Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.

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Collective dynamics of small-world networks

There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

I. Symmetric functions II. Hall polynomials III. HallLittlewood symmetric functions IV. The characters of GLn over a finite field V. The Hecke ring of GLn over a finite field VI. Symmetric functions with two parameters VII. Zonal polynomials

Symmetric functions and Hall polynomials

The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems-be they neurons, power stations or lasers-will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks.

/pdf/exploring-complex-networks-2pdzihykyi.pdf

Exploring complex networks

This book has been written in a frankly partisian spirit-we believe that singularity theory offers an extremely useful approach to bifurcation prob- lems and we hope to convert the reader to this view In this preface we will discuss what we feel are the strengths of the singularity theory approach This discussion then Ieads naturally into a discussion of the contents of the book and the prerequisites for reading it Let us emphasize that our principal contribution in this area has been to apply pre-existing techniques from singularity theory, especially unfolding theory and classification theory, to bifurcation problems Many ofthe ideas in this part of singularity theory were originally proposed by Rene Thom; the subject was then developed rigorously by John Matherand extended by V I Arnold In applying this material to bifurcation problems, we were greatly encouraged by how weil the mathematical ideas of singularity theory meshed with the questions addressed by bifurcation theory Concerning our title, Singularities and Groups in Bifurcation Theory, it should be mentioned that the present text is the first volume in a two-volume sequence In this volume our emphasis is on singularity theory, with group theory playing a subordinate role In Volume II the emphasis will be more balanced Having made these remarks, Iet us set the context for the discussion of the strengths of the singularity theory approach to bifurcation As we use the term, bifurcation theory is the study of equations with multiple solutions

/pdf/singularities-and-groups-in-bifurcation-theory-3jfb7fkeug.pdf

Singularities and groups in bifurcation theory

1. Basic concepts.- 1. Preliminaries.- 2. Nilpotency and solubility.- 3. Subideals.- 4. Derivations.- 5. Classes and closure operations.- 6. Representations and modules.- 7. Chain conditions.- 8. Series.- 2. Soluble subideals.- 1. The circle product.- 2. The Derived Join Theorems.- 3. Coalescent classes of Lie algebras.- 1. An example.- 2. Coalescence of classes with minimal conditions.- 3. Coalescence of classes with maximal conditions.- 4. The local coalescence of D.- 5. A counterexample.- 4. Locally coalescent classes of Lie algebras.- 1. The algebra of formal power series.- 2. Complete and locally coalescent classes.- 3. Acceptable subalgebras.- 5. The Mal'cev correspondence.- 1. The Campbell-Hausdorff formula.- 2. Complete groups.- 3. The matrix version.- 4. Inversion of the Campbell-Hausdorff formula.- 5. The general version.- 6. Explicit descriptions.- 6. Locally nilpotent radicals.- 1. The Hirsch-Plotkin radical.- 2. Baer, Fitting, and Gruenberg radicals.- 3. Behaviour under derivations.- 4. Baer and Fitting algebras.- 5. The Levi?-Tokarenko theorem.- 7. Lie algebras in which every subalgebra is a subideal.- 1. Nilpotent subideals.- 2. The key lemma and some applications.- 3. Engel conditions.- 4. A counterexample.- 5. Unsin's algebras.- 8. Chain conditions for subideals.- 1. Classes related to Min-si.- 2. The structure of algebras in Min-si.- 3. The case of prime characteristic.- 4. Examples of algebras with Min-si.- 5. Min-si in special classes of algebras.- 6. Max-si in special classes of algebras.- 7. Examples of algebras satisfying Max-si.- 9. Chain conditions on ascendant abelian subalgebras.- 1. Maximal conditions.- 2. Minimal conditions.- 3. Applications.- 10. Existence theorems for abelian subalgebras.- 1. Generalised soluble classes.- 2. Locally finite algebras.- 3. Generalisations of Witt algebras.- 11. Finiteness conditions for soluble Lie algebras.- 1. The maximal condition for ideals.- 2. The double chain condition.- 3. Residual finiteness.- 4. Stuntedness.- 12. Frattini theory.- 1. The Frattini subalgebra.- 2. Soluble algebras: preliminary reductions.- 3. Proof of the main theorem.- 4. Nilpotency criteria.- 5. A splitting theorem.- 13. Neoclassical structure theory.- 1. Classical structure theory.- 2. Local subideals.- 3. Radicals in locally finite algebras.- 4. Semisimplicity.- 5. Levi factors.- 14. Varieties.- 1. Verbal properties.- 2. Invariance properties of verbal ideals.- 3. Ellipticity.- 4. Marginal properties.- 5. Hall varieties.- 15. The finite basis problem.- 1. Nilpotent varieties.- 2. Partially well ordered sets.- 3. Metabelian varieties.- 4. Non-finitely based varieties.- 5. Class 2-by-abelian varieties.- 16. Engel conditions.- 1. The second and third Engel conditions.- 2. A non-locally nilpotent Engel algebra.- 3. Finiteness conditions on Engel algebras.- 4. Left and right Engel elements.- 17. Kostrikin's theorem.- 1. The Burnside problem.- 2. Basic computational results.- 3. The existence of an element of order 2.- 4. Elements which generate abelian ideals.- 5. Algebras generated by elements of order 2.- 6. A weakened form of Kostrikin's theorem.- 7. Sketch proof of Kostrikin's theorem.- 18. Razmyslov's theorem.- 1. The construction.- 2. Proof of non-nilpotence.- Some open questions.- References.- Notation index.

Ian Stewart

Papers

Singularities and groups in bifurcation theory

Infinite-dimensional Lie algebras

Catastrophe theory and its applications

Coupled oscillators and biological synchronization.

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