The general unified format alluded to above is in fact a simple rectangular sparse matrix representation. Sparse means that zero entries in the matrix are not stored. A zero entry corresponds to an ordered node pair in the graph for which no arc exists from the first to the second node. An undirected graph corresponds with a symmetric matrix.
The MCL suite uses a slight generalisation of the matrix concept, in that the row and column indices (that is, domains) can be arbitrary lists of nonnegative integers. Usually, but not necessarily, a domain of size N will use the common representation of the list of integers starting at 0 and ending at N-1. The source domain is always associated with the columns of a matrix, and the destination domain is always associated with the rows of a matrix. The matrix format, introduced below, first specifies the two domains. It then represents the nonzero matrix entries (corresponding with graph arc weights) in a column-wise fashion, as a list of lists. For each node from the source domain, it presents the list of all its neighbours in the destination domain along with the corresponding weights. This document describes
2m 2m native matrix format
The format that can be read in by any mcl application expecting a matrix argument. The native format closely resembles the layout of matrices as residing in computer memory. There are two distinct encodings, respectively interchange and binary. Their relative merits are described further below.
2m 2m concatenated native matrix format
This always pertains to matrices in native format concatenated in a single file, refered to as a cat file. It is used for example to encode hierarchical clusterings as generated by mclcm. A cat file either consists of matrices in interchange format or of matrices in binary format.
2m 2m raw intermediate format
This is read by mcxassemble(1).
2m 2m tab format
Used by applications such as mcl(1) and mcxdump(1) to convert between meaningful labels describing the input data and the numerical identifiers used internally.
2m 2m label format
The format used when streaming labels directly into mcl(1) or mcxload(1).
2m 2m transformation syntax
The syntax accepted by mcl(1), mcxalter(1) and many other programs to transform graphs and matrices.
The interchange format is a portable format that can be transmitted across computers and over networks and will work with any version of mcl or its sibling programs. It is documented (here) and very stable. Applications can easily create matrices in this format. The drawback of interchange format is that for very large graphs matrix encodings grow very big and are slow to read.
The binary format is not garantueed to be portable across machines or different versions of mcl or differently compiled versions of mcl. Its distinct advantage is that for very large graphs the speed advantage over interchange format is dramatic.
Conversion between the two formats is easily achieved with mcxconvert. Both mcl(1) and mcxload(1) can save a matrix in either format after constructing it from label input.
The concatenated format is generated e.g. by mclcm(1) and can be transformed by mcxdump(1) using the -imx-cat option. In cat format matrices are simply concatenated, so it is easily generated from the command line if needed. For native binary format it is imperative that no additional bytes are inserted inbetween the matrix encodings. For native interchange format the only requirement is that the last matrix is followed by nothing but white space.
A remark on the sloppy naming conventions used for mcl and its sibling utilities may be in order here. The prefix mcx is used for generic matrix functionality, the prefix clm is used for generic cluster functionaliy. The utility mcx is a general purpose interpreter for manipulating matrices (and grahps, sets, and clusterings). The set of all mcl siblings (cf. mclfamily(7)) is loosely refered to as the mcl family, which makes use of the mcl libraries (rather than the mcx libraries). The full truth is even more horrible, as the mcl/mcx prefix conventions used in the C source code follow still other rules.
In this document, 'MCL' means 'the mcl setting' or 'the mcl family'. An MCL program is one of the programs in the mcl family. The remainder of this document contains the following sections.
There are several aspects to the way in which MCL represents matrices. Internally, indices never act as an ofset in an array, and neither do they participate in ofset computations. This means that they purely act as identifiers. The upshot is that matrices can be handled in which the index domains are non-sequential (more below). Thus one can work with different graphs and matrices all using subsets of the same set of indices/identifiers. This aids in combining data sets in different ways and easily comparing the respective results when experimenting. Secondly, only nonzero values (and their corresponding indices) are stored. Thirdly, MCL stores a matrix as a listing of columns. Iterating over a column is trivial; iterating over a row requires a costly transposition computation. The last two points should matter little to the user of MCL programs.
