Graphs and Networks with Bioconductor

Wolfgang Huber

EMBL/EBI

Bressanone 2005

Based on chapters from "Bioinformatics and Computational Biology Solutions using R and

Bioconductor", Gentleman, Carey, Huber, Irizarry, Dudoit, Springer Verlag, to appear Aug 2005.

Graphs

Set of nodes and set of edges.

Nodes: objects of interest

Edges: relationships between them

A useful abstraction to talk about relationships and interactions (think of integer numbers, apples and fingers)

Edges may have weights, directions, types

Practicalities

As always, need to distinguish between the true, underlying property of nature that you want to measure, and the actual result of a measurement (experiment)

1. False positive edges2. False negative edges (were tested, were not found, but are there in nature)3. Untested edges (were not tested, are not in your data, but are there in nature)

Uncertainty is not usually considered in mainstream graph theory, but cannot be ignored in functional genomics.

Nice application of these concepts to protein interactions: Gentleman and Scholtens, SAGMB 2004

Representation

Node-edge listsAdjacency matrix (straightforward)Adjacency matrix (sparse)From-To matrix

They are equivalent, but may be hugely different in performance and convenience for different applications.

Can coerce between the representations

Algorithms

Bioconductor project emphasizes re-use and interfacing to existing, well-tested software implementations rather than reimplementing everything from scratch ourselves.

RBGL package: interface to Boost Graph Library; started by V. Carey, R. Gentleman, now driven by Li Long.

Example: a pathway

Elementary computations on IMCA pathway

> library("graph")> data("integrinMediatedCellAdhesion")> class(IMCAGraph)> s = acc(IMCAGraph, "SOS")Ha-Ras Raf MEK

1 2 3ERK MYLK MYO4 5 6

F-actin cell proliferation7 5

Machine-readable pathway databases

KEGG

reactome

BioCarta (biocarta.com)

National Cancer Institute cMAP

Gene Ontology (GO)

A structed vocabulary to describe molecular function of gene products, biological processes, and cellular components.

Plus

A set of "is a", "is part of" relationships between these terms

Directed acyclic graph

GO graphs

>tfG=GOGraph("GO:0003700", GOMFPARENTS)

Gene-Literature graphs

DKC1

Graphs: vocabulary

Directed, undirected graphsAdjacent nodesAccessible nodesSelf-loopMulti-edgeNode degreeWalk: alternating sequence of nodes and incident edgesClosed walkDistance between nodes, shortest walkTrail: walk with no repeated edgesPath: trail with no repeated nodes (except possibly first/last)CycleConnected graphWeakly connected directed graph (see next page)

Strong and weak connectivity

Graphs: vocabulary

Cut: remove edges to disconnect a graphCut-set: remove nodes - " -Connectivity of a graphCliques

Special types of graphs

Bipartite graph

Bipartite graphs

AG adjacency matrix (n x m) of a bipartite graph G with node sets U, V

One mode graphs

AU = AGt AG

AV = AG AGt

(Boolean algebra)

Multigraphs

Can have different types of edges

Hypergraphs

:= set of Nodes + set of hyperedges

A hyperedge is a set of nodes (can be more than 2)

A directed hyperedge: pair (tail and head) of sets of nodes

Directed acyclic graphs

Useful for representing hierarchies, partial orderings (e.g. in time, from general to special, from cause to effect)

Many applications:GOMeSHGraphical models

Random Edge Graphs

n nodes, m edgesp(i,j) = 1/m

with high probability:m < n/2: many disconnected componentsm > n/2: one giant connected component: size ~ n.

(next biggest: size ~ log(n)). degrees of separation: log(n).

Erdös and Rényi 1960

Random graphs

Random edge graph: randomEGraph(V, p, edges)V: nodeseither p: probability per edgeor edges: number of edges

Random graph with latent factor: randomGraph(V, M, p, weights=TRUE)V: nodesM: latent factorp: probabilityFor each node, generate a logical vector of length length(M), with P(TRUE)=p. Edges are between nodes that share >= 1 elements. Weights can be generated according to number of shared elements.

Random graph with predefined degree distribution:randomNodeGraph(nodeDegree)

nodeDegree: named integer vectorsum of all node degrees must be even

Random edge graph

100 nodes 50 edges

degree distribution

Random graphs versus permutation graphs

For statistical inference, one can consider null hypotheses based on aforementioned random graph models; and ones based on node permutation of data graphs.

