In this paper, we propose closed-form analytical expressions to determine the minimum number of driver nodes that is needed to control a specific class of networks. We consider swarm signaling networks with regular out-degree distribution where a fraction $p$ of the links is unavailable. We further apply our method to networks with bi-modal out-degree distributions. Our approximations are validated through intensive simulations. Results show that our approximations have high accuracy when compared with simulation results for both types of out-degree distribution.
{"title":"Controllability of a Class of Swarm Signaling Networks","authors":"Peng Sun, R. Kooij, Roland Bouffanais","doi":"10.1093/comnet/cnac054","DOIUrl":"https://doi.org/10.1093/comnet/cnac054","url":null,"abstract":"In this paper, we propose closed-form analytical expressions to determine the minimum number of driver nodes that is needed to control a specific class of networks. We consider swarm signaling networks with regular out-degree distribution where a fraction $p$ of the links is unavailable. We further apply our method to networks with bi-modal out-degree distributions. Our approximations are validated through intensive simulations. Results show that our approximations have high accuracy when compared with simulation results for both types of out-degree distribution.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":"52 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2022-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80293718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Signed networks and balance theory provide a natural setting for real-world scenarios that show polarization dynamics, positive/negative relationships and political partisanship. For example, they have been proven effective in studying the increasing polarization of the votes in the two chambers of the U.S. Congress from World War II on Andris, Lee, Hamilton, Martino, Gunning & Selden (2015, PLoS ONE, 10, 1–14) and Aref & Neal (2020, Sci. Rep., 10, 1–10). To provide further insights into this particular case study, we propose the application of a pipeline to analyze and visualize a signed graphs configuration based on the exploitation of the corresponding Laplacian matrix spectral properties. The overall methodology is comparable with others based on the frustration index, but it has at least two main advantages: first, it requires a much lower computational cost and second, it allows for a quantitative and visual assessment of how arbitrarily small subgraphs (even single nodes) contribute to the overall balance (or unbalance) of the network. The proposed pipeline allows the exploration of polarization dynamics shown by the U.S. Congress from 1945 to 2020 at different resolution scales. In fact, we are able to spot and point out the influence of some (groups of) congressmen in the overall balance, as well as to observe and explore polarizations evolution of both chambers across the years.
{"title":"Analyzing and visualizing polarization and balance with signed networks: the U.S. Congress case study","authors":"A. Capozzi, Alfonso Semeraro, G. Ruffo","doi":"10.1093/comnet/cnad027","DOIUrl":"https://doi.org/10.1093/comnet/cnad027","url":null,"abstract":"\u0000 Signed networks and balance theory provide a natural setting for real-world scenarios that show polarization dynamics, positive/negative relationships and political partisanship. For example, they have been proven effective in studying the increasing polarization of the votes in the two chambers of the U.S. Congress from World War II on Andris, Lee, Hamilton, Martino, Gunning & Selden (2015, PLoS ONE, 10, 1–14) and Aref & Neal (2020, Sci. Rep., 10, 1–10). To provide further insights into this particular case study, we propose the application of a pipeline to analyze and visualize a signed graphs configuration based on the exploitation of the corresponding Laplacian matrix spectral properties. The overall methodology is comparable with others based on the frustration index, but it has at least two main advantages: first, it requires a much lower computational cost and second, it allows for a quantitative and visual assessment of how arbitrarily small subgraphs (even single nodes) contribute to the overall balance (or unbalance) of the network. The proposed pipeline allows the exploration of polarization dynamics shown by the U.S. Congress from 1945 to 2020 at different resolution scales. In fact, we are able to spot and point out the influence of some (groups of) congressmen in the overall balance, as well as to observe and explore polarizations evolution of both chambers across the years.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":"46 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81449196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-29DOI: 10.48550/arXiv.2208.13610
P. Kritzer
Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-$1$ lattice rule to approximate an $s$-dimensional integral is fully specified by its emph{generating vector} $boldsymbol{z} in mathbb{Z}^s$ and its number of points~$N$. While there are many results on the existence of ``good'' rank-$1$ lattice rules, there are no explicit constructions of good generating vectors for dimensions $s ge 3$. This is why one usually resorts to computer search algorithms. In a recent paper by Ebert et al. in the Journal of Complexity, we showed a component-by-component digit-by-digit (CBC-DBD) construction for good generating vectors of rank-1 lattice rules for integration of functions in weighted Korobov classes. However, the result in that paper was limited to product weights. In the present paper, we shall generalize this result to arbitrary positive weights, thereby answering an open question posed in the paper of Ebert et al. We also include a short section on how the algorithm can be implemented in the case of POD weights, by which we see that the CBC-DBD construction is competitive with the classical CBC construction.
