Abstract Higher-order interactions, that is, interactions among the units of group size greater than two, are a fundamental structural feature of a variety of complex systems across the scale. Simplicial complexes are combinatorial objects that can capture and model the higher-order interactions present in a given complex system and thus represent the complex system as a higher-order network comprising simplices. In this work, a given simplicial complex is viewed as a finite union of d-exclusive simplicial complexes. Thus, to represent a complex system as a higher-order network given by a simplicial complex that captures all orders of interactions present in the system, a family of symmetric adjacency tensors A(d) of dimension d + 1 and appropriate order has been used. Each adjacency tensor A(d) represents a d-exclusive simplicial complex and for d≥2 it represents exclusively higher-order interactions of the system. For characterizing the structure of d-exclusive simplicial complexes, the notion of generalized structural centrality indices namely, generalized betweenness centrality and generalized closeness centrality has been established by developing the concepts of generalized walk and generalized distance in the simplicial complex. Generalized centrality indices quantify the contribution of δ-simplices in any d-exclusive simplicial complex Δ, where δ<d and if d≥2, it describes the contribution of δ-faces to the higher-order interactions of Δ. These generalized centrality indices provide local structural descriptions, which lead to mesoscale insights into the simplicial complex that comprises the higher-order network. An important theorem providing a general technique for the characterization of connectedness in d-exclusive simplicial complexes in terms of irreducibility of its adjacency tensor has been established. The concepts developed in this work together with concepts of generalized simplex deletion in d-exclusive simplicial complexes have been illustrated using examples. The effect of deletions on the generalized centralities of the complexes in the examples has been discussed.
{"title":"Some generalized centralities in higher-order networks represented by simplicial complexes","authors":"Udit Raj, Sudeepto Bhattacharya","doi":"10.1093/comnet/cnad032","DOIUrl":"https://doi.org/10.1093/comnet/cnad032","url":null,"abstract":"Abstract Higher-order interactions, that is, interactions among the units of group size greater than two, are a fundamental structural feature of a variety of complex systems across the scale. Simplicial complexes are combinatorial objects that can capture and model the higher-order interactions present in a given complex system and thus represent the complex system as a higher-order network comprising simplices. In this work, a given simplicial complex is viewed as a finite union of d-exclusive simplicial complexes. Thus, to represent a complex system as a higher-order network given by a simplicial complex that captures all orders of interactions present in the system, a family of symmetric adjacency tensors A(d) of dimension d + 1 and appropriate order has been used. Each adjacency tensor A(d) represents a d-exclusive simplicial complex and for d≥2 it represents exclusively higher-order interactions of the system. For characterizing the structure of d-exclusive simplicial complexes, the notion of generalized structural centrality indices namely, generalized betweenness centrality and generalized closeness centrality has been established by developing the concepts of generalized walk and generalized distance in the simplicial complex. Generalized centrality indices quantify the contribution of δ-simplices in any d-exclusive simplicial complex Δ, where δ&lt;d and if d≥2, it describes the contribution of δ-faces to the higher-order interactions of Δ. These generalized centrality indices provide local structural descriptions, which lead to mesoscale insights into the simplicial complex that comprises the higher-order network. An important theorem providing a general technique for the characterization of connectedness in d-exclusive simplicial complexes in terms of irreducibility of its adjacency tensor has been established. The concepts developed in this work together with concepts of generalized simplex deletion in d-exclusive simplicial complexes have been illustrated using examples. The effect of deletions on the generalized centralities of the complexes in the examples has been discussed.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135362900","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}
Abstract The inference of reliable and meaningful connectivity information from weights representing the affinity between nodes in a graph is an outstanding problem in network science. Usually, this is achieved by simply thresholding the edge weights to distinguish true links from false ones and to obtain a sparse set of connections. Tools developed in statistical mechanics have provided particularly effective ways to locate the optimal threshold so as to preserve the statistical properties of the network structure. Thermodynamic analogies together with statistical mechanical ensembles have been proven to be useful in analysing edge-weighted networks. To extend this work, in this article, we use a statistical mechanical model to describe the probability distribution for edge weights. This models the distribution of edge weights using a mixture of Gamma distributions. Using a two-component Gamma mixture model with components describing the edge and non-edge weight distributions, we use the Expectation–Maximization algorithm to estimate the corresponding Gamma distribution parameters and mixing proportions. This gives the optimal threshold to convert weighted networks to sets of binary-valued connections. Numerical analysis shows that it provides a new way to describe the edge weight probability. Furthermore, using a physical analogy in which the weights are the energies of molecules in a solid, the probability density function for nodes is identical to the degree distribution resulting from a uniform weight on edges. This provides an alternative way to study the degree distribution with the nodal probability function in unweighted networks. We observe a phase transition in the low-temperature region, corresponding to a structural transition caused by applying the threshold. Experimental results on real-world weighted and unweighted networks reveal an improved performance for inferring binary edge connections from edge weights.
