16 May 2014
**MCL FAQ**
14-137

mclfaq — faqs and facts about the MCL cluster algorithm.

MCL refers to the generic MCL algorithm and the MCL process on which the
algorithm is based. **mcl** refers to the implementation. This FAQ answers
questions related to both. In some places MCL is written where MCL or mcl
can be read. This is the case for example in
section 3, What kind of graphs.
It should in general be obvious from the context.

This FAQ does not begin to attempt to explain the motivation and mathematics behind the MCL algorithm - the internals are not explained. A broad view is given in faq 1.2, and see also faq 1.5 and section REFERENCES.

Some additional sections preceed the actual faq entries.
The TOC section contains a listing of all questions.
**Clicking on the number of a question**
(where it is answered) will take you to
the corresponding section of the table of contents.

The manual pages for all the utilities that come with **mcl**;
refer to mclfamily for an overview.

See the REFERENCES Section for publications detailing the mathematics behind the MCL algorithm.

General questions

For everybody with an appetite for graph clustering. Regarding the FAQ, I have kept the amount of mathematics as low as possible, as far as matrix analysis is concerned. Inevitably, some terminology pops up and some references are made to the innards of the MCL algorithm, especially in the section on resources and accuracy. Graph terminology is used somewhat more carelessly though. The future might bring definition entries, right now you have to do without. Mathematically inclined people may be interested in the pointers found in the REFERENCES section.

Given this mention of mathematics, let me point out this one time only that
using **mcl** is extremely straightforward anyway. You need only mcl and an
input graph (refer to the mcl manual page), and many people
trained in something else than mathematics are using mcl happily.

**mcl** is what you use for clustering. It implements the MCL algorithm,
which is a cluster algorithm for graphs. The MCL algorithm is basically
a shell in which the MCL process is computed and interpreted. I will
describe them in the natural, reverse, order.

The MCL process generates a sequence of stochastic matrices given some initial
stochastic matrix. The elements with even index are obtained by
*expanding* the previous element, and the elements with odd index are
obtained by *inflating* the previous element given some inflation
constant. Expansion is nothing but normal matrix squaring, and inflation is
a particular way of rescaling the entries of a stochastic matrix such that
it remains stochastic.

The sequence of MCL elements (from the MCL process) is in principle without end,
but what happens is that the elements converge to some specific kind of
matrix, called the *limit* of the process. The heuristic underlying MCL
predicts that the interaction of expansion with inflation will lead to a
limit exhibiting cluster structure in the graph associated with the
initial matrix. This is indeed the case, and several mathematical results
tie MCL iterands and limits and the MCL interpretation together
(REFERENCES).

The MCL algorithm is simply a shell around the MCL process. It transforms an input graph into an initial matrix suitable for starting the process. It sets inflation parameters and stops the MCL process once a limit is reached, i.e. convergence is detected. The result is then interpreted as a clustering.

The **mcl** implementation supplies the functionality of the MCL algorithm,
with some extra facilities for manipulation of the input graph, interpreting
the result, manipulating resources while computing the process, and
monitoring the state of these manipulations.

For *Markov Cluster*. The MCL algorithm is a **cluster** algorithm
that is basically a shell in which an algebraic process is computed.
This process iteratively generates stochastic matrices, also known
as **Markov** matrices, named after the famous Russian
mathematician Andrei Markov.

Sigh. It is a widely known fact that a TLA or Three-Letter-Acronym
is *the canonical self-describing abbreviation for the name
of a species with which computing terminology is infested* (quoted
from the Free Online Dictionary of Computing). Back when I was
thinking of a nice tag for this cute algorithm, I was
totally unaware of this. I naturally dismissed *MC*
(and would still do that today). Then *MCL* occurred
to me, and without giving it much thought I started using it.
A Google search (or was I still using Alta-Vista back then?)
might have kept me from going astray.

