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Sensor Networks with Random Links: Topology Design for Distributed Consensus

Soummya Kar, Jose M. F. Moura

arXiv:0704.0954v1cs.ITcs.LG

TL;DR

The paper asks how to design sensor-network topologies for fast average consensus when links fail randomly and communication is costly and budgeted. It models random links, derives convergence conditions, and formulates topology design as a constrained convex optimization solved with semidefinite programming. The resulting designs substantially improve convergence speed and can approach nonrandom-network performance at a fraction of the communication cost.

  • Problem

    The paper studies how to assign link-reliability probabilities and topology weights to maximize average-consensus convergence under random failures, communication costs, and an overall budget constraint.

  • Method

    The paper models networks with random link failures, establishes convergence conditions, and applies semidefinite programming to a constrained convex topology-design formulation.

  • Results

    The resulting topology design improves convergence speed by about 300% and can achieve nonrandom-network convergence speed at a fraction of the communication cost.

  • Takeaways & Limitations

    Random topology optimization can substantially accelerate average consensus while reducing the communication cost needed to approach nonrandom-network performance.

Abstract

from arXiv · show

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.

I. INTRODUCTION

The paper designs sensor-network topologies that accelerate average consensus despite random link failures, communication costs, and resource constraints. It models these networks probabilistically and applies convex optimization and semidefinite programming to topology design.

  • Problem and novelty: The paper optimizes communication configurations, including link-utilization probabilities and weights, under an overall communication budget.This extends randomized consensus beyond gossip models that activate only one randomly selected sensor pair.
  • Motivation: Random link failures, communication costs, and scarce resources motivate topology design for faster average-consensus convergence.Links are modeled as active independently with fixed formation probabilities across iterations.
  • Convergence analysis: The analysis establishes necessary and sufficient convergence conditions for mean-state, mean-square, and almost-sure consensus under random link failures.The conditions are expressed using the expected algebraic connectivity and the algebraic connectivity of the mean topology.
  • Optimization approach: The topology-design problem is reformulated as a constrained convex optimization problem and solved numerically using semidefinite programming techniques.The paper first introduces the randomized consensus problem with a communication-cost constraint, then develops an alternate tractable formulation.
  • Results: A numerical design can improve convergence rate by a factor of 3 over geometric networks and approach nonrandom-network performance at 50% of the communication cost.The comparison uses geometric networks with communication within a fixed radius as the baseline.

2) Random Topology:

The random-topology model represents realizable links as independently active or failed Bernoulli processes, producing iid graph instances over time. Average consensus then uses synchronously updated, time-varying random weight matrices.

  • 2) Random Topology:: Each realizable link independently follows a Bernoulli process with formation probability P_nl and failure probability 1−P_nl.Link states can change across iterations, while formation probabilities remain fixed.
  • 2) Random Topology:: At each iteration, the active edge set, adjacency matrix, Laplacian, and weight matrix define a random network instance.The resulting edge sets and matrices are iid across distinct iterations.
  • 2) Random Topology:: The edge-formation probability matrix P records link probabilities for realizable edges but is not stochastic because its row and column sums are unnormalized.Its entries lie between 0 and 1, with zeros corresponding to unavailable edges.
  • B. Average Consensus: The paper focuses on equal-weight updates, where the Laplacian and weight matrices inherit the random topology at every iteration.For a fixed connected network, equal weights are represented using a constant α and the graph Laplacian.
  • B. Average Consensus: Random average consensus applies the usual distributed state update with time-dependent random weights determined by the current graph.Nodes exchange states synchronously with their current neighbors.

2) Average consensus: Random topology:

The random-consensus analysis links convergence to properties of random and mean Laplacians. It uses the mean Laplacian as a computable surrogate for expected algebraic connectivity and establishes its spectral structure.

  • 2) Average consensus: Random topology:: The random Laplacians and weight matrices are iid, with their distributions determined by the edge-formation probability matrix P.Their time-independent means support analysis of consensus convergence and cost-constrained design.
  • Connectivity analysis: These Laplacian properties provide the quantities used to analyze convergence of average consensus in random topologies.The convergence analysis considers both expected random algebraic connectivity and the mean-topology connectivity.
  • Mean Laplacian: The mean Laplacian can be interpreted as the weighted Laplacian of a graph whose link weights are the formation probabilities.This connects random-link statistics to a deterministic weighted topology.
  • Mean Laplacian: The mean Laplacian L = E[L(j)] is positive semidefinite and has the normalized all-ones vector associated with its zero eigenvalue.Its eigenvalues therefore retain key Laplacian properties.
  • Connectivity analysis: Algebraic connectivity λ2(L) is concave in L, enabling an upper-bound relationship between expected random connectivity and connectivity of the mean Laplacian.The concavity result follows from the Courant–Fisher characterization and Jensen’s inequality.

