Perhaps the most important structural result about general large dense graphs is the Szemerédi regularity lemma. Here is a standard formulation of that lemma:

Lemma 1 (Szemerédi regularity lemma) Let ${G = (V,E)}$ be a graph on ${n}$ vertices, and let ${\epsilon > 0}$. Then there exists a partition ${V = V_1 \cup \ldots \cup V_M}$ for some ${M \leq M(\epsilon)}$ with the property that for all but at most ${\epsilon M^2}$ of the pairs ${1 \leq i \leq j \leq M}$, the pair ${V_i, V_j}$ is ${\epsilon}$-regular in the sense that

$\displaystyle | d( A, B ) - d( V_i, V_j ) | \leq \epsilon$

whenever ${A \subset V_i, B \subset V_j}$ are such that ${|A| \geq \epsilon |V_i|}$ and ${|B| \geq \epsilon |V_j|}$, and ${d(A,B) := |\{ (a,b) \in A \times B: \{a,b\} \in E \}|/|A| |B|}$ is the edge density between ${A}$ and ${B}$. Furthermore, the partition is equitable in the sense that ${||V_i| - |V_j|| \leq 1}$ for all ${1 \leq i \leq j \leq M}$.

There are many proofs of this lemma, which is actually not that difficult to establish; see for instance these previous blog posts for some examples. In this post I would like to record one further proof, based on the spectral decomposition of the adjacency matrix of ${G}$, which is essentially due to Frieze and Kannan. (Strictly speaking, Frieze and Kannan used a variant of this argument to establish a weaker form of the regularity lemma, but it is not difficult to modify the Frieze-Kannan argument to obtain the usual form of the regularity lemma instead. Some closely related spectral regularity lemmas were also developed by Szegedy.) I found recently (while speaking at the Abel conference in honour of this year’s laureate, Endre Szemerédi) that this particular argument is not as widely known among graph theory experts as I had thought, so I thought I would record it here.

For reasons of exposition, it is convenient to first establish a slightly weaker form of the lemma, in which one drops the hypothesis of equitability (but then has to weight the cells ${V_i}$ by their magnitude when counting bad pairs):

Lemma 2 (Szemerédi regularity lemma, weakened variant) . Let ${G = (V,E)}$ be a graph on ${n}$ vertices, and let ${\epsilon > 0}$. Then there exists a partition ${V = V_1 \cup \ldots \cup V_M}$ for some ${M \leq M(\epsilon)}$ with the property that for all pairs ${(i,j) \in \{1,\ldots,M\}^2}$ outside of an exceptional set ${\Sigma}$, one has

$\displaystyle | E(A,B) - d_{ij} |A| |B| | \ll \epsilon |V_i| |V_j| \ \ \ \ \ (1)$

whenever ${A \subset V_i, B \subset V_j}$, for some real number ${d_{ij}}$, where ${E(A,B) := |\{ (a,b) \in A \times B: \{a,b\} \in E \}|}$ is the number of edges between ${A}$ and ${B}$. Furthermore, we have

$\displaystyle \sum_{(i,j) \in \Sigma} |V_i| |V_j| \ll \epsilon |V|^2. \ \ \ \ \ (2)$

Let us now prove Lemma 2. We enumerate ${V}$ (after relabeling) as ${V = \{1,\ldots,n\}}$. The adjacency matrix ${T}$ of the graph ${G}$ is then a self-adjoint ${n \times n}$ matrix, and thus admits an eigenvalue decomposition

$\displaystyle T = \sum_{i=1}^n \lambda_i u_i^* u_i$

for some orthonormal basis ${u_1,\ldots,u_n}$ of ${{\bf C}^n}$ and some eigenvalues ${\lambda_1,\ldots,\lambda_n \in {\bf R}}$, which we arrange in decreasing order of magnitude:

$\displaystyle |\lambda_1| \geq \ldots \geq |\lambda_n|.$

We can compute the trace of ${T^2}$ as

$\displaystyle \hbox{tr}(T^2) = \sum_{i=1}^n |\lambda_i|^2.$

But we also have ${\hbox{tr}(T^2) = 2|E| \leq n^2}$, so

$\displaystyle \sum_{i=1}^n |\lambda_i|^2 \leq n^2. \ \ \ \ \ (3)$

Among other things, this implies that

$\displaystyle |\lambda_i| \leq \frac{n}{\sqrt{i}} \ \ \ \ \ (4)$

for all ${i \geq 1}$.