Indeed, *MCL* is used as a tag for *Macintosh Common Lisp*,
*Mission Critical Linux*, *Monte Carlo Localization*, *MUD Client
for Linux*, *Movement for Canadian Literacy*, and a gazillion other
things — refer to the file mclmcl.txt. Confusing. It seems that
the three characters `MCL` possess otherworldly magical powers making
them an ever so strange and strong attractor in the space of TLAs. It
probably helps that Em-See-Ell (Em-Say-Ell in Dutch) has some rhythm
to it as well. Anyway MCL stuck, and it's here to stay.

On a more general level, the label *Markov Cluster* is not an entirely
fortunate choice either. Although phrased in the language of stochastic
matrices, MCL theory bears very little relation to Markov theory, and is
much closer to matrix analysis (including Hilbert's distance) and the theory
of dynamical systems. No results have been derived in the latter framework,
but many conjectures are naturally posed in the language of dynamical
systems.

Currently, the most basic explanation of the MCL algorithm is found in the technical report [2]. It contains sections on several other (related) subjects though, and it assumes some working knowledge on graphs, matrix arithmetic, and stochastic matrices.

It should compile and run on virtually any flavour of UNIX (including Linux
and the BSD variants of course). Following the instructions in the INSTALL
file shipped with mcl should be straightforward and sufficient. Courtesy to
Joost van Baal who completely autofooled **mcl**.

Building MCL on Wintel (Windows on Intel chip) should be straightforward if
you use the full suite of cygwin tools. Install cygwin if you do not have it
yet. In the cygwin shell, unpack mcl and simply issue the commands
*./configure, make, make install*, i.e. follow the instructions in
INSTALL.

This MCL implementation should also build successfully on Mac OS X.

The current setup, which I hope to continue, is this. All releases are
identified by a date stamp. For example 02-095 denotes day 95 in the year
2002. This date stamp agrees (as of April 2000) with the (differently
presented) date stamp used in all manual pages shipped with that release.
For example, the date stamp of the FAQ you are reading is **16 May 2014**,
which corresponds with the MCL stamp **14-137**.
The Changelog file contains a list of what's changed/added with each
release. Currently, the date stamp is the primary way of identifying an **mcl**
release. When asked for its version by using **--version**, mcl
outputs both the date stamp and a version tag (see below).

Input format

This is described in the protocols manual page.

What kind of graphs

Any graph (encoded as a matrix of similarities) that is nonnegative, i.e. all similarities are greater than or equal to zero.

Graphs can be weighted, and they should preferably be symmetric. Weights
should carry the meaning of similarity, *not* distance. These weights or
similarities are incorporated into the MCL algorithm in a meaningful way.
Graphs should certainly not contain parts that are (almost) cyclic, although
nothing stops you from experimenting with such input.

Yes, unequivocally. They should preferably be symmetric/undirected though. See entries 3.7 and 3.8.

Such graphs [term] can surely exhibit clear cluster structure. If they do, there is only one way for mcl to find out. You have to change all arcs to edges, i.e. if there is an arc from i to j with similarity s(i,j) — by the DAG property this implies s(j,i) = 0 — then make s(j,i) equal to s(i,j).

This may feel like throwing away valuable information, but in truth the
information that is thrown away (direction) is *not* informative with
respect to the presence of cluster structure. This may well deserve a longer
discussion than would be justified here.

If your graph is directed and acyclic (or parts of it are), you can
transform it before clustering with mcl by using **-tf** **'#max()'**, e.g.

mcl YOUR-GRAPH -I 3.0 -tf '#max()'

Nah, I don't think so. More info at entry 3.7. You could consider the Strahler number, which is numerical measure of branching complexity.

Graphs in which the diameter [term] of (subgraphs induced by) natural clusters is not too large. Additionally, graphs should preferably be (almost) undirected (see entry below) and not so sparse that the cardinality of the edge set is close to the number of nodes.

A class of such very sparse graphs is that of tree graphs. You might look
into *graph visualization* software and research if you are interested
in decomposing trees into 'tight' subtrees.

