Robust Throughput Capacity of Multi-Connectivity Wireless Networks

In this paper, we study the robust throughput capacity of multi-connectivity wireless networks when the network encounters zone node failures. In order to reveal the inherent relationship between the robustness of the network structure and the capability of wireless networks to carry information, robust throughput capacity, which is the product of the fraction of served source and destination (S-D) pairs, the number of S-D pairs and feasible throughput, is defined. It is shown that the robust throughput capacity is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Theta \left ({{\sqrt {\frac {n}{k\log n}}}}\right)$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta \gt 2$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Theta \left ({{\frac {1}{k\log n}\sqrt {\frac {n}{k\log n}}}}\right)$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1 {\lt }\beta \leq 2$ </tex-math></inline-formula>, where n is the number of nodes, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> is the failure exponent and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k\:(\geq 1)$ </tex-math></inline-formula> is the connectivity parameter representing the number of disjoint data paths between any two nodes. To balance the tradeoff between the throughput capacity and the robustness of the network structure, the feasible regions of connectivity parameters, which are limited by the failure exponent, are given for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta \gt 2$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1\lt \beta \leq 2$ </tex-math></inline-formula>, respectively. Correspondingly, the robust throughput capacity is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Theta \left ({{\sqrt {\frac {n^{1-\gamma }}{\log n}}}}\right)$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Theta \left ({{\sqrt {\frac {n^{1-3\gamma }}{\log n}}}}\right)$ </tex-math></inline-formula>, respectively, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\gamma \in [0,1$ </tex-math></inline-formula>) is the robustness exponent. These results can provide guidance for designing network protocols with fault tolerance in large-scale wireless networks.