[Paper Review] Distributed Algorithms for Consensus and Coordination in the Presence of Packet-Dropping Communication Links - Part I: Statistical Moments Analysis Approach

Alejandro D. Domínguez-García, Christoforos N. Hadjicostis+1 more|arXiv (Cornell University)|Sep 29, 2011

Distributed Control Multi-Agent Systems4 citations

TL;DR

This paper proposes a robust distributed algorithm for average consensus in multi-agent systems with unreliable, packet-dropping communication links. By introducing auxiliary variables and analyzing first and second statistical moments, the method ensures almost sure convergence to average consensus even under asymmetric, time-varying, and unreliable communication, with convergence proven via Borel-Cantelli lemmas and moment analysis.

ABSTRACT

This two-part paper discusses robustification methodologies for linear-iterative distributed algorithms for consensus and coordination problems in multicomponent systems, in which unreliable communication links may drop packets. We consider a setup where communication links between components can be asymmetric (i.e., component j might be able to send information to component i, but not necessarily vice-versa), so that the information exchange between components in the system is in general described by a directed graph that is assumed to be strongly connected. In the absence of communication link failures, each component i maintains two auxiliary variables and updates each of their values to be a linear combination of their corresponding previous values and the corresponding previous values of neighboring components (i.e., components that send information to node i). By appropriately initializing these two (decoupled) iterations, the system components can asymptotically calculate variables of interest in a distributed fashion; in particular, the average of the initial conditions can be calculated as a function that involves the ratio of these two auxiliary variables. The focus of this paper to robustify this double-iteration algorithm against communication link failures. We achieve this by modifying the double-iteration algorithm (by introducing some additional auxiliary variables) and prove that the modified double-iteration converges almost surely to average consensus. In the first part of the paper, we study the first and second moments of the two iterations, and use them to establish convergence, and illustrate the performance of the algorithm with several numerical examples. In the second part, in order to establish the convergence of the algorithm, we use coefficients of ergodicity commonly used in analyzing inhomogeneous Markov chains.

Motivation & Objective

• To address the challenge of achieving average consensus in multi-agent systems where communication links may drop packets unpredictably.
• To develop a distributed algorithm that maintains convergence to the average of initial values despite unreliable, asymmetric, and time-varying communication links.
• To ensure robustness in directed, strongly connected networks where traditional consensus algorithms may fail due to link failures.
• To establish theoretical convergence guarantees using statistical moment analysis and probabilistic tools such as the Borel-Cantelli lemmas.
• To provide a foundation for Part II, which uses coefficients of ergodicity for inhomogeneous Markov chains to further validate convergence.

Proposed method

• Introduces a double-iteration algorithm with two auxiliary variables per node, where each variable is updated as a weighted linear combination of its own and neighbors' previous values.
• Modifies the standard consensus algorithm by adding auxiliary variables to handle packet loss, ensuring that the ratio of the two auxiliary variables converges to the average of initial values.
• Analyzes the first and second statistical moments of the auxiliary variables to establish convergence properties under random packet drops.
• Uses the Borel-Cantelli lemma to prove that for each node, the second auxiliary variable $ z_j[k] $ exceeds a positive threshold $ C $ infinitely often with probability one.
• Establishes that the ratio $ y_j[k]/z_j[k] $ converges almost surely to $ 1/ar{ u} $, where $ ar{ u} $ is the average of initial values.
• Employs a probabilistic argument based on the in-degree and maximum out-degree of nodes to bound the probability of mass transfer across the network, ensuring persistent activity in the system.

Experimental results

Research questions

• RQ1 Can a distributed consensus algorithm maintain convergence to the average of initial values when communication links drop packets stochastically and asymmetrically?
• RQ2 How can statistical moment analysis be used to prove almost sure convergence in the presence of unreliable communication?
• RQ3 What conditions ensure that the auxiliary variables in a double-iteration algorithm remain bounded away from zero infinitely often, enabling ratio-based estimation of the average?
• RQ4 Can the Borel-Cantelli lemma be applied to establish that critical thresholds in the auxiliary variables are exceeded infinitely often under packet loss?
• RQ5 How does the structure of the underlying directed graph (strongly connected, bounded out-degree) affect the convergence behavior of the robustified algorithm?

Key findings

• The modified double-iteration algorithm ensures almost sure convergence to average consensus even when communication links drop packets, under the assumption of a strongly connected directed graph.
• For every node $ j $, the event $ z_j[k] o C $ occurs infinitely often with probability one, where $ C = rac{n}{(n+m)( ext{D}_{ ext{max}}^{+})^{n-1}} $, ensuring that the ratio $ y_j[k]/z_j[k] $ can be computed infinitely often.
• The ratio $ y_j[t_n]/z_j[t_n] $ converges almost surely to $ 1/ar{ u} $, where $ ar{ u} $ is the average of the initial values, as $ t_n o ty $, proving convergence to the desired consensus value.
• The analysis shows that $ |ar{ u} y_j[k] - z_j[k]| o 0 $ almost surely, which implies that the error in the ratio estimate vanishes over time.
• The probability of successful mass transfer from any node to any other node within $ n $ steps is bounded below by $ q^n $, where $ q $ depends on the minimum in-degree and network structure.
• The convergence result holds under general conditions: directed, strongly connected graphs with bounded out-degrees, and i.i.d. packet loss on links.