格规则是研究最突出的拟蒙特卡罗方法来近似多元积分。近似$s$维积分的秩- $1$点阵规则由其emph{生成向量}$boldsymbol{z} in mathbb{Z}^s$及其点数$N$完全指定。虽然有许多关于“好”秩- $1$格规则存在的结果，但没有关于维度$s ge 3$的好生成向量的明确构造。这就是为什么人们通常求助于计算机搜索算法。在Ebert等人最近发表在《复杂性杂志》上的一篇论文中，我们展示了一种组件-组件-数字-数字(CBC-DBD)构造，用于加权Korobov类中函数积分的秩-1格规则的良好生成向量。然而，该论文的结果仅限于产品权重。在本文中，我们将这个结果推广到任意正权，从而回答了Ebert等人的论文中提出的一个开放性问题。我们还包括一个关于如何在POD权重的情况下实现算法的简短部分，通过该部分我们可以看到CBC- dbd结构与经典CBC结构相竞争。
{"title":"A note on the CBC-DBD construction of lattice rules with general positive weights","authors":"P. Kritzer","doi":"10.48550/arXiv.2208.13610","DOIUrl":"https://doi.org/10.48550/arXiv.2208.13610","url":null,"abstract":"Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-$1$ lattice rule to approximate an $s$-dimensional integral is fully specified by its emph{generating vector} $boldsymbol{z} in mathbb{Z}^s$ and its number of points~$N$. While there are many results on the existence of ``good'' rank-$1$ lattice rules, there are no explicit constructions of good generating vectors for dimensions $s ge 3$. This is why one usually resorts to computer search algorithms. In a recent paper by Ebert et al. in the Journal of Complexity, we showed a component-by-component digit-by-digit (CBC-DBD) construction for good generating vectors of rank-1 lattice rules for integration of functions in weighted Korobov classes. However, the result in that paper was limited to product weights. In the present paper, we shall generalize this result to arbitrary positive weights, thereby answering an open question posed in the paper of Ebert et al. We also include a short section on how the algorithm can be implemented in the case of POD weights, by which we see that the CBC-DBD construction is competitive with the classical CBC construction.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":"16 1","pages":"101721"},"PeriodicalIF":2.1,"publicationDate":"2022-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75738677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Haji Gul, F. Al-Obeidat, Adnan Amin, Muhammad Mohsin Tahir, Kaizhu Huang
Link prediction in a complex network is a difficult and challenging issue to address. Link prediction tries to better predict relationships, interactions and friendships based on historical knowledge of the complex network graph. Many link prediction techniques exist, including the common neighbour, Adamic-Adar, Katz and Jaccard coefficient, which use node information, local and global routes, and previous knowledge of a complex network to predict the links. These methods are extensively used in various applications because of their interpretability and convenience of use, irrespective of the fact that the majority of these methods were designed for a specific field. This study offers a unique link prediction approach based on the matrix-forest metric and vertex local structural information in a real-world complex network. We empirically examined the proposed link prediction method over 13 real-world network datasets obtained from various sources. Extensive experiments were performed that demonstrated the superior efficacy of the proposed link prediction method compared to other methods and outperformed the existing state-of-the-art in terms of prediction accuracy.
