Multifractality in random networks with power-law decaying bond strengths
Didier A. Vega-Oliveros J. A. Méndez-Bermúdez Francisco A. Rodrigues
























^1Departamento de Computação e Matemáticas, Faculdade de Filosofia Ciências e 
Letras de Ribeirão Preto, Universidade de São Paulo, CEP 14040-901, Ribeirão Preto, SP, Brasil

^2Instituto de Física, Benemérita Universidad Autónoma de Puebla, 
Apartado Postal J-48, 72570 Puebla, México

^3Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas 
e de Computação, Universidade de São Paulo - Campus de São Carlos, CP 668,
13560-970 São Carlos, SP, Brasil






In this paper we demonstrate numerically that random networks whose adjacency matrices A 
are represented by a diluted version of the Power–Law Banded Random Matrix (PBRM) model have 
multifractal eigenfunctions. 
The PBRM model describes one–dimensional samples with random long–range bonds.
The bond strengths of the model, which decay as a power–law, 
are tuned by the parameter μ as A_mn∝ |m-n|^-μ; while the 
sparsity is driven by the average network connectivity α: for α=0 
the vertices in the network are isolated and for α=1 the network is fully 
connected and the PBRM model is recovered. Though it is known that the PBRM model has
multifractal eigenfunctions at the critical value μ=μ_c=1, we clearly 
show [from the scaling of the relative fluctuation of the participation number 
I_2 as well as the scaling of the probability distribution functions P(ln I_2)] 
the existence of the critical value μ_c≡μ_c(α) for α<1.
Moreover, we characterise the multifractality of the eigenfunctions of our
random network model by the use of the corresponding multifractal dimensions D_q,
that we compute from the finite network-size scaling of the typical eigenfunction 
participation numbers exp⟨ln I_q ⟩.


64.60.Aq		
      89.75.Hc		


Multifractality in random networks with power–law decaying bond strengths
    Didier A. Vega-Oliveros,^1 J. A. Méndez-Bermúdez,^2 and Francisco A. Rodrigues^3
    December 30, 2023
====================================================================================









§ INTRODUCTION


Fractality is related to phase transitions in critical phenomena observed in several complex systems <cit.>. Blood vessels, proteins, ocean waves, animal collaboration patterns, and earthquakes exhibit fractality <cit.>. Fractality can also be understood as a signature of the organisation and structure of complex systems, which is far from random or regular <cit.>. Moreover, the structure of complex systems can be mapped to networks, whose structure <cit.> and evolution <cit.> exhibit fractal properties. 

Fractality in networks has been extensively discussed from several perspectives <cit.>. These studies have focused on the structural characterisation of fractal networks <cit.> or networks expressly constructed as fractal objects (deterministic or disordered, e.g. see <cit.>). In this respect some algorithms have been developed and applied to compute the fractal dimension of complex networks, see for example <cit.> and references therein. On the other hand, given a fractal network, there is plenty of works devoted to the signatures of the fractality on the network properties. Among them we can mention the underlying tree structure or skeleton <cit.> as well as dynamical and transport properties, see for example <cit.>.

Here, we approach an alternative but close-related subject: We explore the fractality of the eigenfunctions of the adjacency matrices A of a random network model. Moreover, we demonstrate  that imposing power–law correlations, i.e. A_mn∝ |m-n|^-μ with  μ∼ 1, on a random network model of the Erdös-Rényi–type produces multifractal eigenfunctions.

Therefore, in the following section we first review the Power–Law Banded Random Matrix (PBRM) model; a random matrix model used to study the Anderson metal-to-insulator phase transition, which presents multifractal eigenfunctions at the transition point. Then, we introduce the diluted PBRM (dPBRM) model as an ensemble of adjacency matrices of random networks of the Erdös-Rényi–type.
Using scaling arguments, in Sect. III we show that the dPBRM model also exhibits a metal-to-insulator phase transition where the corresponding eigenfunctions are multifractal objects. Finally, in Sect. IV we draw our conclusions.




