Graphs and Networks with Bioconductor Wolfgang Huber EMBLEBI

Graphs and Networks with Bioconductor Wolfgang Huber EMBL/EBI Bioconductor Conference 2005 Based on chapters from "Bioinformatics and Computational Biology Solutions using R and Bioconductor", Gentleman, Carey, Huber, Irizarry, Dudoit, Springer Verlag.

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 edges 2. 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 lists Adjacency 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("integrin. Mediated. Cell. Adhesion") > class(IMCAGraph) > s = acc(IMCAGraph, "SOS") Ha-Ras Raf MEK 1 2 3 ERK MYLK MYO 4 5 6 F-actin cell proliferation 7 5

Machine-readable pathway databases KEGG reactome Bio. Carta (biocarta. com) National Cancer Institute c. MAP

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 >tf. G=GOGraph("GO: 0003700", GOMFPARENTS)

Gene-Literature graphs DKC 1

Graphs: vocabulary Directed, undirected graphs Adjacent nodes Accessible nodes Self-loop Multi-edge Node degree Walk: alternating sequence of nodes and incident edges Closed walk Distance between nodes, shortest walk Trail: walk with no repeated edges Path: trail with no repeated nodes (except possibly first/last) Cycle Connected graph Weakly connected directed graph (see next page)

Strong and weak connectivity

Graphs: vocabulary Cut: remove edges to disconnect a graph Cut-set: remove nodes - " Connectivity of a graph Cliques

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 = AG t AG AV = A G AG t (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: GO Me. SH Graphical models

Random Edge Graphs n nodes, m edges p(i, j) = 1/m with high probability: m < n/2: many disconnected components m > 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: random. EGraph(V, p, edges) V: nodes either p: probability per edge or edges: number of edges Random graph with latent factor: random. Graph(V, M, p, weights=TRUE) V: nodes M: latent factor p: probability For each node, generate a logical vector of 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: random. Node. Graph(node. Degree) node. Degree: named integer vector sum 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 subgroups For 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) > my. Nodes = c("s", "p", "q", "r") > s p q r my. Edges = list(edges = c("p", "q")), = list(edges = c("p", "r")), = list(edges = c("s"))) > g = new("graph. NEL", nodes = my. Nodes, edge. L = my. Edges, 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) $in. Degree s p q r 1 3 2 1 $out. Degree s p q r 2 2 2 1

Graph manipulation > g 1 <- add. Node("e", g) > g 2 <- remove. Node("d", g) > ## add. Edge(from, to, graph, weights) > g 3 <- add. Edge("e", "a", g 1, pi/2) > ## remove. Edge(from, to, graph) > g 4 <- remove. Edge("e", "a", g 3) > identical(g 4, g 1) [1] TRUE

adjacent and accessible nodes > adj(g, c("b", "c")) $b [1] "b" "c" $c [1] "b" "d" > acc(g, c("b", "c")) $b a c d 3 1 2 $c a b d 2 1 1

Graph representations: from-to-matrix > ft [1, ] [2, ] [3, ] [4, ] [, 1] [, 2] 1 2 2 3 3 1 4 4 > ft. M 2 adj. M(ft) 1 2 3 4 1 0 0 2 0 0 1 0 3 1 0 0 0 4 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="e 1" from="A" to="C"> <attr name="weights"> <int>1</int> </attr> </edge> <edge id="e 2" from="B" to="D"> <attr name="weights"> <int>1</int> </attr> </edge> … </graph> </gxl> from graph/GXL/kmst. Ex. 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 components cc = conn. Comp(rg) table(list. Len(cc)) 1 2 3 4 15 18 36 7 3 2 1 1 rg Choose the largest component wh = which. max(list. Len(cc)) sg = sub. Graph(cc[[wh]], rg) Depth first search dfsres = dfs(sg, node = "N 14") nodes(sg)[dfsres$discovered] [1] "N 14" "N 94" "N 40" "N 69" "N 02" "N 67" "N 45" "N 53" [9] "N 28" "N 46" "N 51" "N 64" "N 07" "N 19" "N 37" "N 35" [17] "N 48" "N 09"

depth / breadth first search dfs(sg, "N 14") bfs(sg, "N 14")

connected components sc = strong. Comp(g 2) nattrs = make. Node. Attrs(g 2, fillcolor="") for(i in 1: length(sc)) nattrs$fillcolor[sc[[i]]] = my. Colors[i] wc = conn. Comp(g 2) plot(g 2, "dot", node. Attrs=nattrs)

shortest path algorithms Different algorithms for different types of graphs o all edge weights the same o positive edge weights o real numbers …and different settings of the problem o single pair o single source o single destination o all pairs Functions bfs dijkstra. sp sp. between johnson. all. pairs. sp

shortest path set. seed(123) rg 2 = random. EGraph(node. Names, edges = 100) from. Node = "N 43" to. Node = "N 81" sp = sp. between(rg 2, from. Node, to. Node) sp[[1]]$path [1] "N 43" "N 08" "N 88" [4] "N 73" "N 50" "N 89" [7] "N 64" "N 93" "N 32" [10] "N 12" "N 81" sp[[1]]$length [1] 10 1

shortest path ap = johnson. all. pairs. sp(rg 2) hist(ap)

minimal spanning tree gr mst = mstree. kruskal(gr)

connectivity Consider 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. > edge. Connectivity(g) $connectivity [1] 2 $min. Discon. Set[[1]] [1] "D" "E" $min. Discon. Set[[2]] [1] "D" "H"

Rgraphviz: the different layout engines dot: 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

Image. Map lg = agopen(g, …) image. Map(lg, con=file("imca-frame 1. html", open="w") tags= list(HREF = href, TITLE = title, TARGET = rep("frame 2", length(Ag. Node(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 n. Pdist: number of common edges as computed by a node label permutation 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 DKC 1

The bipartite gene-literature graph: actor and event size adjustment actors: genes actor size: number of papers that a gene appears in event: paper event 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 c. DNA sequences. PNAS 99: 16899– 903, 2002 cites 15, 000 genes

Are two genes remarkably often co-cited? Note, usually one count (w. l. o. g. n 22) is much larger than everybody else. Test statistics that do not depend on n 22:

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 3 Gene Y Gene 4 Gene 5

A pathway graph

A pathway graph

CGH aberration data Genetic aberrations Tumours From: B. Gunawan et al. , Cancer Res. 63: 6200 -6205 (2003)

Graphical model for CGH aberration data oncotree package by Anja von Heydebreck

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

Further exercises Fine control of graph rendering GOstats example