Explicit Abelian Lifts and Quantum LDPC Codes

For an abelian group H acting on the set [ℓ], an (H,ℓ)-lift of a graph G_0 is a graph obtained by replacing each vertex by ℓ copies, and each edge by a matching corresponding to the action of an element of H. In this work, we show the following explicit constructions of expanders obtained via abelian lifts. For every (transitive) abelian group H ⩽Sym(ℓ), constant degree d ≥ 3 and ϵ > 0, we construct explicit d-regular expander graphs G obtained from an (H,ℓ)-lift of a (suitable) base n-vertex expander G_0 with the following parameters: (i) λ(G) ≤ 2√(d-1) + ϵ, for any lift size ℓ≤ 2^n^δ where δ=δ(d,ϵ), (ii) λ(G) ≤ϵ· d, for any lift size ℓ≤ 2^n^δ_0 for a fixed δ_0 > 0, when d ≥ d_0(ϵ), or (iii) λ(G) ≤O(√(d)), for lift size “exactly” ℓ = 2^Θ(n). As corollaries, we obtain explicit quantum lifted product codes of Panteleev and Kalachev of almost linear distance (and also in a wide range of parameters) and explicit classical quasi-cyclic LDPC codes with wide range of circulant sizes. Items (i) and (ii) above are obtained by extending the techniques of Mohanty, O'Donnell and Paredes [STOC 2020] for 2-lifts to much larger abelian lift sizes (as a byproduct simplifying their construction). This is done by providing a new encoding of special walks arising in the trace power method, carefully "compressing'" depth-first search traversals. Result (iii) is via a simpler proof of Agarwal et al. [SIAM J. Discrete Math 2019] at the expense of polylog factors in the expansion.

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1 Introduction

Graphs are ubiquitous in theoretical computer science and the ability to explicitly construct graphs with special properties can be quite useful. Two such properties are expansion and symmetry. A graph is expanding if it is simultaneously sparse and highly connected (meaning that we need to remove a lot of edges to disconnect a large part of the graph.) The theory of explicit constructions of expander graphs has seen a dramatic development over the past four decades111See [HLW06] for an excellent survey on expander graphs. [LPS88, Mar88, Mor94, RVW00, BL06, BATS08, Coh16, MOP20, OW20, Alo21]. We now have constructions via diverse methods achieving a wide range of expansion guarantees. These range from very explicit algebraic constructions of so-called Ramanujan graphs [LPS88] to recursive combinatorial ones based on the Zig-Zag product [RVW00]. These constructions have a plethora of applications specially to coding theory and pseudorandomness [Vad12]. A highly sought goal is to make the expansion of a family of (bounded) degree

graphs as close to the Ramanujan bound as possible, i.e., having largest non-trivial eigenvalue at most

. The Alon-Boppana bound[Nil91] states that the largest non-trivial eigenvalue is at least , so the Ramanujan bound is in a sense optimal. This goal of achieving strong spectral guarantees has been an important motivation.

Moving beyond spectral guarantees, we can ask for graphs that combine the important property of expansion with additional structure and the one we focus on is symmetry222Informally, we say that has symmetries of if , where denotes the group of all graph isomorphisms to itself. . One of the problems that has been studied in graph theory is to construct graphs with a given automorphism group. Frucht proved in 1939 that for every finite group , we have a graph such that . Babai [Bab74] later showed that there is such a graph on at most vertices333Except for , and .. Thus, we have a natural question

Can we explicitly construct expanding graphs with given symmetries?

While interesting in its own right, the ability to control symmetries also has concrete applications. For example, a very recent work [GW21] constructs many families of expanding asymmetric graphs, i.e., having no symmetries, and shows applications to property testing and other areas. We will focus on an important connection to both quantum and classical codes that was the motivation behind this work.

Low-density parity check (LDPC) codes were first introduced by Gallager [Gal62] in the ’60s and are one of the most popular classes of classical error-correcting codes, both in theory and in practice. LDPC codes are linear codes whose parity check matrices have row and column weights bounded by a constant (which means that each parity check depends only on a constant number of bits). The popularity of this family of codes comes from the fact that there are many known constructions of classical LDPC codes that achieve linear rate and distance that can also be decoded in linear time [RU08].

