# Geometric renormalization of weighted networks

Muhua Zheng  School of Physics and Electronic Engineering, Jiangsu University,
Zhenjiang, Jiangsu, 212013, China  Guillermo García-Pérez  Algorithmiq Ltd,
Kanavakatu 3 C, FI-00160 Helsinki, Finland  Marián Boguñá  Departament de
Física de la Matèria Condensada, Universitat de Barcelona, Martí i Franquès 1,
E-08028 Barcelona, Spain  Universitat de Barcelona Institute of Complex
Systems (UBICS), Universitat de Barcelona, Barcelona, Spain  M. Ángeles
Serrano<EMAIL_ADDRESS>Departament de Física de la Matèria Condensada,
Universitat de Barcelona, Martí i Franquès 1, E-08028 Barcelona, Spain
Universitat de Barcelona Institute of Complex Systems (UBICS), Universitat de
Barcelona, Barcelona, Spain  ICREA, Passeig Lluís Companys 23, E-08010
Barcelona, Spain

###### Abstract

The geometric renormalization technique for complex networks has successfully
revealed the multiscale self-similarity of real network topologies and can be
applied to generate replicas at different length scales. In this letter, we
extend the geometric renormalization framework to weighted networks, where the
intensities of the interactions play a crucial role in their structural
organization and function. Our findings demonstrate that weights in real
networks exhibit multiscale self-similarity under a renormalization protocol
that selects the connections with the maximum weight across increasingly
longer length scales. We present a theory that elucidates this symmetry, and
that sustains the selection of the maximum weight as a meaningful procedure.
Based on our results, scaled-down replicas of weighted networks can be
straightforwardly derived, facilitating the investigation of various size-
dependent phenomena in downstream applications.

Renormalization of real networks García-Pérez  _et al._ (2018); Zheng  _et
al._ (2020); Garuccio  _et al._ (2020); Villegas  _et al._ (2023) can be
performed on a geometric framework García-Pérez  _et al._ (2018) by virtue of
the discovery that their structure is underlain by a latent hyperbolic
geometry Boguñá  _et al._ (2021); Serrano and Boguñá (2022). Distances between
nodes in this space determine the likelihood of connections via a universal
law that operates at all scales and encodes simultaneously short- and long-
range connections. This geometric principle has been able to explain many
features of real networks, including the small-world property, scale-free
degree distributions, and high levels of clustering, as well as fundamental
mechanisms such as preferential attachment in growing networks Papadopoulos
_et al._ (2012), and the emergence of communities García-Pérez  _et al._
(2018b); Zuev  _et al._ (2015). It has also led to embedding techniques that
produce geometric representations of complex network from their topologies
Boguñá  _et al._ (2010); Papadopoulos  _et al._ (2015); Muscoloni  _et al._
(2017); Blasius  _et al._ (2018); García-Pérez  _et al._ (2019).

Weights in real complex networks Barrat  _et al._ (2004); Newman (2004);
Serrano  _et al._ (2006); Mastrandrea  _et al._ (2014); Menichetti  _et al._
(2014) are also amenable to modeling within the hyperbolic network geometry
paradigm. More specifically, the weighted geometric soft configuration model
(W$\mathbb{S}^{D}$) Allard  _et al._ (2017) captures the non-trivial coupling
between network topology and weights, allowing for accurate reproduction of
both the unweighted and the weighted structure of real networks. However, the
geometric renormalization (GR) method only applies to unweighted networks. By
applying coarse graining and rescaling steps to unfold an unweighted network
map into a sequence of scaled-down layers over progressively longer length
scales, GR revealed multiscale self-similarity to be a ubiquitous symmetry in
real networks García-Pérez  _et al._ (2018). This raises the question whether
GR can be generalized to weighted networks as well and whether self-similarity
would be preserved in that case.

Adding to GR, the geometric renormalizaton of weights (GRW) should produce the
multiscale unfolding of a network into a shell of weighted scaled-down layers
that preserve the weighted structure of the network in the flow. Here, we
propose a theory for the renormalization of weighted networks that supports
the selection of the maximum, or supreme, as an effective approximation to
allocate weights in the renormalized layers of real networks. Our theory is
sustained by the renormalizability of the W$\mathbb{S}^{D}$ model, which
entails that the GRW transformation should be a rescaled $p$-norm on the set
of weights to be renormalized. Alternatively, the GR technique was recently
extended to weighted networks using an ad hoc approach that treats weights as
currents or resistances in a parallel circuit—renormalizing by the sum of the
weights or by the inverse of the sum of their inverses, respectively Chen  _et
al._ (2022). The two methods are recovered as particular limits of our theory.

Figure 1: Self-similarity of real weighted networks along GRW flows. The first
row shows the sup-GRW flow of the pdf of weights and their disparity in nodes
in Openflights (a)-(b) and Collaboration (c)-(d). The same in the second row
for the sum-GRW flow. The number of layers in each shell is determined by the
original network size, and $r=2$ in all cases.

To begin with, we provide evidence that self-similarity is a pervasive
symmetry not only in the multiscale organization of real network topologies
but also in the multiscale ulfonding of their weights. To that end, we
implement a GRW transformation, which requires the preliminary application of
the GR technique to unweighted networks García-Pérez  _et al._ (2018).

The GR technique operates on the geometric embedding of a network, as
described in previous works Boguñá  _et al._ (2010); García-Pérez  _et al._
(2019), obtained by maximizing the likelihood that the network topology is
generated by the geometric soft configuration model $\mathbb{S}^{D}$ Serrano
_et al._ (2008). In this model, nodes are assigned coordinates representing
popularity and similarity dimensions, and distances between them determine the
probability of connection $p_{ij}=1/(1+\chi_{ij}^{\beta})$, where
$\chi_{ij}=d_{ij}/(\mu\kappa_{i}\kappa_{j})^{1/D}$. Parameter $\mu$ controls
the average degree, and $\beta>D$ controls the level of clustering and
quantifies the level of coupling between the network topology and the
geometry. The hidden degree $\kappa_{i}$ of node $i\in[1,N]$—equivalent to a
radial coordinate in the hyperbolic plane in the purely geometric formulation
of the model, named $\mathbb{H}^{D+1}$ Krioukov  _et al._ (2009)—measures the
popularity of the node, with higher values indicating a greater likelihood of
connecting to other nodes. In $D=1$, the similarity subspace is represented as
a circle of radius $R=N/(2\pi)$ with unit density. Each node $i$ is assigned
an angular coordinate $\theta_{i}$ in the circle, and angular distances
$d_{ij}=R\Delta\theta_{ij}$ between pairs of nodes account for factors other
than degrees that influence the tendency to form connections. Nodes closer in
the similarity subspace have a higher likelihood of being connected.
Hyperbolic embeddings of unweighted networks can be obtained using the
Mercator mapping tool García-Pérez  _et al._ (2019), which employs statistical
inference techniques to identify the hidden degrees and angular coordinates
while adjusting parameters $\beta$ and $\mu$ accordingly.

Once the geometric map of a real network is generated, GR divides the
similarity circle into non-overlapping blocks of consecutive nodes of size
$r$. These blocks are then coarse grained forming supernodes in a new layer.
Each supernode is positioned within the angular region defined by the
corresponding block, preserving the order of nodes. Any links between nodes in
one supernode and nodes in another are renormalized into a single link
connecting the two supernodes. This way, GR eliminates short-range couplings
and produces a new network topology that is self-similar to the original
except for the average degree, which increases in the renormalization flow
García-Pérez  _et al._ (2018).

The GRW technique involves assigning intensities to the links in the new layer
based on the weights in the original layer, following a specific prescription.
This transformation can be iterated starting from the original network at
layer $l=0$, with the iteration bounded to approximately
$l_{max}\propto\log(N)$ steps due to the finite size of real networks. As a
result, a sequence of self-similar network layers $l$—each $r$ times smaller
than the original one—is produced forming a multiscale weighted shell of the
original network. The process is visually depicted in Fig. S1 of the
Supplemental Material (SM). The crux of GRW lies in how the weights are
renormalized to ensure that their characteristics, such as global and local
weight distributions and the relationship between strength and degree, are
preserved throughout the renormalization flow.

An effective and simple prescription, referred to as sup-GRW, is to define the
weight of the link between two supernodes as the maximum, or supremum, of the
weights in the existing links between their constituent nodes in the original
layer. We applied the sup-GRW technique to $12$ different real weighted
networks from different domains including biology, transportation, knowledge,
and social systems. The networks were processed using blocks of size $r=2$.
Additional details can be found in the SM.

The behavior of the weights in the renormalization flow of two of the networks
are shown in Fig. 1(a)-(d), while Figs. S2-S5 present the corresponding
results for the remaining networks. The relations strength-degree are shown in
Fig. S5. The probability density functions (pdf) of weights and strengths in
the different layers collapse once rescaled by the average weight and average
strength, respectively, in the corresponding layer. Furthermore, the power-law
relations between strength and degree also overlap once the degrees are
rescaled by the average degree of the layer, as demonstrated in Figs. S4 and
S5. To quantify the local heterogeneity of the weights, we measured their
disparity around nodes as a function of the degree, as described in the
Methods section of the SM. The results show, again, statistical invariance
across layers.

Notice that, by construction, the average weight and the average strength in
the sup-GRW layers grows with $l$. While this behavior does not provide
fundamental information for characterizing the description of the weighted
structure of the network, it may still be interesting to understand how
$\langle w\rangle$ and $\langle s\rangle$ depend on the scale of observation
$l$. This is particularly relevant considering that weights in real networks
are often expressed in real-world units. The corresponding results are
presented in Figs. S6 and S7. Furthermore, the sup-GRW transformation exhibits
the semigroup structure with respect to the composition, similar to the
behavior observed in GR for unweighted networks. This means that a certain
number of iterations with a given coarse graining factor are equivalent to a
single transformation with a higher coarse graining factor. The findings shown
in Fig. S8 provide support for this claim.

We also tested an alternative prescription, referred as sum-GRW, where weights
in the new layer are assigned by summing the weights of existing links between
the nodes in supernodes, following the prescription described in Ref. Chen
_et al._ (2022). While this strategy proves effective for many real networks,
there are certain cases in which self-similarity is not maintained in the
renormalization flow. When sum-GRW is applied, the global distribution of
weights, the local heterogeneity of weights in nodes, and the relation between
strength and degree become increasingly heterogeneous compared to the original
graph. This is observed in the Openflights and the scientific collaboration
network, as illustrated in Fig. 1(e)-(h), and in Figs. S9 and S10 for the
remaining networks.

