Surface networks

© Copyright CASA, UCL. The desire to understand and exploit the structure of continuous surfaces is common to researchers in a range of disciplines. Few examples of the varied surfaces forming an integral part of modern subjects include terrain, population density, surface atmospheric pressure, physico-chemical surfaces, computer graphics, and metrological surfaces. The focus of the work here is a group of data structures called Surface Networks, which abstract 2-dimensional surfaces by storing only the most important (also called fundamental, critical or surface-specific) points and lines in the surfaces. Surface networks are intelligent and “natural ” data structures because they store a surface as a framework of “surface ” elements unlike the DEM or TIN data structures. This report presents an overview of the previous works and the ideas being developed by the authors of this report. The research on surface networks has fou

A GDP-driven model for the binary and weighted structure of the International Trade Network

Recent events such as the global financial crisis have renewed the interest in the topic of economic networks. One of the main channels of shock propagation among countries is the International Trade Network (ITN). Two important models for the ITN structure, the classical gravity model of trade (more popular among economists) and the fitness model (more popular among networks scientists), are both limited to the characterization of only one representation of the ITN. The gravity model satisfactorily predicts the volume of trade between connected countries, but cannot reproduce the observed missing links (i.e. the topology). On the other hand, the fitness model can successfully replicate the topology of the ITN, but cannot predict the volumes. This paper tries to make an important step forward in the unification of those two frameworks, by proposing a new GDP-driven model which can simultaneously reproduce the binary and the weighted properties of the ITN. Specifically, we adopt a maximum-entropy approach where both the degree and the strength of each node is preserved. We then identify strong nonlinear relationships between the GDP and the parameters of the model. This ultimately results in a weighted generalization of the fitness model of trade, where the GDP plays the role of a `macroeconomic fitness' shaping the binary and the weighted structure of the ITN simultaneously. Our model mathematically highlights an important asymmetry in the role of binary and weighted network properties, namely the fact that binary properties can be inferred without the knowledge of weighted ones, while the opposite is not true

Distance Preserving Graph Simplification

Large graphs are difficult to represent, visualize, and understand. In this paper, we introduce "gate graph" - a new approach to perform graph simplification. A gate graph provides a simplified topological view of the original graph. Specifically, we construct a gate graph from a large graph so that for any "non-local" vertex pair (distance higher than some threshold) in the original graph, their shortest-path distance can be recovered by consecutive "local" walks through the gate vertices in the gate graph. We perform a theoretical investigation on the gate-vertex set discovery problem. We characterize its computational complexity and reveal the upper bound of minimum gate-vertex set using VC-dimension theory. We propose an efficient mining algorithm to discover a gate-vertex set with guaranteed logarithmic bound. We further present a fast technique for pruning redundant edges in a gate graph. The detailed experimental results using both real and synthetic graphs demonstrate the effectiveness and efficiency of our approach.Comment: A short version of this paper will be published for ICDM'11, December 201

Efficient symbolic computation of approximated small-signal characteristics of analog integrated circuits

A symbolic analysis tool is presented that generates simplified symbolic expressions for the small-signal characteristics of large analog integrated circuits. The expressions are approximated while they are computed, so that only those terms are generated which remain in the final expression. This principle causes drastic savings in CPU time and memory, compared with previous symbolic analysis tools. In this way, the maximum size of circuits that can be analyzed, is largely increased. By taking into account a range for the value of a circuit parameter rather than one single number, the generated expressions are also more generally valid. Mismatch handling is explicitly taken into account in the algorithm. The capabilities of the new tool are illustrated with several experimental result

Relating ordinary and fully simple maps via monotone Hurwitz numbers

A direct relation between the enumeration of ordinary maps and that of fully simple maps first appeared in the work of the first and last authors. The relation is via monotone Hurwitz numbers and was originally proved using Weingarten calculus for matrix integrals. The goal of this paper is to present two independent proofs that are purely combinatorial and generalise in various directions, such as to the setting of stuffed maps and hypermaps. The main motivation to understand the relation between ordinary and fully simple maps is the fact that it could shed light on fundamental, yet still not well-understood, problems in free probability and topological recursion.Comment: 19 pages, 7 figure

A graph-based mathematical morphology reader

This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research