The second is often more appropriate.

Cohesive subgroupsFor data graphs, the concept of clique is usually too restrictive (false negative or untested edges)

n-clique: distance between all members is <=n. (Clique: n=1)

k-plex: maximal subgraph G in which each member is neighbour of at least |G|-k others. (Clique: k=1)

k-core: maximal subgraph G in which each member is neighbour of at least k others. (Clique: k=|G|-1)

After: Social Network Analysis, Wasserman and Faust (1994)

graph, RBGL, Rgraphviz

graph basic class definitions and functionality

RBGL interface to graph algorithms

Rgraphviz rendering functionality Different layout algorithms. Node plotting, line type, color etc. can be controlled by the user.

Creating our first graph

> library("graph"); library(Rgraphviz)

> myNodes = c("s", "p", "q", "r")

> myEdges = list(s = list(edges = c("p", "q")), p = list(edges = c("p", "q")), q = list(edges = c("p", "r")), r = list(edges = c("s")))

> g = new("graphNEL", nodes = myNodes, edgeL = myEdges, edgemode = "directed")

> plot(g)

Querying nodes, edges, degree

> nodes(g)[1] "s" "p" "q" "r"

> edges(g)$s[1] "p" "q"$p[1] "p" "q"$q[1] "p" "r"$r[1] "s"

> degree(g)$inDegrees p q r1 3 2 1$outDegrees p q r2 2 2 1

adjacent and accessible nodes

> adj(g, c("b", "c"))$b[1] "b" "c"$c[1] "b" "d"

> acc(g, c("b", "c"))$ba c d3 1 2

$ca b d2 1 1

Graph manipulation

> g1 <- addNode("e", g)

> g2 <- removeNode("d", g)

> ## addEdge(from, to, graph, weights)

> g3 <- addEdge("e", "a", g1, pi/2)

> ## removeEdge(from, to, graph)

> g4 <- removeEdge("e", "a", g3)

> identical(g4, g1)

[1] TRUE

Graph representations: from-to-matrix

> ft[,1] [,2]

[1,] 1 2[2,] 2 3[3,] 3 1[4,] 4 4

> ftM2adjM(ft)1 2 3 4

1 0 1 0 02 0 0 1 03 1 0 0 04 0 0 0 1

GXL: graph exchange language

<gxl><graph edgemode="directed" id="G"><node id="A"/><node id="B"/><node id="C"/>…<edge id="e1" from="A" to="C"><attr name="weights"><int>1</int></attr></edge><edge id="e2" from="B" to="D"><attr name="weights"><int>1</int></attr></edge>…

</graph></gxl>

from graph/GXL/kmstEx.gxl

GXL (www.gupro.de/GXL)

is "an XML sublanguage

designed to be a standard exchange format for graphs".The graph package

provides tools for im- and exporting

graphs as GXL

RBGL: interface to the Boost Graph Library

Connected componentscc = connComp(rg) table(listLen(cc)) 1 2 3 4 15 18 36 7 3 2 1 1

Choose the largest componentwh = which.max(listLen(cc)) sg = subGraph(cc[[wh]], rg)

Depth first searchdfsres = dfs(sg, node = "N14")nodes(sg)[dfsres$discovered] [1] "N14" "N94" "N40" "N69" "N02" "N67" "N45" "N53" [9] "N28" "N46" "N51" "N64" "N07" "N19" "N37" "N35" [17] "N48" "N09"

rg

depth / breadth first search

dfs(sg, "N14")bfs(sg, "N14")

connected componentssc = strongComp(g2)

nattrs = makeNodeAttrs(g2, fillcolor="")

for(i in 1:length(sc))nattrs$fillcolor[sc[[i]]] =

myColors[i]

plot(g2, "dot", nodeAttrs=nattrs)wc = connComp(g2)

shortest path algorithms

Different algorithms for different types of graphs o all edge weights the sameo positive edge weightso real numbers

…and different settings of the problemo single pairo single sourceo single destinationo all pairs

Functionsbfsdijkstra.spsp.betweenjohnson.all.pairs.sp

shortest path

1

set.seed(123)rg2 = randomEGraph(nodeNames, edges = 100)fromNode = "N43"toNode = "N81"sp = sp.between(rg2,

fromNode, toNode)

sp[[1]]$path [1] "N43" "N08" "N88" [4] "N73" "N50" "N89" [7] "N64" "N93" "N32" [10] "N12" "N81"

sp[[1]]$length [1] 10

shortest path

ap = johnson.all.pairs.sp(rg2)hist(ap)

minimal spanning tree

mst = mstree.kruskal(gr)gr

connectivityConsider graph g with single connected component.Edge connectivity of g: minimum number of edges in g that can be cut to produce a graph with two components. Minimum disconnecting set: the set of edges in this cut.

