What Reasons Might Be Given for the Slower Diffusion of Social Media in Other Countries

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Python for Graph and Network Analysis. 2017 : 165–184.

Information Diffusion in Social Networks

Mohammed Zuhair Al-Taie

^fourteenFaculty of Calculating, Universiti Teknologi Malaysia, Kuala Lumpur, Malaysia

Seifedine Kadry

¹⁵Schoolhouse of Engineering and Engineering science, American University of the Middle East, Kuwait, Kuwait

Abstruse

In this affiliate, we will hash out concepts of data diffusion in social networks. We are interested in knowing how a slice of information (knowledge) is spread through a network. These may be computer viruses spreading on the Internet or a network of computers, diseases through a social network, or rumors and ideas through a social network. Information diffusion methods are ordinarily used in viral marketing, in collaborative filtering systems, in emergency management, in customs detection, and in the report of commendation networks.

Keywords: Social Network, Critical Mass, Betweenness Centrality, Opinion Leader, Data Diffusion

In this chapter, we will hash out concepts of data diffusion in social networks. We are interested in knowing how a piece of information (noesis) is spread through a network. These may be estimator viruses spreading on the Internet or a network of computers, diseases through a social network, or rumors and ideas through a social network. Information improvidence methods are unremarkably used in viral marketing, in collaborative filtering systems, in emergency management, in community detection, and in the study of citation networks.

We will nowadays, in detail, two general types of information diffusion in social networks: improvidence of innovation and epidemics. Diffusion of innovation is studied in many fields, simply in this chapter, we are considering it just from a social network perspective. Other kinds of information diffusion that are not included in this chapter include herd behavior and data cascades.

Diffusion

Diffusion is the process by which data is spread from ane identify to another through interactions. It is a field that encompasses techniques from a plethora of sciences and techniques from different fields such every bit sociology, epidemiology, and ethnography. Of course, everyone is interested in not getting infected by a contagious affliction. The improvidence procedure involves three chief elements as follows:

1. Sender. A sender (or a group of senders) is responsible for initiating the diffusion process.

2. Receiver. A receiver (or a group of receivers) receives the diffusion information from the sender. Commonly, the number of receivers is higher than the number of senders.

3. Medium. This is the channel through which the diffusion information is sent from the sender to the receiver. This tin can be TV, paper, social media (east.g., a tweet on Twitter), social ties, air (in the case of a affliction spreading procedure), etc.

From a network point of view: how is the improvidence process handed over? In fact, social relations play a significant office. They are the channels by which social contagion and persuasion are done. Peculiarly, the structural positions of persons and their personal characteristics make some people more ready to adopt the innovation than others. Networks with dissimilar patterns of connectedness take dissimilar backdrop regarding how things are propagated, which have significant implications for interventions into, for example, rumor propagation.

A diffusion starts with an adopter (or a few number of adopters) who spreads the innovation to others. Innovation typically represents newness, information technology is not the same thing as invention, it is both a procedure and an effect, and it involves discontinuous change.

Those who prefer early on are often besides innovative to be influential in a local network. They contaminate their contacts who in plough contaminate their contacts and so on. The more than people a person is linked to, the greater the chances that that person volition adopt the innovation. At a larger calibration, and since communities are interlinked, it is very likely that an innovation jumps from one community to another via boundary spanners (or bridges) and starts over diffusing again. Information technology is a characteristic of social networks.

However, whatsoever diffusion process can be expedited, delayed, or even stopped if it is discovered that the product (due east.thousand., a video, an audio, a volume, etc.) is faulty, and information technology should be fixed and then released once again. This procedure is chosen an intervention. Intervention can be achieved via several methods such every bit stopping the product of the product, limiting the distribution of the product, restricting the exposure to the product, reducing the involvement in the product, or reducing interactions within the population. In any way, intervention processes tin crusade damage to the piece of work of minor companies every bit many customers will no longer trust the products that are produced past these companies.

Contagion

Another important concept that is sometimes used alongside diffusion is contagion. It describes how a affliction tin can spread quickly in a network. In its biological form, contagion requires close physical contact to propagate. All the same, at times, merely being in the aforementioned identify where infections/germs are nowadays is enough to spread the disease. Fifty-fifty in these cases, at that place are limiting factors that influence whether an exposed person will catch a disease or not such as historic period, gender, immunity, length, weight, overall health, the strength of the virus, and timing of contact with the infected nodes.

