CAMPINAS
Instituto de Matemática, Estatística e
Computação Científica
JOYCE SIMÃO CLIMACO
Synchronization aspects of the Kuramoto
model and some of its generalizations
Aspectos de sincronização do modelo de
Kuramoto e algumas de suas generalizações
Campinas
2019
Synchronization aspects of the Kuramoto model and
some of its generalizations
Aspectos de sincronização do modelo de Kuramoto e
algumas de suas generalizações
Dissertação apresentada ao Instituto de Matemática, Estatística e Computação Cien-tífica da Universidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do título de Mestra em Matemática Aplicada.
Dissertation presented to the Institute of Mathematics, Statistics and Scientific Com-puting of the University of Campinas in par-tial fulfillment of the requirements for the degree of Master in Applied Mathematics.
Supervisor: Alberto Vazquez Saa
Este exemplar corresponde à versão
final da Dissertação defendida pela
aluna Joyce Simão Climaco e
orien-tada pelo Prof. Dr. Alberto Vazquez
Saa.
Campinas
2019
Biblioteca do Instituto de Matemática, Estatística e Computação Científica Ana Regina Machado - CRB 8/5467
Climaco, Joyce Simão,
C613s CliSynchronization aspects of the Kuramoto model and some of its generalizations / Joyce Simão Climaco. – Campinas, SP : [s.n.], 2019.
CliOrientador: Alberto Vazquez Saa.
CliDissertação (mestrado) – Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica.
Cli1. Sincronização. 2. Redes complexas. 3. Kuramoto, Modelo de. I. Saa, Alberto Vazquez, 1966-. II. Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica. III. Título.
Informações para Biblioteca Digital
Título em outro idioma: Aspectos de sincronização do modelo de Kuramoto e algumas de
suas generalizações
Palavras-chave em inglês:
Synchronization Complex networks Kuramoto model
Área de concentração: Matemática Aplicada Titulação: Mestra em Matemática Aplicada Banca examinadora:
Alberto Vazquez Saa [Orientador] Marcus Aloizio Martinez de Aguiar João Batista Florindo
Data de defesa: 13-12-2019
Programa de Pós-Graduação: Matemática Aplicada
Identificação e informações acadêmicas do(a) aluno(a)
- ORCID do autor: https://orcid.org/0000-0002-5943-6840 - Currículo Lattes do autor: http://lattes.cnpq.br/2114449060750696
pela banca examinadora composta pelos Profs. Drs.
Prof. Dr. ALBERTO VAZQUEZ SAA
Prof. Dr. MARCUS ALOIZIO MARTINEZ DE AGUIAR
Prof. Dr. JOÃO BATISTA FLORINDO
A Ata da Defesa, assinada pelos membros da Comissão Examinadora, consta no SIGA/Sistema de Fluxo de Dissertação/Tese e na Secretaria de Pós-Graduação do Instituto de Matemática, Estatística e Computação Científica.
which required an immense amount of help.
Though a great part technical, most of the support that was needed - and happily given - concerned personal matters,
and the difficulties and delights of enduring the constant impermanence in life. For that, I thank every family, friend, acquaintance and stranger
The author acknowledges the financial support of grant no 132043/2017-3,
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and grant no
2013/09357-9, Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), and also sincerely thank A. Saa for all the years of mentorship.
is or is not the sublimity of the intelligence" (Edgar Allan Poe)
O fenômeno de sincronização se mostra presente tanto em sistemas biológicos, quanto físicos ou tecnológicos, como redes de neurônios, oscilações químicas, e redes de transmissão de energia, respectivamente. Consideramos aqui o modelo de Kuramoto em redes complexas com um alto número N de osciladores, e exploramos como variações nesse sistema podem afetar sua sincronização, que é definida como o estado no qual as fases de todos os osciladores evoluem rigidamente concentradas ao redor da fase média ψptq. Sendo assim, iniciamos esse trabalho discutindo como variações na dinâmica ou topologia de rede no modelo Kuramoto podem afetar certos aspectos do comportamento macroscópico do sistema, como transição de fase e sincronização global. Mostramos então como a complexidade de sistemas desse tipo pode ser reduzida através de métodos das coordenadas coletivas, no qual escolhemos novas coordenadas para o sistema que diminuem de N para n ! N o número de equações diferenciais necessárias para descrever seu comportamento. Utilizamos esse em nosso trabalho original para sistemas nos quais um força externa periódica F sin pΩtq é aplicada numa fração f do total de osciladores. A partir dela conseguimos encontrar os limites de f e F a partir dos quais a sincronização global não é possível. Também fomos capazes de estabelecer um algoritmo a partir de critérios analíticos nas novas coordenadas capaz de otimizar a sincronização global do sistema, mostrando que esta é máxima quando escolhermos forçar os osciladores com maior diferença absoluta |ωi´Ω|
entre suas frequências naturais ωi e o período Ω da força externa.
Palavras-chave: sincronização. Kuramoto. otimização. redução dimensional. coordenadas
The synchronization phenomenon may present itself both in biological and in physical or technological systems, such as neuron networks, chemical oscillations and power grids, respectively. We consider here the Kuramoto model in complex networks with a large number N of oscillators, and explore how variations in this system can affect its synchro-nization, which is defined as the state in which the phase evolution of all oscillators are rigidly concentrated around the mean phase ψ. Therefore, we begin this work discussing how some Kuramoto model variations in dynamics or network topology may affect certain aspects of the system’s macroscopic behaviour, such as its phase transitions and global synchronization. We then show how the complexity of systems of this type may be reduced through the collective coordinates method, in which we choose new coordinates for the system to decrease the number of differential equations needed to describe the its behaviour from N to n ! N. We use this method our original work for systems in which an external periodic force F sin pΩtq is applied in a fraction f of the total number of oscillators. Hence, we are able to find thresholds for F and f over which global synchronization is not possible. We also established a hill-climb algorithm from analytic criteria derived in the new coordinates to optimize the system’s synchronization, showing that it reaches its maximum when we choose to force the oscillators with largest absolute difference |ωi´Ω|
between their natural frequency ωi and the external force frequency Ω.
