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Appendix C
Experiments on hierarchical
consensus architectures
This appendix presents several experiments regarding self-refining hierarchical consensus
architectures.
C.1 Configuration of a random hierarchical consensus architecture
In this section, we present some examples that describe, in detail, the configuration process
of random hierarchical consensus architectures (RHCA). The aim is to demostrate how,
given a cluster ensemble size l and a mini-ensemble size b, equations (C.1), (C.2) and (C.3)
allow determining the number of stages s, the number of consensus per stage Ki and the
effective size of each mini-ensemble bij of the corresponding RHCA.
For starters, let us evaluate carefully the three RHCA examples presented in section 3.2.
In these toy examples, the mini-ensemble size is set to b = 2, while the respective cluster
ensembles have l =7, 8 and 9 components. The first step of the RHCA design process
consists of determining the number of stages of the hierarchy, s, accordingtoequation
(C.1).
⎧
⎪⎩
⌊log b (l)⌉ if
⎪⎨
s = ⌊logb (l)⌉−1 if
⌊log b (l)⌉ +1 if
l
b ⌊log b (l)⌉
l
b ⌊log b (l)⌉
l
b ⌊log b (l)⌉
≤ 1and
≤ 1and
> 1
l
b ⌊log b (l)⌉−1
l
b ⌊log b (l)⌉−1
> 1
=1
(C.1)
Table C.1 presents the results of this computation for the three aforementioned examples
(one row per example), specifying the values of the decision factors used for selecting one
of the three options presented in equation (C.1).
Once the number of stages of the RHCA is computed, the next step consists of determining
how many consensus processes are to be executed at each RHCA stage. This factor,
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