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1 hop percolation5/10/2023 Using theories and techniques from continuum percolation, we analytically characterize the connectivity region of the secondary network and reveal the tradeoff between proximity (the number of neighbors) and the occurrence of spectrum opportunities. We introduce the concept of connectivity region defined as the set of density pairs-the density of secondary users and the density of primary transmitters - under which the secondary network is connected. Furthermore, we reveal the tradeoff between proximity (the number of neighbors) and the occurrence of spectrum opportunities by studying the impact of the secondary users ’ transmission power on the connectivity region of the secondary network, and design the transmission power of the secondary users to maximize their tolerance to the primary traffic load.Ībstract-We address the percolation-based connectivity of large-scale ad hoc heterogeneous wireless networks, where secondary users exploit channels temporarily unused by primary users and the existence of a communication link between two secondary users depends on not only the distance between them but also the transmitting and receiving activities of nearby primary users. Using theories and techniques from continuum percolation, we analytically characterize the connectivity region of the secondary network by showing its three basic properties and analyzing its two critical parameters. We introduce the concept of connectivity region defined as the set of density pairs - the density of the secondary users and the density of the primary transmitters - under which the secondary network is connected. We address the connectivity of large-scale ad hoc cognitive radio networks, where secondary users exploit channels temporarily and locally unused by primary users and the existence of a communication link between two secondary users depends not only on the distance between them but also on the transmitting and receiving activities of nearby primary users. Index Terms Cognitive radio network, multihop delay, connectivity, intermittent connectivity, continuum percolation, ergodic theory. When the propagation delay is nonnegligible but small, we show that although the scaling order is always linear, the scaling rate for an instantaneously connected network can be orders of magnitude smaller than the one for an intermittently connected network. Specifically, if the network is instantaneously connected, the minimum multihop delay is asymptotically independent of the distance if the network is only intermittently connected, the minimum multihop delay scales linearly with the distance. When the propagation delay is negligible, we show the starkly different scaling behavior of the minimum multihop delay in instantaneously connected networks as compared to networks that are only intermittently connected due to scarcity of spectrum opportunities. Using theories and techniques from continuum percolation and ergodicity, we establish the scaling law of the minimum multihop delay with respect to the source-destination distance in cognitive radio networks. We analyze the multihop delay of ad hoc cognitive radio networks, where the transmission delay of each hop consists of the propagation delay and the waiting time for the availability of the communication channel (i.e., the occurrence of a spectrum opportunity at this hop). It also serves as an introduction to the field for the other papers in this special issue. This tutorial article surveys some of these techniques, discusses their application to model wireless networks, and presents some of the main results that have appeared in the literature. In this case, different techniques based on stochastic geometry and the theory of random geometric graphs – including point process theory, percolation theory, and probabilistic combinatorics – have led to results on the connectivity, the capacity, the outage probability, and other fundamental limits of wireless networks. Often, the location of the nodes in the network can be modeled as random, following for example a Poisson point process. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communication-theoretic results accounting for the network’s geometrical configuration. Wireless networks are fundamentally limited by the intensity of the received signals and by their interference.
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