16 May 2014
**mcl**
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mcl — The Markov Cluster Algorithm, aka the MCL algorithm.

This program implements **mcl**, a cluster algorithm for graphs. A single
parameter controls the granularity of the output clustering, namely the
**-I** *inflation* option described further below.
In standard usage of the program this parameter is the only one that may
require changing. By default it is set to 2.0 and this is a good way to
start. If you want to explore cluster structure in graphs with MCL, vary
this parameter to obtain clusterings at different levels of granularity. A
good set of starting values is 1.4, 2, 4, and 6.

The program has a rather large set of options. Except for **-I**
none affects the clustering method itself. The other options are for a
variety of aspects, such as study of the underlying MCL process (i.e.
dumping of iterands), network preprocessing (incorporated for efficiency),
resource allocation options (for large-scale analyses), output naming
and placement, output formatting, setting of verbosity levels, and so on.

Network construction and reduction techniques should not be considered as
part of a clustering algorithm. Nevertheless particular techniques may
benefit particular methods or applications. In mcl many transformations are
accessible through the **-tf** option. It can be used for edge
weight transformations and selection, as well as transformations that act on
a graph as a whole.
It is for example possible to remove edges with weight below 0.7 by issuing
**-tf** **'gq(0.7)'**, where the quotes are necessary to prevent the shell
from interpreting the parentheses. The option accepts more complicated
sequences, such as **-tf** **'gq(0.7),add(-0.7)'**. This causes all
remaining edge weights to be shifted to the range [0-0.3], assuming that the
input contains correlations. Many more transformations are supported, as
documented in mcxio. Examples of graph-wide transformations are
`'#knn(<num>)'` and `'#ceilnb(<num>)'`. The first only keeps those edges
that occur in the list of top-`<num>` edges of highest weight in both of
its incident nodes. The second removes edges from nodes of highest degree
first, proceeding until all node degrees satisfy the given threshold.
The **-pi** (pre-inflation) option can be used to increase the
contrast in edge weights. This may be useful when clusterings are coarse and
fine-grained clusterings are difficult to obtain.

There are two main modes of invocation. The most accessible is
*label mode*
which assumes a format alternatively called label input or ABC-format.
The input is then a file or stream in which each
line encodes an edge in terms of two labels (the 'A' and the 'B')
and a numerical value (the 'C'), all separated
by white space. The most basic example of usage is this:

The output is then a file where each line is a cluster of tab-separated labels. If clustering is part of a larger workflow where it is desirable to analyse and compare multiple clusterings, then it is a good idea to use native mode rather than ABC mode. The reason for this is that native mode is understood by all programs in the mcl suite. It is a more stringent and unambiguous format, and hence more suitable for data exchange. The reader is refered to clmprotocols for more information.

The example invocation below assumes matrix input, as explained above
and described in the mcxio section. Switching to label mode requires
the input file to be in ABC-format and the addition of the **--abc**
option.

**mcl** <-|fname>
**[-I** <num> (*inflation*)**]**
**[-o** <str> (*fname*)**]**

These options are sufficient in 95 percent of the cases or more. The first
argument must be the name of a file containing a graph/matrix in the mcl
input format, or a hyphen to read from STDIN. With respect to clustering,
the **-I** option is foremost relevant.

The full listing of **mcl** options is shown further below, separated
into parts corresponding with functional aspects such
as clustering, threading, verbosity, network preprocessing, pruning and resource management,
automatic output naming, and dumping.

**Baseline clustering options**

**[-I** <num> (*inflation*)**]**
**[-o** <fname> (*fname*)**]**

**Output options**

**[-odir** <dname> (*directory*)**]**
**[--d** (*use input directory for output*)**]**

**Input options**

**[--abc** (*expect/write labels*)**]**
**[--sif** (*expect/write labels*)**]**
**[--etc** (*expect/write labels*)**]**
**[--expect-values** (*sif or etc stream contains values*)**]**
**[-use-tab** <fname> (*use mapping to write*)**]**

**Transform options**

**[-tf** <tf-spec> (*transform input matrix values*)**]**
**[-abc-tf** <tf-spec> (*transform input stream values*)**]**
**[--abc-neg-log10** (*take log10 of stream values, negate sign*)**]**
**[--abc-neg-log** (*take log of stream values, negate sign*)**]**
**[-icl** <fname> (*create subgraph on clustering*)**]**

**Cache options**

**[-write-graph** <fname> (*write graph*)**]**
**[-write-graphx** <fname> (*write transformed graph*)**]**
**[-write-expanded** <fname> (*write expanded graph*)**]**
**[--write-limit** (*write mcl process limit*)**]**

**Input manipulation options**

**[-pi** <num> (*pre-inflation*)**]**
**[-ph** <num> (*pre-inflation, max-bound*)**]**
**[-if** <num> (*start-inflation*)**]**
**[--discard-loops=**<y/n> (*discard y/n loops in input*)**]**
**[--sum-loops** (*set loops to sum of other arcs weights*)**]**
**[-c** <num> (*reweight loops*)**]**

**Clustering processing options**

**[-sort** <str> (*sort mode*)**]**
**[-overlap** <str> (*overlap mode*)**]**
**[--force-connected=**<y/n> (*analyze components*)**]**
**[--check-connected=**<y/n> (*analyze components*)**]**
**[--analyze=**<y/n> (*performance criteria*)**]**
**[--show-log=**<y/n> (*show log*)**]**

**Verbosity options**

**[-q** <spec> (*log levels*)**]**
**[-v** <str> (*verbosity type on*)**]**
**[-V** <str> (*verbosity type off*)**]**
**[--show** (*print (small) matrices to screen*)**]**