The reported results are supported by a theoretical framework that clarifies
the conditions under which each of the two weight assignment prescriptions,
selecting the supremum of weights between supernodes or their sum, yields good
performance. Our theory is based on the W$\mathbb{S}^{D}$ model Allard  _et
al._ (2017), that uses the $\mathbb{S}^{D}$ model to mimic the topology of
real networks. In the W$\mathbb{S}^{D}$ model, weights are assigned to
connections between two connected nodes $i$ and $j$ as follows:

$\omega_{ij}=\epsilon_{ij}\frac{\nu\sigma_{i}\sigma_{j}}{\left(\kappa_{i}\kappa_{j}\right)^{1-\alpha/D}d_{ij}^{\alpha}}.$
(1)

Similar to $\bar{k}_{i}\propto\kappa_{i}$ in the $\mathbb{S}^{D}$ model, the
W$\mathbb{S}^{D}$ model ensures that the expected strength of node $i$,
$\bar{s}_{i}$, is proportional to the hidden strength $\sigma_{i}$,
$\bar{s}_{i}\propto\sigma_{i}$. When $\alpha=0$, the weights are independent
of the underlying geometry and primarily influenced by node degrees, while
$\alpha=D$ implies that weights are maximally coupled to the underlying metric
space with no direct contribution of the degrees. Finally, $\epsilon_{ij}$ is
a random variable with mean equal to one and the variance of which regulates
the level of noise in the network. In the subsequent analysis, we assume the
noiseless version of the model to simplify analytical calculations, which
means $\epsilon_{ij}=1\;\forall(i,j)$.

To control the correlation between strength and degree and, consequently,
adjust the strength distribution, we assume a deterministic relation between
hidden variables $\sigma$ and $\kappa$ of the form $\sigma=a\kappa^{\eta}$,
yielding $s(k)\sim ak^{\eta}$ as observed in real complex networks. Working
under this assumption, a valid GRW transformation should preserve the relation
between strength and degree, and in particular the exponent $\eta$, meaning
that the renormalized hidden degree and strength should satisfy
$\sigma^{\prime}=a^{\prime}(\kappa^{\prime})^{\eta}$ (to simplify notation, we
have used prima to denote quantities in the renormalized layer). Using Eq. (1)
and the GR equations for the topological model García-Pérez  _et al._ (2018),
this requirement leads to the following expression for the renormalized
weights

$\omega^{\prime}_{ij}=C\left[\sum\limits_{e=1}^{r^{2}}\left(w_{mn}\right)_{e}^{\phi}\right]^{1/\phi},$
(2)

where the sum runs over the links between nodes within supernodes $i$ and $j$,
derivation in SM. Parameter $\phi\equiv\frac{\beta}{D(\eta-1)+\alpha}$ depends
on both the weighted and unweighted structure of the network, and
$C=\nu^{\prime}/\nu\left(a^{\prime}/a\right)^{2}r^{\alpha/D}$. In practice,
however, we rescale weights by the average weight in each layer, rendering the
constant $C$ irrelevant.

According to the weighted model, for a network with a specific value of
$\phi$, the GRW transformation of weights Eq. (2), denoted as $\phi$-GRW,
preserves the exponent $\eta$ that characterizes the relation between strength
and degree. At the same time, since the distribution of hidden degrees is
assumed to be preserved by GR, the distribution of hidden strengths and the
distribution of weights are also preserved. This is valid as long as
$\beta>(\gamma-1)/2$. Otherwise, the power-law distribution of hidden degrees
looses its self-similarity in the unweighted renormalization flow and this
breaks the self-similarity of weights. Also, note that the $\phi$-GRW
transformation has semigroup structure with respect to the composition,
regardless of the value of $\phi$.

We validated the self-similarity of the $\phi-$GRW transformation in the real
and synthetic networks, Figs. S11-S12 and Figs. S14-S17, respectively,
including its semigroup property. In all cases, the self-similar behavior of
the distribution of weights and strengths, and the power-law relation between
strength and degrees in the renormalization flow is clear across length
scales, which validates our analytic calculations.

Notice that the transformation in Eq. (2) is a $\phi$-norm, which is a
generalization of the Euclidean norm. As $\phi$ increases, the $\phi$-norm
becomes progressively dominated by the supremum of the terms $w_{mn}$ in Eq.
(2) . In fact, the sup-GRW prescription is recovered in the limit
$\phi=\infty$ of $\phi$-GRW. In addition, renormalizing by the sum is
equivalent to setting $\phi=1$, and the renormalization of weights by the
inverse of the sum of inverse values corresponds to $\phi=-1$.

Figure 2:  Asymptotics of the $\phi$-norm. We used the weights
$\\{\omega_{mn}\\}$ in the Openflights and Collaboration networks, and
performed an iteration of $\phi-$GRW to calculate the renormalized weight
$\omega^{\prime}$ with Eq. (2). Note that when $r=2$, the number of links $E$
between the nodes in two supernodes could be $1$, $2$, $3$ or $4$. So we
displayed the renormalized weight $\omega^{\prime}(\phi=1)$ and
$\omega^{\prime}(\phi=\infty)$ versus $\omega^{\prime}(\phi^{*})$ for
different $E$, where $\phi^{*}$ is the inferred value with
$\phi^{*}=\beta/(\eta-1+\alpha)$. Sup-GRW corresponds to the case
$\phi=\infty$ while sum-GRW to $\phi=1$.

To clarify the efficacy of approximating $\phi-$GRW as sup-GRW, we checked the
asymptotic behavior of the $\phi$-norm as a function of the number of elements
$E$ in the set of coarse-grainable weights and of the level of heterogeneity
in the weights, see section VI in SM for more details. Figure 2 shows the
result of applying the supremum and the sum prescriptions as compared with
renormalizing weights using $\phi$-GRW in two of the real networks analyzed in
this letter, see Figs. S20 and S21 for the rest. In synthetic networks, we
simulated weights using a distribution
$p(\omega_{mn})\sim\omega_{mn}^{-\delta}$, where $\delta$ allowed us to tune
the level of heterogeneity, and produced sets of weights that were
renormalized using Eq. (2) with $C=1$ and different values of $\phi$. We also
renormalized the same sets using the alternative sum and supremum
prescriptions, the results are shown in Figs. S18 and S19.

In heterogeneous networks with a markedly scale-free character of the weight
distribution, very small deviation from the supremum are observed and this
occurs primarily for very low values of $\phi$ and low-weight values. As the
number of elements $E$ increases and the degree distribution becomes more
homogeneous, these deviations progressively become larger. As expected, higher
values of $\phi$ reduce the discrepancy between the $\phi$-norm and the
supremum estimator. Nevertheless, across a wide range of parameter values,
which encompass those for realistic networks, there is generally a good
agreement between the $\phi$-norm and the selection of the supremum, with any
existing deviations being quite minor. While for some empirical weight
distributions, sup-GRW and sum-GRW yield the same renormalized weights, e.g.,
the JCN in Figs. S20 and S21, it is important to note that, in general, the
relation between hidden strength and hidden degree is not preserved under sum-
GRW. See Methods section in SM for more details.

Figure 3: Analytic approximation vs simulations. (a) Unrescaled average weight
for different layers $l$. (b) Average strength, $\langle s\rangle$ as a
function of average degree $\langle k\rangle$, in which symbols are the
simulated results and lines indicate the corresponding theoretical analysis
from Eq.(4). The synthetic networks are generated with $\gamma=2.7$,
$\beta=1.5$, $\alpha=0.4$ and $\eta=1.2$ for Net $0$; $\gamma=2.7$,
$\beta=1.5$, $\alpha=0.4$ and $\eta=1.8$ for Net $1$; $\gamma=2.2$,
$\beta=2.0$, $\alpha=0.4$ and $\eta=1.5$ for Net $2$; $\gamma=2.2$,
$\beta=2.0$, $\alpha=0.6$ and $\eta=1.5$ for Net $3$. In all simulations
$N=64000$, $\langle k\rangle=5$, $a=100$, and
$\langle\epsilon^{2}\rangle=1.0$.

The preservation of the relation $\sigma=a\kappa^{\eta}$ allows us to
approximate analytically the flow of the average strength from the flow of the
average degree. In GR, the average degree changes from layer to layer
approximately as $\langle k\rangle^{(l+1)}=r^{\xi}\langle k\rangle^{(l)}$,
with a scaling factor $\xi$ depending on the connectivity structure of the
original network García-Pérez  _et al._ (2018). Combining this with Eq. (2)
and imposing that the rescaling constant of weights $C$ does not change in the
flow, we obtain

$\langle\sigma^{\prime}\rangle=\langle\sigma\rangle r^{\psi},\mbox{
}\psi=\left(\frac{\alpha}{D}+2\eta-1\right)\xi-\frac{\alpha}{D},$ (3)

which, due to the proportionality between observed and hidden strength,
implies that the flow of the average observed strength follows the same
scaling. Therefore, in $D=1$, the strength increases with a scaling factor
that depends on the exponent $\eta$, on the coupling $\alpha$ between topology
and geometry, and on the scaling factor $\xi$ for the flow of the average
degree, see Methods in SM for details. This leads to an analytic approximation
for the growth of the average strength as a function of the average degree

$\langle s\rangle^{(l)}=\langle s\rangle^{(0)}\left(\frac{\langle
k\rangle^{(l)}}{\langle s\rangle^{(0)}}\right)^{\frac{\psi}{\xi}},$ (4)

which agrees with the measurements in synthetic networks where the average
weight may increase, stay flat, or decrease in the flow as shown in Fig. 3.

All together, our results suggest that sup-GRW is a good approximation for
real networks and offers certain advantages over $\phi$-GRW. One advantage is
that it avoids the need to estimate parameters that capture the coupling
between the weighted structure of the network and the underlying geometry,
which can be challenging in practice. Sup-GRW is equivalent to setting
$\phi=\infty$ and, due to the nature of the transformation, it is effectively
reached for relatively low values of $\phi$. In addition, renormalizing by the
sum is equivalent to setting $\phi=1$, which in general does not preserve the
exponent $\eta$ of the relation between $\sigma$ and $\kappa$, see the Methods
section in SM for analytical calculations.

