Markov random geometric graph, MRGG: A growth model for temporal dynamic networks

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.

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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.

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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.

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Directed random geometric graphs

Journal of Complex Networks ◽

10.1093/comnet/cnz006 ◽

2019 ◽

Vol 7 (5) ◽

pp. 792-816

Author(s):

Jesse Michel ◽

Sushruth Reddy ◽

Rikhav Shah ◽

Sandeep Silwal ◽

Ramis Movassagh

Keyword(s):

Real World ◽

Word Association ◽

Graph Model ◽

Small World ◽

Geometric Graph ◽

Clustering Coefficient ◽

Random Geometric Graph ◽

Random Geometric Graphs ◽

Scale Free ◽

Small World Property

Abstract Many real-world networks are intrinsically directed. Such networks include activation of genes, hyperlinks on the internet and the network of followers on Twitter among many others. The challenge, however, is to create a network model that has many of the properties of real-world networks such as power-law degree distributions and the small-world property. To meet these challenges, we introduce the Directed Random Geometric Graph (DRGG) model, which is an extension of the random geometric graph model. We prove that it is scale-free with respect to the indegree distribution, has binomial outdegree distribution, has a high clustering coefficient, has few edges and is likely small-world. These are some of the main features of aforementioned real-world networks. We also empirically observed that word association networks have many of the theoretical properties of the DRGG model.

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Image-Guided Tethering Spine Surgery with Outcome Prediction using Spatio-Temporal Dynamic Networks

IEEE Transactions on Medical Imaging ◽

10.1109/tmi.2020.3030741 ◽

2020 ◽

pp. 1-1

Author(s):

William Mandel ◽

Reda Oulbacha ◽

Marjolaine Roy-Beaudry ◽

Stefan Parent ◽

Samuel Kadoury

Keyword(s):

Spine Surgery ◽

Outcome Prediction ◽

Dynamic Networks ◽

Temporal Dynamic ◽

Image Guided ◽

Spatio Temporal

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Throughput Scalability of Wireless Hybrid Networks over a Random Geometric Graph

Wireless Networks ◽

10.1007/s11276-005-1767-y ◽

2005 ◽

Vol 11 (4) ◽

pp. 435-449 ◽

Cited By ~ 3

Author(s):

Ulaş C. Kozat ◽

Leandros Tassiulas

Keyword(s):

Geometric Graph ◽

Hybrid Networks ◽

Random Geometric Graph

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On the treewidth of random geometric graphs and percolated grids

Advances in Applied Probability ◽

10.1017/apr.2016.78 ◽

2017 ◽

Vol 49 (1) ◽

pp. 49-60 ◽

Cited By ~ 1

Author(s):

Anshui Li ◽

Tobias Müller

Keyword(s):

Critical Radius ◽

Geometric Graph ◽

Giant Component ◽

Bond Percolation ◽

Geometric Graphs ◽

Random Geometric Graph ◽

Random Geometric Graphs

Abstract In this paper we study the treewidth of the random geometric graph, obtained by dropping n points onto the square [0,√n]2 and connecting pairs of points by an edge if their distance is at most r=r(n). We prove a conjecture of Mitsche and Perarnau (2014) stating that, with probability going to 1 as n→∞, the treewidth of the random geometric graph is 𝜣(r√n) when lim inf r>rc, where rc is the critical radius for the appearance of the giant component. The proof makes use of a comparison to standard bond percolation and with a little bit of extra work we are also able to show that, with probability tending to 1 as k→∞, the treewidth of the graph we obtain by retaining each edge of the k×k grid with probability p is 𝜣(k) if p>½ and 𝜣(√log k) if p<½.

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