Keywords: synchronization. Kuramoto. optimization. dimensional reduction. collective
Figure 1 – Visual representation of an undirected graph, with the nodes V rep-resented by circles and assigned an unique number, and the edges E shown as the gray lines between them. . . 21
Figure 2 – Network types, where arrows represent directional edges, and darker edges have more weight (strength) than lighter ones. . . 22
Figure 3 – Types of triplets. . . 24
Figure 4 – Networks with the same number of nodes and edges, but dramatically different clustering coefficients. The regions highlighted in gray represent the closed triplets used to compute clustering in equation 1.8. . . 24
Figure 5 – Schematics of the construction process of a 3-node ER network with
p « 0.66. . . . 25
Figure 6 – Aspect of networks built through the Watts-Strogatz model for different values of the rewiring probability p. From [34]. . . 27
Figure 7 – Node degree ki and respective preferential attachment probability Ppkiq
for a Barabási-Albert network evolution with m “ 2 before and after adding a new node with two new edges. . . 29
Figure 8 – Barabási-Albert networks with N “ 20 for different values of m. . . . . 29
Figure 9 – Energy level and graph schematics of a Bose-Einstein condensation network model with m “ 2, with three initial levels E1, E2 and E3
(pink) with two particles (and therefore two edges) each. A new level
E4 (orange) is introduced, and with it two new edges (dashed lines) are
placed with attachment the probability Ppkq described in 1.20.. . . 30
Figure 10 – Synchronization states. . . 32
Figure 11 – Order parameter evolution in the Kuramoto model applied to all-to-all and small world networks. Adapted from [3]. . . 35
Figure 12 – Supercritical Hopf bifurcation at µ “ 0, when a stable point becomes a stable limit cycle with radius growing as we increase the parameter
µ. Since both the point and the limit cycle are stable, we can also see
nearby trajectories being attracted to them. From [45]. . . 38
Figure 13 – Second order phase transition for the synchronization a m “ 2 Barabási–Albert network of N “ 200 oscillators in the Kuramoto model. . . . 41
Figure 14 – Probability density function of a Γ-distribution for three different shape parameter values, and mean value xλy “ 1. Presented at [22]. . . 42
for a Barabasi-Albert network of 100 Stuart-Landau oscillators with
α “1. We consider four different distributions for the coupling λ:
con-stant (unweighted), and Γ-distributions with shape parameters Θ “ 2, Θ “ 3 and Θ “ 4. The algorithms used in this simulation can be found in the GitHub repository [46], written in Python 3. We used scipy.integrate.odeint to integrate our system of differential equa-tions, as well as the packages numpy, matplotlib, and networkx. Presented at XXXVIII CNMAC [22]. . . 42
Figure 16 – Order parameter diagram for a sequence of networks obtained through increasing the parameter α P r0, 1s to continuously change from a random (ER) to scale-free (BA) network. From [20]. . . 44
Figure 17 – Diagrams of the effective frequency 2.27 of. several oscillators as we increase the coupling strength λ for four members of a network family in which a network is continuously changed from. a random (ER) model to a scale-free (BA) one through decreasing a parameter α P r0, 1s. From [20]. . . 45
Figure 18 – Closed path composed by the curves C1 and C2 in the complex ω
plane to calculate the order parameter integral 3.14 with a residue at
ω “ ω0´ i∆. . . . 52
Figure 19 – Order parameter found through the reduced system equations for the lorentzian network. Adapted from [23]. . . 55
Figure 20 – Phase distribution of a synchronized (¯r “ 0.78) Kuramoto system 4.1 with xωy, λ “ 9.5 for a p “ 0.05 and N “ 200 ER network, with the oscillators in crescent order of their natural frequencies ωi. The
continuous curve represents a smooth cubic function as to show the similarity with the native frequency distribution ωi “ ξ3´0.3ξi with ξ P r0, 1s. From [25]. . . 57
Figure 21 – Order parameter for a N “ 10000 full Kuramoto system 4.1 calculated through numerical integration, as well as from the collective coordinates approach. Adapted from [25]. . . 61
Figure 22 – Saddle node bifurcation in the N=10000 Kuramoto system described by the collective coordinates 4.15. Adapted from [25]. . . 62
Figure 23 – Comparison of the trajectories of the oscillators for a N “ 100 Kuramoto system with coupling strength λ “ 1.5 ą λc between the collective
coordinates ansatz θi “ αωi and the direct numerical integration of the
through direct numerical integration, as well as from the collective coordinates approach for both global and local onsets of synchronization. Adapted from [25]. . . 64
Figure 25 – Order parameter for a N “ 1000 Kuramoto system 4.1 calculated through direct numerical integration, as well as from the collective coordinates approach for both global and local onsets of synchronization. Adapted from [25]. . . 65
Figure 26 – Comparison of the time evolution of individual oscillators in the global synchronization onset (coupling strength λ “ 0.9 ą λc) between the
col-lective coordinates ansatz θi “ αωi and the direct numerical integration
of the simulated system for a Kuramoto network with N “ 1000 and initial conditions θip0q “ α0ωi with α0 “0.5. Adapted from [25]. . . 66
Figure 27 – Comparison of the time evolution of individual oscillators in the partial synchronization onset (coupling strength λ “ 0.6, so that λl ă λ ă λc)
between the collective coordinates ansatz θi “ αωi and the direct
numerical integration of the simulated system for a Kuramoto network with N “ 1000 and initial conditions θip0q “ α0ωi with α0 “ 0.5.
Adapted from [25]. . . 67
Figure 28 – Behaviour of fpα) defined in 5.4 for two different values of coupling strength, λ Ñ 8 and λ “ 0.6, for a Kuramoto system of N “ 1000 oscillators in a Erdös-Renyi network with average degree xky “ pN “ 10 and natural frequencies taken from an uniform distribution in the interval p´1, 1q. From [27]. . . 70
Figure 29 – Order parameter evolution for a Kuramoto system of N “ 1000 os-cillators initially in a Barabási-Albert network with xky “ 6 and a unit natural frequency distribution gpω before and after the network optimization. From [27]. . . 72
Figure 30 – Correlation between the natural frequencies of adjacent oscillators and between the oscillators absolute natural frequencies and node degree |ω| for the optimized Kuramoto system in Figure 29. Adapted from [27]. 73
with 1500 nodes, average degree xdy « 7.7 and with only 96 nodes in the set C, with a uniform frequency distribution over p´1, 1q. The read line is the mean-field prediction 6.13, and the blue circles correspond to the respective numerical values calculated from 6.11. The depicted case corresponds to α “ π , but the overall 4 accuracy is rather insensitive to the specific value of α. On the other hand, the value of NC does play an
important role. The larger NC, the most accurate is the approximation
6.13. The situation for I2pα, βq defined by 6.16 is analogous. For the
case of I3pβqdefined by 6.17, see [14]. Published in [15]. . . 80
Figure 32 – Forbidden region in which global synchronization cannot occur for a partially forced Kuramoto system with Ω “ 1 and uniformly distributed natural frequencies. . . 82
Figure 33 – The pω, θpτq, mod 2πq graphics for a fixed τ in a globally synchronized regime, for an ER random network with 1500 nodes, with 151 of them subjected to the external periodic force. Each point in the graphics corresponds to a Kuramoto oscillator in the network. The drawn line is the simple linear regression of the data. The linear correlation between
ωi and θi is evident, as in ansatz 6.3, with βpτq and αpτq corresponding
to the angular and linear coefficients, respectively. Moreover, we can clearly identify two displaced oscillator populations which similar linear regression slopes. Published in [15]. . . 85
Figure 34 – Synchronization diagrams for r and a as functions of f F
Ω , for
λ
Ω “ 2