> edgeConnectivity(g)$connectivity[1] 2

$minDisconSet$minDisconSet[[1]][1] "D" "E"

$minDisconSet[[2]][1] "D" "H"

Rgraphviz: the different layout enginesdot: directed graphs. Works best on DAGs and other graphs that can be drawn as hierarchies.

neato: undirected graphs using ’spring’ models

twopi: radial layout. One node (‘root’) chosen as the center. Remaining nodes on a sequence of concentric circles about the origin, with radial distance proportional to graph distance. Root can be specified or chosen heuristically.

Rgraphviz: the different layout engines

Rgraphviz: the different layout engines

Combining R graphics and graphviz: custom node drawing functions

Combining: graphviz layout and R plot

ImageMap

lg = agopen(g, …)

imageMap(lg, con=file("imca-frame1.html", open="w")tags= list(HREF = href,

TITLE = title,TARGET = rep("frame2", length(AgNode(nag)))),

imgname=fpng, width=imw, height=imh)

Show drosophila interaction network example

Application: comparing gene co-expression and protein interaction data

Nodes: all yeast genes

Graph 1: co-expression clusters from yeast cell cycle microarray time course

Graph 2: protein interactions reported in the literature

Graph 3: protein interactions found in a yeast-two-hybrid experiment

Questions:Do the graphs overlap more than random? Is there anything special about overlapping edges?

Application: comparing gene co-expression and protein interaction data

Application: comparing gene co-expression and protein interaction data

nPdist: number of common edges as computed by a node label per-mutation model.

Number observed in data: 42

Further questions for exploratory data analysis

• Which expression clusters have intersections with which of the literature clusters?

• Are known cell-cycle regulated protein complexes indeed clustered together in both graphs?

• Are there expression clusters that have a number of literature cluster edges going between them ⇒ suggesting that expression clustering was too fine, or that literature clusters are not cell-cycle regulated.

• Is the expression behavior of genes that are involved in multiple protein complexes different from that of genes that are involved in only one complex?

Generalization

Nothing in the preceding treatment was specific to physical protein interactions or microarray clustering. Can you similar reasoning for many other graphs! - e.g. genomic vicinity, domain composition similarity

Application: Using GO to interprete gene lists

Using GO to interprete gene lists

Packages: Gostats, Rgraphviz

Using GO to interprete gene lists

Gene-Literature graphs

DKC1

The bipartite gene-literature graph: actor and event size adjustment

actors: genesactor size: number of papers that a gene appears inevent: paperevent size: number of genes that appear in a paper

Example: R. Strausberg et al. Generation and initial analysis of more than 15,000 full-length human and mouse cDNA sequences. PNAS 99:16899–903, 2002cites 15,000 genes

Are two genes remarkably often co-cited?

Note, usually one count (w.l.o.g. n22) is much larger than everybody else. Test statistics that do not depend on n22:

Closing gene lists with literature

Boundary of gene list L: set of all genes that have co-citation (above threshold weight) with genes in L.

Gene 1

Gene X

Gene 2

Gene Y

Gene 4

Gene 5

Gene 3

A pathway graph

A pathway graph

CGH aberration data

From: B. Gunawan et al., Cancer Res. 63: 6200-6205 (2003)Tumours

Genetic aberrations

oncotree package by Anja von Heydebreck

Graphical model for CGH aberration data

Summary

Graphs are a natural way to represent relationships, just as numbers are a natural way to represent quantities.

Three main applications: (1) to represent data (e.g. PPI)(2) to represent knowledge (e.g. GO)(3) to represent high-dimensional probability distributions

Bioconductor provides a rich set of tools mainly for (1) and (2). Various parts of R for (3), see also gR project.

There are still many challenges that call for methods to model uncertainty, make inference, and predictions.