δ-exceedance records and random adaptive walks

Abstract

Abstract. We study a modified record process where the k'th record in a series of independent and identically distributed random variables is defined recursively through the condition Y k > Y k−1 − δ k−1 with a deterministic sequence δ k > 0 called the handicap. For constant δ k ≡ δ and exponentially distributed random variables it has been shown in previous work that the process displays a phase transition as a function of δ between a normal phase where the mean record value increases indefinitely and a stationary phase where the mean record value remains bounded and a finite fraction of all entries are records (Park et al 2015 Phys. Rev. E 91 042707). Here we explore the behavior for general probability distributions and decreasing and increasing sequences δ k , focusing in particular on the case when δ k matches the typical spacing between subsequent records in the underlying simple record process without handicap. We find that a continuous phase transition occurs only in the exponential case, but a novel kind of first order transition emerges when δ k is increasing. The problem is partly motivated by the dynamics of evolutionary adaptation in biological fitness landscapes, where δ k corresponds to the change of the deterministic fitness component after k mutational steps. The results for the record process are used to compute the mean number of steps that a population performs in such a landscape before being trapped at a local fitness maximum.

Universality Classes of Interaction Structures for NK Fitness Landscapes

“…Greedy walks reach a local maximum after a finite (small) number of steps, whereas the walk length diverges logarthmically in L for random adaptive walks and linearly for reluctant walks. Analytical results for walk lengths on correlated fitness landscapes are relatively scarce, but some progress has recently been achieved for walks on Rough Mount Fuji landscapes [1,51], a class of models defined by a weighted superposition of an additive fitness landscape and an uncorrelated random (HoC) landscape [65,63,62]. For the discussion of adaptive walks on NK landscapes we start from the observation that the walk length is additive over blocks for the block neighborhood [54,66,79], and therefore…”

Statistical properties of avalanches via the c-record process

“…We keep hearing about record breaking events in sports, in stock prices, in the summer temperature in a given city, in the amount of rainfall in a given place or in the magnitude of earthquakes in a certain geographical zone. The studies on the theory of records were initiated in the statistics literature almost 70 years back [1][2][3][4][5][6] and since then have found numerous applications across disciplines: in sports [7][8][9], in the analysis of climate data [10][11][12][13][14][15][16][17], in fitness models of evolutionary biology [18][19][20][21][22], in condensed matter systems such as spin glasses and high temperature superconductors [23][24][25] and also in models of growing networks [26]. Record statistics have also been studied extensively in various random walk models [27][28][29][30][31][32][33][34] with applications to avalanches and depinning of elastic lines in disordered medium [35], to the analysis of financial data [36][37][38][39] and more recently to active particles with run and tumble dynamics [40][41][42].…”

“…The parameter δ = −c < 0 is negative in our context. In addition, this δ-exceedence model with a negative δ = −c < 0 also appeared in the random adaptive walk (RAW) model to describe biological evolution on a random fitness landscape [20][21][22]. In fact, we use the notation c for −δ in our model following Ref.…”

Exact Statistics of Record Increments of Random Walks and Lévy Flights

“…The statistics of the increments also play an important role in large data analysis, e.g., in the characterization of the practical criteria to decide whether a new entry in a time series is a record (or not). In practice, the record values can be measured only up to a certain precision δ set by the resolution of a detecting instrument [25][26][27][28]. If the increment is smaller than δ the new entry is not counted as a record.…”

“…If the increment is smaller than δ the new entry is not counted as a record. Hence increments also affect the experimentally measured number of records [25][26][27][28].…”