for the ER system described in the text. The blue circles, red crosses, and green stars correspond, respectively, to the optimal, random, and worst subset C of forced nodes. The critical external force Fc, see 6.31,
corresponds precisely to f F
Ω “1 in this case. As one can see, for F ą Fc,
all the oscillators follow the same pace of the external force since a « 1. One can also appreciate that r is indeed enhanced according to our optimization procedure for F ą Fc. Published in [15]. . . 86
We considered the ER network described in the text. The blue circles, red crosses, and green stars correspond, respectively, to the optimal, random, and worst subset C of forced nodes. Although one sees that synchronization (r « 1) can occur for some small values of λ
Ω, the global
synchronization (a « 1) does require a larger value for the coupling constant. However, we can see that the threshold values of λ
Ω for the onset
of global synchronization are also compatible with our optimization procedure, in the sense that the smallest threshold corresponds to the optimal set, and the largest to the worst one. Published in [15]. . . 87
Figure 36 – Synchronization diagrams for r and a as functions of λ
Ω, for
f F
Ω “0.2 in
the ER network described in this section. The blue circles, red crosses, and green stars correspond, respectively, to the optimal, random, and worst subset C of forced nodes. This is a situation where the system is rather intensive to the external force. The synchronization diagrams are similar for the three cases. Notice, in particular, that a « 0, meaning that the synchronized state does not follow the pace of the external force. This situation is essentially the same one of the F “ 0 case discussed in Chapter 5. Published in [15].. . . 88
UNICAMP Universidade Estadual de Campinas (University of Campinas)
IMECC Instituto de Matemática, Estatística e Computação Científica (Institude of Mathematics, Statistics and Cientific Computing) CNMAC Congresso Nacional de Matemática Aplicada e Computacional
(National Congress of Applied and Computational Mathematics) CNPq Conselho Nacional de Desenvolvimento Científico e Tecnológico
(National Council for Scientific and Technological Development)
ER Erdös-Rényi BA Barabási-Albert SW Small-world WS Watts-Strogatz SF Scale-free SL Stuart-Landau
Introduction . . . 18
1 COMPLEX NETWORKS . . . 20
1.1 Graphs . . . 20
1.1.1 Adjacency Matrix . . . 21
1.2 Network Topology . . . 22
1.2.1 Average degree and degree distribution . . . 22
1.2.2 Path length and network diameter . . . 23
1.2.3 Clustering coefficient . . . 24
1.3 Network Models . . . 25
1.3.1 Erdös-Rényil model . . . 25
1.3.2 Watts–Strogatz small-world model . . . 27
1.3.3 Barabási-Albertt model . . . 28
2 SYNCHRONIZATION . . . 31
2.1 The Kuramoto model . . . 31
2.1.1 Order parameter . . . 31
2.1.2 Critical coupling . . . 33
2.2 Stuart-Landau oscillators . . . 36
2.2.1 Stability of the stationary solutions. . . 36
2.2.2 Stuart-Landau networks . . . 38
2.3 Network structure and phase transitions . . . 39
2.3.1 Second order phase transition . . . 40
2.3.2 Random coupling . . . 41
2.3.3 First order phase transition . . . 43
3 THE OTT-ANTONSEN ANSATZ AND DIMENSIONAL REDUC-TION . . . 47
3.1 The ansatz . . . 47
3.1.1 Phase distribution. . . 48
3.1.2 Reduced equations . . . 49
3.1.3 Initial conditions . . . 51
3.2 Lorentzian frequency distribution. . . 52
3.2.1 Initial conditions . . . 53
4.1 Dimensional reduction . . . 56
4.1.1 Collective coordinate ansatz . . . 57
4.1.2 Reduced system . . . 58
4.2 Ansatz results validation. . . 60
4.2.1 Uniform frequency distribution . . . 60
4.2.2 Normal frequency distribution . . . 63
5 DIMENSIONAL REDUCTION AND OPTIMAL SYNCHRONIZA-TION . . . 68
5.1 Optimization model . . . 68
5.1.1 Collective coordinates model . . . 69
5.1.2 The algorithm. . . 71
5.2 Results . . . 71
5.2.1 Optimization effects on synchronization . . . 72
5.2.2 Further applications. . . 73
6 OPTIMAL SYNCHRONIZATION OF PARTIALLY FORCED KU-RAMOTO OSCILLATORS . . . 74
6.1 Forced Kuramoto system . . . 74
6.1.1 The dimensional reduction approach . . . 76
6.1.2 Critical force and synchronization threshold . . . 80
6.1.3 The optimization scheme . . . 82
6.2 Results . . . 84
6.2.1 Numerical simulations . . . 84
6.2.2 Final remarks . . . 88
7 CONCLUSION . . . 90
Introduction
The study of network synchronization as a distinct subject essentially began in 1975 as Yoshiki Kuramoto proposed a mathematical model to describe the macroscopic behaviour of chemical oscillators with the individual dynamics of each oscillatory reaction as a starting point [1]. The expansion of local individual dynamics to a limit in which the system can be described as a collective entity is nothing new in itself, as it is the very principle which connects statistical physics to thermodynamics. In fact, many techniques used in statistical physics, such as the use of order parameters to analyse phase transitions between incoherent and ordered states [2,3].
The phenomena of network synchronization has been extensively studied in the past years, due to being present in a great number of natural and technological systems. Some famous applications include the study of patterns and self-organization in the growth of vertebrates [4], the effect of external stimuli on circadian rhythm [5, 6], and the transmission of eletromagnetic signals in power grids organized in networks of elements such as transformers and generators [7]. An elucidating example on how this topic is relevant for real applications is that of neural networks [8,9], in which both the lack and excess of synchronization in the synapses of neuron clusters [10] may point out to a series of disorders varying from auditory-motor problems [11] and epilepsy [12].
More specifically, one may describe synchronization as a phenomena which emerges spontaneously in populations of connected oscillators. It can be identified as the state in which all oscillators evolve together, closely grouped around the same mean phase. The disposition of their connections are described by complex networks, which are essentially graphs with intricate topological properties that can usually be described by their statistical properties [13]. This system can be generalized in several forms, as by considering the amplitude of oscillations as in Stuart-Landau networks [14], or through the introduction of an external periodic driving force [15, 9] for example. Since this framework is mainly concerned with macroscopic collective behaviour, it is common to take on systems with a large number of oscillators, which in turn increases dramatically the complexity of the mathematical models describing them. The simplification of such models, as well as their use in improving system synchronization is the main topic of this dissertation, as further explained in the overview on how this work is organized.
We start by defining complex networks and discussing its main topological features through a series statistic parameters, and going through the most relevant network evolution models (Erdös-Rényi, Watts-Strogatz and Barabási-Albert) for this study. Chapter 2 follows with an overview on the topic of synchronization, including a detailed description of the Kuramoto model and its main parameters. We also present the framework of Stuart-Landau networks, which also includes the amplitude of the oscillators and is a good mean in which to present ideas of bifurcations, stability analysis [16], and asymmetry induced synchronization (AISync) [17, 18, 19]. The transitions from the incoherent to the ordered (synchronized) states of such systems and their variations are also presented as we discuss first and second order phase transitions, and how correlations in the topology and dynamics of a network may affect or induce one or the other [20, 21]. In particular, Section 2.3.2 focuses on our original work [22], which gives insight over the case where the coupling strength is not the same for every connection, but follows a Γ-distribution instead.