The diameter criterion could be violated by
neighbourhood graphs derived from vector data. In the specific case
of 2 and 3 dimensional data, you might be interested
in *image segmentation* and *boundary detection*, and for
the general case there is a host of other algorithms out there. [add]

In case of weighted graphs, the notion of *diameter* is sometimes not
applicable. Generalizing this notion requires inspecting the *mixing
properties* of a subgraph induced by a natural cluster in terms of its
spectrum. However, the diameter statement is something grounded on heuristic
considerations (confirmed by practical evidence [4])
to begin with, so you should probably forget about mixing properties.

To begin with the last one: you *need not and must not* make the
input graph such that it is stochastic aka Markovian [term]. What you
need to do is make a graph that is preferably symmetric/undirected,
i.e. where s(i,j) = s(j,i) for all nodes i and j. It need not be
perfectly undirected, see the following faq for a discussion of that.
**mcl** will work with the graph of random walks that is associated
with your input graph, and that is the natural state of affairs.

The input graph should preferably be honest in the sense that if `s(x,y)=N`
and `s(x,z)=200N` (i.e. the similarities differ by a factor 200), then
this should really reflect that the similarity of `y` to `x` is neglectible
compared with the similarity of `z` to `x`.

For the rest, anything goes. Try to get a feeling by experimenting. Sometimes it is a good idea to filter out high-frequency and/or low-frequency data, i.e. nodes with either very many neighbours or extremely few neighbours.

It depends. The class of directed graphs can be viewed as a spectrum going
from undirected graphs to uni-directed graphs. *Uni-directed* is
terminology I am inventing here, which I define as the property that
for all node pairs i, j, at least one of s(i,j) or s(j,i) is zero. In other
words, if there is an arc going from i to j in a uni-directed graph, then
there is no arc going from j to i. I call a node pair i, j,
*almost uni-directed* if s(i,j) << s(j,i) or vice versa,
i.e. if the similarities differ by an order of magnitude.

If a graph does not have (large) subparts that are (almost) uni-directed, have a go with mcl. Otherwise, try to make your graph less uni-directed. You are in charge, so do anything with your graph as you see fit, but preferably abstain from feeding mcl uni-directed graphs.

Mathematically, the mcl iterands will be *nice* when the input graph is
symmetric, where *nice* is in this case *diagonally symmetric to a
semi-positive definite matrix* (ignore as needed). For one thing, such nice
matrices can be interpreted as clusterings in a way that generalizes the
interpretation of the mcl limit as a clustering (if you are curious to these
intermediate clusterings, see faq entry 9.3).
See the REFERENCES section for pointers to mathematical
publications.

The reason that mcl dislikes uni-directed graphs is not very mcl specific, it has more to do with the clustering problem itself. Somehow, directionality thwarts the notion of cluster structure. [add].

Whether your graph is created by third-party software or by custom sofware
written by someone you know (e.g. yourself), it is advisable to test whether
the software generates symmetric matrices. This can be done as follows
using the mcxi utility, assuming that you want to test the
matrix stored in file `matrix.mci`. The mcxi utility should be available
on your system if mcl was installed in the normal way.

mcxi /matrix.mci lm tp -1 mul add /check wm

This loads the graph/matrix stored in `matrix.mci` into **mcxi**'s memory with
the **mcxi** *lm* primitive. — the leading slash is how strings are
introduced in the stack language interpreted by **mcxi**. The transpose of
that matrix is then pushed on the stack with the *tp* primitive and
multiplied by minus one. The two matrices are added, and the result is
written to the file `check`.
The transposed matrix is the mirrored version of the original matrix stored
in `matrix.mci`. If a graph/matrix is undirected/symmetric, the mirrored
image is necessarily the same, so if you subtract one from the other it
should yield an all zero matrix.

Thus, the file `check` *should look like this*:

(mclheader
mcltype matrix
dimensions <num>x<num>
)
(mclmatrix
begin
)

Where `<num>` is the same as in the file `matrix.mci`. If this is not
the case, find out what's prohibiting you from feeding mcl symmetric
matrices. Note that any nonzero entries found in the matrix stored as
`check` correspond to node pairs for which the arcs in the two possible
directions have different weight.

Speed and complexity

It's fast - here is how and why. Let `N` be the number of nodes in the input
graph. A straigtforward implementation of MCL will have time and space
complexity respecively `O(N^3)` (i.e. cubic in `N`) and `O(N^2)`
(quadratic in `N`). So you don't want one of those.