Contamination models are similar to improvidence (or social contagion) models merely with some distinctions betwixt the two as follows:

• Contagion is often viewed through the lens of infectious diseases (although sometimes information technology used to describe different phenomena in the marketing and social spaces). Improvidence, on the other hand, is the process or state of something (an idea, innovation, rumor, digital property, and so on) spreading more widely.

• Contagion is commonly initiated by an infection, while diffusion is initiated by traditional media, word of oral cavity, advertising, or industry events, simply to proper noun a few.

• While contagion does not condition a straight contact between the victim and an infected body, diffusion, on the other paw, is concerned with the spread of ideas, innovations, and other concepts that require some straight contact (not necessarily a physical contact).

• Contagion is dependent on physical proximity, regardless whether others are part of a person's social or professional network. The case is different for improvidence which requires some social contact or influence to take root.

• Contagion models do not involve processes such equally decision making, in which an individual makes a decision whether to become infected or not because contagion is considered a random natural process. In improvidence models, diffusion making is typically involved. For example, when a rumor is spreading, the prepare of individuals who receive this rumor would decide whether they are interested in spreading information technology to their neighbors or not.

In the following department, we will discuss diffusion on innovations, which explains how new ideas and practices spread within and betwixt communities.

Diffusion of Innovation

Diffusion of innovation, which is an important social process, describes how an innovation (east.g., product, music, video, fad, opinion, or mental attitude) is handed over from one node (person) to another in a social network over time. Information technology is the theory that has used network principles and perspectives nigh extensively and has provided the theoretical underpinnings to enquiry how networks influence beliefs and behavior change.

New ideas or practices enter communities from external sources such as mass media, labor exchanges, technological innovations, cosmopolitan contact, and many others. Nonetheless, they can likewise exist originated in the same community where they diffuse. In all these cases, diffusions are washed through interpersonal contact networks.

Having roots in anthropology, economics, geography, sociology, marketing, epidemiology, and others, diffusion of innovation is critical for many groups. For example, organizations are interested in the diffusion of information, and opinion makers are interested in the adoption of new products.

Hundreds of studies were conducted in the 1950s and the early 1960s to find answers to why and how innovations spread, the reasons for the diffusion process, why some people prefer a new idea, why some people adopt before than others who wait for a substantial corporeality of time before adopting, and the rate at which an innovation spreads. The research in the field refreshed in recent years with the appearance of more sophisticated network models and technology.

It should be noted that diffusion of innovation typically takes a long time. For example, the telephone took decades earlier information technology became popular in the USA, and the videocassette recorder (or VCR) took a long time to populate. The reason for the improvidence is that information technology is often the result of a network structure, which for i reason or some other inhibits diffusions. In contrast, the advent of calculator technologies and mobile communications has accelerated that rate at which data and other products are adopted. For example, Facebook took only a few years to reach millions of users.

Adoption of Innovations

An adoption is a decision taken by a person (or persons) that includes a full employ of innovation equally the best course of action available. An important observation is that adoption does not happen right after a person first someone learns nigh a new production. Rather, five stage processes of adoption are included in the theory of improvidence of innovations. These stages tin be adopted for use in market partition and for use in measuring the progress toward behavior change.

1. Becoming aware of the product (merely with a limited size of information)

2. Starting to find more information near the product

3. Making a decision to prefer it

4. Trying the product

5. Fully adopting the product

Diffusion of Innovation Models

Social networks allow many new ideas and practices to spread through interpersonal contacts that largely consist of interpersonal communication. Many of the things that spread tin can be modeled in similar means. Hence, knowing how diseases, for instance, propagate through networks will also permit the states know how the rest of things propagate. For example, in retail marketing, data such as reviews and feedback are spread at no cost to the seller via what is chosen viral marketing. Such a mechanism of data spread is important because it provides a manner to know how people are looking at the particular product.

The social media, in most communities, are central to the spreading of information. Hence, several models accept been proposed to correspond the process in which the social media is responsible for the spread of information (or mass communication). In this section, we volition consider one of these models, which is the 2-step flow model (or the multistep period model) which was proposed by Elihu Katz, a professor of advice at the Academy of Pennsylvania. In cooperation with Paul Lazarsfeld and Robert Merton, Professor Katz pioneered many innovations particularly on how radio and television influenced mass audiences.

