Random Networks for Communication

Abstract

When is a random network (almost) connected? How much information can it carry? How can you find a particular destination within the network? And how do you approach these questions - and others - when the network is random? The analysis of communication networks requires a fascinating synthesis of random graph theory, stochastic geometry and percolation theory to provide models for both structure and information flow. This book is the first comprehensive introduction for graduate students and scientists to techniques and problems in the field of spatial random networks. The selection of material is driven by applications arising in engineering, and the treatment is both readable and mathematically rigorous. Though mainly concerned with information-flow-related questions motivated by wireless data networks, the models developed are also of interest in a broader context, ranging from engineering to social networks, biology, and physics.

Impact of boundaries on fully connected random geometric networks

“…(8) in Mao and Anderson [21] which was derived for the specific case of a square domain. Following numerous studies by probabilists and engineers [1,2], these authors however assumed an exponential scaling of system size V with ρ which essentially renders boundary effects negligible. Scaling the system in such a way is a common approach as it corresponds to the limit of infinite density at fixed connection probability, however in practice this limit is approached only for unphysically large volumes.…”

“…Random geometric network models [1,2] comprise a collection of entities called nodes embedded in region of typically two or three dimensions, together with connecting links between pairs of nodes that exist with a probability related to the node locations. They appear in numerous complex systems including in nanoscience [3], epidemiology [4,5], forest fires [6], social networks [7,8], and wireless communications [9][10][11].…”

Strict Inequalities of Critical Values in Continuum Percolation

“…the points in P μ/2,24,25 that are path-connected to V and to T respectively). Let b 1 = b(y, z) and 5) and for any b ∈ D 1 and a ∈ K 1 we have max(|b − y|, |a − z|) ≤ 0.9991. Next take further discs D i = C 0.0001 (b i ) and K i = C 0.0001 (a i ), for 2 ≤ i ≤ 7, such that each of these discs is contained in A 20,24 , and discs D 1 , K 1 , .…”

“…The weak inequality p site c ≥ p bond c is well-known; see for example Chap. 2 of [5]. If G is a rooted tree, then it is easy to see that p site c = p bond c , as each vertex, other than the root, can be uniquely identified by an edge and vice versa.…”

Performance Testing of RNSC and MCL Algorithms on Random Geometric Graphs

“…The random geometric network is now-a-day getting immense popularity in society and nature for the complex system's modelling. Random geometric network models [5,6] consist of a collection of entities called nodes embedded in a region of exclusively two or three dimensions, together with connecting links between pairs of nodes that exist with a probability related to the node locations. These models perform well in demonstrating numerous complex systems including Nano science [7], epidemiology [8,9], forest fires [10], social networks [11,12], and wireless communications [13][14][15].…”