**Thread options**

**[-te** <int> (*#expansion threads*)**]**

**Output file name and annotation options**

**[-o** <str> (*fname*)**]**
**[-ap** <str> (*use str as file name prefix*)**]**
**[-aa** <str> (*append str to suffix*)**]**
**[-az** (*show output file name and exit*)**]**
**[-ax** (*show output suffix and exit*)**]**
**[-annot** <str> (*dummy annotation option*)**]**

**Dump options**

**[-dump-interval** <i:j> (*dump interval*)**]**
**[-dump-modulo** <int> (*dump modulo*)**]**
**[-dump-stem** <stem> (*dump file stem*)**]**
**[-dump** <str> (*type*)**]**
**[-digits** <int> (*printing precision*)**]**
**[--write-binary** (*write matrices in binary format*)**]**

**Info options**

**[--jury-charter** (*explains jury*)**]**
**[--version** (*show version*)**]**
**[-how-much-ram** k (*RAM upper bound*)**]**
**[-h** (*most important options*)**]**
**[--help** (*one-line description for all options*)**]**
**[-z** (*show current settings*)**]**
**[-az** (*show output file name and exit*)**]**
**[-ax** (*show output suffix and exit*)**]**
**[--show-schemes** (*show resource schemes*)**]**

**Implementation options**

**[-sparse** <int> (*sparse matrix multiplication threshold*)**]**

**Pruning options**

The following options all pertain to the various pruning strategies that can
be employed by **mcl**. They are described in the PRUNING OPTIONS
section, accompanied by a description of the mcl pruning strategy.
If your graphs are huge
and you have an appetite for tuning, have a look at the following:

**[-scheme** <int> (*resource scheme*)**]**
**[-resource** <int> (*per-node resource maximum*)**]**
**[-p** <num> (*cutoff*)**]**
**[-P** <int> (*1/cutoff*)**]**
**[-S** <int> (*selection number*)**]**
**[-R** <int> (*recovery number*)**]**
**[-pct** <int> (*recover percentage*)**]**
**[-warn-pct** <int> (*prune warn percentage*)**]**
**[-warn-factor** <int> (*prune warn factor*)**]**

The first argument of **mcl** must be a file name, but some options are allowed
to appear as the first argument instead. These are the options that cause
mcl to print out information of some kind, after which it will gracefully
exit. The full list of these options is

**-z**,
**-h**,
**--help**,
**--version**,
**--show-settings**,
**--show-schemes**,
**--jury-charter**.

**mcl** implements the **MCL algorithm**, short for the **Markov cluster
algorithm**, a cluster algorithm for graphs developed by Stijn van Dongen at
the Centre for Mathematics and Computer Science in Amsterdam, the
Netherlands. The algorithm simulates flow using two simple algebraic
operations on matrices.
The inception of this flow process and the theory behind it are
described elsewhere (see REFERENCES). Frequently asked questions
are answered in the mclfaq section.
The program described here is a fast threaded implementation written by the
algorithm's creator with contributions by several others. Anton Enright
co-implemented threading; see the HISTORY/CREDITS section for a complete
account.
See the APPLICABILITY section for a description of the type of
graph mcl likes best, and for a qualitative assessment of its speed.
**mcl** is accompanied by several other utilities for analyzing clusterings and
performing matrix and graph operations; see the SEE ALSO section.

The first argument is the input file name, or a single hyphen to read from stdin. The rationale for making the name of the input file a fixed parameter is that you typically do several runs with different parameters. In command line mode it is pleasant if you do not have to skip over an immutable parameter all the time.

The **-I** *f* option is the main control,
affecting cluster granularity.
In finding good **mcl** parameter settings for a particular domain,
or in finding cluster structure at different levels of granularity,
one typically runs mcl multiple times for varying values of f (refer
to the **-I** *inflation* option for further information).

**NOTE** MCL interprets the matrix
entries or graph edge weights as **similarities**, and it likes
**undirected input graphs** best. It can handle directed graphs, but any
node pair (i,j) for which w(i,j) is much smaller than w(j,i) or vice versa
will presumably have a slightly negative effect on the clusterings output by
mcl. Many such node pairs will have a distinctly negative effect, so try to
make your input graphs undirected. How your edge weights are computed may
affect mcl's performance. In protein clustering, one way to go is to
choose the negated logarithm of the BLAST probabilities (see
REFERENCES).

**mcl**'s default parameters should make it quite fast under almost all
circumstances. Taking default parameters, mcl has been used to generate
good protein clusters on 133k proteins, taking 10 minutes running time on a
Compaq ES40 system with four alpha EV6.7 processors. It has been applied
(with good results) to graphs with two million nodes, and if you have the memory
(and preferably CPUs as well) nothing should stop you from going further.

For large graphs, there are several groups of parameters available for
tuning the mcl computing process, should it be necessary. The easiest thing
to do is just vary the **-scheme** option. This
triggers different settings for the group of pruning parameters
**-p/-P**, **-R**, **-S**, and
**-pct**. The default setting corresponds with
**-scheme** **6**.
When doing multiple mcl runs for the same graphs with different
**-I** settings (for obtaining clusterings at different levels
of granularity), it can be useful to factor out the first bit
of computation that is common to all runs, by using
the **-write-expanded** option one time
and then using **-if** *inflation* for each run in the set.
Whether mcl considers a graph large depends mainly on the graph
connectivity; a highly connected graph on 50,000 nodes is large to
mcl (so that you might want to tune resources) whereas a sparsely
connected graph on 500,000 nodes may be business as usual.