Beyond theoretical considerations, the practical application of GRW extends to
the generation of scaled-down replicas of weighted networks. These replicas
can serve as valuable testbeds for evaluating the scalability of
computationally intensive protocols or studying processes where the size of a
real network plays a role. The generation of a scaled-down replica involves
obtaining a reduced version of the topology, as described in Ref. García-Pérez
_et al._ (2018), and subsequently rescaling the weights in the renormalized
network layer to mathc the level of the original network. The detailed
procedure can be found in the SM, and the results for the scaled-down replicas
of real weighted networks are presented in Figs. S25-S28.

In summary, the extension of the geometric renormalization framework to
weighted networks demonstrates that multiscale self-similarity characterizes
not only the topology but also the weighted structure of real networks,
provided the appropriate renormalization scheme is applied. Moreover, the
weights in these networks result from processes that determine the intensities
of interactions, and our findings suggest that these processes follow the same
underlying principles across different length scales. Notably, the
transformation implied by the theory is closely approximated by using the
maximum weight prescription, a highly effective approach that can be readily
applied to real networks despite the presence of significant noise affecting
their weights. This observation justifies our confidence that noise will not
fundamentally alter the qualitative results reported in this study.

The present work represents a significant step towards establishing a
comprehensive framework for the renormalization of network structure and opens
up possibilitis for renormalizing dynamical processes on real networks. In
future research, it will be essential to incorporate not only the topology of
connections and their weights but also their directionality, which is crucial
in many real-world processes.

We thank Elisenda Ortiz for helpful discussions. M.Z. acknowledges support
from National Natural Science Foundation of China (Grants No. 12005079), the
Natural Science Foundation of Jiangsu Province (Grant No. BK20220511), the
funding for Scientific Research Startup of Jiangsu University (Grant No.
4111710001), and Jiangsu Specially-Appointed Professor Program. M. A. S and M.
B. acknowledge support from the Agencia Estatal de Investigación project
number PID2019-106290GB-C22 funded by MCIN/AEI/10.13039/501100011033;
Generalitat de Catalunya grant number 2021SGR00856. M. B. acknowledges support
from the ICREA Academia award, funded by the Generalitat de Catalunya.

## References

  * García-Pérez  _et al._ (2018a) G. García-Pérez, M. Boguñá, and M. Á. Serrano, Nature Physics 14, 583 (2018a).
  * Zheng  _et al._ (2020) M. Zheng, A. Allard, P. Hagmann, Y. Alemán-Gómez, and M. Á. Serrano, Proceedings of the National Academy of Sciences 117, 20244 (2020).
  * Garuccio  _et al._ (2020) E. Garuccio, M. Lalli, and D. Garlaschelli, arXiv preprint arXiv:2009.11024 (2020).
  * Villegas  _et al._ (2023) P. Villegas, T. Gili, G. Caldarelli, and A. Gabrielli, Nature Physics 19, 445 (2023).
  * Boguñá  _et al._ (2021) M. Boguñá, I. Bonamassa, M. D. Domenico, S. Havlin, D. Krioukov, and M. A. Serrano, Nature Reviews Physics 3, 114 (2021).
  * Serrano and Boguñá (2022) M. A. Serrano and M. Boguñá,  _The Shortest Path to Network Geometry: A Practical Guide to Basic Models and Applications_, Elements in Structure and Dynamics of Complex Networks (Cambridge University Press, 2022).
  * Papadopoulos  _et al._ (2012) F. Papadopoulos, M. Kitsak, M. Á. Serrano, M. Boguñá, and D. Krioukov, Nature 489, 537 (2012).
  * García-Pérez  _et al._ (2018b) G. García-Pérez, M. Á. Serrano, and M. Boguñá, Journal of Statistical Physics 173, 775 (2018b).
  * Zuev  _et al._ (2015) K. Zuev, M. Boguná, G. Bianconi, and D. Krioukov, Scientific Reports 5, 9421 (2015).
  * Boguñá  _et al._ (2010) M. Boguñá, F. Papadopoulos, and D. Krioukov, Nature Communications 1, 62 (2010).
  * Papadopoulos  _et al._ (2015) F. Papadopoulos, R. Aldecoa, and D. Krioukov, Phys. Rev. E 92, 022807 (2015).
  * Muscoloni  _et al._ (2017) A. Muscoloni, J. M. Thomas, S. Ciucci, G. Bianconi, and C. V. Cannistraci, Nat. Commun. 8, 1615 (2017).
  * Blasius  _et al._ (2018) T. Blasius, T. Friedrich, A. Krohmer, and S. Laue, IEEE/ACM Trans Netw 26, 920 (2018).
  * García-Pérez  _et al._ (2019) G. García-Pérez, A. Allard, M. Á. Serrano, and M. Boguñá, New Journal of Physics 21, 123033 (2019).
  * Barrat  _et al._ (2004) A. Barrat, M. Barthelemy, R. Pastor-Satorras, and A. Vespignani, Proceedings of the National Academy of Sciences 101, 3747 (2004).
  * Newman (2004) M. E. Newman, Physical review E 70, 056131 (2004).
  * Serrano  _et al._ (2006) M. Á. Serrano, M. Boguñá, and R. Pastor-Satorras, Physical Review E 74, 055101 (2006).
  * Mastrandrea  _et al._ (2014) R. Mastrandrea, T. Squartini, G. Fagiolo, and D. Garlaschelli, New Journal of Physics 16, 043022 (2014).
  * Menichetti  _et al._ (2014) G. Menichetti, D. Remondini, P. Panzarasa, R. J. Mondragón, and G. Bianconi, PloS one 9, e97857 (2014).
  * Allard  _et al._ (2017) A. Allard, M. Á. Serrano, G. García-Pérez, and M. Boguñá, Nature Communications 8, 14103 (2017).
  * Chen  _et al._ (2022) D. Chen, H. Su, and Z. Zeng, IEEE Transactions on Computational Social Systems , 1 (2022).
  * Serrano  _et al._ (2008) M. Á. Serrano, D. Krioukov, and M. Boguñá, Physical Review Letters 100, 078701 (2008).
  * Krioukov  _et al._ (2009) D. Krioukov, F. Papadopoulos, A. Vahdat, and M. Boguñá, Physical Review E 80, 035101(R) (2009).

Supplemental Material for
Geometric renormalization of weighted networks
Muhua Zheng1, Guillermo García-Pérez2, Marián Boguñá3,4 & M. Ángeles
Serrano3,4,5∗
1 School of Physics and Electronic Engineering, Jiangsu University, Zhenjiang,
Jiangsu, 212013, China
2Algorithmiq Ltd, Kanavakatu 3 C, FI-00160 Helsinki, Finland
3Departament de Física de la Matèria Condensada,
Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain
4Universitat de Barcelona Institute of Complex Systems (UBICS),
Universitat de Barcelona, Barcelona, Spain
5ICREA, Pg. Lluís Companys 23, E-08010 Barcelona, Spain
*Correspondence and requests for materials should be addressed to M.A.S<EMAIL_ADDRESS>

###### Contents

  1. A Illustration of the geometric renormalization transformation method for weighted networks
  2. B Methods
    1. B.1 Description of empirical data sets
    2. B.2 Network embedding to produce geometric network maps
    3. B.3 The definition of disparity
    4. B.4 Theoretical derivation of the renormalized weights
    5. B.5 Theoretical derivation of the flow of the average strength
    6. B.6 The transformation sum-GRW does not preserve the relation between strength and degree
    7. B.7 Scale down replicas
  3. C Results for sup-GRW
    1. C.1 sup-GRW in empirical data
    2. C.2 Semigroup structure in sup-GRW transformation
  4. D Results for sum-GRW
  5. E Results for $\phi$-GRW
    1. E.1 $\phi$-GRW in empirical data
    2. E.2 $\phi-$GRW in synthetic networks
      1. E.2.1 Semigroup structure
      2. E.2.2 The influence of $\alpha$
      3. E.2.3 The influence of $\eta$
  6. F The asymptotic behavior of the $\phi$-norm
    1. F.0.1 Synthetic weighted distributions
    2. F.0.2 Empirical weighted distributions
  7. G Results for random-GRW
    1. G.1 Random-GRW in synthetic network
    2. G.2 Random-GRW in E. coli network
  8. H Scaled down replicas of weighted networks with sup-GRW

##  A Illustration of the geometric renormalization transformation method for
weighted networks

Figure S1:  Geometric renormalization transformation for weighted networks.
Each layer is obtained after a GRW step with resolution $r$ starting from the
original network in $l=0$. Each node $i$ in red is placed at an angular
position on the similarity circle and has a size proportional to the logarithm
of its hidden degree. Straight solid lines represent the links in each layer
with weights denoted by their thickness. Coarse-graining blocks correspond to
the blue shadowed areas, and dashed lines connect nodes to their supernodes in
layer $l+1$. Two supernodes in layer $l+1$ are connected if and only if some
node of one supernode in layer $l$ is connected to some node of the other,
with the supremum among the weights of links between the constituent nodes as
the weight of the new connection (dark blue links give an example). The GRW
transformation has semigroup structure with respect to the composition. In the
figure, the transformation with $r=4$ goes from $l=0$ to $l=2$ in a single
step.

##  B Methods

###  B.1 Description of empirical data sets

  * •

Cargo ships. The international network of global cargo ship movements consists
of the number of shipping journeys between pairs of major commercial ports in
the world in 2007 Kaluza  _et al._ (2010).

  * •

E. coli. Weights in the metabolic network of the bacteria E. coli K-12 MG1655
consist of the number of different metabolic reactions in which two
metabolites participate Serrano  _et al._ (2012); Orth  _et al._ (2011).

  * •

US commute. The commuting network reflects the daily flow of commuters between
counties in the United States in 2000 Grady  _et al._ (2012).

