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pr.likelihood – $ell^1$-norm of eigenvectors of Erdős-Renyi Graphs

**Setting.** Let $G(n,p)$ denote the habitual Erdős-Renyi (random) graphs. For every such graph there’s an related Laplacian matrix $L = D – A$ the place $D$ collects the levels on the diagonal and $A$ is the adjacency matrix. This matrix has eigenvectors $v_1, dots v_n$ (ordered with respect to rising eigenvalue) and we occupy the eigenvectors are all normalized in $ell^2(mathbb{R}^n)$.

**Localization.** One unaffected query is whether or not any of those eigenvectors can breathe localized and that is typically measured within the sense of $| v_i|_{ell^{infty}}$ being ‘sizable’, one entry being unusually huge. One doesn’t anticipate this to breathe the illustration and it is attention-grabbing to attempt to quantify this understanding (and I cerebrate there are a lot of papers about this). However, there’s twin model: we all know from Hölder that

$$| v_i|_{ell^1} leq sqrt{n} cdot |v|_{ell^2} = sqrt{n}$$

with equality solely attained for the flat vector. So the $ell^1-$norm can too relieve as a touchstone of localization: the extra localized a vector is, the smaller it should breathe.

When I plot the worth of $|v_i|_{ell^1}$ for all of the eigenvectors $i=1,2,dots,n$, I get the next droll round.

This portray exhibits $|v_i|_{ell^1}$ for $i=1,dots, 5000$ for a random realization of $G(n, p)$ for $n=5000$ and $p=0.4$. Knowing the place within the spectrum an eigenvector lies appears to slim down what its $ell^1$-norm can breathe. There too appears to breathe a focus phenomenon: the portray is handsome mighty the identical for every random realization. Here’s a second instance for a random realization of $G(n, p)$ for $n=5000$ and $p=0.01$.

The eigenvectors equivalent to the smallest eigenvalues appear to breathe localized in vertices which have unusually few neighbors and eigenvectors equivalent to the biggest eigenvectors appear to breathe localized in vertices having unusually many neighbors — thus one would maybe some focus within the edges and that might too elucidate why the round is largest within the center and symmetric. But I’m shocked to behold such a pleasant round that appears to refer **uniformly** via the gross spectrum.

**Question 1:** Is this recognized or does it succeed from recognized outcomes? Is there a very good heuristic why this energy breathe undoubted?

Another attention-grabbing query is concerning the most of $| v_i|_{ell^2}$ which appears to breathe assumed someplace within the center.

**Question 2:** What can breathe stated about

$$ 0 leq max_{1 leq i leq n}frac{ | v_i|_{ell^1}}{sqrt{n}} leq 1?$$

Numerically it appears to breathe someplace round 0.8. If we occupy that the entries of a typical flat eigenvector behave love i.i.d. Gaussians, then a primary speculate would breathe that this ratio ought to breathe someplace round $mathbb{E} |X|$ the place $X sim mathcal{N}(0,1)$. We have $mathbb{E} |X| = sqrt{2/pi} sim 0.79788$. Maybe a coincidence?

**Background:** A pair of years in the past, Alex Cloninger and I performed with the issue of looking for construction in Laplacian eigenvectors. We create a system that labored handsome nicely in good settings — we then tried to behold what it might discover on Erdős-Renyi random graphs (which appeared a unaffected take a look at illustration: the eigenvectors ought to breathe largely random and never notably structured). Somehow our system picked out the verge and we did not know why till we create this $ell^1$ anomaly. It’s talked about within the paper (arXiv) however we by no means acquired a random to question lots of people about it.

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