MCL FAQ(7) |
MISCELLANEOUS |
MCL FAQ(7) |

# NAME¶

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 precede the actual faq entries. The TOC section contains a listing of all questions.

# RESOURCES¶

The manual pages for all the utilities that come with **mcl**;
refer to **mclfamily(7)** for an overview.

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

# TOC¶

**1** **General questions**

**1.1** For whom is mcl and for whom is this FAQ?

**1.2** What is the relationship between the MCL process, the MCL
algorithm, and the 'mcl' implementation?

**1.3** What do the letters MCL stand for?

**1.4** How could you be so feebleminded to use MCL as abbreviation? Why is
it labeled 'Markov cluster' anyway?

**1.5** Where can I learn about the innards of the MCL algorithm/process?

**1.6** For which platforms is mcl available?

**1.7** How does mcl's versioning scheme work?

**2** **Input format**

**2.1** How can I get my data into the MCL matrix format?

**3** **What kind of graphs**

**3.1** What is legal input for MCL?

**3.2** What is sensible input for MCL?

**3.3** Does MCL work for weighted graphs?

**3.4** Does MCL work for directed graphs?

**3.5** Can MCL work for lattices / directed acyclic graphs / DAGs?

**3.6** Does MCL work for tree graphs?

**3.7** For what kind of graphs does MCL work well and for which does it
not?

**3.8** What makes a good input graph? How do I construct the similarities?
How to make them satisfy this Markov condition?

**3.9** My input graph is directed. Is that bad?

**3.10** Why does mcl like undirected graphs and why does it dislike
uni-directed graphs so much?

**3.11** How do I check that my graph/matrix is symmetric/undirected?

**4** **Speed and complexity**

**4.1** How fast is mcl/MCL?

**4.2** What statistics are available?

**4.3** Does this implementation need to sort vectors?

**4.4** mcl does not compute the ideal MCL process!

**5** **Comparison with other algorithms**

**5.1** I've read someplace that XYZ is much better than MCL

**5.2** I've read someplace that MCL is slow [compared with XYZ]

**6** **Resource tuning / accuracy**

**6.1** What do you mean by resource tuning?

**6.2** How do I compute the maximum amount of RAM needed by mcl?

**6.3** How much does the mcl clustering differ from the clustering
resulting from a perfectly computed MCL process?

**6.4** How do I know that I am using enough resources?

**6.5** Where is the mathematical analysis of this mcl pruning strategy?

**6.6** What qualitative statements can be made about the effect of
pruning?

**6.7** At different high resource levels my clusterings are not identical.
How can I trust the output clustering?

**7** **Tuning cluster granularity**

**7.1** How do I tune cluster granularity?

**7.2** The effect of inflation on cluster granularity.

**7.3** The effect of node degrees on cluster granularity.

**7.4** The effect of edge weight differentiation on cluster
granularity.

**8** **Implementing the MCL algorithm**

**8.1** How easy is it to implement the MCL algorithm?

**9** **Cluster overlap / MCL iterand cluster
interpretation**

**9.1** Introduction

**9.2** Can the clusterings returned by mcl contain overlap?

**9.3** How do I obtain the clusterings associated with MCL iterands?

**10** **Miscellaneous**

**10.1** How do I find the default settings of mcl?

**10.2** What's next?

# FAQ¶

**General questions**

**1.1** **For whom is mcl and for whom is this FAQ?**

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.

**1.2** **What is the relationship between the MCL process,
the MCL algorithm, and the 'mcl' implementation?**

**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.

**1.3** **What do the letters MCL stand for?**

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.

**1.4** **How could you be so feebleminded to use MCL as
abbreviation? Why** **is it labeled 'Markov cluster' anyway?**

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.

**1.5** **Where can I learn about the innards of the MCL
algorithm/process?**

Currently, the most basic explanation of the MCL algorithm is found in the technical report [3]. It contains sections on several other (related) subjects though, and it assumes some working knowledge on graphs, matrix arithmetic, and stochastic matrices.

**1.6** **For which platforms is mcl available?**

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.

**1.7** **How does mcl's versioning scheme work?**

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 **9 Oct 2022**,
which corresponds with the MCL stamp **22-282**. 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**

**2.1** **How can I get my data into the MCL matrix format?**

This is described in the *protocols manual page*.

