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  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  sec-
          tion  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.

  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 mathemat-
          ics 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?



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   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 result-
          ing 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 similarity distribution homogeneity on cluster  granu-
          larity.

  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 algo-
          rithm,  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



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          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 hap-
          pily.

   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 basi-
          cally 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 start-
          ing the process. It sets inflation parameters and stops the MCL pro-
          cess  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  algo-
          rithm,  with  some  extra  facilities  for manipulation of the input
          graph, interpreting the result, manipulating resources while comput-
          ing 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?



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          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 charac-
          ters 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 [2]. 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  (includ-
          ing  Linux  and  the BSD variants of course). Following the instruc-
          tions 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  straight-
          forward 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



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          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 4 Nov 2009, which corresponds with the MCL stamp  09-308.
          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).

          In early 2002 it occurred to me that mcl should, in addition to time
          stamps,  also  have  something  like version numbers, wanting to use
          those to indicate noteworthy changes. The  April  2002  release  got
          version tag 1.001, in order to celebrate the then-recent addition of
          this FAQ, mcl's new logging facility --log, and clm imac to the  MCL
          distribution. The January 2003 release had its version number bumped
          to 1.002, marking MCL's ability to directly deal with  a  much  more
          general  type  of graph encoding.  1.003 brought mclpipeline, mcxde-
          blast, mclblastline, mcxassemble, clm format  and  automatic  output
          naming.  1.004  revived  binary  format, improved mcxsubs, and had a
          host of other additions. 1.005 removed an IO  bottleneck,  consider-
          ably  speeding  up  matrix  reads.  1.006  featured mcl's ability to
          directly read and produce label output. Currently, the  version  tag
          is  not  used  in the mcl distribution name - only the date stamp is
          used for that.

                                      Input format

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

          One of the easiest ways is if you have a list of  label  pairs  with
          similarities  attached  to  the  pairs.  Put this information into a
          file with each pair of labels and the  associated  similarity  on  a
          single line separated by whitespace. Example:

          apples    oranges    2.14
          oranges   pears      0.78
          oranges   apples     1.57
          pears     apples     1.01

          Suppose  the  file is called mydata.  You can simply cluster it like
          this:

             mcl mydata --abc -I 2.0 -o mydata.cls-I20
             mcl mydata --abc -I 2.4 -o mydata.cls-I24
             mcl mydata --abc -I 2.8 -o mydata.cls-I28

          In this example a few  runs  were  shown  with  different  inflation
          parameters.   This  is nonsensical with this particular 4-node input
          but a good idea in general. mcl provides many options to  cache  the
          intermediate  matrix  input  and the intermediate tab file. Refer to
          the mcl manual  page,  specifically  look  for  the  -write-tab  and
          -write-graph options.

          Alternatively,  use  mcxload(1) to save label input as an mcl matrix



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          and one or more tab files, then apply mcl to the matrix  file  using
          the -use-tab option.

          NOTE
          Simply doing

             mcl mydata --abc -I 2.0
             mcl mydata --abc -I 2.4
             mcl mydata --abc -I 2.8

          will  result in the output files out.mydata.I20, out.mydata.I24, and
          out.mydata.I28.  The automatic naming facility is very convenient  -
          try  it. If you need to script mcl use the -az option to capture the
          name that mcl would generate given the rest of the command-line.

                                  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.

   3.6    Does MCL work for tree graphs?



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          Nah, I don't think so. More info at entry 3.7.

   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) natu-
          ral clusters is not too large. Additionally, graphs  should  prefer-
          ably 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 dimen-
          sional 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.

   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 per-
          fectly 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.



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          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
          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 neces-
          sarily 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:






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          (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 pos-
          sible 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 sec-
          tion 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 max-
          imum 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 squar-
          ing.   There  are  two  sub-algorithms to consider. The first is the
          algorithm responsible for assembling a new vector during matrix mul-
          tiplication.  This  algorithm  has worst case complexity O(K^2). The
          pruning algorithm (which uses heap selection) has  worst  case  com-
          plexity  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 thou-
          sands, 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,



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          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
          iterations are genuinely time consuming; the  first  few  iterations
          (some number below 10) are usually responsible for more than 95 per-
          cent of the running time.