  * •

Facebook like Social Network(Facebook). The Facebook-like Social Network
originate from an online community for students at University of California,
Irvine, in the period between April to October 2004 Panzarasa  _et al._
(2009); Opsahl and Panzarasa (2009). In this network, the nodes are students
and ties are established when online messages are exchanged between the
students. The weight of a directed tie is defined as the number of messages
sent from one student to another. We discard the directions for any link and
preserve the weight $\omega_{ij}$ with the sum of bidirectional messages,
i.e., $\omega_{ij}=\omega_{i\to j}+\omega_{j\to i}$. Notice that we only
consider the giant connected component of the undirected and weighted networks
in this paper.

  * •

Collaboration. This is the co-authorship network of based on preprints posted
to Condensed Matter section of arXiv E-Print Archive between 1995 and 1999
Newman (2001). Authors are identified with nodes, and an edge exists between
two scientists if they have coauthored at least one paper. The weights are the
sum of joint papers. Notice that we only consider the giant connected
component of the undirected and weighted networks in this paper.

  * •

Openflights. Network of flights among all commercial airports in the world, in
2010, derived from the Openflights.org database Opsahl (2011). Nodes represent
the airports. The weights in this network refer to the number of routes
between two airports. We discard the directions for any link and preserve the
weight $\omega_{ij}$ with the sum of bidirectional weights, i.e.,
$\omega_{ij}=\omega_{i\to j}+\omega_{j\to i}$. Notice that we only consider
the giant connected component of the undirected and weighted networks in this
paper.

  * •

Journal Citation Network (JCN). The citation networks from 1900 to 2013 were
reconstructed from data on citations between scientific articles extracted
from the Thomson Reuters Citation Index Hric  _et al._ (2018). A node
corresponds to a journal with publications in the given time period. An edge
is connected from journal $i$ to journal $j$ if an article in journal $i$
cites an article in journal $j$, and the weight of this link is taken to be
the number of such citations. In this work, we use undirected and weighted
networks generated from 3 different time windows, 2008-2013, 1985-1990 and
1965-1975. The data are obtained from Ref. Zheng  _et al._ (2021).

  * •

New Zealand Collaboration Network (NZCN). This is a network of scientific
collaborations among institutions in New Zealand. Nodes are institutions
(universities, organizations, etc.) and edges represent collaborations between
them. In particular, two nodes $i$, $j$ are connected if Scopus lists at least
one publication with authors at institutions $i$ and $j$, in the period
2010-2015. The weights of edges record the number of such collaborations. The
data are obtained from Ref. Aref  _et al._ (2018). Notice that we only
consider the giant connected component of the undirected and weighted networks
in this paper.

  * •

Poppy and foxglove hypocotyl cellular interaction networks. These networks
capture global cellular connectivity within the hypocotyl (embryonic stem) of
poppy and foxglove. Nodes represent cells and edges are their physical
associations in 3D space. Edges are weighted by the size of shared
intercellular interfaces, and nodes annotated with cell type. The data are
obtained from Ref. Jackson  _et al._ (2017).

Network statistics can be found in Table S1.

Table S1: Overview of the considered real-world networks. Columns are: the name of each network (Name), the number of nodes ($N$), the average degree ($\langle k\rangle$), the average local clustering coefficient ($\langle c\rangle$), the hyperbolic embedding parameter $\beta$ and $\mu$, fitting exponent ($\gamma$) in degree distribution, fitting parameters ($a$ and $\eta$) in strength-degree relations, the trade-off between the contribution of degrees and geometry to weights ($\alpha$), the noise ($\langle\epsilon^{2}\rangle$), parameter $\phi=\beta/(\eta-1+\alpha)$, and the references about the data sources (Ref.).  No. | Name | $N$ | $\langle k\rangle$ | $\langle c\rangle$ | $\beta$ | $\mu$ | $\gamma$ | $a$ | $\eta$ | $\alpha$ | $\langle\epsilon^{2}\rangle$ | $\phi$ | Ref.
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | Cargo ships | 821 | 10.58 | 0.51 | 1.88 | 0.028 | 3.3 | 86.10 | 1.04 | 0.66 | 1.50 | 2.68 | Kaluza  _et al._ (2010)
1 | E. coli | 1100 | 6.61 | 0.49 | 1.98 | 0.048 | 2.7 | 1.09 | 1.10 | 0.45 | 1.30 | 3.65 | Serrano  _et al._ (2012); Orth  _et al._ (2011)
2 | US commute | 3025 | 4.36 | 0.40 | 2.08 | 0.076 | 5.5 | 696.78 | 2.06 | 0.57 | 1.50 | 1.28 | Grady  _et al._ (2012)
3 | JCN(2008-2013) | 21460 | 49.79 | 0.59 | 1.70 | 0.005 | 3.4 | 7.35 | 1.41 | 0.62 | 1.30 | 1.65 | Hric  _et al._ (2018); Zheng  _et al._ (2021)
4 | JCN(1985-1990) | 7379 | 28.46 | 0.48 | 1.63 | 0.009 | 2.7 | 9.35 | 1.48 | 0.46 | 1.50 | 1.74 | Hric  _et al._ (2018); Zheng  _et al._ (2021)
5 | JCN(1965-1975) | 4168 | 26.84 | 0.61 | 1.93 | 0.011 | 3.0 | 17.52 | 1.49 | 0.18 | 1.70 | 2.90 | Hric  _et al._ (2018); Zheng  _et al._ (2021)
6 | Facebook | 1893 | 14.62 | 0.14 | 1.01 | 0.001 | 2.9 | 1.97 | 1.20 | 0.45 | 1.10 | 1.56 | Panzarasa  _et al._ (2009); Opsahl and Panzarasa (2009)
7 | Collaboration | 13861 | 6.44 | 0.72 | 5.42 | 0.073 | 3.8 | 1.32 | 1.08 | 0.50 | 1.10 | 9.41 | Newman (2001)
8 | Openflights | 2905 | 10.77 | 0.59 | 1.87 | 0.027 | 1.9 | 2.16 | 1.08 | 0.55 | 1.00 | 2.97 | Opsahl (2011)
9 | NZCN | 1463 | 5.80 | 0.85 | 14.33 | 0.085 | 2.3 | 0.90 | 1.50 | 0.80 | 1.80 | 11.00 | Aref  _et al._ (2018)
10 | Poppy | 2507 | 11.51 | 0.43 | 2.31 | 0.031 | 16.9 | 26.23 | 1.20 | 0.35 | 1.70 | 4.20 | Jackson  _et al._ (2017)
11 | Foxglove | 3005 | 11.34 | 0.43 | 2.33 | 0.032 | 13.0 | 14.69 | 1.46 | 0.39 | 1.70 | 2.72 | Jackson  _et al._ (2017)

###  B.2 Network embedding to produce geometric network maps

We embed each considered network into hyperbolic space using the algorithm
introduced in Ref. García-Pérez  _et al._ (2019), named Mercator. Mercator
takes the network adjacency matrix $A_{ij}$ ($A_{ij}=A_{ji}=1$ if there is a
link between nodes $i$ and $j$, and $A_{ij}=A_{ji}=0$ otherwise) as input and
then returns inferred hidden degrees, angular positions of nodes and global
model parameters. More precisely, the hyperbolic maps were inferred by finding
the hidden degree and angular position of each node, $\\{\kappa_{i}\\}$ and
$\\{\theta_{i}\\}$, that maximize the likelihood $\mathcal{L}$ that the
structure of the network was generated by the $\mathbb{S}^{1}$ model, where

$\displaystyle\mathcal{L}=\prod_{i<j}\left[p_{ij}\right]^{A_{ij}}\left[1-p_{ij}\right]^{1-A_{ij}}\
,$ (5)

and $p_{ij}=1/(1+\chi_{ij}^{\beta})$ is the connected probability.

###  B.3 The definition of disparity

The disparity of nodes. The disparity quantifies the local heterogeneity of
the weights attached to a given node $i$ and is defined as

$\displaystyle
Y(k_{i})=\sum_{j}\left(\frac{\omega_{ij}}{\sum_{j}\omega_{ij}}\right)^{2}$ (6)

where $\omega_{ij}$ is the weight of the link between node $i$ and its
neighbor $j$. From this definition, we see that the disparity scales as $Y\sim
k_{i}^{-1}$, whenever the weights are roughly homogeneously distributed among
the links. Conversely, whenever the disparity decreases slower than
$k_{i}^{-1}$ implies that weights are heterogeneous and that the large
strength of a node is due to a handful of links with large weights.

###  B.4 Theoretical derivation of the renormalized weights

Under GR, the hidden variables of supernodes in the resulting layer,
$\kappa^{\prime}$ and $\theta^{\prime}$, are calculated as a function of the
hidden variables of the constituent nodes as

$\displaystyle\kappa^{\prime}=\left[\sum_{j=1}^{r}(\kappa_{j})^{\beta}\right]^{1/\beta}\text{and
\hskip 2.84544pt
}\theta^{\prime}=\left[\frac{\sum_{j=1}^{r}(\theta_{j}\kappa_{j})^{\beta}}{\sum_{j=1}^{r}(\kappa_{j})^{\beta}}\right]^{1/\beta}.\
$ (7)

The expressions above and Eq. (1) in main text altogether imply that the
renormalized weight should be

$\displaystyle\omega^{\prime}_{ij}$
$\displaystyle=\frac{\nu^{\prime}\sigma^{\prime}_{i}\sigma^{\prime}_{j}}{\left(\kappa^{\prime}_{i}\kappa^{\prime}_{j}\right)^{1-\alpha^{\prime}/D}d_{ij}^{{}^{\prime}\alpha^{\prime}}}=\nu^{\prime}(a^{\prime})^{2}d_{ij}^{{}^{\prime}-\alpha^{\prime}}\left(\kappa^{\prime}_{i}\kappa^{\prime}_{j}\right)^{\eta-1+\alpha^{\prime}/D}$
(8)
$\displaystyle=\epsilon^{\prime}_{ij}\nu^{\prime}(a^{\prime})^{2}d_{ij}^{{}^{\prime}-\alpha^{\prime}}\left[\left(\kappa^{\prime}_{i}\kappa^{\prime}_{j}\right)^{\beta/D}\right]^{\frac{D(\eta-1)+\alpha^{\prime}}{\beta}}$
$\displaystyle=\nu^{\prime}(a^{\prime})^{2}d_{ij}^{{}^{\prime}-\alpha^{\prime}}\left[\sum\limits_{e=1}^{r^{2}}\left(\kappa_{m}\kappa_{n}\right)_{e}^{\beta/D}\right]^{\frac{D(\eta-1)+\alpha^{\prime}}{\beta}}$
$\displaystyle=\nu^{\prime}(a^{\prime})^{2}d_{ij}^{{}^{\prime}-\alpha^{\prime}}\left[\sum\limits_{e=1}^{r^{2}}\left(\frac{w_{mn}}{\nu
a^{2}d_{mn}^{-\alpha}}\right)_{e}^{\frac{\beta}{D(\eta-1)+\alpha}}\right]^{\frac{D(\eta-1)+\alpha^{\prime}}{\beta}}.$