A family of codes that has been extensively studied is cyclic codes, i.e., codes that are invariant under the action of where is the blocklength. This symmetry leads to efficient encoding and decoding algorithms and a major open problem is whether good cyclic codes exist. Babai, Shpilka and Stefankovich [BSS05] showed that cyclic codes cannot be good LDPC codes and this negative result was extended by Kaufman and Wigderson [KW10] to LDPC codes with a transitive action by an arbitrary abelian group.

Quasi-cyclic codes are a generalization of cyclic codes in which symmetry is only under rotations of multiples of a parameter (called index) where . This is equivalent to relaxing the transitivity condition to allow for orbits. Unlike cyclic codes, good quasi-cyclic codes are known to exist as was shown by Chen, Peterson and Weldon [CPW69]. More recently, Bazzi and Mitter [BM06] gave a randomized construction for any constant and showed that it attains Gilbert–Varshamov bound rate . Quasi-cyclic codes have been extensively studied and are very useful in practice (e.g., their LDPC counterparts are part of the 5G standard of mobile communication [LBM18]).

In the realm of quantum computing, the fragility of qubits makes quantum error correcting codes crucial for the realization of scalable quantum computation.

Calderbank-Shor-Steane (CSS) codes are a family of quantum error-correcting codes that was first described in [CS96, Ste96]. A CSS code is defined by a pair of classical linear codes that satisfy an orthogonality condition. The quantum analog of LDPC codes is thus defined as CSS codes where the parity check matrices of both codes have bounded row and column weights.

Constructing quantum LDPC codes of large distance has been active area of research recently. After two decades, [EKZ20] broke the barrier and there was a flurry of activity with [HHO21] extending it to (up to factors). Panteleev and Kalachev [PK21b] came up with another breakthrough construction achieving almost linear distance. Both [HHO21] and [PK21b] are non-explicit constructions crucially relying on symmetries. The construction in [PK21b] interestingly used quasi-cyclic LDPC codes which in turn was constructed using expander graphs with cyclic symmetry. Moreover, Breuckmann and Eberhardt [BE21] introduced a new approach for constructing quantum codes simultaneously generalizing [HHO21] and [PK21b] in order to obtain explicit codes out of a pair of graphs having the symmetries of any group. This provides a very concrete motive to study explicit construction of expander graphs symmetric under various families of groups.

Update

While the current paper was being prepared for publication, Panteleev and Kalachev [PK21a] found a breakthrough construction of explicit good quantum LDPC codes.

Current Techniques

Many of the current known constructions of expanders are Cayley graphs and therefore are highly symmetric but are somewhat rigid in the sense that one may not be able to finely control the symmetries of a given construction. One general approach is to construct an expanding Cayley graph for a given group but the Alon–Roichman theorem [AR94] only guarantees a logarithmic degree which is tight when the group is abelian (and this large degree is undesirable for some application in coding theory). The other technique used to build expanders is via an operation called lifting.

In general form, the random lifting operation takes a lift size parameter , a base expander graph on vertices and a subgroup of the symmetric group, , and constructs a new “lifted” graph on vertices where each vertex of is replaced by -copies and for every edge of a uniformly random element of is sampled and is connected to for . We say that obtained this way is a random -lift of . We call it an unstructured -lift if there is no restriction on the group, i.e., .

Lifting has three very useful properties. One, it preserves the degree of the base graph. Secondly, random lifts preserve expansion444This holds for any lift size in the case of “unstructured” -lifts, but only holds for when is abelian (and transitive).

with high probability. Finally, (and importantly for us), if

is abelian, then the lifted graph inherits symmetries of . The first two properties are clearly useful in constructing larger expanders from a small one, and for this reason, there has been extensive work on lift based constructions.

Bilu and Linial [BL06] introduced 2-lifts in an explicit construction of graphs with expansion for every degree. More recently, Mohanty, O’Donnell and Paredes [MOP20] gave the first explicit construction of near-Ramanujan, i.e., largest non-trivial eigenvalue bounded by , graphs of every degree. The key technique in their work was a finer derandomization of -lifts. Subsequently, Alon [Alo21] gave explicit constructions of near-Ramanujan expanders of every degree and every number of vertices. The work in [MOP20] was also generalized to achieve finer spectral guarantees together with local properties via unstructured -lifts in O’Donnell and Wu [OW20].

