# Randomized detection and detection capacity of multidetector networks

## TL;DR

This paper analyzes the detection capabilities of randomly placed detector networks, establishing the minimum detection time and characterizing schemes that achieve this optimal performance as the number of detectors grows large.

## Contribution

It introduces the concept of capacity achieving detection schemes and provides a complete characterization of these schemes in the context of random detector placement.

## Key findings

- Minimum detection time $T_{cap}$ is determined.
- Existence of randomized schemes that approach $T_{cap}$ for large $n$.
- Complete characterization of all capacity achieving detection schemes.

## Abstract

In this paper, we study the following detection problem. There are $n$ detectors randomly placed in the unit square $S = \left[-\frac{1}{2},\frac{1}{2}\right]^2$ assigned to detect the presence of a source located at the origin. Time is divided into slots of unit length and $D_i(t) \in \{0,1\}$ represents the (random) decision of the $i^{\rm th}$ detector in time slot $t$. The location of the source is unknown to the detectors and the goal is to design schemes that use the decisions $\{D_i(t)\}_{i,t}$ and detect the presence of the source in as short time as possible. We first determine the minimum achievable detection time $T_{cap}$ and show the existence of \emph{randomized} detection schemes that have detection times arbitrarily close to $T_{cap}$ for almost all configuration of detectors, provided the number of detectors $n$ is sufficiently large. We call such schemes as \emph{capacity achieving} and completely characterize all capacity achieving detection schemes.