Epidemic spreading under mutually independent intra- and inter-host pathogen evolution.

The dynamics of epidemic spreading is often reduced to the single control parameter R0 (reproduction-rate), whose value, above or below unity, determines the state of the contagion. If, however, the pathogen evolves as it spreads, R0 may change over time, potentially leading to a mutation-driven spread, in which an initially sub-pandemic pathogen undergoes a breakthrough mutation. To predict the boundaries of this pandemic phase, we introduce here a modeling framework to couple the inter-host network spreading patterns with the intra-host evolutionary dynamics. We find that even in the extreme case when these two process are driven by mutually independent selection forces, mutations can still fundamentally alter the pandemic phase-diagram. The pandemic transitions, we show, are now shaped, not just by R0, but also by the balance between the epidemic and the evolutionary timescales. If mutations are too slow, the pathogen prevalence decays prior to the appearance of a critical mutation. On the other hand, if mutations are too rapid, the pathogen evolution becomes volatile and, once again, it fails to spread. Between these two extremes, however, we identify a broad range of conditions in which an initially sub-pandemic pathogen can breakthrough to gain widespread prevalence.

Distinct spreading patterns induced by coexisting channels in information spreading dynamics.

In modern society, new communication channels and social platforms remarkably change the way of people receiving and sharing information, but the influences of these channels on information spreading dynamics have not been fully explored, especially in the aspects of outbreak patterns. To this end, based on a susceptible-accepted-recovered model, we examined the outbreak patterns of information spreading in a two-layered network with two coexisting channels: the intra-links within a layer and the inter-links across layers. Depending on the inter-layer coupling strength, i.e., average node degree and transmission probability between the two layers, we observed three different spreading patterns: (i) a localized outbreak with weak inter-layer coupling, (ii) two peaks with a time-delay outbreak appear for an intermediate coupling, and (iii) a synchronized outbreak for a strong coupling. Moreover, we showed that even though the average degree between the two layers is small, a large transmission probability still can compensate and promote the information spread from one layer to another, indicating by that the critical average degree decreases as a power law with transmission probability between the two layers. Additionally, we found that a large gap closed to the critical inter-layer average degree appears in the phase space of theoretical analysis, which indicates the emergence of a global large-scope outbreak. Our findings may, therefore, be of significance for understanding the outbreak behaviors of information spreading in real world.

Spatial multi-scaled chimera states of cerebral cortex network and its inherent structure-dynamics relationship in human brain.

Human cerebral cortex displays various dynamics patterns under different states, however the mechanism how such diverse patterns can be supported by the underlying brain network is still not well understood. Human brain has a unique network structure with different regions of interesting to perform cognitive tasks. Using coupled neural mass oscillators on human cortical network and paying attention to both global and local regions, we observe a new feature of chimera states with multiple spatial scales and a positive correlation between the synchronization preference of local region and the degree of symmetry of the connectivity of the region in the network. Further, we use the concept of effective symmetry in the network to build structural and dynamical hierarchical trees and find close matching between them. These results help to explain the multiple brain rhythms observed in experiments and suggest a generic principle for complex brain network as a structure substrate to support diverse functional patterns.

Scaling up real networks by geometric branching growth.

Real networks often grow through the sequential addition of new nodes that connect to older ones in the graph. However, many real systems evolve through the branching of fundamental units, whether those be scientific fields, countries, or species. Here, we provide empirical evidence for self-similar growth of network structure in the evolution of real systems-the journal-citation network and the world trade web-and present the geometric branching growth model, which predicts this evolution and explains the symmetries observed. The model produces multiscale unfolding of a network in a sequence of scaled-up replicas preserving network features, including clustering and community structure, at all scales. Practical applications in real instances include the tuning of network size for best response to external influence and finite-size scaling to assess critical behavior under random link failures.

Network temporality can promote and suppress information spreading.