**mcl** implements a slightly perturbed version of the MCL process,
as discussed in section Resource tuning / accuracy.
Refer to that section for a more extensive discussion of all
the aspects involved. This section is only concerned with the high-level
view of things *and* the nitty gritty complexity details.

While computing the square of a matrix
(the product of that matrix with itself), mcl keeps the matrix sparse
by allowing a certain maximum number of nonzero entries
per stochastic column. The maximum is one of the mcl parameters, and
it is typically set somewhere between 500 and 1500.
Call the maximum `K`.

mcl's time complexity is governed by the complexity of matrix squaring.
There are two sub-algorithms to consider. The first is the
algorithm responsible for assembling a new vector during matrix
multiplication. This algorithm has worst case complexity `O(K^2)`.
The pruning algorithm (which uses heap selection) has worst case complexity
`O(L*log(K))`, where `L` is how large a newly computed matrix column can get
before it is reduced to at most `K` entries. `L` is *bound by* the smallest
of the two numbers `N` and `K^2` (the square of `K`), but on average
`L` will be much smaller than that, as the presence of cluster structure aids in
keeping the factor `L` low. [Related to this is the fact that clustering
algorithms are actually used to compute matrix splittings that minimize
the number of cross-computations when carrying out matrix
multiplication among multiple processors.]
In actual cases of heavy usage, `L` is of order in the tens of thousands, and
`K` is in the order of several hundreds up to a thousand.

It is safe to say that in general the worst case complexity of mcl is of order O(N*K^2); for extremely tight and dense graphs this might become O(N*N*log(K)). Still, these are worst case estimates, and observed running times for actual usage are much better than that. (refer to faq 4.2).

In this analysis, the number of iterations required by mcl was not included. It is nearly always far below 100. Only the first few (less than ten) iterations are genuinely time consuming, and they are usually responsible for more than 95 percent of the running time.

The process of removing the smallest entries of a vector is called pruning. mcl outputs a summary of this once it is done. More information is provided in the pruning section of the mcl manual and Section 6 in this FAQ.

The space complexity is of order `O(N*K)`.

Few. Some experiments are described in [4], and
[5] mentions large graphs being clustered in very reasonable
time. In protein clustering, **mcl** has been applied to graphs with up to one
million nodes, and on high-end hardware such graphs can be clustered within
a few hours.

No, it does not. You might expect that one needs to sort
a vector in order to obtain the `K` largest entries, but a simpler
mechanism called *heap selection* does the job nicely.
Selecting the `K` largest entries from a set of `L` by sorting
would require `O(L*log(L))` operations; heap selection
requires `O(L*log(K))` operations.
Alternatively, the `K` largest entries can be also be
determined in `O(N) + O(K log(K))` asymptotic time by using partition
selection (more here
and there). It is
possible to enable this mode of operaton in mcl with the option
**--partition-selection**. However, benchmarking so far has shown this
to be equivalent in speed to heap selection. This is explained by
the bounded nature of `K` and `L` in practice.

Indeed it does not. What are the ramifications? Several entries in section Resource tuning / accuracy discuss this issue. For a synopsis, consider two ends of a spectrum.

On the one end, a graph that has very strong cluster structure, with clearly (and not necessarity fully) separated clusters. This mcl implementation will certainly retrieve those clusters if the graphs falls into the category of graphs for which mcl is applicable. On the other end, consider a graph that has only weak cluster structure superimposed on a background of a more or less random graph. There might sooner be a difference between the clustering that should ideally result and the one computed by mcl. Such a graph will have a large number of whimsical nodes that might end up either here or there, nodes that are of a peripheral nature, and for which the (cluster) destination is very sensitive to fluctutations in edge weights or algorithm parametrizations (any algorithm, not just mcl).

