Some epidemic spreading models are usually applied to analyze the propagation of opinions or news. However, the dynamics of epidemic spreading and information or behavior spreading are essentially different in many aspects. Centola's experiments [Science 329, 1194[Science 329, (2010] on behavior spreading in online social networks showed that the spreading is faster and broader in regular networks than in random networks. This result contradicts with the former understanding that random networks are preferable for spreading than regular networks. To describe the spreading in online social networks, a unknown-known-approved-exhausted four-status model was proposed, which emphasizes the effect of social reinforcement and assumes that the redundant signals can improve the probability of approval (i.e., the spreading rate). Performing the model on regular and random networks, it is found that our model can well explain the results of Centola's experiments on behavior spreading and some former studies on information spreading in different parameter space. The effects of average degree and network size on behavior spreading process are further analyzed. The results again show the importance of social reinforcement and are accordant with Centola's anticipation that increasing the network size or decreasing the average degree will enlarge the difference of the density of final approved nodes between regular and random networks. Our work complements the former studies on spreading dynamics, especially the spreading in online social networks where the information usually requires individuals' confirmations before being transmitted to others.

Structural connectivity in the brain is typically studied by reducing its observation to a single spatial resolution. However, the brain possesses a rich architecture organized over multiple scales linked to one another. We explored the multiscale organization of human connectomes using datasets of healthy subjects reconstructed at five different resolutions. We found that the structure of the human brain remains self-similar when the resolution of observation is progressively decreased by hierarchical coarse-graining of the anatomical regions. Strikingly, a geometric network model, where distances are not Euclidean, predicts the multiscale properties of connectomes, including self-similarity. The model relies on the application of a geometric renormalization protocol which decreases the resolution by coarse-graining and averaging over short similarity distances. Our results suggest that simple organizing principles underlie the multiscale architecture of human structural brain networks, where the same connectivity law dictates short- and long-range connections between different brain regions over many resolutions. The implications are varied and can be substantial for fundamental debates, such as whether the brain is working near a critical point, as well as for applications including advanced tools to simplify the digital reconstruction and simulation of the brain.

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The study of recurrent epidemic outbreaks has been attracting great attention for decades, but its underlying mechanism is still under debate. Based on a large number of real data from different cities, we find that besides the seasonal periodic outbreaks of influenza, there are also non-periodic outbreaks, i.e. non-seasonal or non-annual behaviors. To understand how the non-periodicity shows up, we present a network model of SIRS epidemic with both time-dependent infection rate and a small possibility of persistent epidemic seeds, representing the influences from the larger annual variation of environment and the infection generated spontaneously in nature, respectively. Our numerical simulations reveal that the model can reproduce the non-periodic outbreaks of recurrent epidemics with the main features of real influenza data. Further, we find that the recurrent outbreaks of epidemic depend not only on the infection rate but also on the density of susceptible agents, indicating that they are both the necessary conditions for the recurrent epidemic patterns with non-periodicity. A theoretical analysis based on Markov dynamics is presented to explain the numerical results. This finding may be of significance to the control of recurrent epidemics.

There are many evidences to show that different pathogens may interplay each other and cause a variety of mutual influences of epidemics in multiplex networks, but it is still lack of a framework to unify all the different dynamic outcomes of the interactions between the pathogens. We here study this problem and first time present the concept of state-dependent infectious rate, in contrast to the constant infectious rate in previous studies. We consider a model consisting of a two-layered network with one pathogen on the first layer and the other on the second layer, and show that all the different influences between the two pathogens can be given by the different range of parameters in the infectious rates, which includes the cases of mutual enhancement, mutual suppression, and even initial cooperation (suppression) induced final suppression (acceleration). A theoretical analysis is present to explain the numerical results.