**mcl** is a memory munger. Its precise appetite depends on the resource
settings. You can get a rough (and usually much too pessimistic) upper
bound for the amount of RAM that is needed by using the
**-how-much-ram** option. The corresponding
entry in this manual page contains the simple formula via which the upper
bound is computed.

Other options of interest are the option to specify threads
**-te**, and the verbosity-related options
**-v** and **-V**.
The actual settings are shown with **-z**, and for graphs with
at most 12 nodes or so you can view the MCL matrix iterands on screen
by supplying **--show** (this may give some
more feeling).

MCL iterands allow a generic interpretation as clusterings as well. The
clusterings associated with early iterands may contain a fair amount of
overlap. Refer to the **-dump** option, the mclfaq
manual, and the **clm imac** utility (Interpret Matrices As Clusterings).
Use **clm imac** only if you have a special reason; the normal usage of **mcl** is
to do multiple runs for varying **-I** parameters and use the
clusterings output by mcl itself.

Under very rare circumstances, **mcl** might get stuck in a seemingly infinite
loop. If the number of iterations exceeds a hundred and the *chaos*
indicator remains nearly constant (presumably around value 0.37), you can
force mcl to stop by sending it the ALRM signal (usually done
by **kill -s ALRM** *pid*). It will finish the current
iteration, and interpret the last iterand as a clustering. Alternatively, you
can wait and mcl might converge by itself or it will certainly stop after
10,000 iterations. The
most probable explanation for such an infinite loop is that the input graph
contains the flip-flop graph of node size three as a subgraph.

The creator of this page feels that manual pages are a valuable resource,
that online html documentation is also a good thing to have, and
that info pages are way *way* ahead of their time. The
NOTES section explains how this page was created.

In the OPTIONS section options are listed in order of importance, with related options grouped together.

Sets the main inflation value to *<num>*. This value is the main handle
for affecting cluster granularity. It is usually chosen somewhere
in the range [1.2-5.0]. **-I** **5.0** will tend to result
in fine-grained clusterings, and **-I** **1.2** will tend to
result in very coarse grained clusterings. Your mileage will vary
depending on the characteristics of your data. That is why it is
a good idea to test the quality and coherency of your clusterings
using **clm dist** and **clm info**. This will most likely reveal that
certain values of **-I** are simply not right for your data. The
**clm dist** section contains a discussion of how to use the cluster
validation tools shipped with **mcl** (see the SEE ALSO section).

With low values for **-I**, like **-I** **1.2**, you should be
prepared to use more resources in order to maintain quality of
clusterings, i.e. increase the argument to the
**-scheme** option.

The default mode of output creation for **mcl** is to create a file name that
uses the input file name stripped of any leading path components, augmented
with a prefix '`out.`' and a suffix encoding pivotal **mcl** parameters.
This will usually be the inflation value which is the argument to the **-I**
option. By default the output file is written in the current directory.
For example, if the input is named `data/small.mci` for example and
inflation is set to three, the output file will be named
`out.small.mci.I30`.

This behaviour can be overridden in various ways. The **-o** option simply
specifies the output file name, which may include path components that
should exist. It is possible to send the clustering to STDOUT by supplying
**-o** **-**. With the **-odir** *<dname>* option **mcl** constructs the
output file name as before, but writes the file in the directory *<dname>*.
Finally, the option **--d** is similar but more specific in that **mcl**
will write the output in the directory specified by the path component of
the input file, that is, the directory in which the input file resides.

If either one of
**--abc**, **--sif**, **--etc** or
**-use-tab** *tab-file* is used the output will be in label format.
Otherwise the clustering is output in the mcl matrix format; see the mcxio
section for more information on this. Refer also to the group of options
discussed at **--abc**.

Look at the **-ap** *prefix* option and its siblings for the
automatic naming constructions employed by **mcl** if the **-o** option is
not used.

With the **-c** *<num>* option, as the final step of loop computation
(i.e. after initialization and shadowing) all loop weights are multiplied by
**<num>**, if supplied.

By default **mcl** will remove any loops that are present in the input. Use
**--discard-loops**=**n** to turn this off. Bear in mind that loops will
still be modified in all cases where the loop weight is not maximal among
the list of edge weights for a given node.

These items all relate to label input and/or label output.
**--abc** tells **mcl** to expect label input and output clusters
in terms of those labels. This simple format expects two or
three fields separated by white space on each line.
The first and second fields are interpreted as labels specifying
source and destination node respectively. The third field, if present,
specifies the weight of the arc connecting the two nodes.

The option **--sif** tells **mcl** to expect SIF (Simple Interaction File)
format. This format is line based. The first two fields
specify the source node (as a label) and the relationship type. An arbitrary number
of fields may follow, each containing a label identifying a destination node.
The second field is simply ignored by **mcl**.
As an extension to the SIF format
weights may optionally follow the labels, separated from them with a colon character.
It is in this case necessary to use the **--expect-values** option.
The **--etc** option expects a format identical in all respects except
that the relationship type is not present, so that all fields after the first
are interpreted as destination labels.

**-use-tab** is only useful when matrix input is used.
It will use the tab file to convert the output to labels; it does
not fail on indices missing from the tab file, but will bind
these to generated dummy labels.

**-tf** transforms the values of the input matrix according
to **<tf-spec>**. **-abc-tf** transforms the stream values
(when **--abc** is used) according to **<tf-spec>**.
**--abc-neg-log** and **--abc-neg-log10**
imply that the stream input values are
replaced by the negation of their log or log10 values, respectively.
The reason for their existence is documented in mcxio.
For a description of the transform language excpected/accepted
in **<tf-spec>** refer to the same.