§ THE RANDOM NETWORK MODEL




 §.§ The Power–Law Banded Random Matrix model


The Power–Law Banded Random Matrix (PBRM) model <cit.> is represented by 
N× N real symmetric matrices whose elements are 
statistically independent random variables drawn from a normal distribution with 
zero mean, ⟨ A_mn⟩=0, and a variance given by 

    ⟨ |A_mn|^2 ⟩ =
       1/2( 1+δ_mm/1+[
       sin( π|m-n|/N )/(π b/N) ]^2μ)  ,

where b and μ are the model parameters. The PBRM model has been 
used to describe one–dimensional tight-binding wires of length N with random 
long–range hoppings. In Eq. (<ref>) the PBRM model is in its periodic version; 
i.e., the one–dimensional wire is in a ring geometry. Theoretical considerations 
<cit.> and detailed numerical investigations 
<cit.> have verified that the PBRM model undergoes a transition at 
μ=1, i.e., from localized eigenfunctions for μ>1 to delocalized eigenfunctions for μ<1. 
This transition shows key features of the disorder driven Anderson metal-insulator 
transition <cit.>, including multifractality of eigenfunctions and non-trivial 
spectral statistics. Thus the PBRM model possesses a line of critical points 
b∈ (0,∞) at μ=μ_c=1. By tuning the parameter b, from b≪ 1 to b≫ 1, the 
eigenfunctions cross over from the strong multifractality (D_q∼ b → 0) which 
corresponds to localized–like or insulator–like eigenfunctions to weak multifractality 
(D_q → 1), showing rather extended, i.e., metallic–like eigenfunctions <cit.>. 
Here, D_q are the eigenfunction's multifractal dimensions (to be defined in Sect. <ref>). 
At the true Anderson transition in d=3 or at the integer 
quantum–Hall transition in d=2 the eigenfunctions belong to the weakly multifractal regime, 
i.e., d-D_2≪ d; it is relevant to note that the PBRM model allows for investigations without such a 
limitation.



 §.§ The diluted Power–Law Banded Random Matrix model


Here we introduce the 
diluted PBRM (dPBRM) model as follows: Starting with the PBRM model, we randomly set off-diagonal matrix elements to zero such that the sparsity (i.e., the average network connectivity) α is defined as the fraction of the N(N-1)/2 independent non-vanishing off-diagonal matrix elements. According to this definition, a diagonal random matrix is obtained for α = 0, whereas the PBRM model is recovered when α = 1. 

The Erdös-Rényi adjacency matrix is considered as a mask to define the nonzero
matrix elements of our dPBRM model. Hence, notice that the dPBRM model of size N works as an ensemble of adjacency matrices of Erdös-Rényi–type networks formed by N vertices. For such networks we allow self-edges and further consider all edges to have random strengths; however, notice that the random strengths are power–law modulated, see Eq. (<ref>). 

The power–law correlations of the dPBRM model are 
tuned by the parameter μ as A_mn∝ |m-n|^-μ, see Eq. (<ref>).
Notice that for μ→∞ the vertices in the
network become isolated since A_mn→ 0; while for μ→ 0 the dPBRM model 
reproduces the Erdös-Rényi random network model with maximal disorder 
(see Refs. <cit.>).
However, here we set μ∼ 1 such that we recover the PBRM model at criticality
(i.e., the PBRM model having multifractal eigenfunctions) when α=1.
Moreover, without loss of generality, we will set the effective bandwidth b of 
the dPBRM model to unity; that is, we use the bandwidth that produces multifractal
eigenfunctions with intermediate fractality, D_2≈ 0.5, in the PBRM model.
Here D_2 is the correlation dimension of the eigenfunctions.

Note that another diluted version of the PBRM model was reported in 
Refs. <cit.> in studies of quantum percolation.

In the following Section we demonstrate that the eigenfunctions of the dPBRM model are multifractal objects. Besides, we share the implementation and analyses of the reported model online [Code available at <https://github.com/didiervega/RandomMatrix>], for easier reproducibility.