In the last step, we have assumed that, for every pair of nodes $(m,n)$, we
can obtain the product $\kappa_{m}\kappa_{n}$ from the corresponding weight
$\omega_{mn}$, which is not true in general, as some links might not exist.
However, this should be a reasonable approximation, since it only misses the
smallest products of hidden degrees. Now, the above transformation cannot be
performed without the precise distances in the embedding, as it depends on
$d_{mn}$, but recalling that $d_{mn}=R\Delta\theta_{mn}$, where
$\Delta\theta_{mn}$ stands for the angular separation between the nodes, and
the fact that all such distances are approximately equal to the angular
separation between the supernodes to which the nodes belong
($\Delta\theta_{mn}\approx\Delta\theta^{\prime}_{ij}$), we can see that fixing
$\alpha^{\prime}=\alpha$ will remove all dependency on the distance,

$\displaystyle\omega^{\prime}_{ij}$
$\displaystyle=\nu^{\prime}(a^{\prime})^{2}d_{ij}^{{}^{\prime}-\alpha}\left[\sum\limits_{e=1}^{r^{2}}\left(\frac{w_{mn}}{\nu
a^{2}d_{mn}^{-\alpha}}\right)_{e}^{\frac{\beta}{D(\eta-1)+\alpha}}\right]^{\frac{D(\eta-1)+\alpha}{\beta}}$
(9)
$\displaystyle=\frac{\nu^{\prime}}{\nu}\left(\frac{a^{\prime}}{a}\right)^{2}\left(\frac{R\Delta\theta^{\prime}_{ij}}{R^{\prime}\Delta\theta^{\prime}_{ij}}\right)^{\alpha}\left[\sum\limits_{e=1}^{r^{2}}\left(w_{mn}\right)_{e}^{\frac{\beta}{D(\eta-1)+\alpha}}\right]^{\frac{D(\eta-1)+\alpha}{\beta}}$
$\displaystyle=\frac{\nu^{\prime}}{\nu}\left(\frac{a^{\prime}}{a}\right)^{2}r^{\alpha/D}\left[\sum\limits_{e=1}^{r^{2}}\left(w_{mn}\right)_{e}^{\frac{\beta}{D(\eta-1)+\alpha}}\right]^{\frac{D(\eta-1)+\alpha}{\beta}},$

where we have used that $R^{\prime}=R/r^{1/D}$.

Finally, we can choose any appropriate relation between primed and unprimed
global parameters leading to

$\omega^{\prime}_{ij}=C\left[\sum\limits_{e=1}^{r^{2}}\left(w_{mn}\right)_{e}^{\phi}\right]^{1/\phi},$
(10)

with $\phi\equiv\frac{\beta}{D(\eta-1)+\alpha}$ and
$C=\frac{\nu^{\prime}}{\nu}\left(\frac{a^{\prime}}{a}\right)^{2}r^{\alpha/D}$.
Therefore, the weighted model predicts that the exponent $\eta$ characterizing
the relation between strength and degree is preserved in the renormalized
network if weights are transformed following Eq. (10) (in the noiseless case)
and the value of $\phi$ that corresponds to the considered network is used.

###  B.5 Theoretical derivation of the flow of the average strength

We start from Eq. (10) ($D=1$) and impose that the rescaling variable

$C=\frac{\nu^{\prime}}{\nu}\left(\frac{a^{\prime}}{a}\right)^{2}r^{\alpha}$

is constant in the flow such that the transformation of weights keeps the same
units in all scales of observation. The transformation of the relation between
hidden strength and hidden degree is

$\frac{a^{\prime}}{a}=\frac{\langle\sigma\rangle^{\prime}}{\langle\sigma\rangle}\frac{\langle\kappa^{\eta}\rangle}{\langle\kappa^{\eta}\rangle^{\prime}}.$

We can also obtain the transformation of the free parameter $\nu$ using its
expression from Allard  _et al._ (2017) and the expression for the parameter
$\mu$,

$\displaystyle\nu=\frac{\Gamma(1/2)}{2\pi^{1/2}\mu^{1-\alpha}I_{2}I_{3}\langle\sigma\rangle}$
$\displaystyle\mu=\frac{\Gamma(1/2)}{2\pi^{1/2}I_{1}\langle k\rangle},$

which leads to

$\frac{\nu^{\prime}}{\nu}=\frac{\langle\sigma\rangle}{\langle\sigma\rangle^{\prime}}\left(\frac{\langle
k\rangle^{\prime}}{\langle k\rangle}\right)^{1-\alpha}$

and therefore to

$C=\frac{\langle\sigma\rangle^{\prime}}{\langle\sigma\rangle}\left(\frac{\langle\kappa^{\eta}\rangle}{\langle\kappa^{\eta}\rangle^{\prime}}\right)^{2}r^{\xi(1-\alpha)+\alpha},$

where we have used the expression for the flow of the average degree. We use
$\langle\kappa^{\eta}\rangle=\frac{\gamma-1}{\gamma-1-\eta}\kappa_{0}^{\eta}$
($\eta<\gamma-1$) to compute its flow, and we obtain

$\frac{\langle\kappa^{\eta}\rangle^{\prime}}{\langle\kappa^{\eta}\rangle}=r^{\xi\eta}.$

Finally,

$C=\frac{\langle\sigma\rangle^{\prime}}{\langle\sigma\rangle}r^{\xi(1-2\eta-\alpha)+\alpha}=\frac{\langle\sigma\rangle^{\prime}}{\langle\sigma\rangle}r^{-\psi},$

and we impose $C=1$ to obtain

$\langle\sigma^{\prime}\rangle=\langle\sigma\rangle r^{\psi},\mbox{
}\psi=\left(\frac{\alpha}{D}+2\eta-1\right)\xi-\frac{\alpha}{D},$ (11)

from which $\psi>0$ implies an increasing average strength in the flow while
it decreases if $\psi<0$.

###  B.6 The transformation sum-GRW does not preserve the relation between
strength and degree

The sum-GRW transformation is

$\displaystyle w^{\prime}_{ij}$
$\displaystyle=\sum\limits_{e=1}^{r^{2}}\epsilon_{mn}w_{mn}$ (12)
$\displaystyle=\nu
d_{mn}^{-\alpha}\sum\limits_{e=1}^{r^{2}}\epsilon_{mn}\sigma_{m}\sigma_{n}(\kappa_{m}\kappa_{n})^{\alpha/D-1},$

where $e$ runs over all pairs of nodes $(m,n)$ with $m$ in supernode $i$ and
$n$ in supernode $j$ and $d_{mn}=R\Delta\theta_{mn}$, where
$\Delta\theta_{mn}$ stands for the angular separation between the nodes. All
such distances are approximately equal to the angular separation between the
supernodes to which the nodes belong
($\Delta\theta_{mn}\approx\Delta\theta^{\prime}_{ij}$), and one can take
$\alpha^{\prime}=\alpha$. Comparing Eq. (1) in main text and (12), we can
write

$\displaystyle\nu^{\prime}d_{ij}^{{}^{\prime}-\alpha^{\prime}}=\nu
d_{ij}^{-\alpha},$ (13)
$\displaystyle\epsilon_{ij}^{\prime}\sigma_{i}^{\prime}\sigma_{j}^{\prime}(\kappa_{i}^{\prime}\kappa_{j}^{\prime})^{\alpha^{\prime}/D-1}$
$\displaystyle=\sum\limits_{e=1}^{r^{2}}\epsilon_{mn}\sigma_{m}\sigma_{n}(\kappa_{m}\kappa_{n})^{\alpha/D-1},$
(14)

and using $R^{\prime}=R/r^{1/D}$ and Eq. (13) altogether, we have

$\displaystyle\nu^{\prime}=\nu r^{-\alpha/D}.$ (15)

Therefore, in the noiseless version ($\epsilon_{mn}=1\;\forall(m,n)$), we can
obtain the hidden strength $\sigma_{i}^{\prime}$ in the supernodes layer as

$\displaystyle\sigma_{i}^{\prime}=\frac{\sum\limits_{m=1}^{r}\sigma_{m}\kappa_{m}^{\alpha/D-1}}{\kappa_{i}^{\prime\alpha/D-1}}\not\approx\kappa_{i}^{\prime\eta},$
(16)

which proves that, in general, the relation between hidden strength and hidden
degree is not preserved under sum-GRW.

###  B.7 Scale down replicas

  1. 1.

We obtain a renormalized network layer by applying the sup-GRW method with a
given value of $r$ and number of iterations to match the target network size.

  2. 2.

Typically, the average degree of the renormalized network layer is higher than
the original one. Thus, to obtain a scaled down network replica of the
topology, we decrease the average degree in the renormalized layer to that in
the original network as explained in Ref. García-Pérez  _et al._ (2018), such
that $\langle k_{\mathrm{new}}^{(l)}\rangle=\langle k^{(0)}\rangle$. The main
idea is to reduce the value of $\mu^{(l)}$ to a new one $\mu_{new}^{(l)}$,
which means that the connection probability of every pair of nodes $(i,j)$,
$p_{ij}^{(l)}$ decreases to $p_{ij,new}^{(l)}$. Therefore, the probability for
a link to exist in the pruned network reads:

$\displaystyle
p_{ij,new}^{(l)}=\frac{1}{1+\left(\frac{d_{ij}}{\mu_{new}^{(l)}\kappa_{i}\kappa_{j}}\right)^{\beta}}.$
(17)

In particular, we prune the links using
$\mu_{\mathrm{new}}^{(l)}=h\frac{\langle k^{(0)}\rangle}{\langle
k^{(l)}\rangle}\mu^{(l)}$ with $h=1$ as initial value. After an iteration for
all the links in the layer, we give $h$ a new value $h(1-0.1u)\to h$ if
$\langle k_{\mathrm{new}}^{(l)}\rangle>\langle k^{(0)}\rangle$, where
$u\in[0,1)$ is a random variable from a uniform distribution. If $\langle
k_{\mathrm{new}}^{(l)}\rangle<\langle k^{(0)}\rangle$, $h(1+0.1u)\to h$. The
procedure stops when $|\langle k_{\mathrm{new}}^{(l)}\rangle-\langle
k^{(0)}\rangle|$ is below a given threshold, that we set to $0.1$.

