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Information Visualization
using graphs algorithms
Symeonidis Alkiviadis
[email protected]
[email protected]
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 r 
for every pair of
genes
xy
N
 2 x 2  2 y 2 
 x 
 y 


N 
N 

xy 
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
v
u

or create them
Store (v,w)
Remove u
v
u
w
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:


Biconnected graph


dfs: O(|E|+\V|)=O(|E|)
Dominated by dfs O(|E|)
Non- biconnected graph

Dominated by extracting block-cut tree O(|E|)
Graph visualization
… until now
Determine groups’ positions
 Determine vertices ordering

۷
۷
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(xnode ,ynode )= r(xgene ,ygene)
 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