In textbook expositions and in many matrix manipulation implementations, matrices are represented with sequentially indexed rows and columns, with the indices usually starting at either zero or one. In the MCL setting, the requirement of sequentiality is dropped, and it follows naturally that no requirement is posed on the first index. The only requirement MCL poses on the indices is that they be nonnegative, and can be represented by the integer type used by MCL. On many machines, the largest allowable integer will be 2147483647.
MCL associates two domains with a matrix M, the row domain and column domain. The matrix M can only have entries M[i,j] where i is in the row domain and j is in the column domain. This is vital when specifying a matrix: it is illegal to specify an entry M[i,j] violating this condition. However, it is not necessary to specify all entries M[i,j] for all possible combinations of i and j. One needs only specify those entries for which the value is nonzero, and only nonzero values will be stored internally. In the MCL matrix format, the matrix domains must be specified explicitly if they are not canonical (more below).
Strictly as an aside, the domains primarily exist to ensure data integrity. When combining matrices with addition or multiplication (e.g. using the mcx utility), MCL will happily combine matrices for which the domains do not match, although it will usually issue a warning. Conceptually, matrices auto-expand to the dimensions required for the operation. Alternatively, a matrix can be viewed as an infinite quadrant, with the domains delimiting the parts in which nonzero entries may exist. In the future, facilities could be added to MCL (c.q. mcx) to allow for placing strict domain requirements on matrices when submitted to binary operations such as addition and multiplication.
From here on, all statements about matrices and graphs are really statements about matrices and graphs in the MCL setting. The specification of a matrix quite closely matches the internal representation.
A matrix M has two domains: the column domain and the row domain. Both simply take the form of a set (represented as an ordered list) of indices. A canonical domain is a domain of some size K where the indices are simply the first K nonnegative integers 0,1..,K-1. The domains dictate which nonzero entries are allowed to occur in a matrix; only entries M[i,j] are allowed where i is in the row domain and j is in the column domain.
The matrix M is specified in three parts, for which the second is optional. The parts are:
2m 2m Header specification
This specifies the dimensions K and L of the matrix, where K is the size of the row domain, and L is the size of the column domain. It looks as follows:
(mclheader mcltype matrix dimensions 9x14 )
This dictates that a matrix will be specified for which the row domain has dimension 9 and the column domain has dimension 14.
2m 2m Domain specification
The domain specification can have various forms: if nothing is specified, the matrix will have canonical domains and a canonical representation, similar to the representation encountered in textbooks. Alternatively, the row and column domains can each be specified separately, and it is also possible to specify only one of them; the other will simply be a canonical domain again. Finally, it is possible to declare the two domains identical and specify them simultaneously. It is perfectly legal in each case to explicitly specify a canonical domain. It is required in each case that the number of indices listed in a domain corresponds with the dimension given in the header.
An example where both a row domain and a column domain are specified:
(mclrows 100 200 300 400 500 600 700 800 900 $ ) (mclcols 30 32 34 36 38 40 42 44 46 48 50 52 56 58 $ )
This example combines with the header given above, as the dimensions fit. Had the row domain specification been omitted, the row domain would automatically be set to the integers 0,1,..8. Had the column specification been omitted, it would be set to 0,1,..13.
Suppose now that the header did specify the dimensions 10x10. Because the dimensions are identical, this raises the possibility that the domains be identical. A valid way to specify the row domain and column domain in one go is this.
