Revisiting small-world network models: Exploring technical realizations and the equivalence of the Newman-Watts and Harary models

We address the relatively less known facts on the equivalence and technical realizations surrounding two network models showing the "small-world" property, namely the Newman-Watts and the Harary models. We provide the most accurate (in terms of faithfulness to the original literature) versions of these models to clarify the deviation from them existing in their variants adopted in one of the most popular network analysis packages. The difference in technical realizations of those models could be conceived as minor details, but we discover significantly notable changes caused by the possibly inadvertent modification. For the Harary model, the stochasticity in the original formulation allows a much wider range of the clustering coefficient and the average shortest path length. For the Newman-Watts model, due to the drastically different degree distributions, the clustering coefficient can also be affected, which is verified by our higher-order analytic derivation. During the process, we discover the equivalence of the Newman-Watts (better known in the network science or physics community) and the Harary (better known in the graph theory or mathematics community) models under a specific condition of restricted parity in variables, which would bridge the two relatively independently developed models in different fields. Our result highlights the importance of each detailed step in constructing network models and the possibility of deeply related models, even if they might initially appear distinct in terms of the time period or the academic disciplines from which they emerged.Comment: 11 pages, 5 figures, 1 table, to appear in J. Korean Phys. So

On approximating shortest paths in weighted triangular tessellations

We study the quality of weighted shortest paths when a continuous 2-dimensional space is discretized by a weighted triangular tessellation. In order to evaluate how well the tessellation approximates the 2-dimensional space, we study three types of shortest paths: a weighted shortest path~$\mathit{SP_w}(s,t)$, which is a shortest path from $s$ to $t$ in the space; a weighted shortest vertex path $\mathit{SVP_w}(s,t)$, which is a shortest path where the vertices of the path are vertices of the tessellation; and a weighted shortest grid path~$\mathit{SGP_w}(s,t)$, which is a shortest path whose edges are edges of the tessellation. The ratios $\frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert}$, $\frac{\lVert \mathit{SVP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert}$, $\frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SVP_w}(s,t)\rVert}$ provide estimates on the quality of the approximation. Given any arbitrary weight assignment to the faces of a triangular tessellation, we prove upper and lower bounds on the estimates that are independent of the weight assignment. Our main result is that $\frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} = \frac{2}{\sqrt{3}} \approx 1.15$ in the worst case, and this is tight.Comment: 17 pages, 10 figure

Evolutionary Construction of Geographical Networks with Nearly Optimal Robustness and Efficient Routing Properties

Robust and efficient design of networks on a realistic geographical space is one of the important issues for the realization of dependable communication systems. In this paper, based on a percolation theory and a geometric graph property, we investigate such a design from the following viewpoints: 1) network evolution according to a spatially heterogeneous population, 2) trimodal low degrees for the tolerant connectivity against both failures and attacks, and 3) decentralized routing within short paths. Furthermore, we point out the weakened tolerance by geographical constraints on local cycles, and propose a practical strategy by adding a small fraction of shortcut links between randomly chosen nodes in order to improve the robustness to a similar level to that of the optimal bimodal networks with a larger degree $O(\sqrt{N})$ for the network size $N$. These properties will be useful for constructing future ad-hoc networks in wide-area communications.Comment: 14 pages, 10 figures, 1 tabl

On approximating shortest paths in weighted triangular tessellations

We study the quality of weighted shortest paths when a continuous 2-dimensional space is discretized by a weighted triangular tessellation. In order to evaluate how well the tessellation approximates the 2-dimensional space, we study three types of shortest paths: a weighted shortest path SPw(s,t) , which is a shortest path from s to t in the space; a weighted shortest vertex path SVPw(s,t) , which is a shortest path where the vertices of the path are vertices of the tessellation; and a weighted shortest grid path SGPw(s,t) , which is a shortest path whose edges are edges of the tessellation. The ratios ¿SGPw(s,t)¿¿SPw(s,t)¿ , ¿SVPw(s,t)¿¿SPw(s,t)¿ , ¿SGPw(s,t)¿¿SVPw(s,t)¿ provide estimates on the quality of the approximation. Given any arbitrary weight assignment to the faces of a triangular tessellation, we prove upper and lower bounds on the estimates that are independent of the weight assignment. Our main result is that ¿SGPw(s,t)¿¿SPw(s,t)¿=23v˜1.15 in the worst case, and this is tight.Peer ReviewedPostprint (author's final draft

On approximating shortest paths in weighted triangular tessellations

© 2023 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We study the quality of weighted shortest paths when a continuous 2-dimensional space is discretized by a weighted triangular tessellation. In order to evaluate how well the tessellation approximates the 2-dimensional space, we study three types of shortest paths: a weighted shortest path , which is a shortest path from s to t in the space; a weighted shortest vertex path , which is an any-angle shortest path; and a weighted shortest grid path , which is a shortest path whose edges are edges of the tessellation. Given any arbitrary weight assignment to the faces of a triangular tessellation, thus extending recent results by Bailey et al. (2021) [6], we prove upper and lower bounds on the ratios , , , which provide estimates on the quality of the approximation. It turns out, surprisingly, that our worst-case bounds are independent of any weight assignment. Our main result is that in the worst case, and this is tight. As a corollary, for the weighted any-angle path we obtain the approximation result .P. B. is partially supported by NSERC. G. E., D. O. and R. I. S. are partially supported by H2020-MSCA-RISE project 734922 - CONNECT and project PID2019-104129GB-I00 funded by MCIN/AEI/10.13039/501100011033. G. E. and D. O. are also supported by PIUAH21/IA-062 and CM/JIN/2021-004. G. E. is also funded by an FPU of the Universidad de Alcalá.Peer ReviewedPostprint (published version