Sparse block-structured random matrices: universality

We study ensembles of sparse block-structured random matrices generated from the adjacency matrix of a Erdös–Renyi random graph with N vertices of average degree Z, inserting a real symmetric d × d random block at each non-vanishing entry. We consider several ensembles of random block matrices with rank r < d and with maximal rank, r = d. The spectral moments of the sparse block-structured random matrix are evaluated for N→∞ , d finite or infinite, and several probability distributions for the blocks (e.g. fixed trace, bounded trace and Gaussian). Because of the concentration of the probability measure in the d→∞ limit, the spectral moments are independent of the probability measure of the blocks (with mild assumptions of isotropy, smoothness and sub-Gaussian tails). The effective medium approximation is the limiting spectral density of the sparse block-structured random ensembles with finite rank. Analogous classes of universality hold for the Laplacian sparse block-structured ensemble. The same limiting distributions are obtained using random regular graphs instead of Erdös–Renyi graphs.


Introduction
The subject of this note is an ensemble of sparse real symmetric block matrices, already studied in recent years: a generic block matrix of the ensemble is generated from a N × N matrix, see equation (1.1), after replacing each entry with a d × d random matrix α ij X ij , i, j = 1, . . . , N, as depicted in equation (1.3), then obtaining a Nd × Nd random block matrix.
The set of random variables {α ij } have distribution probabilities to make the random block matrix sparse, the set of blocks {X ij } are real symmetric random matrices of dimension d × d. Our goal is the limiting, N → ∞, spectral distribution of the adjacency random block matrix A Nd in equation (1.3). The probability distribution of the random matrices {X ij } and their relation to the universality properties of the spectral distribution of the ensemble A Nd are the central topic of this note.
Sparse random matrices are a difficult subject and many well known tools of random matrix theory cannot be used. Possibly the most studied model of sparse random matrix ensemble is the adjacency matrix of a non-oriented random graph with N vertices of average vertex degree or average connectivity Z, 3 The set of N(N − 1)/2 random variables {α i,j }, i > j, is a set of independent identically distributed random variables, each one having the probability density The random matrix ensemble of equations (1.1) and (1.2), was considered a basic model of disordered system in statistical mechanics; it was analyzed for decades from the early days of the replica approach [1] up to more recent cavity methods [2][3][4].
More pertinent to this paper, the moments of the spectral density of the limiting (N → ∞) adjacency matrix A N in equation (1.1) were carefully studied and recursion relations for them were obtained [5,6]. Remarkably, the knowledge of all the spectral moments, at least in principle, was not sufficient to obtain the spectral density, possibly because, as it is indicated in [11], the moments correspond to a class of walks on random trees that was not enumerated.
In recent years, an ensemble of sparse block-structured random matrices was considered, where the entry A i,j of the random matrix is a real symmetric d × d random matrix X i,j , [7][8][9][10][11]: α 1,2 X 1,2 α 1,3 X 1,3 . . . α 1,N X 1,N α 2,1 X 2,1 0 α 2,3 X 2,3 . . . α 2 The generic block X i,j may be considered a matrix weight associated to the non-oriented edge (i, j) of the graph. One may say that the set of random variables α i,j = α j,i encodes the modular structure of the non-oriented graph; in this case, see equation (1.2), it is the Erdös-Renyi random graph, with average vertex degree (or connectivity) Z.
It seems likely that the understanding of the spectral properties of sparse block-structured random matrices of equation (1.3) has relevance on the dynamics of classes of networks. A prominent property of the theory of complex networks is their modular structure and decades of studies were devoted to methods to detect the modular structure from network data [12].
Networks in which modules or communities have a sparse interaction with nodes belonging to different communities represent a feature of many real systems. Furthermore in some models in ecology (plants-pollinators-insects) [13,14] and in deep learning [15], the interactions among the nodes belonging to the same community are absent or irrelevant; this property is called disassortative and the associated graph of interactions is multipartite.
The ensemble of sparse block-structured random matrices we study has N communities (or modules), each one of dimension d which may be constant or going to ∞, in the limit N → ∞, still keeping the ratio d/N → 0. This differs from more usual models of networks where the number of communities is small. Furthermore nodes of one community have a sparse interaction with nodes of a different community, Z is the average number of j-communities which interact with the i-community. No node of the i-community interacts with a node of the same community. The graph corresponding to the interaction of the nodes is multipartite; it corresponds to a complex networks with totally disassortative structure. Finally, the interactions between communities are block matrices of rank r [16]. Explicit results are obtained in the limit N → ∞, d → ∞, Z → ∞, d/N → 0, d/Z fixed, r fixed. In this limit, the dominant class of random walks correspond to non-crossing partitions, that is sequential partitions where the subsequences · · · a · · · b · · · a · · · b are absent) [17][18][19].
