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The last time we conducted a poll to assess Prime Minister Narendra Modi’s approval rating, the results created a bit of a stir on social media platforms such as Twitter. For some commentators, the 74% approval figure was simply too high, especially given the general discourse on social media. If one were to go by commentary on social media, it would appear that a large number of people are dissatisfied with Prime Minister Modi and the National Democratic Alliance (NDA) government.

The latest such poll, conducted once again in association with InstaVaani and with a significantly larger sample size (almost 11,000 respondents), has thrown up a near-identical result, with 74% of respondents approving of the Prime Minister; in fact, this level of approval has been nearly consistent for over a year now.

If the approval rating of the Prime Minister has indeed been consistently high, why is it that social media suggests otherwise? Why is it that what seems to be the “minority opinion" (disapproval of the Prime Minister) looks like the “majority opinion" on social media? In order to understand this, we need to understand better the structure of social networks.

In what is now a famous experiment, psychologist Stanley Milgram sent packages to 160 randomly chosen people living in Omaha, Nebraska, and asked them to forward it to a stockbroker in Boston, with the condition that at each step the holder of the package could only forward it to someone known to him on a first name basis.

Most packages never made it to their destination, but those that did got there after being forwarded six times on average (median). While the term “six degrees of separation" was coined only much later (and not by Milgram), this experiment demonstrated that the world is “small", and that people are connected to each other in far fewer steps than was previously imagined.

Human social networks are an example of what is classified as “small world networks". The “small world" term represents the fact that everyone is connected to everyone else through a “small" number of intermediate connections -- something that Milgram’s experiment illustrated. More formally, a small world network is defined as one where the average distance between two randomly chosen nodes in the network “grows slowly" with the number of nodes in the network.

Studies of various kinds of small world networks, including human social networks and neural networks in the brain, have shown that small world networks have what is known as a “scale free" property. Simply put, what this means is that a small portion of the network displays properties that are similar to the network at large.

An important feature of a scale-free network is that the degree of distribution (in graph theory, “degree" refers to the number of nodes each node is connected to. In real life, think of it as the number of friends) follows a power law distribution (while power law distributions are inherently hard to understand, a good example is the Pareto, or the so-called 80-20 distribution).

If we oversimplify, what this implies is that while a large number of nodes in the network are connected to only few other nodes, there are a few nodes which are connected to large numbers of nodes. Even among these high degree nodes, there is massive inequality - some high degree nodes have higher degrees than other high degree nodes.

This power law distribution is an important condition for the “small world property" to be satisfied - the reason pairs of nodes are connected by a small number of steps is that some of these steps are nodes with large numbers of connections. The power law also ensures that parts of the network resemble the whole.

Research has shown that social networks such as Twitter follow the “scale-free property". In other words, the number of followers of a user or the number of retweets a tweet gets have been both shown to follow power law distributions. Simplifying, a large proportion of users have few followers, while a handful of “Twitter celebrities" have a large number of followers. Retweets are similarly distributed.

Taking this together with the fact that different people tweet to vastly different extents (some put out hundreds of tweets a day while others tweet very intermittently), it can be clear why certain users on Twitter are far more influential than others - heavily retweeted tweets from a prolific user with a large number of followers are seen far more than tweets from either less prolific users, or from users with a small number of followers.

The issue of the Prime Minister’s popularity and approval needs to be looked at from this perspective. When we try to judge the “pulse of the people" based on social media, the power law distribution of follower count and retweets, and differing tweeting frequency means that some people’s opinions implicitly get far more weight than others.

When people’s political opinions are independent of popularity on Twitter, we would expect that the opinions of celebrity users will largely cancel out. However, if there is even a small bias among the “celebrity users" in terms of political opinion, the power law implies that these opinions get vastly magnified. This way, even if a small number of celebrity users of Twitter hold a particular opinion, the impact of these users can get vastly magnified, and this opinion can seem to be the majority opinion.

Thus, it is quite possible that three-fourths of the country actually approve of the Prime Minister, as the various InstaVaani surveys show, but only a small proportion of “celebrity political commentators" on Twitter share this opinion. This could have created the impression that the Prime Minister is unpopular, given the power law, while the reality is otherwise!

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