version 3.696

# Clique -- Compatibility Program

© Copyright 1986-2014 by Joseph Felsenstein. All rights reserved. License terms here.

This program uses the compatibility method for unrooted two-state characters to obtain the largest cliques of characters and the trees which they suggest. This approach originated in the work of Le Quesne (1969), though the algorithms were not precisely specified until the later work of Estabrook, Johnson, and McMorris (1976a, 1976b). These authors proved the theorem that a group of two-state characters which were pairwise compatible would be jointly compatible. This program uses an algorithm inspired by the Kent Fiala - George Estabrook program CLINCH, though closer in detail to the algorithm of Bron and Kerbosch (1973). I am indebted to Kent Fiala for pointing out that paper to me, and to David Penny for decribing to me his branch-and-bound approach to finding the largest cliques, from which I have also borrowed. I am particularly grateful to Kent Fiala for catching a bug in versions 2.0 and 2.1 which resulted in those versions failing to find all of the cliques which they should. The program computes a compatibility matrix for the characters, then uses a recursive procedure to examine all possible cliques of characters.

After one pass through all possible cliques, the program knows the size of the largest clique, and during a second pass it prints out the cliques of the right size. It also, along with each clique, prints out the tree suggested by that clique.

## INPUT, OUTPUT, AND OPTIONS

Input to the algorithm is standard, but the "?", "P", and "B" states are not allowed. This is a serious limitation of this program. If you want to find large cliques in data that has "?" states, I recommend that you use MIX instead with the T (Threshold) option and the value of the threshold set to 2.0. The theory underlying this is given in my paper on character weighting (Felsenstein, 1981b).

The options are chosen from a menu, which looks like this:

    Largest clique program, version 3.69

    Settings for this run:
      A   Use ancestral states in input file?  No
      F              Use factors information?  No
      W                       Sites weighted?  No
      C          Specify minimum clique size?  No
      O                        Outgroup root?  No, use as outgroup species  1
      M           Analyze multiple data sets?  No
      0   Terminal type (IBM PC, ANSI, none)?  ANSI
      1    Print out the data at start of run  No
      2  Print indications of progress of run  Yes
      3        Print out compatibility matrix  No
      4                        Print out tree  Yes
      5       Write out trees onto tree file?  Yes

      Y to accept these or type the letter for one to change

The A (Ancestors), F (Factors), O (Outgroup) ,M (Multiple Data Sets), and W (Weights) options are the usual ones, described in the main documentation file.

If you use option A (Ancestors) you should also choose it in the menu. The compatibility matrix calculation in effect assumes if the Ancestors option is invoked that there is in the data another species that has all the ancestral states. This changes the compatibility patterns in the proper way. The Ancestors option also requires information on the ancestral states of each character to be in the input file.

The O (Outgroup) option will take effect only if the tree is not rooted by the Ancestral States option.

The C (Clique Size) option indicates that you wish to specify a minimum clique size and print out all cliques (and their associated trees) greater than or equal to that size. The program prompts you for the minimum clique size.

Note that this allows you to list all cliques (each with its tree) by simply setting the minimum clique size to 1. If you do one run and find that the largest clique has 23 characters, you can do another run with the minimum clique size set at 18, thus listing all cliques within 5 characters of the largest one.

Output involves a compatibility matrix (using the symbols "." and "1") and the cliques and trees.

If you have used the F option there will be two lists of characters for each clique, one the original multistate characters and the other the binary characters. It is the latter that are shown on the tree. When the F option is not used the output and the cliques reflect only the binary characters.

The trees produced have it indicated on each branch the points at which derived character states arise in the characters that define the clique. There is a legend above the tree showing which binary character is involved. Of course if the tree is unrooted you can read the changes as going in either direction.

The program runs very quickly but if the maximum number of characters is large it will need a good deal of storage, since the compatibility matrix requires ActualChars x ActualChars boolean variables, where ActualChars is the number of characters (in the case of the factors option, the total number of true multistate characters).

## ASSUMPTIONS

Basically the following assumptions are made:

1. Each character evolves independently.
2. Different lineages evolve independently.
3. The ancestral state is not known.
4. Each character has a small chance of being one which evolves so rapidly, or is so thoroughly misinterpreted, that it provides no information on the tree.
5. The probability of a single change in a character (other than in the high rate characters) is low but not as low as the probability of being one of these "bad" characters.
6. The probability of two changes in a low-rate character is much less than the probability that it is a high-rate character.
7. The true tree has segments which are not so unequal in length that two changes in a long are as easy to envisage as one change in a short segment.

The assumptions of compatibility methods have been treated in several of my papers (1978b, 1979, 1981b, 1988b), especially the 1981 paper. For an opposing view arguing that the parsimony methods make no substantive assumptions such as these, see the papers by Farris (1983) and Sober (1983a, 1983b), but also read the exchange between Felsenstein and Sober (1986).

A constant available for alteration at the beginning of the program is the form width, "FormWide", which you may want to change to make it as large as possible consistent with the page width available on your output device, so as to avoid the output of cliques and of trees getting wrapped around unnecessarily.

### TEST DATA SET

         5    6
    Alpha     110110
    Beta      110000
    Gamma     100110
    Delta     001001
    Epsilon   001110

### TEST SET OUTPUT (with all numerical options on)

    Largest clique program, version 3.69

     5 species,   6  characters
    Species  Character states
    -------  --------- ------

    Alpha       11011 0
    Beta        11000 0
    Gamma       10011 0
    Delta       00100 1
    Epsilon     00111 0

    Character Compatibility Matrix (1 if compatible)
    --------- ------------- ------ -- -- -----------

                         111..1
                         111..1
                         111..1
                         ...111
                         ...111
                         111111

    Largest Cliques
    ------- -------

    Characters: (  1  2  3  6)

      Tree and characters:

         2  1  3  6
         0  0  1  1

                 +1-Delta
           +0--1-+
      +--0-+     +--Epsilon
      !    !
      !    +--------Gamma
      !
      +-------------Alpha
      !
      +-------------Beta

    remember: this is an unrooted tree!