Tracing the contagion problems to its source is SIR model proposed by Kermack and McKendrick [17]. The SIR model describes how individual changes when became ill, which are classified as three types. S (Susceptible) means the healthy body with low resistance. I (Infectious) means the ill individual and a source of infection. R (Removed/Recovery) shows that the individual is dead or recovered. Without illness,
recovered bodies can’t infect others, and the probability of being illness again becomes low.
In early studies of contagion, three type groups in SIR model interact with complete random, that equals to investigate contagion in random networks. Although this hypothesis is not meet the reality, but a simplistic model without detailed setting is convenient for researchers. They just have to consider the disease’s ability and the number of infector at initial. Adding to passed studies, a roughly simulation of contagion is then constructed.
However, during the process of contagion, there is a close relationship between the propagation of disease or rumors and different network topology of interpersonal relationship. Thinking about the interaction of humans, in addition to household and friends, we may communicate with a restaurant-owner. Random networks can’t represent whole situation. Besides, there is some difference in detail of contagions.
Thus, different network models are needed for various epidemic problems [18-20].
Generally, the small world model is used to simulate the epidemic propagation and rumors spreading, and scale-free network model for the study of sexual diseases contagions, and computer virus spreading. Therefore, we would discuss contagion problem with these two networks mainly, and compare the results to random networks and regular networks which were not similar to real world.
3 Network Models in the Experiments
According to traditional cellular automata, we used a 100x100 2-D lattice to generate the complex networks as the platform in this paper, and investigated the transformation of these ten thousand individuals. In these virtual societies, all individuals would interact with others by edges only.
Each individual with 4 nearest neighbor links as the level one von Neumann Neighborhood in 2-D lattice was the regular network (figure3a). We thought such the circumstance as the family with five population was similar to reality. The small-world network was generated by adding several shortcuts between random nodes in the regular network (figure 3b). Here we used adding shortcuts instead of rewiring in order to avoid breaking graph [6].
(a) (b)
Fig. 3 Input of the assumption
Furthermore, we used three ways to construct different small-world networks for our investigation of local information. We set a new weighted property d(v) of all
nodes, and let
picked for adding shortcuts. The higher probability an individual had, the easier it was chosen to connect with others. The function of probability was classified into three ways below for closing reality, constant, normal, and uniform distribution. Firstly, constant distribution meant all individuals had equally probability. Secondly, we made the chosen probability of all nodes representing normal distribution. Finally, in uniform distribution, we divided all individuals into three parts with particular probability each.
i
The random network was generated by adding several shortcuts in 2-D lattice without nearest-neighbor links. In order to remain the property of random networks, we would not use the probability when adding nodes. We just picked several node-pairs complete randomly.
Lastly, for constructing the scale-free networks, we set the weighted property as small-world networks for all nodes. At the initial stage, all were set to 1. Then we connected two nodes randomly, and increased the of these two nodes by 1, which made these nodes be picked easier later. Repeating the steps, we had the virtual society of scale-free networks.
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4 Contagion Models
Contagion problems are concerned about how infectious individuals affect accepters via certain infection channel. In epidemiology, infectious individuals are the patients. They make healthy people become infectious by direct contact or air spreading, the routes of infection. When talking about beliefs, missionaries play the role as the infectious ones, spread their beliefs in words or newspaper and magazines, making people inspired and have the belief too. Where words, newspaper and magazines are the routes of infection, and those who are called are the accepters.
From the examples above, we can tell that contagion problems are composed of three individuals, infectious ones, routes of infection, and accepters.
In this article, we use all kind of social network models as the simulation platforms. Owing to the properties will distinct from one kind of contagion problem to another, we will discuss them through from random networks, small world networks, to scale-free networks. Besides, we use SIR model (figure 4-1) to simulate the propagation of ergodic individuals, expecting to show the process of how infectious individual take use of the connections in the networks as their channel.
Fig. 4-1 SIR model
We can tell that when observing the phase transition of each individual, it is possible that a susceptible one (S) will become infectious (I) through the interaction with infectious ones. We call the rate . As time passing, infectious ones will be set as removed or recovery, depending on the . If an infectious one is in a recovery state, it will be harder to be infected again than the susceptible ones because it forms an antibody.
Rateinfect
remove
Rate
It is acceptable and more often used to discuss illnesses spreading by SIR model.
But a question here is that can we use the same model to simulate culture propagating and rumor spreading? Imaging a situation here. During an election, a candidate will do some propaganda. In the beginning, he/she will have some supporters who support his/her politics, we say that they are in the infectious state, that is being infected by the candidate’s allure. These supporters will do more propaganda for the candidate among those they know, say their relatives or friends, which makes it possible for those who do not decide to support a certain candidate yet (Susceptible) to support the
same candidate as their friends or relatives. As for the followers of other candidates, there might be chances for them to give up supporting because of some negative news and become removable
Other than the transition state function in SIR model, the states of individuals are not the same. Just like in the real world, some people have stronger resistance than others, or the contact frequency are not equal. Family members will contact more to each other than the neighbor, that is why there should be some differences in the interaction connections. When it comes to contagion problems in different social networks, what mentioned above is local variances, which we concerned the most here.
5 Local Information Mechanism
Local information mechanism mainly set up the differences between individuals or channels to fit the whole simulation to reality. Thinking about the daily life, we may contact household or friends, such that every node in network models is linked up with each other. But the number of friends depends on person. Some have many friends and some are unsociable. These differences of simulations are interested and studied in detail by researchers.
In this paper, we categorize local information into edge-related and node-related.
Node-related information can be classified into vertex degree and individual attributes in details. The number of friends of each node is what vertex degree dominate, which is controlled by network models’ differences. And individual attribute mainly handle how hard is the node being infectious. Edge-related information then classified into diversity of weights and directions. The difference of weight controls intensity of interaction between the pair. Greater the weight, more interact between the pair. The difference of directions is caused by some specific purposes. Take rumor for example, the hearsay sources just blow about the gossip but not receive anything.
Follows, we would verify the influences of local information on complex networks by experiments below.
6 The Experiments
In the experiments, we used a 100x100 2-D lattice to generate complex networks as the platform. Besides, the average vertex degree was 8 in all models, which included 4 nearest-neighbor links. There were ten individuals set to I-state and others were S-state at initial. During each time step, all individuals interacted with their friends randomly. We traced the number of I-state individuals after 90 time steps and compared the result curves by two directions. Firstly, we would look at the first hill height, appearance time, and its duration. And secondly, we focused on the variance of total number of infectious individuals among the whole world. So, we would show the result curves mainly, and show the accumulative curves on its top.