What makes you, you?
The world around us is brimming with structures consisting of discrete and relatable entities, explains AMSI Summer School 2016 Senior Lecturer Dr Stephen Davis, RMIT University.
Are you really you? I mean, is it really you who determines your political opinions, your religious beliefs and your perception of what a normal healthy weight is, or is it your social context whereby your perception of what is normal and `right’ is dictated by your friends and family? The answer is that your social network probably plays a much larger role than you realise.
In the study of health and disease, the way our social network affects our behaviour has had the profound outcome that diseases such as obesity can be mathematically modelled as if infectious, and obesity spreads through human social networks much like a virus does. Complex Networks is an emerging area in the mathematical sciences that develops and applies mathematical and statistical tools to large network data.
Research into networks from disparate fields, ranging from gene-regulation networks in biology to the Internet to food webs in ecology, led to the stunning discovery that all real complex networks have a common topology: (i) they consist of many nodes with few connections and some nodes that are mega-enormous connected hubs, (ii) if A -> B and A -> C then it is highly likely that B -> C, and (iii) they are all much easier to `get around’ than you’d think, which is the concept popularly known in human society as Six Degrees of Separation.
The consequences for health and disease are again profound, the presence of hubs in sexual networks for example led early theoreticians to the rather panicky conclusion that sexually-transmitted diseases such as HIV AIDS cannot be controlled. In fact, such distributions of connectedness can mean that public health measures can be directed at those most responsible for disease transmission. Even if these individuals are unknown a clever strategy is possible: take a random sample, don’t vaccinate the random individuals but do vaccinate one of their friends, since most edges lead to a highly connected person chances are that you start `hitting’ the hubs in the network.
Connectedness and disease actually do go hand in hand. If you are a highly social person (a hub in your social network) it is now known that you are more likely to get sick and you’ll be one of the first to get sick when something is going around. So if you are a social butterfly, make sure you get the flu vaccine!
YOUR EGO NETWORK
The following exercise will characterise your local network and along the way tell you something about yourself and the way you manage your relationships. Start by drawing a solid circle that represents yourself and then place similar circles around you that represent family, friends or colleagues that you interact with regularly.
Depending on how many there are you might restrict yourself to people you have spoken with for more than one minute in the past week. Next, draw links between yourself and your contacts such that your node looks like the centre of a wheel, with spokes coming outwards. Now the more difficult bit. Draw additional links between your contacts if you think they would know each other on a first-name basis.
The number of links you draw divided by the number of possible links between your contacts puts you on a scale where high values close to 1 indicate a `small- town’ effect (everybody knows everybody) and where a low value (close to 0) indicates just the opposite, that you tend to keep your relationships all apart and separate. If you have m contacts, then the number of possible contacts between them is easily calculated as (1/2) m(m-1).
When this exercise is carried out with a group of people invariably two things happen. Firstly there is a wide range of values, and secondly there is a pause in the class, a moment of self-reflection as people consider what they have just learnt about themselves.
YOUR IMPORTANCE
In January 1994, during a Premiere magazine interview, actor Kevin Bacon commented that he had worked with everybody in Hollywood or someone who’s worked with them. The statement prompted a lengthy newsgroup thread amusingly titled “Kevin Bacon is the Centre of the Hollywood Universe”, Bacon numbers and the trivia game
Six Degrees of Kevin Bacon, popular amongst film aficionados. The latter has your opponent name a Hollywood actor and you must then link the actor to Kevin Bacon via actors who have appeared in the same film.
How to identify `important’ nodes in a network is one of the key questions that arise in Complex Networks. The centrality that Kevin Bacon claimed in the movie-actor network – that every actor is one or two `hops’ away from him – turned out not to be terribly true, he isn’t in the top 100 actors ranked in terms of closeness.
The simplest measure of importance is to count the number of links each node has, which is akin to ranking academics by a raw count of their publications or actors by a raw count of films they appear in. This is dissatisfying at a number of levels and there has long been debate that it is who you are connected to not how many connections you have, or it is the loss of network function that would ensue if you were deleted from the network, or it is the number of times a path between two nodes has to pass through you. Dear Kevin had no idea what sort of can of worms he was opening up!
DEEPER TRUTH
The researchers that first discovered the commonality in topological structure of real networks, i.e. that the presence of hubs was ubiquitous, also argued that a particular mechanism was responsible. First, they noted that all real networks are not static but grow over time. Second, when a new node emerges then it preferentially links to those nodes in the network that are already well connected. This is known as preferential attachment, or the rich get richer.
The concept is biblical – see Matthew 25:29 – but strange to think that things are this simple and that there is no room for any measures of quality. Surely, for example, citations go to high quality papers! We can mathematically prove that preferential attachment does generate realistic networks with so-called heavy-tailed distributions for the number of links each node has, but somewhat reassuringly so does an alternative model, the `good get richer’ model where nodes are allocated an intrinsic fitness. In the latter model the chances of a link appearing depends on some combination of these fitness values.
Perhaps most exciting about the study of real complex networks is their ability to shed light on the fundamental principles that govern self-organising structure, principles as relevant to online social sites as they are to gene regulation and brain function. Indeed, it has been discovered already that all real networks have community structure and all types of networks have sets of motifs, which are small subgraphs that recur repeatedly as building blocks that the larger network is made up of.
It will be fascinating to see what complex networks will next reveal about the world around us, about society and about ourselves.