          The process of removing the smallest entries of a vector  is  called
          pruning.  mcl  provides extensive facilities for monitoring and con-
          trolling the effect of pruning, and it will output statistics and  a
          summary 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 [4], and [5]  mentions  large
          graphs  being clustered in very reasonable time. In protein cluster-
          ing, 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.

   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)) opera-
          tions; heap selection requires O(L*log(K)) operations.

   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 synop-
          sis, 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 ide-
          ally  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).

          One can say that the perturbation  effect  of  the  pruning  process
          applied by mcl is just a small source of noise. Additionally, graphs



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          at the noisy end of the spectrum will generally be very  susceptible
          to  changes in parametrization of the MCL algorithm and process, and
          the perturbation caused by computing an imperfect process will  gen-
          erally  be  small  compared with the effect of changing parametriza-
          tions.

          Of course, 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 possi-
          bility 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 clus-
          ter 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  algo-
          rithm  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 cap-
          tures the notion of cluster structure, let alone best cluster struc-
          ture.  Second, leaving optimization approaches aside, it is not pos-
          sible 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  algo-
          rithm can cater for the broad spectre of problems for which cluster-
          ing 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]

          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 pro-
          cess. And there is always the genuine possibility of a really stupi-
          fyingly  fast algorithm. It is certainly not the case that MCL has a
          time complexity of O(N^3) as is sometimes erroneously stated.




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                               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 mul-
          tiplication, 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  per-
          turbation 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 maxi-
          mum number of neighbours a node is allowed to have during all compu-
          tations.   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 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  specifica-
          tion),  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



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          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 result-
          ing 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,  clus-
          tering, 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/clus-
          ter  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 pertur-
          bation 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 parame-
          ter 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



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          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 ide-
          ally 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 vec-
          tors 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.



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          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 like-
          lihood,  there are anyway nodes which are not in any core of attrac-
          tion, 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. The
          clmdist(1) manual contains an example of doing multiple mcl runs for
          finding granularily different clusterings,  using  the  most  common
          approach, namely that of varying inflation.

   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.   Con-
          ceivable values are from 1.1 to 5.0 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, 5.0 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. The default 2.0 is a  good  value  to
          begin the experimental stage with.



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          For  experiments that are more subtle with respect to inflation, mcl
          provides the -i option in conjunction with the -l (small letter ell)
          option.  Do  this  only  if you have the intention of playing around
          with mcl in order to study the characteristics of the  process  that
          it  computes, and maybe, just maybe, use it in a production environ-
          ment if you find it useful. In the first vein, you may be interested
          to  know  that  mcxi(1) is a stack language/interpreter in which the
          entire MCL algorithm can be written in three lines of code. It  pro-
          vides  comprehensive  access  to the MCL graph and matrix libraries.
          However, the mcxi interface to the MCL pruning facilities is not yet
          satisfactory at this time.

   7.3    The  effect of similarity distribution homogeneity on cluster granu-
          larity.

          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 log-
          arithm, 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 5 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. Implement-
          ing inflation is nearly trivial. The hardest part  may  actually  be



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          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  exer-
          cise.  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 characteris-
          tics.  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  con-
          nected 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 posi-
          tive  semi-definite  -  see [3].  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  degener-
          ated  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



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          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-over-
          lap=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 excep-
          tionally 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 contain-
          ing  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 writ-
          ten). This will cause mcl to write the clustering generically  asso-
          ciated with each iterand to file. The -dumpstem option may be conve-
          nient 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
          -tight parameter which affects the mapping of matrices  to  cluster-
          ings. It takes a value between 0 and 100 as argument. The default is
          100 and corresponds with the strict  mapping.  Lowering  the  -tight
          value  will generally result in clusterings containing more overlap.
          This will have the largest effect for  early  iterands;  its  effect
          will diminish as the iterand index increases.

          When  set to 0, the -tight parameter results in the clustering asso-
          ciated 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 -tight 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).



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   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.

          At some point in the future a second, xml-based,  interchange  input
          format may be introduced.

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

  BUGS
          This FAQ tries to compromise between being  concise  and  comprehen-
          sive. 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 cluster-
          ing 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 utili-
          ties in the mcl family.

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

  REFERENCES
          [1] Stijn van Dongen. Graph Clustering by Flow Simulation.  PhD the-
          sis, 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 Mathe-
          matics 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



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          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  algo-
          rithm  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  conver-
          sion  problems,  while  keeping  some level of sophistication in the
          typesetting.












































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