scholarly journals Random 2-SAT Solution Components and a Fitness Landscape

2011 ◽  
Vol Vol. 13 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Damien Pitman
Keyword(s):  
Fitness Landscape ◽  
Biological Evolution ◽  
Random Variable ◽  
Single Site ◽  
Connected Components ◽  
Number Of Components ◽  
Bounded Number ◽  
International Audience ◽  
Stochastically Bounded ◽  
Probability Range

Graphs and Algorithms International audience We describe a limiting distribution for the number of connected components in the subgraph of the discrete cube induced by the satisfying assignments to a random 2-SAT formula. We show that, for the probability range where formulas are likely to be satisfied, the random number of components converges weakly (in the number of variables) to a distribution determined by a Poisson random variable. The number of satisfying assignments or solutions is known to grow exponentially in the number of variables. Thus, our result implies that exponentially many solutions are organized into a stochastically bounded number of components. We also describe an application to biological evolution; in particular, to a type of fitness landscape where satisfying assignments represent viable genotypes and connectivity of genotypes is limited by single site mutations. The biological result is that, with probability approaching 1, each viable genotype is connected by single site mutations to an exponential number of other viable genotypes while the number of viable clusters is finite.

scholarly journals The Limiting Distribution for the Number of Symbol Comparisons Used by QuickSort is Nondegenerate (Extended Abstract)

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Patrick Bindjeme ◽  
james Allen fill
Keyword(s):  
Large Class ◽  
Continuous Time ◽  
Random Variable ◽  
Limiting Distribution ◽  
International Audience ◽  
The Mean

International audience In a continuous-time setting, Fill (2012) proved, for a large class of probabilistic sources, that the number of symbol comparisons used by $\texttt{QuickSort}$, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution, but proved little about that limiting random variable $Y$—not even that it is nondegenerate. We establish the nondegeneracy of $Y$. The proof is perhaps surprisingly difficult.

scholarly journals On the bialgebra of functional graphs and differential algebras

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Maurice Ginocchio
Keyword(s):  
Dynamical Systems ◽  
Differential Operators ◽  
Discrete Dynamical Systems ◽  
Rooted Trees ◽  
Connected Components ◽  
Differential Algebras ◽  
Noncommutative Polynomials ◽  
International Audience

International audience We develop the bialgebraic structure based on the set of functional graphs, which generalize the case of the forests of rooted trees. We use noncommutative polynomials as generating monomials of the functional graphs, and we introduce circular and arborescent brackets in accordance with the decomposition in connected components of the graph of a mapping of \1, 2, \ldots, n\ in itself as in the frame of the discrete dynamical systems. We give applications fordifferential algebras and algebras of differential operators.

scholarly journals Percolation on a non-homogeneous Poisson blob process

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Fabio P. Machado
Keyword(s):  
Connected Components ◽  
Multi Scale ◽  
International Audience

International audience We present the main results of a study for the existence of vacant and occupied unbounded connected components in a non-homogeneous Poisson blob process. The method used in the proofs is a multi-scale percolation comparison.

Conflict and Compromise in multi-storey timber architecture

2015 ◽  
Vol 19 (3) ◽  
pp. 283-294
Author(s):  
Magnus Larsson ◽  
Alex Kaiser ◽  
Ulf Arne Girhammar
Keyword(s):  
Design Methodology ◽  
Fitness Landscape ◽  
Selection Process ◽  
Biological Evolution ◽  
Timber Structures ◽  
Original Design ◽  
Evolutionary Computations ◽  
Conflicting Objectives ◽  
Research And Design

