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.

/pdf/collective-dynamics-of-small-world-networks-205zhem3gm.pdf

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

抗原变异可使得多种致病微生物易于逃避宿主免疫应答。表达在感染红细胞表面的恶性疟原虫红细胞表面蛋白1（PfPMP1）与感染红细胞、内皮细胞、树突状细胞以及胎盘的单个或多个受体作用，在黏附及免疫逃避中起关键的作用。每个单倍体基因组var基因家族编码约60种成员，通过启动转录不同的var基因变异体为抗原变异提供了分子基础。

疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A

A 2.91-billion base pair (bp) consensus sequence of the euchromatic portion of the human genome was generated by the whole-genome shotgun sequencing method. The 14.8-billion bp DNA sequence was generated over 9 months from 27,271,853 high-quality sequence reads (5.11-fold coverage of the genome) from both ends of plasmid clones made from the DNA of five individuals. Two assembly strategies-a whole-genome assembly and a regional chromosome assembly-were used, each combining sequence data from Celera and the publicly funded genome effort. The public data were shredded into 550-bp segments to create a 2.9-fold coverage of those genome regions that had been sequenced, without including biases inherent in the cloning and assembly procedure used by the publicly funded group. This brought the effective coverage in the assemblies to eightfold, reducing the number and size of gaps in the final assembly over what would be obtained with 5.11-fold coverage. The two assembly strategies yielded very similar results that largely agree with independent mapping data. The assemblies effectively cover the euchromatic regions of the human chromosomes. More than 90% of the genome is in scaffold assemblies of 100,000 bp or more, and 25% of the genome is in scaffolds of 10 million bp or larger. Analysis of the genome sequence revealed 26,588 protein-encoding transcripts for which there was strong corroborating evidence and an additional approximately 12,000 computationally derived genes with mouse matches or other weak supporting evidence. Although gene-dense clusters are obvious, almost half the genes are dispersed in low G+C sequence separated by large tracts of apparently noncoding sequence. Only 1.1% of the genome is spanned by exons, whereas 24% is in introns, with 75% of the genome being intergenic DNA. Duplications of segmental blocks, ranging in size up to chromosomal lengths, are abundant throughout the genome and reveal a complex evolutionary history. Comparative genomic analysis indicates vertebrate expansions of genes associated with neuronal function, with tissue-specific developmental regulation, and with the hemostasis and immune systems. DNA sequence comparisons between the consensus sequence and publicly funded genome data provided locations of 2.1 million single-nucleotide polymorphisms (SNPs). A random pair of human haploid genomes differed at a rate of 1 bp per 1250 on average, but there was marked heterogeneity in the level of polymorphism across the genome. Less than 1% of all SNPs resulted in variation in proteins, but the task of determining which SNPs have functional consequences remains an open challenge.

http://science.sciencemag.org/content/sci/291/5507/1304.full.pdf

The sequence of the human genome.

Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

An extensive statistical survey of universal approximators shows that as the dimension of a typical dissipative dynamical system is increased, the number of positive Lyapunov exponents increases monotonically and the number of parameter windows with periodic behavior decreases. A subset of parameter space remains where noncatastrophic topological change induced by a small parameter variation becomes inevitable. A geometric mechanism depending on dimension and an associated conjecture depict why topological change is expected but not catastrophic, thus providing an explanation of how and why deterministic chaos persists in high dimensions.

/pdf/persistent-chaos-in-high-dimensions-kh7armeszs.pdf

Persistent chaos in high dimensions

The temporal decay of an attractor’s vicinity for a domain‐wall dominated cellular automaton (CA) is studied. Using selected initial pattern ensembles, state space structures in this high‐dimensional nonlinear spatial system can be identified via the resulting decay to its attractors. Considered over a range of lattice sizes, the decay behavior falls into three main classes, each of which shows a characteristic profile. The first consists of even‐size lattices showing a decelerating decay to small nonattracted ensemble fractions. The second class, also for even lattices, is a catastrophic decay to very small or vanishing nonattracted fractions. The third class also shows catastrophic decay and contains all odd‐size lattices. Stochastic models are constructed that mimic the behavior of typical lattices throughout their evolution until finite‐size effects appear. Weak additive noise causes all states on all lattices to fall into the attractor. In the end we find it overwhelmingly likely that the recently proposed attractor‐basin portrait captures the CA’s qualitative dynamics.

