Heterogeneous adoption thresholds exist widely in social contagions, such as behavior spreading, but were always neglected in previous studies. To this end, we introduce heterogeneous adoption threshold distribution into a non-Markovian spreading threshold model, in which an individual adopts a behavior only when the received cumulative pieces of behavioral information from neighbors exceeds his adoption threshold. In order to understand the effects of heterogeneous adoption thresholds quantitatively, an edge-based compartmental theory is developed. A two-state spreading threshold model is taken as an example, in which some individuals have a low adoption threshold (i.e., activists) while the remaining ones hold a relatively higher adoption threshold (i.e., bigots). We find a hierarchical characteristic in adopting behavior, i.e., activists first adopt the behavior and then stimulate bigots to adopt the behavior. Interestingly, two types of crossover phenomena in phase transition occur: for a relatively low adoption threshold of bigots, a change from first-order to secondorder phase transition can be triggered by increasing the fraction of activists; for a relatively higher adoption threshold of bigots, a change from hybrid to second-order phase transition can be induced by varying the fraction of activists, decreasing mean degree or enhancing network heterogeneity. The theoretical predictions based on the suggested theory agree very well with the simulation results.

Epidemic threshold has always been a very hot topic for studying epidemic dynamics on complex networks. The previous studies have provided different theoretical predictions of the epidemic threshold for the susceptible-infected-recovered (SIR) model, but the numerical verification of these theoretical predictions is still lacking. Considering that the large fluctuation of the outbreak size occurs near the epidemic threshold, we propose a novel numerical identification method of SIR epidemic threshold by analyzing the peak of the epidemic variability. Extensive experiments on synthetic and real-world networks demonstrate that the variability measure can successfully give the numerical threshold for the SIR model. The heterogeneous mean-field prediction agrees very well with the numerical threshold, except the case that the networks are disassortative, in which the quenched mean-field prediction is relatively close to the numerical threshold. Moreover, the numerical method presented is also suitable for the susceptible-infected-susceptible model. This work helps to verify the theoretical analysis of epidemic threshold and would promote further studies on the phase transition of epidemic dynamics.

Individuals are always limited by some inelastic resources, such as time and energy, which restrict them to dedicate to social interaction and limit their contact capacities. Contact capacity plays an important role in dynamics of social contagions, which so far has eluded theoretical analysis. In this paper, we first propose a non-Markovian model to understand the effects of contact capacity on social contagions, in which each adopted individual can only contact and transmit the information to a finite number of neighbors. We then develop a heterogeneous edge-based compartmental theory for this model, and a remarkable agreement with simulations is obtained. Through theory and simulations, we find that enlarging the contact capacity makes the network more fragile to behavior spreading. Interestingly, we find that both the continuous and discontinuous dependence of the final adoption size on the information transmission probability can arise. There is a crossover phenomenon between the two types of dependence. More specifically, the crossover phenomenon can be induced by enlarging the contact capacity only when the degree exponent is above a critical degree exponent, while the final behavior adoption size always grows continuously for any contact capacity when degree exponent is below the critical degree exponent.

Accurate identification of effective epidemic threshold is essential for understanding epidemic dynamics on complex networks. The existing studies on the effective epidemic threshold of the susceptible-infected-removed (SIR) model generally assume that all infected nodes immediately recover after the infection process, which more or less does not conform to the realistic situation of disease. In this paper, we systematically study the effect of arbitrary recovery rate on the SIR spreading dynamics on complex networks. We derive the theoretical effective epidemic threshold and final outbreak size based on the edge-based compartmental theory. To validate the proposed theoretical predictions, extensive numerical experiments are implemented by using asynchronous and synchronous updating methods. When asynchronous updating method is used in simulations, recovery rate does not affect the final state of spreading dynamics. But with synchronous updating, we find that the effective epidemic threshold decreases with recovery rate, and final outbreak size increases with recovery rate. A good agreement between the theoretical predictions and numerical results are observed on both synthetic and real-world networks. Our results extend the existing theoretical studies, and help us to understand the phase transition with arbitrary recovery rate. 64.60.Ht How to accurately predict the effective epidemic threshold has attracted increasing attentions. The existing studies on the epidemic threshold generally suppose the recovery process with a constant recovery rate of 1, while the investigation on the effect of recovery rate is still insufficient. Considering the difference of recovery rate between different real diseases and the accompanying effects on the human health, it is very necessary to predict the effective epidemic thresholds with different recovery rates. In this work, the effect of recovery rate on the effective threshold of epidemic outbreak is systematically studied. We first develop a novel theoretical framework based on the edge-based compartmental theory. The developed theory predicts that recovery rate does not affect the spreading dynamics with asynchronous updating, but with synchronous updating, the effective epidemic threshold decreases with the recovery rate, and the final outbreak sizes increases with the recovery rate for a given effective transmission rate. It should be noted that the SIR epidemic of synchronous updating breaks more easily than asynchronous updating. To verify the accuracy of the theoretical predictions, we numerically predict the effective epidemic threshold using the variability measure on random regular networks, where the numerical results agrees well with the theoretical predictions. Moreover, we investigate how the recovery rate affects the epidemic outbreaks with synchronous updating on scale-free networks and real-world networks, and find the same variation trend of effective epidemic threshold. * Electronic address:

