Dual Geometric Graph Network (DG2N) Iterative Network for Deformable Shape Alignment

Abstract We investigate the estimation of the perimeter of a set by a graph cut of a random geometric graph. For Ω ⊆ D = (0, 1)d with d ≥ 2, we are given n random independent and identically distributed points on D whose membership in Ω is known. We consider the sample as a random geometric graph with connection distance ε > 0. We estimate the perimeter of Ω (relative to D) by the, appropriately rescaled, graph cut between the vertices in Ω and the vertices in D ∖ Ω. We obtain bias and variance estimates on the error, which are optimal in scaling with respect to n and ε. We consider two scaling regimes: the dense (when the average degree of the vertices goes to ∞) and the sparse one (when the degree goes to 0). In the dense regime, there is a crossover in the nature of the approximation at dimension d = 5: we show that in low dimensions d = 2, 3, 4 one can obtain confidence intervals for the approximation error, while in higher dimensions one can obtain only error estimates for testing the hypothesis that the perimeter is less than a given number.

Download Full-text

Clique Colourings of Geometric Graphs

The Electronic Journal of Combinatorics ◽

10.37236/7159 ◽

2018 ◽

Vol 25 (4) ◽

Author(s):

Colin McDiarmid ◽

Dieter Mitsche ◽

Pawel Prałat

Keyword(s):

Asymptotic Behaviour ◽

Chromatic Number ◽

Euclidean Distance ◽

High Probability ◽

Maximal Clique ◽

Higher Dimensions ◽

Geometric Graph ◽

Geometric Graphs ◽

Random Geometric Graph ◽

Vertex Set

A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number. Given $n$ points $\mathbf{x}_1, \ldots,\mathbf{x}_n$ in the plane, and a threshold $r>0$, the corresponding geometric graph has vertex set $\{v_1,\ldots,v_n\}$, and distinct $v_i$ and $v_j$ are adjacent when the Euclidean distance between $\mathbf{x}_i$ and $\mathbf{x}_j$ is at most $r$. We investigate the clique chromatic number of such graphs.We first show that the clique chromatic number is at most 9 for any geometric graph in the plane, and briefly consider geometric graphs in higher dimensions. Then we study the asymptotic behaviour of the clique chromatic number for the random geometric graph $\mathcal{G}$ in the plane, where $n$ random points are independently and uniformly distributed in a suitable square. We see that as $r$ increases from 0, with high probability the clique chromatic number is 1 for very small $r$, then 2 for small $r$, then at least 3 for larger $r$, and finally drops back to 2.

Download Full-text

Temporal Matching on Geometric Graph Data

Lecture Notes in Computer Science - Algorithms and Complexity ◽

10.1007/978-3-030-75242-2_28 ◽

2021 ◽

pp. 394-408

Author(s):

Timothe Picavet ◽

Ngoc-Trung Nguyen ◽

Binh-Minh Bui-Xuan

Keyword(s):

Geometric Graph ◽

Graph Data

Download Full-text

A Network-Based Explanation of Perceived Inequality

10.20378/irb-49393 ◽

2021 ◽

Author(s):

Jan Schulz ◽

Daniel Mayerhoffer ◽

Anna Gebhard

Keyword(s):

Public Perception ◽

Small World ◽

Geometric Graph ◽

Public Perceptions ◽

Common Mechanism ◽

Small World Networks ◽

Promising Candidate ◽

Random Geometric Graph ◽

The Public ◽

Individual Perceptions

Across income groups and countries, the public perception of economic inequality and many other macroeconomic variables such as inflation or unemployment rates is spectacularly wrong. These misperceptions have far-reaching consequences, as it is perceived inequality, not actual inequality informing redistributive preferences. The prevalence of this phenomenon is independent of social class and welfare regime, which suggests the existence of a common mechanism behind public perceptions. We propose a network-based explanation of perceived inequality building on recent advances in random geometric graph theory. The literature has identified several stylised facts on how individual perceptions respond to actual inequality and how these biases vary systematically along the income distribution. Our generating mechanism can replicate all of them simultaneously. It also produces social networks that exhibit salient features of real-world networks; namely, they cannot be statistically distinguished from small-world networks, testifying to the robustness of our approach. Our results, therefore, suggest that homophilic segregation is a promising candidate to explain inequality perceptions with strong implications for theories of consumption behaviour.

Download Full-text