§ NETWORK EIGENFUNCTION MULTIFRACTALITY



Given an eigenfunction Ψ it is a common practice (in random matrix models and 
complex Hamiltonian systems) to characterise its complexity by the use 
of the generalised participation numbers

    I_q = ( ∑_i=1^N |Ψ_i|^2q)^-1 ,

where N is the corresponding matrix size. In particular, the participation number I_2 is roughly equal to the number of principal eigenfunction components, and therefore, is a widely accepted measure of the extension of the eigenfunction Ψ in a given basis. 
Participation numbers and also inverse participation ratios (i.e., (I_q)^-1) have 
been already used to characterise the eigenfunctions of the adjacency matrices of random 
network models (see some examples in 
Refs. <cit.>).

In the context of random matrix models showing the metal-to-insulator phase transition,
such as the PBRM model,
it is well established that the distribution functions of the inverse participation ratios
are scale invariant at the transition point <cit.> where the eigenfunctions
are multifractal objects (see also <cit.>). 
The PBRM model with μ=1 is at criticality, however, introducing the
sparsity α may relocate the metal-to-insulator transition point. Therefore,
before talking about multifractality of eigenfunctions for the dPBRM model we first have 
to be sure that the system is at criticality. Thus, to search out the critical points of 
the dPBRM model with α<1 we use the relative fluctuation (the ratio of the standard 
deviation to the mean value) of the participation number I_2,

    η = √(⟨ (I_2)^2 ⟩-⟨ I_2 ⟩^2)/⟨ I_2 ⟩ .

Indeed, this quantity has been used to locate the metal-to-insulator transition point in 
random <cit.> and non-random <cit.> long–range hopping models.

In the following, we use exact numerical diagonalisation to obtain the eigenfunctions Ψ of the adjacency matrices of large ensembles of random networks, represented by 
the dPBRM model (characterized by N and α).

In Fig. <ref> we present the curves of η vs. μ for the dPBRM model with sparsity α. For a given value of the sparsity α we show curves corresponding to different (exponentially growing) network sizes. For all the values of α we consider here, α=[0.02,1], we observe two opposing behaviours: 
When μ≪ 1 [μ≫ 1] the quantity η decreases [increases] for increasing [decreasing] network size N. Moreover, 
we observe that curves for different N have a fixed point at μ=μ_c, revealing the invariance of η and therefore the existence of a metal-to-insulator transition point at μ_c. Then, in Fig. <ref>(a) we plot μ_c vs. α.
From this figure we can see that: for moderate sparsity, i.e., α>0.1, μ_c∼ 1; while for relatively strong sparsity, i.e, α≤ 0.1, μ_c decreases for decreasing α.

As complementary information, in Fig. <ref>(b) we report the values of the relative fluctuation η of the participation number that we found at μ_c. As for μ_c,
here we also observe two different behaviours for η: 
while it decreases for increasing α when  α>0.1;
it is interesting to note that η is approximately constant (η≈ 0.8) for relatively strong sparsity, α≤ 0.1.
 






We stress that from Fig. <ref> we have located the critical points μ_c for different values of α (reported in Fig. <ref>) by the use of the invariance of the relative fluctuation of the participation number I_2. Moreover, we can further verify the existence of  μ_c from the invariance of the probability distribution function (PDF) of I_2 itself (see for example <cit.>). Thus, in Fig. <ref> we show PDFs of the participation number P(ln I_2) for the dPBRM model with sparsity α=0.3, 0.6, and α=0.9. For each α we have selected three values of μ: 
μ=0.9<μ_c, μ=μ_c, and μ=1.2>μ_c. As well as for η, here we identify two behaviours for P(ln I_2) depending on whether μ<μ_c or μ>μ_c:
When μ<μ_c [μ>μ_c] the histograms of P(ln I_2) are narrower [wider] the larger [smaller] the network size. While, as predicted <cit.>, P(ln I_2) is invariant at μ=μ_c and falls on top of a universal PDF when plotted as a function of ln I_2-⟨ln I_2 ⟩. 
We merely want to comment that for small network sizes, N<10^3, at μ=μ_c we observe that P(ln I_2) evolves as a function of N; which is a finite size effect. 
Indeed, as clearly seen in Fig. <ref>, P(ln I_2) is already invariant for N>10^3.