{"title":"Statistical structural inference from edge weights using a mixture of gamma distributions","authors":"Jianjia Wang, Edwin R Hancock","doi":"10.1093/comnet/cnad038","DOIUrl":"https://doi.org/10.1093/comnet/cnad038","url":null,"abstract":"Abstract The inference of reliable and meaningful connectivity information from weights representing the affinity between nodes in a graph is an outstanding problem in network science. Usually, this is achieved by simply thresholding the edge weights to distinguish true links from false ones and to obtain a sparse set of connections. Tools developed in statistical mechanics have provided particularly effective ways to locate the optimal threshold so as to preserve the statistical properties of the network structure. Thermodynamic analogies together with statistical mechanical ensembles have been proven to be useful in analysing edge-weighted networks. To extend this work, in this article, we use a statistical mechanical model to describe the probability distribution for edge weights. This models the distribution of edge weights using a mixture of Gamma distributions. Using a two-component Gamma mixture model with components describing the edge and non-edge weight distributions, we use the Expectation–Maximization algorithm to estimate the corresponding Gamma distribution parameters and mixing proportions. This gives the optimal threshold to convert weighted networks to sets of binary-valued connections. Numerical analysis shows that it provides a new way to describe the edge weight probability. Furthermore, using a physical analogy in which the weights are the energies of molecules in a solid, the probability density function for nodes is identical to the degree distribution resulting from a uniform weight on edges. This provides an alternative way to study the degree distribution with the nodal probability function in unweighted networks. We observe a phase transition in the low-temperature region, corresponding to a structural transition caused by applying the threshold. Experimental results on real-world weighted and unweighted networks reveal an improved performance for inferring binary edge connections from edge weights.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135369554","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}
Abstract The importance of fertilizers to agricultural production is undeniable, and most economies rely on international trade for fertilizer use. The stability of fertilizer trade networks is fundamental to food security. However, quantifying the temporal stability of a fast-growing system, such as the international fertilizer trade, requires a multi-dimensional perception. Therefore, we propose a new method, namely the structural inheritance index, to distinguish the stability of the existing structure from the influence of the growing process. The well-known mutual information and Jaccard index are calculated for comparison. We use the three methods to measure the temporal stability of the overall network and different functional sub-networks of the three fertilizer nutrients N, P and K from 1990 to 2018. The international N, P and K trade systems all have a trend of increasing stability with the process of globalization. The existing structure in the fertilizer trading system has shown high stability since 1990, implying that the instability calculated by the Jaccard index in the early stage comes from the emergence of new trade. The stability of the K trade network is concentrated in large sub-networks, meaning that it is vulnerable to extreme events. The stable medium sub-network helps the N trade become the most stable nutrient trade. The P trade is clearly in the role of a catch-up player. Based on the analysis of the comparisons of three indicators, we concluded that all three nutrient trade networks enter a steady state.
