Evolutionary Fitness Landscape Ruggedness via the Latent Framework
Abstract
Evolutionary fitness landscapes encode how genotypes map to reproductive success. Rugged landscapes—with many local peaks and epistatic interactions—shape adaptation, evolvability, and the predictability of evolutionary paths. Yet quantitative comparisons across biological systems remain fragmented: classical measures (correlation length, number of optima, accessibility) do not share a common geometric backbone that connects sequence space, population genetics, and higher-dimensional phenotypes.
This paper develops a Latent framework for fitness landscapes. We represent landscapes as structured objects embedded in a low-dimensional Latent space whose intrinsic complexity is summarized by the Latent Number \(\rho\) and an effective dimension \(N^\ast\). The Latent Number captures how aggressively the landscape compresses information relative to a reference smooth model, while \(N^\ast\) estimates the number of degrees of freedom required to explain observed fitness variation at fixed accuracy. Together they interpolate between smooth, nearly additive worlds and highly epistatic, glassy regimes.
We instantiate the framework on Kauffman’s NK model with sequence length \(L=10\) and interaction orders \(K\in\{0,2,9\}\). Forty formally verified theorems organize into seven thematic groups spanning spectral structure, ruggedness diagnostics, adaptive walks, NK-specific identities, Latent compression, directed evolution, and cross-domain bridges. Numerical validation on \(2^L\) genotypes (fixed pseudo-random NK draws; seed \(42\) per \(K\) in the bundled validation module) shows \(\rho\) decreasing from \(2.51\) at \(K=0\) to \(1.17\) at \(K=9\), while \(N^\ast/2^L\) rises from about \(0.5\%\) to \(45\%\). Local optima go from \(1\) to \(4\) to \(95\) across the same \(K\) values. Greedy adaptive walks averaged over \(10^4\) random starts (independent walk RNGs) attain mean terminal fitness at \(100\%\), \(98\%\), and \(86\%\) of the global optimum at \(K=0\), \(2\), and \(9\), respectively. All eighteen structural checks in the bundled harness pass.
Novelty
It reframes ruggedness as a paired information-geometry picture—compressibility via a Latent Number ρ and an effective explanatory rank N*—and calibrates how they co-move with epistasis on small exhaustive NK instances, rather than introducing a wholly new biological mechanism.