36 research outputs found
Cluster consensus in discrete-time networks of multi-agents with inter-cluster nonidentical inputs
In this paper, cluster consensus of multi-agent systems is studied via
inter-cluster nonidentical inputs. Here, we consider general graph topologies,
which might be time-varying. The cluster consensus is defined by two aspects:
the intra-cluster synchronization, that the state differences between each pair
of agents in the same cluster converge to zero, and inter-cluster separation,
that the states of the agents in different clusters are separated. For
intra-cluster synchronization, the concepts and theories of consensus including
the spanning trees, scramblingness, infinite stochastic matrix product and
Hajnal inequality, are extended. With them, it is proved that if the graph has
cluster spanning trees and all vertices self-linked, then static linear system
can realize intra-cluster synchronization. For the time-varying coupling cases,
it is proved that if there exists T>0 such that the union graph across any
T-length time interval has cluster spanning trees and all graphs has all
vertices self-linked, then the time-varying linear system can also realize
intra-cluster synchronization. Under the assumption of common inter-cluster
influence, a sort of inter-cluster nonidentical inputs are utilized to realize
inter-cluster separation, that each agent in the same cluster receives the same
inputs and agents in different clusters have different inputs. In addition, the
boundedness of the infinite sum of the inputs can guarantee the boundedness of
the trajectory. As an application, we employ a modified non-Bayesian social
learning model to illustrate the effectiveness of our results.Comment: 13 pages, 4 figure
Stability Conditions for Cluster Synchronization in Networks of Heterogeneous Kuramoto Oscillators
In this paper we study cluster synchronization in networks of oscillators
with heterogenous Kuramoto dynamics, where multiple groups of oscillators with
identical phases coexist in a connected network. Cluster synchronization is at
the basis of several biological and technological processes; yet the underlying
mechanisms to enable cluster synchronization of Kuramoto oscillators have
remained elusive. In this paper we derive quantitative conditions on the
network weights, cluster configuration, and oscillators' natural frequency that
ensure asymptotic stability of the cluster synchronization manifold; that is,
the ability to recover the desired cluster synchronization configuration
following a perturbation of the oscillators' states. Qualitatively, our results
show that cluster synchronization is stable when the intra-cluster coupling is
sufficiently stronger than the inter-cluster coupling, the natural frequencies
of the oscillators in distinct clusters are sufficiently different, or, in the
case of two clusters, when the intra-cluster dynamics is homogeneous. We
illustrate and validate the effectiveness of our theoretical results via
numerical studies.Comment: To apper in IEEE Transactions on Control of Network System
Critical Coupling and Synchronized Clusters in Arbitrary Networks of Kuramoto Oscillators
abstract: The Kuramoto model is an archetypal model for studying synchronization in groups
of nonidentical oscillators where oscillators are imbued with their own frequency and
coupled with other oscillators though a network of interactions. As the coupling
strength increases, there is a bifurcation to complete synchronization where all oscillators
move with the same frequency and show a collective rhythm. Kuramoto-like
dynamics are considered a relevant model for instabilities of the AC-power grid which
operates in synchrony under standard conditions but exhibits, in a state of failure,
segmentation of the grid into desynchronized clusters.
In this dissertation the minimum coupling strength required to ensure total frequency
synchronization in a Kuramoto system, called the critical coupling, is investigated.
For coupling strength below the critical coupling, clusters of oscillators form
where oscillators within a cluster are on average oscillating with the same long-term
frequency. A unified order parameter based approach is developed to create approximations
of the critical coupling. Some of the new approximations provide strict lower
bounds for the critical coupling. In addition, these approximations allow for predictions
of the partially synchronized clusters that emerge in the bifurcation from the
synchronized state.