Temporality is an essential characteristic of many real-world networks and dramatically affects the spreading dynamics on networks. In this paper, we propose an information spreading model on temporal networks with heterogeneous populations. Individuals are divided into activists and bigots to describe the willingness to accept the information. Through a developed discrete Markov chain approach and extensive numerical simulations, we discuss the phase diagram of the model and the effects of network temporality. From the phase diagram, we find that the outbreak phase transition is continuous when bigots are relatively rare, and a hysteresis loop emerges when there are a sufficient number of bigots. The network temporality does not qualitatively alter the phase diagram. However, we find that the network temporality affects the spreading outbreak size by either promoting or suppressing, which relies on the heterogeneities of population and of degree distribution. Specifically, in networks with homogeneous and weak heterogeneous degree distribution, the network temporality suppresses (promotes) the information spreading for small (large) values of information transmission probability. In networks with strong heterogeneous degree distribution, the network temporality always promotes the information spreading when activists dominate the population, or there are relatively fewer activists. Finally, we also find the optimal network evolution scale, under which the network information spreading is maximized.

Geometric renormalization unravels self-similarity of the multiscale human connectome.

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.

Double transition of information spreading in a two-layered network.

A great deal of significant progress has been seen in the study of information spreading on populations of networked individuals. A common point in many of the past studies is that there is only one transition in the phase diagram of the final accepted size versus the transmission probability. However, whether other factors alter this phenomenology is still under debate, especially for the case of information spreading through many channels and platforms. In the present study, we adopt a two-layered network to represent the interactions of multiple channels and propose a Susceptible-Accepted-Recovered information spreading model. Interestingly, our model shows a novel double transition including a continuous transition and a following discontinuous transition in the phase diagram, which originates from two outbreaks between the two layers of the network. Furthermore, we reveal that the key factors are a weak coupling condition between the two layers, a large adoption threshold, and the difference of the degree distributions between the two layers. Moreover, we also test the model in the coupled empirical social networks and find similar results as in the synthetic networks. Then, an edge-based compartmental theory is developed which fully explains all numerical results. Our findings may be of significance for understanding the secondary outbreaks of information in real life.

A paradox of epidemics between the state and parameter spaces.

It is recently revealed from amounts of real data of recurrent epidemics that there is a phenomenon of hysteresis loop in the state space. To understand it, an indirect investigation from the parameter space has been given to qualitatively explain its mechanism but a more convincing study to quantitatively explain the phenomenon directly from the state space is still missing. We here study this phenomenon directly from the state space and find that there is a positive correlation between the size of outbreak and the size of hysteresis loop, implying that the hysteresis is a nature feature of epidemic outbreak in real case. Moreover, we surprisingly find a paradox on the dependence of the size of hysteresis loop on the two parameters of the infectious rate increment and the transient time, i.e. contradictory behaviors between the two spaces, when the evolutionary time of epidemics is long enough. That is, with the increase of the infectious rate increment, the size of hysteresis loop will decrease in the state space but increase in the parameter space. While with the increase of the transient time, the size of hysteresis loop will increase in the state space but decrease in the parameter space. Furthermore, we find that this paradox will disappear when the evolutionary time of epidemics is limited in a fixed period. Some theoretical analysis are presented to both the paradox and other numerical results.

Correlated network of networks enhances robustness against catastrophic failures.

Networks in nature rarely function in isolation but instead interact with one another with a form of a network of networks (NoN). A network of networks with interdependency between distinct networks contains instability of abrupt collapse related to the global rule of activation. As a remedy of the collapse instability, here we investigate a model of correlated NoN. We find that the collapse instability can be removed when hubs provide the majority of interconnections and interconnections are convergent between hubs. Thus, our study identifies a stable structure of correlated NoN against catastrophic failures. Our result further suggests a plausible way to enhance network robustness by manipulating connection patterns, along with other methods such as controlling the state of node based on a local rule.

A model of spreading of sudden events on social networks.

Information spreading has been studied for decades, but its underlying mechanism is still under debate, especially for those ones spreading extremely fast through the Internet. By focusing on the information spreading data of six typical events on Sina Weibo, we surprisingly find that the spreading of modern information shows some new features, i.e., either extremely fast or slow, depending on the individual events. To understand its mechanism, we present a susceptible-accepted-recovered model with both information sensitivity and social reinforcement. Numerical simulations show that the model can reproduce the main spreading patterns of the six typical events. By this model, we further reveal that the spreading can be speeded up by increasing either the strength of information sensitivity or social reinforcement. Depending on the transmission probability and information sensitivity, the final accepted size can change from continuous to discontinuous transition when the strength of the social reinforcement is large. Moreover, an edge-based compartmental theory is presented to explain the numerical results. These findings may be of significance on the control of information spreading in modern society.

