Percolation describes the sudden emergence of large-scale connectivity as edges are added to a lattice or random network. In the Bohman-Frieze-Wormald model (BFW) of percolation, edges sampled from a random graph are considered individually and either added to the graph or rejected provided that the fraction of accepted edges is never smaller than a decreasing function with asymptotic value of $\alpha$, a constant. The BFW process has been studied as a model system for investigating the underlying mechanisms leading to discontinuous phase transitions in percolation. Here we focus on the regime $\alpha \in [0.6,0.95]$ where it is known that only one giant component, denoted ${\mathrm{C}}_1$, initially appears at the discontinuous phase transition. We show that at some point in the supercritical regime ${\mathrm{C}}_1$ stops growing and eventually a second giant component, denoted ${\mathrm{C}}_2$, emerges in a continuous percolation transition. The delay between the emergence of ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ and their asymptotic sizes both depend on the value of $\alpha$ and we establish by several techniques that there exists a bifurcation point ${\alpha }_c=0.763\pm 0.002$. For $\alpha \in [0.6,{\alpha }_c)$, ${\mathrm{C}}_1$ stops growing the instant it emerges and the delay between the emergence of ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ decreases with increasing $\alpha$. For $\alpha \in ({\alpha }_c,0.95]$, in contrast, ${\mathrm{C}}_1$ continues growing into the supercritical regime and the delay between the emergence of ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ increases with increasing $\alpha$. As we show, ${\alpha }_c$ marks the minimal delay possible between the emergence of ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ (i.e., the smallest edge density for which ${\mathrm{C}}_2$ can exist). We also establish many features of the continuous percolation of ${\mathrm{C}}_2$ including scaling exponents and relations. DOI: http://dx.doi.org/10.1103/PhysRevE.87.052130 ©2013 American Physical Society

The location and nature of the percolation transition in random networks is a subject of intense interest. Recently, a series of graph evolution processes have been introduced that lead to discontinuous percolation transitions where the addition of a single edge causes the size of the largest component to exhibit a significant macroscopic jump in the thermodynamic limit. These processes can have additional exotic behaviors, such as displaying a ``Devil's staircase'' of discrete jumps in the supercritical regime. Here we investigate whether the location of the largest jump coincides with the percolation threshold for a range of processes, such as Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyipercolation, percolation via edge competition and via growth by overtaking. We find that the largest jump asymptotically occurs at the percolation transition for Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyiand other processes exhibiting global continuity, including models exhibiting an ``explosive'' transition. However, for percolation processes exhibiting genuine discontinuities, the behavior is substantially richer. In percolation models where the order parameter exhibits a staircase, the largest discontinuity generically does not coincide with the percolation transition. For the generalized Bohman-Frieze-Wormald model, it depends on the model parameter. Distinct parameter regimes well in the supercritical regime feature unstable discontinuous transitions\char22{}a novel and unexpected phenomenon in percolation. We thus demonstrate that seemingly and genuinely discontinuous percolation transitions can involve a rich behavior in supercriticality, a regime that has been largely ignored in percolation.

Vitamin K2, a natural fat-soluble vitamin, is a potent neuroprotective molecule, owing to its antioxidant effect, but its mechanism has not been fully elucidated. Therefore, we stimulated SH-SY5Y cells with 6-hydroxydopamine (6-OHDA) in a proper dose-dependent manner, followed by a treatment of vitamin K2. In the presence of 6-OHDA, cell viability was reduced, the mitochondrial membrane potential was decreased, and the accumulation of reactive oxygen species (ROS) was increased. Moreover, the treatment of 6-OHDA promoted mitochondria-mediated apoptosis and abnormal mitochondrial fission and fusion. However, vitamin K2 significantly suppressed 6-OHDA-induced changes. Vitamin K2 played a significant part in apoptosis by upregulating and downregulating Bcl-2 and Bax protein expressions, respectively, which inhibited mitochondrial depolarization, and ROS accumulation to maintain mitochondrial structure and functional stabilities. Additionally, vitamin K2 significantly inhibited the 6-OHDA-induced downregulation of the MFN1/2 level and upregulation of the DRP1 level, respectively, and this enabled cells to maintain the dynamic balance of mitochondrial fusion and fission. Furthermore, vitamin K2 treatments downregulated the expression level of p62 and upregulated the expression level of LC3A in 6-OHDA-treated cells via the PINK1/Parkin signaling pathway, thereby promoting mitophagy. Moreover, it induced mitochondrial biogenesis in 6-OHDA damaged cells by promoting the expression of PGC-1α, NRF1, and TFAM. These indicated that vitamin K2 can release mitochondrial damage, and that this effect is related to the participation of vitamin K2 in the regulation of the mitochondrial quality-control loop, through the maintenance of the mitochondrial quality-control system, and repair mitochondrial dysfunction, thereby alleviating neuronal cell death mediated by mitochondrial damage.