Convergence Theorems for Some Layout Measures on Random Lattice and Random Geometric Graphs

This work deals with convergence theorems and bounds on the cost of several layout measures for lattice graphs, random lattice graphs and sparse random geometric graphs. Specifically, we consider the following problems: Minimum Linear Arrangement, Cutwidth, Sum Cut, Vertex Separation, Edge Bisection and Vertex Bisection. For full square lattices, we give optimal layouts for the problems still open. For arbitrary lattice graphs, we present best possible bounds disregarding a constant factor. We apply percolation theory to the study of lattice graphs in a probabilistic setting. In particular, we deal with the subcritical regime that this class of graphs exhibits and characterize the behaviour of several layout measures in this space of probability. We extend the results on random lattice graphs to random geometric graphs, which are graphs whose nodes are spread at random in the unit square and whose edges connect pairs of points which are within a given distance. We also characterize the behaviour of several layout measures on random geometric graphs in their subcritical regime. Our main results are convergence theorems that can be viewed as an analogue of the Beardwood, Halton and Hammersley theorem for the Euclidean TSP on random points in the unit square.

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A Random-Line-Graph Approach to Overlapping Line Segments

Journal of Complex Networks ◽

10.1093/comnet/cnaa029 ◽

2020 ◽

Vol 8 (4) ◽

Author(s):

Lucas Böttcher

Keyword(s):

Line Graph ◽

Geometric Graphs ◽

Spatial Network ◽

Graph Structure ◽

Sharp Transition ◽

Random Geometric Graphs ◽

Line Segments ◽

Unit Square ◽

Random Line

Abstract We study graphs that are formed by independently positioned needles (i.e. line segments) in the unit square. To mathematically characterize the graph structure, we derive the probability that two line segments intersect and determine related quantities such as the distribution of intersections, given a certain number of line segments $N$. We interpret intersections between line segments as nodes and connections between them as edges in a spatial network that we refer to as random-line graph (RLG). Using methods from the study of random-geometric graphs, we show that the probability of RLGs to be connected undergoes a sharp transition if the number of lines exceeds a threshold $N^*$.

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Line-of-Sight Networks

Combinatorics Probability Computing ◽

10.1017/s0963548308009334 ◽

2009 ◽

Vol 18 (1-2) ◽

pp. 145-163 ◽

Cited By ~ 11

Author(s):

ALAN FRIEZE ◽

JON KLEINBERG ◽

R. RAVI ◽

WARREN DEBANY

Keyword(s):

Wireless Networks ◽

Shortest Paths ◽

Graph Model ◽

Constant Factor ◽

Line Of Sight ◽

Geometric Graphs ◽

Indoor Environments ◽

Random Graph Model ◽

Random Geometric Graphs ◽

Fixed Distance

Random geometric graphs have been one of the fundamental models for reasoning about wireless networks: one places n points at random in a region of the plane (typically a square or circle), and then connects pairs of points by an edge if they are within a fixed distance of one another. In addition to giving rise to a range of basic theoretical questions, this class of random graphs has been a central analytical tool in the wireless networking community.For many of the primary applications of wireless networks, however, the underlying environment has a large number of obstacles, and communication can only take place among nodes when they are close in space and when they have line-of-sight access to one another – consider, for example, urban settings or large indoor environments. In such domains, the standard model of random geometric graphs is not a good approximation of the true constraints, since it is not designed to capture the line-of-sight restrictions.Here we propose a random-graph model incorporating both range limitations and line-of-sight constraints, and we prove asymptotically tight results for k-connectivity. Specifically, we consider points placed randomly on a grid (or torus), such that each node can see up to a fixed distance along the row and column it belongs to. (We think of the rows and columns as ‘streets’ and ‘avenues’ among a regularly spaced array of obstructions.) Further, we show that when the probability of node placement is a constant factor larger than the threshold for connectivity, near-shortest paths between pairs of nodes can be found, with high probability, by an algorithm using only local information. In addition to analysing connectivity and k-connectivity, we also study the emergence of a giant component, as well an approximation question, in which we seek to connect a set of given nodes in such an environment by adding a small set of additional ‘relay’ nodes.

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Spatial “Artistic” Networks: From Deconstructing Integer-Functions to Visual Arts

Complexity ◽

10.1155/2018/9893867 ◽

2018 ◽

Vol 2018 ◽

pp. 1-8

Author(s):

Ernesto Estrada ◽

Puri Pereira-Ramos

Keyword(s):

Visual Arts ◽

Geometric Graphs ◽

Spatial Networks ◽

Random Geometric Graphs ◽

Unit Square ◽

Graph Theoretic ◽

Network Research ◽

The Aesthetic

Deconstructivism is an aesthetically appealing architectonic style. Here, we identify some general characteristics of this style, such as decomposition of the whole into parts, superposition of layers, and conservation of the memory of the whole. Using these attributes, we propose a method to deconstruct functions based on integers. Using this integer-function deconstruction we generate spatial networks which display a few artistic attributes such as (i) biomorphic shapes, (ii) symmetry, and (iii) beauty. In building these networks, the deconstructed integer-functions are used as the coordinates of the nodes in a unit square, which are then joined according to a given connection radius like in random geometric graphs (RGGs). Some graph-theoretic invariants of these networks are calculated and compared with the classical RGGs. We then show how these networks inspire an artist to create artistic compositions using mixed techniques on canvas and on paper. Finally, we call for avoiding that the applicability of (network) sciences should not go in detriment of curiosity-driven, and aesthetic-driven, researches. We claim that the aesthetic of network research, and not only its applicability, would be an attractor for new minds to this field.

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Normalized Sombor Indices as Complexity Measures of Random Networks

Entropy ◽

10.3390/e23080976 ◽

2021 ◽

Vol 23 (8) ◽

pp. 976

Author(s):

R. Aguilar-Sánchez ◽

J. Méndez-Bermúdez ◽

José Rodríguez ◽

José Sigarreta

Keyword(s):

Random Matrix ◽

Adjacency Matrix ◽

Matrix Theory ◽

Computational Study ◽

Random Networks ◽

Geometric Graphs ◽

Theory Approach ◽

Complexity Measures ◽

Random Geometric Graphs ◽

Highly Correlated

We perform a detailed computational study of the recently introduced Sombor indices on random networks. Specifically, we apply Sombor indices on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the network, scale with the average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random networks and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the adjacency matrix.

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