We are led to these structures for the mathematical tractability and we hope they may be useful. Some understanding of the role of the diagonal blocks in the adjacency random block matrix, which are absent in our work, may be guessed from random matrix theory. In a Wigner matrix of dimension n, the independent off-diagonal entries are often normalized by a factor 1/ √ n. If the diagonal entries have the same normalization, their probability distribution is irrelevant in the n → ∞ limit. However, if they have an explicit normalization dominant on the off-diagonal entries, the limiting distributions are different [20]. A remark on the evaluation of the limiting moments, in presence of diagonal blocks, is provided in section 3.1.
Our goal is the evaluation of the limiting moments µ p of the sparse block-structured random matrix A in equation (1.3), (Tr is the trace in the Nd-dimensional space, tr is the trace in the d-dimensional space, from now on we omit the suffix in A Nd ), Moments µ p are evaluated in terms of weighted paths with p steps on a complete graph. In the present case the weight of any path is the trace of the product of the matrices associated to the edges. Because of the probability distribution of the {α i,j } random variables, equation (1.2), in the limit N → ∞, only closed paths on trees contribute to the limiting moments, [5,6].
This holds true regardless the probability distribution of the random blocks X i,j and their finite dimension d.
For a closed path on a tree, every edge is traversed an even number of times, the limiting moments of odd order vanish. The weight of a path is the product of the weights of the traversed edges; for a closed walk of 2 p steps, the number l of distinct matrices occurring in the weighted path is 1 ⩽ l ⩽ p.
Since the blocks are i.i.d. the identification of a block X i,j in a product, averaged over the joint probability distribution of the factors, is irrelevant. In order to collect equal terms in the moments, it is useful a relabeling of the products of the blocks that only records if the blocks are equal or different to other ones in the product. For instance: (1.5) As the order of the spectral moments increases, the number of relevant products also increases. In appendix A of [11], we listed the analytic contributions up to µ 10 . It holds for any probability choice for the random matrices X i,j and any dimension d, provided that the random block matrix ensemble A has the Erdös-Renyi structure and the matrix blocks are i.i.d.
In the next section we consider ensembles of random block matrices of fixed rank r, parametrized by r random vectors, in which the vectors have uniform distribution on a sphere, uniform distribution in a ball, or a Gaussian measure; then we consider random block matrices of maximal rank, with fixed trace, bounded trace and Gaussian distribution.
We show that the d → ∞ limit with Z d limit of a fixed rank model is the same regardless the class of probability distributions considered for the vectors, and that the d → ∞ limit of the maximal rank model is the same regardless the considered class of probability distributions for the blocks. The proof is analogous in all these cases. This universality can be traced back to known properties of high-dimension probability; we recall the general conditions which lead to the concentration of probability measures and the universality properties of the spectral moments.
In the case of the finite rank models the spectral distribution is the effective medium distribution with parameter t = rZ d .

Expectations
The spectral moments of equation (1.4) are polynomials in the Z variable, where, in an addend, the power of Z is the number of distinct blocks. For instance The highest power of Z in the polynomial multiplies the contribution of the Wigner paths, where each traveled edge is traveled exactly twice. The expectations of each term will be shown to be independent of the probability measure generically (for sub-Gaussian measures), in the d → ∞ limit.

Rank one random blocks X
We begin considering an ensemble of d × d real symmetric random matrices X i,j independent (except for the symmetry We consider three probability measures for the random vector.
• Uniform probability on the sphere Then the random matrix X is a projector into a one-dimensional space. This probability distribution for the set {X i,j } is unusual in random matrix literature 4 . In the limit d → ∞, with t = Z/d fixed, moments of all orders are evaluated [9], the limiting resolvent is solution of a cubic equation, previously obtained in a different model and different approximation, the effective medium approximation [21]. • Uniform probability on the ball where Θ is the step function.