Let ${F: {\bf N} \rightarrow {\bf N}}$ be a function (depending on ${\epsilon}$) to be chosen later, with ${F(i) \geq i}$ for all ${i}$. Applying (3) and the pigeonhole principle (or the finite convergence principle, see this blog post), we can find ${J \leq C(F,\epsilon)}$ such that

$\displaystyle \sum_{J \leq i < F(J)} |\lambda_i|^2 \leq \epsilon^3 n^2.$

(Indeed, the bound on ${J}$ is basically ${F}$ iterated ${1/\epsilon^3}$ times.) We can now split

$\displaystyle T = T_1 + T_2 + T_3, \ \ \ \ \ (5)$

where ${T_1}$ is the “structured” component

$\displaystyle T_1 := \sum_{i < J} \lambda_i u_i^* u_i, \ \ \ \ \ (6)$

${T_2}$ is the “small” component

$\displaystyle T_2 := \sum_{J \leq i < F(J)} \lambda_i u_i^* u_i, \ \ \ \ \ (7)$

and ${T_3}$ is the “pseudorandom” component

$\displaystyle T_3 := \sum_{i > F(J)} \lambda_i u_i^* u_i. \ \ \ \ \ (8)$

We now design a vertex partition to make ${T_1}$ approximately constant on most cells. For each ${i < J}$, we partition ${V}$ into ${O_{J,\epsilon}(1)}$ cells on which ${u_i}$ (viewed as a function from ${V}$ to ${{\bf C}}$) only fluctuates by ${O(\epsilon n^{-1/2} /J)}$, plus an exceptional cell of size ${O( \frac{\epsilon}{J} |V|)}$ coming from the values where ${|u_i|}$ is excessively large (larger than ${\sqrt{\frac{J}{\epsilon}} n^{-1/2}}$). Combining all these partitions together, we can write ${V = V_1 \cup \ldots \cup V_{M-1} \cup V_M}$ for some ${M = O_{J,\epsilon}(1)}$, where ${V_M}$ has cardinality at most ${\epsilon |V|}$, and for all ${1 \leq i \leq M-1}$, the eigenfunctions ${u_1,\ldots,u_{J-1}}$ all fluctuate by at most ${O(\epsilon/J)}$. In particular, if ${1 \leq i,j \leq M-1}$, then (by (4) and (6)) the entries of ${T_1}$ fluctuate by at most ${O(\epsilon)}$ on each block ${V_i \times V_j}$. If we let ${d_{ij}}$ be the mean value of these entries on ${V_i \times V_j}$, we thus have

$\displaystyle 1_B^* T_1 1_A = d_{ij} |A| |B| + O( \epsilon |V_i| |V_j| ) \ \ \ \ \ (9)$

for any ${1 \leq i,j \leq M-1}$ and ${A \subset V_i, B \subset V_j}$, where we view the indicator functions ${1_A, 1_B}$ as column vectors of dimension ${n}$.

Next, we observe from (3) and (7) that ${\hbox{tr} T_2^2 \leq \epsilon^3 n^2}$. If we let ${x_{ab}}$ be the coefficients of ${T_2}$, we thus have

$\displaystyle \sum_{a,b \in V} |x_{ab}|^2 \leq \epsilon^3 n^2$

and hence by Markov’s inequality we have

$\displaystyle \sum_{a \in V_i} \sum_{b \in V_j} |x_{ab}|^2 \leq \epsilon^2 |V_i| |V_j| \ \ \ \ \ (10)$

for all pairs ${(i,j) \in \{1,\ldots,M-1\}^2}$ outside of an exceptional set ${\Sigma_1}$ with

$\displaystyle \sum_{(i,j) \in \Sigma_1} |V_i| |V_j| \leq \epsilon |V|^2.$

If ${(i,j) \in \{1,\ldots,M-1\}^2}$ avoids ${\Sigma_1}$, we thus have

$\displaystyle 1_B^* T_2 1_A = O( \epsilon |V_i| |V_j| ) \ \ \ \ \ (11)$

for any ${A \subset V_i, B \subset V_j}$, by (10) and the Cauchy-Schwarz inequality.

