Now, when we go on holiday, I come home surprised when we haven’t bumped into someone we know. Partly that’s because it’s happened a lot – on the streets and in the hotels of Hong Kong, Tokyo, Costa Rica, San Francisco, Cape Town, Lisbon. The weirdest was at a bus stop on a highway in the middle of Israel, far from the nearest town, when an old school-friend of my wife pulled up in her car next to me to drop off a hitchhiker.

Partly, though, it’s because I understand the maths. I know lots of people and we all travel a lot. The number of opportunities to bump into one of them is vast, so it’s bound to happen sooner or later – particularly in concentrated tourist centres such as hotels and restaurants. Even the weird story is not that improbable. There are thousands of weird combinations of people and places that could have involved me over thousands of hours of travel, so if it wasn’t this one, it would likely have been another. All of these coincidences are simply a result of the law of truly large numbers – with a large enough number of opportunities, any outrageous thing is likely to happen (David Hand).

Yet even as I feel I understand this principle, there are still times when it catches me out.

Like the time I met Steve McManaman.

For those unfamiliar with him, Steve McManaman is a football pundit who used to play midfield for the team I support – Liverpool – as well as for England. Two seasons ago I arrived in Rome to watch Liverpool play in the Champions League semi-final and, as I was exiting the airport, I saw him standing there, looking down at his phone. Now it’s not lost on me that celebrities probably don’t like being approached by strangers for idle conversation, especially if they are busy on their phones. But for me anyway, Steve McManaman is different – *because I know his brother-in-law*. So, I wander up to him, declare that I know his brother-in-law and try to engage him in idle conversation. One eye on his phone and one eye on me, he responds politely but I could tell, he just wasn’t that interested. After a very short while he tells me to enjoy the game and turns his full attention back to his phone. ‘How odd,’ I thought, ‘there’s an amazing coincidence here. Of all the hundreds of people I know, only one of them is Steve McManaman’s brother-in-law. But Steve McManaman doesn’t seem that bothered.’

It was only afterwards that I considered the situation from his perspective. Suppose he knows a thousand or so people (although in all likelihood a decorated former footballer like him knows a few more). And suppose each of them knows a thousand people. That’s a million people in the world with a friend in common with Steve McManaman. Given those numbers, it’s quite possible that he has someone approach him every day with the same pitch in an attempt to make idle conversation.

Not that I learned. Three weeks later I was on a plane to Kiev to attend the final of the Champions League. And who should walk up the aisle and squeeze into the seat next to me, but Damian Lewis. ‘Incredible,’ I thought to myself, ‘Damian Lewis no doubt thinks I’m a complete stranger. But I happen to know his old school friend, Inigo…’

The purpose of these stories is to illustrate two powerful features of networks both of which have far-reaching consequences for the world we live in.

## Six degrees of separation

The first is just how quickly connections proliferate.

Everybody knows the notion of six degrees of separation – although six degrees turns out to be conservative. If everybody knows a thousand people, you can hit the population of the world in less than four steps. Clearly there is some overlap in friendship circles which dampens that kind of reach, but Facebook has shown that among its 1.6 billion users (as of four years ago) there are an average of 3.57 degrees of separation.

The reason this works is that connections open up so many billions of routes between people that the law of truly large numbers renders it inevitable that one of them – just a single route out of the many billions available – will connect two specific people.

But sometimes we overlook the depth of networks and end up underestimating the number of connections available. This stems from our tendency to think much more about the *nodes* in a network rather than their *edges* (or links). Which is only natural. ‘It’s a boy!’ a new parent will declare. Rather than, ‘It’s an additional three (or four) connections in our immediate family network.’ The latter formulation may not be the first that comes to mind, but it is quite powerful. Not only is the number of connections higher than the number of nodes, but it increases quadratically as new nodes are added. A family of three embeds three relationships; a family of four embeds six; a family of five, ten and a family of six, fifteen. Double the number of family members and the number of connections goes up roughly four-fold each time.

At one level this is quite obvious. But its consequences are less so. Such thinking can lead us to underestimate the probability of an event, and therefore overestimate the occurrence of coincidences. The best example is the Birthday Problem. In a class of thirty children, chances are that two of them will share a birthday. Incredible, some say, how can that be given there are 365 possible birthdays? The answer lies in the number of connections. In a class of thirty there are 435 possible birthday pairs [the formula is N(N-1)/2]. The chances that two of them are the same is therefore quite high. If I were to specify in advance that the shared birthday had to be 6^{th} October, say, the odds would evaporate. Just like if I specified before going to Tokyo that I would bump into Simon there. But I’m not specifying anything beforehand; I’ll take any matching birthday and any combination of people and place.

In his book, *Fluke*, mathematician Joseph Mazur suggests that the Birthday Problem is a mathematical proxy for understanding coincidences. That, and the Monkey Problem, which asks if given a large enough amount of time could a money type out the works of Shakespeare? The monkey problem captures the law of truly large numbers through time, the birthday problem through combinations.

The number of connections can also impose an upper limit on a network. In her book *The Art of the Gathering*, author Priya Parker identifies several ‘magic numbers’ for the optimal size of a social group. The smallest is six. “*Groups of this rough size are wonderfully conducive to intimacy, high levels of sharing, and discussion through storytelling.*” The reason is that in order to function properly as a group, one-to-one conversations need to be suppressed and with just 15 of them possible in a group of six, that’s totally realistic. Invite ten people round for dinner and 45 two-way conversations need to be suppressed if an inclusive conversation is what you want. Her other magic numbers are 12-15, 30 and 150. Even the largest – known as the Dunbar number – owes something to the limits imposed by one-to-one behaviour. Parker cites a Belgian hotelier who felt that 150 was the size a party could be for everyone to see one another at the same time.