With this option **mcl** will subcluster the provided clustering.
It does so by removing, first of all, all edges from the input
graph that connect different clusters.
The resulting graph consists of different
components, at least as many as there are clusters in
the input clustering. This graph is then subjected to transformations,
if any are specified, and then clustered.
The output name is constructed by appending the normal mcl-created
file name suffix to the name of the input clustering.

The first two options are somewhat outdated, in that the prefered way of
loading networks is by using mcxload. The option
**-write-expanded** can be useful for exploring more complicated input
transformations that incorporate an expansion step, but is not really
relevant for production use. The last option is mainly educational and for
analyzing the **mcl** process itself.

There are currently seven different resource schemes, indexed 1..7.
High schemes result in more expensive computations that may possibly be
more accurate. The default scheme is 4. When **mcl** is done, it will give a
grade (the so called *jury synopsis*) to the appropriateness of the
scheme used. *A low grade does not necessarily imply that the
resulting clustering is bad* - but anyway, a low grade should be reason
to try for a higher scheme.

Use the **-resource** *<num>* option to cap for each nodes the number of
neighbours tracked during computation at *<num>* nodes.

The PRUNING OPTIONS section contains an elaborate description
of the way **mcl** manages resources, should you be interested.
In case you are worried about the validation of the resulting
clusterings, the mclfaq section
has several entries discussing this issue. The bottom line is
that you have to compare the clusterings resulting from different
schemes (and otherwise identical parameters) using utilities
such as **clm dist**, **clm info** on the one hand, and your
own sound judgment on the other hand.

If your input graph is extremely dense, with an average node degree
(i.e. the number of neighbours per node) that is somewhere above
500, you may need to filter the input graph by removing edges,
for example by using one of **-tf** **'#ceilnb()'**
or **-tf** **'#knn()'**.

Shows the explicit settings to which the different preset schemes correspond.

The characteristics are written in the same format (more or less) as
the output triggered by **-v** **pruning**.

See the **-v** option below.

These are the different verbosity modes:

**pruning**

**explain**

**cls**

**all**

To make mcl as quiet as can be, add
**-q** **x** **-V** **all** to the command line.

The **-q** option governs a general logging mechanism.
The format accepted is described in the tingea.log manual page.

The other options govern verbosity levels specific to mcl. **-v** **all**
turns them all on, **-V** **all** turns them all off. **-v** *str* and
**-V** *str* turn on/off the single mode *str* (for *str*
equal to one of **pruning**, **cls**, or **explain**). Each verbosity
mode is given its own entry below.

This mode causes the output of explanatory headers illuminating the
output generated with the **pruning** verbosity mode.

This mode causes output of resource-related quantities. It has a separate entry in the PRUNING OPTIONS section.

This mode (on by default) prints a terse list of characteristics of the
clusterings associated with intermediate iterands. The characteristics are
**E/V**, **cls**, **olap**, and **dd**. They respectively stand for the
number of outgoing arcs per node (as an average), the number of clusters in
the overlapping clustering associated with the iterand, the number of nodes
in overlap, and the *dag depth* associated with the DAG (directed acyclic
graph) associated with the iterand. For more information on this DAG refer
to the **-dump** option description in this manual and also
mclfaq.

**Standard log information**

m-ie

This gives the ratio of (1) the number of edges after initial expansion, before pruning, to
(2) the number of edges of the current iterand.

m-ex

This gives the ratio of (1) the number of edges after expansion (including pruning), to
(2) the number of edges of the current iterand.

i-ex

This gives the ratio of (1) the number of edges after expansion (including pruning), to
(2) the number of edges of the original input graph.

fmv

This gives the percentage of nodes (matrix columns) for which full matrix/vector
computation was used (as opposed to using a sparse technique).

See the **-ap** option below.

If the **-o** *fname* option is not used,
**mcl** will create a file name (for writing output to) that
should uniquely characterize the important parameters used in the
current invocation of mcl. The default format is **out.fname.suf**,
where **out** is simply the literal string `out`, **fname** is the
first argument containing the name of the file (with the graph) to be
clustered, and where **suf** is the suffix encoding a set of parameters
(described further below).

The **-ap** *str* option specifies a prefix to use
rather than **out.fname** as sketched above.
However, **mcl** will interpret the character '=', if present
in *str*, as a placeholder for the input file name.

If the **-aa** *str* option is used, **mcl** will append
**str** to the suffix **suf** created by itself.
You can use this if you need to encode some extra information in the
file name suffix.

The suffix is constructed as follows. The **-I** *f*
and **-scheme** parameter are always encoded.
Other options, such as **-pi** *f* and **-knn**
are only encoded if they are used. Any real argument *f*
is encoded using *exactly one* trailing digit behind the decimal
separator (which itself is not written). The setting **-I** **3.14**
is thus encoded as I31. The **-scheme** option is encoded using the
letter 's', all other options mentioned here are encoded as themselves
(stripped of the hyphen). For example

mcl small.mci -I 3 -c 2.5 -pi 0.8 -scheme 5

results in the file name `out.small.mci.I30s5c25pi08`.
If you want to know beforehand what file name will be produced,
use the **-az** option.

If **mcl** automatically constructs a file name, it can be helpful to known
beforehand what that file name will be. Use **-az** and mcl will
write the file name to STDOUT and exit. This can be used if mcl is
integrated into other software for which the automatic creation of
unique file names is convenient.