When one restricts to be abelian, Agarwal et al. [ACKM19] showed that random -lifts (also known as shift lifts) are expanding. Motivated by the applications of these lifts to codes, we obtain explicit constructions of expanding abelian lifts, for a wide range of lift sizes.

1.1 Our Results and Techniques

Our construction of the lifts (and the expansion thereof) vary based on the parameter and we make the following classification for ease in presenting the results. Let be given.

  • Sub-Exponential - This is the regime where . The exponent goes to zero as the degree () increases or vanishes.

  • Moderately-Exponential - This is when . The exponent is some fixed universal constant .

  • Exactly-Exponential - This is the regime where .

Our first main result shows explicit constructions in the sub-exponential and moderately exponential regimes.

Theorem 1.1.

For large enough and constant degree , given such that , the generating elements of a transitive abelian group , and any fixed constant , we can construct in deterministic polynomial time, a -regular graph on vertices such that

  • is -lift of a graph on vertices.

  • (Sub-Exponential) If , then .

  • (Moderately-Exponential) If and also , then .

The bulk of the technical work is in the proof of Theorem 1.1. For this, we build on the techniques of [MOP20] for derandomizing -lifts via the trace power method. When analyzing larger lift sizes (required in our derandomization of quantum and classical codes), we are led to consider much larger walk lengths in the trace power method. A central technical component in their work is the counting of some special walks which ultimately governs the final spectral bound of the construction. For lift sizes larger than , their counting trivializes no longer implying expansion of the construction. Our main technical contribution consists in providing alternative ways of counting such special walks by carefully compressing the traversal of the depth-first search (DFS) algorithm.

We are able to extend the near-Ramanujan guarantee for -lifts from [MOP20] to the entire sub-exponential regime of lift sizes . In the moderately exponential regime, the walks are too long and we resort to another counting that can only guarantee an expansion of . Theorem 1.1 can be seen (slight) simplification of the construction in [MOP20] since we can now do a single large lift instead of performing a sequence of -lifts as in their work555Performing a single lift also has the advantage of having to meet a technical condition (bicycle-freeness) only once instead of at each lift operation..

Let us now formally state the results of Agarwal et al. in Theorem 1.2 showing randomized constructions of abelian lifts.

Theorem 1.2 (Agarwal et al. [Ackm19], Theorem 1.2).

Let be a -regular -vertex graph, where . Let be a random -lift of . Then

with probability . Moreover, if , then no abelian lift has .

This result is based on discrepancy methods building on the work of Bilu and Linial [BL06] and gives lower and upper bounds that are tight up to a factor of in the exponent.

Theorem 1.1 can be seen as a (derandomization of the parameters) in Theorem 1.2 for every constant degree and lift size from all the way to . In the sub-exponential regime, our result improves their spectral guarantee from to .

Our second main result shows explicit constructions in the exponential regime. While it is not hard to observe that one can derandomize the exponential lift by using off-the-shelf tools, we give a short proof via a key lemma of Bilu and Linial [BL06]

that is a converse of the expander mixing lemma. Although it gives a spectral guarantee that is weaker by a log factor, it yields an accessible proof and moreover, interpolates the exponent from

all the way to the barrier of thereby bridging the -gap.

Theorem 1.3 (Exactly Exponential Lifts).

For any positive integers and every constant degree , given , the generating elements of a transitive abelian group , there exists a deterministic time algorithm that constructs a -regular graph on vertices such that

  • is -lift of a graph on vertices, and

  • If , then .

  • If for , then .

In particular, we have explicit polynomial time construction of a lift when .

1.2 Derandomized Quantum and Classical Codes

We first state the code constructions in [PK21b] and then show how large explicit abelian lifts derandomize their codes.

Theorem 1.4 ([PK21b]).

Let be a -regular graph on -vertices such that has a symmetry666To be more precise, acts freely on . of and . Then we can construct the following,

  • A good quasi-cyclic LDPC code of block length and index .

  • A quantum LDPC code which has distance and dimension .

Panteleev and Kalachev use the aforementioned randomized construction of abelian lifted expanders by Agarwal et al. [ACKM19], where each edge of the base graph is a associated with an element in sampled uniformly. When is in the exponential regime they obtain quantum LDPC codes with almost linear distance, i.e., .