  3. 3.

Finally, we rescale the weights in the resulting network by a global factor to
match the average weight of the original network. Specifically, we calculate
the average weight $\langle w_{\mathrm{new}}^{(l)}\rangle$ of the resulting
network from step (2) and the average weight $\langle w^{(0)}\rangle$ in the
original network. Then we rescale the weight of each link by the factor
$c=\frac{\langle w^{(0)}\rangle}{\langle w_{\mathrm{new}}^{(l)}\rangle}$.

##  C Results for sup-GRW

###  C.1 sup-GRW in empirical data

Figure S2: Network properties for sup-GRW in different empirical networks.
First column: complementary cumulative degree distribution
$P_{c}^{(l)}(k_{res}^{(l)})$, Second column: the degree-dependent clustering
coefficient $\bar{c}^{(l)}(k_{res}^{(l)})$, Last column: normalized average
nearest-neighbour degree
$\bar{k}_{nn,n}^{(l)}(k_{res}^{(l)})=\bar{k}_{nn}^{(l)}(k_{res}^{(l)})\langle
k^{(l)}\rangle/\langle(k^{(l)})^{2}\rangle$ of rescaled degrees
$k^{(l)}_{res}$ for different layers $l$. Each row indicates an empirical
network. Figure S3: Network properties for sup-GRW in different empirical
networks. First column: complementary cumulative degree distribution
$P_{c}^{(l)}(k_{res}^{(l)})$, Second column: the degree-dependent clustering
coefficient $\bar{c}^{(l)}(k_{res}^{(l)})$, Last column: normalized average
nearest-neighbour degree
$\bar{k}_{nn,n}^{(l)}(k_{res}^{(l)})=\bar{k}_{nn}^{(l)}(k_{res}^{(l)})\langle
k^{(l)}\rangle/\langle(k^{(l)})^{2}\rangle$ of rescaled degrees
$k^{(l)}_{res}$ for different layers $l$. Each row indicates an empirical
network. Figure S4: Network properties for sup-GRW in different empirical
networks. First column: complementary cumulative weight distributions
$P_{c}^{(l)}(w_{res}^{(l)})$ of rescaled weights
$w^{(l)}_{res}=w^{(l)}/\langle w^{(l)}\rangle$ for different layers $l$.
Second column: complementary cumulative strength distributions
$P_{c}^{(l)}(s_{res}^{(l)})$ of rescaled strengths
$s^{(l)}_{res}=s^{(l)}/\langle s^{(l)}\rangle$ for different layers $l$. Third
column: average rescaled strengths as a function of rescaled degrees, i.e.,
$\bar{s}_{res}^{(l)}(k_{res}^{(l)})=\bar{s}^{(l)}(k)/\langle s\rangle^{(l)}$.
Inset shows average strength as a function of degree. Last column: disparity
of nodes as a function of their degree. Each row indicates an empirical
network. Figure S5: Network properties for sup-GRW in different empirical
networks. First column: complementary cumulative weight distributions
$P_{c}^{(l)}(w_{res}^{(l)})$ of rescaled weights
$w^{(l)}_{res}=w^{(l)}/\langle w^{(l)}\rangle$ for different layers $l$.
Second column: complementary cumulative strength distributions
$P_{c}^{(l)}(s_{res}^{(l)})$ of rescaled strengths
$s^{(l)}_{res}=s^{(l)}/\langle s^{(l)}\rangle$ for different layers $l$. Third
column: average rescaled strengths as a function of rescaled degrees, i.e.,
$\bar{s}_{res}^{(l)}(k_{res}^{(l)})=\bar{s}^{(l)}(k)/\langle s\rangle^{(l)}$.
Inset shows average strength as a function of degree. Last column: disparity
of nodes as a function of their degree. Each row indicates an empirical
network. Figure S6: (a) average clustering coefficient, (b) average degree,
(c) rescaled average weight and (d) corresponding average strength, (e)
unrescaled average weight and (f) corresponding average strength for different
layers $l$.  Figure S7: (a) average clustering coefficient, (b) average
degree, (c) rescaled average weight and (d) corresponding average strength,
(e) unrescaled average weight and (f) corresponding average strength for
different layers $l$.

###  C.2 Semigroup structure in sup-GRW transformation

The geometric renormalization transformation has Abelian semigroup structure
with respect to the composition, meaning that a certain number of iterations
of a given resolution are equivalent to a single transformation of higher
resolution. We here validated the semigroup structure in sup-GRW
transformation with synthetic and empirical networks. Given an original
network, we performed the sup-GRW with $r=2$ and $r=4$, respectively. When the
geometric renormalization transformation to the same network size, we compared
the their network properties. Figure S8 shows the results for a representative
synthetic network.

Figure S8: Semigroup structure of sup-GRW in a synthetic network with $r=2$
and $r=4$. (a) Complementary cumulative degree distribution
$P_{c}^{(l)}(k_{res}^{(l)})$, (b) The degree-dependent clustering coefficient
$\bar{c}^{(l)}(k_{res}^{(l)})$, (c) Normalized average nearest-neighbour
degree
$\bar{k}_{nn,n}^{(l)}(k_{res}^{(l)})=\bar{k}_{nn}^{(l)}(k_{res}^{(l)})\langle
k^{(l)}\rangle/\langle(k^{(l)})^{2}\rangle$ of rescaled degrees
$k^{(l)}_{res}$ for different layers $l$. (d) Disparity of nodes as a function
of their degree. (e) Complementary cumulative weight distributions
$P_{c}^{(l)}(w_{res}^{(l)})$ of rescaled weights
$w^{(l)}_{res}=w^{(l)}/\langle w^{(l)}\rangle$ for different layers $l$. (f)
Complementary cumulative strength distributions $P_{c}^{(l)}(s_{res}^{(l)})$
of rescaled strengths $s^{(l)}_{res}=s^{(l)}/\langle s^{(l)}\rangle$ for
different layers $l$. (g) average rescaled strengths as a function of rescaled
degrees, i.e., $\bar{s}_{res}^{(l)}(k_{res}^{(l)})=\bar{s}^{(l)}(k)/\langle
s\rangle^{(l)}$. Inset shows average strength as a function of degree. (h)
Average strength $\langle s\rangle$ as a function of the network size $N$. The
parameters are $N=58411$, $\beta=1.5$, $\mu=0.0413$, $a=100$, $\eta=1.5$,
$\alpha=0.40$, $\gamma=2.5$, and $\langle\epsilon^{2}\rangle=1.0$,
$\phi=1.67$. Here symbols show the case of $r=2$ and lines are the one of
$r=4$.

##  D Results for sum-GRW

Figure S9: Network properties for sum-GRW in different empirical networks.
First column: complementary cumulative weight distributions
$P_{c}^{(l)}(w_{res}^{(l)})$ of rescaled weights
$w^{(l)}_{res}=w^{(l)}/\langle w^{(l)}\rangle$ for different layers $l$.
Second column: complementary cumulative strength distributions
$P_{c}^{(l)}(s_{res}^{(l)})$ of rescaled strengths
$s^{(l)}_{res}=s^{(l)}/\langle s^{(l)}\rangle$ for different layers $l$. Third
column: average rescaled strengths as a function of rescaled degrees, i.e.,
$\bar{s}_{res}^{(l)}(k_{res}^{(l)})=\bar{s}^{(l)}(k)/\langle s\rangle^{(l)}$.
Inset shows average strength as a function of degree. Last column: disparity
of nodes as a function of their degree. Each row indicates an empirical
network. Figure S10: Network properties for sum-GRW in different empirical
networks. First column: complementary cumulative weight distributions
$P_{c}^{(l)}(w_{res}^{(l)})$ of rescaled weights
$w^{(l)}_{res}=w^{(l)}/\langle w^{(l)}\rangle$ for different layers $l$.
Second column: complementary cumulative strength distributions
$P_{c}^{(l)}(s_{res}^{(l)})$ of rescaled strengths
$s^{(l)}_{res}=s^{(l)}/\langle s^{(l)}\rangle$ for different layers $l$. Third
column: average rescaled strengths as a function of rescaled degrees, i.e.,
$\bar{s}_{res}^{(l)}(k_{res}^{(l)})=\bar{s}^{(l)}(k)/\langle s\rangle^{(l)}$.
Inset shows average strength as a function of degree. Last column: disparity
of nodes as a function of their degree. Each row indicates an empirical
network.

##  E Results for $\phi$-GRW

###  E.1 $\phi$-GRW in empirical data

Figure S11: Network properties for $\phi$-GRW in different empirical networks.
First column: complementary cumulative weight distributions
$P_{c}^{(l)}(w_{res}^{(l)})$ of rescaled weights
$w^{(l)}_{res}=w^{(l)}/\langle w^{(l)}\rangle$ for different layers $l$.
Second column: complementary cumulative strength distributions
$P_{c}^{(l)}(s_{res}^{(l)})$ of rescaled strengths
$s^{(l)}_{res}=s^{(l)}/\langle s^{(l)}\rangle$ for different layers $l$. Third
column: average rescaled strengths as a function of rescaled degrees, i.e.,
$\bar{s}_{res}^{(l)}(k_{res}^{(l)})=\bar{s}^{(l)}(k)/\langle s\rangle^{(l)}$.
Inset shows average strength as a function of degree. Last column: disparity
of nodes as a function of their degree. Each row indicates an empirical
network. Figure S12: Network properties for $\phi$-GRW in different empirical
networks. First column: complementary cumulative weight distributions
$P_{c}^{(l)}(w_{res}^{(l)})$ of rescaled weights
$w^{(l)}_{res}=w^{(l)}/\langle w^{(l)}\rangle$ for different layers $l$.
Second column: complementary cumulative strength distributions
$P_{c}^{(l)}(s_{res}^{(l)})$ of rescaled strengths
$s^{(l)}_{res}=s^{(l)}/\langle s^{(l)}\rangle$ for different layers $l$. Third
column: average rescaled strengths as a function of rescaled degrees, i.e.,
$\bar{s}_{res}^{(l)}(k_{res}^{(l)})=\bar{s}^{(l)}(k)/\langle s\rangle^{(l)}$.
Inset shows average strength as a function of degree. Last column: disparity
of nodes as a function of their degree. Each row indicates an empirical
network.

