not the case that if you have 100 times more contacts then you will also spread a disease to 100 times more people, and we study the effects of this,” he says. The project has largely focused on the role of people in spreading disease, but Professor Lengler also intends to look at the importance of events and meetings in future. “One type of meeting would be between close-by vertices, where one vertex invites others, and usually they know each other,” he continues. “Another type of event is where vertices which are very different from each other come together.”
Model behaviour in social networks
Epidemiology
Example of a randomly generated GIRG instance. Those networks combine geometric structure with inhomogeneous degrees of the nodes. Image created by Joost Jorritsma.
Background image created by Joost Jorritsma.
New network models have been developed over recent years that could help researchers gain deeper insights into the spread of disease. We spoke to Professor Johannes Lengler about his work in refining geometric inhomogenous random graphs (GIRGs), and how they could improve epidemic modelling and inform more effective quarantine strategies.
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A variety of highly sophisticated network models have been developed over the years to look at the dynamics of social interaction, for example the way that rumours filter through a social network and how diseases spread across populations. One prominent type are inhomogenous network models, which can generate clear results. “For example, we see with inhomogenous network models that information can spread very fast,” outlines Johannes Lengler, Professor of Computer Science at ETH Zurich. Another type are geometric models, which while capturing certain aspects of social networks in the real world, may generate results contradictory to those from inhomogenous network models. “For example, if you look at the spread of epidemics, then it’s extremely fast if you look at the inhomogenous models, then it’s very slow if you look at the other type, the geometric models. So this raises the question, which is the right one?” asks Professor Lengler.
DynaGIRG project This is a topic Professor Lengler is exploring further in the DynaGIRG project, an initiative backed by the Swiss National Science Foundation (SNSF), in which he and his colleagues are looking at the dynamics of social interaction in network models that combine these characteristics. This research centres around geometric inhomogenous random graphs (GIRGs), a new type of network model which Professor Lengler says is highly flexible. “There is a lot of freedom in the model definition, and it’s quite a robust model,” he explains. In the project Professor Lengler is working to refine these GIRGs and build a deeper understanding of their component structure. “I am taking the GIRG model, trying to improve it, and to make it even more flexible,” he continues. “Another part involves looking at certain processes on the model, including bootstrap percolation, routing and the progression of epidemics.
The project’s agenda also includes research into how information is transmitted through social networks, a topic addressed in Stanley Milgram’s famous ‘small-world’ experiment in 1967. In this experiment two people were randomly selected in two entirely different locations, and one was asked to send a letter to another, with only limited information. “They knew their address and a little personal information, but person A was not allowed to send the letter to person B directly. They were only allowed to send the letter to one of their friends, who was then asked to send a letter on to one of their friends, and so on. Was it possible with such a chain to reach B?” says Professor Lengler. This was indeed found to be possible, and letters needed just six hops on average before they reached their intended recipient. “Even if you have only a little information, you will be able to find a friend who is better at finding the intended recipient,” continues Professor Lengler.
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This routing process has been modelled on GIRGs, and researchers have been able to essentially reconstruct the chain of intermediaries which passed on the letters, as well as to analyse what happens in each of the different phases. As part of his work in the project, Professor Lengler is now looking to check whether these insights also hold in terms of how information filters through social networks. “The most obvious question is, how many steps does it take to get to the intended recipient? We can also look at the route of messages and what happens when a lot of messages are sent. Do all these paths go through the same few vertices in the model, or are all these paths very different from each other?” says Professor Lengler. “We have found that these messages do indeed all go through a very small set of well-connected vertices, which is what we would expect to see in social networks.” A parallel can be drawn here with the epidemiology case, where a single individual may play a particularly significant role in the spread of a disease, a so-called superspreader. The first step towards becoming a super-spreader is to meet a lot of people, yet Professor Lengler says other factors are also involved. “A person can be a super-spreader if they have a high viral load. However, it’s
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The project’s research holds wider relevance in terms of understanding how diseases spread and in developing quarantine strategies to control or limit their transmission. Epidemiological models were used extensively during the Covid-19 pandemic for example to assess the likely impact of restrictions on the spread of the disease. “If you impose restrictions on flights, if you try to force vertices not to move very far but otherwise don’t restrict them very much, does this change the dynamics of disease spread?” outlines Professor Lengler. The GIRG models can provide a more detailed picture than standard epidemic models in this respect. “Standard epidemic models
Simulation of four different epidemics on the same network. The colors encode the order in which vertices are infected. Blue: early infection, red: late infection. The pattern and the speed of dissemination changes with the characteristics of the infection. Image created by Zylan Benjert.
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