The idea that one part of a system, in this case connections, scales non-linearly with another part, in this case people, has deep implications across all sorts of phenomena. Many are discussed in the book *Scale* by physicist Geoffrey West. One is the size of cities.

West observes that GDP of cities scales at a rate 15 percent higher than population growth. In fact, he shows that this rule holds for a range of socioeconomic quantities like wages, crime, restaurants, patents. Double the size of a city and wages, wealth, innovation, crime, pollution, disease, all increase by approximately 15 percent per capita. Incredibly, that relationship holds up across cities all over the world, whatever country they are in and however large they are.

The reason lies in the interaction between two kinds of network at the heart of a city – social networks on the one hand, and infrastructure networks on the other. Socioeconomic factors such as innovation, crime and disease are driven by relationships embedded in social networks. And, as we saw above, these networks scale with an exponent of 2 – double the population and you basically quadruple the number of relationships (or, equivalently, double them per capita). The reason why the scaling factor here is lower than 2 is because of the physical constraints the city imposes.

No socioeconomic benefit came from my all-too-brief encounter with Steve McManaman but after some reinforcement from Damian Lewis, at least the ideas of network degree were driven home.

## The rich get richer

The second observation gleaned from my rendezvous with Steve McManaman is the inherent asymmetry in it. Steve McManaman didn’t stop me to tell me that we have his brother-in-law in common. I stopped him. This reflects a feature of social networks. Unlike random networks where each node has approximately the same number of links to other nodes, most real-world networks are skewed – some nodes have a very large number of connections and some nodes have very few. Social networks are no exception. And quite simply, Steve McManaman’s node sees more traffic than mine.

Such asymmetry is present at all scales. There’s a so-called friendship paradox which states that on average your friends have more friends than you. It arises because the most popular people appear on many other people’s friendship lists, while the least popular appear on relatively few lists. In other words, the people with many friends are overrepresented, and the people with few friends are underrepresented.

Twitter makes it clearer. Sadly, Steve McManaman is not active on Twitter. Damian Lewis, however, is and he has over 200,000 followers, which is a lot more than me. In fact, when I go through the list of accounts I follow, most have more followers than me. And I’m not alone. A study found that 99 percent of Twitter users are less popular than the people they follow. (This gives a new slant on the representation of the elite as ‘the one percent’.)

The friendship paradox has all sorts of real-world implications. Matthew Jackson talks about them in his book, *The Human Network*. Most directly they relate to peer influence within social groups. It doesn’t take a majority of the group to sway opinion or behaviour, just a few of the more visible members. Which is why students often overestimate the proportion of their peers who consume alcohol or take drugs as well as the frequency with which they do so. And why they underestimate the time spent studying or undertaking solitary pursuits. A few students go to all the parties and the way they behave colours the views of the entire student population.

The chart of my Twitter follows above highlights the shape of skew between the number of connections people have. That shape reflects a Pareto-like distribution, named after the father of the ‘80/20’ rule. In my case, just ten accounts out of the 500 I follow have 80 percent of aggregated follows. A distribution like this is evident across many real-world situations where networks play a role. Studies have shown that 80 percent of links on the web point to only 15 percent of webpages; 80 percent of citations go to only 38 percent of scientists; and 80 percent of links in Hollywood are connected to 30 percent of actors. The chart below shows that 80 percent of worldwide box office receipts in 2019 were generated by just 7 percent of movies.

This distribution emerges from a mechanism called preferential attachment, aka cumulative advantage, the rich-get-richer and the Matthew Effect. As they expand some networks gain new edges in proportion to the number they already have. Hence the Matthew Effect: “*For to everyone who has will more be given, and he will have abundance..*.”

In many cases, it is easy to see why someone or something with lots of connections is well-placed to gather more. Faced with a wide choice of prospects, people will reduce the cost of search by copying what other people do. So, movie-goers watch the same movies others watch, authors of a new webpage borrow links from other pages on related topics and scientists decide what to read and cite by following the references from the papers they have read. In social networks the more acquaintances a person has the higher the chance they will be introduced to new people by their existing acquaintances – in other words we ‘copy’ the friends of our friends.

In a growing network, some nodes have a time advantage on others over which to accrue more links. When I look at my list of Twitter follows a correlation is evident between how long ago someone joined Twitter and how many followers they have, albeit it is quite weak. This is why businesses like Uber whose strategy is to monetise network effects are in such a rush to grow quickly and spend huge amounts of money seeding new customers. In a system where preferential attachment is a feature it is easy to place a high value on those first few connections.

Damian Lewis’ Twitter account is a product of preferential attachment at play in multiple interlocking networks. First, the more shows he appears in, the more familiar a casting director becomes with his skills. In network parlance the higher his ‘degree’ in the actor network, the higher the chances he will be considered for a new role. Then, as he gets cast, the more people who watch his shows, the more others will watch them too. Finally, as his public exposure grows, his celebrity transfers into the world of social media and the more Twitter followers he has the more inclined other Twitter users are to follow him.

Social media has raised the profile of networks as a lens through which to see how the world operates. The lens can be used to understand business strategy particularly in tech-enabled industries, financial crises like in 2008 when interconnectedness of financial institutions was underestimated, and even history where Niall Ferguson has written a book, *The Square and the Tower*. As this piece demonstrates it can also be used to understand social phenomena like the odds of bumping into Steve McManaman and why Damian Lewis has more followers than me.