Eigenvalues of Euclidean Random Matrices

We study the spectral measure of large Euclidean random matrices. The entries of these matrices are determined by the relative position of $n$ random points in a compact set $\Omega_n$ of $\R^d$. Under various assumptions we establish the almost sure convergence of the limiting spectral measure as the number of points goes to infinity. The moments of the limiting distribution are computed, and we prove that the limit of this limiting distribution as the density of points goes to infinity has a nice expression. We apply our results to the adjacency matrix of the geometric graph.Comment: 16 pages, 1 figur

Assessing Simulations of Imperial Dynamics and Conflict in the Ancient World

The development of models to capture large-scale dynamics in human history is one of the core contributions of cliodynamics. Most often, these models are assessed by their predictive capability on some macro-scale and aggregated measure and compared to manually curated historical data. In this report, we consider the model from Turchin et al. (2013), where the evaluation is done on the prediction of "imperial density": the relative frequency with which a geographical area belonged to large-scale polities over a certain time window. We implement the model and release both code and data for reproducibility. We then assess its behaviour against three historical data sets: the relative size of simulated polities vs historical ones; the spatial correlation of simulated imperial density with historical population density; the spatial correlation of simulated conflict vs historical conflict. At the global level, we show good agreement with population density ($R^2 < 0.75$), and some agreement with historical conflict in Europe ($R^2 < 0.42$). The model instead fails to reproduce the historical shape of individual polities. Finally, we tweak the model to behave greedily by having polities preferentially attacking weaker neighbours. Results significantly degrade, suggesting that random attacks are a key trait of the original model. We conclude by proposing a way forward by matching the probabilistic imperial strength from simulations to inferred networked communities from real settlement data

Binary indices at various densities

Binary similarity indices are numerical analysis methods used to compare data involving two binary vectors (lists). The scope of this project involved comparing 54 binary similarity indices methods in relationship to binary vector density using the R programming language. Matrices were created of various vector data. The matrices were then scrambled to represent random data. Finally, the data was analyzed and plotted. Vector density variation can result in large differences - in both rate of change relative to density and magnitude. Awareness of these differences is important when selecting an analysis method and understanding the effects of changing vector density on analysis of results