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Redundancy analysis using lcm-filtrations: networks, system signature and sensitivity evaluation

Fatemeh Mohammadi, Eduardo Sáenz-de-Cabezón, Henry Wynn

TL;DR

The paper addresses the computational redundancy arising in the least common multiple (${\rm lcm}$) structure of monomial ideals by introducing the stepwise ${\rm lcm}$-filtration and connecting it to a simplicial framework via Stanley-Reisner theory. It provides formal definitions, compatibility results, and comparative analyses with the usual ${\rm lcm}$-filtration, including phase-transition findings for cut ideals in random graphs and applications to simultaneous failures in coherent systems and sensitivity analysis. Key contributions include a simplicial interpretation of the stepwise filtration, phase-transition results distinguishing dense vs. sparse graph regimes, and practical guidance on when to deploy each filtration in networks, reliability, and model-uncertainty analyses. The work delivers scalable algebraic-combinatorial tools for studying interactions among minimal generators and their impact on simplicial complexes, with potential broad impact in reliability engineering, network analysis, and algebraic topology-informed sensitivity analysis.

Abstract

We introduce the lcm-filtration and stepwise filtration, comparing their performance across various scenarios in terms of computational complexity, efficiency, and redundancy. The lcm-filtration often involves identical steps or ideals, leading to unnecessary computations. To address this, we analyse how stepwise filtration can effectively compute only the non-identical steps, offering a more efficient approach. We compare these filtrations in applications to networks, system signatures, and sensitivity analysis.

Redundancy analysis using lcm-filtrations: networks, system signature and sensitivity evaluation

TL;DR

The paper addresses the computational redundancy arising in the least common multiple () structure of monomial ideals by introducing the stepwise -filtration and connecting it to a simplicial framework via Stanley-Reisner theory. It provides formal definitions, compatibility results, and comparative analyses with the usual -filtration, including phase-transition findings for cut ideals in random graphs and applications to simultaneous failures in coherent systems and sensitivity analysis. Key contributions include a simplicial interpretation of the stepwise filtration, phase-transition results distinguishing dense vs. sparse graph regimes, and practical guidance on when to deploy each filtration in networks, reliability, and model-uncertainty analyses. The work delivers scalable algebraic-combinatorial tools for studying interactions among minimal generators and their impact on simplicial complexes, with potential broad impact in reliability engineering, network analysis, and algebraic topology-informed sensitivity analysis.

Abstract

We introduce the lcm-filtration and stepwise filtration, comparing their performance across various scenarios in terms of computational complexity, efficiency, and redundancy. The lcm-filtration often involves identical steps or ideals, leading to unnecessary computations. To address this, we analyse how stepwise filtration can effectively compute only the non-identical steps, offering a more efficient approach. We compare these filtrations in applications to networks, system signatures, and sensitivity analysis.
Paper Structure (12 sections, 8 theorems, 25 equations, 5 figures, 2 tables)

This paper contains 12 sections, 8 theorems, 25 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Main algebraic concepts
  3. Stanley-Reisner ideals of simplicial complexes
  4. The lcm-lattice
  5. The lcm-filtrations of a monomial ideal
  6. The simplicial form of the stepwise ${\rm lcm}$-filtration
  7. lcm-filtration versus stepwise lcm-filtration
  8. The cut ideals of graphs
  9. Density regime in cut ideals of random graphs
  10. Simultaneous failures in coherent systems
  11. Sensitivity analysis
  12. Conclusions

Key Result

Theorem 2.4

Let $\underline{\Delta}$ be a simplicial complex on $\{1,\ldots,n\}$ with the associated Stanley-Reisner ideal $I_{\underline{\Delta}}=\underline{I}_1$. Then for all $k$ we have $I_{\underline{\Delta}_{k}}=\underline{I}_k$.

Figures (5)

  • Figure 1: Graph density vs. size of lcm-lattice for cut ideals of subgraphs of the complete graph on $n$ vertices, $n = 5, \dots, 8$.
  • Figure 2: Size of the lcm-lattice vs. $k$ for $k$-out-of-$15$ ideals, and their linear and circular consecutive variants.
  • Figure 3: Four different simplicial complexes on $7$ vertices with same f-vector and Betti numbers.
  • Figure 4: Persistence diagrams of four simplicial complexes with same f-vector and Betti numbers (see Fig. \ref{['fig:complexesSameF-B']}). Diagrams obtained using the lcm-filtration.
  • Figure 5: Persistence diagrams of four simplicial complexes with same f-vector and Betti numbers (see Fig. \ref{['fig:complexesSameF-B']}). Diagrams obtained using the stepwise lcm-filtration.

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 1
  • Theorem 2.4
  • proof
  • Example 2
  • Definition 3.1
  • Corollary 3.2
  • Lemma 3.3
  • ...and 12 more