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In this paper, we study parameter-independent stability in qualitativelyheterogeneous passive networked systems containing damped and undamped nodes.Given the graph topology and a set of damped nodes, we ask if output consensusis achieved for all system parameter values. For given parameter values, aneigenspace analysis is used to determine output consensus. The extension toparameter-independent stability is characterized by a coloring problem, namedthe richly balanced coloring (RBC) problem. The RBC problem asks if all nodesof the graph can be colored red, blue and black in such a way that (i) everydamped node is black, (ii) every black node has blue neighbors if and only ifit has red neighbors, and (iii) not all nodes in the graph are black. Such acolored graph is referred to as a richly balanced colored graph.Parameter-independent stability is guaranteed if there does not exist a richlybalanced coloring. The RBC problem is shown to cover another well-known graphcoloring scheme known as zero forcing sets. That is, if the damped nodes form azero forcing set in the graph, then a richly balanced coloring does not existand thus, parameter-independent stability is guaranteed. However, the fullequivalence of zero forcing sets and parameter-independent stability holds onlytrue for tree graphs. For more general graphs with few fundamental cycles analgorithm, named chord node coloring, is proposed that significantlyoutperforms a brute-force search for solving the NP-complete RBC problem.

Topological and Graph-coloring Conditions on the Parameter-independent Stability of Second-order Networked Systems