In short, the perturbation effect of the pruning process applied by mcl is a
source of noise. It is small compared to the effect of
changing the inflation parametrization or perturbing the edge weights. If
the change is larger, this is because the computed process tends to converge
prematurely, leading to finer-grained clusterings. As a result the
clustering will be close to a *subclustering* of the clustering resulting
from more conservative resource settings, and in that respect be consistent.
All this can be measured using the program
clm dist. It is possible to
offset such a change by slightly lowering the inflation parameter.

There is the issue of very large and very dense graphs. The act of pruning will have a larger impact as graphs grow larger and denser. Obviously, mcl will have trouble dealing with such very large and very dense graphs — so will other methods.

Finally, there is the engineering approach, which offers the possibility of
pruning a whole lot of speculation. Do the experiments with **mcl**, try it
out, and see what's there to like and dislike.

Comparison with other algorithms

XYZ might well be the bees knees of all things clustering. Bear in mind
though that comparing cluster algorithms is a very tricky affair.
One particular trap is the following. Sometimes a new cluster algorithm is proposed based
on some optimization criterion. The algorithm is then compared with
previous algorithms (e.g. MCL). But how to compare? Quite often the
comparison will be done by computing a criterion and astoundingly,
quite often the chosen criterion is simply the optimization criterion again.
*Of course* XYZ will do very well. It would be a very poor algorithm
it if did not score well on its own optimization criterion, and it
would be a very poor algorithm if it did not perform better than other
algorithms which are built on different principles.

There are some further issues that have to be considered here. First, there is not a single optimization criterion that fully captures the notion of cluster structure, let alone best cluster structure. Second, leaving optimization approaches aside, it is not possible to speak of a best clustering. Best always depends on context - application field, data characteristics, scale (granularity), and practitioner to name but a few aspects. Accordingly, the best a clustering algorithm can hope for is to be a good fit for a certain class of problems. The class should not be too narrow, but no algorithm can cater for the broad spectre of problems for which clustering solutions are sought. The class of problems to which MCL is applicable is discussed in section What kind of graphs.

Presumably, they did not know mcl, and did not read the parts
in [1] and [2] that discuss implementation. Perhaps
they assume or insist that the only way to implement MCL is to implement the
ideal process. And there is always the genuine possibility
of a *really* stupifyingly fast algorithm. It is certainly not the
case that MCL has a time complexity of `O(N^3)` as is sometimes erroneously
stated.

Resource tuning / accuracy

**mcl** computes a process in which stochastic matrices are alternately
expanded and inflated. Expansion is nothing but standard matrix
multiplication, inflation is a particular way of rescaling the matrix
entries.

Expansion causes problems in terms of both time and space. mcl works with
matrices of dimension `N`, where `N` is the number of nodes in the input graph.
If no precautions are taken, the number of entries in the mcl iterands
(which are stochastic matrices) will soon approach the square of `N`. The
time it takes to compute such a matrix will be proportional to the cube of
`N`. If your input graph has 100.000 nodes, the memory requirements become
infeasible and the time requirements become impossible.

What mcl does is perturbing the process it computes a little
by removing the smallest entries — it keeps its matrices *sparse*.
This is a natural thing to do, because the matrices are sparse in
a weighted sense (a very high proportion of the stochastic mass
is contained in relatively few entries), and the process converges
to matrices that are extremely sparse, with usually no more than `N` entries.
It is thus known that the MCL iterands are sparse in a weighted
sense and are usually very close to truly sparse matrices.
The way mcl perturbs its matrices is by the strategy
of pruning, selection, and recovery that is extensively described
in the mcl manual page.
The question then is: What is the effect of this perturbation
on the resulting clustering, i.e. how would the clustering
resulting from a *perfectly computed* mcl process compare with
the clustering I have on disk?
Faq entry 6.3 discusses this issue.

The amount of *resources* used by mcl is bounded in terms of the maximum
number of neighbours a node is allowed to have during all computations.
Equivalently, this is the maximum number of nonzero entries a matrix column
can possibly have. This number, finally, is the maximum of the
the values corresponding with the **-S** and **-R** options.
The latter two are listed when using the **-z** option
(see faq 10.1).

