{'fraction_non_alphanumeric': 0.023088023088023088, 'fraction_numerical': 0.004329004329004329, 'mean_word_length': 5.004329004329004, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Critical Density for Connectivity in 2D and 3D Wireless Multi-Hop Networks

In this paper we investigate the critical node density required to ensure that an arbitrary node in a large-scale wireless multi-hop network is connected (via multi-hop path) to infinitely many other nodes with a positive probability. Specifically we consider a wireless multi-hop network where nodes are distributed in \mathbb{R}^d (d = 2,3) following a homogeneous Poisson point process. The establishment of a direct connection between any two nodes is independent of connections between other pairs of nodes and its probability satisfies some intuitively reasonable conditions, viz. rotational and translational invariance, non-increasing monotonicity, and integral boundedness. Under the above random connection model we first obtain analytically the upper and lower bounds for the critical density. Then we compare the new bounds with other existing bounds in the literature under the unit disk model and the log-normal model which are special cases of the random connection model. The comparison shows that our bounds are either close to or tighter than the known ones. To the best of our knowledge, this is the first result for the random connection model in both 2D and 3D networks. The result is of practical use for designing large-scale wireless multi-hop networks such as wireless sensor networks.