**What kind of graphs**

**3.1** **What is legal input for MCL?**

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

**3.2** **What is sensible input for MCL?**

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.

**3.3** **Does MCL work for weighted graphs?**

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

**3.4** **Does MCL work for directed graphs?**

Maybe, with a big caveat. See entries 3.8 and 3.9.

**3.5** **Can MCL work for lattices / directed acyclic graphs
/ DAGs?**

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()'

**3.6** **Does MCL work for tree graphs?**

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.

**3.7** **For what kind of graphs does MCL work well and for
which does it not?**

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 [5])
to begin with, so you should probably forget about mixing properties.

**3.8** **What makes a good input graph?** **How do I
construct the similarities?** **How to make them satisfy this Markov
condition?**

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.

**3.9** **My input graph is directed. Is that bad?**

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.

**3.10** **Why does mcl like undirected graphs and why does
it** **dislike uni-directed graphs so much?**

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].

**3.11** **How do I check that my graph/matrix is
symmetric/undirected?**

Whether your graph is created by third-party software or by custom
software 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**

**4.1** **How fast is mcl/MCL?**

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)*.*

**4.2** **What statistics are available?**

Few. Some experiments are described in [5], and [6] mentions large
graphs being clustered in very reasonable time. In protein clustering,
**mcl** has been applied to graphs with over seven million nodes, and on
high-end hardware such graphs can be clustered within a few hours, using
mcl's -t** threads option.**

**4.3** **Does this implementation need to sort
vectors?**

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.**

**4.4** **mcl does not compute the ideal MCL process!**

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**

**5.1** **I've read someplace that XYZ is much better than MCL**

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*.

**5.2** **I've read someplace that MCL is slow [compared with
XYZ]**

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**

**6.1** **What do you mean by resource tuning?**

**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).

**6.2** **How do I compute the maximum amount of RAM needed by
mcl?**

It is roughly 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.

**6.3** **How much does the mcl clustering differ from the
clustering resulting** **from a perfectly computed MCL process?**

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 [5].

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.

**6.4** **How do I know that I am using enough
resources?**

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.

**6.5** **Where is the mathematical analysis of this mcl
pruning strategy?**

There is none. [add]

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

**6.6** **What qualitative statements can be made about the
effect of pruning?**

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.

**6.7** **At different high resource levels my clusterings are
not identical.** **How can I trust the output clustering?**

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**

**7.1** **How do I tune cluster granularity?**

There are several ways for influencing cluster granularity. These
ways and their relative merits are successively discussed below. Reading
**clmprotocols(5)** is also a good idea.

**7.2** **The effect of inflation on cluster
granularity.**

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.

**7.3** **The effect of node degrees on cluster
granularity.**

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(5)**. Briefly, they are the transformations #knn()** and
#ceilnb()** available to **mcl**, **mcx alter** and several
more programs.

**7.4** **The effect of edge weight differentiation on cluster
granularity.**

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**

**8.1** **How easy is it to implement 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**

**9.1** **Introduction**

A natural mapping exists of MCL iterands to DAGs (directed acyclic
graphs). This is because MCL iterands are generally *diagonally positive
semi-definite* - see [1]. Such a DAG can be interpreted as a clustering,
simply by taking as cores all endnodes (sinks) of the DAG, and by attaching
to each core all the nodes that reach it. This procedure may result in
clusterings containing overlap.

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**.

**9.2** **Can the clusterings returned by mcl contain
overlap?**

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.

**9.3** **How do I obtain the clusterings associated with MCL
iterands?**

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 [1]. This DAG is pruned (thus possibly resulting in less
overlap in the clustering) by increasing the **-strict** parameter.
[add]

**Miscellaneous**

**10.1** **How do I find the default settings of mcl?**

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).

**10.2** **What's next?**

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.

# BUGS¶

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.

# AUTHOR¶

Stijn van Dongen.

# SEE ALSO¶

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

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

# REFERENCES¶

[1] 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

[2] 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

[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 and Cei Abreu-Goodger, *Using mcl to
extract clusters from networks*, in *Bacterial Molecular Networks:
Methods and Protocols*, Methods in Molecular Biology, Vol 804, pages
281-295 (2012). PMID 22144159.

http://www.ncbi.nlm.nih.gov/pubmed/22144159

[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).

# NOTES¶

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.

9 Oct 2022 | MCL FAQ 22-282 |