Related papers: Generalised central limit theorems for growth rate…
The rate equations are used to study the scale-free behavior of the weight distribution in evolving networks whose topology is determined only by degrees of preexisting vertices. An analysis of these equations shows that the degree…
The scaling law, a cornerstone of Large Language Model (LLM) development, predicts improvements in model performance with increasing computational resources. Yet, while empirically validated, its theoretical underpinnings remain poorly…
Understanding cities is central to addressing major global challenges from climate and health to economic resilience. Although increasingly perceived as fundamental socio-economic units, the detailed fabric of urban economic activities is…
We study the distributions of money in a simple closed economic system for different types of monetary transactions. We know that for arbitrary and random sharing but locally conserving money transactions, the money distribution goes to the…
Using high precision Monte Carlo simulations and a mean-field theory, we explore coarsening phenomena in a simple driven diffusive system. The model is reminiscent of vehicular traffic on a two-lane ring road. At sufficiently high density,…
A class of generalized non-minimal coupling theories is investigated, in search of scaling attractors able to provide an accelerated expansion at the present time. Solutions are found in the strong coupling regime and when the coupling…
Suppose $B_i:= B(p,r_i)$ are nested balls of radius $r_i$ about a point $p$ in a dynamical system $(T,X,\mu)$. The question of whether $T^i x\in B_i$ infinitely often (i. o.) for $\mu$ a.e.\ $x$ is often called the shrinking target problem.…
We propose a dynamical scheme for the combined processes of fragmentation and merging as a model system for cluster dynamics in nature and society displaying scale invariant properties. The clusters merge and fragment with rates…
We study largest singular values of large random matrices, each with mean of a fixed rank $K$. Our main result is a limit theorem as the number of rows and columns approach infinity, while their ratio approaches a positive constant. It…
The asymptotics of the first rows and columns of random Young diagrams corresponding to extremal characters of the infinite symmetric group is studied. We consider rows and columns with linear growth in $n$, the number of boxes of random…
Scale independence is a ubiquitous feature of complex systems which implies a highly skewed distribution of resources with no characteristic scale. Research has long focused on why systems as varied as protein networks, evolution and stock…
This paper reviews selected aspects of the growth of cosmological structure, covering the following general areas: (1) expected characteristics of linear density perturbations according to various candidate theories for the origin of…
Real complex systems are not rigidly structured; no clear rules or blueprints exist for their construction. Yet, amidst their apparent randomness, complex structural properties universally emerge. We propose that an important class of…
We introduce a family of one-dimensional geometric growth models, constructed iteratively by locally optimizing the tradeoffs between two competing metrics, and show that this family is equivalent to a family of preferential attachment…
Recent theories suggest that Neural Scaling Laws arise whenever the task is linearly decomposed into power-law distributed units. Alternatively, scaling laws also emerge when data exhibit a hierarchically compositional structure, as is…
We propose a bare-bones stochastic model that takes into account both the geographical distribution of people within a country and their complex network of connections. The model, which is designed to give rise to a scale-free network of…
We build networks of genetic similarity in which the nodes are organisms sampled from biological populations. The procedure is illustrated by constructing networks from genetic data of a marine clonal plant. An important feature in the…
Power law scaling is observed in many physical, biological and socio-economical complex systems and is now considered as an important property of these systems. In general, power law exists in the central part of the distribution. It has…
We introduce and solve a model which considers two coupled networks growing simultaneously. The dynamics of the networks is governed by the new arrival of network elements (nodes) making preferential attachments to pre-existing nodes in…
We use a model whose rules were inspired by population genetics, the random capability growth model, to describe the statistical details observed in experiments of fragmentation of brittle platelike objects, and in particular the existence…