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The Relationship between Factor Regression Model, Factor Model and Regression Model

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			'''Synchronization networks''' are also often known as "networks of coupled [[dynamical systems]]". Both of these refer to networks connecting oscillators, where oscillators are nodes that emit a signal with somewhat regular (possibly variable) frequency, and are also capable of receiving a signal.

			Particularly interesting is the [[phase transition]] where the entire network (or a very large percentage) of oscillators begins pulsing at the same frequency, known as synchronization.

			The synchronization network then becomes the substrate through which synchronization of these oscillators travels.

			Since there is no central authority organizing nodes, this is a form of [[self organizing system]].

			== Definition ==

			Generally, [[oscillators]] can be biological, electronic, or physical. Some examples are [[fireflies]], [[crickets]], [[cardiac muscle\|heart cells]], [[lasers]], [[microwave]] oscillators, and [[neurons]]. Further example can be found in many domains.

			In a particular system, oscillators may be identical or non-identical. That is, either the network is made up of homogeneous or heterogeneous nodes.

			Properties of oscillators include: [[frequency]], [[Phase (waves)\|phase]] and [[resonant frequency\|natural frequency]].

			Network edges describe couplings between oscillators. Couplings may be physical attachment, or consist of some proximity measure through a medium such as air or space.

			Networks have several properties, including: number of nodes (oscillators), [[network topology]], and coupling strength between oscillators.

			== Kuramoto Model ==

			{{main\|Kuramoto Model}}

			Kuramoto developed a major analytical framework for coupled [[dynamical systems]], as follows:
			<ref>Steven H. Strogatz (March 2001). "Exploring complex networks". Nature 410 (6825).</ref><ref>Y. Kuramoto AND I. Nishikawa, Statistical macrodynamics of large dynamical systems. Case of a
			phase transition in oscillator communities, J. Statist. Phys., 49 (1987)</ref><ref>Mirollo, R. E., Steven H. Strogatz (December 1990). "Synchronization of pulse-coupled biological oscillators". SIAM Journal on Applied Mathematics 50</ref><ref>Steven H. Strogatz (September 2000). "From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators". Physica D: Nonlinear Phenomena 143</ref><ref>Strogatz, Steven (2003). Sync : the emerging science of spontaneous order. Hyperion. ISBN 978-0-7868-6844-5. OCLC 50511177</ref>

			A network of oscillators with varied natural frequencies will be incoherent while the coupling strength is weak.

			Letting <math> \theta_i(t)</math> be the phase of the <math>i</math>th oscillator and <math>\omega_i</math> be its [[natural frequency]], randomly selected from a [[Cauchy distribution\|Cauchy-Lorentz distribution]] as follows,

			<math> g(\omega) = \frac{\gamma}{\pi[\gamma^2 + (\omega - \omega_0)^2)}</math>, having width <math> \gamma </math> and mean <math> \omega_0 </math>,

			we obtain a description of collective synchronization:

			<math>\frac{d\theta_i}{dt} = \omega_i + \frac{K_{ij}}{N} \sum^N_{j = 1} \sin(\theta_j - \theta_i), i = 1, ..., N </math>,

			where <math>N</math> is the number of nodes (oscillators), and <math>K</math> is the coupling strength between nodes <math>i</math> and <math>j</math>.

			Kuramoto has also developed an "[[order parameter]]", which measures synchronization between nodes:

			<math> r(t) = \bigg\| \frac{1}{N} \sum^N_{j = 1} e^{i\theta_j(t)}\bigg\| </math>

			This leads to the asymptotic definition of <math>K_c</math>, the critical coupling strength, as <math>N \to \infty</math> and <math>t \to \infty</math>

			<math>
			r =
			\begin{cases}
			0, & K < K_c \\
			\sqrt{1-(K_c/K)}, & K \ge K_c
			\end{cases}
			</math>

			with <math>K_c = 2 \gamma</math>.

			Note that <math> r = 0 \to </math> no synchronization,

			<math> r = e^{i\theta} \to</math> perfect synchronization.

			Beyond <math>K_c</math>, each oscillator will belong to one of two groups:

			* a group that is synchronized.
			* a group that will never synchronize, since their natural frequencies vary too greatly from the synchronization frequency.

			== Network Topology ==

			Synchronization Networks may have one of many topologies. [[network topology\|Topology]] may have a great deal of influence over the spread of dynamics.<ref>Nature, Vol. 393, No. 6684. (4 June 1998), pp. 440-442</ref>

			Some major topologies are listed below:

			*[[Regular graph\|Regular Network]] This describes networks where every node has the same number of links. [[Lattice graph\|Lattices]], [[Network_topology#Ring\|rings]], and fully connected networks are some examples of this topology. {{main\|Regular graph}}

			*[[Random graph\|Random Graphs]] Developed by [[Paul Erdős\|Erdős]] and [[Alfréd Rényi\|Rényi]], these graphs are characterized by a constant probability of a link existing between any two nodes. {{main\|Random graph}}

			*[[Small world networks\|Small World Networks]] These networks are the result of rewiring a certain number of edges in regular lattice networks. The resulting networks have much smaller average path length than the original networks. {{main\|Small world networks}}

			*[[Scale-free network\|Scale-Free Networks]] Found ubiquitously in naturally-occurring systems, scale free networks are characterized by a large number of high-degree nodes. In particular, the degree distribution follows a power-law. {{main\|Scale-free network}}

			==History==

			[[Coupled_oscillation#Coupled_oscillations\|Coupled oscillators]] have been studied for many years, at least since the [[Wilberforce pendulum]] in 1896.
			In particular, pulse coupled oscillators were pioneered by Peskin in 1975 with his study of cardiac cells.<ref>Peskin, C. S., Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York University, New York, 1975</ref>
			Winfree developed a mean-field approach to synchronization in 1967, which was developed further in the [[Kuramoto model]] in the 1970s and 1980s to describe large systems of coupled oscillators.<ref>Winfree, A. T., Biological rhythms and the behavior of populations of coupled oscillators, J. Theoret. Biol., 16 (1967)</ref>
			Crawford brought the tools of manifold theory and bifurcation theory to bear on the stability of synchronization with his work in the mid-1990s.<ref>J.D. Crawford, J. Statist. Phys. 74 (1994) 1047.</ref>
			These works coincided with the development of a more general theory of coupled dynamical systems and popularization by Strogatz ''et al.'' in 1990, continuing through the early 2000s.

			== See also ==

			* [[Kuramoto model]]

			* [[Complex Networks]]

			* [[Coupled_oscillation#Coupled_oscillations\|Coupled Oscillators]]

			* [[Dynamical Systems]]

			* [[Statistical Physics]]

			* [[Self-organizing system\|Self-Organizing Systems]]

			== References ==

			<references/>

			== External links ==
			* [http://www.ted.com/talks/lang/eng/steven_strogatz_on_sync.html Steven Strogatz Ted Talk]
			* [http://tam.cornell.edu/faculty-bio.cfm?NetID=shs7 Strogatz @ Cornell]
			* [http://www.eecs.harvard.edu/ssr/projects/sync/ Self Organizing Systems Research Group at Harvard]


			[[Category:Oscillation]]

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