Breuckmann and Eberhardt [BE21] gave a derandomization of [PK21b] in a more restricted parameter regime by observing that the Ramanujan graph construction by Lubotsky, Philips and Sarnak [LPS88] of size has a (free) action of By Theorem 1.4, we have an explicit quantum LDPC code of distance under the notion of distance777[BE21] state their result for a slightly different notion of a quantum codes called subsystems codes for which the corresponding distance (also known as dressed distance) is larger. in [PK21b, HHO21].

As a direct corollary of Theorem 1.3, we have a complete derandomization of [PK21b] yielding explicit quantum LDPC codes of almost linear distance. This greatly improves the distance of the existing explicit construction. We also get good quasi-cyclic LDPC codes of almost linear circulant size. Moreover, the ability to construct a wide range of lift sizes from Theorem 1.1 lets us control the circulant size which can be useful in practice. By controlling the lift size, we can also directly amplify the rate of their quantum LDPC codes (without resorting to the product of complexes). To summarize,

Corollary 1.5 ([PK21b], Theorem 1.1 ,Theorem 1.3).

We have explicit polynomial time construction of each of the following,

  • Good quasi-cyclic LDPC code of block length and any circulant size up to or .

  • Quantum LDPC code with distance and dimension .

  • Quantum LDPC code with distance and dimension for every constant .

Further Directions

Our work also leads to several natural avenues for further exploration.

  1. More Symmetries - While these lift-based constructions yield graphs with symmetries arising from abelian groups, it is interesting to understand whether one can construct sparse graphs with symmetries corresponding to other families of groups. Such constructions may require new ways of using the symmetry groups, in ways other than in lifts of a base graph. More generally, it may be useful to investigate other ways of exploiting graph symmetry, beyond their applications to codes.

  2. Better notions of explicitness - It is a very interesting problem to find strongly explicit constructions of lifted abelian expander. Even making the running time closer to linear would be interesting. Also, since quasicyclic codes are widely used in practice, it may be helpful to find explicit constructions which are efficiently implementable.

  3. Complete Range - Can we derandomize abelian lifts for in between and ? Can we extend the near-Ramanujan bound beyond the subexponential range?

2 Preliminaries

For an operator , let its eigenvalues be ordered such that . We define . For an an -vertex graph , we denote by , where is its adjacency operator.

We assume that we have an an ordering on and by convention, if .

A character888The definition we give is that of a linear character. We use the term character as we work only with abelian groups. of a group is a map that respects group multiplication, i.e., . For a finite group for every . The trivial character is the one which has for every . The rest of the characters we call non-trivial.

The action of a group on a set of elements is defined by a map which satisfies . Since we only care about the action of the group, we will assume that our input is actually and the action is the natural one.

Definition 2.1 (-lift of a graph).

An -signing of an undirected graph is a function . The lifted graph is a graph on copies of the vertices where for every edge we have

We will restrict to analyzing abelian and the most important case to consider is when , i.e. the cyclic group. A necessary condition for the lift to be expanding is for it to be connected. A subgroup is transitive if for every , there exists such that . Lifts of non-transitive subgroups are disconnected because if the pair violate the condition then any pair and are disconnected. Thus, we will assume henceforth that we work with transitive abelian subgroups.

Let denote the set of directed edges i.e. . We extend the signing to such that for an edge , .

Definition 2.2 (Non-backtracking walk operator).

For an extended signing and a character of , the signed non-backtracking walk matrix is a non-symmetric matrix of size in which the entry corresponding to the pair of edges is if , and zero otherwise.

The unsigned variant is obtained by taking the trivial character in the definition above. Let the non-backtracking walk matrix of be and the lifted graph with respect to a signing be . We use the following standard facts.

Fact 2.3.

Let be the non-backtracking walk matrix of a -regular graph . Then,

Fact 2.4.

If is abelian, then there exist characters 999These need not be distinct. For example if is trivial, then all the are trivial such that we have . If is transitive, then exactly one of the characters is trivial.

3 Proof Strategy

We give an overview of the proof of Theorem 1.1. As mentioned earlier, our results build on the work of Mohanty, O’Donnell and Paredes [MOP20], so we briefly recall notions and ideas from their work that we will need.

