scholarly journals Optimal paths on the space-time SINR random graph

2011 ◽  
Vol 43 (1) ◽  
pp. 131-150 ◽  
Author(s):  
François Baccelli ◽  
Bartłomiej Błaszczyszyn ◽  
Mir-Omid Haji-Mirsadeghi
Keyword(s):  
Wireless Networks ◽  
Point Process ◽  
Random Graph ◽  
Time Constant ◽  
Random Graphs ◽  
Wireless Channels ◽  
Poisson Point Process ◽  
Wireless Channel ◽  
Space And Time ◽  
Optimal Paths

We analyze a class of signal-to-interference-and-noise-ratio (SINR) random graphs. These random graphs arise in the modeling packet transmissions in wireless networks. In contrast to previous studies on SINR graphs, we consider both a space and a time dimension. The spatial aspect originates from the random locations of the network nodes in the Euclidean plane. The time aspect stems from the random transmission policy followed by each network node and from the time variations of the wireless channel characteristics. The combination of these random space and time aspects leads to fluctuations of the SINR experienced by the wireless channels, which in turn determine the progression of packets in space and time in such a network. In this paper we study optimal paths in such wireless networks in terms of first passage percolation on this random graph. We establish both ‘positive’ and ‘negative’ results on the associated time constant. The latter determines the asymptotics of the minimum delay required by a packet to progress from a source node to a destination node when the Euclidean distance between the two tends to ∞. The main negative result states that this time constant is infinite on the random graph associated with a Poisson point process under natural assumptions on the wireless channels. The main positive result states that, when adding a periodic node infrastructure of arbitrarily small intensity to the Poisson point process, the time constant is positive and finite.

scholarly journals Optimal paths on the space-time SINR random graph

2011 ◽  
Vol 43 (01) ◽  
pp. 131-150 ◽  
Author(s):  
François Baccelli ◽  
Bartłomiej Błaszczyszyn ◽  
Mir-Omid Haji-Mirsadeghi
Keyword(s):  
Wireless Networks ◽  
Point Process ◽  
Random Graph ◽  
Time Constant ◽  
Random Graphs ◽  
Wireless Channels ◽  
Poisson Point Process ◽  
Wireless Channel ◽  
Space And Time ◽  
Optimal Paths

We analyze a class of signal-to-interference-and-noise-ratio (SINR) random graphs. These random graphs arise in the modeling packet transmissions in wireless networks. In contrast to previous studies on SINR graphs, we consider both a space and a time dimension. The spatial aspect originates from the random locations of the network nodes in the Euclidean plane. The time aspect stems from the random transmission policy followed by each network node and from the time variations of the wireless channel characteristics. The combination of these random space and time aspects leads to fluctuations of the SINR experienced by the wireless channels, which in turn determine the progression of packets in space and time in such a network. In this paper we study optimal paths in such wireless networks in terms of first passage percolation on this random graph. We establish both ‘positive’ and ‘negative’ results on the associated time constant. The latter determines the asymptotics of the minimum delay required by a packet to progress from a source node to a destination node when the Euclidean distance between the two tends to ∞. The main negative result states that this time constant is infinite on the random graph associated with a Poisson point process under natural assumptions on the wireless channels. The main positive result states that, when adding a periodic node infrastructure of arbitrarily small intensity to the Poisson point process, the time constant is positive and finite.

scholarly journals SYNCHRONIZATION IN RANDOM GEOMETRIC GRAPHS

2009 ◽  
Vol 19 (02) ◽  
pp. 687-693 ◽  
Author(s):  
ALBERT DÍAZ-GUILERA ◽  
JESÚS GÓMEZ-GARDEÑES ◽  
YAMIR MORENO ◽  
MAZIAR NEKOVEE
Keyword(s):  
Wireless Networks ◽  
Complex Networks ◽  
Random Graph ◽  
Random Graphs ◽  
Coupling Strength ◽  
Order Parameter ◽  
Geometric Graphs ◽  
Random Geometric Graphs ◽  
Distributed Synchronization ◽  
The Stability

