Broadcast Gossip Algorithms for Consensus on Strongly Connected Digraphs

We study a general framework for broadcast gossip algorithms which use companion variables to solve the average consensus problem. Each node maintains an initial state and a companion variable. Iterative updates are performed asynchronously whereby one random node broadcasts its current state and companion variable and all other nodes receiving the broadcast update their state and companion variable. We provide conditions under which this scheme is guaranteed to converge to a consensus solution, where all nodes have the same limiting values, on any strongly connected directed graph. Under stronger conditions, which are reasonable when the underlying communication graph is undirected, we guarantee that the consensus value is equal to the average, both in expectation and in the mean-squared sense. Our analysis uses tools from non-negative matrix theory and perturbation theory. The perturbation results rely on a parameter being sufficiently small. We characterize the allowable upper bound as well as the optimal setting for the perturbation parameter as a function of the network topology, and this allows us to characterize the worst-case rate of convergence. Simulations illustrate that, in comparison to existing broadcast gossip algorithms, the approaches proposed in this paper have the advantage that they simultaneously can be guaranteed to converge to the average consensus and they converge in a small number of broadcasts.

💡 Research Summary

The paper introduces a novel broadcast gossip framework that solves the average‑consensus problem on arbitrary strongly‑connected directed graphs by augmenting each node with a companion variable. In the proposed scheme every node i holds an initial state x_i(0) and a companion variable y_i(0). At each asynchronous iteration a randomly selected node k broadcasts the pair (x_k, y_k) to the whole network. All other nodes i update both their state and companion variable using two pre‑defined weight matrices W and V and a scalar perturbation parameter ε>0:

x_i⁺ = x_i + w_{ik}(x_k – x_i) + ε·v_{ik}(y_k – y_i)
y_i⁺ = y_i + ε·v_{ik}(x_k – x_i) + w_{ik}(y_k – y_i).

The broadcasting node updates in the same way. This simultaneous update of two coupled variables creates a block‑matrix dynamics

M(ε)=⎡W εV⎤
  ⎣εV W⎦,

which is a non‑negative matrix. The authors first prove that for any strongly‑connected digraph there exists a sufficiently small ε such that M(ε) is primitive. By the Perron–Frobenius theorem the dominant eigenvalue is 1, simple, and its associated eigenvector has strictly positive entries; consequently all trajectories converge to a common limit (global consensus).

The second major contribution is to identify conditions under which the limiting consensus value equals the true arithmetic average of the initial states. When the underlying communication graph is undirected (so that W and V can be chosen symmetric and doubly‑stochastic) the eigenvector associated with eigenvalue 1 is proportional to the all‑ones vector. In this case the authors show, both in expectation and in the mean‑squared sense, that

E