Analysis of properties of Ising and Kuramoto models that are preserved in networks constructed by visualization algorithms

Recently it has been shown that building networks from time series allows to study complex systems to characterize them when they go through a phase transition. This give us the opportunity to study this systems from a entire new point of view. In the present work we have used the natural and horizontal visualization algorithms to built networks of two popular models, which present phase transitions: the Ising model and the Kuramoto model. By measuring some topological quantities as the average degree, or the clustering coefficient, it was found that the networks retain the capability of detecting the phase transition of the system. From our results it is possible to establish that both visibility algorithms are capable of detecting the critical control parameter, as in every quantity analyzed (the average degree, the average path and the clustering coefficient) there is a minimum or a maximum value. In the case of the natural visualization algorithm, the average path results are much more noisy than in the other quantities in the study. Specially for the Kuramoto Model, which in this case does not allow a detection of the critical point at plain sight as for the other quantities. The horizontal visualization algorithm has proven to be more explicit in every quantity, as every one of them show a clear change of behavior before and after the critical point of the transition.

[1]  Zhu Su,et al.  Multifractality and Network Analysis of Phase Transition , 2016, PloS one.

[2]  S. Carpenter,et al.  Rising variance: a leading indicator of ecological transition. , 2006, Ecology letters.

[3]  S. Strogatz.  From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators , 2000 .

[4]  Mark Newman,et al.  Networks: An Introduction , 2010 .

[5]  Yue Yang,et al.  Complex network-based time series analysis , 2008 .

[6]  B. Luque,et al.  Horizontal visibility graphs: exact results for random time series. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Lucas Lacasa,et al.  Visibility Algorithms: A Short Review , 2012 .

[8]  Elias B. Kosmatopoulos,et al.  Adaptive hybrid synchronisation in uncertain Kuramoto networks with limited information , 2019, IET Control Theory & Applications.

[9]  Zhao Yang,et al.  A Comparative Analysis of Community Detection Algorithms on Artificial Networks , 2016, Scientific Reports.

[10]  L. Amaral,et al.  Duality between Time Series and Networks , 2011, PloS one.

[11]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[12]  I. Morales,et al.  Enhancement of early warning properties in the Kuramoto model and in an atrial fibrillation model due to an external perturbation of the system , 2017, PloS one.

[13]  Ludvig Bohlin,et al.  Community detection and visualization of networks with the map equation framework , 2014 .

[14]  Nicanor Quijano,et al.  Synchronisation of heterogeneous Kuramoto oscillators with sampled information and a constant leader , 2019, Int. J. Control.

[15]  L. Kadanoff.  Spin-spin correlations in the two-dimensional ising model , 1966 .

[16]  V. Guttal,et al.  Changing skewness: an early warning signal of regime shifts in ecosystems. , 2008, Ecology letters.

[17]  R. Spigler,et al.  The Kuramoto model: A simple paradigm for synchronization phenomena , 2005 .

[18]  Lucas Lacasa,et al.  From time series to complex networks: The visibility graph , 2008, Proceedings of the National Academy of Sciences.

[19]  Bosiljka Tadić,et al.  Hidden geometry of traffic jamming. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Bryan C. Daniels.  Synchronization of Globally Coupled Nonlinear Oscillators: the Rich Behavior of the Kuramoto Model , 2005 .

[21]  S. Carpenter,et al.  Early-warning signals for critical transitions , 2009, Nature.

[22]  M. Scheffer,et al.  Slowing down as an early warning signal for abrupt climate change , 2008, Proceedings of the National Academy of Sciences.

[23]  Nauenberg,et al.  Distribution of fractal clusters and scaling in the Ising model. , 1986, Physical review. B, Condensed matter.

[24]  Hermann Held,et al.  The potential role of spectral properties in detecting thresholds in the Earth system: application to the thermohaline circulation , 2003 .

[25]  Florian Dörfler,et al.  Synchronization in complex networks of phase oscillators: A survey , 2014, Autom..

[26]  I. Morales,et al.  Behavior of Early Warnings near the Critical Temperature in the Two-Dimensional Ising Model , 2015, PloS one.

[27]  Helge Dietert,et al.  Stability and bifurcation for the Kuramoto model , 2014, 1411.3752.

[28]  S. Carpenter,et al.  Methods for Detecting Early Warnings of Critical Transitions in Time Series Illustrated Using Simulated Ecological Data , 2012, PloS one.

[29]  Sabrina Maniscalco,et al.  Fluctuation relations for driven coupled classical two-level systems with incomplete measurements. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.