The participation of individuals in multi-layer networks allows for feedback between network layers, opening new possibilities to mitigate epidemic spreading. For instance, the spread of a biological disease such as Ebola in a physical contact network may trigger the propagation of the information related to this disease in a communication network, e.g. an online social network. The information propagated in the communication network may increase the awareness of some individuals, resulting in them avoiding contact with their infected neighbors in the physical contact network, which might protect the population from the infection. In this work, we aim to understand how the time scale γ of the information propagation (speed that information is spread and forgotten) in the communication network relative to that of the epidemic spread (speed that an epidemic is spread and cured) in the physical contact network influences such mitigation using awareness information. We begin by proposing a model of the interaction between information propagation and epidemic spread, taking into account the relative time scale γ. We analytically derive the average fraction of infected nodes in the meta-stable state for this model (i) by developing an individual-based mean-field approximation (IBMFA) method and (ii) by extending the microscopic Markov chain approach (MMCA). We show that when the time scale γ of the information spread relative to the epidemic spread is large, our IBMFA approximation is better compared to MMCA near the epidemic threshold, whereas MMCA performs better when the prevalence of the epidemic is high. Furthermore, we find that an optimal mitigation exists that leads to a minimal fraction of infected nodes. The optimal mitigation is achieved at a non-trivial relative time scale γ, which depends on the rate at which an infected individual becomes aware. Contrary to our intuition, information spread too fast in the communication network could reduce the mitigation effect. Finally, our finding has been validated in the real-world two-layer network obtained from the location-based social network Brightkite.@en

In the recent decades, various dynamic process models on complex networks have been built to study the mechanisms by which an opinion, a disease or generally the information spreads in real-world networks. For example, opinion models are developed to illustrate the competition of opinions in a population, and epidemic models are used to describe, e.g. how an epidemic spreads in a social contact network or how information propagates in an online social network. Classic models always assume the homogeneous interactions. For example, the infection rates are the same for all pairs of nodes. However, the infection rates between different pairs of nodes which may depend on e.g. interaction frequencies are usually different , thus heterogeneous. In this thesis, we aim to explore the influence of heterogeneity on dynamic processes especially on the prevalence of an epidemic or opinion. We consider two types of dynamic processes: the Non- Consensus Opinion (NCO) model and the Susceptible-Infected-Susceptible (SIS) model. This thesis is mainly devoted to the latter one. We investigate the heterogeneity in both network topology models, e.g. directed networks, and dynamic process models, such as heterogeneous infection rates. @en

Traces of user activities recorded in online social networks open new possibilities to systematically understand the information diffusion process on social networks. From the online social network WeChat, we collected a large number of information cascade trees, each of which tells the spreading trajectory of a message/information such as which user creates the information and which users view or forward the information shared by which neighbors. In this work, we propose two heterogeneous non-linear models, one for the topologies of the information cascade trees and the other for the stochastic process of information diffusion on a social network. Both models are validated by the WeChat data in reproducing and explaining key features of cascade trees. Specifically, we apply the Random Recursive Tree (RRT) to model the growth of cascade trees. The RRT model could capture key features, i.e. the average path length and degree variance of a cascade tree in relation to the number of nodes (size) of the tree. Its single identified parameter quantifies the relative depth or broadness of the cascade trees and indicates that information propagates via a star-like broadcasting or viral-like hop by hop spreading. The RRT model explains the appearance of hubs, thus a possibly smaller average path length as the cascade size increases, as observed in WeChat. We further propose the stochastic Susceptible View Forward Removed (SVFR) model to depict the dynamic user behavior including creating, viewing, forwarding and ignoring a message on a given social network. Beside the average path length and degree variance of the cascade trees in relation to their sizes, the SVFR model could further explain the power-law cascade size distribution in WeChat and unravel that a user with a large number of friends may actually have a smaller probability to read a message (s)he receives due to limited attention.@en

The prevalence, which is the average fraction of infected nodes, has been studied to evaluate the robustness of a network subject to the spread of epidemics. We explore the vulnerability (infection probability) of each node in the metastable state with a given effective infection rate τ. Specifically, we investigate the ranking of the nodal vulnerability subject to a susceptible-infected-susceptible epidemic, motivated by the fact that the ranking can be crucial for a network operator to assess which nodes are more vulnerable. Via both theoretical and numerical approaches, we unveil that the ranking of nodal vulnerability tends to change more significantly as τ varies when τ is smaller or in Barabási-Albert than Erdos-Rényi random graphs.@en

