Complex time-varying networks are prominent models for a wide variety of spatiotemporal phenomena. The functioning of networks depends crucially on their connectivity, yet reliable techniques for determining communities in spacetime networks remain elusive. We adapt successful spectral techniques from continuous-time dynamics on manifolds to the graph setting to fill this gap. We formulate an {it inflated dynamic Laplacian} for graphs and develop a spectral theory to underpin the corresponding algorithmic realisations. We develop spectral clustering approaches for both multiplex and non-multiplex networks, based on the eigenvectors of the inflated dynamic Laplacian and specialised Sparse EigenBasis Approximation (SEBA) post-processing of these eigenvectors. We demonstrate that our approach can outperform the Leiden algorithm applied both in spacetime and layer-by-layer, and we analyse voting data from the US senate (where senators come and go as congresses evolve) to quantify increasing polarisation in time.
{"title":"Spectral clustering of time-evolving networks using the inflated dynamic Laplacian for graphs","authors":"Gary Froyland, Manu Kalia, Peter Koltai","doi":"arxiv-2409.11984","DOIUrl":"https://doi.org/arxiv-2409.11984","url":null,"abstract":"Complex time-varying networks are prominent models for a wide variety of\u0000spatiotemporal phenomena. The functioning of networks depends crucially on\u0000their connectivity, yet reliable techniques for determining communities in\u0000spacetime networks remain elusive. We adapt successful spectral techniques from\u0000continuous-time dynamics on manifolds to the graph setting to fill this gap. We\u0000formulate an {it inflated dynamic Laplacian} for graphs and develop a spectral\u0000theory to underpin the corresponding algorithmic realisations. We develop\u0000spectral clustering approaches for both multiplex and non-multiplex networks,\u0000based on the eigenvectors of the inflated dynamic Laplacian and specialised\u0000Sparse EigenBasis Approximation (SEBA) post-processing of these eigenvectors.\u0000We demonstrate that our approach can outperform the Leiden algorithm applied\u0000both in spacetime and layer-by-layer, and we analyse voting data from the US\u0000senate (where senators come and go as congresses evolve) to quantify increasing\u0000polarisation in time.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we consider an initial-boundary value problem related to some network dynamics where the underlying graph has unbounded edges. We show that there exists a C0-semigroup for this problem using a general result from the literature. We also find an explicit formula for this semigroup. This is achieved using the method of characteristics and then showing that the Laplace transform of the solution is equal to the resolvent operator of the generator.
{"title":"Existence and explicit formula for a semigroup related to some network problems with unbounded edges","authors":"Adam Błoch","doi":"arxiv-2409.11903","DOIUrl":"https://doi.org/arxiv-2409.11903","url":null,"abstract":"In this paper we consider an initial-boundary value problem related to some\u0000network dynamics where the underlying graph has unbounded edges. We show that\u0000there exists a C0-semigroup for this problem using a general result from the\u0000literature. We also find an explicit formula for this semigroup. This is\u0000achieved using the method of characteristics and then showing that the Laplace\u0000transform of the solution is equal to the resolvent operator of the generator.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"191 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
the action of the function on its Julia set is still ergodic if some, but not all of the asymptotic values land on infinity, and the remaining ones land on a compact repeller. In this paper, we complete the characterization of ergodicity for Nevanlinna functions but proving that if all the asymptotic values land on infinity, then the Julia set is the whole sphere and the action of the map there is non-ergodic.
如果部分渐近值落在无穷大上，而不是所有渐近值都落在无穷大上，并且剩余的渐近值落在一个紧密排斥者上，那么函数在其 Julia 集上的作用仍然是遍历性的。在本文中，我们完成了对 Nevanlinna 函数遍历性的表征，但证明了如果所有渐近值都落在无穷大上，那么 Julia 集就是整个球面，并且该球面的作用是非遍历性的。
{"title":"Meromorphic functions whose action on their Julia sets is Non-Ergodic","authors":"Tao Chen, Yunping Jiang, Linda Keen","doi":"arxiv-2409.12127","DOIUrl":"https://doi.org/arxiv-2409.12127","url":null,"abstract":"the action of the function on its Julia set is still ergodic if some, but not\u0000all of the asymptotic values land on infinity, and the remaining ones land on a\u0000compact repeller. In this paper, we complete the characterization of ergodicity\u0000for Nevanlinna functions but proving that if all the asymptotic values land on\u0000infinity, then the Julia set is the whole sphere and the action of the map\u0000there is non-ergodic.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that skew products with the cocycle given by the function $f(x)=a(x-1/2)$ with $aneq 0$ are ergodic for every ergodic symmetric IET in the base, thus giving the full characterization of ergodic extensions in this family. Moreover, we prove that under an additional natural assumption of unique ergodicity on the IET, we can replace $f$ with any differentiable function with a non-zero sum of jumps. Finally, by considering weakly mixing IETs instead of just ergodic, we show that the skew products with cocycle given by $f$ have infinite ergodic index.