By default mcl incorporates the input file name into the output file
name and appends a short suffix describing the most important
option settings. Use **-ax** to find out what that suffix is.
This can be useful in wrapper pipeline scripts such as clxcoarse.

**mcl** writes the command line with which it was invoked to the output
clustering file. Use this option to include any additional
information. MCL does nothing with this option except copying
it as just described.

Threading is useful if you have a multi-processor system. **mcl** will
spawn *k* threads of computation. If these are computed
in parallel (this depends on the number of CPUs available to the
mcl process) it will speed up the process accordingly.

When threading, it is best not to turn on pruning verbosity
mode if you are letting mcl run unattended, unless you want to
scrutinize its output later. This is because it makes **mcl** run
somewhat slower, although the difference is not dramatic.

If used, **mcl** will apply inflation one time to the input graph
before entering the main process. This can be useful for
making the edge weights in a graph either more homogeneous (which
may result in less granular clusterings) or more heterogeneous
(which may result in more granular clusterings).
Homogeneity is achieved for values *<num>* less than one,
heterogeneity for values larger than one.
Values to try are normally in the range `[2.0,10.0]`.

The **-ph** option is special in that it does not rescale
columns to be stochastic. Instead, it rescales columns so that
the maximum value found in the column stays the same after
inflation was applied. There is little significance to this,
and what little there is is undocumented.

If used, **mcl** will apply inflation one time to the input graph
before entering the main process. The difference with
**-pi** is that with the latter option mcl may apply
certain transformations after reading in the matrix such
as adding or modifying loops. The purpose of
the **-if** (mnemonic for *inflation-first*)
option is to use it on graphs saved
with the **--write-expanded** option and convey
to mcl that it should not apply those transformations.

Dump during iterations i..j-1. Use *all* to dump in all
iterations. See the **-dump** *str* option below.

Sampling rate: select only these iterations in the dump interval.
See the **-dump** *str* option below.

Set the the stem for file names of dumped
objects (default *mcl*). See the **-dump** *str* option below.

*str* is checked for substring occurrences of the following entries.
Repeated use of **-dump** is also allowed.

**ite**

**dag**

**cls**

**chr**

**lines**

**cat**

**lines** and **cat** change the mode of dumping. The first
changes the dump format to a line based pairwise format rather
than the default mcl matrix format. The second causes all
dumped items to be dumped to the default stream used for the
output clustering, which is appended at the end.

The **ite** option writes **mcl** iterands to file. The **cls**
option writes clusterings associated with mcl iterands to file.
These clusters are obtained from a particular directed acyclic graph
(abbreviated as DAG) associated with each iterand. The **dag** option
writes that DAG to file. The DAG can optionally be further
pruned and then again be interpreted as a
clustering using **clm imac**, and **clm imac** can also
work with the matrices written using the **ite** option.
It should be noted that clusterings associated with intermediate
iterands may contain overlap, which is interesting in
many applications. For more information
refer to mclfaq and the REFERENCES section below.

The **result** option dumps the usual MCL clustering.

The **chr** option says, for each iterand I, to output a matrix C with
characteristics of I. C has the same number of columns as I. For each
column k in C, row entry 0 is the diagonal or 'loop' value of column k in
I *after* expansion and pruning, and *before* inflation and
rescaling. Entry 1 is the loop value *after* inflation and rescaling.
Entry 2 is the center of column k (the sum of its entries squared)
computed *after* expansion and *before* pruning, entry 3 is the
maximum value found in that column at the same time. Entry 4 is the
amount of mass kept for that column *after pruning*.

The **-ds** option sets the stem for file names of dumped
objects (default *mcl*). The **-di** and **-dm**
options allow a selection of iterands to be made.

This has two completely different uses. It sets
the number of decimals used for pretty-printing **mcl** iterands
when using the **--show** option (see below),
and it sets the number of decimals used for writing
the expanded matrix when using the **-write-expanded** option.

Print matrices to screen. The number of significant digits to be
printed can be tuned with **-digits** *n*. An 80-column screen
allows graphs (matrices) of size up to 12(x12) to be printed with
three digits precision (behind the comma), and of size up to 14(x14)
with two digits. This can give you an idea of how **mcl** operates,
and what the effect of pruning is.
Use e.g. **-S** **6** for such
a small graph and view the MCL matrix iterands with **--show**.

Write matrix dump output in binary mcl format rather
than interchange mcl format (the default). Note that **mcxconvert**
can be used to convert each one into the other.
See mcxio and mcx for more information.

*str* can be one of **lex**, **size**, **revsize**,
or **none**. The default is 'revsize', in which the largest
clusters come first. If the mode is 'size', smallest clusters
come first, if the mode is 'lex', clusters are ordered
lexicographically, and if the mode is 'none', the order
is the same as produced by the procedure used by mcl to
map matrices onto clusterings.

Mode *keep* causes mcl to retain overlap should this improbable event
occur. In theory, **mcl** may generate a clustering that contains overlap,
although this almost never happens in practice, as it requires some
particular type of symmetry to be present in the input graph (not just any
symmetry will do). Mathematically speaking, this is a conjecture and not a
theorem, but the present author wil eat his shoe if it fails to be true (for
marzipan values of shoe). It is easy though to construct an input graph for
which certain mcl settings result in overlap - for example a line graph on
an odd number of nodes. The default is to excise overlapping parts and
introduce them as clusters in their own right. It is possible to allocate
nodes in overlap to the first cluster in which they occur (i.e. rather
arbitrarily), corresponding with mode *cut*.