Ii-Footstep Flow Model

The two-stride flow model is one of the simplest models to model the diffusion of the innovation process. The model considered the effects of mass media on many behaviors including customer beliefs and voting patterns. The model proposed the idea that media effects were mediated by interpersonal influence, as opposed a previous view that the mass media had impacted people directly.

In this model, which is consistent with a network approach, mass communication processes are divided into 2 stages.

1. In the first phase, mass media influence opinion leaders. Opinion leaders are individuals who commonly have a high influence on the behavior of the rest in his or her network by virtue of age, feel, charisma, embeddedness, and perceived homophily. They use both types of ties (silent social relations and advice and friendship relations) to tell their contacts nigh how important the innovation is and why they should prefer it.

2. In the second phase, opinion leaders influence potential adopters. People who were exposed to letters from mass media did not automatically believe them. Considering opinion leaders were more exposed to media and more aware of the current trends, they were able to persuade others to follow their views with the help of media communications to support their arguments. This model, and considering media influence stance leaders who in turn influence others that in turn influence others, can as well be called a multistep menstruum model (Fig. 8.one).

   

Here is a simple case of how the two-step flow model works. Suppose that a visitor is trying to market a new software program in a network of users. The company did not classify plenty upkeep for the entrada. So, what information technology did is that information technology selected a small number of users (shop owners) to promote its software, store owners who are central in their local networks with the highest in-degree scores. They can talk to their friends, customers, and social connections about the software and why information technology is a new trend in the market place. Their firsthand neighbors, in turn, will talk well-nigh the product to their neighbors and then on until finally the news about the product is spread to a large population of users in the network. In this simple example, the company is the mass media, shop owners are the opinion leaders, and neighbors are the potential adopters.

Social Contagion

The process past which an innovation is passed from 1 necktie to another is chosen social contamination. This process is similar in many ways to the spread of an infectious disease. The model, which is a chain-similar form, starts with a small number of people who prefer the innovation. This number increases in the second stage when many other people prefer the innovation. Although the number of adopters increases largely in the second phase, the growth rate decreases, particularly afterward 10–twenty% of the actors have adopted. In the third stage, the number of people who adopt the innovation decreases until the diffusion process slowly reaches its end.

Those who decide to adopt the innovation are doing and then either dependently (i.e., they received information that other people had adopted the innovation) or independently. Those who adopt an innovation dependently are bold that because others also adopted the innovation, it is a strong signal of its value. The information they receive tin be either local (information about the behavior of his or her immediate neighbors, e.thou., coworkers in a company) or global (information about the behavior of all the individuals in the extended network, e.thou., a union).

The process of adopting an innovation tin can be represented every bit a graph called the diffusion bend. In this curve, which has the sigmoid function shape (S), the x-axis is the lifetime of a diffusion, and the y-axis is the percentage of innovation adopters.

Information technology should exist noted that the size of adoption and adoption speed are bigger in dense networks than in sparse networks, unconnected networks, and networks with cut points or bridges. An individual with a larger number of neighbors is likely to adopt earlier than a person with few neighbors. A improvidence that starts with a person having a fundamental position in the network tin be faster than with a person positioned at the network periphery.

Adoption Rate

The adoption charge per unit is a metric used to measure out the speed of the diffusion process at a item time. This metric represents the number of new people who adopt the innovation at a given time. Clearly, network construction has a loftier impact on the diffusion process such that a improvidence process, which starts with a node in a fundamental position, can reach a significant amount of the desired population in a shorter time. However, the network structure is not the only parameter. The type of innovation itself and personal characteristics of individuals also influence the flow of adoption.

Adoption Categories and Thresholds

Not everybody tin go infected if only his or her neighbor is infected. Friendship alone is not enough to persuade someone to adopt something. This is because some people are less receptive to innovation than other people. One fashion to explain this observation is through what is called adoption categories.

According to the adoption categories, people tin can be classified based on their adoption time in relevance to all other adopters. For example, in marketing, adopters are categorized into four categories: early adopters, who constitute 16%; early majority, who found 34%; late majority, who institute 34%; and late adopters or laggards, who constitute 16% of all adopters. This type of nomenclature is important for marketers because information technology helps them identify the social and demographic characteristics of early on adopters.