我们证明，由函数$f(x)=a(x-1/2)$与$a/neq 0$给出的循环的斜积对于基内的每一个遍历对称IET都是遍历的，从而给出了这个家族中遍历扩展的全部特征。此外，我们还证明，在 IET 唯一遍历性的额外自然假设下，我们可以用任何具有非零跳跃之和的可微函数来代替 $f$。最后，通过考虑弱混合 IET 而不是仅仅考虑遍历性，我们证明了由 $f$ 给定循环的斜积具有无限遍历指数。
{"title":"Ergodic properties of infinite extension of symmetric interval exchange transformations","authors":"Przemysław Berk, Frank Trujillo, Hao Wu","doi":"arxiv-2409.12168","DOIUrl":"https://doi.org/arxiv-2409.12168","url":null,"abstract":"We prove that skew products with the cocycle given by the function\u0000$f(x)=a(x-1/2)$ with $aneq 0$ are ergodic for every ergodic symmetric IET in\u0000the base, thus giving the full characterization of ergodic extensions in this\u0000family. Moreover, we prove that under an additional natural assumption of\u0000unique ergodicity on the IET, we can replace $f$ with any differentiable\u0000function with a non-zero sum of jumps. Finally, by considering weakly mixing\u0000IETs instead of just ergodic, we show that the skew products with cocycle given\u0000by $f$ have infinite ergodic index.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the computational complexity theory of smooth, finite-dimensional dynamical systems. Building off of previous work, we give definitions for what it means for a smooth dynamical system to simulate a Turing machine. We then show that 'chaotic' dynamical systems (more precisely, Axiom A systems) and 'integrable' dynamical systems (more generally, measure-preserving systems) cannot robustly simulate universal Turing machines, although such machines can be robustly simulated by other kinds of dynamical systems. Subsequently, we show that any Turing machine that can be encoded into a structurally stable one-dimensional dynamical system must have a decidable halting problem, and moreover an explicit time complexity bound in instances where it does halt. More broadly, our work elucidates what it means for one 'machine' to simulate another, and emphasizes the necessity of defining low-complexity 'encoders' and 'decoders' to translate between the dynamics of the simulation and the system being simulated. We highlight how the notion of a computational dynamical system leads to questions at the intersection of computational complexity theory, dynamical systems theory, and real algebraic geometry.
{"title":"Computational Dynamical Systems","authors":"Jordan Cotler, Semon Rezchikov","doi":"arxiv-2409.12179","DOIUrl":"https://doi.org/arxiv-2409.12179","url":null,"abstract":"We study the computational complexity theory of smooth, finite-dimensional\u0000dynamical systems. Building off of previous work, we give definitions for what\u0000it means for a smooth dynamical system to simulate a Turing machine. We then\u0000show that 'chaotic' dynamical systems (more precisely, Axiom A systems) and\u0000'integrable' dynamical systems (more generally, measure-preserving systems)\u0000cannot robustly simulate universal Turing machines, although such machines can\u0000be robustly simulated by other kinds of dynamical systems. Subsequently, we\u0000show that any Turing machine that can be encoded into a structurally stable\u0000one-dimensional dynamical system must have a decidable halting problem, and\u0000moreover an explicit time complexity bound in instances where it does halt.\u0000More broadly, our work elucidates what it means for one 'machine' to simulate\u0000another, and emphasizes the necessity of defining low-complexity 'encoders' and\u0000'decoders' to translate between the dynamics of the simulation and the system\u0000being simulated. We highlight how the notion of a computational dynamical\u0000system leads to questions at the intersection of computational complexity\u0000theory, dynamical systems theory, and real algebraic geometry.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Juan Paz, Camilo Rocha, Luis Tobòn, Frank Valencia
Interest is growing in social learning models where users share opinions and adjust their beliefs in response to others. This paper introduces generalized-bias opinion models, an extension of the DeGroot model, that captures a broader range of cognitive biases. These models can capture, among others, dynamic (changing) influences as well as ingroup favoritism and out-group hostility, a bias where agents may react differently to opinions from members of their own group compared to those from outside. The reactions are formalized as arbitrary functions that depend, not only on opinion difference, but also on the particular opinions of the individuals interacting. Under certain reasonable conditions, all agents (despite their biases) will converge to a consensus if the influence graph is strongly connected, as in the original DeGroot model. The proposed approach combines different biases, providing deeper insights into the mechanics of opinion dynamics and influence within social networks.
