The process of information diffusion has attracted wide attention. Many researchers have devoted themselves to investigating how information propagates from several initial spreaders to the population.^[1,2] Different models have been put forward to describe local behaviors and analyze the dynamic diffusion process in various kinds of networks.^[3,4] These models make a connection between local diffusion actions and the macroscopic state. Analysis and simulations have been conducted to reveal the dependence on model parameters.^[5,6] As a typical representative, epidemic models reflect the mechanism of infection and recovery for diseases, and can be used to characterize information diffusion, such as the susceptible-infected (SI) model,^[7,8] susceptible-infected-susceptible model (SIS),^[9] susceptible-infected-refractory model (SIR),^[10–12] etc. Rumor models consider that the rumor may be clarified when two spreaders contact each other.^[13] The underlying networks are of great importance for the diffusion process, determining the propagation velocity and extent.^[14,15] A propagation model in two-layer multiplex networks was presented, and results proved that interactions between layers enhanced the diffusion process.^[16] Moreover, in conjoint social-physical networks,^[17] the scale of information diffusion is increased dramatically. In Ref. [18], the effect of community structure was studied in the presence of social reinforcement. Meanwhile, some studies extracted the most influential nodes in the diffusion process.^[19,20] In Ref. [21], compared with the degree, the coreness constitutes a good topological descriptor to identify influential spreaders in complex networks, if epidemic models are used. When the diffusion process is characterized by rumor models, the coreness does not determine the spreading ability of nodes, but it reflects whether a node can suppress the rumor.^[22]

With the development of Web 2.0, online social networks have become one of the most important ways for people to gain and share information. A lot of researches focused on information diffusion in online social networks.^[23,24] Power-law degree distribution in online social networks yields an optimal structure for information diffusion.^[25] The spread of online user behaviors was explored, and it was found that clustered-lattice networks accelerated behavior spreading.^[26] Besides, weak ties also greatly affect the spreading dynamics. In Ref. [27], selecting weak ties preferentially cannot make information diffuse quickly, but they act as bridges connecting isolated local communities. The environment in online social networks is more complex than real society, so researchers try to characterize the elaborate diffusion process.^[28] In Ref. [29], the simultaneous diffusion of negative and positive information was considered, and more states of user behaviors were introduced into the diffusion model. Based on the model, in Ref. [30] two methods of restraining rumor spreading were compared, and it was found that the method of the truth clarification had a long-term performance while blocking rumors at the nodes with a large degree only took effect in the early stage. To describe online user decisions and actions, an evolutionary game theoretic framework^[31] was proposed to model the diffusion dynamics, and the model could be explained by differential equations.

Current studies often assume that agents focus their attention on the topic all the time, and they continue to forward information and express their ideas. In current models, agents have the ability to spread information once they are infected by it. In fact, even if agents in online social networks are convinced by spreaders, they have no influence on neighbors until they publish or republish the information. After the forwarding event, susceptible neighbors can contact the information. As depicted in Ref. [29], the mechanism of forwarding information for agents was introduced into the rumor model. Their model is only suitable for the environment where two opposite kinds of information coexist, and the forwarding event is individual-related, so the spreading threshold cannot be derived analytically. Furthermore, users in online social networks do not always keep active. After a diffusion action, they may enter into a waiting period of stochastic length where no further action takes place. Therefore, individual latency generally exists in consecutive user behaviors. In Ref. [32], latency of voters was introduced into opinion dynamic models, and results showed that the macroscopic state was changed by latency. In these models it was assumed that agents have initial opinions towards a topic, and information is expected to reach all the population before the dynamics. However, whether latency in information diffusion affects the spreading scale and velocity still needs further exploring, and this problem is important for understanding the real intrinsic diffusion dynamics. In this paper, we propose a diffusion model with node latency. Spreaders may enter into the latent state and do not spread the information temporarily, but they may be reactivated again. We conduct mean-field analysis and numerical simulations of our model. We find that the node latency does not change the spreading threshold, but determines the final diffusion extent.

The rest of this paper is structured as follows. In Section 2, an information diffusion model with node latency for the online social medium is presented. In Section 3, the mean-field analyses of the model in homogeneous networks and heterogeneous networks are conducted. In Section 4 numerical simulations and a discussion about the model are given. Concluding remarks are given in Section 5.

In online social networks, users spread information to others by sending a message or publishing a post. They are not active all the time. After they take a diffusion action, their activity decreases and they temporarily keep away from the diffusion process. With the time elapsing, latent users may be reactivated, and they will interact with neighbors. Therefore, the activity of users is heterogeneous and dynamic, and it affects the process of information diffusion. We introduce latency of nodes into the epidemic model. After a node takes a diffusion action, it enters into a waiting period of stochastic length.