{"title":"Efficient link prediction model for real-world complex networks using matrix-forest metric with local similarity features","authors":"Haji Gul, F. Al-Obeidat, Adnan Amin, Muhammad Mohsin Tahir, Kaizhu Huang","doi":"10.1093/comnet/cnac039","DOIUrl":"https://doi.org/10.1093/comnet/cnac039","url":null,"abstract":"\u0000 Link prediction in a complex network is a difficult and challenging issue to address. Link prediction tries to better predict relationships, interactions and friendships based on historical knowledge of the complex network graph. Many link prediction techniques exist, including the common neighbour, Adamic-Adar, Katz and Jaccard coefficient, which use node information, local and global routes, and previous knowledge of a complex network to predict the links. These methods are extensively used in various applications because of their interpretability and convenience of use, irrespective of the fact that the majority of these methods were designed for a specific field. This study offers a unique link prediction approach based on the matrix-forest metric and vertex local structural information in a real-world complex network. We empirically examined the proposed link prediction method over 13 real-world network datasets obtained from various sources. Extensive experiments were performed that demonstrated the superior efficacy of the proposed link prediction method compared to other methods and outperformed the existing state-of-the-art in terms of prediction accuracy.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":"17 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85414906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Estimating the number of eigenvalues $mu_{[a,b]}$ of a network’s adjacency matrix in a given interval $[a,b]$ is essential in several fields. The straightforward approach consists of calculating all the eigenvalues in $O(n^3)$ (where $n$ is the number of nodes in the network) and then counting the ones that belong to the interval $[a,b]$. Another approach is to use Sylvester’s law of inertia, which also requires $O(n^3)$. Although both methods provide the exact number of eigenvalues in $[a,b]$, their application for large networks is computationally infeasible. Sometimes, an approximation of $mu_{[a,b]}$ is enough. In this case, Chebyshev’s method approximates $mu_{[a,b]}$ in $O(|E|)$ (where $|E|$ is the number of edges). This study presents two alternatives to compute $mu_{[a,b]}$ for locally tree-like networks: edge- and degree-based algorithms. The former presented a better accuracy than Chebyshev’s method. It runs in $O(d|E|)$, where $d$ is the number of iterations. The latter presented slightly lower accuracy but ran linearly ($O(n)$).
{"title":"Efficient eigenvalue counts for tree-like networks","authors":"Grover E. C. Guzman, P. Stadler, André Fujita","doi":"10.1093/comnet/cnac040","DOIUrl":"https://doi.org/10.1093/comnet/cnac040","url":null,"abstract":"\u0000 Estimating the number of eigenvalues $mu_{[a,b]}$ of a network’s adjacency matrix in a given interval $[a,b]$ is essential in several fields. The straightforward approach consists of calculating all the eigenvalues in $O(n^3)$ (where $n$ is the number of nodes in the network) and then counting the ones that belong to the interval $[a,b]$. Another approach is to use Sylvester’s law of inertia, which also requires $O(n^3)$. Although both methods provide the exact number of eigenvalues in $[a,b]$, their application for large networks is computationally infeasible. Sometimes, an approximation of $mu_{[a,b]}$ is enough. In this case, Chebyshev’s method approximates $mu_{[a,b]}$ in $O(|E|)$ (where $|E|$ is the number of edges). This study presents two alternatives to compute $mu_{[a,b]}$ for locally tree-like networks: edge- and degree-based algorithms. The former presented a better accuracy than Chebyshev’s method. It runs in $O(d|E|)$, where $d$ is the number of iterations. The latter presented slightly lower accuracy but ran linearly ($O(n)$).","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":"31 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84515635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The stochastic block model (SBM) is a popular model for community detecting problems. Many community detecting approaches have been proposed, and most of them assume that the number of communities is given previously. However, in practice, the number of communities is often unknown. Plenty of approaches were proposed to estimate the number of communities, but most of them were computationally intensive. Moreover, when outliers exist, there are no approaches to consistently estimate the number of communities. In this article, we propose a fast method based on the eigenvalues of the regularized and normalized adjacency matrix to estimate the number of communities under the SBM with outliers. We show that our method can consistently estimate the number of communities when outliers exist. Moreover, we extend our method to the degree-corrected SBM. We show that our approach is comparable to the other existing approaches in simulations. We also illustrate our approach on four real-world networks.