{"title":"Consensus in Models for Opinion Dynamics with Generalized-Bias","authors":"Juan Paz, Camilo Rocha, Luis Tobòn, Frank Valencia","doi":"arxiv-2409.10809","DOIUrl":"https://doi.org/arxiv-2409.10809","url":null,"abstract":"Interest is growing in social learning models where users share opinions and\u0000adjust their beliefs in response to others. This paper introduces\u0000generalized-bias opinion models, an extension of the DeGroot model, that\u0000captures a broader range of cognitive biases. These models can capture, among\u0000others, dynamic (changing) influences as well as ingroup favoritism and\u0000out-group hostility, a bias where agents may react differently to opinions from\u0000members of their own group compared to those from outside. The reactions are\u0000formalized as arbitrary functions that depend, not only on opinion difference,\u0000but also on the particular opinions of the individuals interacting. Under\u0000certain reasonable conditions, all agents (despite their biases) will converge\u0000to a consensus if the influence graph is strongly connected, as in the original\u0000DeGroot model. The proposed approach combines different biases, providing\u0000deeper insights into the mechanics of opinion dynamics and influence within\u0000social networks.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250408","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jan Lorenz Svensen, Nicola Cantisani, Wilson Ricardo Leal da Silva, Javier Pigazo Merino, Dinesh Sampath, John Bagterp Jørgensen
We provide a cyclone model for dynamical simulations in the pyro-process of cement production. The model is given as an index-1 differential-algebraic equation (DAE) model based on first engineering principle. Using a systematic approach, the model integrates cyclone geometry, thermo-physical aspects, stoichiometry and kinetics, mass and energy balances, and algebraic equations for volume and internal energy. The paper provides simulation results that fit expected dynamics. The cyclone model is part of an overall model for dynamical simulations of the pyro-process in a cement plant. This model can be used in the design of control and optimization systems to improve energy efficiency and reduce CO2 emission.
{"title":"A first engineering principles model for dynamical simulation of cement pyro-process cyclones","authors":"Jan Lorenz Svensen, Nicola Cantisani, Wilson Ricardo Leal da Silva, Javier Pigazo Merino, Dinesh Sampath, John Bagterp Jørgensen","doi":"arxiv-2409.10916","DOIUrl":"https://doi.org/arxiv-2409.10916","url":null,"abstract":"We provide a cyclone model for dynamical simulations in the pyro-process of\u0000cement production. The model is given as an index-1 differential-algebraic\u0000equation (DAE) model based on first engineering principle. Using a systematic\u0000approach, the model integrates cyclone geometry, thermo-physical aspects,\u0000stoichiometry and kinetics, mass and energy balances, and algebraic equations\u0000for volume and internal energy. The paper provides simulation results that fit expected dynamics. The cyclone model is part of an overall model for dynamical simulations of\u0000the pyro-process in a cement plant. This model can be used in the design of\u0000control and optimization systems to improve energy efficiency and reduce CO2\u0000emission.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we extend the ideas of certain notions that one studies in thermodynamic formalism of maps to the context when the dynamics in the phase space evolves by complex holomorphic correspondences. Towards that end, we define the topological entropy of holomorphic correspondences using spanning sets. We then, define the pressure of a real-valued continuous function defined on the Riemann sphere and investigate the Ruelle operator with respect to the H"{o}lder continuous function, however restricted on the support of the Dinh-Sibony measure.
