control//multi-agent control//consensus protocol//resilient consensus

2026-08-20

Resilient consensus is a family of consensus protocols in which each agent discards suspicious neighbour values before averaging, so that the healthy agents still agree on a value inside the range of their own initial values even when some agents broadcast wrong values, on purpose or by fault. It answers the weakness of the linear consensus protocol: one node that keeps sending a fixed value (a stuck sensor, a captured drone) drags the whole network to that value.

The best-known rule, W-MSR (weighted mean-subsequence-reduced), is simple. At every step agent iii sorts the values it received, removes the FFF largest of those above its own value and the FFF smallest of those below it, and applies the ordinary consensus update to what is left. A liar survives the filter only if its value sits between values that honest neighbours also sent, and from there it cannot pull anyone outside the honest range. FFF is the number of faulty neighbours the design assumes.

Throwing away extremes only works on a graph with enough redundant paths.

Connectivity alone falls short: an honest agent whose only link to the rest carries values that get filtered out is cut off. The graph must be robust, a stronger property that costs links, and links cost bandwidth.

Graph robustness (r-robustness) is the redundancy behind the guarantee: of any two disjoint groups of agents, at least one contains an agent with rrr or more neighbours outside its group, so information always has rrr ways in. With up to FFF malicious agents in the whole network, (2F+1)(2F+1)(2F+1)-robustness is the standard sufficient condition.

The price is precision and speed. Honest extreme values are discarded too, so the agreed value lands somewhere in the honest range instead of on its exact average, and convergence is slower.

It sits beside the computing bound: agreement among replicas with fff arbitrary faults needs 3f+13f+13f+1 nodes (Byzantine fault tolerance). Both are expensive, which is why identity comes first: signed messages stop an outsider, and a Sybil attack that fakes many identities defeats any FFF counted in nodes.

The loop-level view of the same problem, a controller that must keep working while some of its inputs lie, is resilient control; the attacks themselves are in cyber-physical security.