Convergence of Maximum Bisection Ratio of Sparse Random Graphs

We consider sequences of large sparse random graphs whose degree distribution approaches a limit with finite mean. This model includes both the random regular graphs and the Erd\"os-Renyi graphs of constant average degree. We prove that the maximum bisection ratio of such a graph sequence converges almost surely to a deterministic limit. We extend this result to so-called 2-spin spin glasses in the paramagnetic to ferromagnetic regime. Our work generalizes the graph interpolation method to some non-additive graph parameters.


Introduction
The interpolation method is used in a remarkable paper by Guerra and Toninelli [5] to prove the existence of an infinite volume limit of thermodynamic quantities. In this method, a system of size n is compared, by a sequence of interpolating systems, to a pair of independent systems of size n 1 and n 2 , where n 1 + n 2 = n. If, at each step of the interpolation, the parameter of interest increases, then the parameter is subadditive in n, and therefore converges when divided by n.
Bayati, Gamarnik, and Tetali [1] adapted this technique in a combinatorial setting as graph interpolation. Using graph interpolation, [1] proved that in both the sparse Erdös-Renyi and d-regular random graph models, several graph parameters, including independence number and maxcut size, converge when divided by n. Gamarnik [4] showed an analogous result for log-partition functions in the context of right-convergence of graphs, and found that the subadditivity required for graph interpolation follows from a concavity property of the graph parameter.
In a recent synthesis, Salez [8] further generalized these results by identifying the properties of these parameters that permit interpolation; Salez proved that an interpolation argument succeeds whenever the graph parameter satisfies additivity, Lipschitz, and concavity conditions. Moreover, [8] generalized the d-regular random graph model of [1] to graphs with arbitrary degree distribution generated by a configuration model.
The interpolation arguments in the literature all depend on an additivity property of the graph parameter -that if G is the vertex-disjoint union of graphs G 1 , G 2 , the graph parameter f satisfies f (G) = f (G 1 ) + f (G 2 ). While many graph parameters of interest, such as independence number, maxcut, K-SAT, and log-partition functions all have this property, other graph parameters, such as maximum bisection, do not.
In this paper, we show that the maximum bisection parameter in the arbitrary degree sequence model converges when divided by n. The random regular graph case of our result resolves an open problem on spin glasses [6,Problem 2.3]. The analogue of this problem for Erdös-Renyi random graphs was resolved in an unpublished result of Gamarnik and Tetali; this result is also implied by our result.
We then consider a type of p-hybrid bisections for p ∈ [0, 1], interpolating between the maximum and minimum bisections. These are the maximum bisections of the "2-spin spin glass" model studied by Franz and Leone in [2], where the parameter p determines the ferromagnetism of the system. We show that for p ≥ 1 2 , the p-hybrid bisection in the arbitrary degree sequence model also converges when divided by n. In other words, the maximum bisection of the 2-spin spin glass model has a scaling limit in the paramagnetic to ferromagnetic regime.
The key idea allowing us to extend the results in [1] and [8] to maximum bisection and maximum p-hybrid bisection, which are not additive, is to consider (A, B)bisections, bisections that also bisect two given sets A, B that partition V (G). This added constraint allows us to decompose a system into two parts, an operation that previously depended on additivity. By showing a form of subadditivity on maximum (A, B)-bisections, we can show subadditivity on maximum bisection and establish the existence of a scaling limit.
Acknowledgements. The author gratefully acknowledges Mustazee Rahman for many insightful conversations, and for bringing much of the relevant literature to the author's attention. The author also thanks the MIT Math Department's Undergraduate Research Opportunities Program, in which this work was completed.

Preliminaries
2.1. Random Graphs with Given Degree Sequence. Throughout this paper, we will work with finite, undirected graphs, where loops and multiple edges are allowed.
We will work with the following random graph model.  We let G d denote the distribution of G[m], where m is a uniformly random complete matching on H d . Note that when d is a constant function with value r, G d is the random r-regular graph model, and when d is sampled from the degree distribution of an Erdös-Renyi random graph, the doubly-random G d is the corresponding Erdös-Renyi random graph model.
We say a sequence {d n : [n] → N} n≥1 converges in distribution to a probability measure µ : N → [0, 1] with finite mean µ if for all k ∈ N, The results in this paper are concerned with families of random graphs {G dn } n≥1 , where each G dn is sampled independently, and where the degree functions d n converge in distribution to a measure µ with finite mean.

Graph Parameters.
A graph parameter is a real-valued, isomorphism-invariant function on graphs. Given a graph parameter f and a graph G, define ∆ G,f as the matrix given by We say a graph parameter f is additive if when G is the disjoint union of G 1 and G 2 . We say f is 1-Lipschitz if for all G, and all i, j ∈ V (G), where is the all-1 vector on V (G). The most general result on graph parameters is due to Salez.
[8] Let f be an additive, κ-Lipschitz, concave graph parameter, and let {d n } n≥1 converge in distribution to a measure µ with finite mean. Then, the sequence of independent samples converges almost surely to a limit Ψ(µ) as n → ∞. Moreover, the scaling limit exists and equals Ψ(µ).
Bayati, Gamarnik, and Tetali [1] showed, before Salez, that the scaling limit (2.8) exists for the max-cut, independence number, K-SAT, and not-all-equal K-SAT parameters, in the random r-regular graph and Erdös-Renyi random graph models. As these parameters all satisfy the hypothesis of Theorem 2.1, and the random regular graph and Erdös-Renyi random graph models are special cases of the arbitrary degree sequence model, this result is a consequence of Theorem 2.1.

