clm_dist - compute the distance between two or more partitions (clusterings).

The distance that is computed can be any of

*split/join distance*,

*variance of information*, or

*Mirkin metric*.

clmdist is not in actual fact a program. This manual page documents the
behaviour and options of the clm program when invoked in mode

*dist*. The
options

**-h**,

**--apropos**,

**--version**,

**-set**,

**--nop** are accessible in all

**clm** modes. They are described in the

**clm** manual page.

**clm dist** [options] <file name> <file name>+

**clm dist** **[-mode** <sj|vi|mk|sc> (

*distance type*)

**]**
**[-o** fname (

*output file*)

**]** **[--chain** (

*only compare
consecutive clusterings*)

**]** **[--one-to-many** (

*compare first
clustering to all others*)

**]** **[--sort** (

*sort clusterings
based on coarseness*)

**]** **[--index** (

*output Rand, adjusted
Rand and Jaccard indices*)

**]** **[-digits** k (

*output
decimals*)

**]** **[-h** (

*print synopsis, exit*)

**]**
**[--apropos** (

*print synopsis, exit*)

**]** **[--version**
(

*print version, exit*)

**]** <file name> <file name>+

**clm dist** computes distances between clusterings. It can compute the

*split/join distance* (described below), the

*variance of
information* *measure*, and the

*Mirkin metric*. By default it
computes the chosen distance for all pairs of distances in the clusterings
provided. Clusterings must be in the mcl matrix format (cf.

**mcxio(5)**),
and are supplied on the command line as the names of the files in which they
are stored. It is possible to compare only consecutive clusterings by using
the

**--chain** option.

Currently,

**clm dist** cannot compute different distance types
simultaneously.

The output is linewise, each line giving information about the distance between
a pair of clusterings. A line has the following format:

d d1 d2 N v name1 name2 [v50,v75,v90,v95,v99]

where dT is the distance between the two clusterings, d1 is the distance from
the first clustering to the greatest common subclustering (alternatively
called GCS, intersection, or meet) of the two clusterings, d2 is similarly the
distance from the second clustering to the GCS, N is the number of nodes in
the set over which the clusterings are defined, name1 is the name of the file
containing the first clustering, name2 is the name of the file containing the
second clustering, and vXX is the number of

*volatile nodes* at
stringency factor 0.XX (i.e. 0.5 for v50). Refer to

**clm vol** for
a definition of

*volatile node*.

**-mode** <sj|vi|mk> (

*distance type*)

Use

**sj** for the

*split/join distance* (described below),

**vi**
for the

*variance of information measure* and

**mk** for the

*Mirkin
metric*.

**--chain** (

*only compare consecutive clusterings*)

This option can be used if you know that the clusterings are nested clusterings
(or appoximately so) and ordered from coarse to fine-grained or vice versa. An
example of this is the set of clusterings resulting from applying

**mcl**
with a range of inflation parameters.

**--one-to-many** (

*compare first clustering to all others*)

Use this option for example to compare a gold standard classification to a
collection of clusterings. Bear in mind that sub-clustering and
super-clustering are also ways for a clustering to be compatible with a gold
standard. This means that the simple numerical criterion of distance between
clusters (by whatever method) is only partially informative. For the Mirkin,
variation of information and split/join metrics it pays to take into account
the constituent distances d1 and d2 (see above). Assuming that the first
clustering given as argument represents a gold standard, a small value for d1
implies that the second clustering is (nearly) a superclustering, and
similarly a small value for d2 implies that it is (nearly) a subclustering.

**--sort** (

*sort clusterings based on coarseness*)

This option can be useful in conjunction with the

**--chain** option, in case
the list of clusterings supplied is not necessarily ordered by granularity.

**--index** (

*output Rand, adjusted Rand and Jaccard indices*)

As described.

**-o** fname (

*output file*)

**-digits** k (

*output decimals*)

The number of decimals printed when using the variance of information measure.