From the Stone Age to the Silicon Age, humanity’s relationship with materials has framed our scientific and cultural evolution. Despite recent scientific advances that open the laboratory doors to a future of material experimentation, the building industry remains stuck in the Concrete Age. The next material age is likely to be a Timber Age, as engineered timber finally comes into its own as a structural material suitable for multi-storey buildings.The research and design methodology of our practice can be summarised as an essay in conflict and compromise; a continual infliction of self-imposed constraints that allows us to discover hitherto unimaginable paths through the available options. We achieve this by searching through a space of possibilities demarcated by the properties and performative logic of the material at hand. First, we create an abundant field of alternatives, then we often use evolutionary computations to find our way through this field towards a visionary and original design.All design involves conflicting objectives. The best schemes offer the best compromise between desires. This article discusses how evolutionary solvers can be used as a tool for material-based architectural optimisation of geometries and structures, and how we have used them in designs for the imminent Timber Age. As one potentially conflicting objective is weighed against another, we move closer and closer to a tradeoff: the fitness of cooperating, opposing or unconnected variables.A population of possible design responses is visualised as a ‘fitness landscape’. Inspired by the role of natural selection in biological evolution, we use evolutionary algorithms to perform a selection process in which the ‘most fit’ members of such a population survive, while the ‘least fit’ members are eliminated, with the selection process guiding the algorithm towards ever-better solutions. The resulting timber structures would have made Darwin proud.

scholarly journals Statistical Properties of Similarity Score Functions

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Jérémie Bourdon ◽  
Alban Mancheron
Keyword(s):  
Probabilistic Model ◽  
Random Variable ◽  
Similarity Score ◽  
Longest Common Subsequence ◽  
Score Functions ◽  
Common Subsequence ◽  
International Audience ◽  
The Mean ◽  
Classical Behavior ◽  
Or Genes

International audience In computational biology, a large amount of problems, such as pattern discovery, deals with the comparison of several sequences (of nucleotides, proteins or genes for instance). Very often, algorithms that address this problem use score functions that reflect a notion of similarity between the sequences. The most efficient methods take benefit from theoretical knowledge of the classical behavior of these score functions such as their mean, their variance, and sometime their asymptotic distribution in a given probabilistic model. In this paper, we study a recent family of score functions introduced in Mancheron 2003, which allows to compare two words having the same length. Here, the similarity takes into account all matches and mismatches between two sequences and not only the longest common subsequence as in the case of classical algorithms such as BLAST or FASTA. Based on generating functions, we provide closed formulas for the mean and the variance of these functions in an independent probabilistic model. Finally, we prove that every function in this family asymptotically behaves as a Gaussian random variable.

scholarly journals Punctuated equilibrium as the default mode of evolution of large populations on fitness landscapes dominated by saddle points in the weak-mutation limit

2020 ◽  
Author(s):  
Yuri Bakhtin ◽  
Mikhail I. Katsnelson ◽  
Yuri I. Wolf ◽  
Eugene V. Koonin
Keyword(s):  
Fitness Landscape ◽  
Large Population ◽  
Saddle Points ◽  
Biological Evolution ◽  
Punctuated Equilibrium ◽  
Fitness Landscapes ◽  
Phenotypic Change ◽  
Beneficial Mutation ◽  
Effective Population ◽  
Default Mode

AbstractPunctuated equilibrium is a mode of evolution in which phenetic change occurs in rapid bursts that are separated by much longer intervals of stasis during which mutations accumulate but no major phenotypic change occurs. Punctuated equilibrium has been originally proposed within the framework of paleobiology, to explain the lack of transitional forms that is typical of the fossil record. Theoretically, punctuated equilibrium has been linked to self-organized criticality (SOC), a model in which the size of ‘avalanches’ in an evolving system is power-law distributed, resulting in increasing rarity of major events. We show here that, under the weak-mutation limit, a large population would spend most of the time in stasis in the vicinity of saddle points in the fitness landscape. The periods of stasis are punctuated by fast transitions, in lnNe time (Ne, effective population size), when a new beneficial mutation is fixed in the evolving population, which moves to a different saddle, or on much rarer occasions, from a saddle to a local peak. Thus, punctuated equilibrium is the default mode of evolution under a simple model that does not involve SOC or other special conditions.SignificanceThe gradual character of evolution is a key feature of the Darwinian worldview. However, macroevolutionary events are often thought to occur in a non-gradualist manner, in a regime known as punctuated equilibrium, whereby extended periods of evolutionary stasis are punctuated by rapid transitions between states. Here we analyze a mathematical model of population evolution on fitness landscapes and show that, for a large population in the weak-mutation limit, the process of adaptive evolution consists of extended periods of stasis, which the population spends around saddle points on the landscape, interrupted by rapid transitions to new saddle points when a beneficial mutation is fixed. Thus, punctuated equilibrium appears to be the default regime of biological evolution.