It is rougly equal to

2 * s * K * N

bytes, where 2 is the number of matrices held in memory by **mcl**, s is the
size of a single cell (c.q. matrix entry or node/arc specification), `N` is
the number of nodes in the input graph, and where `K` is the maximum of the
values corresponding with the **-S** and **-R** options (and this
assumes that the average node degree in the input graph does not exceed `K`
either). The value of s can be found by using the **-z** option. It
is listed in one of the first lines of the resulting output. s equals the
size of an int plus the size of a float, which will be 8 on most systems.
The estimate above will in most cases be way too pessimistic (meaning
you do not need that amount of memory).

The **-how-much-ram** option is provided by mcl for computing
the bound given above. This options takes as argument the number of
nodes in the input graph.

The theoretically more precise upper bound is slightly larger due to overhead. It is something like

( 2 * s * (K + c)) * N

where c is 5 or so, but one should not pay attention to such a small
difference anyway.
For graphs with up until a few thousand nodes a *perfectly computed*
MCL process can be achieved by abstaining from pruning and doing
full-blown matrix arithmetic. Of course, this still leaves the
issue of machine precision, but let us wholeheartedly ignore that.

Such experiments give evidence (albeit incidental) that pruning is indeed really what it is thought to be - a small perturbation. In many cases, the 'approximated' clustering is identical to the 'exact' clustering. In other cases, they are very close to each other in terms of the metric split/join distance as computed by clm dist. Some experiments with randomly generated test graphs, clustering, and pruning are described in [4].

On a different level of abstraction, note that perturbations of the inflation parameter will also lead to perturbations in the resulting clusterings, and surely, large changes in the inflation parameter will in general lead to large shifts in the clusterings. Node/cluster pairs that are different for the approximated and the exact clustering will very likely correspond with nodes that are in a boundary region between two or more clusters anyway, as the perturbation is not likely to move a node from one core of attraction to another.

Faq entry 6.6 has more to say about this subject.

In **mcl** parlance, this becomes *how do I know that my* **-scheme**
*parameter is high enough* or more elaborately *how do I know
that the values of the {-P, -S, -R, -pct} combo are high enough?*

There are several aspects. First, watch the *jury marks* reported by **mcl**
when it's done.
The jury marks are three grades, each out of 100. They indicate how well
pruning went. If the marks are in the seventies, eighties, or nineties, mcl
is probably doing fine. If they are in the eighties or lower, try to see if
you can get the marks higher by spending more resources (e.g. increase the
parameter to the **-scheme** option).

Second, you can do multiple **mcl** runs for different resource schemes,
and compare the resulting clusterings using **clm dist**. See
the clmdist manual for a case study.

There is none. [add]

Ok, the next entry gives an engineer's rule of thumb.

The more severe pruning is, the more the computed process will tend to
converge prematurely. This will generally lead to finer-grained clusterings.
In cases where pruning was severe, the **mcl** clustering will likely be closer
to a clustering ideally resulting from another MCL process with higher
inflation value, than to the clustering ideally resulting from the same MCL
process. Strong support for this is found in a general observation
illustrated by the following example. Suppose u is a stochastic vector
resulting from expansion:

u = 0.300 0.200 0.200 0.100 0.050 0.050 0.050 0.050

Applying inflation with inflation value 2.0 to u gives

v = 0.474 0.211 0.211 0.053 0.013 0.013 0.013 0.013

Now suppose we first apply pruning to u such that the 3 largest entries 0.300, 0.200 and 0.200 survive, throwing away 30 percent of the stochastic mass (which is quite a lot by all means). We rescale those three entries and obtain

u' = 0.429 0.286 0.286 0.000 0.000 0.000 0.000 0.000

Applying inflation with inflation value 2.0 to u' gives

v' = 0.529 0.235 0.235 0.000 0.000 0.000 0.000 0.000

If we had applied inflation with inflation value 2.5 to u, we would have obtained

v'' = 0.531 0.201 0.201 0.038 0.007 0.007 0.007 0.007

The vectors v' and v'' are much closer to each other than the vectors v' and v, illustrating the general idea.

In practice, **mcl** should (on average) do much better than in this
example.

