Information Visualization using graphs algorithms Symeonidis Alkiviadis

Contents Preliminaries Gene clustering Graph extraction from biological data Graph visualization Open issues Discussion

Preliminaries Visualize clusters of genes produced by clustering over gene expressions Gene expression: set of values of genes over a set of patients

Preliminaries Graph G(V,E) : set of vertices, with edges joining vertices Each vertex represents a gene Each edge represents strong correlation Clustering => groups of vertices

Contents Preliminaries Gene clustering Graph extraction from biological data Graph visualization Open issues Discussion

Gene clustering Correlation Compute Pearson's correlation coefficient for every pair of genes

Gene clustering Greedy clustering for every unclassified gene x create a cluster which includes it add all genes y with correlation > threshold Cost: O(|genes| 2 )

Contents Preliminaries Gene clustering Graph extraction from biological data Graph visualization Open issues Discussion

Graph extraction from biological data Genes → vertices ۷ Clusters→ groups ۷ Edges ?

Graph extraction from biological data In-cluster relation Mean value of correlation coefficients for all genes in a cluster All pairs of genes with correlation higher than threshold* mean are considered highly correlated Edge meaning: (Very) strong correlation

Graph extraction from biological data Inter-cluster relation Mean value of correlation coefficients for each cluster All pairs of genes with correlation higher than threshold* (mean1+mean2)/2 are considered highly correlated Edge meaning: Possibly wrong classification

Graph extraction from biological data Genes → vertices ۷ Clusters→ groups ۷ Edges ۷ all highly correlated pairs of genes

Contents Preliminaries Gene clustering Graph extraction from biological data Graph visualization Open issues Discussion

Graph visualization Gene → Vertex → circle High correlation → Edge → line Cluster → Group → Circle with respective genes - vertices on its periphery

Graph visualization Place groups Determine ordering of vertices in group Try to reduce crossings

Graph visualization placing groups Force - directed method over groups

Graph visualization Place groups Determine ordering of vertices in group Try to reduce crossings

Graph visualization Determine ordering of vertices in group(tree) Tree depth first search discovery time

Graph visualization Determine ordering of vertices in group(bicon) Biconnected graph: Remains connected after removing one(any) vertex/edge

Graph visualization Determine ordering of vertices in group(bicon) For every node u identify triangles or create them Store (v,w) Remove u u v w u v w

Graph visualization Determine ordering of vertices in group(bicon) Restore graph Remove all stored edges Perform dfs, compute longest path and place it

Graph visualization Determine ordering of vertices in group(bicon) Place any remaining vertices Next to 2 neighbors Next to 1 neighbor Next to 0 neighbors

Graph visualization Determine ordering of vertices in group(n-bic) Non-biconnected graph … under development There is a vertex whose removal disconnects the graph Decompose into bicon. components get articulation points vertices responsible for non-biconnectivity

Graph visualization Determine ordering of vertices in group(n-bic) Decompose into bicon. components biconnected subgraphs get articulation points vertices responsible for non-biconnectivity

Graph visualization Determine ordering of vertices in group(n-bic) Articulation points + biconnected components Block - cut - point tree -Dfs on block cut point=> relative ordering of components - For each biconnected component act as before

Graph visualization Determine ordering of vertices in group Cost Tree: dfs: O(|E|+\V|)=O(|E|) Biconnected graph Dominated by dfs O(|E|) Non- biconnected graph Dominated by extracting block-cut tree O(|E|)

Graph visualization … until now Determine groups’ positions ۷ Determine vertices ordering۷

Graph visualization Place groups ۷ Determine ordering in group ۷ Try to reduce crossings

Graph visualization reduce crossings Spin groups trying to minimize energy

Graph visualization edge coloring Each edge is assigned a weight weight(x node,y node )= r(x gene,y gene ) The color of each edge reflects its weight brighter color → stronger correlation In- group edges have different color than inter-group edges

Graph visualization Overall Initially…

Graph visualization overall Finally…

Open issues Clustering Edge translation Visualize large data sets Zoom Layered drawing Scrollbars