(mcldoms 11 22 33 44 55 66 77 88 99 100 $ )
2m 2m Matrix specification
The matrix specification starts with the sequence
The 'begin' keyword in the '(mclmatrix' part is followed by a list of listings, where the primary list ranges over all column indices in M (i.e. indices in the column domain), and where each secondary lists encodes all positive entries in the corresponding column. A secondary list (or matrix column) starts with the index c of the column, and then contains a listing of all row entries in c (these are matrix entries M[r,c] for varying r). The entry M[r,c] is specified either as 'r' or as 'r:f', where f is a float. In the first case, the entry M[r,c] defaults to 1.0, in the second case, it is set to f. The secondary list is closed with the `$' character. A full fledged examples thus looks as follows:
(mclheader mcltype matrix dimensions 12x3 ) (mclrows 11 22 33 44 55 66 77 88 99 123 456 2147483647 $ ) (mclcols 0 1 2 $ ) (mclmatrix begin 0 44 88 99 456 2147483647 $ 1 11 66 77 123 $ 2 22 33 55 $ )
Note that the column domain is canonical; its specifiation could have been omitted. In this example, no values were specified. See below for more. A graph is simply a matrix where the row domain is the same as the column domain. Graphs should have positive entries only. Example:
(mclheader mcltype matrix dimensions 12x12 ) (mcldoms 11 22 33 44 55 66 77 88 99 123 456 2147483647 $ ) (mclmatrix begin 11 22:2 66:3.4 77:3 123:8 $ 22 11:2 33:3.8 55:8.1 $ 33 22:3.8 44:7 55:6.2 $ 44 33:7 88:5.7 99:7.0 456:3 $ 55 22:8.1 33:6.2 77:2.9 88:3.0 $ 66 11:3.4 123:5.1 $ 77 11:3 55:2.9 123:1.5 $ 88 44:5.7 55:3.0 99:3.0 456:4.2 $ 99 44:7.0 88:3.0 456:1.8 2147483647:3.9 $ 123 11:8 66:5.1 77:1.5 $ 456 44:3 88:4.2 99:1.8 2147483647:6.3 $ 2147483647 99:3.9 456:6.3 $ )
Incidentally, clustering this graph with mcl, using default parameters, yields a cluster that is represented by the 12x3 matrix shown earlier.
The following example shows the same graph, now represented on a canonical domain, and with all values implicitly set to 1.0:
(mclheader mcltype matrix dimensions 12x12 ) (mclmatrix begin 0 1 5 6 9 $ 1 0 2 4 $ 2 1 3 4 $ 3 2 7 8 10 $ 4 1 2 6 7 $ 5 0 9 $ 6 0 4 9 $ 7 3 4 8 10 $ 8 3 7 10 11 $ 9 0 5 6 $ 10 3 7 8 11 $ 11 8 10 $ )
There are few restrictions on the format that one might actually expect. Vectors and entries may occur in any order and need not be sorted. Repeated entries and repeated vectors are allowed but are always discarded while an error message is emitted.
If you want functionally interesting behaviour in combining repeated vectors and repeated entries, have a look at the next section and at mcxassemble(1).
Within the vector listing, the '#' is a token that introduces a comment until the end of line.
A file in raw format is simply a listing of vectors without any sectioning structure. No header specification, no domain specification, and no matrix introduction syntax is used - these are supplied to the processing application by other means. The end-of-vector token '$' must still be used, and the comment token '#' is still valid. mcxassemble(1) imports a file in raw format, creates a native matrix from the data therein, and writes the matrix to (a different) file. It allows customizable behaviour in how to combine repeated entries and repeated vectors. This is typically used in the following procedure. A) Do a one-pass-parse on some external cooccurrence file/format, generate raw data during the parse and write it to file (without needing to build a huge data structure in memory). B) mcxassemble takes the raw data and assembles it according to instruction into a native mcl matrix. Several mcl programs accept options such as -tab, -tabc, -tabr, -use-tab, -strict-tab, and -extend-tab. The argument to these options is invariably the name of a so-called tab file. Tab files are used to convert between labels (describing entities in the data) and indices as used in the mcl matrix format. In a tab file each line starts with a unique number which presumably corresponds to an index used in a matrix file. The rest of the line contains a descriptive string associated with the number. It is required that each string is unique, although not all mcl programs enforce this at the time of writing. The string may contain spaces. Lines starting with # are considered comment and are disregarded.
The ordered set of indices found in the tab file is called the tab domain.
