Problems of Information Transmission, Vol. 37, No. 2, 2001, pp. 140–154. Translated from Problemy Peredachi Informatsii, No. 2, 2001, pp. 62–76.
Original Russian Text Copyright
2001 by Polesskii.
Untying of Clutter-Family Supports and Its Role
in the Monotone-Structure Reliability Theory
V. P. Polesskii
Received December 14, 2000
Abstract—We suggest a transformation of supports of a clutter family. It is a natural gener-
alization of the untying transformation of a one-clutter support previously introduced by the
author. We show that the new transformation does not decrease the reliability of the corre-
sponding clutter sum. We give new bounds for the clutter-sum reliability. We demonstrate that
the untying transformation of clutter-family supports and the factor transformation introduced
by McDiarmid provide a combinatorial basis for the monotone-structure reliability theory.
Monotone structures under consideration can be in two states, namely, either they function
(i.e., are in order) or fail to function (i.e., are faulty). Analogously, their components can be in
two states only, either they are good or faulty. Monotonicity means that any replacement of faulty
elements with nonfaulty ones does not impair the system operation. Such a monotone structure
can be deﬁned in various ways, for example, by a set of its components (elements) and a monotone
increasing family S of its subsets that characterize the system operability. This means that the
system works if and only if all elements of some subset from S function. For a monotone structure,
the state of any of its elements is a random event independent of the states of other elements.
By the reliability of such a structure, we mean the probability that the structure functions. This
elementary model of reliability for various compound technical systems was introduced in . The
monotone-structure reliability is a generalization of many parameters of the network reliability, in
particular, the two-terminal reliability, connectedness probability, and terminal reliability.
The ﬁrst exposition of the monotone-structure reliability theory was given in . Later, the
monotone-structure reliability was studied in percolation theory (see ) under the name of the
probability of clutter percolation. Presently, a number of fundamental results has been obtained
in the monotone-structure reliability theory. Their proofs use various techniques, for example,
the FKG inequality, correlation inequalities for monotone events, McDiarmid’s “clutter percolation
The purpose of this paper is to demonstrate that the combinatorial foundation of these results
may consist of two types of transformations only. These are untying transformations of clutter-
family supports introduced below (its particular case was earlier introduced by the author in )
and factor transformations of a clutter introduced by McDiarmid .
As for the idea of an untying transformation, which, surprisingly, remained unnoticed in the
monotone-structure reliability theory, we can ﬁnd its evident hint in [6, p. 195]; furthermore, unty-
ing transformations of a one-clutter support introduced by the author in  are nothing else than
speciﬁc inverse transformations for the clutter factor transformation introduced by McDiarmid
in . However, no results in the monotone-structure reliability has been derived from these trans-
formations. McDiarmid only applied the clutter factor transformation to investigate interrelations
between some reliability characteristics of random graphs and digraphs (see ).
2001 MAIK “Nauka/Interperiodica”