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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
Martina Contisciani;Hadiseh Safdari;Caterina De Bacco
To unravel the driving patterns of networks, the most popular models rely on community detection algorithms. However, these approaches are generally unable to reproduce the structural features of the network. Therefore, attempts are always made to develop models that incorporate these network properties beside the community structure. In this article, we present a probabilistic generative model and an efficient algorithm to both perform community detection and capture reciprocity in networks. Our approach jointly models pairs of edges with exact two-edge joint distributions. In addition, it provides closed-form analytical expressions for both marginal and conditional distributions. We validate our model on synthetic data in recovering communities, edge prediction tasks and generating synthetic networks that replicate the reciprocity values observed in real networks. We also highlight these findings on two real datasets that are relevant for social scientists and behavioural ecologists. Our method overcomes the limitations of both standard algorithms and recent models that incorporate reciprocity through a pseudo-likelihood approximation. The inference of the model parameters is implemented by the efficient and scalable expectation–maximization algorithm, as it exploits the sparsity of the dataset. We provide an open-source implementation of the code online.
{"title":"Community detection and reciprocity in networks by jointly modelling pairs of edges","authors":"Martina Contisciani;Hadiseh Safdari;Caterina De Bacco","doi":"10.1093/comnet/cnac034","DOIUrl":"https://doi.org/10.1093/comnet/cnac034","url":null,"abstract":"To unravel the driving patterns of networks, the most popular models rely on community detection algorithms. However, these approaches are generally unable to reproduce the structural features of the network. Therefore, attempts are always made to develop models that incorporate these network properties beside the community structure. In this article, we present a probabilistic generative model and an efficient algorithm to both perform community detection and capture reciprocity in networks. Our approach jointly models pairs of edges with exact two-edge joint distributions. In addition, it provides closed-form analytical expressions for both marginal and conditional distributions. We validate our model on synthetic data in recovering communities, edge prediction tasks and generating synthetic networks that replicate the reciprocity values observed in real networks. We also highlight these findings on two real datasets that are relevant for social scientists and behavioural ecologists. Our method overcomes the limitations of both standard algorithms and recent models that incorporate reciprocity through a pseudo-likelihood approximation. The inference of the model parameters is implemented by the efficient and scalable expectation–maximization algorithm, as it exploits the sparsity of the dataset. We provide an open-source implementation of the code online.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/iel7/8016804/10070447/10070458.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49943942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recent developments in network neuroscience have highlighted the importance of developing techniques for analysing and modelling brain networks. A particularly powerful approach for studying complex neural systems is to formulate generative models that use wiring rules to synthesize networks closely resembling the topology of a given connectome. Successful models can highlight the principles by which a network is organized (identify structural features that arise from wiring rules versus those that emerge) and potentially uncover the mechanisms by which it grows and develops. Previous research has shown that such models can validate the effectiveness of spatial embedding and other (non-spatial) wiring rules in shaping the network topology of the human connectome. In this research, we propose variants of the action-based model that combine a variety of generative factors capable of explaining the topology of the human connectome. We test the descriptive validity of our models by evaluating their ability to explain between-subject variability. Our analysis provides evidence that geometric constraints are vital for connectivity between brain regions, and an action-based model relying on both topological and geometric properties can account for between-subject variability in structural network properties. Further, we test correlations between parameters of subject-optimized models and various measures of cognitive ability and find that higher cognitive ability is associated with an individual's tendency to form long-range or non-local connections.
{"title":"Investigating cognitive ability using action-based models of structural brain networks","authors":"Viplove Arora;Enrico Amico;Joaquín Goñi;Mario Ventresca","doi":"10.1093/comnet/cnac037","DOIUrl":"https://doi.org/10.1093/comnet/cnac037","url":null,"abstract":"Recent developments in network neuroscience have highlighted the importance of developing techniques for analysing and modelling brain networks. A particularly powerful approach for studying complex neural systems is to formulate generative models that use wiring rules to synthesize networks closely resembling the topology of a given connectome. Successful models can highlight the principles by which a network is organized (identify structural features that arise from wiring rules versus those that emerge) and potentially uncover the mechanisms by which it grows and develops. Previous research has shown that such models can validate the effectiveness of spatial embedding and other (non-spatial) wiring rules in shaping the network topology of the human connectome. In this research, we propose variants of the action-based model that combine a variety of generative factors capable of explaining the topology of the human connectome. We test the descriptive validity of our models by evaluating their ability to explain between-subject variability. Our analysis provides evidence that geometric constraints are vital for connectivity between brain regions, and an action-based model relying on both topological and geometric properties can account for between-subject variability in structural network properties. Further, we test correlations between parameters of subject-optimized models and various measures of cognitive ability and find that higher cognitive ability is associated with an individual's tendency to form long-range or non-local connections.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49943939","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}
In the Internet era, rumours will spread rapidly in the network and hinder the development of all aspects of society. To create a harmonious network environment, it is essential to take punitive measures against malicious rumour mongers on social platforms. Take the measure of forbidden as an example. The forbidden one may stop spreading rumours because of being punished, or he may become a disseminator again because of paranoia. Other people who know rumours may become alert and stop propagating rumours or temporarily forget rumours. And therefore, the forbidden state is added to describe the above phenomenon, and the SIFR (Ignorant–Disseminator–Forbidden–Restorer) model is proposed. Taking the vigilance and paranoia derived from punishment measures into account, the connection edges from the forbidden to the disseminator and from the disseminator to the restorer are increased in this model. And then, the stability of SIFR model is proved by using the basic regeneration number and Routh–Hurwitz stability theorem. The simulation results demonstrate that individual paranoia may do harm to the control of rumour dissemination. While the punishment mechanism, individual forgetting mechanism and vigilance can effectively curb the spread of rumours.