{"title":"Estimating the number of communities in the stochastic block model with outliers","authors":"Jingsong Xiao, Fei Ye, Weidong Ma, Ying Yang","doi":"10.1093/comnet/cnac042","DOIUrl":"https://doi.org/10.1093/comnet/cnac042","url":null,"abstract":"\u0000 The stochastic block model (SBM) is a popular model for community detecting problems. Many community detecting approaches have been proposed, and most of them assume that the number of communities is given previously. However, in practice, the number of communities is often unknown. Plenty of approaches were proposed to estimate the number of communities, but most of them were computationally intensive. Moreover, when outliers exist, there are no approaches to consistently estimate the number of communities. In this article, we propose a fast method based on the eigenvalues of the regularized and normalized adjacency matrix to estimate the number of communities under the SBM with outliers. We show that our method can consistently estimate the number of communities when outliers exist. Moreover, we extend our method to the degree-corrected SBM. We show that our approach is comparable to the other existing approaches in simulations. We also illustrate our approach on four real-world networks.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":"7 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84026306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-19DOI: 10.48550/arXiv.2208.09108
D. Dung
We investigate the approximation of weighted integrals over $mathbb{R}^d$ for integrands from weighted Sobolev spaces of mixed smoothness. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to $n$ integration nodes for functions from these spaces. In the one-dimensional case $(d=1)$, we obtain the right convergence rate of optimal quadratures. For $d ge 2$, the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain $mathbb{R}^d$.
{"title":"Numerical weighted integration of functions having mixed smoothness","authors":"D. Dung","doi":"10.48550/arXiv.2208.09108","DOIUrl":"https://doi.org/10.48550/arXiv.2208.09108","url":null,"abstract":"We investigate the approximation of weighted integrals over $mathbb{R}^d$ for integrands from weighted Sobolev spaces of mixed smoothness. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to $n$ integration nodes for functions from these spaces. In the one-dimensional case $(d=1)$, we obtain the right convergence rate of optimal quadratures. For $d ge 2$, the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain $mathbb{R}^d$.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":"132 1","pages":"101757"},"PeriodicalIF":2.1,"publicationDate":"2022-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79655330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-03DOI: 10.48550/arXiv.2208.02113
F. Dai, A. Prymak, A. Shadrin, V. Temlyakov, S. Tikhonov
A set $Q$ in $mathbb{Z}_+^d$ is a lower set if $(k_1,dots,k_d)in Q$ implies $(l_1,dots,l_d)in Q$ whenever $0le l_ile k_i$ for all $i$. We derive new and refine known results regarding the cardinality of the lower sets of size $n$ in $mathbb{Z}_+^d$. Next we apply these results for universal discretization of the $L_2$-norm of elements from $n$-dimensional subspaces of trigonometric polynomials generated by lower sets.