Merging the order parameter approach with graph theoretical concepts leads to a
characterization of this bifurcation as a weighted graph partitioning problem on an
arbitrary networks which then leads to an optimization problem that can efficiently
estimate the partially synchronized clusters. Numerical experiments on random Kuramoto
systems show the high accuracy of these methods. An interpretation of the
methods in the context of power systems is provided.Dissertation/ThesisDoctoral Dissertation Applied Mathematics 201
Cluster Synchronization of Kuramoto Oscillators and Brain Functional Connectivity
The recent progress of functional magnetic resonance imaging techniques has
unveiled that human brains exhibit clustered correlation patterns of their
spontaneous activities. It is important to understand the mechanism of cluster
synchronization phenomena since it may reflect the underlying brain functions
and brain diseases. In this paper, we investigate cluster synchronization
conditions for networks of Kuramoto oscillators. The key analytical tool that
we use is the method of averaging, and we provide a unified framework of
stability analysis for cluster synchronization. The main results show that
cluster synchronization is achieved if (i) the inter-cluster coupling strengths
are sufficiently weak and/or (ii) the natural frequencies are largely different
among clusters. Moreover, we apply our theoretical findings to empirical brain
networks. Discussions on how to understand brain functional connectivity and
further directions to investigate neuroscientific questions are provided
Resilient Learning-Based Control for Synchronization of Passive Multi-Agent Systems under Attack
In this paper, we show synchronization for a group of output passive agents
that communicate with each other according to an underlying communication graph
to achieve a common goal. We propose a distributed event-triggered control
framework that will guarantee synchronization and considerably decrease the
required communication load on the band-limited network. We define a general
Byzantine attack on the event-triggered multi-agent network system and
characterize its negative effects on synchronization. The Byzantine agents are
capable of intelligently falsifying their data and manipulating the underlying
communication graph by altering their respective control feedback weights. We
introduce a decentralized detection framework and analyze its steady-state and
transient performances. We propose a way of identifying individual Byzantine
neighbors and a learning-based method of estimating the attack parameters.
Lastly, we propose learning-based control approaches to mitigate the negative
effects of the adversarial attack
The Kuramoto model in complex networks
181 pages, 48 figures. In Press, Accepted Manuscript, Physics Reports 2015 Acknowledgments We are indebted with B. Sonnenschein, E. R. dos Santos, P. Schultz, C. Grabow, M. Ha and C. Choi for insightful and helpful discussions. T.P. acknowledges FAPESP (No. 2012/22160-7 and No. 2015/02486-3) and IRTG 1740. P.J. thanks founding from the China Scholarship Council (CSC). F.A.R. acknowledges CNPq (Grant No. 305940/2010-4) and FAPESP (Grants No. 2011/50761-2 and No. 2013/26416-9) for financial support. J.K. would like to acknowledge IRTG 1740 (DFG and FAPESP).Peer reviewedPreprin
From Dynamics to Structure of Complex Networks: Exploiting Heterogeneity in the Sakaguchi-Kuramoto Model
[eng] Most of the real-world complex systems are best described as complex networks and can be mathematically described as oscillatory systems, coupled with the neighbours through the connections of the network. The flashing of fireflies, the neuronal brain signals or the energy flow through the power grid are some examples. Yoshiki Kuramoto came up with a tractable mathematical model that could capture the phenomenology of collective synchronization by suggesting that oscillators were coupled by a sinusoidal function of their phase differences. Later, Yoshiki Kuramoto together with Hidetsugu Sakaguchi presented a generalization of the previous limit-cycle set of oscillators Kuramoto’s model which incorporated a constant phase lag between oscillators. Subsequent studies of the model included the network structure within the model together with the global shift. For a wide range of the phase lag values, the system becomes synchronized to a resulting frequency, i.e., the dynamics reaches a stationary state.
In the original work of Kuramoto and Sakaguchi and in most of the consequent later studies, a uniform distribution of phase lag parameters is customarily assumed. However, the intrinsic properties of nodes – that assuredly represent the constituents of real systems – do not need be identical but distributed non-homogeneously among the population. This thesis contributes to the understanding of the Kuramoto-Sakaguchi model with a generalization for nonhomogeneous phase lag parameter distribution. Considering different scenarios concerning the distribution of the frustration parameter among the oscillators represents a major step towards the extension of the original model and provides significant novel insights into the structure and function of the considered network.