{"title":"A Ruelle operator for holomorphic correspondences","authors":"Shrihari Sridharan, Subith G","doi":"arxiv-2409.11085","DOIUrl":"https://doi.org/arxiv-2409.11085","url":null,"abstract":"In this paper, we extend the ideas of certain notions that one studies in\u0000thermodynamic formalism of maps to the context when the dynamics in the phase\u0000space evolves by complex holomorphic correspondences. Towards that end, we\u0000define the topological entropy of holomorphic correspondences using spanning\u0000sets. We then, define the pressure of a real-valued continuous function defined\u0000on the Riemann sphere and investigate the Ruelle operator with respect to the\u0000H\"{o}lder continuous function, however restricted on the support of the\u0000Dinh-Sibony measure.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the specification property implies infinite topological entropy for operators acting on infinite dimensional $F$-spaces. Furthermore, we establish compact operators acting on Banach spaces exhibit finite entropy and the entropy depends exclusively on the operator's point spectrum. Additionally, we prove that the variational principle does not hold for compact operators acting on Banach spaces.
{"title":"Entropy for compact operators and results on entropy and specification","authors":"Paulo Lupatini, Felipe Silva, Régis Varão","doi":"arxiv-2409.10844","DOIUrl":"https://doi.org/arxiv-2409.10844","url":null,"abstract":"We prove that the specification property implies infinite topological entropy\u0000for operators acting on infinite dimensional $F$-spaces. Furthermore, we\u0000establish compact operators acting on Banach spaces exhibit finite entropy and\u0000the entropy depends exclusively on the operator's point spectrum. Additionally,\u0000we prove that the variational principle does not hold for compact operators\u0000acting on Banach spaces.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pedro C. C. R. Pereira, Mike R. Jeffrey, Douglas D. Novaes
When a dynamical system is subject to a periodic perturbation, the averaging method can be applied to obtain an autonomous leading order `guiding system', placing the time dependence at higher orders. Recent research focused on investigating invariant structures in non-autonomous differential systems arising from hyperbolic structures in the guiding system, such as periodic orbits and invariant tori. The effect that bifurcations in the guiding system have on the original non-autonomous one has also been recently explored. This paper extends the study by providing a broader description of the dynamics that can emerge from non-hyperbolic structures of the guiding system. Specifically, we prove here that $K$-universal bifurcations in the guiding system persist in the original non-autonomous one, while non-versal bifurcations, such as the transcritical and pitchfork, do not, being instead perturbed into stable bifurcation families. We illustrate the results on examples of a fold, a transcritical, a pitchfork, and a saddle-focus. By applying these results to the physical scenario of systems with time-varying parameters, we show that the average parameter value becomes a bifurcation parameter of the averaged system.
{"title":"Averaging theory and catastrophes: The persistence of bifurcations under time-varying perturbations","authors":"Pedro C. C. R. Pereira, Mike R. Jeffrey, Douglas D. Novaes","doi":"arxiv-2409.11054","DOIUrl":"https://doi.org/arxiv-2409.11054","url":null,"abstract":"When a dynamical system is subject to a periodic perturbation, the averaging\u0000method can be applied to obtain an autonomous leading order `guiding system',\u0000placing the time dependence at higher orders. Recent research focused on\u0000investigating invariant structures in non-autonomous differential systems\u0000arising from hyperbolic structures in the guiding system, such as periodic\u0000orbits and invariant tori. The effect that bifurcations in the guiding system\u0000have on the original non-autonomous one has also been recently explored. This\u0000paper extends the study by providing a broader description of the dynamics that\u0000can emerge from non-hyperbolic structures of the guiding system. Specifically,\u0000we prove here that $K$-universal bifurcations in the guiding system persist in\u0000the original non-autonomous one, while non-versal bifurcations, such as the\u0000transcritical and pitchfork, do not, being instead perturbed into stable\u0000bifurcation families. We illustrate the results on examples of a fold, a\u0000transcritical, a pitchfork, and a saddle-focus. By applying these results to\u0000the physical scenario of systems with time-varying parameters, we show that the\u0000average parameter value becomes a bifurcation parameter of the averaged system.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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