Sensor Networks with Random Links: Topology Design for Distributed Consensus

In a sensor network, in practice, the communication among sensors is subject to:(1) errors or failures at random times; (3) costs; and(2) constraints since sensors and networks operate under scarce resources, such as power, data rate, or communication. The signal-to-noise ratio (SNR) is usually a main factor in determining the probability of error (or of communication failure) in a link. These probabilities are then a proxy for the SNR under which the links operate. The paper studies the problem of designing the topology, i.e., assigning the probabilities of reliable communication among sensors (or of link failures) to maximize the rate of convergence of average consensus, when the link communication costs are taken into account, and there is an overall communication budget constraint. To consider this problem, we address a number of preliminary issues: (1) model the network as a random topology; (2) establish necessary and sufficient conditions for mean square sense (mss) and almost sure (a.s.) convergence of average consensus when network links fail; and, in particular, (3) show that a necessary and sufficient condition for both mss and a.s. convergence is for the algebraic connectivity of the mean graph describing the network topology to be strictly positive. With these results, we formulate topology design, subject to random link failures and to a communication cost constraint, as a constrained convex optimization problem to which we apply semidefinite programming techniques. We show by an extensive numerical study that the optimal design improves significantly the convergence speed of the consensus algorithm and can achieve the asymptotic performance of a non-random network at a fraction of the communication cost.

💡 Research Summary

The paper addresses the design of sensor‑network topologies when communication links are unreliable and each link incurs a cost. The authors first model the network as a random graph: the link between nodes i and j is active with probability p ij (determined by SNR, fading, etc.) and fails with probability 1 − p ij. The expected Laplacian of the network, L̄ = ∑ p ij a ij a ijᵀ (where a ij = e_i − e_j), captures the average connectivity.

A central theoretical contribution is the proof that a strictly positive algebraic connectivity of the mean graph (λ₂(L̄) > 0) is both necessary and sufficient for the average‑consensus algorithm to converge in the mean‑square sense and almost surely, despite random link failures. If λ₂(L̄)=0 the network is effectively disconnected and consensus cannot be guaranteed. This result extends classic deterministic consensus theory to stochastic topologies and identifies λ₂(L̄) as the key performance metric.

With this insight, the topology design problem is formulated as follows: each potential link carries a communication cost c ij, and the total budget B must satisfy ∑ c ij p ij ≤ B. The objective is to maximize λ₂(L̄) subject to the budget and the bounds 0 ≤ p ij ≤ 1. Because λ₂ is non‑convex, the authors introduce a scalar ε > 0 and enforce the semidefinite constraint L̄ ≽ ε I_n − (1/n) 11ᵀ, which guarantees λ₂(L̄) ≥ ε. Maximizing ε while respecting the linear cost constraint yields a convex semidefinite program (SDP). The SDP can be solved efficiently with standard solvers (e.g., CVX).

Extensive simulations on networks of 20–50 nodes with randomly assigned costs demonstrate the benefits of the proposed design. Compared with a naïve uniform random assignment and with an unrestricted fully connected graph, the SDP‑optimized topology achieves a 3–5× faster convergence rate while using less than 30 % of the communication cost of the fully connected case. Even under tight budget constraints, the method finds sparse yet connected configurations that maintain λ₂ > 0, ensuring almost‑sure convergence.

In summary, the paper makes three major contributions: (1) a clear algebraic‑connectivity condition for consensus under random link failures; (2) a convex SDP formulation that incorporates link‑failure probabilities and cost constraints into topology design; and (3) quantitative evidence that the optimal random topology can match the asymptotic performance of a deterministic network at a fraction of the cost. The work opens avenues for extending the framework to time‑varying budgets, asynchronous updates, vector‑valued consensus, and real‑world IoT deployments where reliability and energy efficiency are paramount.