Many thanks for sharing this, @sinanaral and @rodrikdani.
Paper here bengolub.net/papers/naivele…, and tl;dr version here bengolub.net/papers/naivele…
and an even shorter version below.
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Paper here bengolub.net/papers/naivele…, and tl;dr version here bengolub.net/papers/naivele…
and an even shorter version below.
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https://twitter.com/rodrikdani/status/1315373828713975808
We look at a society where people update their opinions according to the _DeGroot model of updating_. It says you decide what to think tomorrow by taking a weighted average of what you and your friends think today.
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Despite its simplicity and strong assumptions, DeGroot's model has been a surprisingly helpful workhorse in networks.
We ask: suppose initial estimates are centered at the truth θ and conditionally independent.
Do we get a "wisdom of crowds" in the long run? More precisely...
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We ask: suppose initial estimates are centered at the truth θ and conditionally independent.
Do we get a "wisdom of crowds" in the long run? More precisely...
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... given that initial estimates b(0) are as described above, will opinions after enough rounds of updating get very close to the truth?
We give a necessary and sufficient condition for this to happen in terms of the network that mediates agents' updating.
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We give a necessary and sufficient condition for this to happen in terms of the network that mediates agents' updating.
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To give the characterization, we define a measure of how influential people are in the network. Here the network is given by the matrix T defining the updating rule.
The measure of influence we need is a standard one - eigenvector centrality.
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The measure of influence we need is a standard one - eigenvector centrality.
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Our result says that we will see "the wisdom of crowds"
-- convergence to the truth in the long run -- if we have a network where nobody has too high a centrality.
Here's an example of a network like this: most nodes are pretty much equal, and a few have low centralities.
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-- convergence to the truth in the long run -- if we have a network where nobody has too high a centrality.
Here's an example of a network like this: most nodes are pretty much equal, and a few have low centralities.
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You can prove that the expected squared error of anyone's long-run estimate will be ≤ the maximum of the centralities.
If centralities get small, estimates get precise.
In the above example, all the nodes in the core have similar centralities, so nobody's can be too big.
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If centralities get small, estimates get precise.
In the above example, all the nodes in the core have similar centralities, so nobody's can be too big.
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On the other hand, in a network like this, the center gets a lot of centrality no matter how many peripheral nodes there are, and so we won't get wisdom.
The reason is intuitive: idiosyncratic errors in the initial estimate of the central node sway everyone and persist.
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The reason is intuitive: idiosyncratic errors in the initial estimate of the central node sway everyone and persist.
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That is the content of the characterization in the title tweet.
My guess is that the real contribution of the paper was to show that network structure can play a central (no pun intended) and nontrivial role in answering whether good long-run learning happens.
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My guess is that the real contribution of the paper was to show that network structure can play a central (no pun intended) and nontrivial role in answering whether good long-run learning happens.
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There are many limitations both in the model of learning used here and the answer that we give. Fortunately a thriving literature has made a lot of progress on these.
Here is a survey covering some of that work. 10/10
bengolub.net/papers/survey.…
Here is a survey covering some of that work. 10/10
bengolub.net/papers/survey.…
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