Multiple peaks patterns of epidemic spreading in multi-layer networks.

The study of epidemic spreading on populations of networked individuals has seen recently a great deal of significant progresses. A common point in many of past studies is, however, that there is only one peak of infected density in each single epidemic spreading episode. At variance, real data from different cities over the world suggest that, besides a major single peak trait of infected density, a finite probability exists for a pattern made of two (or multiple) peaks. We show that such a latter feature is distinctive of a multilayered network of interactions, and reveal that a two peaks pattern may emerge from different time delays at which the epidemic spreads in between the two layers. Further, we show that the essential ingredient is a weak coupling condition between the layers themselves, while different degree distributions in the two layers are also helpful. Moreover, an edge-based theory is developed which fully explains all numerical results. Our findings may therefore be of significance for protecting secondary disasters of epidemics, which are definitely undesired in real life.

A dynamic vaccination strategy to suppress the recurrent epidemic outbreaks.

Efficient vaccination strategy is crucial for controlling recurrent epidemic spreading on networks. In this paper, based on the analysis of real epidemic data and simulations, it's found that the risk indicator of recurrent epidemic outbreaks could be determined by the ratio of the epidemic infection rate of the year to the average infected density of the former year. According to the risk indicator, the dynamic vaccination probability of each year can be designed to suppress the epidemic outbreaks. Our simulation results show that the dynamic vaccination strategy could effectively decrease the maximal and average infected density, and meanwhile increase the time intervals of epidemic outbreaks and individuals attacked by epidemic. In addition, our results indicate that to depress the influenza outbreaks, it is not necessary to keep the vaccination probability high every year; and adjusting the vaccination probability at right time could decrease the outbreak risks with lower costs. Our findings may present a theoretical guidance for the government and the public to control the recurrent epidemic outbreaks.

Synchronized and mixed outbreaks of coupled recurrent epidemics.

Epidemic spreading has been studied for a long time and most of them are focused on the growing aspect of a single epidemic outbreak. Recently, we extended the study to the case of recurrent epidemics (Sci. Rep. 5, 16010 (2015)) but limited only to a single network. We here report from the real data of coupled regions or cities that the recurrent epidemics in two coupled networks are closely related to each other and can show either synchronized outbreak pattern where outbreaks occur simultaneously in both networks or mixed outbreak pattern where outbreaks occur in one network but do not in another one. To reveal the underlying mechanism, we present a two-layered network model of coupled recurrent epidemics to reproduce the synchronized and mixed outbreak patterns. We show that the synchronized outbreak pattern is preferred to be triggered in two coupled networks with the same average degree while the mixed outbreak pattern is likely to show for the case with different average degrees. Further, we show that the coupling between the two layers tends to suppress the mixed outbreak pattern but enhance the synchronized outbreak pattern. A theoretical analysis based on microscopic Markov-chain approach is presented to explain the numerical results. This finding opens a new window for studying the recurrent epidemics in multi-layered networks.

Hysteresis loop of nonperiodic outbreaks of recurrent epidemics.

Most of the studies on epidemics so far have focused on the growing phase, such as how an epidemic spreads and what are the conditions for an epidemic to break out in a variety of cases. However, we discover from real data that on a large scale, the spread of an epidemic is in fact a recurrent event with distinctive growing and recovering phases, i.e., a hysteresis loop. We show here that the hysteresis loop can be reproduced in epidemic models provided that the infectious rate is adiabatically increased or decreased before the system reaches its stationary state. Two ways to the hysteresis loop are revealed, which is helpful in understanding the mechanics of infections in real evolution. Moreover, a theoretical analysis is presented to explain the mechanism of the hysteresis loop.

Non-periodic outbreaks of recurrent epidemics and its network modelling.

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.

A unified framework of mutual influence between two pathogens in multiplex networks.

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.

Spreading in online social networks: the role of social reinforcement.

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 (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.