scholarly journals A Driven Disordered Systems Approach to Biological Evolution in Changing Environments

2021 ◽  
Author(s):  
Suman Gaurab Das ◽  
Joachim Krug ◽  
Muhittin Mungan
Keyword(s):  
Disordered Systems ◽  
Evolutionary Dynamics ◽  
Systems Approach ◽  
Fitness Landscape ◽  
Biological Evolution ◽  
Hysteresis Loops ◽  
Antibiotic Concentration ◽  
Physical Systems ◽  
Resistance Evolution ◽  
Changing Environments

Biological evolution of a population is governed by the fitness landscape, which is a map from genotype to fitness. However, a fitness landscape depends on the organism's environment, and evolution in changing environments is still poorly understood. We study a particular model of antibiotic resistance evolution in bacteria where the antibiotic concentration is an environmental parameter and the fitness landscapes incorporate tradeoffs between adaptation to low and high antibiotic concentration. With evolutionary dynamics that follow fitness gradients, the evolution of the system under slowly changing antibiotic concentration resembles the athermal dynamics of disordered physical systems under quasistatic external drives. Specifically, our model can be described as a system with interacting hysteretic elements, and it exhibits effects such as hysteresis loops and memory formation under antibiotic concentration cycling. Using methods familiar from studies in this field, we derive a number of analytical and numerical results. Our approach provides a general framework for studying motifs of evolutionary dynamics in biological systems in a changing environment.

scholarly journals Decomposable graphs and definitions with no quantifier alternation

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Oleg Pikhurko ◽  
Joel Spencer ◽  
Oleg Verbitsky
Keyword(s):  
The Other ◽  
Connected Components ◽  
Large Graphs ◽  
First Order ◽  
Minimum Number ◽  
International Audience ◽  
Decomposable Graphs ◽  
Number Of Iterations ◽  
Quantifier Alternation ◽  
Binary Logarithm

International audience Let $D(G)$ be the minimum quantifier depth of a first order sentence $\Phi$ that defines a graph $G$ up to isomorphism in terms of the adjacency and the equality relations. Let $D_0(G)$ be a variant of $D(G)$ where we do not allow quantifier alternations in $\Phi$. Using large graphs decomposable in complement-connected components by a short sequence of serial and parallel decompositions, we show examples of $G$ on $n$ vertices with $D_0(G) \leq 2 \log^{\ast}n+O(1)$. On the other hand, we prove a lower bound $D_0(G) \geq \log^{\ast}n-\log^{\ast}\log^{\ast}n-O(1)$ for all $G$. Here $\log^{\ast}n$ is equal to the minimum number of iterations of the binary logarithm needed to bring $n$ below $1$.

scholarly journals A simple formula for bipartite and quasi-bipartite maps with boundaries

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Gwendal Collet ◽  
Eric Fusy
Keyword(s):  
Generating Function ◽  
Simple Formula ◽  
Connected Components ◽  
Nous Obtenons ◽  
Single Tree ◽  
International Audience ◽  
Planar Maps

International audience We obtain a very simple formula for the generating function of bipartite (resp. quasi-bipartite) planar maps with boundaries (holes) of prescribed lengths, which generalizes certain expressions obtained by Eynard in a book to appear. The formula is derived from a bijection due to Bouttier, Di Francesco and Guitter combined with a process (reminiscent of a construction of Pitman) of aggregating connected components of a forest into a single tree. Nous obtenons une formule très simple pour la série génératrice des cartes biparties ayant des bords (trous) de tailles fixées, généralisant certaines expressions obtenues par Eynard dans un livre à paraître. Nous obtenons la formule à partir d'une bijection due à Bouttier, Di Francesco et Guitter, combinée avec un processus (dans l'esprit d'une construction due à Pitman) pour agréger les composantes connexes d'une forêt en un unique arbre.

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