Let us consider the trace of the product of P random matrices X, where r 1 = m 1 + m 2 + . . . is the sum of the powers of the matrix X 1 in the product, r 2 = n 1 + n 2 + . . . is the sum of the powers of the matrix X 2 in the product, etc s is the number of distinct matrices in the product, P = s k=1 r k , each r k is an even integer: 3 . . . . Expectations of traces of products of the random matrices {X} with the three probability measures equations (2.2)-(2.4) are straightforward. We merely quote here two equations which compare expectations of the uniform probability on the ball with the Gaussian measure and the uniform probability on the sphere with the Gaussian measure. They are valid for any R and d and may be obtained by Laplace transform, as it is described in the third subsection. The method and the results are a simple multi-matrix generalization of well known fact, beginning with Rosenzweig and Bronk [22], later elaborated and generalized [23][24][25][26], the three probability measures, P δ , P Θ , P Gauss , obtain the same expectations for any multi-matrix product in the d → ∞ limit. The fact that the volume probability distribution P Θ obtains the same expectations of the surface probability distribution P δ , is the most simple example of concentration of the probability measure in spaces of high dimension. Any probability measure for the components of the random vector, such that the probability on the tails is bound by a Gaussian function (the sub-Gaussian distributions) would also lead to the concentration of the measure in the d → ∞ limit [27,28].
With the probability P δ , it was proved in [8] for moments of low order and in [9] for moments of every order that in the limit d → ∞ and Z → ∞ with fixed ratio t = Z/d the products of multi-matrices associated with non-crossing partitions have finite limit whereas the products of multi-matrices associated with crossing partitions vanish. This allows the determination of the non random spectral distribution of the matrix A in this limit, the resolvent obeys a cubic equation, sometimes called effective medium approximation, quoted in equation (2.12). In the same limit, it was shown that the spectral density of the associated random block Laplacian is the Marchenko-Pastur density, quoted in equation (2.13). Our work was confirmed by an independent derivation; indeed Dembczak-Kolodziejczyk and Lytova [29] studied the same ensembles of random block matrices with the method of resolvent, or Stieltjes transform. Their theorems (1.4) and (1.5) confirm our statements on the limiting spectral distribution for the adjacency block matrices and for the Laplacian block matrices. These derivations by different techniques (resolvent or cavity) are very valuable because in our moment's method, the limit N → ∞ is taken first. This leads to consider only closed paths on tree graphs. The further parameters, d, Z, may be fixed constants or large. Of course when we consider these parameters in the limit d → ∞ and or Z → ∞, they must be much less then N, for consistency. But our method does not provide a quantitative comparison 5 .
We may now assert, by the measure concentration, that infinitely many probability distributions of the random blocks {X} lead, in this limit, to the same spectral distribution for the sparse random block matrix A, the crucial feature being the blocks having finite rank, as it is shown in next subsection.

Random blocks X of rank r
The derivations related to the random blocks X of unit rank, may be generalized to random blocks X of any rank, provided it remains finite, in the limit d → ∞, Z → ∞, t = Z/d fixed.
We consider an ensemble of d × d real symmetric random matrices X i,j independent (except for the symmetry We consider the joint probability measure for the random vectors This probability measure will exhibit the consequences of the finite rank r and the large dimensionality d → ∞. Integration over sets of orthogonal vectors is mentioned in chap. 21 of [30] and in the paper [31]. In the limit d → ∞ the expectations are evaluated in a way completely analogous to the case of the probability density in equation (2.2) and in the paper [9], by distinguishing contributions to the moments associated to non-crossing partitions from those associated to the crossing partitions. For instance, all the terms, except one, in equation (2.1) are associated to non-crossing partitions. They are evaluated by repeated use of factorization 6 . For instance The only term in equation (2.1) associated to crossing partitions may be neglected, in the d → ∞ limit because it involves higher number of internal products < ⃗ v|⃗ w > between distinct vectors All the non-crossing contributions may be evaluated, as in the case of rank one blocks. The spectral moments of the sparse random blocks matrix A, in the limit d → ∞, with the ratio Z/d fixed, reproduce the moments of the effective medium approximation, with the parameter t = rZ/d, We describe now a similar ensemble of random blocks X each one made of r independent random vectors ⃗ v a ∈ R d , without the constraint of being an orthogonal set. Since for d → ∞ any two random vectors tend to be orthogonal, one expects that this ensemble, for d → ∞, reproduces the result of the ensemble of orthogonal vectors and it is easier for simulations.