The following two chapters then present the dimensional reduction approach on dealing with large populations of oscillators. Chapter 3 presents the Ott-Antonsen ansatz [23,24], which is able to dramatically reduce the number of differential equations necessary to describe both the collective and individual behaviour of a fully connected Kuramoto network by cleverly restricting its initial conditions. This approach was then expanded by Gottwald in [25] (see also [26]) as he introduced the concept of collective coordinates, which sets new coordinates to describe the system based on an approximated parametrization of its already know results, as we see in Chapter 4. The Gottwald ansatz allows us to describing the system synchronization (and even some of its individual behaviour) with very few equations, which then allowed R. Pinto and A. Saa [27] to develop the optimization scheme to improve synchronization, as shown in Chapter 5. This optimization technique rewires the network connections based on the maximization of the quantity ωTLω, where
L is the system’s Laplacian and ω its natural frequencies vector.
Finally, in Chapter 6 we present our original work published in [15] which tackles the problem of global synchronization in partially forced Kuramoto networks. We use the tools from the previous chapters to choose collective coordinates for the system throughout which we are able to draw certain thresholds relating the possibility of network syncrhonization with the external force magnitude and the fraction of forced oscillators. The main result of this work then takes such findings to craft an optimization scheme which successfully improves network synchronization simply by rewiring its connections based on the maximization of a quadratic function.
1 Complex Networks
The technological advancement of the past years has dramatically increased our capability of dealing with complex sets of data and information over systems. The structure of complex networks has been particularly important for its ability in modeling systems with entities connected to one another in a way that is neither completely regular or random. For example, a social network in which people are connected by whether they performed some economic transaction with one another has an obvious degree of randomness, but can also present connection patterns through geographic proximity, age, and other demographic factors.
Complex networks are described by graphs, which are objects in discrete mathematics used precisely to describe systems of a series of connected entities. As the networks being described acquired a more complex topology, the use statistical tools to analyse such systems became indispensable, as they already presented a great a degree of success in physical frameworks such as Bose-Einstein condensation. Eventually the compilation of the many tools used to study this systems gave rise to the field we know today as complex networks, which nowadays has applications spread throughout many different areas, from the social and biological study of epidemics up to the modeling of the Internet itself. In this chapter, we aim to explain the concept of complex networks, starting by the definition of graphs and continuing by demonstrating its main topological/statistical features, providing examples, and presenting the most relevant network models in the context of this project (Erdös-Rényi, Watts-Strogatz and Barabási-Albert).
1.1
Graphs
Networks are represented by the mathematical objects known as graphs. A graph GpV, Eq is composed of a set V of vertices (or nodes) and the connections E between them, which are called edges. E is therefore described as a subset of ordered pairs of elements of V , making these two sets necessarily disjoint. A visual representation of a graph can be seen in Figure 1.
As it has already been pointed out, the structure of nodes and edges accom-modates a wide variety of real phenomena. For example, a network of protein-protein interactions has the proteins themselves as the vertices, while their edges represent the pairwise biochemical interactions that regulate the proteins’ production and degradation [28]. Other real network examples may be found in Table 1.
Figure 1 – Visual representation of an undirected graph, with the nodes V represented by circles and assigned an unique number, and the edges E shown as the gray lines between them.
Examples Nodes Edges
WWW webpages hyperlnks
power grids generators/transformers/substations transmission lines
neural networks neurons synapses/gap junctions
cellular networks substrates (AT P, ADP, H2O) chemical reactions
Table 1 – Table of real network examples with their corresponding graph elements.
1.1.1
Adjacency Matrix
A set of N nodes may be represented by a N-dimensional vector, with each node uniquely assigned to an index. Therefore, the edges of a graph may be represented by a matrix Aij such that:
Aij “
$ &
%
1 if nodes i and j are connected
0 otherwise . (1.1)
Aij is called the adjacency matrix, and it mathematically represents the
net-work’s topology. When A is symmetric (Aij “ Aji) the network is undirected, since there is
no preferential direction in the connection between any two nodes. If A is not symmetric, the network is then said to be directed. One may notice, however, that form presented in
1.1only works if the connection between any two nodes has the same strength, since all nonzero elements of A are unitary. Such networks are classified as unweighted. Networks in which some edges have different weights than others are called weighted networks, and requires us to generalize the definition of the adjacency matrix for its mathematical representation
Aij “weight of connection i Ñ j. (1.2)
(a) Unweighted and undirected
(b) Unweighted and di-rected
(c) Weighted and undi-rected
(d) Weighted and
di-rected
Figure 2 – Network types, where arrows represent directional edges, and darker edges have more weight (strength) than lighter ones.
1.2
Network Topology
The topology of a network may be analysed through a variety of what are called network measurements, each of them holding different information over how the network is organized. The most relevant measurements to this project shall be stated throughout the chapter. For more information on the subject, see [29].
1.2.1
Average degree and degree distribution
The first measurement we shall turn our attention to is the node degree ki,
defined as the number of edges coming from node i and given by
ki “ N
ÿ
j“1
for undirected networks. An individual node degree measures only a single node and its surroundings. To get a glimpse of the network’s global picture, we must analyse the average
node degree xky “ 1 N ÿ j kj “ 1 N ÿ ij Aij, (1.4)
which gives us the average number of connections per node in a network. Despite its usefulness, the information held by xky is still insufficient to provide a good overview of the network, since it tells nothing about how the accounted edges are distributed. The same average degree can represent both a network with all edges having xky connections, and one with a few highly connected nodes while having most nodes holding little to no edges at all. To account for this heterogeneity on edge distribution, we also describe the network by its degree distribution P pkq, where
P pkq “probability that a randomly selected node has degree k, (1.5)
so that 0 ď P pkq ď 1. We shall take a deeper look into this property and its significance in a later section, as we dive more specifically into a couple of network models.
1.2.2
Path length and network diameter
The geodesic distance dij between two nodes is defined as the smaller number of
edges one must go through to from node i to node j. It is worth noticing that for directed networks, dij ‰ dji, whereas undirected ones obviously have dij “ dji. Since the number of
possible edges in a network of N nodes is NpN ´ 1q, we may calculate the average path
length as ` “ 1 N pN ´1q ÿ i‰j dij. (1.6)
A shorter ` gives us what is called small world effect, where most nodes are closely related to each other, as in the famous six degrees of separation theory proposed by Stanley Milgram [30]. The distance between the two most distant nodes is called network diameter, and essentially represents the linear size of the network. The difference between the diameter and the average node degree of a network also gives some insight over how homogeneous (or not) the network might be.
1.2.3
Clustering coefficient
Another interesting oversight one can have in this context has to do with how
clustered the network is. To get a better idea about what that means, we first introduce
the local clustering coefficient
Ci “ 2e i
kipki´1q
, (1.7)
where ei is the number of (undirected) edges between the neighbors of node i, and kipki´1q
is number of possible node pairs that can be formed with two nodes that are both connected to i. Hence, Ci accounts for the chance of two nodes connected to the same central node
being connected to each other, forming a closed triplet as shown in Figure 3(a).
(a) Closed (b) Open
Figure 3 – Types of triplets.