We introduce our diffusion model. In the model, a node may be in one of four potential states, i.e., the susceptible state, the infected state, the latent state, and the refractory state. The susceptible state means that agents do not see any information, and have no idea about the topic. Susceptible nodes may be infected if they receive the information from their neighbors. The infected state means that nodes are convinced of the information and are willing to propagate it. However, they do not have influence on neighbors until they forward the information. Latent users have accepted the information, but they freeze it and do not take diffusion actions. Refractory nodes absolutely lose their interest in the topic, and drop out of interactions.

Nodes interact with neighbors in terms of their states. Each time, infected nodes may send a message to their neighbors with the probability p_lat and then enter into the latent state. Susceptible neighbors receive the message and read it, and they may be infected with the spreading probability λ. Meanwhile, the latent nodes may recover their activity and enter into the infected state again with the probability p_inf, or else, they may become refractory and withdraw from interactions with the probability δ. We provide the state transition graph for an arbitrary node in Fig. 1.

Fig. 1.

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	Fig. 1. State transition graph of a node in the topology.

From the above model, after nodes temporarily stop their diffusion actions, they may become active again with the probability p_inf, and they may also lose their activity permanently with the probability (1 − p_inf) δ. In the model, only the refractory state is stable, and infected and latent nodes will become extinct in the diffusion process. The final system evolves into the regime in which all nodes become refractory or the susceptible and refractory states coexist with different densities. From Fig. 1, the latent state can be treated as an intermediate state between diffusion actions and withdrawal from interactions. Although latent nodes trust the topic, they currently do not diffuse information to others. Therefore, the effect of node latency on the diffusion process cannot be ignored. The threshold of information diffusion is also related to the latent state. We will conduct mean-field analysis of the spreading dynamics.

If the underlying network is a heterogeneous network, node degrees exhibit distribution P(k). The average degree of the network is

3.1. Homogeneous network

First, we consider homogeneous networks to explore basic features of the diffusion model. In the networks, degree fluctuations are very small with the same degree k̄, and there are no degree correlations. Therefore, ρ_s (k,t), ρ_i (k,t), ρ_l (k,t) and ρ_r (k,t) are simply written as ρ_s (t), ρ_i (t), ρ_l (t), and ρ_r (t). Equations (6)–(8) are reduced to

Assume that information propagates in the global scope after time τ. When t < τ, information is trapped in the local area. Thus, we have ρ_s (τ) ≈ ρ_s (0), ρ_i (τ) ≈ ρ_i (0), and ρ_l (τ) ≈ ρ_l (0). In the beginning, only one infected node is assigned in the network, so we have the following initial condition:

When information propagates beyond the local region, the density of refractory agents varies beyond 0. In the situation, ∂ ρ_i (t)/∂t > 0 should hold for t = τ + Δt. Since Δt → 0 and the increase of ρ_l (t) is usually later than the increase of ρ_i (t), we have ∂ ρ_l (τ + Δt)/∂t ≈ 0. After some elementary manipulations of the third equation in Eq. (9), we have the following equation with the above condition:

The threshold condition of the second equation in Eq. (9) can also be written as

Inserting Eq. (11) into Eq. (12), we obtain

Since ρ_s (τ + Δt) ≈ 1, we find the final condition

From Eq. (14), the threshold of spreading probability correlates with the average node degree k̄, and refractory probability δ and the reactivating probability p_inf. However, the probability of becoming latent p_lat does not affect the spreading threshold.

Figure 2 illustrates the solution of Eq. (9). From Fig. 2, a threshold of spreading probability for information diffusion exists clearly, and the value of the threshold approaches Eq. (14), which is in accordance with our analysis. Above the threshold, ρ_r (∞) increases drastically with λ increasing, leading to a phase transition. In the left panel of Fig. 2, large average degree and reactivating probability p_inf advance the diffusion process and reduce the threshold. We notice that when k̄ = 4, the information for λ ≥ 0.8 invades the whole system, but when k̄ = 2, some nodes are still unaware of the information for any λ. As shown in the right panel of Fig. 2, the latency parameter p_lat is unrelated to threshold, which is in consistence with the analysis of Eq. [14]. However, when information can propagate in the system, ρ_r (∞) varies slightly with the latency parameter. The larger refractory probability increases the threshold a little and reduces the diffusion extent, but its effect also depends on p_inf. When p_inf is large enough, δ does not have an obvious influence.

Fig. 2.

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	Fig. 2. Plots of final density of refractory nodes ρr (∞) versus λ, given by Eqs. (9) in homogeneous networks, with ρi (0) = 1/2500. (a) δ = 0.5, and plat = 0.5; (b) k = 4, and pinf = 0.3.