{"title":"Quantifying the temporal stability of international fertilizer trade networks","authors":"Mu-Yao Li, Li Wang, Wen-Jie Xie, Wei-Xing Zhou","doi":"10.1093/comnet/cnad037","DOIUrl":"https://doi.org/10.1093/comnet/cnad037","url":null,"abstract":"Abstract The importance of fertilizers to agricultural production is undeniable, and most economies rely on international trade for fertilizer use. The stability of fertilizer trade networks is fundamental to food security. However, quantifying the temporal stability of a fast-growing system, such as the international fertilizer trade, requires a multi-dimensional perception. Therefore, we propose a new method, namely the structural inheritance index, to distinguish the stability of the existing structure from the influence of the growing process. The well-known mutual information and Jaccard index are calculated for comparison. We use the three methods to measure the temporal stability of the overall network and different functional sub-networks of the three fertilizer nutrients N, P and K from 1990 to 2018. The international N, P and K trade systems all have a trend of increasing stability with the process of globalization. The existing structure in the fertilizer trading system has shown high stability since 1990, implying that the instability calculated by the Jaccard index in the early stage comes from the emergence of new trade. The stability of the K trade network is concentrated in large sub-networks, meaning that it is vulnerable to extreme events. The stable medium sub-network helps the N trade become the most stable nutrient trade. The P trade is clearly in the role of a catch-up player. Based on the analysis of the comparisons of three indicators, we concluded that all three nutrient trade networks enter a steady state.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135364425","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}
Aruane M Pineda, Francisco A Rodrigues, Caroline L Alves, Michael Möckel, Thaise G L de O Toutain, Joel Augusto Moura Porto
Abstract Heart disease, also known as cardiovascular disease, encompasses a variety of heart conditions that can result in sudden death for many people. Examples include high blood pressure, ischaemia, irregular heartbeats and pericardial effusion. Electrocardiogram (ECG) signal analysis is frequently used to diagnose heart diseases, providing crucial information on how the heart functions. To analyse ECG signals, quantile graphs (QGs) is a method that maps a time series into a network based on the time-series fluctuation proprieties. Here, we demonstrate that the QG methodology can differentiate younger and older patients. Furthermore, we construct networks from the QG method and use machine-learning algorithms to perform the automatic diagnosis, obtaining high accuracy. Indeed, we verify that this method can automatically detect changes in the ECG of elderly and young subjects, with the highest classification performance for the adjacency matrix with a mean area under the receiver operating characteristic curve close to one. The findings reported here confirm the QG method’s utility in deciphering intricate, nonlinear signals like those produced by patient ECGs. Furthermore, we find a more significant, more connected and lower distribution of information networks associated with the networks from ECG data of the elderly compared with younger subjects. Finally, this methodology can be applied to other ECG data related to other diseases, such as ischaemia.
{"title":"Analysis of quantile graphs in EGC data from elderly and young individuals using machine learning and deep learning","authors":"Aruane M Pineda, Francisco A Rodrigues, Caroline L Alves, Michael Möckel, Thaise G L de O Toutain, Joel Augusto Moura Porto","doi":"10.1093/comnet/cnad030","DOIUrl":"https://doi.org/10.1093/comnet/cnad030","url":null,"abstract":"Abstract Heart disease, also known as cardiovascular disease, encompasses a variety of heart conditions that can result in sudden death for many people. Examples include high blood pressure, ischaemia, irregular heartbeats and pericardial effusion. Electrocardiogram (ECG) signal analysis is frequently used to diagnose heart diseases, providing crucial information on how the heart functions. To analyse ECG signals, quantile graphs (QGs) is a method that maps a time series into a network based on the time-series fluctuation proprieties. Here, we demonstrate that the QG methodology can differentiate younger and older patients. Furthermore, we construct networks from the QG method and use machine-learning algorithms to perform the automatic diagnosis, obtaining high accuracy. Indeed, we verify that this method can automatically detect changes in the ECG of elderly and young subjects, with the highest classification performance for the adjacency matrix with a mean area under the receiver operating characteristic curve close to one. The findings reported here confirm the QG method’s utility in deciphering intricate, nonlinear signals like those produced by patient ECGs. Furthermore, we find a more significant, more connected and lower distribution of information networks associated with the networks from ECG data of the elderly compared with younger subjects. Finally, this methodology can be applied to other ECG data related to other diseases, such as ischaemia.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135253942","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}
Abstract We consider several families of network centrality measures induced by graph kernels, which include some well-known measures and many new ones. The Self-consistency and Bridge axioms, which appeared earlier in the literature, are closely related to certain kernels and one of the families. We obtain a necessary and sufficient condition for Self-consistency, a sufficient condition for the Bridge axiom, indicate specific measures that satisfy these axioms and show that under some additional conditions they are incompatible. PageRank centrality applied to undirected networks violates most conditions under study and has a property that according to some authors is ‘hard to imagine’ for a centrality measure. We explain this phenomenon. Adopting the Self-consistency or Bridge axiom leads to a drastic reduction in survey time in the culling method designed to select the most appropriate centrality measures.
