Random Schreier graphs as expanders

TL;DR

This work investigates randomized Schreier-graph constructions for expander graphs, aiming to reduce randomness by using linear actions from $\mathrm{GL}_k(\mathbb{F}_q)$ and Toeplitz matrices. Through extensive experiments on undirected regular and bipartite biregular graphs, the authors show that the second eigenvalue $\mu_2$ frequently attains near-Ramanujan values $\mu_2 \approx 2\sqrt{d-1}$ (for regular graphs) or $\sqrt{d_L-1}+\sqrt{d_R-1}$ (for bipartite cases), closely matching the performance of fully random permutations. They supplement the empirical results with computer-assisted theoretical bounds via the trace method, deriving computable, though weak, estimates for $\mathbb{E}[|\mu_2|]$ and extending them to Toeplitz-based constructions. The findings suggest a practical pathway to generate large, well-mibrating expanders with substantially fewer random bits, with Toeplitz matrices offering a promising balance between randomness reduction and spectral quality, while highlighting the need for deeper theoretical explanations of the observed phenomena.

Abstract

Expander graphs, due to their mixing properties, are useful in many algorithms and combinatorial constructions. One can produce an expander graph with high probability by taking a random graph (e.g., the union of $d$ random bijections for a bipartite graph of degree $d$). This construction is much simpler than all known explicit constructions of expanders and gives graphs with good mixing properties (small second largest eigenvalue) with high probability. However, from the practical viewpoint, it uses too many random bits, so it is difficult to generate and store these bits for large graphs. The natural idea is to restrict the class of the bijections that we use. For example, if both sides are linear spaces $\mathbb{F}_q^k$ over a finite field $\mathbb{F}_q$, we may consider only \emph{linear} bijections, making the number of random bits polynomial in $k$ (and not $q^k$). In this paper we provide some experimental data that shows that this approach conserves the mixing properties (the second eigenvalue) for several types of graphs (undirected regular and biregular bipartite graphs). We also prove some upper bounds for the second eigenvalue (though they are quite weak compared with the experimental results). Finally, we discuss the possibility to decrease the number of random bits further by using Toeplitz matrices; our experiments show that this change makes the mixing properties only marginally worse while the number of random bits decreases significantly.

Figures (7)

• Figure 1: All the eigenvalues (horizontal axis) are divided by $2\sqrt{d-1}$. The two upper curves (quite close to each other) show the empirical distributions for random permutation model (dashed line) and random linear permutation model for graphs of degree $d=30$ ($15$ permutations) and size $16383=2^{14}-1$. The third curve (wider green one in the bottom part) shows the distribution for random linear permutations on $5$-tuples from a $7$-element field, so the graph has $7^5-1=16806$ vertices (and the same degree $30$, obtained with $15$ permutations). For each distribution $5000$ experiments were made; the eigenvalues are grouped in $40$ bins.
• Figure 2: On the left: second largest eigenvalue distributions for $\textit{GL}_{14}(\mathbb{F}_2)$ Schreier graphs of degrees $10, 30, 60$ and $120$. On the right: instead of the degree, we vary the dimension of the matrices and keep the same field $\mathbb{F}_2$.
• Figure 3: Here, n1 is the size of each partition. The dashed line (blue) shows the distribution of second eigenvalues for a bipartite graph constructed from $30$ random permutations of $16383$-element set (nonzero bit vectors of size $14$). The orange line (close to the first one) shows the same for $30$linear permutations (over $\mathbb{F}_2$). The third line (green, in the bottom part) shows the distribution for $30$ permutations of a $16806$-element set that are randomly chosen among the $\mathbb{F}_7$-linear permutations of nonzero $5$-tuples with elements in $\mathbb{F}_7$.
• Figure 4: The dashed line (blue) shows the distribution of the second eigenvalues for $5000$ experiments when we take $10$ random permutations of a set with $2^{14}-1=16383$ elements, construct a bipartite graph with degree $10$ in both sides and then merge triples of vertices on one side thus getting degree $30$ and size $16383/3=5461$. The solid (orange) line replaces random permutations by random linear (over $\mathbb{F}_2$) permutations of non-zero vectors in $\mathbb{F}_2^{14}$. Finally, the line with dots (green) shows the same distribution if we take $10$ linear permutations of non-zero vectors in $\mathbb{F}_7^5$.
• Figure 5: On the left: distributions of second largest eigenvalues of 5000 graphs obtained from Toeplitz matrices (TP) for $\mathbb{F}_2^{14}$ and $\mathbb{F}_7^5$ compared with the corresponding empirical distributions for random linear permutations in the same vector spaces (GL). On the right, all graphs are from Toeplitz matrices over $\mathbb{F}_2$.

Theorems & Definitions (21)

• proposition 1
• theorem 1
• theorem 2
• proposition 2
• lemma 1
• proof
• definition 1: free and forced step
• lemma 2
• proof
• lemma 3