We conduct Monte-Carlo simulations to find the role of latency in the process of information diffusion, and the spreading thresholds in different networks are also investigated. In the beginning, a node is selected at random and it is set to be in the infected state, while all the other nodes are susceptible. Simulations are implemented synchronously. Each time, all infected nodes make a decision to take actions and spread information to neighbors, and then they may enter into the latent state. All latent nodes may recover their activity or withdraw from interactions. The dynamics continues until no more updates take place. Homogeneous networks and Barabasi–Albert scale-free networks mediate the diffusion process, and a real social network is also used as the underlying topology.

Figure 3 shows the densities of nodes in different states each as a function of time. The underlying networks are homogeneous networks. The density of latent nodes ρ_l (t) stays a little smaller than that of infected nodes ρ_i (t), and ρ_i (t) increases earlier. Latent nodes and infected nodes can interconvert into each other, and will become extinct together. Increasing λ raises the peaks of ρ_l (t), ρ_i (t), and ρ_r (∞). For larger λ, ρ_l (t) and ρ_i (t) increase rapidly in the early stage and reach their peaks within less time, but then they drop relatively slowly. ρ_s (t) becomes stable more rapidly than the densities of nodes in other states. Although infected nodes still exist in the system, all of their neighboring nodes enter into the refractory state and ρ_s (t) is prevented from decreasing. Larger λ accelerates the evolution of ρ_s (t), but it does not reduce the relaxation times of ρ_i (t) and ρ_r (t).

Fig. 3.

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	Fig. 3. Time evolutions of the density of susceptible, infected, latent, refractory nodes in homogeneous networks with the degree k̄ = 4. The system size is 2500, δ = 0.2, pinf = 0.5, and plat = 0.5. (a) λ = 0.2; (b) λ = 0.5.

To demonstrate the threshold for the diffusion process, we calculate the final densities of refractory nodes with different spreading probabilities as shown in Fig. 4. Information diffusion witnesses a distinct threshold in both networks. In the left panel, the threshold in a homogeneous network is nearly the same as that in Fig. 2 with the equal values of parameters, and it accords with the analytical result. The parameter p_lat does not change the threshold, but it affects the final diffusion extent, which is in accordance with the mean-field analysis of Section 3. In a scale-free network, ρ_r (∞) becomes slightly larger than in a homogeneous network due to the promotion of central nodes. In addition, the threshold in a scale-free network is approximately half that in a homogeneous network. In heterogeneous networks, the threshold depends on , as given in Eq. (25). However, in homogeneous networks, the threshold is proportional to 1/k̄. For a Barabasi-Albert scale-free network, holds true. Therefore, we have .

Fig. 4.

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	Fig. 4. Plots of ρr (∞) versus λ, when N = 2500, δ = 0.5, and pinf = 0.3. (a) The underlying network is a homogeneous network with k̄ = 4; (b) a scale-free network mediates the diffusion process. The results are averaged over 100 different realizations.

Figure 5 shows the effects of latency probability p_lat on the diffusion extent. In the left panel, the underlying topology is the same scale-free network as that in Fig. 4. In the right panel, we use a real online network extracted from the Sina website that is a famous micro-blog in China. The real social network contains 5906 nodes with the average degree k̄ = 4.5008. The average shortest path length of the network is 4.99, and the clustering coefficient is 0.0772. From Fig. 5, it is dramatic that ρ_r (∞) first decreases with the increase of p_lat, and then it increases. Around p_lat = 0.5, ρ_r (∞) reaches the minimum. For small p_lat, infected nodes become latent with a small chance. Since only latent nodes may enter into the refractory state and thoroughly stop their interactions, the diffusion process will last a long time. More infected nodes exist in the system with small p_lat, and they have the ability to spread information, leading to a larger final diffusion. For large p_lat, although infected nodes become latent more quickly, they may forward more messages to their neighbors before they pause their diffusion actions. After infected nodes forward a message, their susceptible neighbors can see the message and have the opportunity to be infected. Therefore, large p_lat speeds up the diffusion process, and also increases ρ_r (∞). Obviously, the maximal density of infected nodes over time decreases with the increase of p_lat, but the change for the density of latent nodes is opposite. We also calculate the maximum sum of densities of infected and latent nodes over time. When λ = 0.2, δ = 0.2, and p_inf = 0.5, the maximum sums in a scale-free network are 0.2792 for p_lat = 0.1, 0.2573 for p_lat = 0.5, 0.3071 for p_lat = 0.9, respectively. We can see the bottom of the sum occurs around p_lat = 0.5. Therefore, p_lat has a subtle role in the diffusion, and both large and small p_lat lead to a larger diffusion extent. Increasing p_inf makes latent nodes easier to retrieve their activity, and causes more spreaders, so ρ_r (∞) with large p_inf increases markedly. In addition, more nodes are infected in a scale-free network than in the real social network, since the scale-free network has a smaller average shortest path length 4.37. Furthermore, p_inf in a scale-free network exerts a more obvious effect on the diffusion process.