/pdf/attractor-vicinity-decay-for-a-cellular-automaton-2prxl5d1fy.pdf

Attractor vicinity decay for a cellular automaton.

Key to biological success, the requisite variety that confronts an adaptive organism is the set of detectable, accessible, and controllable states in its environment. We analyze its role in the thermodynamic functioning of information ratchets---a form of autonomous Maxwellian Demon capable of exploiting fluctuations in an external information reservoir to harvest useful work from a thermal bath. This establishes a quantitative paradigm for understanding how adaptive agents leverage structured thermal environments for their own thermodynamic benefit. General ratchets behave as memoryful communication channels, interacting with their environment sequentially and storing results to an output. The bulk of thermal ratchets analyzed to date, however, assume memoryless environments that generate input signals without temporal correlations. Employing computational mechanics and a new information-processing Second Law of Thermodynamics (IPSL) we remove these restrictions, analyzing general finite-state ratchets interacting with structured environments that generate correlated input signals. On the one hand, we demonstrate that a ratchet need not have memory to exploit an uncorrelated environment. On the other, and more appropriate to biological adaptation, we show that a ratchet must have memory to most effectively leverage structure and correlation in its environment. The lesson is that to optimally harvest work a ratchet's memory must reflect the input generator's memory. Finally, we investigate achieving the IPSL bounds on the amount of work a ratchet can extract from its environment, discovering that finite-state, optimal ratchets are unable to reach these bounds. In contrast, we show that infinite-state ratchets can go well beyond these bounds by utilizing their own infinite "negentropy". We conclude with an outline of the collective thermodynamics of information-ratchet swarms.

Leveraging Environmental Correlations: The Thermodynamics of Requisite Variety

In our work we are studying how genetic algorithms (GAs) can evolve cellular automata (CAs) to perform computations that require global coordination The "evolving cellular automata" framework is an idealized means for studying how evolution (natural or computational) can create systems that perform emergent computation, in which the actions of simple computation, in which the actions of simple components with local information and communication give rise to coordinated global information processing

/pdf/embedded-particle-computation-in-evolved-cellular-automata-39jppzmylx.pdf

Embedded particle computation in evolved cellular automata

Maintained by environmental fluxes, biological systems are thermodynamic processes that operate far from equilibrium without detailed-balanced dynamics Yet, they often exhibit well defined nonequilibrium steady states (NESSs) More importantly, critical thermodynamic functionality arises directly from transitions among their NESSs, driven by environmental switching Here, we identify the constraints on excess heat and dissipated work necessary to control a system that is kept far from equilibrium by background, uncontrolled “housekeeping” forces We do this by extending the Crooks fluctuation theorem to transitions among NESSs, without invoking an unphysical dual dynamics This and corresponding integral fluctuation theorems determine how much work must be expended when controlling systems maintained far from equilibrium This generalizes thermodynamic feedback control theory, showing that Maxwellian Demons can leverage mesoscopic-state information to take advantage of the excess energetics in NESS transitions We also generalize an approach recently used to determine the work dissipated when driving between functionally relevant configurations of an active energy-consuming complex system Altogether, these results highlight universal thermodynamic laws that apply to the accessible degrees of freedom within the effective dynamic at any emergent level of hierarchical organization By way of illustration, we analyze a voltage-gated sodium ion channel whose molecular conformational dynamics play a critical functional role in propagating action potentials in mammalian neuronal membranes

/pdf/fluctuations-when-driving-between-nonequilibrium-steady-3qanwc5irb.pdf

James P. Crutchfield

Papers

Persistent chaos in high dimensions

Attractor vicinity decay for a cellular automaton.

Leveraging Environmental Correlations: The Thermodynamics of Requisite Variety

Embedded particle computation in evolved cellular automata

Fluctuations When Driving Between Nonequilibrium Steady States