10 March 2025
A collaboration between network scientist Dion O’Neale, modeller and analyst Emily Harvey, and illustrator Hanna Breurkes. Edited by Jonathan Burgess.
One day in 1967, strange letters began arriving in letterboxes in Omaha, Nebraska. The letters had the name of a stockbroker in Boston, Massachusetts and instructions to send them on to a friend or acquaintance who might get the letter one step closer to that stockbroker.
This was part of a series of experiments by sociologist Stanley Milgram to explore how connected America was. Each time an envelope was passed on, the name of the new participant was recorded. Milgram found that it took only an average of six transfers between people for the message to navigate its way across the country between two complete strangers.
This finding gave us the well-known concept that a random pair of people in the world are separated by only six degrees of separation.
We now know that social networks have interesting structural features that are different from people being simply connected at random. These structural features have consequences for how things might spread on the network, whether that be information – like fake news or hilarious cat gifs – or communicable disease such as COVID-19 or HIV.
One of the many interesting structural features in social networks is known as a “heavy-tailed degree distribution”. Degree is network scientist speak for the number of connections that a node in the network has. In this case, each node is a person, and the degree is the number of friends each person has in a social network.
Facebook is a very countable example of a “friendship” network with explicit connections. The degree distribution is when you count up the number of people in a network who each have a certain number of connections. You might find that in a particular network ten people each have three connections, six people have four connections, five people have nine connections, and so on – until you reach that one person who seems to know almost everybody.
When we put the numbers from this degree distribution on a chart we find that the chart continues far out to the right – with a small but significant number of people having a large number of connections – we say that the degree distribution is heavy-tailed. These charts end up looking a little bit like the back half of a brontosaurus.
If a social network has a heavy-tailed degree distribution, it will contain some people who are very highly connected. When something spreading on the network arrives at one of these people, they have a large number of potential people to pass it on to. Highly connected people link up parts of the network that would otherwise be far apart.
The potential for highly connected individuals to spread things showed up in Milgram’s letter passing experiments. In one case, 24 of the letters that made their way from Omaha to Boston were passed to the target person at his home address, with 16 of those coming from the same preceding person. And over half of the letters that reached the target at his office came from only two other people in the preceding hop. The letters had found their way to the same small number of highly connected individuals who were then able to pass them on to their intended location in a single step.
The fact that most people have a connection to someone highly connected is almost an inevitable consequence of how social networks are connected. This is related to the friendship paradox, which says that your friends probably have more friends than you do.
The maths to prove that this is a general feature of most different types of networks can get a bit complicated, but it’s true enough in general that we can use the consequences of the friendship paradox in practical ways to monitor or influence things that might be spreading on a network.
Let’s say you want to detect whether the outbreak of some disease is coming. You could do this by simply selecting people at random, monitoring when they get sick, and watching to see when cases start to climb rapidly. A better choice would be to select some highly-connected individuals and monitor their health, since their larger number of connections means that they are likely to be infected sooner.
We usually don’t know in advance who these highly connected people are going to be. This is where things like the friendship paradox can help us. If we start with a randomly selected set of people and ask them each to name a friend then – on average – those friends are going to be more highly connected than the set of people that we started with. Even without knowing the entire structure of a network, we can use ideas like the friendship paradox to make useful guesses about the local connections around a specific point in the network.
And this isn’t something that’s just true in theory. Network scientists showed in a study involving an influenza outbreak at Harvard College that by picking students at random and then asking those students to name a friend, they were able to detect an influenza outbreak almost two weeks early by monitoring the set of friends instead of the randomly selected individuals.
We can also use this property of network structure to reduce the spread of disease, by targeting highly-connected people for interventions like vaccination. Doing so can slow the spread of disease by meaning that transmission pathways have to be much longer to reach people, giving more time to vaccinate, or otherwise protect others.
When you’re making decisions about something that spreads – whether it’s disease, ideas, money or electricity – considering the structure of the network that it will spread on is important for understanding the outcomes of that spreading process and can help you to make a decision with the best possible information.
Although they are everywhere in our lives, our intuition can fail us when it comes to thinking about networks. Remember – your friends probably have more friends than you do.
Read more about how something spreads across a network
Dion O’Neale co-leads the spreading processes on (multilayer and multiplex) networks project at Te Pūnaha Matatini.
Emily Harvey is a researcher on the spreading processes on (multilayer and multiplex) networks project at Te Pūnaha Matatini.
Hanna is a designer and illustrator who is passionate about designing to improve wellbeing and is inspired by nature.