In this work, we aim to understand the influence of the heterogeneity of infection rates on the Susceptible-Infected-Susceptible (SIS) epidemic spreading. Employing the classic SIS model as the benchmark, we study the influence of the independently identically distributed infection rates on the average fraction of infected nodes in the metastable state. The log-normal, gamma and a newly designed distributions are considered for infection rates. We find that, when the recovery rate is small, i.e., the epidemic spreads out in both homogeneous and heterogeneous cases: 1) the heterogeneity of infection rates on average retards the virus spreading, and 2) a larger even-order moment of the infection rates leads to a smaller average fraction of infected nodes, but the odd-order moments contribute in the opposite way; when the recovery rate is large, i.e., the epidemic may die out or infect a small fraction of the population, the heterogeneity of infection rates may enhance the probability that the epidemic spreads out. Finally, we verify our conclusions via real-world networks with their heterogeneous infection rates. Our results suggest that, in reality the epidemic spread may not be so severe as the classic SIS model indicates, but to eliminate the epidemic is probably more difficult.@en

Online social networks can detailedly and accurately record the activities of human beings and the trajectories of information dissemination over time, which provides us an opportunity to understand the information diffusion process from a renewed, more realistic, data driven modeling dimensionality. In consideration of two fundamental behaviors (viewing and sharing) involved in information diffusion, we propose a stochastic, heterogeneous, continuous-time delay Unknown-View-Share-Removed (UVSR) model to characterize the information diffusion process. The UVSR model introduces four parameters to describe the diffusion probability and speed: viewing/sharing probability/delay. These parameters are subject to some sort of distributions from the actual data, or based on empirical assumptions. To validate the model, we collect and analyze large number of information cascades (tree structure) diffused in WeChat network. We find that the viewing delay and sharing delay are approximately subject to log-normal and power-law distributions respectively, and the sharing probability follows a Gaussian distribution. Driven by these empirical findings and a constant viewing probability assumption, our model can reproduce numerous key features of information diffusion process in both topology and temporal dynamics, such as cascade size distribution, structural virality, life span distribution and relative propagation speed. Our work contributes to a better understanding of the topological features and temporal dynamics of information diffusion from a continuous time, stochastic modeling view.@en

The epidemic spreading has been widely studied in homogeneous cases, where each node may get infected by an infected neighbor with the same rate. However, the infection rate between a pair of nodes, which may depend on e.g. their interaction frequency, is usually heterogeneous and even correlated with their nodal degrees in the contact network. In this paper, we aim to understand how such correlated heterogeneous infection rates influence the epidemic spreading on different network topologies. Motivated by real-world datasets, we propose a Correlated heterogeneous Susceptible–Infected–Susceptible (CSIS) model which assumes that the infection rate βij(=βji) between node i and j is correlated with the degree of the two end nodes: βij=c(didj)α, where α indicates the strength of the correlation between the infection rates and nodal degrees, and c is selected so that the average infection rate is 1 in this work. The recovery rate is the same for all the nodes. In order to understand the effect of such correlation on epidemic spreading, we consider as well the corresponding uncorrected but still heterogeneous infection rate scenario as a reference, where the original correlated infection rates in our CSIS model are shuffled and reallocated to the links of the same network topology. We compare these two scenarios in the average fraction of infected nodes in the metastable state on Erdös–Rényi (ER) and scale-free (SF) networks with a similar average degree. Through the continuous-time simulations, we find that, when the recovery rate is small, the negative correlation is more likely to help the epidemic spread and the positive correlation prohibits the spreading; as the recovery rate increases to be larger than a critical value, the positive but not negative correlation tends to help the spreading. Our findings are further analytically proved in a wheel network (one central node connects with each of the nodes in a ring) and validated on real-world networks with correlated heterogeneous interaction frequencies.@en

The nodes in communication networks are possibly and most likely equipped with different recovery resources, which allow them to recover from a virus with different rates. In this paper, we aim to understand know how to allocate the limited recovery resources to efficiently prevent the spreading of epidemics. We study the susceptible-infected-susceptible (SIS) epidemic model on directed scale-free networks. In the classic SIS model, a susceptible node can be infected by an infected neighbor with the infection rate β and an infected node can be recovered to be susceptible again with the recovery rate δ. In the steady state a fraction y∞ of nodes are infected, which shows how severely the network is infected. Instead of considering the same recovery rate for all the nodes in the SIS model, we propose to allocate the recovery rate δi for node i according to its indegree and outdegree-δi∼ kα in i, in kα out i, out given the finite average recovery rate (δ) representing the limited recovery resources over the whole network. We consider directed networks with the same power law indegree and outdegree distribution, but different values of the directionality ξ (the proportion of unidirectional links) and linear correlation coefficients ρ between the indegree and outdegree. We find that, by tuning the two scaling exponents αin and αout, we can always reduce the infection fraction y∞ thus reducing the extent of infections, comparing to the homogeneous recovery rates allocation. Moreover, we can find our optimal strategy via the optimal choice of the exponent αin and αout. Our optimal strategy indicates that when the recovery resources are sufficient, more resources should be allocated to the nodes with a larger indegree or outdegree, but when the recovery resource is very limited, only the nodes with a larger outdegree should be equipped with more resources. We also find that our optimal strategy works better when the recovery resources are sufficient but not yet able to make the epidemic die out, and when the indegree outdegree correlation is small.@en