{"title":"Selection of centrality measures using Self-consistency and Bridge axioms","authors":"Pavel Chebotarev","doi":"10.1093/comnet/cnad035","DOIUrl":"https://doi.org/10.1093/comnet/cnad035","url":null,"abstract":"Abstract We consider several families of network centrality measures induced by graph kernels, which include some well-known measures and many new ones. The Self-consistency and Bridge axioms, which appeared earlier in the literature, are closely related to certain kernels and one of the families. We obtain a necessary and sufficient condition for Self-consistency, a sufficient condition for the Bridge axiom, indicate specific measures that satisfy these axioms and show that under some additional conditions they are incompatible. PageRank centrality applied to undirected networks violates most conditions under study and has a property that according to some authors is ‘hard to imagine’ for a centrality measure. We explain this phenomenon. Adopting the Self-consistency or Bridge axiom leads to a drastic reduction in survey time in the culling method designed to select the most appropriate centrality measures.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135364437","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}
Shenwen Chen, Yisen Wang, Ziquan Liu, Wenbo Du, Lei Zheng, Runran Liu
Abstract Scientific collaboration is an essential aspect of the educational field, offering significant reference value in resource sharing and policy making. With the increasing diversity and inter-disciplinary nature of educational research, understanding scientific collaboration within and between various subfields is crucial for its development. This article employs topic modelling to extract educational research topics from publication metadata obtained from 265 scientific journals spanning the period from 2000 to 2021. We construct a multilayer co-authorship network whose layers represent the scientific collaboration in different subfields. The topological properties of the layers are compared, highlighting the differences and common features of scientific collaboration between hot and cold topics, with the main difference being the existence of a significant largest connected component. Further, the cross-layer cooperation behaviour is investigated by studying the structural measures of the multilayer network and reveals authors’ inclination to collaborate with familiar individuals in familiar subfields. Moreover, the relationships between the authors’ features on the network topology and their H-index are investigated. The results emphasize the significance of establishing a clear research direction to enhance the academic reputation of authors, as well as the importance of cross-layer collaboration for expanding their research groups. Finally, based on the above results, we propose a multilayer network generation model of scientific collaboration and verify its validity.
{"title":"Analysing educational scientific collaboration through multilayer networks: patterns, impact and network generation model","authors":"Shenwen Chen, Yisen Wang, Ziquan Liu, Wenbo Du, Lei Zheng, Runran Liu","doi":"10.1093/comnet/cnad033","DOIUrl":"https://doi.org/10.1093/comnet/cnad033","url":null,"abstract":"Abstract Scientific collaboration is an essential aspect of the educational field, offering significant reference value in resource sharing and policy making. With the increasing diversity and inter-disciplinary nature of educational research, understanding scientific collaboration within and between various subfields is crucial for its development. This article employs topic modelling to extract educational research topics from publication metadata obtained from 265 scientific journals spanning the period from 2000 to 2021. We construct a multilayer co-authorship network whose layers represent the scientific collaboration in different subfields. The topological properties of the layers are compared, highlighting the differences and common features of scientific collaboration between hot and cold topics, with the main difference being the existence of a significant largest connected component. Further, the cross-layer cooperation behaviour is investigated by studying the structural measures of the multilayer network and reveals authors’ inclination to collaborate with familiar individuals in familiar subfields. Moreover, the relationships between the authors’ features on the network topology and their H-index are investigated. The results emphasize the significance of establishing a clear research direction to enhance the academic reputation of authors, as well as the importance of cross-layer collaboration for expanding their research groups. Finally, based on the above results, we propose a multilayer network generation model of scientific collaboration and verify its validity.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135363082","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}
Abstract Temporal networks are known as the most important tools for representing and storing dynamic systems. This type of network accurately demonstrates all the dynamic changes that occur in a dynamic system. In different applications of dynamic systems, different representation of network models has been used to represent temporal networks. In the last decade, controllability in dynamic systems has become one of the most important challenges in this field. Controllability means the transfer of the network from an initial state to a desired final state in a certain period of time. The most common representation of network model used in control processes is the layered model. But this model has a high overhead, and on the other hand, it slows down the network control processes. In this article, we have proposed a new model for storing and representing temporal networks, which uses a tree structure to save all dynamics of network. Considering that in the proposed model only essential network control information is stored, this model has a very low data overhead compared to the layered model, and this makes the control processes run at a higher speed.
