A Network-Based Explanation of Perceived Inequality

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.

Cliques in high-dimensional random geometric graphs

Random geometric graphs have become now a popular object of research. Defined rather simply, these graphs describe real networks much better than classical Erdős–Rényi graphs due to their ability to produce tightly connected communities. The n vertices of a random geometric graph are points in d-dimensional Euclidean space, and two vertices are adjacent if they are close to each other. Many properties of these graphs have been revealed in the case when d is fixed. However, the case of growing dimension d is practically unexplored. This regime corresponds to a real-life situation when one has a data set of n observations with a significant number of features, a quite common case in data science today. In this paper, we study the clique structure of random geometric graphs when $$n\rightarrow \infty$$ n → ∞ , and $$d \rightarrow \infty$$ d → ∞ , and average vertex degree grows significantly slower than n. We show that under these conditions, random geometric graphs do not contain cliques of size 4 a. s. if only $$d \gg \log ^{1 + \epsilon } n$$ d ≫ log 1 + ϵ n . As for the cliques of size 3, we present new bounds on the expected number of triangles in the case $$\log ^2 n \ll d \ll \log ^3 n$$ log 2 n ≪ d ≪ log 3 n that improve previously known results. In addition, we provide new numerical results showing that the underlying geometry can be detected using the number of triangles even for small n.

Plane and planarity thresholds for random geometric graphs

A random geometric graph, [Formula: see text], is formed by choosing [Formula: see text] points independently and uniformly at random in a unit square; two points are connected by a straight-line edge if they are at Euclidean distance at most [Formula: see text]. For a given constant [Formula: see text], we show that [Formula: see text] is a distance threshold function for [Formula: see text] to have a connected subgraph on [Formula: see text] points. Based on this, we show that [Formula: see text] is a distance threshold for [Formula: see text] to be plane, and [Formula: see text] is a distance threshold to be planar. We also investigate distance thresholds for [Formula: see text] to have a non-crossing edge, a clique of a given size, and an independent set of a given size.

Directed random geometric graphs

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.

Clique Colourings of Geometric Graphs

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.

On the critical threshold for continuum AB percolation

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ,μ. For any λ>0 we consider the percolation threshold μc(λ) associated to the parameter μ. Denoting by λc the percolation threshold for the standard Poisson Boolean model with radii r, we show the lower bound μc(λ)≥clog(c∕(λ−λc)) for any λ>λc with c>0 a fixed constant. In particular, there is no phase transition in μ at the critical value of λ, that is, μc(λc) =∞.

АЛГОРИТМ ОЦЕНКИ ПРОПУСКНОЙ СПОСОБНОСТИ ПРИ УПРАВЛЕНИИ ТРАФИКОМ БЕСПИЛОТНЫХ ЛЕТАТЕЛЬНЫХ АППАРАТОВ

The subject of the study in the article is the processes of assessing the airship throughput in controlling the unmanned aerial vehicles (UAV) traffic management. The goal is to improve the quality of air traffic control, taking into account the avoidance of conflicts involving three or more UAV. Problems: to develop a mathematical model of the probabilistic traffic map, as well as to formalize the construction of a random geometric graph model for the estimation of alleged UAVs conflicts and collisions; To implement algorithms given models construction for airship throughput automation. The models used: Poisson process whose intensity model is used for building a probabilistic traffic map, random geometric graph model is used for calculate the number of possible conflicts involving the UAV. The following results are obtained. A formalized model of the UAV location map has been created taking into account: the given region with the specified population density and the expected number of operations during the specified time interval. This model was used in the construction of a random geometric graph, in which, taking into account the minimum distance possible for the approximation of two UAVs, an estimation of the probability of conflicts and collisions was conducted. The model is the basis for obtaining an algorithm for estimating the factors limiting the capacity of the airspace, as a result of the occurrence of difficult solvable conflicts. The scientific novelty of the obtained results is as follows: The random geometric graph model is improved by formalizing the position of the vertices. The vertices, taking into account the law of the Poisson process, are placed in the cells of a given region. This allows us to obtain an objective picture of the location of the UAV in the city's airspace. Two-dimensional models of probabilistic traffic maps (Dutch model "Metropolis", model Cal) have been further developed, due to the formalization of the initial UAV placement, taking into account the law of the Poisson process. This will help to determine the technical requirements for ensuring uninterrupted operation of small unmanned aerial vehicles in the urban airspace