To put it another way, the regularity lemma tells us that every large dense adjacency matrix is in some sense a “combination” of a bounded rank matrix (which divides up into a bounded number of blocks) and a random matrix; the “block-diagonal” matrices and the Erdős–Rényi matrices reflect the two possible extremes of behaviour, and every other graph is in some sense a mixture of these two extremes.

]]>I wanted to make sense of Szemerédi regularity lemma (SRL) for Erdős–Rényi random graph G(n,p).

If I understood correctly the SRL states that any random dense graph (the adjacency matrix) can be “approximately” partitioned into block-diagonal structures (after proper rearrangement) .

Lets generate G(n=10000,p=.1), a dense random network butthe corresponding adjacency matrix A it can NOT be represented as a block-diagonal form (whatever rearrangement we do). Then my question is, how to interpret SRL in thisset-up ? What I’m missing ?

Thank you so much for your time and apology for my ignorance.

]]>Otherwise it seems that partitioning the graph into one part V1 = V, would trivially satisfy the conclusions of the lemma.

*[Hmm, you’re right. Thanks for the correction! – T.]*

Such an O(n) algorithm appears (explicitly) in the following paper of mine with Fischer and Matsliach.

http://www.cs.tau.ac.il/~asafico/regalg.pdf

That algorithm actually has the added advantage of being able to find (more or less) the smallest regular partition in the input.

]]>