Determining r- and (r,s)-robustness of digraphs using mixed integer linear programming

Abstract

There has been an increase in the use of resilient control algorithms based on the graph theoretic properties of r-and (r, s)robustness. These algorithms guarantee consensus of normally behaving agents in the presence of a bounded number of arbitrarily misbehaving agents if the values of the integers r and s are sufficiently large. However, determining an arbitrary graph's robustness is a highly nontrivial problem. This paper introduces a novel method for determining the r-and (r, s)robustness of digraphs using mixed integer linear programming; to the best of the authors' knowledge it is the first time that mixed integer programming methods have been applied to the robustness determination problem. The approach only requires knowledge of the graph Laplacian matrix, and can be formulated with binary integer variables. Mixed integer programming algorithms such as branch-and-bound are used to iteratively tighten the lower and upper bounds on r and s. Simulations are presented which compare the performance of this approach to prior robustness determination algorithms.

Resilient Consensus with Multi-hop Communication

“…Moreover, the graph property called robustness is shown to be critical for the network structure, guaranteeing the success of resilient consensus algorithms in static networks [9], [12] as well as time-varying networks [23]. A recent work [24] attempts to check robustness of given graphs using mixed integer linear programming. Nevertheless, such robustness requires the networks to be relatively dense and complex.…”