{"title":"On cardinality of the lower sets and universal discretization","authors":"F. Dai, A. Prymak, A. Shadrin, V. Temlyakov, S. Tikhonov","doi":"10.48550/arXiv.2208.02113","DOIUrl":"https://doi.org/10.48550/arXiv.2208.02113","url":null,"abstract":"A set $Q$ in $mathbb{Z}_+^d$ is a lower set if $(k_1,dots,k_d)in Q$ implies $(l_1,dots,l_d)in Q$ whenever $0le l_ile k_i$ for all $i$. We derive new and refine known results regarding the cardinality of the lower sets of size $n$ in $mathbb{Z}_+^d$. Next we apply these results for universal discretization of the $L_2$-norm of elements from $n$-dimensional subspaces of trigonometric polynomials generated by lower sets.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":"18 1","pages":"101726"},"PeriodicalIF":2.1,"publicationDate":"2022-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88535279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-14DOI: 10.48550/arXiv.2207.06775
Martin Keller-Ressel, Stephanie Nargang
We introduce L-hydra (landmarked hyperbolic distance recovery and approximation), a method for embedding network- or distance-based data into hyperbolic space, which requires only the distance measurements to a few ‘landmark nodes’. This landmark heuristic makes L-hydra applicable to large-scale graphs and improves upon previously introduced methods. As a mathematical justification, we show that a point configuration in $d$-dimensional hyperbolic space can be perfectly recovered (up to isometry) from distance measurements to just $d+1$ landmarks. We also show that L-hydra solves a two-stage strain-minimization problem, similar to our previous (unlandmarked) method ‘hydra’. Testing on real network data, we show that L-hydra is an order of magnitude faster than the existing hyperbolic embedding methods and scales linearly in the number of nodes. While the embedding error of L-hydra is higher than the error of the existing methods, we introduce an extension, L-hydra+, which outperforms the existing methods in both runtime and embedding quality.
{"title":"Strain-Minimizing Hyperbolic Network Embeddings with Landmarks","authors":"Martin Keller-Ressel, Stephanie Nargang","doi":"10.48550/arXiv.2207.06775","DOIUrl":"https://doi.org/10.48550/arXiv.2207.06775","url":null,"abstract":"\u0000 We introduce L-hydra (landmarked hyperbolic distance recovery and approximation), a method for embedding network- or distance-based data into hyperbolic space, which requires only the distance measurements to a few ‘landmark nodes’. This landmark heuristic makes L-hydra applicable to large-scale graphs and improves upon previously introduced methods. As a mathematical justification, we show that a point configuration in $d$-dimensional hyperbolic space can be perfectly recovered (up to isometry) from distance measurements to just $d+1$ landmarks. We also show that L-hydra solves a two-stage strain-minimization problem, similar to our previous (unlandmarked) method ‘hydra’. Testing on real network data, we show that L-hydra is an order of magnitude faster than the existing hyperbolic embedding methods and scales linearly in the number of nodes. While the embedding error of L-hydra is higher than the error of the existing methods, we introduce an extension, L-hydra+, which outperforms the existing methods in both runtime and embedding quality.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":"30 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2022-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86713283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper introduces Haros graphs, a construction which provides a graph-theoretical representation of real numbers in the unit interval reached via paths in the Farey binary tree. We show how the topological structure of Haros graphs yields a natural classification of the reals numbers into a hierarchy of families. To unveil such classification, we introduce an entropic functional on these graphs and show that it can be expressed, thanks to its fractal nature, in terms of a generalised de Rham curve. We show that this entropy reaches a global maximum at the reciprocal of the Golden number and otherwise displays a rich hierarchy of local maxima and minima that relate to specific families of irrationals (noble numbers) and rationals, overall providing an exotic classification and representation of the reals numbers according to entropic principles. We close the paper with a number of conjectures and outline a research programme on Haros graphs.
{"title":"Haros graphs: an exotic representation of real numbers","authors":"Jorge Calero-Sanz, B. Luque, L. Lacasa","doi":"10.1093/comnet/cnac043","DOIUrl":"https://doi.org/10.1093/comnet/cnac043","url":null,"abstract":"This paper introduces Haros graphs, a construction which provides a graph-theoretical representation of real numbers in the unit interval reached via paths in the Farey binary tree. We show how the topological structure of Haros graphs yields a natural classification of the reals numbers into a hierarchy of families. To unveil such classification, we introduce an entropic functional on these graphs and show that it can be expressed, thanks to its fractal nature, in terms of a generalised de Rham curve. We show that this entropy reaches a global maximum at the reciprocal of the Golden number and otherwise displays a rich hierarchy of local maxima and minima that relate to specific families of irrationals (noble numbers) and rationals, overall providing an exotic classification and representation of the reals numbers according to entropic principles. We close the paper with a number of conjectures and outline a research programme on Haros graphs.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":"650 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2022-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77291359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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