# Optimal Paths on the Space-Time SINR Random Graph - Mathematics > Probability

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Abstract: We analyze a class of Signal-to-Interference-and-Noise-Ratio SINR randomgraphs. These random graphs arise in the modeling packet transmissions inwireless networks. In contrast to previous studies on the SINR graphs, weconsider both a space and a time dimension. The spatial aspect originates fromthe random locations of the network nodes in the Euclidean plane. The timeaspect stems from the random transmission policy followed by each network nodeand from the time variations of the wireless channel characteristics. Thecombination of these random space and time aspects leads to fluctuations of theSINR experienced by the wireless channels, which in turn determine theprogression of packets in space and time in such a network. This paper studiesoptimal paths in such wireless networks in terms of first passage percolationon this random graph. We establish both -positive- and -negative- results onthe associated time constant. The latter determines the asymptotics of theminimum delay required by a packet to progress from a source node to adestination node when the Euclidean distance between the two tends to infinity.The main negative result states that this time constant is infinite on therandom graph associated with a Poisson point process under natural assumptionson the wireless channels. The main positive result states that when adding aperiodic node infrastructure of arbitrarily small intensity to the Poissonpoint process, the time constant is positive and finite.

Author: ** Francois Baccelli, Bartlomiej Blaszczyszyn, Mir Omid Haji Mirsadeghi**

Source: https://arxiv.org/