Introduction

With the advent of various online Social Networking Services (SNS), the dissemination of information through Internet, also known as word-of-mouth or peer-to-peer spreading, has attracted much attention for researchers in recent years [1], [2]. Following the rapid development of database technology and computational power, various spreading phenomena on large-scale social networks, such as the news [3], rumors [4], [5], innovation [6], behavior [7], [8], culture [9], viral marketing [10], cooperative behaviors [11] etc, can now be deeply studied in terms of theoretical models as well as empirical analyses.

Generally, the information spreading dynamics is usually studied under the framework of epidemic spreading [12]. Therefore, in many studies, the process of information diffusion is regarded equally as the disease propagation, which informs agents to transmit information to their neighbors via social connections [13]. Among them, the Susceptible-Infected-Recovered (SIR) model is the most commonly used method to describe the information spreading process, where individuals would lose interest of further contribution due to a variety of less predictable factors, which is very similar to the state of the SIR model in epidemic spreading. The interplay between the spreading dynamics and network structure is a key insight in the study on network spreading dynamics [14], [15]. The network structure affects both the spreading speed and prevalence through features such as the shortest path length, degree distribution, degree correlations, and so on. In addition, the upper bound of informed proportion is proved to be approximate to 80% for the SIR model on random networks when the population size is infinite [16]. Previous works have also revealed that there indeed exists a propagation threshold of information spreading on the small-world network [17], [18], and the spreading is much faster and broader than that on regular network for the existence of the long-range edges [19]. Moreno et al. [5], [20] studied the rumor spreading on scale-free networks, and found that the existence of hub nodes can enhance the spreading speed rather than the influenced scope. Recently, there is a vast class of studies focusing on the spreading dynamics on interconnected networks [21], and there is a mixed phase below the critical infection strength in weakly coupled networks [22]. The study of cooperative behaviour spreading across the interdependent networks [23] indicated that interdependent links are more likely to connect two cooperators than the regular links. Furthermore, some interesting phenomena are discovered based on the information diffusion on real social systems, such as the telephone interactions [24], [25], tweets [26], [27] and emails [24]. Empirical results indicate that the human active patterns [28], [29] and the role of weak ties [30] would strongly affect the information spreading.

However, there are also plenty of researches arguing that the underlying mechanism of information diffusion should be fundamentally different from that of epidemic spreading. Lü et al. [31] summarized the significant differences between them, and concluded that the information, by considering the social enhancement, would spread more effectively in regular networks than that in random networks. This would to some extent support the real human experiment reported by Centola [7]. Most theoretical models assumed that the informed agents would transmit the information to all their neighbors [32], [33] or just one randomly chosen neighbor [34] in one single time step. However, since passing messages along would take a perceived transmission cost [35] in real societies, the diffusion targets would be selected among individuals with potential interests [36]. Thus, the information flow would travel through social connections thereby depending on the properties of observed networks [37].

In this paper, we propose a theoretical model with considering the effect of partial interaction, where the number of interacted neighbors is in proportion to the spreader's degree. Simulation results show that the spreading process percolates on all three classical networks (ER, BA and WS networks), which is quite similar with the site percolation in statistical physics [38]. In addition, it is observed that the percolation peaks later on WS network than other two networks, which could be regarded as a good explanation of why information can spread more widely on WS network. Furthermore, it is also astonishing to find that the size distribution of unreachable clusters, namely the Information Blind Area, exhibits the power-law property, which is the universal phenomenon in the community distribution [39], [40].

Models

In this paper, we consider a synthetic network , where is the number of nodes and is the number of links, representing the individuals and their interactions, respectively. Analogous with the SIR model, every individual would be and only be at one of the three following states during the process of information spreading,

• Uninformed (S). The individual has not yet received the information, and is analogous to the susceptible state of the SIR model;
• Informed (I). The individual is aware of the information but has not transmitted it, and is analogous to the infected state in the SIR model;
• Exhausted (R). After transmitting the information, the individual will probably lose interest and no longer transmit it [41], thus is analogous to the recovered state of the SIR model.