For each pair of clusterings

**C1**,

**C2**, two numbers are given, say

**d1** and

**d2**. Then

**d1** +

**d2** equals the number of nodes
that have to be exchanged in order to transform any of the two clusterings
into the other, and you can think of (

**d1**+

**d2**)/

**2N** as the
percentage that the two clusterings differ. The split/join distance has a
linearity property with respect to the meet of

**C1** and

**C2**, see
below.

The split/join distance

**sjd** is very handy in computing the consistency of
two or more clusterings of the same graph, or comparing clusterings made with
different resource (but otherwise identical) parameters. The latter is for
finding out whether you can settle for cheaper mcl settings, or whether you
need to switch to more expensive settings. The former is for finding out
whether clusterings are identical, conflicting, or whether one is (almost) a
subclustering of the other - mostly for comparing a set of clusterings of
different granularity, made by letting the mcl parameter

**-I** vary. The

**EXAMPLES** section contains examples of all these

**clm dist** uses,
and the use of

**clm info** and

**clm meet** is also discussed there.

**sjd** is a metric distance on the space of partitions of a set of a given
fixed cardinality. It has the following linearity property. Let

**P1** and

**P2** be partitions, then

**sjd**(

**P1**,

**P2**) =

**sjd**(

**P1**,

**D**) +

**sjd**(

**P2**,

**D**)

where

**D** (for Dutch Doorsnede) is the intersection of

**P1** and

**P2**, i.e. the unique clustering that is both a subclustering of

**P1** and

**P2** *and* a superclustering of all other
subclusterings of

**P1** and

**P2**. Sloppily worded,

**D** is the
largest subclustering of both

**P1** and

**P2**. See the

**REFERENCES** section for a pointer to the technical report in which

**sjd** was first defined (and in which the non-trivial triangle inequality
is proven).

Because it is useful to know whether one partition (or clustering) is almost a
subclustering of the other,

**clm dist** returns the two constituents

**sjd**(

**P1**,

**D**) and

**sjd**(

**P2**,

**D**).

Let

**P1** and

**P2** be two clusterings of a graph of cardinality

**N**, and suppose

**clm dist** returns the integers

**d1** and

**d2**. You can think of

**100 * (d1 + d2) / N** as the percentage that

**P1** and

**P2** differ. This interpretation is in fact slightly
conservative. The numerator is the number of nodes that need to be exchanged
in order to transform one into the other. This number may grow as large as

**2*N - 2*sqrt(N)**, so it would be justified to take 50 as a scaling
factor rather than 100.

For example, if

**A** and

**B** are both clusterings of a graph on a set
of 9058 nodes and

**clm dist** returns [38, 2096], this conveys that

**A** is almost a subclustering of

**B** (by splitting 38 nodes in

**A** we obtain a clustering

**D** that is a subclustering of

**B**),
and that

**B** is much less granular than

**A**. The latter is because
we can obtain

**B** from

**D** by

*joining* 2096 nodes in some way.

The following is an example of several mcl validation tools applied to a set of
clusterings on a protein graph of 9058 nodes. In the first experiment, six
different clusterings were generated for different values of the inflation
parameter, which was respectively set to 1.2, 1.6, 2.0, 2.4, 2.8, and 3.2. It
should be noted that protein graphs seem somewhat special in that an inflation
parameter setting as low as 1.2 still produces a very acceptable clustering.
The six clusterings are scrutinized using

**clm dist**,

**clm info**,
and

**clm meet**. In the second experiment, four different clusterings were
generated with identical flow (i.e. inflation) parameter, but with different
resource parameters.

**clm dist** is used to choose a sufficient resource
level.

High

**-P/-S/-R** values make

**mcl** more accurate but also more time and
memory consuming. Run

**mcl** with different settings for these parameters,
holding other parameters fixed. If the expensive and supposedly more accurate
clusterings are very similar to the clusterings resulting from cheaper
settings, the cheaper setting is sufficient. If the distances between cheaper
clusterings and more expensive clusterings are large, this is an indication
that you need the expensive settings. In that case, you may want to increase
the

**-P/-S/-R** parameters (or simply the

**-scheme** parameter) until
associated clusterings at nearby resource levels are very similar.

In this particular example, the validation tools do not reveal that one
clustering in particular can be chosen as 'best', because all clusterings seem
at least acceptable. They do aid however in showing the relative merits of
each clusterings. The most important issue in this respect is cluster
granularity. The table below shows the output of

**clm info**.

Efficiency Mass frac Area frac Cl weight Mx link weight
1.2 0.42364 0.98690 0.02616 52.06002 50.82800
1.6 0.58297 0.95441 0.01353 55.40282 50.82800
2.0 0.63279 0.92386 0.01171 58.09409 50.82800
2.4 0.65532 0.90702 0.01091 59.58283 50.82800
2.8 0.66854 0.84954 0.00940 63.19183 50.82800
3.2 0.67674 0.82275 0.00845 66.10831 50.82800

