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Complex systems can be described at myriad different scales, and their causal workings often have multiscale structure (e.g., a computer can be described at the microscale of its hardware circuitry, the mesoscale of its machine code, and…
Discrete Markov random fields are undirected graphical models that capture complex conditional dependencies between discrete variables. Conducting exact posterior inference in these models is often computationally challenging because…
Urban scaling theory posits that urban indicators follow power-law relations with population, yet the evolution of these patterns - and the role of regional differences in settings marked by social inequalities and unplanned urbanization -…
Probabilistic networks display a wide range of high average clustering coefficients independent of the number of nodes in the network. In particular, the local clustering coefficient decreases with the degree of the subtending node in a…
We study an asymptotic behavior of the return probability for the critical random matrix ensemble in the regime of strong multifractality. The return probability is expected to show critical scaling in the limit of large time or large…
Downstream scaling laws aim to predict task performance at larger scales from the model's performance at smaller scales. Whether such prediction should be possible is unclear: some works discover clear linear scaling trends after simple…
We consider the estimation of densities in multiple subpopulations, where the available sample size in each subpopulation greatly varies. This problem occurs in epidemiology, for example, where different diseases may share similar…
Recently, the concept of geometric renormalization group provides a good approach for studying the structural symmetry and functional invariance of complex networks. Along this line, we systematically investigate the finite-size scaling of…
We report on the exact treatment of a random-matrix representation of bond percolation model on a square lattice in two dimensions with occupation probability $p$. The percolation problem is mapped onto a random complex matrix composed of…
We present a numerical study of anisotropic statistical fluctuations in homogeneous turbulent flows. We give an argument to predict the dimensional scaling exponents, (p+j)/3, for the projections of p-th order structure function in the j-th…
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…
A longstanding puzzle in urban science is whether there's an intrinsic match between human populations and the mass of their built environments. Previous findings have revealed various urban properties scaling nonlinearly with population,…
We construct a new framework for accelerating Markov chain Monte Carlo in posterior sampling problems where standard methods are limited by the computational cost of the likelihood, or of numerical models embedded therein. Our approach…
A major driver of AI products today is the fact that new skills emerge in language models when their parameter set and training corpora are scaled up. This phenomenon is poorly understood, and a mechanistic explanation via mathematical…
Urban scaling analysis, the study of how aggregated urban features vary with the population of an urban area, provides a promising framework for discovering commonalities across cities and uncovering dynamics shared by cities across time…
We study extremal statistics and return intervals in stationary long-range correlated sequences for which the underlying probability density function is bounded and uniform. The extremal statistics we consider e.g., maximum relative to…
We consider settings where the observations are drawn from a zero-mean multivariate (real or complex) normal distribution with the population covariance matrix having eigenvalues of arbitrary multiplicity. We assume that the eigenvectors of…
Many complex systems--from social and communication networks to biological networks and the Internet--are thought to exhibit scale-free structure. However, prevailing explanations rely on the constant addition of new nodes, an assumption…
Given the variety of the visual world there is not one true scale for recognition: objects may appear at drastically different sizes across the visual field. Rather than enumerate variations across filter channels or pyramid levels, dynamic…
We consider a class of real numbers, a subset of irrational numbers and certain mathematical constants, for which the elements in the simple continued fraction appears to be random. As an illustrative example, one can consider $\pi = \{x_0,…