Using Sparse Matrix
Reordering Algorithms for
Cluster Identification
Chris Mueller
Dec 9, 2004
Visualizing a Graph as a Matrix
Each row and column in the matrix corresponds to
a node in the graph. The nodes are ordered the
same in the rows and columns, so node 10 is
represented by
row=10
and
col=10.
Each
edge
between
two
nodes
(a,b)
is
rendered
as
a
dot
at
(i,j)
where
i
is
the
row
for
a
and
j
is
the
column
for
b
.
The
solid
diagonal
shows
the
identity
relationship
for
each
node
.
Undirected graphs can be rendered as
lower triangles, with each edge is
displayed so that
i <= j.
Visually Identifying Clusters
Reordering the nodes (rows/cols)
can reduce the noise in the display
and highlight clusters.
Dense areas in the matrix reveal potential
clusters.
Some dense areas may be in the same
row or column as others, suggesting a
relationship.
(Some) Previous Work
The basic idea of visualizing relational data as a reordered matrix has been
around since the early days of computer science. Some examples are:
Bertin (1981),
Graphics and Graphic Information Processing
. From http://www.math.yorku.ca/SCS/Gallery/bright

ideas.html
The Reorderable Matrix (Bertin, 1981)
Block Clustering (Hartigan, 1972)
GAP Generalized Association Plots (Chen, 2002)
www.stat.sinica.edu.tw/SLR/PDF/
吳
漢銘

Cluster_Lecture_040206

new.pdf
Sparse Matrices
Sparse matrices can be stored in memory
in data structures that are more compact
that 2D arrays:
The banded representation stores only the
diagonals that have values:
Matrices are the basic data structure for
most numerical computations:
0 1 2 3
9 3 0 3
4 8 0 1
3 5 8 0
1
9 3 0
4 0 1
8 0
Sparse matrices
are matrices that do not
need explicit values for each element:
Note that zeros may be
important and cannot always
be excluded from that matrix.
1
3 9 0
4 0 1
8 0
1
9 3 0
4 0 1
8 0
[
1 0 1 n 3 0 0 9 n 8 4 n
]
The
bandwidth
is the number
of diagonals required to store
the matrix. In this example,
the bandwidth is 4.
Sparse matrix reordering algorithms reorder
the elements in the matrix to achieve better
use of memory or computational resources:
[
n 0 1 1 9 0 0 3 4 8
]
Swapping column 1 and 2
reduced the bandwidth to 3,
decreased the amount of
storage required by 2
elements, and removed 2
empty elements.
Sparse Matrix Reordering Algorithms
Bandwidth Minimization: Reverse Cuthill

McKee and King’s Algorithm
RCM(matrix):
Represent the matrix as a graph
Choose a suitable starting node
For each node reachable from the current node:
Output the node
Find all unvisited neighbors
Order them based on increasing degree
Visit them in that order
1
2
3
4
5
6
7
Minimizing Non

Zero Structure: Modified Minimum Degree
MMD(matrix):
Represent the matrix as a graph
Order nodes based on degree
1
2
3
4
5
7
6
8
9
Note that these algorithms are stochastic in the choice of starting nodes and ordering for nodes
with the same degree.
King’s algorithm is similar but it orders based on edges out of the current cluster rather than total edges.
Reordering the COG Database
Basic Protocol:
1.
Filter edges based on FASTA score
1.
cmp2 is original data, cmp90, cmp200 are filtered
2.
Shuffle the data
3.
For each sorted and shuffled graph
1.
Identify the connected components
?
2.
Apply RCM and King’s algorithm to each component
3.
Apply MMD to the entire graph
Results by the Numbers
(but the pictures show sooo much more…)
Visualization Key
Red dots are edges
Both axes have the nodes in the same order
Blue dots are the
COG families for the
node in column j.
Green lines show the
extent of a COG family.
Black dots show the
elements in the family.
Discussion
•
All algorithms worked as expected
•
However, the matrix ordering goals were too simple
to yield good cluster clusters.
•
Possible Future Work
–
Extended algorithms that allow more information
to be used
–
Exploit features of ordering strategies to do a
second pass that generates better clusters?
–
Hypergraph reordering
•
(demo of reordering by hand)
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