Spectral properties, localization transition and multifractal eigenvectors of the Laplacian on heterogeneous networks

1- Clear, understandable and original
2- Clever use of an analytically solvable case and sophisticated cavity method techniques
3- Interesting exposition of the strongly multifractal regime around the $\lambda = 0$ states of the weighted Laplacian matrix.
4- Contrast with the more traditionally localized case and the extended case make the paper accessible to those unfamiliar with more exotic phenomenology

1- A "weakly multifractal" phase for $\lambda \neq 0$ is likely to be there, but this is not really commented upon or explored.
2- What makes the results for the Laplacian distinct from those of the adjacency matrix could be more thoroughly commented upon
3- The approximation of the graph Laplacian by a weighted fully-connected graph is only tested using the average DoS. A numerical test comparing the full statistics of the LDoS (or the multifractal exponents) for actual configuration model networks and the "annealed" approximation might be a good idea.

The manuscript by da Silva, Tapias, Sollich and Metz is a nice collaboration combining ideas from the authors' previous works. Tapias and Sollich recently described the strongly multifractal states around $\lambda = 0$ for the adjacency matrix of weighted configuration model networks (notably following da Silva and Metz in using the gamma distribution of rescaled degrees). Together with da Silva and Metz, the ensemble of authors have now exploited some clever new tricks, in combination with the trusty cavity method, to gain insight into the Laplacian matrix of such networks.

The work is clear, suitably pedagogical, and gives the reader a rather intuitive picture of why we ought to expect novel behaviour of the eigenstates in this model: In the case with no off-diagonal disorder, the Laplacian has only diagonal disorder (dictated by the degree distribution). Naturally, this leads to localized states, with an average DoS dictated by the degree distribution. When all nodes have the same degree but off-diagonal disorder is allowed, there is no diagonal disorder and the states are extended, with a DoS that follows a shifted version of (what I think is) Fyodorov's result. When we have both diagonal and off-diagonal disorder, one therefore expects that there could be an interesting mix between the two extremes.

A thorough analysis is provided, showing the behaviour of the multifractal exponents, the singularity spectrum and the local and average DoS in selected cases. Particular attention is paid to the $\lambda = 0$, where one finds a novel localization transition (as a function of the degree heterogeneity) between localized multifractal states and non-ergodic delocalized states with weak multifractality. This also coincides with a transition from singular to finite DoS at $\lambda = 0$.

I would recommend publication after minor revision. I have only the following comments and queries.

1- Have the authors also considered the case of the normalized Laplacian (where the rows are divided by the degree of the node in question, such that the diagonal elements are all equal to unity -- see e.g. Ref [29] of the manuscript)? In many applications, it is the normalized version that is used, but I can imagine that the phenomenology might differ quite a lot to that of the version used here. Certainly, the DoS is different (one obtains the Wigner semi-circle in the dense limit), and in the fully localized phase with $J_1= 0$ the states would become degenerate. I of course do not recommend that the same thorough analysis be carried out for the normalized version, but a comment on this could be elucidating for readers.

2- As is also discussed in Ref. [59] in the case of sparse ER graphs, the $\lambda = 0$ case is seen to be special. It also gives the strongly multifractal behaviour mentioned here, and is related to leaves of the network (i.e. low degree nodes). In Ref. [59] however, the "weakly multifractal" phase for $\lambda\neq 0$ is also explored. Is it also present here?

3- I think it is important to clarify some of the definitions used here. It would be helpful in particular to emphasize earlier and more clearly (i.e. perhaps in Section 2) that $\tau (q) = q-1$ for all q connotes extended states, that deviations from this behaviour indicate multifractality, and that $\tau(q) = 0$ for higher moments (i.e. q greater than some lower bound) indicates localization. Please correct me if I got any of the aforementioned items wrong. Incidentally, the behaviour in Eq. (44) is typical of Anderson localization on well-connected tree-like graphs, but I believe it fits the definition of "strongly multifractal" in Evers and Mirlin Mod Rev Phys (2008) in the same way that Eq. (56) does. Do the authors agree with this assessment?

4- The approximation given in Eq. (29) is tested against the average DoS of bona fide weighted configuration model Laplacians in Fig. 1. Thereafter, only Eq. (29) is used to check analytical results against numerics. However, it strikes me that a very similar approximation is made in the analytical procedure. Is this the case?

Regardless, perhaps it would be sensible also to check the results of direct diagonalization of configuration model networks for a more complicated observable such as the local DoS distribution at least once (it seems from Ref [59] that this should be within the realm of computational possibility).

5 - Related to this, I believe the "annealed network approximation", whereby one replaces the network by a weighted fully-connected graph with appropriately modified statistics (as is encapsulated by Eq. (29)), has already been given for adjacency matrices with $J_0 \neq 0$ and $J_1 \neq 0$ in Baron Phys Rev E (2022). The annealed network approximation has also been used in many other contexts (see e.g. Dorogovtsev, Goltsev, and Mendes Rev. Mod. Phys. (2008) or Carro, Toral, San Miguel Sci. Rep. (2016)). Although the use of the approximation to perform numerics in the case of Laplacian matrices is new here, I believe these related works should be referenced.