{"title":"Online dynamic rumour propagation model considering punishment mechanism and individual personality characteristics","authors":"Chengai Sun;Donghang Qiao;Liqing Qiu","doi":"10.1093/comnet/cnac038","DOIUrl":"https://doi.org/10.1093/comnet/cnac038","url":null,"abstract":"In the Internet era, rumours will spread rapidly in the network and hinder the development of all aspects of society. To create a harmonious network environment, it is essential to take punitive measures against malicious rumour mongers on social platforms. Take the measure of forbidden as an example. The forbidden one may stop spreading rumours because of being punished, or he may become a disseminator again because of paranoia. Other people who know rumours may become alert and stop propagating rumours or temporarily forget rumours. And therefore, the forbidden state is added to describe the above phenomenon, and the SIFR (Ignorant–Disseminator–Forbidden–Restorer) model is proposed. Taking the vigilance and paranoia derived from punishment measures into account, the connection edges from the forbidden to the disseminator and from the disseminator to the restorer are increased in this model. And then, the stability of SIFR model is proved by using the basic regeneration number and Routh–Hurwitz stability theorem. The simulation results demonstrate that individual paranoia may do harm to the control of rumour dissemination. While the punishment mechanism, individual forgetting mechanism and vigilance can effectively curb the spread of rumours.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49943397","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}
We introduce hub centrality and study the relation between hub centrality and the degree of each node in the networks. We discover and verify a universal relation between them in various networks generated by the growth method, but the relation is not applied to real-world networks due to the rich-club phenomenon and the presence of local hubs. Through the study of a targeted attack and overload cascading failure, we prove that hub centrality is a meaningful parameter that gives extra insight beyond degree in real-world networks. Especially, we show that the local hubs occupy key positions in real-world networks with higher probabilities to incur global cascading failure. Therefore, we conclude that networks generated by the growth method, which do not include local hubs, have inevitable limitations to describe real-world networks.
{"title":"Universal behaviour of the growth method and importance of local hubs in cascading failure","authors":"Wonhee Jeong;Unjong Yu","doi":"10.1093/comnet/cnac028","DOIUrl":"https://doi.org/10.1093/comnet/cnac028","url":null,"abstract":"We introduce hub centrality and study the relation between hub centrality and the degree of each node in the networks. We discover and verify a universal relation between them in various networks generated by the growth method, but the relation is not applied to real-world networks due to the rich-club phenomenon and the presence of local hubs. Through the study of a targeted attack and overload cascading failure, we prove that hub centrality is a meaningful parameter that gives extra insight beyond degree in real-world networks. Especially, we show that the local hubs occupy key positions in real-world networks with higher probabilities to incur global cascading failure. Therefore, we conclude that networks generated by the growth method, which do not include local hubs, have inevitable limitations to describe real-world networks.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49943398","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}
Centrality measures are used in network science to assess the centrality of vertices or the position they occupy in a network. There are a large number of centrality measures according to some criterion. However, the generalizations of the most well-known centrality measures for weighted networks, degree centrality, closeness centrality and betweenness centrality have solely assumed the edge weights to be constants. This article proposes a methodology to generalize degree, closeness and betweenness centralities taking into account the variability of edge weights in the form of closed intervals (interval-weighted networks, IWN). We apply our centrality measures approach to two real-world IWN. The first is a commuter network in mainland Portugal, between the 23 NUTS 3 Regions. The second focuses on annual merchandise trade between 28 European countries, from 2003 to 2015.
