Broadcast gossip averaging algorithms: interference and asymptotical error in large networks

In this paper we study two related iterative randomized algorithms for distributed computation of averages. The first one is the recently proposed Broadcast Gossip Algorithm, in which at each iteration one randomly selected node broadcasts its own state to its neighbors. The second algorithm is a novel de-synchronized version of the previous one, in which at each iteration every node is allowed to broadcast, with a given probability: hence this algorithm is affected by interference among messages. Both algorithms are proved to converge, and their performance is evaluated in terms of rate of convergence and asymptotical error: focusing on the behavior for large networks, we highlight the role of topology and design parameters on the performance. Namely, we show that on fully-connected graphs the rate is bounded away from one, whereas the asymptotical error is bounded away from zero. On the contrary, on a wide class of locally-connected graphs, the rate goes to one and the asymptotical error goes to zero, as the size of the network grows larger.

💡 Research Summary

The paper investigates two randomized distributed averaging algorithms: the Broadcast Gossip Algorithm (BGA) and a novel desynchronized variant called Probabilistic Broadcast Gossip (PBG). In BGA, at each discrete time step a single node is selected uniformly at random, broadcasts its current scalar state to all of its neighbors, and each neighbor updates its own state by taking a weighted average of its previous value and the received value. This process can be modeled as a stochastic matrix product, and its expected transition matrix is directly linked to the underlying graph Laplacian. By examining the second largest eigenvalue of the expected matrix, the authors derive the convergence rate, showing that the rate is bounded away from one on fully‑connected graphs.

PBG extends BGA by allowing every node to attempt a broadcast independently with probability p. When multiple nodes broadcast simultaneously, interference (collision) occurs. The authors consider two collision models: (i) complete loss of all messages, and (ii) reception of a noisy aggregate. Collisions are captured by an additional random diagonal matrix that multiplies the usual gossip matrix, thereby introducing extra variance into the dynamics. The expected transition matrix now contains terms proportional to p and to the node degree d, reflecting the likelihood of collisions.

Both algorithms preserve the global average in expectation, guaranteeing that the Markov chain’s stationary distribution corresponds to the true average of the initial states. However, the asymptotic mean‑square error (MSE) behaves differently. For BGA on a complete graph K_n, the MSE converges to a constant that does not vanish as n grows, because each broadcast instantly reaches all nodes, limiting the benefit of averaging over many iterations. In contrast, for locally‑connected graphs (e.g., k‑nearest‑neighbor rings or 2‑D grids), the second eigenvalue approaches one as the network size N increases, causing the convergence rate to slow down, but the MSE decays like O(1/N) because information diffuses through many hops and collisions become less significant in the large‑N limit.

The authors provide a detailed spectral analysis. For BGA, the expected matrix is ( \mathbb{E}