In mode *split* mcl will put all nodes in overlap into
separate clusters. These clusters are chosen such that
two nodes are put in the same new cluster if and only if
they always occur paired in the clusters of the
overlapping clustering.

This option has no effect on the clusterings that are
output when using **-dump** *cls* -
the default for those is that overlap is not touched,
and this default can not yet be overridden.

If the input graph has strong bipartite characteristics, mcl may yield clusters that do not correspond to connected components in the input graph. Turn one of these modes on to analyze the resultant clustering.

If loose clusters are found
they will be split into subclusters corresponding to
connected components.
With **--force-connected**=*y* mcl will write the
corrected clustering to the normal output file, and the old clustering
to the same file with suffix `orig`.
With **--check-connected**=*y* mcl will write the
loose clustering to the normal output file, and the corrected clustering
to the same file with suffix `coco`.

These options are not on by default, as the analysis is currently (overly) time-consuming and mcl's behaviour actually makes some sense (when taking bipartite characteristics into account).

With this mode turned on, **mcl** will reread the input matrix
and compute a few performance criteria and attach them to
the output file. Off by default.

Shows the log with process characteristics on STDOUT. By default, this mode is off.

Explains how the jury synopsis is computed from the jury marks.

Show version.

**<int>** is interpreted as the number of nodes of an input graph.
mcl will print the maximum amount of RAM it needs for its computations.
The formula for this number in bytes is:

2 * c * k * <int>
2 : two matrices are concurrently held in memory.
c : mcl cell size (as shown by -z).
<int>: graph cardinality (number of nodes).
k : MAX(s, r).
s : select number (-S, -scheme options).
r : recover number (-R, -scheme options).

This estimate will usually be too pessimistic. It does assume though that
the average node degree of the input graph does not exceed k. The
**-how-much-ram** option takes other command-line arguments into
account (such as **-S** and **-R**), and it expresses the
amount of RAM in megabyte units.

Shows a selection of the most important **mcl** options.

Gives a one-line description for all options.

Show current settings for tunable parameters.
**--show-settings** is a synonym.

After computing a new (column stochastic) matrix vector during expansion
(which is matrix multiplication c.q. squaring), the vector is
successively exposed to different pruning strategies. The intent of
pruning is that many small entries are removed while retaining much of
the stochastic mass of the original vector. After pruning, vectors are
rescaled to be stochastic again. MCL iterands are theoretically known to
be sparse in a weighted sense, and this manoever effectively perturbs the
MCL process a little in order to obtain matrices that are genuinely
sparse, thus keeping the computation tractable. An example of monitoring
pruning can be found in the discussion of
**-v** **pruning**
at the end of this section.

**mcl** proceeds as follows. First, entries that are smaller than
*cutoff* are removed, resulting in a vector with at most
*1/cutoff* entries. The cutoff can be supplied either by
**-p**, or as the inverse value by **-P**. The latter is more
intuitive, if your intuition is like mine (P stands for precision or pruning).
The cutoff just described is rigid; it is the same for all vectors. The
**--adapt** option causes the computation of a
cutoff that depends on a vector's homogeneity properties, and this option
may or may not speed up mcl.

Second, if the remaining stochastic mass (i.e. the sum of all remaining
entries) is less than *<pct>*/100 and the number of remaining
entries is less than *<r>* (as specified by the **-R** flag),
**mcl** will try to regain ground by recovering the largest discarded
entries. The total number of entries is not allowed to grow larger than
*<r>*.
If recovery was not necessary, mcl tries to prune the vector further
down to at most *s* entries (if applicable), as specified by the
**-S** flag. If this results in a vector that satisfies the recovery
condition then recovery is attempted, exactly as described above. The
latter will not occur of course if *<r>* <= *<s>*.

The default setting is something like **-P** **4000** **-S** **500**
**-R** **600**. Check the **-z** flag to be sure. There is a set
of precomposed settings, which can be triggered with the
**-scheme** *k* option. *k*=4 is the default
scheme; higher values for *k* result in costlier and more accurate
computations (vice versa for lower, cheaper, and less accurate).
The schemes are listed using the **--show-schemes** option. It is
advisable to use the **-scheme** option only in interactive mode,
and to use the explicit expressions when doing batch processing. The
reason is that there is *no guarantee whatsoever* that the schemes
will not change between different releases. This is because the scheme
options should reflect good general purpose settings, and it may become
appararent that other schemes are better.

Note that 'less accurate' or 'more accurate' computations may still
generate the same output clusterings. Use **clm dist** to compare output
clusterings for different resource parameters. Refer to clm dist
for a discussion of this issue.

The two options **-warn-pct** and **-warn-factor** relate to
warnings that may be triggered once the *initial* pruning of a vector
is completed. The idea is to issue warnings if initial pruning almost
completely destroys a computed vector, as this may be a sign that the
pruning parameters should be changed. It depends on the mass remaining
after initial pruning whether a warning will be issued. If that mass is
less than *warn-pct* or if the number of remaining entries is smaller
by a factor *warn-factor* than both the number of entries originally
computed *and* the recovery number, in that case, **mcl** will issue a
warning.

**-warn-pct** takes an integer between 0 and 100 as parameter,
**-warn-factor** takes a real positive number. They default to
something like 30 and 50.0. If you want to see less warnings, decrease
*warn-pct* and increase *warn-factor*. Set *warn-factor* to zero
if you want no warnings.

Pruning verbosity mode causes **mcl** to emit several statistics related to
the pruning process, each of which is described below. Use
**-v** **explain** to get explanatory headers in the output as well
(or simply use **-v** **all**).