Amount of Exposure

The amount of exposure, for a particular person in a network at a given time, is the portion of a person's neighbors who have adopted the innovation earlier that fourth dimension. Once exposure reaches its required level for that person, he or she volition corroborate the innovation and get-go infecting others. This model treats time more explicitly in the sense that information technology models what happens at the microlevel at each signal in the fourth dimension during a diffusion. Some researchers refer to the amount of exposure as network exposure, referring to the influence of a person's social network and measured with the following equation:

where W is the network weight matrix that represents the direct contacts of ane person and y is a vector of adoption beliefs. In the instance of egoistic networks, network exposure model can be weighted by many factors such as the frequency of interaction and the similarity between ego and his or her neighbors. With sociometric data, many types of social influence can exist used to weight network exposure model.

Suppose that a person has four friends in his social network. Network exposure (or E _i ) is the proportion of those who have adopted an innovation (Fig. 8.2).

Amount of exposure at four-time steps

The above figure shows network exposure for a person with four friends at four points in time. At the start fourth dimension point, this person has i adopter, and so exposure is 25%. At the second fourth dimension point, there are two adopters, so network exposure is 50%. At the next time point, there are three adopters so exposure is 75%. Afterward all four friends adopt, exposure is 100%.

Adoption of innovation is not a straightforward process every bit it depends on a person'southward characteristics and features of the innovation. Even though some people are non close enough to the source of improvidence, they all the same prefer early. This is because the amount of exposure varies over time and among people. Knowledge about a friend's adoption or that other persons with similar network positions accept adopted may persuade a person to adopt.

For any innovation to be adopted, the expected adopter should be in agreement with the relevance, content, benefit, immediacy, and source of the innovation. This ways that innovations should be appreciable, have a relative advantage over electric current practices, be compatible with the sociocultural paradigm, and not be highly complex.

It is possible to calculate adoption fourth dimension for each person at each signal in fourth dimension. This procedure is called upshot history assay. This is important because, at the time that we accept a improvidence of a new production, we can clarify for each person whether he (or she) has adopted the production and how many of his or her neighbors have adopted it too.

Adopters and Adoption

People show different levels of exposure before they adopt. For case, some people tin can be hands persuaded such that they only demand to know someone who had already adopted the idea, whereas others would take a longer time to prefer. Too, some people receive more exposure, whether from media or social ties, than others. This is referred to as threshold of exposure, which represents the level of exposure that an private needs to adopt an innovation. Identifying these thresholds (also chosen individual tipping points) is of import for researchers because it enables them to understand the different types of adopters and the low- and high-threshold ones.

Early adopters are those who are more vulnerable to innovations such that they will exercise and so when only a few people in the network have already adopted. In other words, their thresholds are depression. Such people do no wait for a majority of their network to adopt the innovation before they are willing to. They take risks and prefer new behaviors before their peers are ready to do and so.

Late adopters (or laggards), on the other hand, are hard to persuade and will only prefer once most others in their network accept adopted. They are typically embedded in sparse social networks, have lower social status, are less exposed to mass media, and tend to learn about new ideas or products from interpersonal channels, particularly trusted peers.

People with lower threshold values are more than innovative and are expected to adopt an innovation before than noninnovative people. We look that those people turn to the media to learn about new ideas and trends. This occurs because they have few peers to take advice from about the new idea or trend. When the adoption occurs, they transport the new idea or tendency to their local community, acting as bridges.

This positive relation between low exposure thresholds and innovativeness (adopting the innovation earlier than other actors in his or her social circle) weather the presence of all-encompassing media use, many contacts outside the local community, a high level of education, and a high socioeconomic condition.

Dorsum to our adoption model, the first adopters cannot be exposed to before adopters. Therefore, their thresholds are zero. On the other hand, the last adopters are very probable to be connected to earlier adopters. Therefore their exposure and thresholds are high at the time of adoption.

Instance

Let us take a look at what a simple improvidence process, which is related to the adoption of innovation in a pocket-size network, might look like (Fig. eight.3):

Here is a instance where we have the following criteria:

• The network is directed. Nodes (actors) and edges describe the communication channels between actors. A node tin only have an effect on the node (or nodes) that information technology is continued to. Once a node is activated, information technology can activate its neighboring nodes.