###  E.2 $\phi-$GRW in synthetic networks

####  E.2.1 Semigroup structure

Figure S13: Semigroup structure of $\phi-$GRW in synthetic network with $r=2$
and $r=4$(a) Complementary cumulative degree distribution
$P_{c}^{(l)}(k_{res}^{(l)})$, (b) The degree-dependent clustering coefficient
$\bar{c}^{(l)}(k_{res}^{(l)})$, (c) Normalized average nearest-neighbour
degree
$\bar{k}_{nn,n}^{(l)}(k_{res}^{(l)})=\bar{k}_{nn}^{(l)}(k_{res}^{(l)})\langle
k^{(l)}\rangle/\langle(k^{(l)})^{2}\rangle$ of rescaled degrees
$k^{(l)}_{res}$ for different layers $l$. (d) Disparity of nodes as a function
of their degree. (e) Complementary cumulative weight distributions
$P_{c}^{(l)}(w_{res}^{(l)})$ of rescaled weights
$w^{(l)}_{res}=w^{(l)}/\langle w^{(l)}\rangle$ for different layers $l$. (f)
Complementary cumulative strength distributions $P_{c}^{(l)}(s_{res}^{(l)})$
of rescaled strengths $s^{(l)}_{res}=s^{(l)}/\langle s^{(l)}\rangle$ for
different layers $l$. (g) average rescaled strengths as a function of rescaled
degrees, i.e., $\bar{s}_{res}^{(l)}(k_{res}^{(l)})=\bar{s}^{(l)}(k)/\langle
s\rangle^{(l)}$. Inset shows average strength as a function of degree. (h)
Average strength $\langle s\rangle$ as a function of the network size $N$. The
parameters are $N=58411$, $\beta=1.5$, $\mu=0.0413$, $a=100$, $\eta=1.5$,
$\alpha=0.40$, $\gamma=2.5$, and $\langle\epsilon^{2}\rangle=1.0$,
$\phi=1.67$. Here symbols show the case of $r=2$ and lines are the one of
$r=4$.

####  E.2.2 The influence of $\alpha$

Figure S14: $\phi$-GRW in synthetic network with different $\alpha$. First
column: complementary cumulative degree distribution
$P_{c}^{(l)}(k_{res}^{(l)})$, Second column: the degree-dependent clustering
coefficient $\bar{c}^{(l)}(k_{res}^{(l)})$, Last column: normalized average
nearest-neighbour degree
$\bar{k}_{nn,n}^{(l)}(k_{res}^{(l)})=\bar{k}_{nn}^{(l)}(k_{res}^{(l)})\langle
k^{(l)}\rangle/\langle(k^{(l)})^{2}\rangle$ of rescaled degrees
$k^{(l)}_{res}$ for different layers $l$. Each row indicates a synthetic
network. The parameters are $N=58948$, $\beta=1.5$, $\mu=0.0413$, $a=100$,
$\eta=1.0$, $\gamma=2.5$ and $\langle\epsilon^{2}\rangle=1.0$. Figure S15:
$\phi$-GRW in synthetic network with different $\alpha$. First column:
complementary cumulative weight distributions $P_{c}^{(l)}(w_{res}^{(l)})$ of
rescaled weights $w^{(l)}_{res}=w^{(l)}/\langle w^{(l)}\rangle$ for different
layers $l$. Second column: complementary cumulative strength distributions
$P_{c}^{(l)}(s_{res}^{(l)})$ of rescaled strengths
$s^{(l)}_{res}=s^{(l)}/\langle s^{(l)}\rangle$ for different layers $l$. Third
column: average rescaled strengths as a function of rescaled degrees, i.e.,
$\bar{s}_{res}^{(l)}(k_{res}^{(l)})=\bar{s}^{(l)}(k)/\langle s\rangle^{(l)}$.
Inset shows average strength as a function of degree. Last column: disparity
of nodes as a function of their degree. Each row indicates a synthetic
network.The parameters are $N=58948$, $\beta=1.5$, $\mu=0.0413$, $a=100$,
$\eta=1.0$, $\gamma=2.5$ and $\langle\epsilon^{2}\rangle=1.0$.

####  E.2.3 The influence of $\eta$

Figure S16: $\phi$-GRW in synthetic network with different $\eta$. First
column: complementary cumulative degree distribution
$P_{c}^{(l)}(k_{res}^{(l)})$, Second column: the degree-dependent clustering
coefficient $\bar{c}^{(l)}(k_{res}^{(l)})$, Last column: normalized average
nearest-neighbour degree
$\bar{k}_{nn,n}^{(l)}(k_{res}^{(l)})=\bar{k}_{nn}^{(l)}(k_{res}^{(l)})\langle
k^{(l)}\rangle/\langle(k^{(l)})^{2}\rangle$ of rescaled degrees
$k^{(l)}_{res}$ for different layers $l$. Each row indicates a synthetic
network. The parameters are $N=58948$, $\beta=1.5$, $\mu=0.0413$, $a=100$,
$\eta=1.0$, $\gamma=2.5$ and $\langle\epsilon^{2}\rangle=1.0$. Figure S17:
$\phi$-GRW in synthetic network with different $\eta$. First column:
complementary cumulative weight distributions $P_{c}^{(l)}(w_{res}^{(l)})$ of
rescaled weights $w^{(l)}_{res}=w^{(l)}/\langle w^{(l)}\rangle$ for different
layers $l$. Second column: complementary cumulative strength distributions
$P_{c}^{(l)}(s_{res}^{(l)})$ of rescaled strengths
$s^{(l)}_{res}=s^{(l)}/\langle s^{(l)}\rangle$ for different layers $l$. Third
column: average rescaled strengths as a function of rescaled degrees, i.e.,
$\bar{s}_{res}^{(l)}(k_{res}^{(l)})=\bar{s}^{(l)}(k)/\langle s\rangle^{(l)}$.
Inset shows average strength as a function of degree. Last column: disparity
of nodes as a function of their degree. Each row indicates a synthetic
network.The parameters are $N=58948$, $\beta=1.5$, $\mu=0.0413$, $a=100$,
$\eta=1.0$, $\gamma=2.5$ and $\langle\epsilon^{2}\rangle=1.0$.

##  F The asymptotic behavior of the $\phi$-norm

To better understand the behavior of $\phi$ normalization in different
weighted distributions, we perform the following test. We generate the sampled
weights $\omega_{mn}$ according to the weight distribution
$p(\omega_{mn})\sim\omega_{mn}^{-\delta}$. The smaller values of the exponent
$\delta$, the more heterogeneous of the distributions.With the sampled weights
$\omega_{mn}$ on hand, we can normalize the weight as:

$\omega^{\prime}(\phi)=\left[\sum\limits_{e=1}^{E}\left(\omega_{mn}\right)^{\phi}\right]^{1/\phi},$
(18)

where $E$ is the number of sampled weights combining into new weight
$\omega^{\prime}$. The implementation process is as follows.

(1) We firstly generate a weight list $w_{list}$ with sampled length $N=20000$
from distribution $p(\omega_{mn})\sim\omega_{mn}^{-\delta}$.

(2) We divide the weight list $w_{list}$ into non-overlapping groups in
sequence, where each group’s size equals $E$. In other words, each group has
$E$ samples $\omega_{mn}$. We then calculate $\omega^{\prime}(\phi)$ wit
Eq.(18) in each group for different $\phi$.

(3) We compare $\omega^{\prime}(\phi)$ with $\omega^{\prime}(\phi=1)$ and
$\omega^{\prime}(\phi=\infty)$. Note that sum-GRW corresponds to the case
$\phi=1$ while sup-GRW to $\phi=\infty$.

We have two ways to check the asymptotic behavior of the $\phi$-norm in the
empirical weighted distributions. The first one is as simple as the one in
synthetic distribution. We only need to replace the weight list $w_{list}$
with the empirical data. In this case, the samples $\omega_{mn}$ in each group
are uncorrelated. However, in the $\phi$-GRW process, the weights in the same
group may relate to the coordinates of sub-nodes $m$ and $n$. Therefore, we
implement the second way to check the asymptotic behavior.

(1) We implement the $\phi$-GRW process with one layer. There may have $E$
weights links between the constituent nodes of two supernodes. Note that when
$r=2$, the number of links $E$ could be $1$, $2$, $3$ or $4$. So, we divide
the empirical weights into different group, where each group’s size equals
$E$.

(2) We then calculate $\omega^{\prime}(\phi)$ wit Eq.(18) in each group for
different $\phi$.

(3) We compare $\omega^{\prime}(\phi)$ with $\omega^{\prime}(\phi=1)$ and
$\omega^{\prime}(\phi=\infty)$. Note that sum-GRW corresponds to the case
$\phi=1$ while sup-GRW to $\phi=\infty$.

In the end, we find that the results obtained by these two ways are robust. We
only show the results for the sets following the coarse-graining (i.e., the
second way) in this paper.

####  F.0.1 Synthetic weighted distributions

Figure S18: The normalized weight $\omega^{\prime}(\phi=1)$ and
$\omega^{\prime}(\phi=\infty)$ versus $\omega^{\prime}(\phi)$ for different
$E$ and $\phi$. Note that sum-GRW corresponds to the case $\phi=1$ while sup-
GRW to $\phi=\infty$. Figure S19: The normalized weight
$\omega^{\prime}(\phi=1)$ and $\omega^{\prime}(\phi=\infty)$ versus
$\omega^{\prime}(\phi)$ for different $E$ and $\phi$. Note that sum-GRW
corresponds to the case $\phi=1$ while sup-GRW to $\phi=\infty$.

