Spectral analysis reveals network dynamics and reliability over finite ring graphs
Spectral Characterization and Network Systems Dynamics of Zero-Divisor Topologies over $\mathbb{F}_p[x]/\langle x^4\rangle$: Consensus, Reliability, and Transport
Information Theory
Summary
This paper studies special networks formed from algebraic structures called zero-divisor graphs over a mathematical ring. The authors analyze properties like how fast these networks reach agreement (consensus), how reliable they are against failures, and how signals or traffic move through them. They find exact formulas describing modes of error decay, network connectivity, and stress on nodes during random walks, helping us understand the behavior of these networks as their size grows. This work connects deep algebraic properties with practical network measures.
What this means in practice
- •For network engineers: Analyze and predict consensus speed and fault tolerance in communication networks modeled by algebraic-structure-based topologies.
- •For transportation planners: Assess routing stress and traffic concentration patterns in networks with algebraic graph structures to improve load balancing.
Authors
Apurba Sarkar, Kalyan Hansda, Makhan Maji
Abstract
For an odd prime $p$, let $Γ(R)$ denote the zero-divisor graph associated with the finite local ring $R= \mathbb{F}_p[x]/\langle x^4 \rangle$, having $p^3 - 1$ vertices and size $|E(Γ(R))| = \frac{1}{2}(3p^4 - 4p^3 - p^2 + 2)$. Using an equitable 4-cell partition induced by the ideal filtration $\langle u^3 \rangle \subset \langle u^2 \rangle \subset \langle u \rangle$, we prove that $\operatorname{Spec}_A(Γ(R))$ decomposes into discrete levels $λ\in \{0, -1\}$ with multiplicities $p^3 - p^2 - 1$ and $p^2 - 3$, respectively, alongside three irrational roots of an irreducible cubic polynomial $P_3(λ) = 0$, yielding adjacency energy $ε(Γ(R)) = 4p^2 + \mathcal{O}(p)$. In contrast, the Laplacian spectrum is entirely integral: $\operatorname{Spec}_L(Γ(R)) = \{0^1, (p - 1)^{p^3 - p^2}, (p^2 - 1)^{p^2 - p - 1}, (p^3 - 1)^{p - 1}\}$. Consequently, the tree entropy satisfies $z(Γ(R)) = \ln p - \frac{1}{p} + \frac{\ln p}{p} + \mathcal{O}(\frac{\ln p}{p^2})$, with small-failure unreliability bounded by $p^2(p - 1)(1 - q)^{p-1}$. In linear consensus $\dot{x} = -Lx$, error trajectories decouple into three modal timescales $\{ (p^3 - 1)^{-1}, (p^2 - 1)^{-1}, (p - 1)^{-1} \}$ with algebraic connectivity $α(Γ(R)) = p - 1$ and steady-state $H_2$ error variance $H_2^2 = \frac{Kf(Γ(R))}{2n^2} = \frac{1}{2p} + \mathcal{O}(p^{-3})$. Furthermore, the Kirchhoff index yields a vanishing average resistance distance $\overline{r}(Γ(R)) = \frac{2}{p} + \mathcal{O}(p^{-3}) \to 0$, while unbiased Markovian random walks induce an asymptotic traffic concentration vector $Π= (\frac{1}{3}, 0, \frac{1}{3}, \frac{1}{3})$ with quadratic per-node core routing stress $π_{\mathrm{core}}/π_{\mathrm{periph}} \sim p^2 + p + 1$.