Subsequently, we are mainly interested in the effect of partial interaction. That is to say, each infected node, so-called spreader [42], will only influence a certain fraction of its neighbors. Thus, the model can be described as follows.

• Initially, one arbitrary node is randomly picked as the Information Seed (I-state) and the rest remain uninformed (S-state).
• Then, the seed will transmit information to fraction of its neighbors and then becomes exhausted (R-state), where ;
• For simplicity, we assume that all individuals fully trust their social connections. Consequently, each individual will approve the information once s/he receives it;
• After approval, s/he becomes an informed individual or a spreader (I-state), and will transmit the information to all fraction of her/his neighbors and becomes exhausted (R-state);
• The above process will repeat until there is no individual transmits the information any more.

In this model, the popular individuals (nodes with large degree) tend to interact with more neighbors when they approve the information. However, since there is transmission cost in the spreading process [35], that is to say, individuals would not be able to interact with all their own neighbours. Therefore, we propose a tunable parameter , representing the interaction strength. Thus, the number of neighbors will be informed, , is proportional to the spreader's degree , neglecting the neighbors' states. And the probability of each neighbor of being selected is at each round, and the repeating selection is prohibited. Figure 1 shows the proposed spreading rule. By setting  = 0.6, the informed node can only transmit information to three neighbors, two uninformed nodes , and one exhausted node . Therefore, the present rule implies a different feature of information spreading, Partial Interaction, which is usually neglected in the standard SIR model and its variants for information spreading. In addition, the immediate influence in the present model corresponds to the two parameters in traditional SIR model, the infected probability and the recovered probability , are both equal to one in the proposed model.

To better investigate the effect of the present model, we perform analyses on three kinds of network: (i) ER network [43]: a random graph where nodes connected by edges which are chosen randomly from all the possible edges; (ii) BA network [44]: a growing network where each newly added node connects to old nodes by preferential attachment mechanism; (iii) WS network [45]: Randomly reshuffle links of a regular network with probability and result in the small-world network. As a consequence, we generate corresponding BA, ER and WS networks with the same network size  = 10000,  = 3,  = 0.5 and the average degree  = 6. In addition, to alleviate the effect of randomly selecting the spreading , all simulation results are obtained by averaging over 1000 independent realizations.

Results and Analysis

Conclusions & Discussion

In this paper, we have applied the partial interaction effect in the classical SIR model where three types of node states are considered: (1) Uninformed (S); (2) Informed (I); (3) Exhausted (R). Subsequently, we adopt it in an information spreading scenario, where the interaction strength is proportional to each spreader's degree. Numerical experiments show that there is a clearly positive relationship between the interaction strength and final coverage of spreading. In addition, we find that the spreading effect is highly influenced by the network structure. One interesting property will result from the spreading process, where the network is divided into several information-unreachable areas, namely Information Blind Areas. Analysis reveals that a phase transition of such blind areas' size distribution will emerge when the fraction of R-state individuals grows to a certain critical point. In spite of quite similar results from site percolation analysis, detailed experiments show that the diffusion process is significantly different from random selection. Further numerical analyses on three representative networks (BA, ER and WS) demonstrate that, for the spreading process, the more heterogeneous the network is, the smaller the critical point will be. By contrast, the random labeling process of site percolation is completely opposite. Moreover, the critical value of network-based diffusion is smaller than that of purely random selection. Those findings can be regarded as additional explanation why small-world network yields the most efficient information/epdemic spreading in previous studies [19], [31].

Recently, the research of both contagion-based spreading models and applications has attracted more and more attention [24], [37]. Numerical results in this paper demonstrate that, due to the various network structure, there are always unreachable individuals. Human communication pattern analysis [25], [53] would be a promising method to help in understanding how to lighten the Dark Corners. In addition, activation of Long-tail individuals and products would also enhance the efficiency of Information Filtering [54] in the era of big data to solve the cold-start dilemma [55], [56].