For an analysis of the network as a whole, we may consider the average xCiy,
or the global clustering coefficient (also referred to as transitivity), which is defined as
C “ number of closed triplets
number of all triplets (open and closed). (1.8)
Regions with a high number of closed triplets close to each other are called clusters, as the one in the center of Figure 4(b), which intends to give visual insight over the influence of clustering in the network’s organization.
(a) Smaller clustering coef-ficient
(b) Higher clustering coef-ficient
Figure 4 – Networks with the same number of nodes and edges, but dramatically different clustering coefficients. The regions highlighted in gray represent the closed triplets used to compute clustering in equation 1.8.
1.3
Network Models
Now that we’ve stated some of the main topological features of graphs, we may gain some more insight about what a complex network is. As it was already anticipated in the beginning of this chapter, complex networks are generally defined as networks with non-trivial topology. This means essentially that we are able to find a mathematical model to describe the network’s evolution, i.e., how the network is formed, which can result in finding expressions to the measurements from the previous section. The inverse reasoning, however, where an evolution model is proposed to describe already known topological features, is what mainly happens when trying to describe real-life networks. The most important models for this project shall be explored throughout this section, which will exemplify and hopefully elucidate the concept of complex networks.
1.3.1
Erdös-Rényil model
Generally speaking, a random graph is generated by a process in which its nodes, edges and/or connection patterns are defined throughout some random process. In a mathematical context, however, random-graphs almost exclusively refers to the Erdös-Renyi (ER) model [31] with a fixed number of nodes, while the presence of an edge between them is determined by a fixed probability. This construction method generates several interesting topological features represented by a set of expressions for the network measures, which shall be demonstrated shortly. The construction process for an ER network with N nodes and connection probability p goes as follows:
1. Start with a set of N disconnected nodes
2. Go through each of the N pN ´2 1q pairs of nodes and place (or not) an edge between them with probability p, independently of the rest of the network.
Therefore, constructing the graph in such manner gives an approximate number of pN pN ´1q 2
edges. For a visual example of this process, see the Figure 5.
Each node has a probability p of connecting with any of the N ´1 other vertices, so he average node degree for an ER network has the following expression
xky “ ppN ´1q « pN. (1.9)
The global clustering coefficient is essentially the probability that two nodes that are connected to a third node, are connected to each other. Since the probability that two neighbors have an edge between them in a random graph is p, we write
C “ p “ xky
N , (1.10)
which is a good approximation only for large networks, where the number of nodes is a big enough for the probability p to effectively represent the network structure. Most real networks, however, seem to have an almost independent relationship between the clustering coefficient and the network size specially for large N which should motivate us to keep looking into other network evolution models to better encompass real-life applications. In contrast, ER networks actually capture the average path length of many real networks [2], as (for a certain range of xky), ` scales as
` „ ln N
ln xky. (1.11)
One of the most interesting features of this model is its degree distribution. For a node with degree k, there are Ck
N ´1 possible pairings with other nodes, with each
pair having probability p of being connected, and p1 ´ pq of not having any edges between them. Hence, the degree distribution for an AR network is given by a binomial distribution
P pkq “ CN ´1k pkp1 ´ pqN ´1´k. (1.12)
In the limit for small p and large N, expression 1.12 becomes a Poisson distribution
P pkq “ exp
´xky
xkyk
k! . (1.13)
Random networks also present a second order percolation phase transition in xky “ 1, since for xky ą 1 the network stops behaving as a composition of trees and we have the appearance of a giant cluster. Furthermore, when xky ě lnpNq the graph becomes totally connected [32].
1.3.2
Watts–Strogatz small-world model
As the name suggests, small-world networks reproduce the so called small-world phenomena, and therefore shold present a small average path length, so that every node has a reasonably low geodesic distant to every other node. With the intention of capturing this low path length of real networks as well as their network size independent clustering coefficient, a new one was proposed by Watts and Strogatz [33]. This innovative small world model produces networks with more closed triplets to increase clustering by placing itself in between the fully structured topology of a regular lattice and that of a random graph. The construction process of a Watts-Strogatz network with N nodes, κN edges and rewiring probability p can be summarized as:
1. Start with a regular ring with each node connected to its first k neighbors 2. Randomly choose a edges to be rewired with probability p
Hence, p “ 0 describes a regular network, which becomes more random as p approaches 1, turning into a completely random graph for p “ 1 as Figure 6.
Figure 6 – Aspect of networks built through the Watts-Strogatz model for different values of the rewiring probability p. From [34].
The average path length ` in the WS model grows linearly with N for small networks, and logarithmically for high N [2]. To check whether the clustering coefficient is independent of the network size as we wished, we can compute C1
ppq. Starting with the regular network at p “ 0 with clustering Cp0q, we have that a closed triplet from the regular lattice will remain a closed as we increase p only if all three edges remain intact. Since the probability of an edge not being rewired in a WS process is p1 ´ pq, we find that
As it was shown in [35], this formula describes well the clustering behavior of WS networks, presenting only small deviations that go to zero as the network size increases. Hence, this model produces both the logarithmic ` for high N and linear for low N, and the high clustering coefficient (as in social networks, for example) that are usually present in networks.
1.3.3
Barabási-Albertt model
Despite this partial success of the Watts Strogatz model in depicting some network measures like clustering and path length aligned with the general observations of real life networks, they still fails to present the power-law degree distribution that is so predominant in real systems. In fact, power-law distributed networks are so abundantly found they earned their own name: scale free (SF) networks. Systems that present this scale-free structure include the modeling of electric power grids [36], epidemic spreading [37] and infection processes [38], gene and protein interaction analysis in cell-biology [39], social networks [40] and cooperation [41], as well as the documents and links of the World Wide Web [42].
Seeking a greater correspondence to reality, Barabási and Réka Albert proposed a SF network model [43] in which the network grows (or evolves) as it acquires more nodes/edges at each step of the process, dramatically diverging from other models (WS and ER, for example) in which the number of vertices and edges remains fixed. The construction of a BA network is divided into two parts, as it follows:
1. Growth: Start with a small number m ď m0 of nodes, and add a new node with m
edges at each step
2. Preferential attachment: The m edges of each new node will be connected to the existent vertices with the so called preferential attachment, where the probability P of connection depends on properties of the targeted node. For the classic BA model, the probability Ppkiq of connecting the new node to an already present node i is
proportional to the degree ki of that node is
Ppkiq “ ki N ÿ j kj . (1.15)
Figure 7 illustrates the probability Ppkiq of connection for each node before and after a
new node is added to the network. Some examples of BA networks for different values of
(a) Before adding the eighth
node to the network (b) After adding the new node(in orange)
Figure 7 – Node degree ki and respective preferential attachment probability Ppkiq for a
Barabási-Albert network evolution with m “ 2 before and after adding a new node with two new edges.
The first measure for the BA networks that we choose to highlight here is the average path length
` “ Aln N ´ B ` C, (1.16)
which is smaller than for random networks, and therefore more efficient in bringing nodes together.
