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Critical transmission ranges (or radii) in wireless ad-hoc and sensor networks have been extensively investigated for various performance metrics such as connectivity, coverage, power assignment and energy consumption. However, the regions on which the networks are distributed are typically either squares or disks in existing works, which seriously limits the usage in real-life applications. In this article, we consider a convex region (i.e., a generalisation of squares and disks) on which wireless nodes are uniformly distributed. We have investigated two types of critical transmission radii, defined in terms of k-connectivity and the minimum vertex degree, respectively, and have also established their precise asymptotic distributions. These make the previous results obtained under the circumstance of squares or disks special cases of this work. More importantly, our results reveal how the region shape impacts on the critical transmission ranges: it is the length of the boundary of the (fixed-area) region that completely determines the transmission ranges. Furthermore, by isodiametric inequality, the smallest critical transmission ranges are achieved when regions are disks only.

This paper, based on $k$-NN graph, presents symmetric $(k,j)$-NN graph $(1 \leq j < k)$, a brand new topology which could be adopted by a series of network-based structures. We show that the $k$ nearest neighbors of a node exert disparate influence on guaranteeing network connectivity, and connections with the farthest $j$ ones among these $k$ neighbors are competent to build up a connected network, contrast to the current popular strategy of connecting all these $k$ neighbors. In particular, for a network with node amount $n$ up to $10^3$, as experiments demonstrate, connecting with the farthest three, rather than all, of the five nearest neighbor nodes, i.e. $(k,j)=(5,3)$, can guarantee the network connectivity in high probabilities. We further reveal that more than $0.75n$ links or edges in $5$-NN graph are not necessary for the connectivity. Moreover, a composite topology combining symmetric $(k,j)$-NN and random geometric graph (RGG) is constructed for constrained transmission radii in wireless sensor networks (WSNs) application.