Let be a base expander graph and be a signing that defines a lift. It is convenient to first think that the signing is chosen uniformly at random and later see which properties were indeed used so that an appropriate derandomization tool may be used. Using well known facts (creftype 2.3 and creftype 2.4) they reduce the problem of analyzing the expansion of the lifted graph to that of bounding the spectral radius of the non-backtracking operator .

The MOP Argument: A common technique to bound the spectral radius is the trace power method which in our case amounts to counting special non-backtracking walks. This is the motivation for using the non-backtracking operator instead of the more common adjacency operator which require counting closed walks (which is potentially harder). Another standard fact101010To avoid discussing some unimportant technicalities, we will make some simplifications in this high-level overview. is that

The above expression greatly simplifies when we take the expectation over a uniformly random signing since only walks in which every edge occurs at least twice stand a chance of surviving the expectation. These walks are called singleton free in [MOP20]. We have

reducing the problem of bounding the spectral radius to a counting problem of these special walks. In the hypothetical (idealized) scenario of being Ramanujan and the counting on the RHS above being , we would have a Ramanujan lift. The above expression also hints that -bias distributions might be a useful derandomization tool here. This idealized scenario can be too optimistic and the count of has additional factors, but they remain small after taking a -th root (when is neither too small or large)

One of the main technical contributions in [MOP20] is the counting of -length singleton free non-backtracking walks in , which they call hikes. For the sake of intuition, we will assume that has girth , but it is not hard to modify the argument when has at most one cycle around any neighborhood of radius centered at vertex in (the bicycle freeness property). They view the vertices and edges visited in a hike as forming a hike graph . Assuming that , if is not too large, then looks like a tree possibly with a few additional edges forming cycles as established by Alon, Hoory and Linial in [AHL02] (and generalized in [MOP20] to bicycle-free radius from girth).

Assuming that the hike is singleton free, we can have at most steps that visit an edge that was not previously visited. This implies that the hike graph has at most edges and at most vertices (since it is connected). They count the number of these special walks by directly specifying an encoding for the hike. Up to negligible factors (after -th root for not too small), they show that there are at most

singleton free hikes of length (see [MOP20, Theorem 3.9] for precise details). This bound trivializes, i.e., it becomes at least , for . This means that we cannot use their bound for very long walks and this in turn prevents us from getting lift sizes larger than from their results.

Our Approach: Now, let’s consider lifts for large . The spectral radius of each individual can be analyzed in a similar fashion as above via the trace power method. However, we need to bound all of them simultaneously. We know no better way than a simple union bound over the cases, but this will force us to obtain a much better concentration guarantee out of the trace power method which in turn entails having to consider much larger walk lengths.

Instead of encoding a hike directly as in [MOP20], we will first encode the subgraph of traversed by the hike, which we call hike graph, and then encode the hike having the full hike graph at our disposal. We will give two different encodings for the hike graph. The first one is simpler and can encode an arbitrary graph. The second encoding uses the special structure of the hike graph, namely, having few vertices of degree greater than . Both encodings are based on the traversal history of the simple depth-first search (DFS) algorithm. Let be the hike graph on edges and vertices. As DFS traverses , each of its edges will be visited twice: first “forward” via a recursive call and later “backwards” via a backtracking operation. We view each step of the DFS traversal as being associated with an edge that is being currently traversed and the associated type of traversal: recursive (R) or backtracking (B). A key observation is that only for the recursive traversals we need to know the next neighbor out of possibilities (except for the first step). For the backtracking steps, we can rely on the current stack of DFS. Thus, if we are given a starting vertex from , a binary string in and a next neighbor for each recursive step, we can reconstruct . Note that there at most

such encodings. Having access to the hike graph and again assuming that the graph has girth (similarly, bicycle freeness is also enough). Using the locally tree-like structure, a -length hike can be specified by splitting it into segments of length , by specifying the starting vertex of the first segment and the ending vertex of each segment, we have enough information to recover the full hike. Note that there are at most

ways of encoding a hike. Then, the number of -hikes in is at most

Now we can take for a sufficiently small and obtain, after taking the -th root of the above quantity,

when is sufficiently large and is sufficiently small. The extra factor prevent us from obtaining near-Ramanujan bounds with this counting. Nonetheless, the simple counting already allows us to obtain expansion for lifts sizes as large as . Moreover, by weakening the expansion guarantee we can obtain lift sizes as large as from this counting and obtain part of Theorem 1.1. If we insist on getting a near-Ramanujan bound, we need to compress the traversal history further since storing a string is too costly and leads to this factor of . Note that this string has an equal number of and symbols, so it cannot be naively compressed.

