Performance analysis of connectivity-based geolocation in ultra-dense networks (UDNs) is a very important task. Although several performance analyses have been presented for range-free localization, determining the best achievable positioning accuracy of range-free localization remains an open problem. In this paper, we first derive the Cramer-Rao lower bound (CRLB) for the performance evaluation of range-free localization. All the current performance analyses in the literature for range-free localization are used to evaluate the real performance of a given algorithm, whereas the proposed CRLB provides a benchmark to evaluate the performance of any unbiased range-free location algorithm and determines the physical impossibility of the variance of an unbiased estimator being less than the bound. To the best of our knowledge, this is the first time in the literature that the CRLB for range-free localization has been derived. Second, the theoretical variance of centroid-based localization (CL) with an arbitrary node distribution is derived in this paper. In contrast to the existing theoretical variance of CL for uniform node distribution, the proposed theoretical variance can be used to evaluate the performance of CL in the case of an arbitrary node distribution. Additionally, characteristics of the proposed CRLB and theoretical variance are given in this paper. Finally, an optimal estimator based on a maximum likelihood estimator (MLE) is proposed to improve positioning accuracy. Since both prior information on the spatial node distribution and the connectivity property are effectively utilized in our algorithm, the proposed method performs better than the CL method and can asymptotically attain the CRLB. INDEX TERMS Connectivity-based localization, Cramer-Rao lower bound (CRLB), theoretical variance, maximum likelihood estimator (MLE), centroid-based localization (CL).

Nuclear receptors (NRs) are ligand-dependent transcription factors that regulate the transcription of target genes. Bile acids (BAs) can be used as effector molecules to regulate physiological processes in the gut, and NRs are important receptors for bile acid signaling. Relevant studies have shown that NRs are closely related to the occurrence of Crohn’s disease (CD). Although the mechanism of NRs in CD has not been clarified completely, growing evidence shows that NRs play an important role in regulating intestinal immunity, mucosal barrier, and intestinal flora. NRs can participate in the progress of CD by mediating inflammation, immunity, and autophagy. As the important parts of traditional Chinese medicine (TCM) therapy, acupuncture and moxibustion in the treatment of CD curative mechanism can get a lot of research support. At the same time, acupuncture and moxibustion can regulate the changes of related NRs. Therefore, to explore whether acupuncture can regulate BA circulation and NRs expression and then participate in the disease progression of CD, a new theoretical basis for acupuncture treatment of CD is provided.

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