Once we know the position of μ_c for the dPBRM model, we can characterise the 
multifractality of the corresponding eigenfunctions through the eigenfunction multifractal 
dimensions D_q, which are defined by the scaling of the typical 
participation numbers 

    I_q^typ≡exp⟨ln I_q ⟩

as a function of N:

    I_q^typ∝ N^(q-1)D_q .

The multifractal dimensions D_q can also be extracted from the scaling of 
the average participation numbers, ⟨ I_q ⟩∝ N^(q-1)D_q,
however here we choose to use typical participation numbers. 
We recall that for strongly localized eigenfunctions the corresponding 
I_q^typ does not scale with the system size: I_q^typ∼ 1 and 
D_q→ 0 for all q. This situation corresponds to an insulating regime. 
While extended eigenfunctions always feel the entire system. Thus, a signature of the metallic regime is given by 
I_q^typ∝ N and D_q→ d. Moreover, multifractal eigenfunctions 
should be described by the series of D_q, which are nonlinear functions of the 
index q.

We extract the multifractal dimensions D_q from the linear fit of the logarithm of the typical participation numbers ⟨ln I_q ⟩ versus the logarithm of N (see Eq. (<ref>)). We use N=2^n, 7≤ n≤ 12. 
The average was performed over 2^n-3 eigenfunctions with eigenvalues around the band centre with 2^18-n realisations of our dPBRM model.
As examples, in Fig. <ref> we present the scalings of ⟨ln I_2 ⟩ 
vs. ln N for selected values of sparsity. Therefore, the correlation dimension D_2 is
extracted from the linear fits to the data (see dashed lines).
We note the remarkably clean linear scaling of ⟨ln I_2 ⟩ vs. ln N.

Finally, in Fig. <ref>(a) we report the multifractal dimensions D_q as a function of q for the dPBRM model with selected values of α (to avoid figure saturation). The nonlinearity of the curves D_q vs. q is the signature of the multifractality of eigenfunctions of our network model. Also, as a reference, in Fig. <ref>(a) we include the values of D_q for the PBRM model (i.e. the values of D_q for the dPBRM model with α=1). Additionally, in Fig. <ref>(b) we show D_q vs. α for selected values of q, in particular we include the information dimension D_1 and the correlation dimension D_2.
From this figure we observe two behaviours: an initial decrease of D_q for decreasing α, for relatively large values of α (α≥ 0.2); while, remarkably, the further decreasing of α (i.e., α<0.08) makes the multifractality of the eigenfunctions of the dPBRM model to grow to values close to those for weak sparsity. 




§ DISCUSSION AND CONCLUSIONS


In this paper we consider random networks whose adjacency matrices A are represented by a sparse version of the Power–Law Banded Random Matrix (PBRM) model, therefore having a power–law structure, A_mn∝ |m-n|^-μ, tuned by the parameter μ (see Eq. (<ref>)). We call this random network model the diluted PBRM (dPBRM) model.
We would like to emphasise that the dPBRM model belongs to the same universality class than the PBRM model, as discussed in <cit.> where more general long-range quantum 
hopping models in one-dimension have been studied.
 
The sparsity of the dPBRM model is driven by the average network connectivity α: 
for α=0 the vertices in the network are isolated and for α=1 the network 
is fully connected. Notice that the original PBRM model is recovered for α=1,
which is known to have multifractal eigenfunctions at the critical value μ=μ_c=1
where a metal-to-insulator phase transition takes place.
Here, we show that the dPBRM model exhibits a critical value μ_c≡μ_c(α) 
for α<1, as reported in Fig. <ref>. 
Moreover, we found that μ_c∼ 1 for α>0.1; while 
for relatively strong sparsity, ⟨k||≪⟩N or α≪ 1 (since ⟨k||≡⟩α N,
where ⟨k||$⟩ is the average degree),μ_cdecreases for decreasingα. 

