The Zigzag Graph Product CS 229 r October

The Zig-zag Graph Product CS 229 r October 20, 2020

Zig-Zag Product – First Intuition • Tensoring: G 1 G 2 contains independent copies, G 1 and G 2. (u, i) is connected to (v, j) if: i u G 2 G 1 j v • Tempting to use “dependent copies”: if G 2 has deg(G 1) vertices, use j to choose an edge in G 1

Zig-Zag Product - Construction Vertices of G 1 z G 2: pairs (vertex u of G 1, edge (label) i leaving u) 2 G 1 1 = vertex of G 1 z G 2 u 3

Zig-Zag Product – Construction (cont. ) Vertices of G 1 z G 2: pairs (vertex u of G 1, edge (label) i leaving u) G 1 (u, 2) (u, 1) = vertex of G 1 z G 2 (u, 3)

Zig-Zag Product – Construction (cont. ) Vertices of G 1 z G 2: pairs (vertex u of G 1, edge (label) i leaving u) (u, 2) (u, 1) = vertex of G 1 z G 2 (u, 3)

Zig-Zag Product – Construction (cont. ) Edges of G 1 z G 2: when to connect (u, i) and (v, j)? • G 2 has deg(G 1) vertices can place copy of G 2 on each “cloud. ” G 2 G 1 (u, i) (v, j)

… Edges of G 1 z G 2 • Connect (u, i) and (v, j) iff i , j such that 1. i and i connected in G 2 2. (u, i ) and (v, j ) correspond to same edge of G 1 3. j and j connected in G 2 G 1 (u, i) (v, j) (u, i ) z = edge of G 1 z G 2

Zig-Zag Product – Construction (cont. ) … Edges of G 1 z G 2 • The four neighbors of (u, i): (u, i) = edge of G 1 z G 2

Zig-Zag Graph Product Intuition • Need to show: Random step on G 1 G 2 makes nonuniform probability distributions closer to uniform. • Random step on G 1 G 2 – 1. random step within “cloud”, – 2. jump between clouds. – 3. random step within new cloud. Case I: very non-uniform within “clouds”. • Step 1 makes more uniform (by expansion of G 2 ) • Steps 2 & 3 cannot make less uniform.

Zig-zag Graph Product Intuition (cont. ) Case II: uniform within clouds. • Step 1 (within cloud) does not change . • Step 2: Jump between clouds random step on G 1 Distribution on clouds themselves becomes more uniform. (by expansion of G 1 ) [randomness “transferred” from distrib. within clouds] • Step 3 (within cloud) makes distribution within clouds more uniform. (by expansion of G 2 )