Codes identifying sets of vertices in random networks

Abstract

In this paper we deal with codes identifying sets of vertices in random networks; that is, ( 1 , ⩽ ℓ ) -identifying codes. These codes enable us to detect sets of faulty processors in a multiprocessor system, assuming that the maximum number of faulty processors is bounded by a fixed constant ℓ . The ( 1 , ⩽ 1 ) -identifying codes are of special interest. For random graphs we use the model G ( n , p ) , in which each one of the ( n 2 ) possible edges exists with probability p . We give upper and lower bounds on the minimum cardinality of a ( 1 , ⩽ ℓ ) -identifying code in a random graph, as well as threshold functions for the property of admitting such a code. We derive existence results from probabilistic constructions. A connection between identifying codes and superimposed codes is also established.