• TE numbers on the chart are the threshold of exposure values, which represents the level of exposure that an private needs to prefer an innovation. For case, nodes Chiliad and F both have a threshold of exposure equals 0.8. Each of the remaining nodes has a threshold of exposure equals 0.6.

• The numbers on the edges correspond the degree of relative influence (network exposure such as trust, persuasion, or bullheaded simulated) i node has on another. For instance, node A, which is the source of the improvidence, has relative influence levels of 0.8, 0.three, and 0.3 on nodes B, F, and E, respectively.

• Node B has already adopted the innovation considering the relative influence of node A on node B exceeded the threshold of exposure of node B, which is 0.6.

• Node G is thinking of adopting the innovation. It is receiving an adequate accumulative support from its neighbors that exceeds its threshold of exposure, which is 0.8. Note that knowing that node B has adopted the innovation was non sufficient for node Grand to adopt the innovation. It needed further support from i of the 2 remaining nodes before feeling confident plenty to make the adoption.

• None of the nodes C, D, E, and F has received the adequate level of influence from its neighbors that satisfied the requirements of its threshold of exposure.

Critical Mass

A diffusion process may succeed if almost everybody in the target grouping adopts the innovation. However, it may fail if only a few people adopt and spread the innovation. This observation tin by justified by what is called critical mass, which represents the minimum number of adopters needed to sustain a diffusion process. Once the critical mass is achieved, it has ongoing momentum that keeps the diffusion going and is hard to reverse.

The critical mass concept (also called critical level or the tipping point) applies to many phenomena and seems nearly ubiquitous. It explains how opinion leaders take a potent effect on others' behavior where this effect can be scaled up to the national or international level.

Nonetheless, why this happen? I mean why the critical mass causes the number of adopters to flourish suddenly? The answer to this question has two slots. The first one looks at the process as purely quantitative. When a sufficient number of well-connected people are achieved, other people become exposed to the innovation, after which fifty-fifty more people are exposed. The second slot looks at the process as a qualitative change to the organization, namely, a sudden lowering of individual thresholds that makes some people feel confident or even obliged to adopt the innovation.

Betweenness centrality measure out is typically linked to critical mass. Targeting those with high betweenness centralities in the network is a practiced strategy for launching a successful innovation.

The diffusion process requires, in its kickoff stages, outside assist in the form of, for case, an advertisement entrada. However, in later stages, the diffusion process tin sustain or even accelerate itself without help from outside, specially when a sufficient number of opinion leaders prefer the innovation. Social contagion and then ensures wide and rapid diffusion.

If a diffusion fails to reach the disquisitional mass within a certain amount of time, its adoption rates level off, and the diffusion somewhen dies. In dissimilarity, a improvidence may reach its critical mass, and the adoption grows exponentially until it reaches what is called saturation betoken—a point in which almost all those who received the exposure have adopted the innovation. From that bespeak, the improvidence volition start to pass up and finally dies.

The pinpoint of the moment when critical mass is reached is non easy. We may demand to have detailed information about the effects of some events such as media campaigns and social contagions on the diffusion process. Ii approaches are proposed by researchers in an endeavor to know if a critical mass is reached or not: the first arroyo uses a dominion of thumb and assumes that a detail phenomenon occurs when the innovation has been adopted by 16 (or 10–20) percent of all people who will adopt eventually. The other perspective assumes that a diffusion procedure attains its disquisitional mass when the nearly central people accept adopted. At that betoken, so many actors in the network are exposed to adopters, and many of them have accomplished their exposure thresholds.

Example

Suppose that we want to describe the diffusion curve for a new social media platform. The website administrators tried to build local market place that spans the entire country. They first bars to a small dense local community but expanded later to include all other communities in the state (Fig. 8.4).

Improvidence curve for the new social media platform

The innovation adoption model, shown in a higher place, starts with a limited number of people. The x-centrality indicates the time of adoption and the y-centrality, thresholds, the per centum of network contacts who take adopted. We can come across, for example, that fifty% of the community had adopted when roughly half the fourth dimension had passed. Nosotros can run across, however, that many people have thresholds in a higher place and beneath this value.

