Preferential attachment is an appealing mechanism for modeling power-law behavior of the degree distributions in directed social networks. In this paper, we consider methods for fitting a 5-parameter linear preferential model to network data under two data scenarios. In the case where full history of the network formation is given, we derive the maximum likelihood estimator of the parameters and show that it is strongly consistent and asymptotically normal. In the case where only a single-time snapshot of the network is available, we propose an estimation method which combines method of moments with an approximation to the likelihood. The resulting estimator is also strongly consistent and performs quite well compared to the MLE estimator. We illustrate both estimation procedures through simulated data, and explore the usage of this model in a real data example.

Chronic lead exposure is associated with the development of neurodegenerative diseases, characterized by the long-term memory decline. However, whether this pathogenesis could be prevented through adjusting gut microbiota is not yet understood. To address the issue, pregnant rats and their female offspring were treated with lead (125 ppm) or separately the extra probiotics (1010 organisms/rat/day) till adulthood. For results, memory dysfunction was alleviated by the treatment of multispecies probiotics. Meanwhile, the gut microbiota composition was partially normalized against lead-exposed rats, which in turn mediated the memory repairment via fecal transplantation trials. In the molecular aspect, the decreased H3K27me3 (trimethylation of histone H3 Lys 27) in the adult hippocampus was restored with probiotic intervention, an epigenetic event mediated by EZH2 (enhancer of zeste homolog 2) at early developmental stage. In a neural cellular model, EZH2 overexpression showed the similar rescue effect with probiotics, whereas its blockade led to the neural re-damages. Regarding the gut–brain inflammatory mediators, the disrupted IL-6 (interleukin 6) expression was resumed by probiotic treatment. Intraperitoneal injection of tocilizumab, an IL-6 receptor antagonist, upregulated the hippocampal EZH2 level and consequently alleviated the memory injuries. In conclusion, reshaping gut microbiota could mitigate memory dysfunction caused by chronic lead exposure, wherein the inflammation–hippocampal epigenetic pathway of IL-6-EZH2-H3K27me3, was first proposed to mediate the studied gut–brain communication. These findings provided insight with epigenetic mechanisms underlying a unique gut–brain interaction, shedding light on the safe and non-invasive treatment of neurodegenerative disorders with environmental etiology.

Preferential attachment is an appealing edge generating mechanism for modeling social networks. It provides both an intuitive description of network growth and an explanation for the observed power laws in degree distributions. However, there are often limitations in fitting parametric network models to data due to the complex nature of real-world networks. In this paper, we consider a semi-parametric estimation approach by looking at only the nodes with large in-or out-degrees of the network. This method examines the tail behavior of both the marginal and joint degree distributions and is based on extreme value theory. We compare it with the existing parametric approaches and demonstrate how it can provide more robust estimates of parameters associated with the network when the data are corrupted or when the model is misspecified.

Preferential attachment is widely used to model power-law behavior of degree distributions in both directed and undirected networks. Practical analyses on the tail exponent of the power-law degree distribution use Hill estimator as one of the key summary statistics, whose consistency is justified mostly for iid data. The major goal in this paper is to answer the question whether the Hill estimator is still consistent when applied to non-iid network data. To do this, we first derive the asymptotic behavior of the degree sequence via embedding the degree growth of a fixed node into a birth immigration process. We also need to show the convergence of the tail empirical measure, from which the consistency of Hill estimators is obtained. This step requires checking the concentration of degree counts. We give a proof for a particular linear preferential attachment model and use simulation results as an illustration in other choices of models.

Power-law distributions have been widely observed in different areas of scientific research. Practical estimation issues include how to select a threshold above which observations follow a power-law distribution and then how to estimate the power-law tail index. A minimum distance selection procedure (MDSP) is proposed in [8] and has been widely adopted in practice, especially in the analyses of social networks. However, theoretical justifications for this selection procedure remain scant. In this paper, we study the asymptotic behavior of the selected threshold and the corresponding power-law index given by the MDSP. We find that the MDSP tends to choose too high a threshold level and leads to Hill estimates with large variances and root mean squared errors for simulated data with Pareto-like tails.