Finally, to control ${T_3}$ we see from (4) and (8) that ${T_3}$ has an operator norm of at most ${n/\sqrt{F(J)}}$. In particular, we have from the Cauchy-Schwarz inequality that

$\displaystyle 1_B^* T_3 1_A = O( n^2 / \sqrt{F(J)} ) \ \ \ \ \ (12)$

for any ${A, B \subset V}$.

Let ${\Sigma}$ be the set of all pairs ${(i,j) \in \{1,\ldots,M\}^2}$ where either ${(i,j) \in \Sigma_1}$, ${i = M}$, ${j=M}$, or

$\displaystyle \min(|V_i|, |V_j|) \leq \frac{\epsilon}{M} n.$

One easily verifies that (2) holds. If ${(i,j) \in \{1,\ldots,M\}^2}$ is not in ${\Sigma}$, then by summing (9), (11), (12) and using (5), we see that

$\displaystyle 1_B^* T 1_A = d_{ij} |A| |B| + O( \epsilon |V_i| |V_j| ) + O( n^2 / \sqrt{F(J)} ) \ \ \ \ \ (13)$

for all ${A \subset V_i, B \subset V_j}$. The left-hand side is just ${E(A,B)}$. As ${(i,j) \not \in \Sigma}$, we have

$\displaystyle |V_i|, |V_j| > \frac{\epsilon}{M} n$

and so (since ${M = O_{J,\epsilon}(1)}$)

$\displaystyle n^2 / \sqrt{F(J)} \ll_{J,\epsilon} |V_i| |V_j| / \sqrt{F(J)}.$

If we let ${F}$ be a sufficiently rapidly growing function of ${J}$ that depends on ${\epsilon}$, the second error term in (13) can be absorbed in the first, and (1) follows. This concludes the proof of Lemma 2.

To prove Lemma 1, one argues similarly (after modifying ${\epsilon}$ as necessary), except that the initial partition ${V_1,\ldots,V_M}$ of ${V}$ constructed above needs to be subdivided further into equitable components (of size ${\epsilon |V|/M+O(1)}$), plus some remainder sets which can be aggregated into an exceptional component of size ${O( \epsilon |V| )}$ (and which can then be redistributed amongst the other components to arrive at a truly equitable partition). We omit the details.

Remark 1 It is easy to verify that ${F}$ needs to be growing exponentially in ${J}$ in order for the above argument to work, which leads to tower-exponential bounds in the number of cells ${M}$ in the partition. It was shown by Gowers that a tower-exponential bound is actually necessary here. By varying ${F}$, one basically obtains the strong regularity lemma first established by Alon, Fischer, Krivelevich, and Szegedy; in the opposite direction, setting ${F(J) := J}$ essentially gives the weak regularity lemma of Frieze and Kannan.

Remark 2 If we specialise to a Cayley graph, in which ${V = (V,+)}$ is a finite abelian group and ${E = \{ (a,b): a-b \in A \}}$ for some (symmetric) subset ${A}$ of ${V}$, then the eigenvectors are characters, and one essentially recovers the arithmetic regularity lemma of Green, in which the vertex partition classes ${V_i}$ are given by Bohr sets (and one can then place additional regularity properties on these Bohr sets with some additional arguments). The components ${T_1, T_2, T_3}$ of ${T}$, representing high, medium, and low eigenvalues of ${T}$, then become a decomposition associated to high, medium, and low Fourier coefficients of ${A}$.

Remark 3 The use of spectral theory here is parallel to the use of Fourier analysis to establish results such as Roth’s theorem on arithmetic progressions of length three. In analogy with this, one could view hypergraph regularity as being a sort of “higher order spectral theory”, although this spectral perspective is not as convenient as it is in the graph case.