3. Maximal cliques
Maximal cliques are groups of co-occurring taxa not contained in any larger group of co-occurring
taxa. The maximal cliques are candidates for the status of unitary associations, but will be further
processed below. In PAST, maximal cliques receive a number and are also named after a maximal
horizon in the original data set which is identical to, or contained in (marked with asterisk), the
4. Superposition of maximal cliques
The superpositional relationships between maximal cliques are decided by inspecting the
superpositional relationships between their constituent taxa, as computed in step 2. Contradictions
(some taxa in clique A occur below some taxa in clique B, and vice versa) are resolved by a 'majority
vote'. The contradictions between cliques can be viewed in PAST.
The superpositions and co-occurrences of cliques can be viewed in the maximal clique graph. In this
graph, cliques are coded as numbers. Co-occurrences between pairs of cliques are shown as solid
blue lines. Superpositions are shown as dashed red lines, with long dashes from the above-occurring
clique and short dashes from the below-occurring clique. Also, cycles between maximal cliques (see
below) can be viewed as green lines.
5. Resolving cycles
It will sometimes be the case that maximal cliques are now ordered in cycles: A is below B, which is
below C, which is below A again. This is clearly contradictory. The 'weakest link' (superpositional
relationship supported by fewest taxa) in such cycles is destroyed.
6. Reduction to unique path
At this stage, we should ideally have a single path (chain) of superpositional relationships between
maximal cliques, from bottom to top. This is however often not the case, for example if A and B are
below C, which is below D, or if we have isolated paths without any relationships (A below B and C
below D). To produce a single path, it is necessary to merge cliques according to special rules.
7. Post-processing of maximal cliques
Finally, a number of minor manipulations are carried out to 'polish' the result: Generation of the
'consecutive ones' property, reinsertion of residual virtual co-occurrences and superpositions, and
compaction to remove any generated non-maximal cliques. For details on these procedures, see
Guex (1991). At last, we now have the Unitary Associations, which can be viewed in PAST.