rcode{to-gene} 
and the strength of interaction \Rcode{weight} for each of the tissues studied.

We proceed to load the data for male and female mice. 

<<loadData>>=
data("esetsFemale")
data("esetsMale")
names(esetsFemale)
head(esetsFemale$brain)
@

The \Robject{esetsFemale} and \Robject{esetsMale} objects are a named \Robject{list}
of data frames corresponding to the data obtained from adipose, brain, liver and 
muscle tissues for the male and female mice that were studied. Each data frame
has a from, to and a weight column corresponding to the from and to genes that
were studied and weight representing the strength of interaction of the 
corresponding genes.

We proceed to create \Rclass{MultiGraph} objects for the male and female data sets 
by making use of the \Rclass{MultiGraph} constructor, which directly accepts 
a named list of data frames as the input and returns a MultiGraph with edgeSets
corresponding to the names of the data frames. 

<<createMultiGraphs>>=
female  <- MultiGraph(edgeSets = esetsFemale, directed = TRUE)
male  <- MultiGraph(edgeSets = esetsMale, directed = TRUE )
male
female
@

We then select a particular gene of interest in this network and proceed to 
identify its neighboring genes connected to this gene in terms of the maximum sum
of weights along the path that connects the genes for the brain edge set. 

We are interested in the  gene "10024416717" and the sum of the weights along the
path that connects this genes to the other genes for the brain tissue. Since the 
algorithms in the \Rpackage{RBGL} package that we will use to find the edges that
are connected to the gene "10024416717" do not work directly with 
\Rpackage{MultiGraph} objects, we proceed to create \Rcode{graphBAM} objects from 
the male and female edge sets for the brain tissue. 

\Rpackage{MultiGraph} objects can be converted to a named list of \Robject{graphBAM} 
objects using the \Rmethod{graphBAM} method.

<<graphBAMs>>=
maleBrain <- extractGraphBAM(male, "brain")[["brain"]]
maleBrain
femaleBrain <-  extractGraphBAM(female, "brain")[["brain"]]
@

We then identify the genes connected to gene "10024416717" as well as the sum of 
the weights along the path that connect the identified genes using the function 
\Rfunction{bellman.ford.sp} function from the \Rpackage{RBGL} package. 

<<edgeDistance>>=
maleWt <- bellman.ford.sp(maleBrain, start = c("10024416717"))$distance
maleWt <- maleWt[maleWt != Inf & maleWt != 0]
maleWt

femaleWt <- bellman.ford.sp(femaleBrain, start = c("10024416717"))$distance
femaleWt <- femaleWt[femaleWt != Inf & femaleWt != 0]
femaleWt
@

For the subset of genes we identified, we proceed to add node attributes to our
original \Robject{MultiGraph} objects for the male and female data. The node 
"10024416717" and all its connected nodes are assigned a color attribute "red" 
 while  the rest of the nodes are assigned a color color attribute of "gray".

<<nodeAttr>>=
nodeDataDefaults(male, attr = "color") <- "gray"
nodeData(male , n = c("10024416717", names(maleWt)), attr = "color" ) <- c("red")

nodeDataDefaults(female, attr = "color") <- "gray"
nodeData(female , n = c("10024416717", names(femaleWt)), attr = "color" ) <- c("red")
@

Our \Robject{MultiGraph} objects now contain the required node attributes for the
subset of genes that we have narrowed our selection to.

For the \Robject{MultiGraph} objects for male and female, we are also interested 
in the genes that are common to both \Robject{MultiGraph}s. This can be calculated
using the \Rfunction{graphIntersect} method.

<<nodeSub>>=
resInt <- graphIntersect(male, female)
resInt
@

The operations we have dealt with so far only deal with manipulation of \Rclass{MultiGraph}
objects. Additional functions will need to be implemented for the visualization
 of the \Rclass{MultiGraph} objects.

\end{document}