B. Weight matrices

The paper studies spectral properties of random and mean weight matrices to characterize consensus convergence and establish convexity results used in topology optimization.

  • B. Weight matrices: The convergence factor depends on the equal weight α and the network Laplacian, with the non-consensus eigenvalues determining contraction.The weight matrix preserves the all-ones eigenvector with eigenvalue 1.
  • B. Weight matrices: For symmetric weight matrices, the spectral radius equals the matrix 2-norm, allowing convergence analysis through spectral norms.The relevant eigenvalues exclude the consensus eigenvalue associated with the all-ones vector.
  • B. Weight matrices: The spectral norm is convex as a function of α and the Laplacian L.This convexity supports optimization over the equal weight and topology variables.
  • B. Weight matrices: Expected spectral norms over the Laplacian distribution are also treated as convex quantities for convergence analysis.The paper derives these results using the corresponding random-matrix lemmas and Jensen’s inequality.

IV. CONVERGENCE OF AVERAGE CONSENSUS: RANDOM TOPOLOGY

The paper analyzes average-consensus convergence under random topologies through mean-vector convergence and derives necessary and sufficient conditions for selecting a convergent weight.

  • Convergence framework: The analysis separates convergence of the expected state vector from mean-square and almost sure convergence.Mean convergence is treated first, followed by topology and convergence-rate analysis.
  • Mean convergence: The sequence of expected state vectors converges under the condition stated in the paper’s convergence theorem.
  • Mean convergence: The mean-state evolution is characterized through a lemma, reducing mean convergence to deterministic distributed average consensus.
  • Topology and rate: The optimal topology minimizes the convergence factor, while the admissible weight α is determined by the graph’s spectral conditions.
  • Convergence conditions: The section derives necessary and sufficient convergence conditions for the mean and provides values of α that guarantee convergence when λ2 satisfies the required condition.

B. Mean Square Convergence

Mean-square convergence is characterized by a condition on the algebraic connectivity of the mean Laplacian and by a convergence factor depending on the random Laplacian distribution and weight α.

  • Convergence rate: The mss convergence factor depends on the probability distribution of the Laplacian and the constant weight α.
  • Relation to mean convergence: Mean-square convergence cannot be faster than convergence of the mean vector.
  • Convergence rate: For a fixed random-topology distribution, smaller mss convergence factors correspond to faster mean-square convergence.
  • Convergence condition: A weight α exists for mean-square convergence if and only if the algebraic connectivity λ2 of the mean Laplacian is strictly positive.
  • Topology interpretation: The condition λ2 > 0 is straightforward to check from the edge-formation probability matrix P.

C. Almost Sure Convergence

Almost sure convergence is governed by the same strict-positivity condition on the mean graph’s algebraic connectivity that characterizes mean-square convergence.

  • Convergence condition: A necessary and sufficient condition for almost sure convergence is that the mean graph’s algebraic connectivity λ2 be strictly positive.
  • Convergence condition: When λ2 > 0, choosing α = αmss = 1/2dmax yields almost sure convergence to the average.
  • Proof strategy: The proof establishes convergence through convergence in probability, a convergent subsequence, and properties of non-increasing random-variable sequences.
  • Boundary condition: If the mean Laplacian has zero algebraic connectivity, the network separates into components with zero-probability communication between them, preventing almost sure convergence.
  • Weight assignments: The necessary condition λ2 > 0 extends to link-weight assignments that allow different weights, although the theorems analyze equal weights.

V. MSS CONVERGENCE RATE

The paper defines the best achievable mean-square convergence rate by optimizing the constant weight for a given random Laplacian distribution.

  • Rate objective: The convergence rate is measured using the mss convergence factor, with smaller factors indicating faster convergence.
  • Weight optimization: For a fixed edge-formation probability distribution P, the optimal α minimizes the mss convergence factor.
  • Optimization: The best achievable mss convergence rate is reported as S∗ after optimizing the weight.
  • Numerical optimization: The minimization depends on the probability distribution of the Laplacian and can be attained using numerical procedures.
  • Optimization constraints: The optimal α lies in a bounded range constrained by the convergence conditions for the mean and mean-square senses.