To obtain a near-Ramanujan graph, we will take advantage of the special structure of the hike graph (when the walk length is large but not too large) in which most of its vertices have degree exactly . These degree vertices are particularly simple to handle in a DFS traversal. For them, we only need to store the next neighbor out of possibilities in (except possibly for the first step). In a sequence of backtrackings, if the top of the DFS stack is a degree vertex we know that we are done processing it since no further recursive call will be initiated from it. Then, we simply pop it from the stack. It is for the “rare” at most vertices of degree that we need to store how many extra recursive calls we issue from and a tuple of additional next neighbors . The total number of such encodings is at most

which combined with the same previous way of encoding a hike given its graph results in a total number of hike encodings of of at most

By choosing sufficiently small and taking sufficiently large, we obtain after taking the -th root

indeed leading to a near-Ramanujan bound for lifts as large as in Theorem 1.1.

Now we briefly explain how to handle the union bound to ensure that is simultaneously small for all non-trivial characters (in the decomposition of creftype 2.4). This union bound is standard when using the trace power method, what is relevant is the trade-off between lift size and walk length. To obtain a high probability guarantee from a guarantee on expectation, it is standard to consider larger walk lengths from which concentration follows from a simple Markov inequality. More precisely, if for some function , , then by Markov’s inequality,

Therefore, for sufficiently large, we can union bound over all characters and obtain similar bounds as before. As alluded above, this lower bound on the length of the walk depending on the lift size is the reason why we are led to consider much longer walks. To conclude this proof sketch, we need to replace a random signing by a pseudorandom random one. As in [MOP20], we use -biased distributions but suitably generalized to abelian groups, e.g., the one111111For our application, it suffices to have the support size of the -biased distribution polynomial in . by Jalan and Moshkovitz in [JM21]. We may be taking very large walks on the base graph , so the error of the generator needs to be smaller than , where can be as large as . We note that as long as the degree is a constant this quantity is at most a polynomial in the size of the final lifted graph since walks of length suffice for any lift size up to full extent of , for which abelian lifts can be expanding.

The above argument covers Theorem 1.1, namely, the sub-exponential and moderately-exponential abelian lift sizes. The “exact” exponential regime of Theorem 1.3 relies on an elegant converse of the expander mixing lemma by Bilu and Linial [BL06]. Since this regime is simpler, we defer the details to Section 6, where it is formally presented.

4 A New Encoding for Special Walks

In this section we will count the total number of singleton-free hikes of a given length on a fixed graph, . We split the count into two parts. First, we count the number of possible hike graphs and then, for a given hike graph , we count the number of hikes that can i.e., yield on traversal. Each of these counts is via an encoding argument and therefore we have two kinds of encoding. One for graphs and the other for hikes. In the first part of the section we give two ways of encoding graphs, and in the other half, we encode hikes. Since the first section is a general encoding for subgraphs, we relegate formal definitions related to hikes to a later section.

4.1 Graph Encoding

Let be a subgraph of a fixed -regular graph . We wish to encode in a succinct way such that given the encoding and , we can recover uniquely. We will give two ways of encoding . The first one will be generic that works for any subgraph of a -regular graph. The second encoding takes advantage of the special sparse structure (not too many vertices of degree greater than two). We assume that we have an order on the neighbors of every vertex, and thus, given , we can access the neighbor of efficiently.

We will do this by encoding a DFS based-traversal of it from a given start vertex . Here, we really need our DFS traversal to be optimal in the sense that the number of times each edge is traversed is at most two and not any higher. We, therefore, include precise details of our implementation in  Appendix B.

To reconstruct the graph, we reconstruct the traversal and so need access to two types of data before every step - (1) Is this step recursive or backtracking (2) If it is a recursive step, then which neighbour do we recurse to.