Graph Bisections.
Define the maximum bisection of a graph G by where e(V 1 , V 2 ) is the number of edges between V 1 and V 2 in G. Observe that the maximum bisection is not additive, and therefore Theorem 2.1 does not apply.
The first result of this paper is: where each G dn is sampled independently, converges almost surely as n → ∞. Moreover, the scaling limit exists.
Whether the same result holds for the minimum bisection graph parameter is an open problem. In fact, the random regular graph case of this problem is implied by the following stronger conjecture.

Conjecture 3.2. [9]
Let M C and mB denote, respectively, the max-cut and minbisection parameters, and let G(n, r) be a random r-regular graph on n vertices. Then, where |E| = 1 2 nr is the number of edges in G(n, r).

Hybrid Bisections.
We define the p-hybrid bisection HB p of a graph G as follows. Let Ω be a labeling of the edges of G, with each edge independently labeled +1 with probability p, and −1 with probability 1 − p, and let G(Ω) denote G with the labeling Ω. In the statistical physics literature (cf. [2], [3], [7]), the graph G(Ω) is a 2-spin spin glass, with the parameter p determining the system's magnetism: the system is ferromagnetic at p = 1, paramagnetic at p = 1 2 , and antiferromagnetic at p = 0.
We define where the expectation is over the randomness of Ω. Note that when p = 1, a p-hybrid bisection is a max bisection, and when p = 0, a p-hybrid bisection is a min bisection. Our main result is: 1 2 , and let {d n } n≥1 converge in distribution to a measure µ with finite mean. Then, where each G dn is sampled independently, converges almost surely as n → ∞. Moreover, the scaling limit exists.
As Theorem 3.1 is a special case of Theorem 3.3, the rest of this paper will be devoted to proving Theorem 3.3.
Remark 3.4. Let α ∈ (0, 1). We can define an α-cut of G as a cut that partitions V (G) into the ratio α : (1 − α). In particular, a bisection is a 1 2 -cut. Theorems 3.1 and 3.3 remain true when "bisection" is replaced by "α-cut," and their proofs are analogous. where Ω is defined as before. Note that both M B A,B and HB A,B p are graph pseudoparameters, and that they are additive in the following sense. When there are no edges from A and B, will imply that HB p is subadditive.

Graph Interpolation. Fix a partition A, B of [n] and a function d : [n] → N.
Say that an edge in a matching m ∈ H d is an A-edge if both of its endpoints are in A; define a B-edge analogously. Say an edge is a cross-edge if it has an endpoint in each of A and B.
For a graph pseudo-parameter g and feasible (α, β, γ), define be the total degree of the sets A and B. The following result is the proof of Theorem 3 of [8]; while this was proved for graph parameters, its proof extends directly to pseudo-parameters.

Proposition 4.2. [8]
Let g be a graph pseudo-parameter obeying the following conditions.
Then, for any γ ≤ min(d(A), d(B)), , γ is the expected value of g(G) for G sampled from G d , conditioned on G having γ cross-edges.
Thus, by taking a weighted average of (4.10) over γ of the correct parity, we get the following result. Corollary 4.3. Let g be a graph pseudo-parameter obeying (4.8) and (4.9). Then,
We will first show that for any p ≥ 1 2 , HB A,B p satisfies the hypotheses of Corollary 4.3. For any feasible (α, β, γ) and (α ′ , β ′ , γ ′ ), Proof. It suffices to prove this for the case when (α, β, γ) and (α ′ , β ′ , γ ′ ) differ by 1 in exactly one coordinate. Let m be uniformly sampled from M(α, β, γ), and m ′ be obtained from m by adding a uniformly random A-edge. The argument for B-edges and cross-edges is analogous.
Proof. We will prove a stronger claim: for any m ∈ M(α, β, γ), and any labeling Ω of G[m], If we add a (+1)-labeled edge e + to G[m](Ω), its maxcut increases if and only if e + crosses some cut in C * ; equivalently, the endpoints of e + must be in different equivalence classes.
The first main observation is that and analogously So, The second main observation is that the left-hand side is a linear function of p, so verifying (5.12) at p = 1 and p = 1 2 is sufficient. At p = 1, (5.12) follows from: At p = 1 2 , (5.12) follows from: We are now ready to prove Proposition 4.1.
Proof of Proposition 4. , 0 have no cross-edges. So, an optimal (A, B)-bisection of these graphs is the sum of an optimal bisection of A and an optimal bisection of B. Thus, as desired.
6. Proof of Theorem 3.3 On the space of probability measures on N with finite mean, define the Wasserstein distance We will use the following result from [8].
Throughout this proof, let G IID µ,n denote the random graph on n vertices, where each vertex's degree is sampled i.i.d. from the distribution µ. Fix a distribution µ on N with finite mean. By averaging the above inequality over d whose values are sampled i.i.d. from µ, we have (6.4) E HB p (G IID µ,|A| ) + E HB p (G IID µ,|B| ) ≤ E HB p (G IID µ,n ) + ψ where we have used Jensen's Inequality on the concavity of ψ. Note that ψ( 1 2 µn) = o n 2/3 . By Fekete's Subadditivity Lemma, this implies that the scaling limit (6.5) lim n → ∞ 1 n E HB p (G IID µ,n ) exists. Let this limit equal Ψ(µ).
Since HB p is Lipschitz, Proposition 6.1 applies. By setting d = d n , sampling each d ′ (1), . . . , d ′ (n) uniformly from µ, and taking the limit as n → ∞, we get as well. Moreover, as HB p is Lipschitz, Azuma-Hoeffding's inequality implies the concentration inequality .
Since d converges in distribution to µ, the Borel-Cantelli Lemma implies the almostsure convergence The result follows.