{"title":"Novel network representation model for improving controllability processes on temporal networks","authors":"Yan Liu, Jianhang Zeng, Yue Xu","doi":"10.1093/comnet/cnad036","DOIUrl":"https://doi.org/10.1093/comnet/cnad036","url":null,"abstract":"Abstract Temporal networks are known as the most important tools for representing and storing dynamic systems. This type of network accurately demonstrates all the dynamic changes that occur in a dynamic system. In different applications of dynamic systems, different representation of network models has been used to represent temporal networks. In the last decade, controllability in dynamic systems has become one of the most important challenges in this field. Controllability means the transfer of the network from an initial state to a desired final state in a certain period of time. The most common representation of network model used in control processes is the layered model. But this model has a high overhead, and on the other hand, it slows down the network control processes. In this article, we have proposed a new model for storing and representing temporal networks, which uses a tree structure to save all dynamics of network. Considering that in the proposed model only essential network control information is stored, this model has a very low data overhead compared to the layered model, and this makes the control processes run at a higher speed.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135364422","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}
{"title":"Correction to: Analysis of quantile graphs in EGC data from elderly and young individuals using machine learning and deep learning","authors":"","doi":"10.1093/comnet/cnad041","DOIUrl":"https://doi.org/10.1093/comnet/cnad041","url":null,"abstract":"","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135368983","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 : 2023-07-01DOI: 10.48550/arXiv.2307.12461
Tong Mao, Ding-Xuan Zhou
Neural networks activated by the rectified linear unit (ReLU) play a central role in the recent development of deep learning. The topic of approximating functions from H"older spaces by these networks is crucial for understanding the efficiency of the induced learning algorithms. Although the topic has been well investigated in the setting of deep neural networks with many layers of hidden neurons, it is still open for shallow networks having only one hidden layer. In this paper, we provide rates of uniform approximation by these networks. We show that ReLU shallow neural networks with $m$ hidden neurons can uniformly approximate functions from the H"older space $W_infty^r([-1, 1]^d)$ with rates $O((log m)^{frac{1}{2} +d}m^{-frac{r}{d}frac{d+2}{d+4}})$ when $r
{"title":"Rates of Approximation by ReLU Shallow Neural Networks","authors":"Tong Mao, Ding-Xuan Zhou","doi":"10.48550/arXiv.2307.12461","DOIUrl":"https://doi.org/10.48550/arXiv.2307.12461","url":null,"abstract":"Neural networks activated by the rectified linear unit (ReLU) play a central role in the recent development of deep learning. The topic of approximating functions from H\"older spaces by these networks is crucial for understanding the efficiency of the induced learning algorithms. Although the topic has been well investigated in the setting of deep neural networks with many layers of hidden neurons, it is still open for shallow networks having only one hidden layer. In this paper, we provide rates of uniform approximation by these networks. We show that ReLU shallow neural networks with $m$ hidden neurons can uniformly approximate functions from the H\"older space $W_infty^r([-1, 1]^d)$ with rates $O((log m)^{frac{1}{2} +d}m^{-frac{r}{d}frac{d+2}{d+4}})$ when $r<d/2 +2$. Such rates are very close to the optimal one $O(m^{-frac{r}{d}})$ in the sense that $frac{d+2}{d+4}$ is close to $1$, when the dimension $d$ is large.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80563859","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}
Zhao-Yan Li, Jing Zhang, Guozhong Zheng, Li Chen, Jiqiang Zhang, Weiran Cai
While network science has become an indispensable tool for studying complex systems, the conventional use of pairwise links often shows limitations in describing high-order interactions properly. Hypergraphs, where each edge can connect more than two nodes, have thus become a new paradigm in network science. Yet, we are still in lack of models linking network growth and hyperedge expansion, both of which are commonly observable in the real world. Here, we propose a generative hypergraph model by employing the preferential attachment mechanism in both nodes and hyperedge formation. The model can produce bi-heterogeneity, exhibiting scale-free distributions in both hyperdegree and hyperedge size. We provide a mean-field treatment that gives the expression of the two scaling exponents, which agree with the numerical simulations. Our model may help to understand the networked systems showing both types of heterogeneity and facilitate the study of complex dynamics thereon.
{"title":"A generative hypergraph model for double heterogeneity","authors":"Zhao-Yan Li, Jing Zhang, Guozhong Zheng, Li Chen, Jiqiang Zhang, Weiran Cai","doi":"10.1093/comnet/cnad048","DOIUrl":"https://doi.org/10.1093/comnet/cnad048","url":null,"abstract":"While network science has become an indispensable tool for studying complex systems, the conventional use of pairwise links often shows limitations in describing high-order interactions properly. Hypergraphs, where each edge can connect more than two nodes, have thus become a new paradigm in network science. Yet, we are still in lack of models linking network growth and hyperedge expansion, both of which are commonly observable in the real world. Here, we propose a generative hypergraph model by employing the preferential attachment mechanism in both nodes and hyperedge formation. The model can produce bi-heterogeneity, exhibiting scale-free distributions in both hyperdegree and hyperedge size. We provide a mean-field treatment that gives the expression of the two scaling exponents, which agree with the numerical simulations. Our model may help to understand the networked systems showing both types of heterogeneity and facilitate the study of complex dynamics thereon.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139368856","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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