The nodes in communication networks are possibly and most likely equipped with different recovery resources, which allow them to recover from a virus with different rates. In this paper, we aim to understand know how to allocate the limited recovery resources to efficiently prevent the spreading of epidemics. We study the susceptible-infected-susceptible (SIS) epidemic model on directed scale-free networks. In the classic SIS model, a susceptible node can be infected by an infected neighbor with the infection rate β and an infected node can be recovered to be susceptible again with the recovery rate δ. In the steady state a fraction y∞ of nodes are infected, which shows how severely the network is infected. Instead of considering the same recovery rate for all the nodes in the SIS model, we propose to allocate the recovery rate δi for node i according to its indegree and outdegree-δi∼ kα in i, in kα out i, out given the finite average recovery rate (δ) representing the limited recovery resources over the whole network. We consider directed networks with the same power law indegree and outdegree distribution, but different values of the directionality ξ (the proportion of unidirectional links) and linear correlation coefficients ρ between the indegree and outdegree. We find that, by tuning the two scaling exponents αin and αout, we can always reduce the infection fraction y∞ thus reducing the extent of infections, comparing to the homogeneous recovery rates allocation. Moreover, we can find our optimal strategy via the optimal choice of the exponent αin and αout. Our optimal strategy indicates that when the recovery resources are sufficient, more resources should be allocated to the nodes with a larger indegree or outdegree, but when the recovery resource is very limited, only the nodes with a larger outdegree should be equipped with more resources. We also find that our optimal strategy works better when the recovery resources are sufficient but not yet able to make the epidemic die out, and when the indegree outdegree correlation is small.@en

The nodes in communication networks are possibly and most likely equipped with different recovery resources, which allow them to recover from a virus with different rates. In this paper, we aim to understand know how to allocate the limited recovery resources to efficiently prevent the spreading of epidemics. We study the susceptible-infected-susceptible (SIS) epidemic model on directed scale-free networks. In the classic SIS model, a susceptible node can be infected by an infected neighbor with the infection rate β and an infected node can be recovered to be susceptible again with the recovery rate δ. In the steady state a fraction y∞ of nodes are infected, which shows how severely the network is infected. Instead of considering the same recovery rate for all the nodes in the SIS model, we propose to allocate the recovery rate δi for node i according to its indegree and outdegree-δi∼ kα in i, in kα out i, out given the finite average recovery rate (δ) representing the limited recovery resources over the whole network. We consider directed networks with the same power law indegree and outdegree distribution, but different values of the directionality ξ (the proportion of unidirectional links) and linear correlation coefficients ρ between the indegree and outdegree. We find that, by tuning the two scaling exponents αin and αout, we can always reduce the infection fraction y∞ thus reducing the extent of infections, comparing to the homogeneous recovery rates allocation. Moreover, we can find our optimal strategy via the optimal choice of the exponent αin and αout. Our optimal strategy indicates that when the recovery resources are sufficient, more resources should be allocated to the nodes with a larger indegree or outdegree, but when the recovery resource is very limited, only the nodes with a larger outdegree should be equipped with more resources. We also find that our optimal strategy works better when the recovery resources are sufficient but not yet able to make the epidemic die out, and when the indegree outdegree correlation is small.@en

The epidemic spreading over a network has been studied for years by applying the mean-field approach in both homogeneous case, where each node may get infected by an infected neighbor with the same rate, and heterogeneous case, where the infection rates between different pairs of nodes are also different. Researchers have discussed whether the mean-field approaches could accurately describe the epidemic spreading for the homogeneous cases but not for the heterogeneous cases. In this paper, we explore if and under what conditions the mean-field approach could perform well when the infection rates are heterogeneous. In particular, we employ the Susceptible-Infected-Susceptible (SIS) model and compare the average fraction of infected nodes in the metastable state, where the fraction of infected nodes remains stable for a long time, obtained by the continuous-time simulation and the mean-field approximation. We concentrate on an individual-based mean-field approximation called the N-intertwined Mean Field Approximation (NIMFA), which is an advanced approach considered the underlying network topology. Moreover, for the heterogeneity of the infection rates, we consider not only the independent and identically distributed (i.i.d.) infection rate but also the infection rate correlated with the degree of the two end nodes. We conclude that NIMFA is generally more accurate when the prevalence of the epidemic is higher. Given the same effective infection rate, NIMFA is less accurate when the variance of the i.i.d. infection rate or the correlation between the infection rate and the nodal degree leads to a lower prevalence. Moreover, given the same actual prevalence, NIMFA performs better in the cases: 1) when the variance of the i.i.d. infection rates is smaller (while the average is unchanged); 2) when the correlation between the infection rate and the nodal degree is positive. Our work suggests the conditions when the mean-field approach, in particular NIMFA, is more accurate in the approximation of the SIS epidemic with heterogeneous infection rates.@en