This data shows that there is exceptionally strong cluster structure present in
the input graph. The 1.2 clustering captures almost all edge mass using only
2.5 percent of 'area'. The 3.2 clustering still captures 82 percent of the
mass using less than 1 percent of area. We continue with looking at the mutual
consistency of the six clusterings. Below is a table that shows all pairwise
distances between the clusterings.

| 1.6 | 2.0 | 2.4 | 2.8 | 3.2 | 3.6
-----------------------------------------------------------.
1.2 |2096,38 |2728,41 |3045,48 |3404,45 |3621,43 |3800, 42 |
-----------------------------------------------------------|
1.6 | | 797,72 |1204,76 |1638,78 |1919,70 |2167, 69 |
-----------------------------------------------------------|
2.0 | | | 477,68 | 936,78 |1235,85 |1504, 88 |
-----------------------------------------------------------|
2.4 | | | | 498,64 | 836,91 |1124,103 |
-----------------------------------------------------------|
2.8 | | | | | 384,95 | 688,119 |
-----------------------------------------------------------|
3.2 | | | | | | 350,110 |
-----------------------------------------------------------.

The table shows that the different clusterings are pretty consistent with each
other, because for two different clusterings it is generally true that one is
almost a subclustering of the other. The interpretation for the distance
between the 1.6 and the 3.2 clustering for example, is that by rearranging 43
nodes in the 3.2 clustering, we obtain a subclustering of the 1.6 clustering.
The table shows that for any pair of clusterings, at most 119 entries need to
be rearranged in order to make one a subclustering of the other.

The overall consistency becomes all the more clear by looking at the meet of all
the clusterings:

clm meet -o meet out12 out16 out20 out24 out28 out32
clm dist meet out12 out16 out20 out24 out28 out32

results in the following distances between the respective clusterings and their
meet.

| 1.2 | 1.6 | 2.0 | 2.4 | 2.8 | 3.2 |
-------------- --------------------------------------------.
meet| 0,3663| 0,1972| 0,1321 | 0,958 | 0,559 | 0,283 |
-------------- --------------------------------------------.

This shows that by rearranging only 283 nodes in the 3.2 clustering, one obtains
a subclustering of all other clusterings.

In the last experiment,

**mcl** was run with inflation parameter 1.4, for
each of the four different preset pruning schemes k=1,2,3,4. The

**clm
dist** distances between the different clusterings are shown below.

| k=2 | k=3 | k=4 |
-------------------------------.
k=1 | 17,17 | 16,16 | 16,16 |
-------------------------------.
k=2 | | 3,3 | 5,5 |
-------------------------------.
k=3 | | | 4,4 |
-------------------------------.

This example is a little boring in that the cheapest scheme seems adequate. If
anything, the gaps between the k=1 scheme and the rest are a little larger
than the three gaps between the k=2, k=3, and k=4 clusterings. Had all
distances been much larger, then such an observation would be reason to choose
the k=2 setting.

Note that you need not feel uncomfortable with the clusterings still being
different at high resource levels, if ever so slightly. In all likelihood,
there are anyway nodes which are not in any core of attraction, and that are
on the boundary between two or more clusterings. They may go one way or
another, and these are the nodes which will go different ways even at high
resource levels. Such nodes may be stable in clusterings obtained for lower
inflation values (i.e. coarser clusterings), in which the different clusters
to which they are attracted are merged.

Stijn van Dongen.

**mclfamily(7)** for an overview of all the documentation and the utilities
in the mcl family.

Stijn van Dongen.

*Performance criteria for graph clustering and Markov*
*cluster experiments*. Technical Report INS-R0012, National Research
Institute for Mathematics and Computer Science in the Netherlands, Amsterdam,
May 2000.

http://www.cwi.nl/ftp/CWIreports/INS/INS-R0012.ps.Z

Marina Meila.

*Comparing Clusterings - An Axiomatic View*. In

*Proceedings of the 22nd International Conference on Machine Learning*,
Bonn, Germany, 2005.

Marina Meila.

*Comparing Clusterings*, UW Statistics Technical Report 418.

http://www.stat.washington.edu/www/research/reports/2002/tr418.ps