Representation of Archimedean Networks and Inclusion: Computational Applications to Percolation and Network Transitions

TL;DR

This paper introduces a new geometric labeling method for Archimedean lattices that improves computational modeling and reveals how percolation phase diagrams relate to lattice inclusion and topology.

Contribution

It presents a systematic labeling framework for Archimedean lattices and demonstrates its application in analyzing percolation phase diagrams and lattice inclusion relations.

Findings

Phase diagrams ordered by lattice inclusion.

Inversion of percolation thresholds observed.

Phase diagram crossings linked to topology.

Abstract

We present an alternative geometric representation for the eleven Archimedean lattices, in which each site and bond is uniquely labeled by an ordered pair of integers and characterized via a modular function. This structured labeling enables efficient and systematic implementation of computational models on these lattices, without relying on ad hoc indexing, and provides a versatile framework for future studies on regular tilings. As an application, we obtain, for each Archimedean lattice, the phase diagrams generated by monomer deposition for the site-bond percolation models known in the literature as $S \cup B$ and $S \cap B$. We show that these diagrams are ordered in phase space according to the partial and total inclusion relations among the lattices, as demonstrated by Parviainen et al. (2003). Furthermore, for the lattice pairs (3.6.3.6)/(3.4.6.4) and $(3^3.4^2)/(3^2.4.3.4)$, we observe an inversion between the pure site and bond percolation thresholds. We show that this phenomenon is linked to the crossing of their respective phase diagrams, highlighting the sensitivity of critical behavior to lattice topology.