Gossip Propagation Models

More than ten years ago, back when I was at IIT Madras, I considered myself to be a clearinghouse of gossip. Every evening after dinner I would walk across to Sri Gurunath Patisserie, and plonk myself at one of the tables there with a Rs. 5 Nescafe instant coffee. And there I would meet people. Sometimes we would discuss ideas (while these discussions were rare, they were most fulfilling). Other times we would discuss events. Most of the time, and in conversations that would be entertaining if not fulfilling, we discussed people.

Constant participation in such discussions made sure that any gossip generated anywhere on campus would reach me, and to fill time in subsequent similar conversations I would propagate them. I soon got to know about random details of random people on campus who I hardly cared about. Such information was important purely because someone else might find it interesting. Apart from the joy of learning such gossip, however, I didn’t get remunerated for my services as clearinghouse.

I was thinking about this topic earlier today while reading this studmax post that the wife has written about gossip distribution models. In it she writes:

This confirmed my earlier hypothesis that gossip follows a power law distribution – very few people hold all the enormous hoards of information while the large majority of people have almost negligible information. Gossip primarily follows a hub and spoke model (eg. when someone shares inappropriate pictures of others on a whatsapp group) and in some rare cases especially in private circles (best friends, etc.), it’s point to point.

For starters, if you plot the amount of gossip that is propagated by different people (if a particular quantum of gossip is propagated to two different people, we will count it twice), it is very well possible that it follows a power law distribution. This well follows from the now well-known result that degree distribution in real-world social networks follows a power law distribution. On top of this if you assume that some people are much more likely to propagate quantums of gossip they know to other people, and that such propensity for propagation is usually correlated with the person’s “degree” (number of connections), the above result is not hard to show.

The next question is on the way gossip actually propagates. The wife looks at the possibilities through two discrete models – hub-and-spoke and peer-to-peer. In the hub-and-spoke models, gossip is likely to spread along the spokes. Let us assume that the high-degree people are the hubs (intuitive), and according to this model, these people collect gossip from spokes (low degree people) and transmit it to others. In this model, gossip seldom propagates directly between two low-degree people.

At the other end is the peer-to-peer model where the likelihood of gossip spreading along an edge (connection between two people) is independent of the nature of the nodes at the end of the edge. In this kind of a model, gossip is equally likely to flow across any edge. However, if you overlay the (scale free/ power law) network structure over this model, then it will start appearing to be like a hub and spoke model!

In reality, neither of these models is strictly true since we also need to consider each person’s propensity to propagate gossip. There are some people who are extremely “sadhu” and politically correct, who think it is morally wrong to propagate unsubstantiated stories. They are sinks as far as any gossip is concerned. The amount of gossip that reaches them is also lower because their friends know that they’re not interested in either knowing or propagating it. On the other hand you have people (like I used to be) who have a higher propensity of propagating gossip. This also results in their receiving more gossip, and they end up propagating more.

So does gossip propagation follow the hub-and-spoke model or peer-to-peer model? The answer is “somewhere in between”, and a function of the correlation between the likelihood of a node propagating gossip and the degree of the node. If the two are uncorrelated (not unreasonable), then the flow will be closer to peer-to-peer (though degree distribution being a power law makes it appear as if it is hub-and-spoke). If there is very high positive correlation between likelihood of propagation and node degree, the model is very close to hub-and-spoke, since the likelihood of gossip flowing between low degree nodes in such a case is very very low, and thus most of the gossip flow happens through one of the hubs. And if the correlation between likelihood of propagation and node degree is low (negative), then it is likely to lead to a flow that is definitely peer-to-peer.

I plan to set up some simulations to actually study the above possibilities and further model how gossip flows!