6- Is Eq. (46) what is called the Fyodorov distribution in Ref. [29], or is it different (aside from the shift due to the non-zero mean $J_0 \neq 0$)?

7- In general, it would be nice to see a bit more direct comparison to closely related results of previous works. That is, what are the main differences between the results (particularly $\tau(q)$ and $f(\alpha)$) seen here and those found in the case of (a) the adjacency matrix of sparse networks in Ref. [59] and (b) the dense but heterogeneous adjacency matrix in Ref [35]?

Ask for minor revision

1. Interesting and very well supported results,
2. Clarity of the exposition.

Most of the technical breakthroughs had already been presented in previous publications.

In this manuscript, the authors consider the Laplacian matrix on highly connected networks with a gamma distribution of the degrees, and with random weights. The spectral and localization properties of the model are encoded in the solution of the resolvent (cavity) distributional equations, which is here extracted analytically in the large-connectivity limit. This allows to analyze in great depth not only the spectral density of the model, but also the distribution of the LDoS, which in turn gives access to the localization properties of the eigenvectors. The singularity spectrum and multifractal exponents are then evaluated as a function of the (inverse) variance $\gamma$ of the rescaled degrees, separately for the case of constant or random coupling strengths. While in the former case the eigenvectors in the bulk of the spectrum are always localized, the latter case exhibits a transition from non-ergodic delocalized states to localized eigenvectors when the degree fluctuations become sufficiently strong (which happens at $\gamma=1/2$). This is accompanied by the divergence of the spectral density in the bulk of the spectrum.

I found the paper very well written and pedagogical in its exposition, and I think that it provides a thorough and comprehensive analysis of the model under consideration.
I would not hesitate to recommend its publication in SciPost Physics, but in fact I’m not yet entirely convinced of the originality of these results. Indeed, the analytic solution of the cavity equations in the high-connectivity limit was first presented in Ref. [34] (by authors 1 and 4 of the present manuscript), although for the case of the adjacency matrix (instead of the Laplacian). Such solution allowed for the analysis of the spectral density and IPR on heterogeneous random graphs. This study did not include the characterization of the multifractal exponents, which was later addressed in Ref. [35] (by authors 2 and 3), where the mechanism of “statistical localization” was also unveiled and described. The authors themselves state all of this very clearly already in the Introduction, and also elsewhere in the present manuscript.
Although I did appreciate a lot the organic and self-contained nature of this work, I'd encourage the authors to comment clearly on how it distinguishes itself from Refs. [34-35], apart from of course replacing the adjacency matrix by the Laplacian matrix. For example, in the present case the value $\gamma=1/2$ marks an interesting transition from non-ergodic delocalized to localized states: did the same happen already for the case of the adjacency matrix?

Below I add a few remarks and suggestions.

1. On p. 5, there is probably an “as” missing from “the entire degree distribution acts a control parameter”.

2. On p. 6, the sentence “the empirical spectral density in Eq. (4) is the first moment of $P_\lambda (y)$” might be promoted as a displayed equation; this would allow to refer to it later, e.g. in the discussion that follows Eq. (35).

3. In Eq. (21) and in Appendix A, can the meaning of the integration over $H^+$ be spelled out?

4. On p. 9, “Eqs. (23-25) exhibit a universal behaviour with respect to the fluctuations of the coupling strengths, since they depend on the distribution $p_J$ only through its first two moments.” However, earlier on p. 5 it was assumed “that higher-order moments of $p_J$ are proportional to $1/c^\beta$, with $\beta>1$”. Aren’t these two points related? Is it surprising that only the first two moments of the distribution intervene into Eqs. (23-25)?

5. Right panel of Fig. 1: why in this case $\rho_\epsilon$ was plotted instead of $\rho$, that appears instead on the left panel? Can the authors comment on the slight discrepancy observed in the middle of the spectrum?

6. In Fig. 4, the yellow line is not really “dash-dotted”: does it still correspond to Eq. (43)? Why is the agreement with the numerical data getting worse at $\gamma=1$, whereas in Fig. 9(left) the opposite seems to occur?

7. In Eq. (46), it may be worth recalling that $w$ is the Faddeeva function introduced earlier in Eq. (27). Moreover, it may be worth mentioning that $w$ is in fact the Cauchy-Stieltjes transform associated to a Gaussian distribution, which is after all the reason why Eq. (46) suggests a free additive convolution.

8. In Sec. 4.2.2, the strong degree fluctuations for $\gamma= 1/3$ unfortunately affect negatively the numerical measurements. Still, a way to further support the correctness of the singularity spectrum depicted in Fig. 9(left) may be to check if $f(\alpha)$ satisfies the very general symmetry proposed in [PRL 97, 046803 (2006)].

Ask for minor revision