(a) m “ 1 (b) m “ 2 (c) m “ 3
Figure 8 – Barabási-Albert networks with N “ 20 for different values of m. The clustering coefficient presents a slower decay with the network size N than the N´1 rate of the ER model, being
C „ N´0.75. (1.17)
The true power of the BA model, however, is shown through it’s degree-distribution, which corresponds to a scale-free network with
Bki
Bt “ mPpkiq ñ P pkq „2 ?
where γ “ 3. As it was already discussed in the beginning of this section, SF networks correspond to a wide variety of real systems, thus the importance of being able to describe its evolution process. Different values of γ in the power-law distribution may be obtained by generalizing the preferential attachment rule. For example, we may write
Ppkq “ A ` kα, (1.19)
where A is called initial attractiveness. The addition of A assures that Ppkq ‰ 0 even if
k “ 0, i.e., no node that starts unconnected will stay unconnected.
Another great example of this type of network evolution is the Bose-Einstein condensation, where nodes are energy levels Ek, an edge between i and j correspond to a
pair of non-interacting particles, one with energy Ei and the other with energy at Ej.
(a) Energy level representation, where each circle corresponds to a particle with its corresponding energy, while the boxes symbolises the edges represented by each of the surrounded pairs
(b) Graph representation correspond-ing to the particle-energy distribution in 9(a)
Figure 9 – Energy level and graph schematics of a Bose-Einstein condensation network model with m “ 2, with three initial levels E1, E2 and E3 (pink) with two
particles (and therefore two edges) each. A new level E4 (orange) is introduced,
and with it two new edges (dashed lines) are placed with attachment the probability Ppkq described in 1.20.
Adding a new node to the system means adding a new energy level El and 2m new particles,
m of those being in level El, as. it’s shown in Figure 9. The preferential attachment
probability for such system is
Ppkq “ e
´βik
i
ř e´βi, (1.20)
for which the energy level occupation npEq assertively obeys the Bose statistics as t Ñ 8 [2]:
npEq “ 1
2 Synchronization
Synchronization is a phenomenon that may emerge in populations capable of dynamically interaction between its members, and can be broadly defined as the process throughout which all elements of the system start presenting their action patterns at the same time or rate. Though it may happen for other types of dynamical systems, as for instance the chaotic systems approached in the iconic work of Percora, Caroll and Thomas [44], in this project we shall focus solely on the synchronization of interconnected oscillators. Some chemical reactions present a phenomenon known as chemical oscillations, which happens when the concentration of one or more components present a periodic behaviour to maintain equilibrium, as in the Briggs–Rauscher and the Belousov–Zhabotinsky reactions for example. For the concentration as a whole to oscillate, every individual reaction in the flask must share a common phase and frequency, i.e., their oscillation must be synchronized. Motivated by chemical oscillations, in 1975 Yoshiki Kuramoto [1] proposed to model such systems as a network of interconnected harmonic oscillators as means to study its synchronization.
2.1
The Kuramoto model
The equation for a single harmonic oscillator with phase θ and natural frequency
ω is 9θ “ ω, so N oscillators coupled through a sinusoidal function may be described by
9 θi “ ωi` λ N N ÿ j“1 Aijsin pθj ´ θiq, (2.1)
where Aij is the network’s adjacency matrix, and λ the coupling strength.
2.1.1
Order parameter
The synchronization of this system is quantified by the order parameter r and mean phase ψ, which are calculated as
rptqeiψptq“ 1 N N ÿ j“1 eiθjptq. (2.2)
In this expression, every oscillator in the system has its phase represented in the complex unit circle (eiθj), so that the complex number rptqeiψptq represents the centroid of this
system. The order parameter has bounds 0 ď r ď 1 such that r Ñ 0 (Figure 10(a)) means the oscillators are evenly spread throughout the unit circle, and therefore presents no
synchronization whatsoever. As r Ñ 1 (Figure 10(c)) we approach a globally synchronized system, since it would mean all oscillators share approximately the same phase, while
r ă1 (Figure 10(b)) means the system is only partially synchronized.
(a) r Ñ 0 (b) 0 ă r ă 1
(c) r Ñ 1
Figure 10 – Synchronization states.
The natural frequencies ωi are usually drawn from a distribution gpωq, which is
often is symmetric, and shall be considered as such for this study. Also, we can always shift the frequency distribution so that its average is zero without loss of generality. Furthermore, by assuming gpωq symmetric with xωy “ 0 in 2.2 one should easily notice that r Ñ 0 for small λ, i.e., synchronization does not occur if the coupling strength is too small, which suggests the existence of a synchronization threshold for λ. In practice, the value of r for isolated time instants t holds little useful information, since the behaviour of subsequent instants might be completely different from t. Hence, for relevant synchronization data, we take the average of r over a time interval, which is usually a period of oscillations after a long time has passed and the system has stabilized
xry “ 1 ∆t
żt0`∆t
t0
2.1.2
Critical coupling
Throughout his pioneer work [1], Kuramoto considered a all-to-all network, i.e., a network where all nodes are connected to each other. Therefore, the adjacency matrix obeys Aij “1 @i, j , i ‰ j and may be omitted from2.1 so we can substitute the
expression 2.2 for r in 2.1 to write the ODE’s governing the system as
dθi
dt “ ωi` λ r sin pψ ´ θiq. (2.4)
To improve our analytical approach to this system, we define the probability density
f pθ, ω, tq as the probability that an oscillator of natural frequency ω has phase θ at time t.
Thus, from definition we know that f must be normalized ż2π
0
f pθ, ω, tq dθ “ gpωq. (2.5)
Considering that the network structure should not be altered throughout our analysis, i.e., no nodes/edges are added, removed, or changed, we conclude that f must also obey the continuity equation defined by
Bf Bt `
B
Bθ tvf u “0, (2.6)
where vpθ, ω, tq “ ω ` λr sin pψ ´ θq, as in the Kuramoto expression 2.4 for the angular speed 9θ. Taking advantage of f, the continuous limit for the order parameter turns to
re´iψptq “ żπ ´π ż8 ´8 eiθf pω, θ, tq dωdθ. (2.7)
By setting r “ 0 in equation 2.7, we find that the fully incoherent state of this system corresponds to an uniform phase distribution
f pθ, ω, tq “ gpωq
2π . (2.8)
For the partial synchronization regime, we solve the continuity equation 2.6for 0 ără 1. Since we’re looking for the stabilized state of system, we consider the stationary regime Bf
Bt “0, which allows us to find an analytical expression for the phase probability
distribution f pθ, ωq “ gpωq $ ’ & ’ % δ ! θ ´ ψ ´arcsin ´ω λr ¯) if |ω| ď λr C |ω ´ λr sin pθ ´ ψq| otherwise , (2.9)
where C “ ?ω2´λ2r2
2π . Expression2.9divides the oscillators into two groups: the synchronized
oscillators with |ω| ď λr that evolve with the same mean phase ψ, and the drifting oscillators |ω| ą λr which keep its phase incoherence throughout time.