We now define the d × d blocks X of rank r and a factorized joint probability distribution We begin by examining the expectation of the trace of the power of a single block in the limit d → ∞ The term represents products of pairs of different vectors times products of pairs of the same vectors. We recall (see for instance equations (9)-(11) in [9]) that expectations of scalar products vanish in the limit d → ∞. Then (2.10) A mild generalization of the second part of proposition 1 in [9]. allows to prove that the expectation of the trace of a product of any number of blocks, which correspond to a non-crossing partition, gives the contribution t m , where m is the number of distinct blocks. 6 In the paper [11], where the blocks {X j } are member of the Gaussian Orthogonal Ensemble, we recalled the relevance of factorization to evaluate expectations of multi-matrix products for d finite or infinite. If the entries of the pair of matrices A, B, are stochastically independent Next the isotropy relation Indeed in a non-crossing term, there exist at least a block X j appearing as (X j ) k in one position and not elsewhere. Averaging over the vectors of the block X j we find, as in equation (2.9) The resulting product of blocks is still non-crossing, the steps may be repeated with a new block X i , until one is left with a single block, like in equation (2.10).
Finally it remains to show that the average of the trace of a product of blocks, which is crossing, is negligible in the d → ∞ limit.
A generic product of blocks, which is crossing, may contain one or more blocks X j appearing as powers (X j ) k in one position of the product and nowhere else. By repeating the steps of equation (2.11) one eventually obtains the expectation of a reduced product of blocks, still crossing, where no block occurs in a single position. Its asymptotic behavior in the d → ∞ limit, is the same of the original product.
Every block (X i ) m i in the reduced product occurs at least in two non consecutive positions and its random vectors form scalar products with a larger number of distinct vectors than it would happen in case of consecutive positions. Then all crossing contributions may be neglected in the limit d → ∞, Z → ∞, with fixed ratio Z/d. A detailed evaluation of the contributions is done in the first part of proposition 1 in [9].

Remarks
The spectral moments µ 2k of the sparse random block matrix, with R = 1, in the limit d → ∞, Z → ∞, with fixed ratio Z/d, are those of the effective medium approximation. The generating function of the moments, g(z) = k=0 µ 2k z 2k+1 , is solution of the cubic equation The spectral moments of the sparse random block Laplacian matrix, in the same limit are the spectral moments of the Marchenko-Pastur distribution ρ MP (x) We performed a few simulations with random block matrices with blocks with dimension d between 2 and 16, both for the adjacency and the Laplacian sparse block ensembles, with blocks of rank r = 1 with the measure with the Gaussian probability (see figure 1), and with blocks of rank r = 2, with independent random vectors with uniform probability on the sphere (see figure 2). They support the analysis and the conclusions of this note. The spectral distribution approaches the limiting distribution as d increases, but slower than in the case r = 1 with uniform probability on the sphere and same t (see figure 2 in [8]). For sake of simplicity we considered in equations (2.7) and (2.8) only the fixed length probability of the random vectors. Also the bounded length or the Gaussian length may be considered. The blocks {X} are sum of r contributions, each one being of the form considered in rank one blocks subsection. The equivalence of the different measures in the d → ∞ limit is established by the method discussed there.
The fixed length probability of the random vectors considered in equation (2.8) may be generalized by associating different R a to the vectors ⃗ v (a) and the limiting resolvent of the adjacency block ensemble satisfies a polynomial equation of order higher than 3. The use of the non-crossing partition transform, described in section 4 of [11] is convenient: only paths associated to non-crossing partitions are relevant, the generating function is (2.14) Using the non-crossing partition transform, the generating functions of the moments is (2.15) We did not attempt to compute the spectral distribution in this more general case.

Random blocks X of maximum rank
We consider ensembles of random real symmetric matrices {X} of order d, with three probability measures analogous to the ones in equations (2.2)-(2.4) • Fixed trace We quote here two equations which compare expectations of the trace of a generic multi-matrix product for the fixed trace probability in equation (2.16) with the Gaussian measure in equation (2.18) and for the bounded trace probability in equation (2.17) with the Gaussian measure (2.18). They are valid for any R and d. (2.20) An outline of the derivation of equations (2.19) and (2.20) is the following. One inserts the integral representation for each of the distinct blocks, j = 1, 2, . . . , s, occurring in the integral One performs the Gaussian multi-matrix integral over the distinct blocks X j , next the integration of the z j variables in the complex plane; finally the division of the s normalization factors Z Θ lead to equation (2.19). In the same way one obtains equations (2.5), (2.6) and (2.20). They are straightforward multi-matrix generalizations of one matrix equations known long ago [22][23][24][25][26]. Analogous equations hold for complex Hermitian blocks X i,j .
