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Afterwards, we show percolation in the PMLG via coupling it with percolation in the discrete lattice and study connectivity in the presence and absence of primary users.

For both cases, we define and characterize connectivity in terms of the available number of channels, deployment densities, number of simultaneous transmissions per node, and communication range.

In the absence of the primary users, we find the upper bound for the number of channels at which phase transition occurs i. In the presence of the primary users, we characterize connectivity in terms of the primary density and identify two ranges that have very different effects on connectivity. In the first range, we show how the connectivity increases as the primary density increases. The two ranges are partitioned by an optimal point where the connectivity is maximized.

In the second range, we show how the connectivity decreases as the primary density grows. With the insights from the naive rendezvous protocol, we analyze the effects of advanced rendezvous protocols on the connectivity.

We also find the minimum deployment density needed to maintain connectivity for a given rendezvous probability. Finally, we interpret our findings in terms of achieving connectivity using the minimal deployment density, number of simultaneous transmissions per node, and scanning time.

However, the literature is rich with classical results for connectivity of both homogeneous [8], [17], [25] and heterogeneous [13], [14], [23], [27] networks.

In [8], percolation was used to show that even when the SINR model is considered, the resulting connectivity is similar to that of a Boolean model given that the interference from other nodes can be sufficiently reduced at each receiving node.

In [17], continuum percolation was used to obtain the critical transmission range of the asymptotic connectivity of i. In [25], the k-connectivity of a random graph is linked with the minimum node degree for graphs with high number of nodes.

In [13], the k-connectivity of CRN is studied using percolation and asymptotic-connectivity where the critical secondary density is found. In [14], the percolation degree of the secondary users and its relation with the k-disjoint paths is studied in large scale CRNs. In [23], the activity of the primary users is assumed to be dynamic; its effects on the available number of channels for the secondary network is studied using percolation theory in which the critical primary user densities were illustrated.

In [27], percolation was used to study the connectivity of heterogeneous networks via identifying connectivity regions which were characterized by a set of primary and secondary networks in which percolation occurs. All the mentioned works provide useful theoretical insights. However, none of these i studied the impact of rendezvous on connectivity, ii considered that a radio can communicate simultaneously on finite multiple channels, iii accounted for the scanning limitations. Our arguments are motivated by [7], [8], [12], [23], [24].

All users are distributed uniformly randomly with a coverage radius of rs. The two networks are embedded in R2 with a total of N channels vacant and nonvacant each of which could be accessed by primary users with equal probability i.

Due to primary activity and spatial diversity, all secondaries do not necessarily observe the same number of vacant channels. Commonly used notations are shown in Table I. We emphasize that although the sensing method will not change the core idea rather than increasing or decreasing the value of M. Therefore, at any point of time a secondary user Si can communicate over at maximum any M of its ni available channels which are determined by the sensing process.

This is the disk graph model where nodes within the disk centered at xi are considered neighbors of Si. In our context, the term advanced refers to any rendezvous protocol that guarantees a common channel within a certain period of time bounded time which makes the rendezvous probability strictly greater than zero.

Such rendezvous techniques are considered later in section V.

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Note, the naive protocol does not guarantee common channel s between two neighboring nodes in a bounded time interval; however it is the easiest to implement as compared to the advanced ones whose performance vary and depends on their complexity. El grupo de estudio fue alimentado pre- cozmente con leche humana para estimulo trofico y lue- go fortificada con fortificadores de leche humana.

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