Path A shows a rapid and consummate adoption by a population. After reaching the required critical mass in the second stage, the number of adopters increases sharply, but the concluding part of it witnesses a decrease in the growth rate. Afterward reaching saturation in the last phase, the number of adopters decreases, and the diffusion model dies. The curve (growth of population) witnesses what is fluctuation points, times where the bend accelerated or deaccelerated dramatically. What happened is that the new social media platform is especially prone to critical mass effects because when this innovation was adopted by a largely sufficient number of people, it became too hard for them to defect to another medium. Although the adoption curve A shows a elementary natural growth of adoption, information technology does not tell us why some people have adopted the product, why do some people practise so much later than others, or why some people never adopt it at all.

Other scenarios are besides probable for the product adoption. For instance, path B shows a like design to path A but follows a longer lag phase. In path C, the adoption failed to achieve the critical mass (which is needed to convince the majority of the audience of the utility of the product), while path D witnesses a sudden decline for an unknown reason although the critical mass threshold is achieved.

Motivations of innovation adoption are quite different before and afterward a system has reached its disquisitional mass such that adoption earlier the critical mass carries more run a risk.

Various types of models have been developed, such as game theoretical, epidemic, threshold, and pour, to study the data diffusion in social networks. In the following section, we volition consider information diffusion in epidemics.

Epidemics

An epidemic is a affliction outbreak such as malaria, bubonic plague, or AIDS that spreads widely via some spreading mechanism such every bit breathing, blood transfusion, drinking, eating, or sexual activity, within a population of hosts such as humans, animals, or plants. Although the term is commonly used in the context of diseases and their spreading throughout a population, it tin can also be used to describe the spread of perceived problems in a club or the adoption of a product.

Understanding the potential of an epidemic requires that we fully understand the biological process inside each host, the immune system procedure, interactions amid individuals, and social and cultural attributes.

We besides need, from a network perspective, to empathize the structure of the network and the significance of nodes (persons) within the network. By exposing the structure of the underlying network, it becomes highly possible to proceeds considerable insight into the potential spread of the diffusion (disease) and whether a new diffusion will succeed or not.

• Highly influential nodes (hubs) in the network enable a rapid spread of a disease through the network. In contrast, poorly connected or periphery nodes would dull the spreading of information and permit only a pocket-sized portion of the network to become exposed to the diffusion.

• Networks with lots of localized clusters (i.e., a highly fragmented low-density network) may limit the spread of a disease and will have difficulty in establishing any momentum, whereas dumbo networks with few gaps (i.e., few purlieus spanners) would promote the dissemination of the disease.

The transmission of a disease in a network can be stopped via some outside intervention such as simple abstention, vaccination, or isolation. Such procedures are highly capable of reducing and perhaps halting the spread of the disease.

Epidemic Models

Epidemic modeling has been an agile inquiry area for researchers working on network-based dynamic process models. The Black Expiry epidemic in the thirteenth century, the Keen Plague of London, the smallpox epidemic in the seventeenth century, and contempo epidemics such equally HIV/AIDS, SARS, and H5N1 motivated the study of epidemics and introduction of the epidemic models.

An epidemic model is a simplified means of describing the transmission of an infectious disease through individuals. Various models have been developed to study the mechanisms past which diseases spread, to predict the future course of diverse real-world outbreaks, and to evaluate strategies to control an epidemic. In the following section, we will take a look at a group of traditional (i.e., nonnetwork) epidemic models.

Deterministic Compartmental Models

In the deterministic compartmental model, individuals in the population are assigned to dissimilar subgroups or compartments such that each represents a specific stage of the epidemic. This model is useful for dealing with large populations, such as in the case of tuberculosis. Information technology categorizes people according to their state on the disease. All people are in ane of these 3 states:

1. Susceptible (S): not infected yet but susceptible to catch the disease

2. Infected (I): defenseless the disease and contagious

3. Recovered (R): no longer contagious or susceptible to reinfection

A combination of these three messages gives several disease models. The common ones are SIR, Sister, SI, and SIRS. All of these models were originally designed to report how infectious diseases intermission out and spread over a population. They too describe the disease bicycle in a host, using a combination of terms (I, R, S) to characterize each stage. In the following department, we will focus on one of these models, which is the SIR model.

