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Invariant Properties of Linear-Iterative Distributed Averaging Algorithms and Application to Error Detection

Christoforos N. Hadjicostis, Alejandro D. Dominguez-Garcia

TL;DR

This paper reports on another invariance property for the class of averaging algorithms that reflects the conservation of certain quantities capturing an aggregate of all the values received by a node from its in-neighbors and all the values sent by said node to its out-neighbors throughout the execution of the averaging algorithm.

Abstract

We consider the problem of average consensus in a distributed system comprising a set of nodes that can exchange information among themselves. We focus on a class of algorithms for solving such a problem whereby each node maintains a state and updates it iteratively as a linear combination of the states maintained by its in-neighbors, i.e., nodes from which it receives information directly. Averaging algorithms within this class can be thought of as discrete-time linear time-varying systems without external driving inputs and whose state matrix is column stochastic. As a result, the algorithms exhibit a global invariance property in that the sum of the state variables remains constant at all times. In this paper, we report on another invariance property for the aforementioned class of averaging algorithms. This property is local to each node and reflects the conservation of certain quantities capturing an aggregate of all the values received by a node from its in-neighbors and all the values sent by said node to its out-neighbors (i.e., nodes to which it sends information directly) throughout the execution of the averaging algorithm. We show how this newly-discovered invariant can be leveraged for detecting errors while executing the averaging algorithm.

Invariant Properties of Linear-Iterative Distributed Averaging Algorithms and Application to Error Detection

TL;DR

This paper reports on another invariance property for the class of averaging algorithms that reflects the conservation of certain quantities capturing an aggregate of all the values received by a node from its in-neighbors and all the values sent by said node to its out-neighbors throughout the execution of the averaging algorithm.

Abstract

We consider the problem of average consensus in a distributed system comprising a set of nodes that can exchange information among themselves. We focus on a class of algorithms for solving such a problem whereby each node maintains a state and updates it iteratively as a linear combination of the states maintained by its in-neighbors, i.e., nodes from which it receives information directly. Averaging algorithms within this class can be thought of as discrete-time linear time-varying systems without external driving inputs and whose state matrix is column stochastic. As a result, the algorithms exhibit a global invariance property in that the sum of the state variables remains constant at all times. In this paper, we report on another invariance property for the aforementioned class of averaging algorithms. This property is local to each node and reflects the conservation of certain quantities capturing an aggregate of all the values received by a node from its in-neighbors and all the values sent by said node to its out-neighbors (i.e., nodes to which it sends information directly) throughout the execution of the averaging algorithm. We show how this newly-discovered invariant can be leveraged for detecting errors while executing the averaging algorithm.
Paper Structure (15 sections, 4 theorems, 28 equations)

This paper contains 15 sections, 4 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Communication Topology Model
  3. Distributed Averaging via Linear Iterations
  4. Weight Choice
  5. Ratio-Consensus Algorithm
  6. Generalized Push-Sum
  7. Convergence Analysis
  8. Invariant Properties of Linear-Iterative Distributed Averaging Algorithms
  9. Global Invariant
  10. Local Invariant
  11. Error Detection
  12. Existing Techniques for Error Detection/Correction
  13. Any-Time Consistency Checking
  14. Invariant-Based Error Detection
  15. Concluding Remarks

Key Result

Lemma 1

Consider alg_x with the $w_{ji}[k]$'s chosen so that the weight matrix $W[k]= [w_{j,i}[k]] \in \mathbb{R}^{N \times N}$ in alg_x_matrix satisfies Properties W1--W3. Then, for all $k=0,1,2,\dots$.

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof