Combinatorial Landscapes

Reidys, Christian M.; Stadler, Peter F.

doi:10.1137/s0036144501395952

Cited by 214 publications

(127 citation statements)

References 191 publications

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“…From this point of view, equation (3) becomes equivalent to a search for the ground-state energy, and the characteristic valuation can be viewed as the ground-state configuration. This analogy is straightforward if one realizes that equation (3) is equivalent to the minimization of the weighted cut of the entire networked system (whose adjacency matrix G is defined in equation (1)), and that the minimum of this function corresponds to the ground state of a wide class of energy functions 24 and fitness landscapes 25 . These include, among others, the energy associated with the Ising spin models 26 and cost functions of combinatorial optimization problems, such as the travelling salesman problem 27 .…”

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confidence: 99%

Abrupt transition in the structural formation of interconnected networks

2013

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Our world is linked by a complex mesh of networks through which information, people and goods flow. These networks are interdependent on each other, and present structural and dynamical features 1-6 different from those observed in isolated networks 7-9 . Although examples of such dissimilar properties are becoming more abundant-such as in diffusion, robustness and competition-it is not yet clear where these differences are rooted. Here we show that the process of building independent networks into an interconnected network of networks undergoes a structurally sharp transition as the interconnections are formed. Depending on the relative importance of inter-and intra-layer connections, we find that the entire interdependent system can be tuned between two regimes: in one regime, the various layers are structurally decoupled and they act as independent entities; in the other regime, network layers are indistinguishable and the whole system behaves as a singlelevel network. We analytically show that the transition between the two regimes is discontinuous even for finite-size networks. Thus, any real-world interconnected system is potentially at risk of abrupt changes in its structure, which may manifest new dynamical properties.Interacting, interdependent or multiplex networks are different ways of naming the same class of complex systems where networks are not considered as isolated entities but interacting with each other. In multiplex, the nodes at each network are instances of the same entity; thus, the networks are representing simply different categorical relationships between entities, and usually categories are represented by layers. Interdependent networks is a more general framework where nodes can be different at each network.Many, if not all, real networks are coupled with other real networks. Examples can be found in several domains: social networks (for example, Facebook, Twitter and so on) are coupled because they share the same actors 10 ; multimodal transportation networks are composed of different layers (for example, bus, subway and so on) that share the same locations 11 ; the functioning of communication and power grid systems depends one on the other 1 . So far, all phenomena that have been studied on interdependent networks, including percolation 1,3 , epidemics 4 and linear dynamical systems 5 , have provided results that differ much from those valid in the case of isolated complex networks. Sometimes the difference is radical: for example, whereas isolated scale-free networks are robust against failures of their nodes or edges 12 , scale-free interdependent networks are instead very fragile 1,3 .Given such observations, two fundamentally important theoretical questions are in order: Why do dynamical and critical phenomena running on interdependent network models differ

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confidence: 99%

Abrupt transition in the structural formation of interconnected networks

2013

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“…If, instead, we are interested in the human genome merely as a sample from a larger population of extant or possible vertebrate genomes, then this larger population would need to be precisely defined (not an easy task), and the danger of unwittingly considering an ill-posed (insufficiently specified) or biologically irrelevant problem should be kept in mind. Furthermore, we have not found quantitative results that would allow us to estimate 'expected' changes of peak number in a generic or rugged landscape as it widens or narrows during evolution (see however Reidys and Stadler 2002 for research that may lead in this direction).…”

Section: Comments On the Significance Of Multimodalitymentioning

confidence: 91%

Compositional Gene Landscapes in Vertebrates

Cruveiller

Jabbari²,

Clay³

et al. 2004

Genome Res.

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The existence of a well conserved linear relationship between GC levels of genes' second and third codon positions (GC2, GC3) prompted us to focus on the landscape, or joint distribution, spanned by these two variables. In human, well curated coding sequences now cover at least 15%-30% of the estimated total gene set. Our analysis of the landscape defined by this gene set revealed not only the well documented linear crest, but also the presence of several peaks and valleys along that crest, a property that was also indicated in two other warm-blooded vertebrates represented by large gene databases, that is, mouse and chicken. GC2 is the sum of eight amino acid frequencies, whereas GC3 is linearly related to the GC level of the chromosomal region containing the gene. The landscapes therefore portray relations between proteins and the DNA environments of the genes that encode them.Two-dimensional frequency distributions of the GC levels of second and third codon positions of protein-coding genes reveal compositional constraints and selection pressures: those acting on proteins on one hand, and those acting on DNA, RNA, and possibly translational accuracy on the other hand. Indeed, GC levels in third positions (GC3) are almost free of constraints at the amino acid level, whereas those in second positions (GC2) are almost completely determined by the gene product. Their joint distribution, or landscape, therefore displays relations between the DNA and the proteins that its embedded genes encode.Among taxa that are well represented in sequence databases, genic GC2 and GC3 levels exhibit a tendency to cluster along a widely conserved, straight line in the landscape: the landscape's major axis or orthogonal regression line. The correlation to which this linearity corresponds is found in species as distant as human and Escherichia coli, and the major axis is consistently close to the line GC3 = 6 GC2 ‫מ‬ 200%. In other words, a 1% change in GC2 corresponds roughly to a 6% change in GC3, and the two codon positions have similar GC levels around 40%. The intragenomic correlation, and the correlation between first/ second and third codon positions (GC1 + 2 vs.

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“…Fitness and energy landscapes have become a unifying theme in fields as diverse as drug design, spin glass physics, molecular structure, protein folding, combinatorial optimization, and evolutionary theory, see e.g. [60] for a recent review. In each case, there is a function f , e.g., a molecular index, a Hamiltonian, a cost-function, or a fitness, that evaluates each member x ∈ X of a (usually very large) configuration set X.…”

Section: Fitness and Fitness Landscapesmentioning

confidence: 99%

The Topology of Evolutionary Biology

Stadler

2004

Natural Computing Series

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Summary. Central notions in evolutionary biology are intrinsically topological. This claim is maybe most obvious for the discontinuities associated with punctuated equilibria. Recently, a mathematical framework has been developed that derives the concepts of phenotypic characters and homology from the topological structure of the phenotype space. This structure in turn is determined by the genetic operators and their interplay with the properties of the genotype-phenotype map.

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Combinatorial Landscapes

Cited by 214 publications

References 191 publications

Abrupt transition in the structural formation of interconnected networks

Abrupt transition in the structural formation of interconnected networks

Compositional Gene Landscapes in Vertebrates

The Topology of Evolutionary Biology

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