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Weak ties play a significant role in the structures and the dynamics of community networks. Based on the contact process, we study numerically how weak ties influence the predictability of epidemic dynamics. We first investigate the effects of the degree of bridge nodes on the variabilities of both the arrival time and the prevalence of disease, and find out that the bridge node with a small degree can enhance the predictability of epidemic spreading. Once weak ties are settled, the variability of the prevalence will display a complete opposite trend to that of the arrival time, as the distance from the initial seed to the bridge node or the degree of the initial seed increases. More specifically, the further distance and the larger degree of the initial seed can induce the better predictability of the arrival time and the worse predictability of the prevalence. Moreover, we discuss the effects of the number of weak ties on the epidemic variability. As the community strength becomes very strong, which is caused by the decrease of the number of weak ties, the epidemic variability will change dramatically. Compared with the case of the hub seed and the random seed, the bridge seed can result in the worst predictability of the arrival time and the best predictability of the prevalence.

Although an increasing amount of research is being done on the dynamical processes on interdependent spatial networks, knowledge of how interdependent spatial networks influence the dynamics of social contagion in them is sparse. Here we present a novel non-Markovian social contagion model on interdependent spatial networks composed of two identical two-dimensional lattices. We compare the dynamics of social contagion on networks with different fractions of dependency links and find that the density of final recovered nodes increases as the number of dependency links is increased. We use a finite-size analysis method to identify the type of phase transition in the giant connected components (GCC) of the final adopted nodes and find that as we increase the fraction of dependency links, the phase transition switches from second-order to first-order. In strong interdependent spatial networks with abundant dependency links, increasing the fraction of initial adopted nodes can induce the switch from a first-order to second-order phase transition associated with social contagion dynamics. In networks with a small number of dependency links, the phase transition remains second-order. In addition, both the second-order and first-order phase transition points can be decreased by increasing the fraction of dependency links or the number of initially-adopted nodes.

Background Panax stipuleanatus represents a folk medicine for treatment of inflammation. However, lack of experimental data does not confirm its function. This article aims to investigate the analgesic and anti-inflammatory activities of triterpenoid saponins isolated from P. stipuleanatus . Methods The chemical characterization of P. stipuleanatus allowed the identification and quantitation of two major compounds. Analgesic effects of triterpenoid saponins were evaluated in two models of thermal- and chemical-stimulated acute pain. Anti-inflammatory effects of triterpenoid saponins were also evaluated using four models of acetic acid–induced vascular permeability, xylene-induced ear edema, carrageenan-induced paw edema, and cotton pellet–induced granuloma in mice. Results Two triterpenoid saponins of stipuleanosides R 1 (SP-R 1 ) and R 2 (SP-R 2 ) were isolated and identified from P. stipuleanatus. The results showed that SP-R 1 and SP-R 2 significantly increased the latency time to thermal pain in the hot plate test and reduced the writhing response in the acetic acid–induced writhing test. SP-R 1 and SP-R 2 caused a significant decrease in vascular permeability, ear edema, paw edema, and granuloma formation in inflammatory models. Further studies showed that the levels of inflammatory mediators, nitric oxide, malondialdehyde, tumor necrosis factor-α, and interleukin 6 in paw tissues were downregulated by SP-R 1 and SP-R 2 . In addition, the rational harvest of three- to five-year-old P. stipuleanatus was preferable to obtain a higher level of triterpenoid saponins. SP-R 2 showed the highest content in P. stipuleanatus , which had potential as a chemical marker for quality control of P. stipuleanatus . Conclusion This study provides important basic information about utilization of P. stipuleanatus resources for production of active triterpenoid saponins.