Why it matters

Expander graphs enable derand. Understanding shapes theory.

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The architecture

Every small vertex subset has many neighbors.

Constant-degree.

Spectral gap property.

Expander graph propertiesVertex expansion|N(S)| >= alpha |S|Constant degreesparseSpectral gapsecond eigenvalue smallRamanujan graphs optimal; Zig-zag construction + PPSZ derandomization
Expander graphs.
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How it works end to end

Ramanujan graphs (optimal).

Zig-zag construction.

Applications: derand + error correction + Markov chain mixing.