Since lim d→∞ F Θ (d) = 1, lim d→∞ F δ (d) = 1, the three probability measures, P δ , P Θ , P Gauss , equations (2.16)-(2.18), lead to the same expectations for any multi-matrix product in the d → ∞ limit. At the time of the derivation of equal results for any moment, for the one matrix case, in the d → ∞ limit, it did seem related to the special analytic relations between Gaussian probability and restricted trace probabilities. Now it is understood as an example of concentration of the probability measure.

Classes of universality and other ensembles
In this note we call universality the property that the spectral moments of the sparse block-structured random matrix A converge to the same deterministic limit, in the limit: lim d→∞ lim N→∞ , regardless the probability measure of the entries of the blocks X i,j .
Concentration inequalities proved in spaces of high dimensionality lead to these results [27,28]. We merely recall the concentration inequality for the squared norm of a n dimensional random vector V which evaluates the probability that it deviates from its average. If the random vector has independent components, each with a normal distribution, the Euclidean squared norm is a sub-exponential random variable. Its deviation from the average is bound by the Bernstein inequality where c is a positive constant.
The two terms on the right side refer to small and large deviations (see chapter 3 in [28]). The inequality states that the random vector is concentrated on a thin spherical shell. It is usually said that a normal distribution is similar to the uniform distribution, in a space of high dimension.
An analogous bound exists for the norm of the n dimensional random vector. The inequalities are valid for every isotropic random vector where the components have distributions with sub-Gaussian tails.
The three probability densities P δ (x) in equation (2.16), P Θ (x) in equation (2.17), P Gauss (x) in equation (2.18) are functions of trX 2 = d i =1 (X i,i ) 2 + 2 i<j (X i,j ) 2 , which may be seen as the squared norm of a Euclidean random vector ⃗ V in a space with d(d + 1)/2 dimensions, after a trivial rescaling of the diagonal entries. For the probability density P Gauss (x) in equation (2.18), the components of the random vector are independent centered normal variables. For every d, ⟨trX 2 ⟩ Gauss = R 2 (d + 1), the sub-exponential random variable (1/d) trX 2 concentrates around a thin shell around R 2 . The equivalence of the three probability densities in the limit of large dimension of the space, which was known since the early days of random matrix theory, is now seen as a property of the probability in high dimensional spaces.
The expectation of each trace of product of blocks and their sum, all the spectral moments of the sparse block matrix A obtain limiting values independent of the probability distribution and the limiting spectral density of the matrix itself has this universality property. If the blocks X are random matrix with full rank, we do not know the limiting spectral density. If the blocks X have finite rank r ⩾ 1, the set of walks on trees on the random graph which contribute to the moments is the set associated to non-crossing partitions. In the most simple cases, with the spherical symmetry considered in this note, the limiting universal density is the effective medium approximation with a parameter t = rZ/d.
The limiting deterministic spectral density may depend on the symmetries of the joint probability distribution of the entries of the random matrices. A simple example is provided by the rank-one blocks X = |⃗ x⟩⟨⃗ x| where the random d-vector ⃗ x has uniform distribution in the cube [−R, R] d . This probability measure concentrates to the 'skin' of the cube, that is its surface.
It seems proper to consider classes of universality, since infinite probability distributions of the matrix entries, with the same rank of the random matrix, the same symmetry properties and sub-Gaussian tails of the distributions are expected to concentrate to the same joint probability distribution in the d → ∞ limit, then obtaining the same limiting expectations.
Analogous results are obtained for Laplacian sparse random blocks. Universality of the spectral moments is assured by the sub-Gaussian distribution of the entries of the blocks X. The limiting spectral function, the Marchenko-Pastur distribution, is obtained if the blocks have any finite rank r ⩾ 1, see [8,9,29] for the case r = 1 with random vectors with uniform distribution on the sphere.
In appendix A of [11] and in section VI A of [32], it was mentioned how to modify the multiplicities of the products of blocks corresponding to walks on trees, in order to obtain the moments of regular random block matrices. Here too, universality of the expectations implies that the limiting spectral density of the random regular block ensemble is not dependent on the probability distribution of the block entries (in the case d = 1 the moments are those of the Kesten-McKay distribution [33]).
