Complexity Analysis of “Small-World Networks” and Spanning Tree Entropy

Mokhlissi, Raihana; Lotfi, Dounia; Debnath, Joyati; Marraki, Mohamed El

doi:10.1007/978-3-319-50901-3_16

Cited by 2 publications

(2 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The novelty of our work is to analytically investigate two generalized families of small-world networks, called the Small-World Exponential network. See, e.g., Mokhlissi, Lotfi, Debnath and El Marraki [13] and Liu, Dolgushev, Qi and Zhang [14], and the Koch network. See, e.g., Zhang, Zhou, Xie, Chen, Lin and Guan [15] and Zhang, Gao, Chen, Zhou, Zhang, and Guan [16].…”

Section: Introductionmentioning

confidence: 99%

The Evaluation of the Number and the Entropy of Spanning Trees on Generalized Small-World Networks

Mokhlissi

Lotfi

Debnath

et al. 2018

Journal of Applied Mathematics

Self Cite

View full text Add to dashboard Cite

Spanning trees have been widely investigated in many aspects of mathematics: theoretical computer science, combinatorics, so on. An important issue is to compute the number of these spanning trees. This number remains a challenge, particularly for large and complex networks. As a model of complex networks, we study two families of generalized small-world networks, namely, the Small-World Exponential and the Koch networks, by changing the size and the dimension of the cyclic subgraphs. We introduce their construction and their structural properties which are built in an iterative way. We propose a decomposition method for counting their number of spanning trees and we obtain the exact formulas, which are then verified by numerical simulations. From this number, we find their spanning tree entropy, which is lower than that of the other networks having the same average degree. This entropy allows quantifying the robustness of the networks and characterizing their structures.

show abstract

Section: Introductionmentioning

confidence: 99%

The Evaluation of the Number and the Entropy of Spanning Trees on Generalized Small-World Networks

Mokhlissi

Lotfi

Debnath

et al. 2018

Journal of Applied Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…For example, Lyons, Peled, and Schramm [14] study the asymptotic growth of the complexity of the largest connected component, known as the giant component, in an Erdős-Rényi random network. Also, in [16] Mokhlissi, Lotfi, Debnath, and Marraki study the complexity of special classes of small-world networks. Small-world networks have been shown to provide good models for many real-world phenomena [2].…”

Section: Introductionmentioning

confidence: 99%

Volume Bounds for the Phase-locking Region in the Kuramoto Model with Asymmetric Coupling

Ferguson¹

2018

Preprint

View full text Add to dashboard Cite

The Kuramoto model is a system of nonlinear differential equations that models networks of coupled oscillators and is often used to study synchronization among them. It has been observed that if the natural frequencies of the oscillators are similar they will phase-lock, meaning that they oscillate at a common frequency with fixed phase differences. Conversely, we do not observe this behavior when the natural frequencies are very dissimilar. In [7] Bronski and the author gave upper and lower bounds for the volume of the set of frequencies exhibiting phase-locking behavior. This was done under the assumption that any two oscillators affect each other with equal strength. In this paper the author generalizes these upper and lower bounds by removing this assumption. Similar to [7] where the upper and lower bounds are sums over spanning trees of the network, our generalized upper and lower bounds are sums over certain directed subgraphs of the network. In particular, our lower bound is a sum over directed spanning trees. Finally, we numerically simulate the dependence of the number of directed spanning trees on the presence of certain two edge motifs in the network and compare this dependence with that of the synchronization of oscillator models found in [26].

show abstract

Complexity Analysis of “Small-World Networks” and Spanning Tree Entropy

Cited by 2 publications

References 14 publications

The Evaluation of the Number and the Entropy of Spanning Trees on Generalized Small-World Networks

The Evaluation of the Number and the Entropy of Spanning Trees on Generalized Small-World Networks

Volume Bounds for the Phase-locking Region in the Kuramoto Model with Asymmetric Coupling

Contact Info

Product

Resources

About