VI. CONSENSUS WITH COMMUNICATION CONSTRAINTS: TOPOLOGY OPTIMIZATION

The section formulates topology optimization for distributed average consensus under communication costs, considering fixed and random topologies with equal or heterogeneous link costs. The general randomized problem optimizes link-formation probabilities and consensus weights under an average cost constraint.

  • The design goal is a connectivity graph that yields the fastest consensus convergence under a total communication cost per-iteration constraint.
  • Fixed topology with equal costs: The fixed equal-cost case reduces to constraining the number of links, with non-bipartite Ramanujan graphs forming the optimal topology class.
  • Fixed topology with different costs (FCCC): With heterogeneous fixed link costs, fastest-convergence topology design is a difficult combinatorial optimization problem without a general closed-form solution.
  • Random topology with different costs (RCCC): RCCC relaxes FCCC because binary entries of P recover deterministic topologies, while fractional probabilities allow randomized link use under the same cost framework.
  • Random topology with different costs (RCCC): The RCCC problem jointly designs the probability matrix P and equal weight α to maximize mean-square convergence under an average communication-cost constraint.
  • Random topology with different costs (RCCC): The matrix P represents link-formation probabilities, which can also be interpreted as link-use frequencies or as probabilities of reliable communication.

B. Alternate Randomized Consensus under Communication Cost Constraints (ARCCC)

Because RCCC jointly optimizes α and P and is difficult to solve, the section introduces ARCCC as a convex surrogate. ARCCC is efficiently numerically solvable, approximates RCCC, and produces topologies with fast convergence in simulations.

  • ARCCC is introduced as a convex alternate formulation of RCCC that can be solved by fast numerical optimization procedures.
  • ARCCC is designed to approximate RCCC while retaining the communication-cost constraint and targeting good mean-square consensus convergence.
  • RCCC’s difficulty stems from joint optimization over the weight α and the topology distribution represented through P.
  • A. ARCCC as a Good Approximation to RCCC: The surrogate replaces maximizing the convergence rate with maximizing expected algebraic connectivity, then replaces that expectation with a tractable quantity involving the mean Laplacian.
  • A. ARCCC as a Good Approximation to RCCC: The convergence-rate simulation uses 500 sensors, seven average-degree values from 10 to 40, 200 random graphs per degree, and 400 Laplacian realizations per probability matrix.
  • A. ARCCC as a Good Approximation to RCCC: Fig. 1 shows remarkably similar trends for the convergence rate and the surrogate quantity, supporting ARCCC topologies as good RCCC topologies.

B. ARCCC: Performance Analysis

The performance analysis characterizes ARCCC’s optimal value as a concave function of the communication budget and derives bounds relative to full use of realizable links. It also explains why randomized designs can outperform cost-proportional expectations.

  • The ARCCC optimal value φ(U) is concave in the communication-cost budget U.
  • Ctot is the per-iteration communication cost when every realizable link is used.
  • When U is at least the total cost Ctot of all realizable links, the optimal value equals λ2(LE); for a complete graph, it equals N.
  • The bound from concavity shows that ARCCC performance can exceed the fraction of full-topology performance suggested by the fraction of communication cost used.

C. Numerical Studies: ARCCC

The ARCCC study designs random link probabilities under communication costs and compares the resulting topology with fixed-radius connectivity. Across the reported example, ARCCC converges faster and reaches non-random-network performance at substantially lower cost.

  • Results: ARCCC converges much faster than FRC, with the improvement becoming more significant at medium to lower communication costs.The reported convergence-gain curves show ARCCC above FRC as a function of the cost constraint U.
  • Results: 3.3 times faster: ARCCC reaches convergence rate c Sg = .505 versus FRC’s Sg = .152.For this example, ARCCC has total communication cost Ctot = 14.7×10^4 and achieves nonrandom-network asymptotic performance using less than 50% of the communication cost.
  • Analysis: Necessary and sufficient convergence conditions are expressed through expected algebraic connectivity of the random graph and algebraic connectivity of the average topology.These conditions support the subsequent topology-design optimization.
  • Optimization: Semidefinite programming solves the ARCCC topology-design problem subject to the communication-cost constraint.The optimization determines the link error probabilities for the realizable links.
  • Setup: The network assigns each realizable link a probability of error or expected activity, producing a random topology under communication costs.The topology is optimized subject to an overall communication-cost constraint.
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