This amounts to replacing, for a walk on E distinct edges, the factor Z E with a monic polynomial in Z of degree E. In the limit Z → ∞, with Z d fixed, the contribution to the moment is the same as on random Erdös-Renyi graphs.
Here too, universality of the expectations implies that the limiting spectral density of the random regular block ensemble is not dependent on the probability distribution of the block entries. In the case of blocks of finite rank r ⩾ 1, the limit d → ∞ is performed with the degree Z → ∞ and fixed ratio t = rZ/d. The limiting spectral function is then the same effective medium approximation obtained for the sparse block-structured random ensemble A. If the blocks have maximum rank, the limiting spectral density of the random regular block ensemble is universal, but not known.

Remark on adding diagonal blocks
As it was mentioned in the introduction, in random block models the diagonal blocks are often considered to be random blocks, representing the interaction within a module or community. For instance in several brain networks it seems that nodes are partitioned into densely connected communities separated by sparse inter-community connectivity [34].
If the diagonal terms are sufficiently sparse, they are not expected to change the moments in the large N limit. Let us mention how the computation of the moments is modified by the presence of diagonal blocks.
Adding diagonal terms to the adjacency block matrix, of the form α i,i X i,i , where {α i,i } is a set of i.i.d. random variables with the probability density and {X i,i } is a set of i.i.d. random blocks with probability distribution possibly different from the one of the off-diagonal blocks, does not change the moments of the random block matrix in the limit N → ∞ with Z, Z d , d fixed, in any of the models considered in this paper. The argument is very close to the one used in [9] to show that only walks on tree graphs contribute for N → ∞.
Consider the moment µ k ; where the sums are from 1 to N. Starting from the left, split the sums on an index in the cases in which it is equal to a previous index, and in an index different from the previous ones, obtaining in the end a sum of terms, each one being a sum with V ⩽ k indices different from each other: the sum on I 2 is separated in the case I 2 = i 1 and in the sum on i 2 ̸ = i 1 ; the sum on I 3 is separated in the case I 3 = i 1 and, if i 2 is present, in the case I 3 = i 2 , and in an index i 3 ̸ = i 1 (and i 3 ̸ = i 2 if i 2 is present); and so on. Since the averages are on i.i.d. random variables, the average of the addends in the sums all give the same contribution, with multiplicity the falling factorial N V . For example, the following term appears in the computation of dµ 3 Consider one of these terms, having V index sums; associate a simple graph to this term, where to each index a vertex is associated; to each distinct A i,j , with i ̸ = j, associate an edge with endpoints i, j; to each distinct A i,i associate a loop at the node i. Let there be E distinct off-diagonal blocks and E d distinct diagonal blocks. The averages of the random variables α give a factor ( Z N ) E ( Z d N ) E d . Therefore the contribution of this term to µ k goes as N V−E−1 N −E d . Since V − E − 1 ⩽ 0, if E d > 0 this term vanishes for N → ∞, with Z, Z d , d fixed. Therefore the moments of the random block matrix are not modified in this limit by the addition of diagonal terms α i,i X i,i . Replacing in the above α i,i with 1, the diagonal terms give nonvanishing contribution to the moments in the limit N → ∞.

Conclusions
By some explicit analytic evaluations and some implications of the theory of probability in high dimensional spaces, the spectral moments of ensembles of sparse block-structured random matrices are shown to be independent of the probability measure of the blocks in the double limit N → ∞ first and d → ∞ next. The evaluations crucially depend on the rank of the random blocks. Our evaluations show a crucial difference between the case of blocks with finite rank r ⩾ 1 and the case of maximal rank r = d.
It is remarkable that in the most simple formulations of ensembles of sparse block-structured random matrices, with blocks of finite rank, the limiting spectral distribution is the effective medium approximation.
The symmetries of the joint probability distribution of the random matrix entries are relevant for the concentration of the probability measure, then it seems proper to study classes of universality.
The method of asserting this universality on each contribution of the spectral moments is rather general and we commented its validity on other ensembles where the tree structure of the random graph is known to dominate the limit N → ∞. Further ensembles where loops cannot be neglected may also be addressed with this method.
The double limit considered in this note is often studied with the cavity method. Future work to elucidate the common features of the two methods would be relevant. It would be helpful for applications to the very large area of network models.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).