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This paper explores the conditions for determining fixed nodes in structured networks, specifically focusing on directed acyclic graphs (DAGs). We introduce several necessary and sufficient conditions for determining fixed nodes in $p$-layered DAGs. This is accomplished by defining the problem of maximum disjoint stems, based on the observation that all DAGs can be represented as hierarchical structures with a unique label for each layer. For structured networks, we discuss the importance of fixed nodes by considering their controllability against the variations of network parameters. Moreover, we present an efficient algorithm that simultaneously performs labeling and fixed node search for $p$-layered DAGs with an analysis of its time complexity. The results presented in this paper have implications for the analysis of controllability at the individual node level in structured networks.

This paper presents new results and reinterpretation of existing conditions for strong structural controllability in a structured network determined by the zero/non-zero patterns of edges. For diffusively-coupled networks with self-loops, we first establish a necessary and sufficient condition for strong structural controllability, based on the concepts of dedicated and sharing nodes. Subsequently, we define several conditions for strong structural controllability across various graph types by decomposing them into disjoint path graphs. We further extend our findings by introducing a composition rule, facilitating the analysis of strong structural controllability in larger networks. This rule allows us to determine the strong structural controllability of connected graphs called pactus graphs (a generalization of the well-known cactus graph) by consideration of the strong structural controllability of its disjoint component graphs. In this process, we introduce the notion of a component input node, which is a state node that functions identically to an external input node. Based on this concept, we present an algorithm with approximate polynomial complexity to determine the minimum number of external input nodes required to maintain strong structural controllability in a diffusively-coupled network with self-loops., Comment: 11 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:2205.05275

Within the context of structured networks, this paper introduces the concept of the Fixed Strongly Structurally Controllable Subspace (FSSCS), enabling a comprehensive characterization of controllable subspaces. From a graph-theoretical viewpoint, the paper defines Fixed Strongly Structurally Controllable (FSSC) nodes based on the FSSCS concept and establishes the necessary and sufficient conditions for their identification. This paper proposes a method for determining the exact dimension of the Strongly Structurally Controllable Subspace (SSCS) in hierarchical directed acyclic graphs, employing a blend of graph-theoretical approaches and controllability matrix analyses. This approach not only facilitates the identification of FSSC nodes but also enhances our understanding of the robustness of node controllability against variations in network parameters within structured networks, marking a significant advancement in the field of strong structural controllability of individual nodes., Comment: The proofs of Lemma 1 and Theorem 3 have errors

This paper presents several conditions to determine strong sign controllability for diffusively-coupled undirected networks. The strong sign controllability is determined by the sign patterns (positive, negative, zero) of the edges. We first provide the necessary and sufficient conditions for strong sign controllability of basic components, such as path, cycle, and tree. Next, we propose a merging process to extend the basic componenets to a larger graph based on the conditions of the strong sign controllability. Furthermore, we develop an algorithm of polynomial complexity to find the minimum number of external input nodes while maintaining the strong sign controllability of a network.