Tab files are almost always employed in conjunction with an mcl matrix file. mcxdump(1) and clmformat(1) require by default that the tab domain coincides with the matrix domain (either row or column or both) to which they will be applied. This can be relaxed for either by supplying the --lazy-tab option.
mcl provides explicit modes for dealing with tab structures by means of the -extend-tab, -restrict-tab and -strict-tab options. Refer to the mcl(1) documentation. Label input is a line based input where two nodes and an optional value are specified on each line. The nodes should be specified by labels. If the labels contain spaces they should be separated by tabs (and the value if present should be separated from the second label by a tab as well). The parse code will assume tab-separated labels if it sees a tab character in the input, otherwise it will split the input on any kind of whitespace. Any line where the first non-whitespace character is the octothorp (#) is ignored. The following is an example of label input.
mcl(1) can read in label input and cluster it when it is given the --abc option. It can optionally save the input graph in native format and save the label information in a tab file with the -save-graph and -save-tab options.
Refer to the MCL getting started and MCL manual examples sections for more information on how MCL deals with label input.
mcxload(1) is a general purpose program for reading in label data and other stream formats. It encodes them in native mcl format and tab files. It allows intermediate transformations on the values. mcl(1), mcxload(1), mcxsubs(1), mcxassemble(1) and mcxalter(1) all accept the same transformation language in their respective tf-type options and mcxsub's val specification.
A statement in this language is simply a comma-separated list of functions accepting a single numerical value. The syntax of a function invocation in general is func(arg). The functions exp, log, neglog can also be given an empty parameter list, indicating that e is taken as the exponent base. In this case, the invocation looks like func(). Functions with names that start with # operate on graphs in their entirety. For example, #knn(50) indicates the k-Nearest Neighbour transformation for k=50. All other names encode functions that operate directly on edges. Functions with names that start with #arc operate on directed graphs and yield directed graphs. Most of the other # functions either expect an undirected graph (such as #knn()) or yield an undirected graph (such as #add() and #max(). The following are supported.
9m 9m lt Filter out values greater than or equal to arg.
9m 9m lq Filter out values greater than arg.
9m 9m gq Filter out values less than arg.
9m 9m gt Filter out values less than or equal to arg.
9m 9m ceil Set everything higher than arg to arg.
9m 9m floor Set everything lower than arg to arg.
9m 9m mul Multiply by arg.
9m 9m add Add arg to it.
9m 9m power Raise to power arg.
9m 9m exp Raise arg (e if omitted) to value.
9m 9m log Take log in base arg (e if omitted).
9m 9m neglog Take minus log in base arg (e if omitted).
9m 9m #knn k-Nearest Neighbour reduction with k=arg.
9m 9m #ceilnb Cap neighbours at arg at most.
9m 9m #mcl Cluster with inflation=arg, use induced graph.
9m 9m #add Convert two arcs to one edge using addition.
9m 9m #min Convert two arcs to one edge using minimum.
9m 9m #max Convert two arcs to one edge using maximum.
9m 9m #mul Convert two arcs to one edge using multiplication.
9m 9m #rev Encode graph in reverse direction.
9m 9m #tp Same as above in matrix speak (transpose).
9m 9m #tug Perturb edge weights with factor arg.
9m 9m #rug Randomly perturb edge weights with factor arg.
9m 9m #step Use the k-step relation, where k=arg.
9m 9m #thread Set thread pool size to arg.
9m 9m #arcmax Only keep largest arc between two nodes.
9m 9m #arcsub Replace G by G - G^T.
9m 9m #arcmcl As #mcl, use symmetrised graph for clustering.
mcl(1) accepts --abc-neg-log and --abc-neg-log10 to specify log transformations. Similarly, mcxload(1) accepts --stream-log, --stream-neg-log, and --stream-neg-log10. The reason is that probabilities are sometimes encoded below the precision dictated by the IEEE (32 bit) float specification. This poses a problem as the mcl applications encode values by default as floats, and the transformation specifications are always applied to the mcl encoding. The options just mentioned are applied after a value has been read from an input stream and before it is converted to the native encoding. mcxassemble(1), and mclfamily(7) for an overview of all the documentation and the utilities in the mcl family. Stijn van Dongen.