Related papers: Simple models for scaling in phylogenetic trees
We revisit the number theoretic division model of self-organized criticality [Phys. Rev. Lett. 101, 158702 (2008)]. The model consists of a pool of $M-1$ ordered integers $\{2, 3, \cdots, M\}$, and the aim is to dynamically form a primitive…
On large scales, the Lyman-$\alpha$ forest provides insights into the expansion history of the Universe, while on small scales, it imposes strict constraints on the growth history, the nature of dark matter, and the sum of neutrino masses.…
We study the properties of random walks on complex trees. We observe that the absence of loops reflects in physical observables showing large differences with respect to their looped counterparts. First, both the vertex discovery rate and…
Neural scaling laws are observed in a range of domains, to date with no universal understanding of why they occur. Recent theories suggest that loss power laws arise from Zipf's law, a power law observed in domains like natural language.…
Neural scaling laws aim to characterize how out-of-sample error behaves as a function of model and training dataset size. Such scaling laws guide allocation of a computational resources between model and data processing to minimize error.…
Hierarchies can be modeled by a set of exponential functions, from which we can derive a set of power laws indicative of scaling. The solution to a scaling relation equation is always a power law. The scaling laws are followed by many…
Language models (LMs) require robust episodic grounding-the capacity to learn from and apply past experiences-to excel at physical planning tasks. Current episodic grounding approaches struggle with scalability and integration, limiting…
Scale transformations have played an extremely successful role in studies of cosmological large-scale structure by relating the non-linear spectrum of cosmological density fluctuations to the linear primordial power at longer wavelengths.…
Many biological networks have been labelled scale-free as their degree distribution can be approximately described by a powerlaw distribution. While the degree distribution does not summarize all aspects of a network it has often been…
Phylogenetic networks are a generalization of phylogenetic trees allowing for the representation of non-treelike evolutionary events such as hybridization. Typically, such networks have been analyzed based on their `level', i.e. based on…
On the rooted $k$-ary tree we consider a 0-1 kinetically constrained spin model in which the occupancy variable at each node is re-sampled with rate one from the Bernoulli(p) measure iff all its children are empty. For this process the…
Looping, reusing a block of layers across depth, and depth growing, training shallow-to-deep models by duplicating middle layers, have both been linked to stronger reasoning, but their relationship remains unclear. We provide a mechanistic…
Open string amplitudes at tree level have been studied for over fifty years, but there is no known analytic form for general $n$-point amplitudes, and their conventional representation in terms of worldsheet integrals does not make many of…
We investigate the statistics of trees grown from some initial tree by attaching links to preexisting vertices, with attachment probabilities depending only on the valence of these vertices. We consider the asymptotic mass distribution that…
Experimental studies on the irreversible growth of field-induced chains of dipolar particles suggest an asymptotic power-law behavior of several relevant quantities. We introduce a Monte Carlo model of chain growth that explicitly…
The body plan of the fruit fly is determined by the expression of just a handful of genes. We show that the spatial patterns of expression for several of these genes scale precisely with the size of the embryo. Concretely, discrete…
Neural scaling laws have driven significant advancements in machine learning, particularly in domains like language modeling and computer vision. However, the exploration of neural scaling laws within robotics has remained relatively…
We present a new technique for proving logarithmic upper bounds for diameters of evolving random graph models, which is based on defining a coupling between random graphs and variants of random recursive trees. The advantage of the…
The comprehensive characterization of the structure of complex networks is essential to understand the dynamical processes which guide their evolution. The discovery of the scale-free distribution and the small world property of real…
In biology, a phylogenetic tree is a tool to represent the evolutionary relationship between species. Unfortunately, the classical Schr\"oder tree model is not adapted to take into account the chronology between the branching nodes. In…