This value (by default set to 10) determines when mcl switches to sparse matrix/vector multiplication.
For a given column stochastic vector (corresponding with all the neighbours
of a given node `v` according to the current mcl iterand) the sum `S` of neighbour counts
of all neighbours of `v` is computed, counting duplicates. This is exactly the
number of matrix entries involved in the computation of the new column vector
for the matrix product. If `S` times *<int>* does not exceed the
number of nodes in the graph (equal to both column and row dimension of
the matrices used) then a sparse implementation is used. Otherwise
an optimized regular implementation is used. Intuitively, this option can
be thought of as the estimated overhead per matrix floating point operation incurred
by the sparse implementation compared with the regular implementation.
MCL uses this estimated overhead to determine which implementation is likely
to be quicker. Testing has shown this strategy works very well for graphs of a wide
range of sizes, including graphs with up to 3 million nodes and 500 million edges.

**NOTE**

The effectiveness of this option is influenced by hardware-specific properties
such as the CPU L2 cache size. The default value should work reasonably well
across a wide variety of scenarios, but it may be possible to squeeze
faster run times out of mcl by tuning this parameter to the graphs that are
specific for your application domain.

The following is an example of label input

---8<------8<------8<------8<------8<---
cat hat 0.2
hat bat 0.16
bat cat 1.0
bat bit 0.125
bit fit 0.25
fit hit 0.5
hit bit 0.16
--->8------>8------>8------>8------>8---

It can be clustered like this:

mcl cathat --abc -o out.cathat

The file out.cathat should now like like this

---8<------8<------8<------8<------8<---
cat hat bat
bit fit hit
--->8------>8------>8------>8------>8---

A few things to note. First, MCL will symmetrize any
arrow it finds. If it sees `bat cat 1.0` it will act as if it also
saw `cat bat 1.0`. You can explicitly specify `cat bat 1.0`,
mcl will in the first parse stage simply end up with duplicate
entries. Second, MCL deduplicates repeated edges by taking the
one with the maximum value. So,

---8<------8<------8<------8<------8<---
cat hat 0.2
hat cat 0.16
hat cat 0.8
--->8------>8------>8------>8------>8---

Will result in two arrows `cat-hat` and `hat-cat` both
with value 0.8.

**mcl** will work very well for graphs in which the diameter of the natural
clusters is not too large. The presence of many edges between different
clusters is not problematic; as long as there is cluster structure, mcl
will find it. It is less likely to work well for graphs with clusters
(inducing subgraphs) of large diameter, e.g. grid-like graphs derived from
Euclidean data. So mcl in its canonical form is certainly not fit for
boundary detection or image segmentation. I experimented with a modified
mcl and boundary detection in the thesis pointed to below (see
REFERENCES). This was fun and not entirely unsuccesful, but not
something to be pursued further.

**mcl** likes *undirected input graphs best*, and it really dislikes graphs
with node pairs (i,j) for which an arc going from i to j is present and the
counter-arc from j to i is absent. Try to make your input graph undirected.
Furthermore, mcl interprets edge weights in graphs as similarities. If you
are used to working with dissimilarities, you will have to convert those to
similarities using some conversion formula. The most important thing is
that you feel confident that the similarities are reasonable, i.e. if X is
similar to Y with weight 2, and X is similar to Z with weight 200, then this
should mean that the similarity of Y (to X) is neglectible compared with the
similarity of Z (to X).

**mcl** is probably not suited for clustering *tree graphs*. This is because
mcl works best if there are multiple paths between different nodes in the
natural clusters, but in tree graphs there is only one path between any pair
of nodes. Trees are too sparse a structure for mcl to work on.

**mcl** may well be suited for clustering *lattices*. It will depend
on the density characteristics of the lattice, and the conditions for
success are the same as those for clustering graphs in general: The
diameter of the natural clusters should not be too large.
**NOTE** when clustering a lattice, you *have* to cluster
the underlying undirected graph, and not the directed graph that represents
the lattice itself. The reason is that one has to allow mcl (or any other
cluster algorithm) to 'look back in time', so to speak. Clustering and
directionality bite each other (long discussion omitted).

**mcl** has a worst-case time complexity O(N*k^2), where N is the number of
nodes in the graph, and k is the maximum number of neighbours tracked during
computations. k depends on the **-P** and **-S** options. If the
**-S** option is used (which is the default setting) then k equals the
value corresponding with this option. Typical values for k are in the range
500..1000. The average case is much better than the worst case though, as
cluster structure itself has the effect of helping mcl's pruning schemes,
certainly if the diameter of natural clusters is not large.

There are currently no resource nor configuration files. The mcl matrix format is described in the mcxio section.

MCLXIODIGITS

When writing matrices in interchange format, mcl will use this variable (if present) as the precision (number of digits) for printing the fractional part of values.

MCLXIOVERBOSITY

MCL and its sibling applications will usually report about matrix input/output from/to disk. The verbosity level can be regulated via MCLXIOVERBOSITY. These are the levels it can currently be set to.

1

Silent but applications may alter this.

2

Silent and applications can not alter this.

4

Verbose but applications may alter this.

8

Verbose and applications can not alter this (default).

MCLXIOFORMAT

MCL and its sibling applications will by default output matrices in interchange format rather than binary format (cf. mcxio). The desired format can be controlled via the variable MCLXIOFORMAT. These are the levels it can currently be set to.

1

Interchange format but applications may alter this.

2

Interchange format and applications can not alter this (default).

4

Binary format but applications may alter this.

8

Binary format and applications can not alter this.

