Small-World Networks: a Summary & Evaluation of Watts & Strogatz & Weeden & Cornwell

Introduction

Different articles provide different statistic concepts that help in making decisions or helping in navigating various problems. It is important to understand the findings of different papers and articles so that one can apply the concepts found in the different papers. This report will provide a summary of two papers; 'The paper by Watts and Strogatz and ‘the paper by Weeden and Cornwell on small-world networks’. The report will also provide an evaluation on the issue of small-world networks as discussed by Weeden and Cornwell and Watts and Strogatz

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Summary

The Small-World Network of College Classes: Implications for Epidemic Spread on a University Campus

Many universities and colleges have shifted to online instruction to minimize the spread of Coronavirus among learners. However, many universities are trying to find a way on how they can resume in-person instruction of the learners as it is more effective than online instruction. The purpose of this paper by Weeden and Cornwell was to utilize transcript data from a mid-sized American university to discuss three networks of enrolment that connect instructors and students thus creating social networks for the spread of the infectious disease. The transcript data was collected from Cornell University, which has more than 15,000 students (Weeden and Cornwell 223). The data covered undergraduate, graduate and post-graduate students who had enrolled in the university in the fall of 2019 but excluding those who spent this semester in a study abroad program. The three networks used in the research include an undergraduate-only network, a liberal arts college network, and a university-wide network. The three networks are referred to as ‘small worlds’ as they are lead to high clustering, shorter average path paths and multiple independent paths that connect the students.

The researchers found out that course enrolments using different networks exposed students to a large number of other students by creating shorter chains of connection between the learners, and this is what potentially results to the spread of the virus through different campuses of the university (Weeden and Cornwell 224). They also found out that all the three networks examined exhibited 'small world' characteristics which result in a high number of clustering of the students. The study further showed that students' connectivity is greater in the liberal arts college networks and the undergraduate networks when compared to networks involving graduate students.

The researchers concluded that online instruction had led small-world world' networks which increase clustering and social experiences among the students among different campus. As a result, small-world world' networks have a possibility of increasing the risk of spreading the virus of infectious disease (Weeden and Cornwell 440). However, the researchers have also concluded that hybrid models of instruction involving more aggressive course enrolments improve the paths for the students' connectivity and this reduces clustering, and therefore, when used, they can reduce the risk of spreading the virus of the infectious disease through small-worldworld' networks that have been created by universities to provide online instruction.

Summary

‘Collective Dynamics of ‘Small-world’ networks’ by Watts and Strogatz

Networks that have been coupled by dynamistic systems have been useful in the modelling of biological oscillators such genetic control networks, Josephson junction arrays, spatial games, excitable media, neural networks, among many other self-organizing systems (Watts and Strogatz 441). The connection topology of these dynamical systems is ordinarily assumed to be either completely random or completely regular. However, many of the biological, social and topological networks lie somewhere between the two extremes, that is between being completely regular or completely random.

The aim of this paper was exploring the simple models of networks that can be remodelled through completely regular networks to increase the number of disorders. The researchers found out that these dynamical systems can by highly clustered such as regular lattices. However, the researchers further noted that although the systems displayed features of being highly clustered, they have small characteristic path lengths such as random graphs. These dynamical systems are what the researchers termed them as 'small world' networks as the analogy has a small-world phenomenon. Some of the good examples of the 'small world' networks presented by the researchers of this paper include the collaboration graph of film actors, the neural network of the worm Caenorhabditis elegans, and the power grid of the western United States (Watts and Strogatz 442)

According to the results obtained in the study, the critical infectiousness of disease decreased rapidly when there was a small probability of the disease to infect half of the population. The researchers noted that infectious diseases like Coronavirus spread much faster and easily in small worlds. Therefore, creating short cuts in the world will create small worlds that would then increase the spread of infectious disease in the whole population. The researchers have also noted that many other models indicate that the structure of networks influence how quicker and extent that a disease will be transmitted. The researchers study, however, aimed at showing the relationship of the dynamical systems rather than concentrating on a few topological systems such as random graphs, chains and stars. The researchers used an analysis model that connected all its graphs, and thus the predicted changes in the spreading dynamics are as a result of structural characteristics rather than the connectedness of the graph.

The researcher concluded that Small world networks have a distinct combination of high clustering and short path length (Watts and Strogatz 443). This makes it hard to capture them using traditional approximations. However, small-world architecture is given less attention. It may, however, prove to be widespread in social, biological and man-made systems with vital dynamic consequences.

Critical Evaluation of the Small World Networks

Networks can play a significant role in the description of the interrelationships and interconnection between persons, which play a significant role in increasing the spread of infectious diseases like HIV and most recent COVID-19 pandemic. The small-world networks have been found to have a significant effect in the spread of epidemics in both single small-world networks and the interconnected small-world networks. Unlike single small-world networks, the epidemic threshold of in interconnected networks usually declines when there is increased rewiring of the probability of the small world networks raises. This means that when the infection rate of a pandemic is low, the rewriting probability will impact the global steady-state infection density. In contrast, when the infection rate is high, the infection density becomes insensitive of the rewriting probability. Therefore, in the interconnected small-world networks, the spread of infectious disease at different speed rely on the rewiring probability. Based on the information provided in the papers summarized above, small-world networks, whether single or interconnected have the ability to reduce the spread of infectious diseases, but they are unable to eliminate the diseases. That is why many universities have created small world networks aiming to reduce the spread of Coronavirus among the students (Weeden and Cornwell 233). However, the interconnected small-world networks continue to spread the virus due to social interconnections.

Conclusion

In conclusion, small-world networks help to reduce the spread of infectious diseases. However, the spread of the infections dependents on rewiring probability. It is the rewriting probability that impact the spread of infectious diseases in interconnected small-world networks and the two papers summarized in this report have demonstrated that.

Works Cited

Watts, Duncan J., and Steven H. Strogatz. "Collective dynamics of ‘small-world’networks." nature 393.6684 (1998): 440-442.

Weeden, Kim A., and Ben Cornwell. "The Small-World Network of College Classes: Implications for Epidemic Spread on a University Campus." Sociological Science 7 (2020): 222-241.