An expression for the order parameter is found by simply substituting equation
2.9 for fpθ, ωq in 2.7. We keep considering a symmetric distribution of natural frequen-cies gpωq “ ´gpωq, while noticing that f is periodic in θ, i.e. fpθ ` π, ´ωq “ fpθ, ωq. Consequently, the drifting term of r vanishes, which gives
r “ λr
żπ{2
´π{2
gpλr sin θq cos2θ dθ. (2.10)
Our main interest with this expression lies in figuring out how r behaves in terms of the coupling strength λ. For λ “ 0 we naturally have r “ 0 as one would expect since the non interaction between the oscillators makes it impossible for them to synchronize. Since the network is fully connected and doesn’t change its topology throughout this process, we may also expect r Ñ 1 as λ Ñ 8. The function should also be crescent, as physically speaking a greater connection strength in this context also means all oscillators have more influence over all the others, which enhances their collective behaviour. We therefore define the critical coupling λc as the coupling strength value for which the order parameter r
starts to increase, i.e.
rpλcq “ lim
rÑ0`rpλq. (2.11) Making r Ñ 0` in equation 2.10 we find that
λc “ 2
πgp0q. (2.12)
Assuming g”p0q ă 0 and expanding r in powers of λr, we find that near the critical point,
r grows as r „ pλ ´ λcqβ with β “ 1{2 [3], representing a second order phase transition,
which will be better explained further on. The behavior of r can then be represented in a
r ˆ λ graph, as it is shown in Figure 11for both the analytical and computer simulation
results for the globally connected case we are studying in this section, but also for some small-world networks. One can easily see that in all the cases displayed, r Ñ 0 for small λ, and after a critical value λc our order parameter starts increasing with rate depending
on the system’s properties such as frequency distribution an network topology, and even reaches r Ñ 1 in most cases (global synchronization might also be possible on the others, but for grater values of λ than were shown in the graph).
Figure 11 – Order parameter evolution in the Kuramoto model applied to all-to-all and small world networks. Adapted from [3].
Particularly in the globally coupled (all-to-all network) case, since all oscillator are connected to each other the first oscillators to synchronize are the ones with natural frequency ω closest to the mean natural frequency xωy, which we set as zero earlier in this section. As we increase the coupling λ the synchronizing population expands also prioritizing frequencies closer to xωy, making the incoherent group smaller at each step until every oscillator is synchronized. We emphasize here that the joining preference for nodes with natural frequency closest to xωy occurs almost exclusively in the all-to-all network with the traditional Kuramoto dynamics described in 2.1, as in other systems the different network topology or dynamics may play a very important role when determining which oscillators will remain incoherent or not.
This natural frequency order in the attachment to the synchronized group, or even the fact that such group maintains the same mean phase as it grows, relies not only on the simple dynamic of the traditional Kuramoto model 2.4 considered, but also on the globally coupled topology of the network. The same dynamics in a network with several clusters, for example, might synchronize each cluster close to its own mean frequency before all clusters synchronize among themselves. The foundation provided by Kuramoto can be therefore expanded to accommodate more complex systems and hopefully uncover more interesting and real-life applicable phenomena regarding synchronization.
2.2
Stuart-Landau oscillators
Stuart-Landau (SL) oscillators differ from simple harmonic oscillators on the sense that their amplitude fptq is also a function of time, not just the phase θ as in the case of the Kuramoto model. We will first consider a single SL oscillator, but in Section
2.2.2we will couple them in a network, similar to the Kuramoto system. Since we must also incorporate the amplitude ρ in our description, it is only natural to express the state of an SL oscillator as a complex number z “ ρkeiθ, θ corresponding to its phase. The SL
model is then described by the equation 9
zptq “ pµ ` iωqz ´ pβ ` ibq|z|2z. (2.13)
where ω is the natural frequency of the oscillator, while µ and b are simply control parameters. In terms of ρ and θ, this translates to
$ ’ & ’ % 9 ρ “ µρ ´ βρ3 “ ρβˆ µ β ´ ρ 2 ˙ 9 θ “ ω ´ bρ2 , (2.14)
which gives us the Jacobian
J pρ, θq “ « µ ´3βρ2 0 2bρ 0 ff . (2.15)
The SL equations are in fact the general equations for the vicinity of a bifurca-tion, as it shall be more deeply explored in the following subsecbifurca-tion, which occurs when a stability point turns into a limit cycle by changing a control parameter. For this to be better understood, we should go through how the stability of this system can be analyzed.
2.2.1
Stability of the stationary solutions
The stability of the stationary solutions of a system may be determined by calculating its Lyapunov exponent Λ, which intends to characterize the maximum expo-nential rate for which trajectories in the phase space distance or approach each other locally. For a system
9
x “ f px, tq, (2.16)
if J0 is the Jacobian of fpx, tq at a stationary point x0, then the distance δx between two
trajectories near x0 may be locally taken from the linearized system as
which obviously holds the solution δx0 “ eJ0t. Therefore, along the jth eigenvector (with
correspondent eigenvalue λj) of J0, two trajectories will locally deviate from each other as
||δx0|| „ eλjt. (2.18)
We choose λ1 “0 to be the eigenvalue correspondent to the synchronized state, so the
deviation from such state depends only on the eigenvalues λj with j ě 2. Hence, equation
2.18 shows that the largest local deviation from synchronization will correspond to the eigenvalue Λ with the largest real part, i.e.
Λ ” max
jě2 Repλjq, (2.19)
If Λ ă 0, then since all other eigenvalues are smaller (more negative) than Λ, all nearby trajectories will inevitably approach x0 at an exponential rate, so x0 is said to be a
stable equilibrium point. Otherwise, for Λ ą 0, the equilibrium point is unstable, since the
neighborhood paths exponentially deviate from it.
Going back to the SL system, in the stationary point z0 “0, the Jacobian has
eigenvalues λ1 “ 0 and λ2 “ µ. By treating µ as a control parameter, we see that for
µ ă 0 z0 is a stable point, whereas at µ “ 0 the system crosses a threshold and z0 becomes
unstable for µ ą 0. A limit cycle of a given dynamical system corresponds to a closed trajectory in the phase space such that all nearby trajectories converge to it as t Ñ 8 if the cycle is stable, or as t Ñ ´8 if it’s unstable, meaning that either all trajectories are attracted to this periodic orbit, or are repelled from it. The SL system has a limit cycle given by |zα| “ α2, where α “ µ{β. The Jacobian for this orbit has eigenvalues λ1 “ 0
and λ2 “ ´2µ, and therefore we have two cases: β ą 0 and β ă 0. For β ą 0, the limit
cycle only exists for µ ą 0, since α2 must be positive, and hence it’s stable. Thus, as
we increase the control parameter µ, the stable equilibrium point z0 turns unstable after
µ “ 0, where the stable spiral zα appears around it. The system is then said to have gone
through a supercritical bifurcation at µ “ 0, since a stable point gave rise to a stable cycle and became unstable at the bifurcation point.
For β ă 0, the closed orbit only exists for µ ă 0, and therefore we have the opposite case: as µ decreases, the unstable point becomes stable and gives rise to an unstable spiral around itself at the supercritical Hopf bifurcation at µ “ 0. A Hopf
bifurcation is defined by a point changing stability at the critical point while also creating
a spiral of opposite stability around it. The supercritical case refers to when the spiral created at the bifurcation point is stable. If the created cycle is unstable, we say the (Hopf) bifurcation is subcritical.