####  F.0.2 Empirical weighted distributions

Figure S20: The normalized weight $\omega^{\prime}(\phi=1)$ and
$\omega^{\prime}(\phi=\infty)$ versus $\omega^{\prime}(\phi)$ for different
$E$ and inferred $\phi$. Note that sum-GRW corresponds to the case $\phi=1$
while sup-GRW to $\phi=\infty$. Figure S21: The normalized weight
$\omega^{\prime}(\phi=1)$ and $\omega^{\prime}(\phi=\infty)$ versus
$\omega^{\prime}(\phi)$ for different $E$ and inferred $\phi$. Note that sum-
GRW corresponds to the case $\phi=1$ while sup-GRW to $\phi=\infty$.

##  G Results for random-GRW

A prescription selecting the weight between two supernodes at random from the
coarse-grainable set would always result in the self-similarity of the
distribution of weights if the selection set is supplemented with the links
between nodes in the same supernode. However, those links are coarse-grained
in the renormalization process and the balance between the weights of links
inside supernodes and of links between nodes in different supernodes dictates
in which situations the random selection works. Experiments in synthetic
networks, Figs. S22 and S23, prove that the heterogeneity of the distribution
of weights favors a better self-similar scaling. Decreasing the coupling of
weights with topology and geometry in the W$\mathbb{S}^{D}$ model produces
more homogeneous distributions of weights, which causes the loss of self-
similarity in the flow, see SI Fig. S24 for results relative to the metabolic
network of E. coli.

###  G.1 Random-GRW in synthetic network

Figure S22: Random-GRW in synthetic network with different $\alpha$. First
column: complementary cumulative weight distributions
$P_{c}^{(l)}(w_{res}^{(l)})$ of rescaled weights
$w^{(l)}_{res}=w^{(l)}/\langle w^{(l)}\rangle$ for different layers $l$.
Second column: complementary cumulative strength distributions
$P_{c}^{(l)}(s_{res}^{(l)})$ of rescaled strengths
$s^{(l)}_{res}=s^{(l)}/\langle s^{(l)}\rangle$ for different layers $l$. Third
column: average rescaled strengths as a function of rescaled degrees, i.e.,
$\bar{s}_{res}^{(l)}(k_{res}^{(l)})=\bar{s}^{(l)}(k)/\langle s\rangle^{(l)}$.
Inset shows average strength as a function of degree. Last column: disparity
of nodes as a function of their degree. Each row indicates a synthetic
network.The parameters are The parameters are $N=64000$, $\beta=1.5$,
$\mu=0.0059$, $a=100$, $\eta=1.0$, $\gamma=2.5$, $\langle k\rangle=35$ and
$\langle\epsilon^{2}\rangle=1.0$. Figure S23: Random-GRW in synthetic network
with different $\eta$. First column: complementary cumulative weight
distributions $P_{c}^{(l)}(w_{res}^{(l)})$ of rescaled weights
$w^{(l)}_{res}=w^{(l)}/\langle w^{(l)}\rangle$ for different layers $l$.
Second column: complementary cumulative strength distributions
$P_{c}^{(l)}(s_{res}^{(l)})$ of rescaled strengths
$s^{(l)}_{res}=s^{(l)}/\langle s^{(l)}\rangle$ for different layers $l$. Third
column: average rescaled strengths as a function of rescaled degrees, i.e.,
$\bar{s}_{res}^{(l)}(k_{res}^{(l)})=\bar{s}^{(l)}(k)/\langle s\rangle^{(l)}$.
Inset shows average strength as a function of degree. Last column: disparity
of nodes as a function of their degree. Each row indicates a synthetic
network.The parameters are The parameters are $N=64000$, $\beta=1.5$,
$\mu=0.0059$, $a=100$, $\alpha=0.40$, $\gamma=2.5$, $\langle k\rangle=35$ and
$\langle\epsilon^{2}\rangle=1.0$.

###  G.2 Random-GRW in E. coli network

Figure S24: random-GRW in E. coli network. (a) Complementary cumulative weight
distributions $P_{c}^{(l)}(w_{res}^{(l)})$ of rescaled weights
$w^{(l)}_{res}=w^{(l)}/\langle w^{(l)}\rangle$ for different layers $l$. (b)
Complementary cumulative strength distributions $P_{c}^{(l)}(s_{res}^{(l)})$
of rescaled strengths $s^{(l)}_{res}=s^{(l)}/\langle s^{(l)}\rangle$ for
different layers $l$. (c) average rescaled strengths as a function of rescaled
degrees, i.e., $\bar{s}_{res}^{(l)}(k_{res}^{(l)})=\bar{s}^{(l)}(k)/\langle
s\rangle^{(l)}$. Inset shows average strength as a function of degree. (d)
Disparity of nodes as a function of their degree.

##  H Scaled down replicas of weighted networks with sup-GRW

Figure S25: Scaled down replicas with sup-GRW in different empirical networks.
First column: complementary cumulative degree distribution
$P_{c}^{(l)}(k^{(l)})$, Second column: the degree-dependent clustering
coefficient $\bar{c}^{(l)}(k^{(l)})$, Last column: normalized average nearest-
neighbour degree
$\bar{k}_{nn,n}^{(l)}(k^{(l)})=\bar{k}_{nn}^{(l)}(k^{(l)})\langle
k^{(l)}\rangle/\langle(k^{(l)})^{2}\rangle$ of degrees $k^{(l)}$ for different
layers $l$. Each row indicates an empirical network. Figure S26: Scaled down
replicas with sup-GRW in different empirical networks. First column:
complementary cumulative degree distribution $P_{c}^{(l)}(k^{(l)})$, Second
column: the degree-dependent clustering coefficient $\bar{c}^{(l)}(k^{(l)})$,
Last column: normalized average nearest-neighbour degree
$\bar{k}_{nn,n}^{(l)}(k^{(l)})=\bar{k}_{nn}^{(l)}(k^{(l)})\langle
k^{(l)}\rangle/\langle(k^{(l)})^{2}\rangle$ of degrees $k^{(l)}$ for different
layers $l$. Each row indicates an empirical network. Figure S27: Scaled down
replicas with sup-GRW in different empirical networks. First column:
complementary cumulative weight distributions $P_{c}^{(l)}(w^{(l)})$ for
different layers $l$. Second column: complementary cumulative strength
distributions $P_{c}^{(l)}(s^{(l)})$ for different layers $l$. Third column:
average strength $\bar{s}^{(l)}(k^{(l)})$ as a function of degree $k^{(l)}$.
Last column: disparity of nodes as a function of their degree. Each row
indicates an empirical network. Figure S28: Scaled down replicas with sup-GRW
in different empirical networks. First column: complementary cumulative weight
distributions $P_{c}^{(l)}(w^{(l)})$ for different layers $l$. Second column:
complementary cumulative strength distributions $P_{c}^{(l)}(s^{(l)})$ for
different layers $l$. Third column: average strength $\bar{s}^{(l)}(k^{(l)})$
as a function of degree $k^{(l)}$. Last column: disparity of nodes as a
function of their degree. Each row indicates an empirical network.

## References

  * Kaluza  _et al._ (2010) P. Kaluza, A. Kölzsch, M. T. Gastner, and B. Blasius, The complex network of global cargo ship movements, Journal of the Royal Society Interface 7, 1093 (2010).
  * Serrano  _et al._ (2012) M. Á. Serrano, M. Boguná, and F. Sagués, Uncovering the hidden geometry behind metabolic networks, Molecular Biosystems 8, 843 (2012).
  * Orth  _et al._ (2011) J. D. Orth, T. M. Conrad, J. Na, J. A. Lerman, H. Nam, A. M. Feist, and B. Ø. Palsson, A comprehensive genome-scale reconstruction of escherichia coli metabolismâ€”2011, Molecular Systems Biology 7, 535 (2011).
  * Grady  _et al._ (2012) D. Grady, C. Thiemann, and D. Brockmann, Robust classification of salient links in complex networks, Nature Communications 3, 1 (2012).
  * Panzarasa  _et al._ (2009) P. Panzarasa, T. Opsahl, and K. M. Carley, Patterns and dynamics of users’ behavior and interaction: Network analysis of an online community, Journal of the American Society for Information Science and Technology 60, 911 (2009).
  * Opsahl and Panzarasa (2009) T. Opsahl and P. Panzarasa, Clustering in weighted networks, Social Networks 31, 155 (2009).
  * Newman (2001) M. E. Newman, The structure of scientific collaboration networks, Proceedings of the National Academy of Sciences 98, 404 (2001).
  * Opsahl (2011) T. Opsahl, Why anchorage is not (that) important: Binary ties and sample selection, http://wp.me/poFcY-Vw (2011), accessed: 2022-3-1.
  * Hric  _et al._ (2018) D. Hric, K. Kaski, and M. Kivelä, Stochastic block model reveals maps of citation patterns and their evolution in time, Journal of Informetrics 12, 757 (2018).
  * Zheng  _et al._ (2021) M. Zheng, G. García-Pérez, M. Boguñá, and M. Á. Serrano, Scaling up real networks by geometric branching growth, Proceedings of the National Academy of Sciences 118 (2021).
  * Aref  _et al._ (2018) S. Aref, D. Friggens, and S. Hendy, Analysing scientific collaborations of new zealand institutions using scopus bibliometric data, in  _Proceedings of the Australasian Computer Science Week Multiconference_ (2018) pp. 1–10.
  * Jackson  _et al._ (2017) M. D. Jackson, H. Xu, S. Duran-Nebreda, P. Stamm, and G. W. Bassel, Topological analysis of multicellular complexity in the plant hypocotyl, Elife 6, e26023 (2017).
  * García-Pérez  _et al._ (2019) G. García-Pérez, A. Allard, M. Á. Serrano, and M. Boguñá, Mercator: uncovering faithful hyperbolic embeddings of complex networks, New Journal of Physics 21, 123033 (2019).
  * García-Pérez  _et al._ (2018) G. García-Pérez, M. Boguñá, and M. Á. Serrano, Multiscale unfolding of real networks by geometric renormalization, Nature Physics 14, 583 (2018).
  * Allard  _et al._ (2017) A. Allard, M. Á. Serrano, G. García-Pérez, and M. Boguñá, The geometric nature of weights in real complex networks, Nature Communications 8, 14103 (2017).