Mathematics of how rumours spread R. Ramanujam, The Institute of Mathematical Sciences, Chennai How many of you are on facebook, whatsapp and other social media? When you hear a rumour, say "tomorrow is a holiday for all schools", or "a cyclone is on its way", what do you do? Do you immediately forward it to all your friends? Gossip and spreading rumours have always been a favourite of human beings. In villages everyone very quickly finds out what happens to any one. But in recent times, the Internet and the social media (like facebook, whatsapp, etc.) have led to rumours spreading very fast indeed. They are also used a lot in political campaigns these days too. This can lead to people being highly misinformed, and at some stage, people stop distinguishing what is unreliable rumour and what is reliable information. Is there a scientific way to study how rumours spread? Perhaps if we understand how misinformation spreads, we can also try and counter it. We might be able to also use the science to help spread information fast: for instance, when a tsunami is about to strike our coast, it is critical to inform everyone living close to the coast as quickly as possible. There is indeed a way, and the answer is mathematical. Suppose there are four persons: Alka, Balu, Chetan, Devi. Let us say that each of them has a secret, and can only communicate pairwise, and at once immediately learn each other's secret. How many communications are sufficient for everyone to learn each other's secrets? It is easy to see that 6 calls will do the job for sure: Alka - Balu, Chetan - Devi, Alka - Chetan, Alka - Devi, Balu - Chetan, Balu - Devi. Can you do with fewer calls? Of course, if we assume that during the call each learns not only the other's secret but whatever each person knows (about others' secrets as well), then fewer calls should do. (Five? Four? Fewer?) When you have a population of thousands, you cannot analyse information spreading like this. For this, mathematicians proceed differently. Let us assume that every one in the population is of three types: 1. Spreaders: Anyone, who when they hear the rumour, passes it on to everyone they are connected to. 2. Ignorants: Those who have not heard the rumour yet. 3. Stiflers: Those who have heard the rumour but do not pass it on. We can now try and describe how rumours spread. (a) Initially there are only Ignorants and Spreaders. (b) Ignorants turn into Spreaders as and when they first hear the rumour. (c) Spreaders tell the rumour to everyone they meet until they come across either another Spreader, or a Stifler — at this point they decide that the rumour is known and become a Stifler. (d) The rumour ends when there are only Ignorants and Stiflers remaining. All this is done with formulas, and numbers like 0.7 etc (probabilities for what is the chance of a person of one type meeting another of the same type or a different type). Some equations are also used to describe the rate at which "interactions" happen. With all this one can try and estimate what percentage of the population the rumour reaches. With just about a small percentage of spreaders initially, this number comes to about 80%! Perhaps also surprisingly it comes to only 85% even if half the population were spreaders initially. A paper written last year by A J Ganesh, a mathematician in the UK, studies all this in what is called the "random graph" model (where connections between people form by random processes). He shows that if every one is as likely to be connected to every one else (which is of course unrealistic, but mathematically simple to analyse) then among n persons, 2 log n communications are enough to spread the rumour fully (in technical terms, to "saturate the network"). If n is the human population, and each communication takes place over one day, it would only take 45 days for all the world to know the rumour! ------- BOX log n stands for "natural logarithm" or logarithm to the base 2 here. If 2 ^k = n then log n = k. ------- What about finding the source of a rumour? In 2010, Devavrat Shah and Tauhid Zaman of the Massachusetts Institute of Technology, USA have come up with an algorithm that sets out to do just that. They show that if the network is "complex enough" then there is a one in three chance of detecting the source. But their algorithm worked only for networks of particular shape ("tree-like"). A few months ago, Lei Ying and Kai Zhou of Arizona State University have come up with an algorithm on random graphs, which scientists believe is very general. Why bother to study these? With such mathematics, we can also learn how viruses multiply, epidemics spread, .... And that is very important to our health!