Figure 12 – Supercritical Hopf bifurcation at µ “ 0, when a stable point becomes a stable limit cycle with radius growing as we increase the parameter µ. Since both the point and the limit cycle are stable, we can also see nearby trajectories being attracted to them. From [45].
2.2.2
Stuart-Landau networks
We may without loos of generality re-scale ρ the original expression to fix β “ 1 so that for a network of N coupled SL oscillators we have
9 zkpz, tq “ tα2` iωk´ |zk|2uzk` λ N ÿ j“1 Aijpzj ´ zkq, (2.20)
where ωi is the natural frequency of oscillator i, α is the SL parameter that defines the
limit cycle |zk|2 “ α2 and its stability, Aij is the adjacency matrix of the network, and λ
is the coupling strength. Note that we also set b “ 0 from the original expression so that the ρ2 term would not appear in the 9θ expression. In this case, synchronization is only
possible if we induce asymmetry in the system by giving oscillators different values for µ. This phenomena called AISync (Asymmetry Induced Synchronization) [17, 18, 19] can be regarded as the converse of symmetry breaking, since in the latter a symmetry in the system causes asymmetry in the dynamics (chimera states).
To find the SL network ODEs in polar coordinates, we apply the transformation
zk“ ρkeiθk in equation 2.20 and obtain
9 ρipρ, θ, tq “ ρipα2´ ρ2iq ` λ N ÿ j“1 Aijtρjcos pθj´ θiq ´ ρiu , (2.21) 9 θipρ, θ, tq “ ωi` λ N ÿ j“1 Aij ρj ρi sin pθj´ θiq. (2.22)
The Jacobian then has a zero eigenvalue corresponding to the synchronous state, and its elements for a network with no self-edges (Ajj “0 @j) are
$ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % B 9ρi Bρj “ pα 2
´3ρ2i ´ λAijqδij ` λAijcos pθj´ θiq B 9ρi Bθj “ ´λAijρjsin pθj´ θiq B 9θi Bρj “ λ ρiAijsin pθj ´ θiq B 9θi Bρj “ λAij ρj ρi cos pθj´ θiq . (2.23)
We may therefore apply the stability analysis to the network and get a sense on the behavior of the synchronized state.
2.3
Network structure and phase transitions
As it was discussed before, one of the aspects that can be funther generalized from the Kuramoto model is the network itself. Even considering the same model of harmonic oscillators with sinosoidal coupling while switching from the previous all-to-all network to more complex structures as the network models introduced in Chapter 1 may hold interesting changes in the system behavior. Here, we shall deepen the understanding of the relationship between network structure and synchronization in the context of phase transitions, as means to elucidate this other important topic on synchronizing systems as well.
Phase transitions are generally associated with order, as the system goes from a disordered to an ordered state or vice versa, and can classified as first order or second order. The transition between such states can be measure according to an order parameter, which as the name suggests should quantify how ordered your system is as another parameter is varied. The behavior of the order parameter when transitioning from one state to the other dictates whether the transition is of first or second order, as we shall explain shortly. In the context of complex networks of oscillators, we will be dealing with mainly two phases: the synchronized (r Ñ 1) and the incoherent (not synchronized) states, the first corresponding to the state where all (or at least the majority) of oscillators share a common phase (phase
lock), while the second depicts just the opposite of that. Both these states are measured
It’s worth noticing that other states where coherence and incoherence coexist not as a transients between the incoherent and the fully synchronized states, but rather being a stable state itself, also exist and are called chimera states. The appearance of chimera states within a perfectly symmetric network structure constitutes the symmetry
breaking which characterizes the phenomena known as Asymmetry induced symmetry
(AISync), which proved itself to be quite abundant in nature and for that has motivated several studies such as [17,18,19]. However, our focus when talking about phase transitions here will be mainly to refer to the stable incoherent and synchronized states, leaving the research of this hybrid phase to other capable hands.
2.3.1
Second order phase transition
A phase transition is said to be of second order if the order parameter evolves continuously from one state to the other. A well known example is the ferromagnetic phase transition, where the order parameter is the magnetization Mz (assuming ˆz as the
direction of the total magnetic moment) measured in terms of the temperature T . The material is classified based on how aligned the magnetic moments of its atoms are, such that if all neighboring magnetic spins point in a common direction the material is said to be ferromagnetic, whereas the case in which the magnetic moments directions are mostly random (in the absence of an external magnetic field) is classified as paramagnetic. At lower temperatures, the materials is in the ferromagnetic state and presents a magnetization
Mz ą0. As we increase the temperature, the atoms get more agitated and Mz continually
decreases until reaching a critical temperature TC, also know as the Curie temperature,
after which Mz “0. Since there’s a direct correlation between the temperature, the atoms
alignment and the total magnetic moment Mz, the phase transition is continuous in the
Mz ˆ T diagram, and is therefore of second order.
As we saw in Figure 11 for the all-to-all and small network Kuramoto models, many systems of connected oscillators exhibit a transition of second order when going from the incoherent to the synchronized state, in which the order parameter curve in the
r ˆ λ diagram presents the format presented in Figure 13. In this case, increasing the
coupling strength gradually makes the oscillators phases approach the common mean, until they are finally synchronized (r Ñ 1). Even though the shape of the second order transition curve keeps its shape, some of its characteristics change based on the network’s topology, not only the critical coupling itself but also how fast does the curve increases until it asymptotes to the synchronized state. In Figure 11 for example, we see that the first network to synchronize is the completely connected one, whereas the SW networks synchronize faster the more random they are, i.e., the curve ascends faster for larger values of p.
Figure 13 – Second order phase transition for the synchronization a m “ 2 Barabási–Albert network of N “ 200 oscillators in the Kuramoto model.
Besides the type of network, we can also influence its topology by assigning different weights to each of the connections. In the case of a Stuart-Landau network, this means we should consider the system as
9 zk“ pα2` iωk´ |zk|2qzk` N ÿ j“1 λkjAkjpzj ´ zkq, (2.24)
where λij represents the coupling strength of the connection between i and j. It should
therefore be interesting to check how different sets of λkj may affect the synchronization,
specially in cases where λkj are picked from well-known distributions with parameters that
can be quantified for the purposes of our studies.
2.3.2
Random coupling
Let us suppose the coupling strength is a random variable with a Γ-distribution. Its probability density function is
f pλ; s, θq “ λ
s´1e´λ{Θ
ΘsΓpsq , (2.25)
where s is called the scale parameter, and Θ the shape parameter, which relate to the form of the distribution as shown in Figure 14. For our original work presented in [22], we chose the Γ-distribution, since this function is always positive. Connections with negative
λ discourage the synchronization between the nodes connected by them and therefore the
problem it presents escapes form the intended scope of this work.
The presence of both scale and shape parameters is also a great advantage in this study, once they not only provide insight on what the Γ-distribution looks like, but also relate to its mean value xλy and standard deviation σλ simply by
$ & % xλy “ sΘ σλ “ sΘ2 . (2.26)