Fig. 5.

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	Fig. 5. Plots of ρr (∞) versus plat, when λ = 0.2 and δ = 0.2. (a) A scale-free network is used; (b) a real social network is used. Each plot represents the results averaged over 100 different realizations.

Now, we concentrate on the dependence on the refractory probability δ. Intuitively, by increasing δ, more nodes thoroughly drop out of interactions, and information becomes less easy to propagate. As shown in Fig. 6, the final density of refractory nodes nearly decays with the increase of δ as a power law, that is, ρ_r (∞) ∝ δ^−γ. The power exponent is γ = − 0.6963 ± 0.0718 in a scale-free network, while it is γ = − 0.7307 ± 0.0406 in the real network. When δ is small, the information in a scale-free network almost occupies the absolute majority of the system, while in the real network, many susceptible nodes survive in the end.

We measure the influence of nodes in the diffusion process, and try to find the effective topological descriptor for influence maximization. We use the real social network as interaction topology. Then, we define C(t) = ρ_i (t) + ρ_l (t) + ρ_r (t), where C(t) denotes the density of nodes that have ever been influenced by the information, and therefore, it can be used to measure the influence of initial spreaders at time t. We use three topological descriptors to identify influential nodes, i.e., the degree, k-core index, and betweenness.

Fig. 6.

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	Fig. 6. Plots of ρr (∞) versus δ, when λ = 0.2, pinf = 0.5, and plat = 0.5. Each plot represents the results averaged over 100 different realizations.

Fig. 7.

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	Fig. 7. Variations of C(t) with time in the real network, when δ = 0.2, pinf = 0.5, and plat = 0.5. Initial infected nodes are assigned according to the degree and betweenness.

Fig. 8.

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	Fig. 8. Variations of C(t) with time in the real network, when δ = 0.2, pinf = 0.5, and plat = 0.5. Initial infected nodes are assigned according to the coreness and betweenness.

We compare the performance of the degree with betweenness in Fig. 7, and compare the k-core index with betweenness in Fig. 8. In Fig. 7, we sort nodes according to the degree and betweenness, respectively, and select the top 10 nodes from each rank. To obtain a distinct result, we delete the common nodes occurring simultaneously in the top 10 of the degree and betweenness rank. For each topological descriptor, we retain 4 nodes as initial infected nodes. In Fig. 8, initial infected nodes are assigned similarly. From Fig. 7, the betweenness outperforms the degree a little in grasping the influential nodes in our diffusion model, despite the parameters. Moreover, the descriptor of betweenness has an obvious advantage compared with the k-core index from Fig. 8. Therefore, the betweenness is the best descriptor to mimic the node influence in information diffusion with latency.

In this paper, we study the effect of latency on the information diffusion process. We put forward a diffusion model that includes four node states, i.e., the susceptible, infected, latent and refractory state. After infected nodes take diffuse actions, they enter into the latent state and stop spreading information temporarily. Latent nodes may recover their activity, or they may become refractory and withdraw from interactions. We implement mean-field analysis and Monte-Carlo simulations to investigate the spreading threshold and final density of refractory nodes.

The results show that the threshold for information diffusion depends on the spreading probability, refractory probability, and reactivating probability, but the probability of becoming latent is unrelated to the threshold. The threshold in scale-free networks is nearly 4/log (N) times as large as that in homogeneous networks. Small or large latency leads to a larger diffusion extent, but the essential causes for these two situations are different. Moreover, the final density of refractory nodes increases markedly with the reactivating probability increasing. We also detect influential nodes in the diffuse model. From the result, the spreading capabilities of nodes mainly correlate with their betweenness, while the coreness does not perform well for identifying node influence.

In online social networks, there are two cases of popular information. In the first case, users forward information rapidly, and the information propagates on a large scale within a short time, implying large latency probability in the model. In the second case, users forward information and become latent slowly, but the diffusion process lasts a long time. Many users are influenced by the information finally. From the results, if one hopes to restrain the spread of a rumor, the method of creating another topic to distract user attention may not be so effective in online social networks. Under the measure, active users enter into the latent state temporarily but they do not withdraw from the diffusion of the rumor. The rumor can propagate slowly and can cause a long-term impact gradually. A better way to restrain the diffusion may be to separate the rumor from neighbors and to clarify the truth at the nodes with large betweenness.