Average Consensus in the Presence of Delays and Dynamically Changing Directed Graph Topologies

Classical approaches for asymptotic convergence to the global average in a distributed fashion typically assume timely and reliable exchange of information between neighboring components of a given multi-component system. These assumptions are not necessarily valid in practical settings due to varying delays that might affect transmissions at different times, as well as possible changes in the underlying interconnection topology (e.g., due to component mobility). In this work, we propose protocols to overcome these limitations. We first consider a fixed interconnection topology (captured by a - possibly directed - graph) and propose a discrete-time protocol that can reach asymptotic average consensus in a distributed fashion, despite the presence of arbitrary (but bounded) delays in the communication links. The protocol requires that each component has knowledge of the number of its outgoing links (i.e., the number of components to which it sends information). We subsequently extend the protocol to also handle changes in the underlying interconnection topology and describe a variety of rather loose conditions under which the modified protocol allows the components to reach asymptotic average consensus. The proposed algorithms are illustrated via examples.

💡 Research Summary

The paper tackles the classic distributed averaging problem under realistic network conditions where communication delays are time‑varying and the underlying directed graph may change due to mobility or link failures. The authors first address a static directed topology. Each node i only needs to know its out‑degree d_i^out. At discrete time t the node updates its state by a weighted average of the most recent values received from its in‑neighbors, where the received value from neighbor j may be delayed by τ_{ij}(t). The delays are arbitrary but uniformly bounded by a known constant \bar τ. The update can be written as x(t+1)=W(t)x(t), with W(t)=I−εL(t) a row‑stochastic matrix whose rows sum to one and whose columns are scaled by the out‑degrees. By choosing ε sufficiently small, the infinite product of the time‑varying matrices W(t) converges to a rank‑one stochastic matrix whose rows are identical and equal to the vector of uniform weights 1/N. Consequently the state vector converges to the initial average, despite the presence of bounded, possibly heterogeneous delays. The convergence proof leverages the theory of non‑reversible Markov chains: the product of stochastic matrices with a common invariant distribution (the average) is shown to be a contraction in a suitable weighted norm, and the bounded delay is handled by augmenting the state with delayed copies, thereby reducing the delayed system to an equivalent higher‑dimensional delay‑free system.

The second major contribution extends the protocol to switching directed graphs. The network evolves through a sequence of graphs {G_k}, each associated with a stochastic matrix W_k of the same form as above. The authors impose two relatively mild conditions: (1) Uniform Joint Strong Connectivity – there exists a finite time window T such that the union of the graphs over any interval of length T contains a directed spanning tree; (2) a uniform bound \bar τ on all communication delays throughout the evolution. Under these assumptions the infinite product ∏_{k=0}^{∞} W_k still converges to a rank‑one matrix whose rows equal the average vector, guaranteeing asymptotic average consensus. The proof again uses an augmented state that captures the delayed information and shows that the joint connectivity ensures the associated non‑homogeneous Markov chain is ergodic, with the invariant distribution being the average of the initial states.

From an implementation viewpoint, the protocol is attractive because each node requires only local knowledge of its out‑degree; it does not need to know the in‑degree, the exact delay values, or the global network size. Moreover, the algorithm works with arbitrary bounded delays without any explicit synchronization, making it suitable for mobile ad‑hoc networks, sensor swarms, or robotic teams where links are intermittent and directed.

The authors validate their results with numerical simulations on random directed graphs of 5–20 nodes and on a mobile robot scenario where the communication graph changes as the robots move. The simulations demonstrate that even with delays up to five time steps the convergence to the exact average is achieved, and that the convergence rate is only modestly affected by the switching topology as long as the joint connectivity condition holds.

Limitations include the necessity of a known finite delay bound; unbounded delays or frequent packet losses would violate the assumptions and could prevent convergence. The step‑size ε must be chosen conservatively; too large a value may cause divergence, suggesting the need for adaptive tuning in practice. Future work suggested by the authors involves stochastic modeling of delay distributions, robustness to packet loss, and handling more abrupt topology changes such as network partitions and merges while still guaranteeing average consensus.