To determine the neighbor of the current vertex we need to move to in a recursive call we need to specify one out possibilities (except in the first step which has possibilities). This can be specified by a tuple of indicating the neighbor. For a backtracking step, we just pop the stack and thus don’t need any additional data.

We use two ways to figure out whether a step is recursive or backtracking. The direct way is to just record the sequence in a binary string of length . A neighbour of is called recursive if the edge is visited by a recursive call from . A simple observation about backtracking sequences is that – It starts when we encounter a vertex that has already been visited or we reach a degree one vertex and ends when we see a visited vertex that has unvisited recursive neighbors. Therefore, we store a string in which denotes the number of recursive neighbors of the visited vertex. To summarize,
 
:

  1. Starting vertex

  2. A sequence of degrees

  3. Either (Encoding I) or (Encoding II)

Algorithm 4.1 (Unpacking Algorithm for GraphEnc).
Input
Output
  • [topsep=0.3ex,itemsep=0.4ex,parsep=0.7ex,label=,leftmargin = 0.4cm]

  • Initialize DFS stack with

  • Initialize

  • Initialize     // count visited vertices, recursive steps and total steps

  • Initialize

  • While :

    • [label=,parsep=0.7ex]

    • Let be the top vertex on the stack

    • If (recursive):

      • [label=]

      • Assign to be neighbor of and increment

      • Add edge to

      • If is unvisited :

        • [label=]

        • Add vertex to

      • Else if is visited, increment     // Next step is backtracking

    • If (backtracking):

      • [label=]

      • pop(S)

  • return

Algorithm 4.2 (StepType).
Input
Output

Note - The subroutine to detect the type of step depends on the encoding string .

  • [topsep=0.3ex, itemsep=0.4ex,parsep=0.7ex,label=,leftmargin = 0.4cm]

  • If is from Encoding I, return

  • Else, let

    • [label=]

    • If     //Check if there are any remaining recursive neighbours

      • [label=]

      • Decrement

      • return

    • Else, return

4.1.1 Counting the encodings

For the first kind of encoding of type, we have strings of length over . The second encoding might seem wasteful in general but it is much better when the graph has special structure that our hike graph will satisfy. We first note that for any vertex , the number of recursive neighbours (or if ).

Definition 4.3 (Excess).

The excess of is defined as .

Definition 4.4 (Excess Set).

We define a vertex to be an excess vertex in if and we define the excess set to be the set consisting of such vertices i.e

Lemma 4.5.

Let be a fixed -regular graph on vertices. The total number of connected subgraphs of having at most edges is at most

Moreover, if is constrained to have at most two vertices of degree one121212We will see later that hike graphs satisfy this strange property and , the count is at most

Proof.

We first fix the number of edges as and we will then sum up the expression for . Algorithm 4.1 unambiguously recovers the graph and therefore the number of possible graphs can be counted by counting the number of possible inputs. The number of degree sequences and start vertices are . The number of -strings of encoding I are . Therefore for a given , we have and summing this gives the first claim.

In the second case, the key idea is that for every vertex (except the start) of degree , must be . Since , almost all of the string is filled by .

We first pick the number of vertices, say . There are at most choices for this. Then, we let the number of excess vertices be . Summing over all possible , the number of -strings of length is .

Here the first term counts the ways or having or up to two vertices of degree 1, the second counts the ways to choose the excess vertices and the third counts the number of their recursive neighbours. In the last inequality we used that .

The complete expression for the number of graphs would then be

 

4.2 Bounding Singleton Free Hikes

Following [MOP20], we make the following useful definitions,

Definition 4.6 (Singleton-free hikes).

A -hike is a closed walk of -steps131313That is sequence of such that and in in which every step except possibly the is non-backtracking. A hike is singleton-free if no edge is traversed exactly once.

Definition 4.7 (Bicycle free radius [Mop20]).

A graph is said to have a bicycle-free radius at radius if the subgraph of distance- neighborhood of every vertex has .

We will work with singleton-free hikes in this section. A singleton-free -hike on defines a subgraph such that there at most two vertices of degree (the start vertex and the middle vertex) and the number of edges is at most as every edge is traversed at least twice. The goal now is to count the possible number of singleton-free -hikes that yield a fixed subgraph . Having access to , we will need to encode the hike in a way similar to the encoding of stale stretches in [MOP20].