Did you read all other entries in this section? That should have reassured you somewhat, except perhaps for Faq answer 6.5.

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.

By the way, you do know all about clm dist, don't you? Because the
statement that clusterings are not identical should be quantified: *How
much do they differ?* This issue is discussed in the clm dist manual
page — **clm dist** gives you a robust measure for the distance (dissimilarity)
between two clusterings.

There are other means of gaining trust in a clustering, and there are
different issues at play. There is the matter of how accurately this **mcl**
computed the mcl process, and there is the matter of how well the chosen
inflation parameter fits the data. The first can be judged by looking at
the jury marks (faq 6.4)
and applying **clm dist** to different clusterings. The
second can be judged by measurement (e.g. use clm info) and/or
inspection (use your judgment).

Tuning cluster granularity

There are several ways for influencing cluster granularity. These ways and their relative merits are successively discussed below. Reading clmprotocols is also a good idea.

The main handle for changing inflation is the **-I** option. This is
also *the* principal handle for regulating cluster granularity. Unless
you are mangling huge graphs it could be the only **mcl** option you ever need
besides the output redirection option **-o**.

Increasing the value of **-I** will increase cluster granularity.
Conceivable values are from 1.1 to 10.0 or so, but the range of suitable
values will certainly depend on your input graph. For many graphs, 1.1 will
be far too low, and for many other graphs, 8.0 will be far too high. You
will have to find the right value or range of values by experimenting, using
your judgment, and using measurement tools such as clm dist and
clm info. A good set of values to start with is 1.4, 2 and 6.

Preferably the network should not have nodes of very high degree,
that is, with exorbitantly many neighbours. Such nodes tend to
obscure cluster structure and contribute to coarse clusters.
The ways to combat this using **mcl** and sibling programs are documented
in clmprotocols. Briefly, they are the
transformations `#knn()` and `#ceilnb()` available
to **mcl**, **mcx alter** and several more programs.

How similarities in the input graph were derived, constructed, adapted, filtered (et cetera) will affect cluster granularity. It is important that the similarities are honest; refer to faq 3.8.

Another issue is that homogeneous similarities tend to result in more coarse-grained clusterings. You can make a set of similarities more homogeneous by applying some function to all of them, e.g. for all pairs of nodes (x y) replace S(x,y) by the square root, the logarithm, or some other convex function. Note that you need not worry about scaling, i.e. the possibly large changes in magnitude of the similarities. MCL is not affected by absolute magnitudes, it is only affected by magnitudes taken relative to each other.

As of version 03-154, mcl supports the pre-inflation **-pi** *f* option.
Make a graph more homogeneous with respect to the weight
function by using **-pi** with argument *f* somewhere
in the interval [0,1] — 0.5 can be considered a reasonable first try.
Make it less homogeneous by setting *f* somewhere in the interval [1,10].
In this case 3 is a reasonable starting point.

Implementing the MCL algorithm

Very easy, if you will be doing small graphs only, say up to a few thousand entries at most. These are the basic ingredients:

o

Adding loops to the input graph, conversion to a stochastic matrix.

o

Matrix multiplication and matrix inflation.

o

The interpretation function mapping MCL limits onto clusterings.

These must be wrapped in a program that does graph input and cluster output, alternates multiplication (i.e. expansion) and inflation in a loop, monitors the matrix iterands thus found, quits the loop when convergence is detected, and interprets the last iterand.

Implementing matrix muliplication is a standard exercise. Implementing inflation is nearly trivial. The hardest part may actually be the interpretation function, because you need to cover the corner cases of overlap and attractor systems of cardinality greater than one. Note that MCL does not use intricate and expensive operations such as matrix inversion or matrix reductions.

In Mathematica or Maple, mcl should be doable in at most 100 lines of code. For perl you may need twice that amount. In lower level languages such as C or Fortran a basic MCL program may need a few hundred lines, but the largest part will probably be input/output and interpretation.

To illustrate all these points, mcl now ships with minimcl, a small perl script that implements mcl for educational purposes. Its structure is very simple and should be easy to follow.

