# A Stochastic Process on a Network with Connections to Laplacian Systems of Equations

## TL;DR

This paper analyzes a queueing network model for multi-hop sensor data collection, revealing critical data rates that determine system stability and connecting the process to Laplacian systems relevant for distributed algorithms.

## Contribution

It introduces a stochastic process model for sensor networks, establishes a phase transition based on data rate, and links the process to Laplacian systems for computational applications.

## Key findings

- Existence of a critical data rate separating ergodic and non-ergodic regimes.
- Geometric convergence to stationarity in the sub-critical regime.
- Connections to Laplacian systems for efficient distributed algorithms.

## Abstract

We study an open discrete-time queueing network that models the collection of data in a multi-hop sensor network. We assume data is generated at the sensor nodes as a discrete-time Bernoulli process. All nodes in the network maintain a queue and relay data, which is to be finally collected by a designated sink. We prove that the resulting multi-dimensional Markov chain representing the queue size of nodes has two behavior regimes depending on the value of the rate of data generation. In particular, we show that there is a non-trivial critical value of data rate below which the chain is ergodic and converges to a stationary distribution and above which it is non-ergodic, i.e., the queues at the nodes grow in an unbounded manner. We show that the rate of convergence to stationarity is geometric in the sub-critical regime. We also show the connections of this process to a class of Laplacian systems of equations whose solutions include the important problem of finding the effective resistance between two nodes, a subroutine that has been widely used to develop efficient algorithms for a number of computational problems. Hence our work provides the theoretical basis for a new class of distributed algorithms for these problems.