HikeEnc:

  1. ,where and is the bicycle free radius of .

  2. . Here, denotes the number of times the unique cycle (in the neighborhood of ) is to be traversed and the sign indicates the orientation. Since each stretch is of length and each cycle of length at least we can traverse a cycle at most times.

Claim 4.8.

For any graph that is bicycle free at radius , the number of simple singleton-free -hikes that have as their hike graph is at most .

Proof.

Follows from the possible values the encoding HikeEnc can take.      

We use a generalization of the bound of Alon et al. [AHL02] on the excess number (originally involving the girth), extended to bicycle-free radius in [MOP20].

Theorem 4.9.

[MOP20, Theorem 2.13] Let be a bicycle free graph of radius . Then

Corollary 4.10.

Let be a regular graph on vertices bicycle free at radius . Let be a subgraph with at most two vertices of degree one on vertices where for some . Then,

Lemma 4.11.

Let be a regular graph, with , on vertices bicycle free at radius . Then, the total number of singleton free -hikes on is at most

If we assume that , then it is at most

Proof.

Any singleton-free -hike defines a connected graph with at most edges and therefore at most vertices. If there is no backtracking step then all vertices except the start have degree at least two. Else, the end point of one of the backtracking step may have degree . Thus there are at most 2 vertices of degree one. When is unbounded, we use the bound from the first encoding i.e. Lemma 4.5 and combine it with the number of possible hikes on this from creftype 4.8 to get

The assumption on lets us use Corollary 4.10 which when combined with Lemma 4.5 gives us the bound on the number of such graphs as . Combining with the number of possible hikes on this from creftype 4.8, we get the total number of singleton-free -hikes bounded by

 

5 Instantiation of The First Two Main Results

In this section, we will use the bound on singleton-free hikes obtained in the last section to bound the eigenvalue of the lifted graph. We first handle non-singleton free hikes and show that they can be easily bounded by the -biased property of the distribution of the signings. We then formalize the construction by instantiating it using an expander from MOP having large bicycle-free radius and then bring the bounds together.

5.1 A Simple Generalization of The Trace Power Method in MOP

We now show that the problem of bounding the spectral radius of the signed non-backtracking operator reduces to counting singleton-free hikes. This reduction is a straightforward generalization of the argument [MOP20, Prop. 3.3] for to any abelian group.

Let (as defined in Definition 2.2) be the signed non-backtracking operator with respect to a signing and a non-trivial character and denote its spectral radius. The goal is to bound the largest eigenvalue of . The trace method is the name for utilizing the following inequality,

The signing is drawn from some distribution and we wish to show via the probabilistic method that there exists a signing in for which is small for any set of non-trivial characters . We will use a first-order Markov argument and therefore wish to bound . Writing it out we get,

Notice that don’t appear in the term and so we define as the multiset of all tuples appearing in the support of this summation. We denote each term in the summation above by where . It follows directly from the definition that each defines a -hike. Also observe that, any tuple appears at most times as given a tuple , we have at most choices for each . Let denote the singleton-free hikes in . We can split where

We now define -biased distributions that will be the key pseudorandomness tool.

Definition 5.1 (Bias).

Given a distribution on a group and a character , we can define the bias of with respect to as and the bias of as , where the maximization is over non-trivial characters.

Lemma 5.2.

Let be an -biased distribution and let be a singleton-hike i.e. there is an edge that is travelled exactly once. Then, .

Proof.

Let the set of distinct edges in be and let edge be travelled times where takes the sign into account.141414 Let appear times in the first steps and times in the next steps. Similarly let which is the reverse direction of appear times in the first steps and times in the next steps. Then, .Let be the edge traversed exactly once. Then, . Now, we can rewrite and it can be extended to a character on . Since , this character is non-trivial and the claim follows from the -biased property.      

Lemma 5.3 (Analog of Corr. 3.11 in [Mop20]).

Let be a -regular graph on -vertices, be a fixed constant, be a parameter, be an abelian group and be an -biased distribution such that .

Assume that the number of singleton-free -hikes is bounded by . Then for any non-trivial character of , we have that except with probability at most over , where .

Proof.

By the decomposition above, we have . As each term in the expression is of the form and as remarked earlier, all the characters are roots of unity so . Thus,