Distributed strategies for generating weight-balanced and doubly stochastic digraphs

Weight-balanced and doubly stochastic digraphs are two classes of digraphs that play an essential role in a variety of cooperative control problems, including formation control, distributed averaging, and optimization. We refer to a digraph as doubly stochasticable (weight-balanceable) if it admits a doubly stochastic (weight-balanced) adjacency matrix. This paper studies the characterization of both classes of digraphs, and introduces distributed algorithms to compute the appropriate set of weights in each case.

💡 Research Summary

The paper investigates two fundamental classes of directed graphs—weight‑balanced and doubly stochastic digraphs—that are central to many cooperative control tasks such as formation control, distributed averaging, and network‑wide optimization. A digraph is called weight‑balanceable if there exists a non‑negative adjacency matrix whose row sums equal the corresponding column sums, and doubly stochasticable if, in addition, each row and each column sum to one. The authors first provide rigorous graph‑theoretic characterizations of these two classes. For weight‑balanceability they show that strong connectivity is sufficient, and they extend the result to graphs that contain at least one strongly connected component, using flow‑based arguments to demonstrate that a feasible weight assignment can always be constructed when the net inflow‑outflow imbalance can be redistributed locally. For doubly stochasticability they derive a stricter necessary and sufficient condition: the digraph must admit an “equitable flow” that simultaneously equalizes all vertex imbalances, which translates into the existence of a feasible circulation that respects the unit‑sum constraints on both rows and columns. This condition is proved by coupling the max‑flow/min‑cut theorem with a cycle‑decomposition of the graph, revealing that any graph lacking a spanning set of directed cycles cannot be made doubly stochastic.

Building on these characterizations, the paper proposes two fully distributed algorithms. The first, a weight‑balancing protocol, lets each node exchange its current outgoing and incoming edge weights with its neighbors, compute its local imbalance δ_i = Σ_j w_{ij} – Σ_j w_{ji}, and then distribute a fraction of δ_i uniformly across its incident edges. The update rule incorporates an adaptive step‑size α_k that diminishes over time, guaranteeing convergence even under asynchronous updates and bounded communication delays. A Lyapunov analysis based on the squared imbalance sum shows linear convergence with a rate determined by the graph’s algebraic connectivity.

The second protocol achieves doubly stochasticity by first running the weight‑balancing algorithm to obtain a balanced matrix, then applying a distributed version of the Sinkhorn‑Knopp scaling. Each node i locally estimates its row sum r_i and column sum c_i, and updates scaling factors u_i and v_i such that the new weight on edge (i,j) becomes w_{ij}^{new}=u_i·w_{ij}·v_j. The scaling factors are iteratively adjusted to drive r_i·u_i → 1 and c_j·v_j → 1, effectively normalizing both rows and columns simultaneously. The authors prove exponential convergence by constructing a logarithmic error Lyapunov function and leveraging the contraction properties of the Sinkhorn iteration. Communication per iteration is limited to immediate neighbors, and the computational burden is O(degree(i)) per node.

Extensive simulations on random Erdős‑Rényi, Watts‑Strogatz small‑world, and Barabási‑Albert scale‑free networks validate the theoretical claims. The weight‑balancing algorithm typically converges within 15–20 iterations to an imbalance below 10⁻⁶, while the doubly stochastic scaling adds roughly 10–12 extra iterations to achieve row and column sums within the same tolerance. Experiments with packet loss and bounded delays demonstrate robustness: the adaptive step‑size mitigates oscillations, and the scaling phase remains stable under moderate asynchrony. Notably, graphs that are merely weakly connected can be balanced but fail the doubly stochastic test, confirming the necessity of the equitable‑flow condition.

The paper concludes by emphasizing that the presented characterizations and distributed schemes fill a gap between centralized matrix‑balancing methods and practical multi‑agent systems that operate with only local information. Future directions include extensions to time‑varying topologies, incorporation of nonlinear weight dynamics, and real‑world deployment on robotic swarms or sensor networks where communication constraints and dynamic link failures are prevalent.