In addition, we demonstrate the multifractality of the eigenfunctions of our random 
network model atμ_cby the calculation of the corresponding multifractal dimensionsD_q. Indeed, we observed from Fig. <ref> that the multifractality of the 
eigenfunctions of the dPBRM model can be effectively tuned by the average network 
connectivityα.

We emphasize that the calculation ofD_qfrom the finite network-size scaling of the typical eigenfunction participation numbers, see Eq. (<ref>), is equivalent to a standard box covering algorithm (where the network sizeNworks as the box size). However, due to the normalised nature of the eigenfunctions, the scaling of⟨lnI_q ⟩vs.lnNis very stable, as clearly shown in Fig. <ref>, providing quite precise values of multifractal dimensions.

Our approach may be used to investigate the multifractality of eigenfunctions in other random network models. 
Indeed, similar studies have been already performed to explore the multifractality of eigenfunctions of the Anderson 
model on Cayley trees (AMCT) <cit.> and random graphs <cit.>.
It is relevant to stress that there are three important differences between the network model studied here and the 
AMCT studied in Refs. <cit.>:
(i) Cayley trees have a fixed degree (the AMCT in <cit.> is characterized byk=3), while 
due to the random-network nature of the dPBRM model the degree is defined as an average quantity here.
(ii) The dPBRM model represents networks with randomly-weighted bond strengths between vertices, while 
the AMCT in <cit.> is defined as a network with constant bond strengths.
(iii) The dPBRM model possesses an infinite line of critical points characterized by the parameterb∈(0,∞)(that we did not examine here since we fixedb=1in Eq. (<ref>) as a representative case), whereas
the AMCT has a single critical point for a given on-site disorder strength.
Thus, even though it may be expected that the dPBRM with⟨k| |≈⟩3should show similar properties
than the AMCT in <cit.>, this must be properly verified, given the differences between both models. 
Moreover, inspired by <cit.> it should also be interesting to explore the eigenfunction statistics of the 
dPBRM model off criticality, i.e.,μ1in Eq. (<ref>).

The relation between the fractality of networks (in networks  specifically constructed as deterministic or disordered 
fractal objects) and the (possible) fractality of the eigenfunctions of the corresponding adjacency matrix is also another 
important subject to be explored.

We would like to add that the dPBRM model, when interpreted as a model for one-dimensional quantum
chains with long–range interactions, has characteristics proper of models currently
used in the study of excitation transport <cit.>: disorder and power–law decaying bond strengths.
Furthermore, these characteristics can presumably be implemented and tuned in state-of-the-art ion-chain 
experiments; thus the dPBRM model may find applications related to quantum 
transport with high efficiencies <cit.>.




Research carried out using the computational resources of the Center for Mathematical 
Sciences Applied to Industry (CeMEAI) funded by FAPESP (Fundação de 
Amparo à Pesquisa do Estado de São Paulo, Grant No. 2013/07375-0).
J.A.M.-B. acknowledges support from VIEP-BUAP (Grant No. MEBJ-EXC18-G), 
Fondo Institucional PIFCA (Grant No. BUAP-CA-169), and CONACyT (Grant No. CB-2013/220624).
F.A.R. acknowledges CNPq (Grant No. 305940/2010-4) and FAPESP (Grant No. 13/26416-9). 
D.A.VO. acknowledges CNPq (Grant 140688/2013-7) and FAPESP (Grant No. 2016/23698-1).

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Phys. Rev. Lett.  118, 166801 (2017).

NP18
N. Trautmann and P. Hauke,
Phys. Rev. A  97, 023606 (2018).

TM19
K. S. Tikhonov and A. D. Mirlin
Phys. Rev. B  99, 024202 (2019).