{"title":"Centrality measures in interval-weighted networks","authors":"Hélder Alves;Paula Brito;Pedro Campos","doi":"10.1093/comnet/cnac031","DOIUrl":"https://doi.org/10.1093/comnet/cnac031","url":null,"abstract":"Centrality measures are used in network science to assess the centrality of vertices or the position they occupy in a network. There are a large number of centrality measures according to some criterion. However, the generalizations of the most well-known centrality measures for weighted networks, degree centrality, closeness centrality and betweenness centrality have solely assumed the edge weights to be constants. This article proposes a methodology to generalize degree, closeness and betweenness centralities taking into account the variability of edge weights in the form of closed intervals (interval-weighted networks, IWN). We apply our centrality measures approach to two real-world IWN. The first is a commuter network in mainland Portugal, between the 23 NUTS 3 Regions. The second focuses on annual merchandise trade between 28 European countries, from 2003 to 2015.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49943937","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}
We introduce a kernel Lasso (kLasso) approach which is a type of sparse optimization that simultaneously accounts for spatial regularity and structural sparsity to reconstruct spatially embedded complex networks from time-series data about nodal states. Through the design of a spatial kernel function motivated by real-world network features, the proposed kLasso approach exploits spatial embedding distances to penalize overabundance of spatially long-distance connections. Examples of both random geometric graphs and real-world transportation networks show that the proposed method improves significantly upon existing network reconstruction techniques that mainly concern sparsity but not spatial regularity. Our results highlight the promise of data and information fusion in the reconstruction of complex networks, by utilizing both microscopic node-level dynamics (e.g. time series data) and macroscopic network-level information (metadata or other prior information).
{"title":"Data fusion reconstruction of spatially embedded complex networks","authors":"Jie Sun;Fernando J Quevedo;Erik M Bollt","doi":"10.1093/comnet/cnac032","DOIUrl":"https://doi.org/10.1093/comnet/cnac032","url":null,"abstract":"We introduce a kernel Lasso (kLasso) approach which is a type of sparse optimization that simultaneously accounts for spatial regularity and structural sparsity to reconstruct spatially embedded complex networks from time-series data about nodal states. Through the design of a spatial kernel function motivated by real-world network features, the proposed kLasso approach exploits spatial embedding distances to penalize overabundance of spatially long-distance connections. Examples of both random geometric graphs and real-world transportation networks show that the proposed method improves significantly upon existing network reconstruction techniques that mainly concern sparsity but not spatial regularity. Our results highlight the promise of data and information fusion in the reconstruction of complex networks, by utilizing both microscopic node-level dynamics (e.g. time series data) and macroscopic network-level information (metadata or other prior information).","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49943945","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}
Many complex engineering systems network together functional elements to balance demand spikes but suffer from stability issues due to cascades. The research challenge is to prove the stability conditions for any arbitrarily large and dynamic network topology with any complex balancing function. Most current analyses linearize the system around fixed equilibrium solutions. This approach is insufficient for dynamic networks with multiple equilibria, for example, with different initial conditions or perturbations. Region of attraction (ROA) estimation is needed in order to ensure that the desirable equilibria are reached. This is challenging because a networked system of non-linear dynamics requires compression to obtain a tractable ROA analysis. Here, we employ master stability-inspired method to reveal that the extreme eigenvalues of the Laplacian are explicitly linked to the ROA. This novel relationship between the ROA and the largest eigenvalue in turn provides a pathway to augmenting the network structure to improve stability. We demonstrate using a case study on how the network with multiple equilibria can be optimized to ensure stability.
{"title":"Analysing region of attraction of load balancing on complex network","authors":"Mengbang Zou;Weisi Guo","doi":"10.1093/comnet/cnac025","DOIUrl":"https://doi.org/10.1093/comnet/cnac025","url":null,"abstract":"Many complex engineering systems network together functional elements to balance demand spikes but suffer from stability issues due to cascades. The research challenge is to prove the stability conditions for any arbitrarily large and dynamic network topology with any complex balancing function. Most current analyses linearize the system around fixed equilibrium solutions. This approach is insufficient for dynamic networks with multiple equilibria, for example, with different initial conditions or perturbations. Region of attraction (ROA) estimation is needed in order to ensure that the desirable equilibria are reached. This is challenging because a networked system of non-linear dynamics requires compression to obtain a tractable ROA analysis. Here, we employ master stability-inspired method to reveal that the extreme eigenvalues of the Laplacian are explicitly linked to the ROA. This novel relationship between the ROA and the largest eigenvalue in turn provides a pathway to augmenting the network structure to improve stability. We demonstrate using a case study on how the network with multiple equilibria can be optimized to ensure stability.","PeriodicalId":15442,"journal":{"name":"Journal of complex networks","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49943943","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：