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Contact tracing via digital tracking applications installed on mobile phones is an important tool for controlling epidemic spreading. Its effectivity can be quantified by modifying the standard methodology for analyzing percolation and connectivity of contact networks. We apply this framework to networks with varying degree distributions, numbers of application users, and probabilities of quarantine failures. Further, we study structured populations with homophily and heterophily and the possibility of degree-targeted application distribution. Our results are based on a combination of explicit simulations and mean-field analysis. They indicate that there can be major differences in the epidemic size and epidemic probabilities which are equivalent in the normal susceptible-infectious-recovered (SIR) processes. Further, degree heterogeneity is seen to be especially important for the epidemic threshold but not as much for the epidemic size. The probability that tracing leads to quarantines is not as important as the application adoption rate. Finally, both strong homophily and especially heterophily with regard to application adoption can be detrimental. Overall, epidemic dynamics are very sensitive to all of the parameter values we tested out, which makes the problem of estimating the effect of digital contact tracing an inherently multidimensional problem.

Experimental and computational studies provide compelling evidence that neuronal systems are characterized by power-law distributions of neuronal avalanche sizes. This fact is interpreted as an indication that these systems are operating near criticality, and, in turn, typical properties of critical dynamical processes, such as optimal information transmission and stability, are attributed to neuronal systems. The purpose of this Rapid Communication is to show that the presence of power-law distributions for the size of neuronal avalanches is not a sufficient condition for the system to operate near criticality. Specifically, we consider a simplistic model of neuronal dynamics on networks and show that the degree distribution of the underlying neuronal network may trigger power-law distributions for neuronal avalanches even when the system is not in its critical regime. To certify and explain our findings we develop an analytical approach based on percolation theory and branching processes techniques.

Background: Facet radiofrequency denervation is a prevalent procedure used to try and relieve back pain. Despite the increasing use of this treatment, its effectiveness has been questioned. In consideration of the conflicting reports in the literature, we sought to conduct a trial to study the short-term effect of facet denervation in patients undergoing lumbar laminectomy(s) to determine the short-term effect of adding facet denervation to patients undergoing lumbar laminectomy(s) where the anatomy was exposed, allowing an open technique to be used for the denervation., Methods: Sixty patients with a diagnosis of degenerative lumbar spinal stenosis who complained of neurogenic claudication and back pain for at least 3 months were randomized to undergo a lumbar laminectomy(s) either with or without facet joint denervation. Pain and self-reported function using a 10 cm visual analogue scale (VAS) and the Roland-Morris Disability Questionnaire (RMDQ) were measured before surgery and at patients' 6-, 12- and 24-week follow-up clinic visit. Various parametric and non-parametric tests including the Chi-square, independent samples t -tests, the Mann Whitney U, Wilcoxon sign ranks, one-way ANOVA with a Bonferroni post hoc test were used to analyze the data. The RMDQ scores were analyzed between groups and within groups over time., Results: No differences in pain or functional ability were seen between groups. Both groups significantly improved in both pain and function from baseline to follow up times., Conclusions: Our findings do not support the addition of facet denervation for short-term treatment of back pain in patients undergoing lumbar laminectomy(s) for spinal stenosis within our study design., Competing Interests: Conflicts of Interest: The authors have no conflicts of interest to declare.

We show that the community structure of a network can be used as a coarse version of its embedding in a hidden space with hyperbolic geometry. The finding emerges from a systematic analysis of several real-world and synthetic networks. We take advantage of the analogy for reinterpreting results originally obtained through network hyperbolic embedding in terms of community structure only. First, we show that the robustness of a multiplex network can be controlled by tuning the correlation between the community structures across different layers. Second, we deploy an efficient greedy protocol for network navigability that makes use of routing tables based on community structure.

Optimal percolation is the problem of finding the minimal set of nodes whose removal from a network fragments the system into non-extensive disconnected clusters. The solution to this problem is important for strategies of immunization in disease spreading, and influence maximization in opinion dynamics. Optimal percolation has received considerable attention in the context of isolated networks. However, its generalization to multiplex networks has not yet been considered. Here we show that approximating the solution of the optimal percolation problem on a multiplex network with solutions valid for single-layer networks extracted from the multiplex may have serious consequences in the characterization of the true robustness of the system. We reach this conclusion by extending many of the methods for finding approximate solutions of the optimal percolation problem from single-layer to multiplex networks, and performing a systematic analysis on synthetic and real-world multiplex networks.

It is commonly assumed in percolation theories that at most one percolating cluster can exist in a network. We show that several coexisting percolating clusters (CPCs) can emerge in networks due to limited mixing, i.e., a finite and sufficiently small number of interlinks between network modules. We develop an approach called modular message passing (MMP) to describe and verify these observations. We demonstrate that the appearance of CPCs is an important source of inaccuracy in previously introduced percolation theories, such as the message passing (MP) approach, which is a state-of-the-art theory based on the belief propagation method. Moreover, we show that the MMP theory improves significantly over the predictions of MP for percolation on synthetic networks with limited mixing and also on several real-world networks. These findings have important implications for understanding the robustness of networks and in quantifying epidemic outbreaks in the susceptible-infected-recovered (SIR) model of disease spread.

We introduce network L-cloning, a technique for creating ensembles of random networks from any given real-world or artificial network. Each member of the ensemble is an L-cloned network constructed from L copies of the original network. The degree distribution of an L-cloned network and, more importantly, the degree-degree correlation between and beyond nearest neighbors are identical to those of the original network. The density of triangles in an L-cloned network, and hence its clustering coefficient, is reduced by a factor of L compared to those of the original network. Furthermore, the density of loops of any fixed length approaches zero for sufficiently large values of L. Other variants of L-cloning allow us to keep intact the short loops of certain lengths. As an application, we employ these network cloning methods to investigate the effect of short loops on dynamical processes running on networks and to inspect the accuracy of corresponding tree-based theories. We demonstrate that dynamics on L-cloned networks (with sufficiently large L) are accurately described by the so-called adjacency tree-based theories, examples of which include the message passing technique, some pair approximation methods, and the belief propagation algorithm used respectively to study bond percolation, SI epidemics, and the Ising model.