Implementing the basic MCL algorithm makes a nice programming exercise. However, if you need an implementation that scales to several hundreds of thousands of nodes and possibly beyond, then your duties become much heavier. This is because one needs to prune MCL iterands (c.q. matrices) such that they remain sparse. This must be done carefully and preferably in such a way that a trade-off between speed, memory usage, and potential losses or gains in accuracy can be controlled via monitoring and logging of relevant characteristics. Some other points are i) support for threading via pthreads, openMP, or some other parallel programming API. ii) a robust and generic interpretation function is written in terms of weakly connected components.

Cluster overlap / MCL iterand cluster interpretation

In the MCL limit, the associated DAG has in general a very degenerated form, which induces overlap only on very rare occasions (see faq entry 9.2).

Interpreting **mcl** iterands as clusterings may well be interesting.
Few experiments have been done so far. It is clear though that
early iterands generally contain the most overlap (when interpreted
as clusterings). Overlap dissappears soon as the iterand
index increases. For more information, consult the other entries
in this section and the clmimac manual page.

No. Clusterings resulting from the abstract MCL algorithm may in theory
contain overlap, but the default behaviour in **mcl** is to remove it should it
occur, by allocating the nodes in overlap to the first cluster in which they
are seen. **mcl** will warn you if this occurs. This behaviour is switched
off by supplying **--keep-overlap=yes**.

Do note that overlap is mostly a theoretical possibility.
It is conjectured that it requires the presence of very strong
symmetries in the input graph, to the extent that there *exists
an automorphism of the input graph mapping the overlapping part
onto itself*.

It is possible to construct (highly symmetric) input graphs leading to cluster overlap. Examples of overlap in which a few nodes are involved are easy to construct; examples with many nodes are exceptionally hard to construct.

Clusterings associated with intermediate/early MCL iterands may very well contain overlap, see the introduction in this section and other entries.

There are two options. If you are interested in clusterings containing overlap, you should go for the second. If not, use the first, but beware that the resulting clusterings may contain overlap.

The first solution is to use **-dump** **cls** (probably in conjunction
with either **-L** or **-dumpi** in order to limit the number of
matrices written). This will cause **mcl** to write the clustering generically
associated with each iterand to file. The **-dumpstem** option may be
convenient as well.

The second solution is to use the **-dump** **ite** option
(**-dumpi** and **-dumpstem** may be of use again). This will
cause **mcl** to write the intermediate iterands to file. After that, you can
apply clm imac (interpret matrix as clustering) to those iterands. **clm imac**
has a **-strict** parameter which affects the mapping of matrices to
clusterings. It takes a value between 0.0 and 1.0 as argument. The default is
0.001 and corresponds with promoting overlap. Increasing the **-strict**
value will generally result in clusterings containing less overlap. This
will have the largest effect for early iterands; its effect will diminish as
the iterand index increases.

When set to 0, the **-strict** parameter results in the clustering
associated with the DAG associated with an MCL iterand as described
in [3]. This DAG is pruned (thus possibly resulting
in less overlap in the clustering) by increasing the **-strict**
parameter. [add]

Miscellaneous

Use **-z** to find out the actual settings - it shows
the settings as resulting from the command line options (e.g. the default
settings if no other options are given).

I'd like to port MCL to cluster computing, using one of the PVM, MPI, or openMP frameworks. For the 1.002 release, mcl's internals were rewritten to allow more general matrix computations. Among other things, mcl's data structures and primitive operations are now more suited to be employed in a distributed computing environment. However, much remains to be done before mcl can operate in such an environment.

If you feel that mcl should support some other standard matrix format, let us know.

This FAQ tries to compromise between being concise and comprehensive. The collection of answers should preferably cover the universe of questions at a pleasant level of semantic granularity without too much overlap. It should offer value to people interested in clustering but without sound mathematical training. Therefore, if this FAQ has not failed somewhere, it must have failed.

Send criticism and missing questions for consideration to mcl-faq at micans.org.

Stijn van Dongen.

mclfamily for an overview of all the documentation and the utilities in the mcl family.

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. *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

[3]
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

[4]
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

[5]
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).

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