Timeline for $\ell^1$-norm of eigenvectors of Erdős-Renyi Graphs
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Dec 19, 2020 at 18:01 | history | bounty ended | Stefan Steinerberger | ||
| Dec 15, 2020 at 7:19 | history | edited | ofer zeitouni | CC BY-SA 4.0 |
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| Dec 15, 2020 at 7:17 | comment | added | ofer zeitouni | There is indeed a difference between adjacency matrices (which my answer discussed) and the (normalized) Laplacian. See the edit to my answer, and in particular what you get for the second eigenvector of the Laplacian. | |
| Dec 15, 2020 at 7:08 | history | edited | ofer zeitouni | CC BY-SA 4.0 |
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| Dec 15, 2020 at 0:11 | comment | added | Stefan Steinerberger | The curve suggests that the smallest values are on the edge. I had a look at the smallest nontrivial eigenvector and it does seem to localize in the vertex with the fewest neighbors. (I don't know what happens if there is more than one such vertex). The question is then somehow whether it localizes quickly enough to have a small $\ell^1$ norm. Some basic numbers seem to suggest that $\| v_2\|_{\ell^1} / \sqrt{n}$ is decaying as $n$ increases. I get $\sim 0.34$ for $n=500$ and $\sim 0.3$ for $n=5000$. | |
| Dec 13, 2020 at 21:37 | history | edited | ofer zeitouni | CC BY-SA 4.0 |
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| Dec 13, 2020 at 18:59 | history | answered | ofer zeitouni | CC BY-SA 4.0 |