In this paper, we study the synchronization properties of random geometric graphs. We show that the onset of synchronization takes place roughly at the same value of the order parameter as a random graph with the same size and average connectivity. However, the dependence of the order parameter on the coupling strength indicates that the fully synchronized state is more easily attained in random graphs. We next focus on the complete synchronized state and show that this state is less stable for random geometric graphs than for other kinds of complex networks. Finally, a rewiring mechanism is proposed as a way to improve the stability of the fully synchronized state as well as to lower the value of the coupling strength at which it is achieved. Our work has important implications for the synchronization of wireless networks, and should provide valuable insights for the development and deployment of more efficient and robust distributed synchronization protocols for these systems.

scholarly journals Invariant colorings of random planar maps

2010 ◽  
Vol 31 (2) ◽  
pp. 549-562 ◽  
Author(s):  
ÁDÁM TIMÁR
Keyword(s):  
Point Process ◽  
Random Graph ◽  
Poisson Point Process ◽  
Invariant Distribution ◽  
Locally Finite ◽  
Mild Assumption ◽  
Voronoi Map ◽  
Planar Maps ◽  
Map Coloring

AbstractWe show that every locally finite random graph embedded in the plane with an isometry-invariant distribution can be five-colored in an invariant and deterministic way, under some non-triviality assumption and a mild assumption on the tail of edge lengths. The assumptions hold for any Voronoi map on a point process that has no non-trivial symmetries almost surely, hence we improve and generalize previous results on six-coloring the Voronoi map on a Poisson point process (see Angel, Benjamini, Gurel-Gurevich, Mayerovitch and Peled [Stationary map coloring. Preprint, 2008]).

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

scholarly journals Pathwise optimality of the exponential scheduling rule for wireless channels

2004 ◽  
Vol 36 (04) ◽  
pp. 1021-1045 ◽  
Author(s):  
Sanjay Shakkottai ◽  
R. Srikant ◽  
Alexander L. Stolyar
Keyword(s):  
Queue Length ◽  
Wireless Channels ◽  
Unique Feature ◽  
Heavy Traffic ◽  
Wireless Channel ◽  
Service Rate ◽  
Resource Pooling ◽  
Scheduling Policy ◽  
Multiple Data ◽  
Heavy Traffic Limit

We consider the problem of scheduling the transmissions of multiple data users (flows) sharing the same wireless channel (server). The unique feature of this problem is the fact that the capacity (service rate) of the channel varies randomly with time and asynchronously for different users. We study a scheduling policy called the exponential scheduling rule, which was introduced in an earlier paper. Given a system withNusers, and any set of positive numbers {an},n= 1, 2,…,N, we show that in a heavy-traffic limit, under a nonrestrictive ‘complete resource pooling’ condition, this algorithm has the property that, for each timet, it (asymptotically) minimizes maxnanq̃n(t), whereq̃n(t) is the queue length of usernin the heavy-traffic regime.

scholarly journals The Total Acquisition Number of Random Graphs

10.37236/5327 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Deepak Bal ◽  
Patrick Bennett ◽  
Andrzej Dudek ◽  
Paweł Prałat
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
High Probability ◽  
Sharp Threshold ◽  
Positive Weight ◽  
Minimum Value ◽  
Almost All

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum possible size of the set of vertices with positive weight at the end of the process.LeSaulnier, Prince, Wenger, West, and Worah asked for the minimum value of $p=p(n)$ such that $a_t(\mathcal{G}(n,p)) = 1$ with high probability, where $\mathcal{G}(n,p)$ is a binomial random graph. We show that $p = \frac{\log_2 n}{n} \approx 1.4427 \ \frac{\log n}{n}$ is a sharp threshold for this property. We also show that almost all trees $T$ satisfy $a_t(T) = \Theta(n)$, confirming a conjecture of West.

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