Detection Performance of the Majority Dominance Rule in $m$ -Ary Relay Trees With Node and Link Failures

We study detection performance of the majority dominance rule in an m-ary relay tree with node and link failures. The leaves of the tree represent N identical and independent sensors, the root of the tree represents a fusion center that makes the final detection decision, each of the other nodes in the tree is a relay node that combines m binary messages from its immediate child nodes and forms a new binary message using the majority dominance rule. We first provide the limit performance of the detection error probability at the fusion center for the case when N goes to infinity. Then, we derive upper and lower bounds for the detection error probability at the fusion center as explicit functions of N. These bounds also characterize the asymptotic decay rate of the detection error probability as N goes to infinity, and show that the decay rate in the failure case is not faster than that in the nonfailure case. Furthermore, we derive a necessary and sufficient condition in terms of the decay rate of the local failure probability (combination of node failure probability and link failure probability) at each level to ensure that the detection error probability of the tree with node and link failures decays as fast as that of the tree without failures.