For more on this and several other applications, check out the preprint of the survey on my website.

Hope you enjoy reading.

https://bengolub.net/eigenvalues_ams_r

Formally, look for a vector c of scores with

λcᵢ = Σⱼ cⱼMⱼᵢ.

In words, the centrality of i is a weighted sum of the centralities of nodes pointing to i; λ is the common scaling (you need this flexibility to have any hope of finding a solution).

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Equivalently: there's no leader who retains nontrivial influence.

Spare model, sharp moral: the crowd is wise only when influence is diffuse.

The eigenvector centrality notion captures how position becomes weight.

(This small result was my undergraduate thesis.)

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A priori, it is not clear that you could ever find such a c.

The deceptively technical Perron-Frobenius Theorem comes to the rescue.

It implies: In a strongly connected M, there is a (unique!) positive vector c satisfying the centrality equation.

Now a simple demo of how this concept is useful in a behavioral model.

In the DeGroot model, each agent's opinion today is the average of neighbors' opinions yesterday.

x(t+1)=M x(t),

where x(t) = vector of opinions at t Mᵢⱼ ≥ 0 is how much i listens to j (rows sum to 1)

The uniqueness is remarkable:

You might have thought that many different assignments of centrality would satisfy the high-school principle.

But no: Importance is determined internally, by the pattern of weighted arrows.

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If M is strongly connected and aperiodic, x(t) converges to a consensus a·1 (everyone has the same opinion a).

That consensus is a weighted average of initial opinions,

a = Σᵢ cᵢxᵢ(0),

where c is the left eigenvector for eigenvalue 1 that Perron-Frobenius gave us!

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Key insight:

Why is c the natural weight?

Influence is recursive: an agent's weight in the consensus comes from being listened to, and counts for more when influential agents do the listening.

That's the centrality equation from tweet 6!

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The consensus is the centrality-weighted average

a=Σᵢ cᵢxᵢ(0).

A weighted average of independent noise concentrates on μ only when the largest weight vanishes: a→μ in probability exactly when maxᵢ cᵢ→0.

(This is a weighted law of large numbers.)

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Now a "wisdom-of-crowds" application.

Grow the society: a sequence of larger and larger networks, each agent's initial opinion a noisy signal about a common truth μ (the xᵢ(0) independent, mean μ, finite variance).

Does the consensus recover μ as n→∞?

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