The first setting that the present thesis considers consists in perturbing the stationary state of the system by introducing a non-zero phase lag shift into the dynamics of a single node. The aim of this work is to sort the nodes by their potential effect on the whole network when a change on their individual dynamics spreads over the entire oscillatory system by disrupting the otherwise synchronized state. In particular, we define functionability, a novel centrality measure that addresses the question of which are the nodes that, when individually perturbed, are best able to move the system away from the fully synchronized state. This issue may be relevant for the identification of critical nodes that are either beneficial – by enabling access to a broader spectrum of states – or harmful – by destroying the overall synchronization.
The second scenario that the present thesis addresses considers a more general configuration in which the phase lag parameter is an intrinsic property of each node, not necessarily zero, and hence exploring the potential heterogeneity of the frustration among oscillators. We obtain the analytical
solution of frustration parameters so as to achieve any phase configuration, by linearizing the most general model. We also address the fact that the question ’among all the possible solutions, which is the one that makes the system achieve a particular phase configuration with the minimum required cost?’ is of particular relevance when we consider the plausible real nature of the system.
Finally, the homogenous distribution of phase lag parameters is revisited in the last scenario. As studied in the literature, a certain degree of symmetry is an attribute of real-world networks. Nevertheless, beyond structural or topological symmetry, one should consider the fact that real- world networks are exposed to fluctuations or errors, as well as mistaken insertions or removals. In the present thesis, we provide an alternative notion to approximate symmetries, which we call ‘Quasi-Symmetries’ and are defined such that they remain free to impose any invariance of a particular network property and are obtained from the stationary state of the Kuramoto-Sakaguchi model with a homogeneous phase lag distribution. A first contribution is exploring the distributions of structural similarity among all pairs of nodes. Secondly, we define the ‘dual network’, a weighted network –and its corresponding binarized counterpart– that effectively encloses all the information of quasi-symmetries in the original one.[cat] La major part dels sistemes complexos presents en la natura i la societat es poden descriure com a xarxes complexes. Molts d’aquests sistemes es poden modelitzar matemàticament com un sistema oscil·latori, on les unitats queden acoblades amb els components veïns a través de les connexions de la xarxa. Yoshiki Kuramoto i Hidetsugu Sakaguchi van presentar la generalització del ben conegut model d’oscil·ladors de Kuramoto, on s’incorporava un terme de desfasament entre parelles d’oscil·ladors. Aquesta tesi contribueix en la comprensió d’aquest model, tot considerant una distribució no homogènia d’aquest paràmetre de desfasament o frustració. S’han considerat tres escenaris diferents, tots ells donant lloc a resultats que permeten una millor descripció de l’estructura i funció de la xarxa que s’està considerant.
Una primera configuració consisteix en pertorbar l’estat estacionari tot introduint un desfasament en la dinàmica d’un node de manera aïllada. Seguidament, definim la funcionabilitat, una mesura de centralitat única que respon a la pregunta de, quins nodes, quan són pertorbats individualment, són més capaços d’allunyar el sistema de l’estat sincronitzat. Aquest fet podria suposar un comportament beneficiós o perjudicial per sistemes reals.
El segon escenari considera la configuració més flexible, explorant la potencial heterogeneïtat dels paràmetres de frustració dels diferents nodes. Obtenim la solució analítica d’aquesta distribució per tal d’assolir qualsevol configuració de les fases dels oscil·ladors, a través de la linearització del model. També contestem a la pregunta: “de totes les possibles solucions, quina és la que implica un menor cost per tal d’assolir una configuració en particular?”.
Finalment, en l’últim escenari, proporcionem una definició alternativa al concepte de simetria aproximada d’una xarxa, i que anomenem “Quasi simetries”. Aquestes són definides sense imposar invariàncies en les propietats del sistema, sinó que s’obtenen de l’estat estacionari del model de Kuramoto-Sakaguchi model, tot considerant una distribució homogènia dels paràmetres de frustració