Structural dependence of network coherence in odd-polygon chain networks

Building upon cyclic pentagon networks, this paper proposes a family of odd-polygon chain networks parameterized by a structural index, where the fundamental building blocks extend from pentagons to generic odd-edged cyclic modules. For these architectures, we characterize the network coherence measuring the extent of consensus under the additive noise. By constructing recursive Laplacian matrices and exploiting the spectral interplay between the eigenvalues and the coefficients of their characteristic polynomials, we derive an exact expression for coherence as a function of this structural index. Our results quantify how network coherence scales with odd-polygon connectivity patterns and network size, uncovering fundamental coherence-structure relationships in these cyclic network configurations.