SIR Model

This susceptible-infected-recovered (SIR) model, which was kickoff introduced by Kermack and McKendrick in 1932, describes a disease where a person is susceptible, gets infected, recovers, and becomes finally allowed to the illness. This model is the one most normally used class of the continuous-time epidemic models. Examples of such diseases include polio, measles, mumps, and rubella. In the kickoff phase, people start out to be initially susceptible to the affliction. In the second phase, they may go infected past contact with another infected person. They can also infect others in this phase. In the third phase, they become recovered and cannot get infected again.

The SIR model was successfully used on large real-globe networks to explore how the structure of the underlying network affects the diffusion process. Information technology was likewise used to report the data spread taking place in conjoint frameworks. Information technology was also used to explore the spread of violence or extremist topics in social media.

Three components are included in this model: the nodes that are continued to each other, the paths that the disease takes to spread, and the way that these nodes become infected and then recover.

At whatsoever time step, only infected nodes can infect any of the neighboring nodes which are in a susceptible state to the disease with some probability β. Afterwards that time step, the node that was previously in the infected state moves into a recovered state with probability α and is no longer able to infect others or go infected. Figure 8.5 above depicts the SIR model.

SIR model showing the three phases

Instance

The following figure shows a hypothetical diffusion of the epidemic through a pocket-size network consisting of 8 nodes. These nodes are connected through symmetric ties that allow for the spread of infection from one node to another. Nodes that go infected can infect others but somewhen recover. We are bold that one infected neighbor is enough to get susceptible to the disease. The figure shows four rounds of diffusion, showtime with two infected nodes, A and B. Each round represents the network status at a given period (Fig. viii.half dozen).

Hypothetical diffusion of epidemic

1. Nodes A and B in circular 1 have cognition nearly the data (i.e., they are infected) and can transmit the information to their neighbors. Nonetheless, they recover from the infection in the next round. When they recover, they forget all about this information (infection) or lose interest in information technology. They cannot acquire the aforementioned infection for the second time.

2. Nodes C, D, and East do not know yet about the data (infection), but they are susceptible in round 1 due to their connections to the infected nodes, although non infected yet. This does non mean that the infection is inevitable, every bit many factors will influence whether an exposed person will take hold of a disease or not. These nodes get infected in circular 2, recover in round 3, and never become infected or contagious once more.

3. Nodes F, G, and H in round i are neither infected nor susceptible. However, they go susceptible in circular 2, infected in round 3, and finally recover in round 4.

Properties of the SIR Model

Some of the properties that we should put in mind regarding the functionality of the SIR model:

• The SIR model puts no consideration to the level of susceptibility or what makes some people more or less likely to catch a disease. This means that a person is either susceptible or not.

• The processes included in this model are deterministic (as opposed to stochastic) which means this model assumes that any susceptible person who is exposed to an infection will get infected likewise. This assumption is not very realistic in real life because some people are more robust to diseases compared to some others who are genetically more likely to get ill. Also, illness spread at different rates and individuals have various levels of exposure to them.

• The model assumes that after the recovery from a affliction, the person will no longer become susceptible. In the susceptible-infected-recover-susceptible (SIRS) model, an individual is susceptible, becomes sick, recovers, enjoys a menstruum of immunity, and finally becomes susceptible again.

• In the SIR model, when a person recovers from a illness, he or she becomes immune to the illness. In contrast, according to the susceptible-infected-susceptible (Sis) model, the infected person does non become immune to the disease after the infection. On the other hand, the susceptible-infected (SI) model describes fatal diseases where the infected person never returns to the recovered or susceptible states.

Example

Nosotros will run into how we tin model epidemic spreading on a network. The following example gives credit to the volume.

The network structure influences infection such that nodes with a high degree will potentially infect neighbors and also get infected.

We define a function that, given β and α, makes an SIR model part and returns it for afterwards employ:

Now, nosotros define the spreading process as a office that takes a graph and a model function and applies the model to every node in the graph:

Nosotros will also define a function that applies the same model dynamics repeatedly for a number of iterations:

At present, we create the model. Nosotros will use the Erdos-Renyi ER model:

Nosotros volition visualize the model using the spring layout (Fig. viii.7).

Epidemic spreading on a network

We then network the model with a certain proportion of ill individuals, with the rest being susceptible:

Finally, we create the network dynamics:

We now run the model:

Nosotros can also inquire what percentage of the model is infected:

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