MCLXICFLAGS

If matrices are output in interchange format, by default empty vectors
will not be listed. Equivalently (during input time),
vectors for which no listing is present are understood to be empty -
note that the *presence* of a vector is established using
the domain information found in the header part.
It is possible to enforce listing of empty vectors by
setting bit '1' in the variable MCLXICFLAGS.

MCLXIOUNCHECKED

MCL and its sibling applications will always check a matrix for consistency while it is being read. If this variable is set, the consistency check is omitted. For large graphs the speed up can be considerable. However, if the input graph is not conforming it will likely crash the application that is using it.

If **mcl** issues a diagnostic error, it will most likely be
because the input matrix could not be parsed succesfully.
**mcl** tries to be helpful in describing the kind of parse error.
The mcl matrix format is described in the mcxio section.

No known bugs at this time.

Stijn van Dongen.

The MCL algorithm was conceived in spring 1996 by the present author. The first implementation of the MCL algorithm followed that spring and summer. It was written in Perl and proved the viability of the algorithm. The implementation described here began its life in autumn 1997. The first versions of the vital matrix library were designed jointly by Stijn van Dongen and Annius Groenink in the period Oktober 1997 - May 1999. The efficient matrix-vector multiplication routine was written by Annius. This routine is without significant changes still one of the cornerstones of this MCL implementation.

Since May 1999 all MCL libraries have seen much development and redesign by the present author. Matrix-matrix multiplication has been rewritten several times to take full advantage of the sparseness properties of the stochastic matrices brought forth by the MCL algorithm. This mostly concerns the issue of pruning — removal of small elements in a stochastic column in order to keep matrices sparse.

Very instructive was that around April 2001 Rob Koopman pointed out that selecting the k largest elements out of a collection of n is best done using a min-heap. This was the key to the second major rewrite (now counting three) of the MCL pruning schemes, resulting in much faster code, generally producing a more accurate computation of the MCL process.

In May 2001 Anton Enright initiated the parallellization of the
**mcl** code and threaded inflation. From this example, Stijn threaded
expansion. This was great, as the MCL data structures and operands
(normal matrix multiplication and Hadamard multiplication) just beg
for parallellization.

Onwards. The January 2003 03-010 release introduced support for sparsely enumerated (i.e. indices need not be sequential) graphs and matrices, the result of a major overhaul of the matrix library and most higher layers. Conceptually, the library now sees matrices as infinite quadrants of which only finite subsections happen to have nonzero entries.

The June 2003 03-154 release introduced unix-type pipelines for clustering, including the BLAST parser mcxdeblast and the mclblastline script. The April 2004 04-105 release revived binary format, which has been a first class citizen every since.

With the March 2005 05-090 release mcxsubs finally acquired a sane specification syntax. The November 2005 05-314 release brought the ability to stream label input directly into mcl. The subsequent release introduced a transformation language shared by various mcl siblings that allows arbitrary progressions of transformations to be applied to either stream values or matrix values.

Joost van Baal set up the mcl CVS tree and packaged mcl for Debian GNU/Linux. He completely autotooled the sources, so much so that at first I found it hard to find them back amidst bootstrap, aclocal.m4, depcomp, and other beauties.

Jan van der Steen shared his elegant mempool code. Philip Lijnzaad gave useful comments. Philip, Shawn Hoon, Abel Ureta-Vidal, and Martin Mokrejs sent helpful bug reports.

Abel Ureta-Vidal and Dinakarpandian Deendayal commented on and contributed to mcxdeblast and mcxassemble.

Tim Hughes contributed several good bug reports for mcxassemble,
mcxdeblast and zoem (a workhorse for **clm format**).

mclfaq - Frequently Asked Questions.

mcxio - a description of the mcl matrix format.

There are many more utilities. Consult mclfamily for an overview of and links to all the documentation and the utilities in the mcl family.

**minimcl** is a 200-line perl implementation of mcl. It is shipped
in the mcl distribution and can be found online at
http://micans.org/mcl.

mcl's home at http://micans.org/mcl/.

[1]
Stijn van Dongen, *Graph Clustering by Flow Simulation*.
PhD thesis, University of Utrecht, May 2000.

http://www.library.uu.nl/digiarchief/dip/diss/1895620/inhoud.htm

[2]
Stijn van Dongen, *Graph Clustering Via a Discrete Uncoupling Process*,
SIAM Journal on Matrix Analysis and Applications, 30(1):121-141, 2008.
http://link.aip.org/link/?SJMAEL/30/121/1

[3]
Stijn van Dongen. *A cluster algorithm for graphs*.
Technical Report INS-R0010, National Research Institute for Mathematics and
Computer Science in the Netherlands, Amsterdam, May 2000.

http://www.cwi.nl/ftp/CWIreports/INS/INS-R0010.ps.Z

[4]
Stijn van Dongen. *A stochastic uncoupling process for graphs*.
Technical Report INS-R0011, National Research Institute for Mathematics and
Computer Science in the Netherlands, Amsterdam, May 2000.

http://www.cwi.nl/ftp/CWIreports/INS/INS-R0011.ps.Z

[5]
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

[6]
Enright A.J., Van Dongen S., Ouzounis C.A.
*An efficient algorithm for large-scale detection of protein families*,
Nucleic Acids Research 30(7):1575-1584 (2002).

This page was generated from **ZOEM** manual macros,
http://micans.org/zoem. Both html and roff pages can be created
from the same source without having to bother with all the usual conversion
problems, while keeping some level of sophistication in the typesetting.
This is the PostScript derived from the zoem troff
output.