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研究

Frequency-based brain networks:
From a multiplex framework to a
full multilayer description

Javier M. Buldú1,2 and Mason A. Porter3,4,5

1Laboratory of Biological Networks, Center for Biomedical Technology (UPM), Pozuelo de Alarcón, Madrid, 西班牙
2Complex Systems Group & G.I.S.C., Universidad Rey Juan Carlos, Móstoles, Madrid, 西班牙
3Department of Mathematics, University of California Los Angeles, 天使们, CA, 美国
4Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, 牛津大学, 牛津, 英国
5CABDyN Complexity Centre, 牛津大学, 牛津, 英国

开放访问

杂志

关键词: Functional brain networks, 脑磁图, Multilayer networks, Multiplex
网络, Algebraic connectivity

抽象的

We explore how to study dynamical interactions between brain regions by using functional
multilayer networks whose layers represent different frequency bands at which a brain
运作. Speciﬁcally, we investigate the consequences of considering the brain as (我) A
multilayer network, in which all brain regions can interact with each other at different
频带; 并作为 (二) a multiplex network, in which interactions between different
frequency bands are allowed only within each brain region and not between them. 我们
study the second-smallest eigenvalue λ
2 of the combinatorial supra-Laplacian matrix of
both the multiplex and multilayer networks, as λ
of network synchronizability and as a biomarker for several brain diseases. 我们表明
the heterogeneity of interlayer edge weights and, 尤其, the fraction of missing edges
crucially modify the value of λ
2, and we illustrate our results with both synthetic network
models and real data obtained from resting-state magnetoencephalography. Our work
highlights the differences between using a multiplex approach and a full multilayer approach
when studying frequency-based multilayer brain networks.

2 has been used previously as an indicator

作者总结

For more than a decade, network analysis has been used to investigate the organization
and function of the human brain. 然而, applications of multilayer network analysis to
neuronal networks are still at a preliminary stage, in part because of the difﬁculties of
adequately representing brain-imaging data in the form of multilayer networks. 在这个
学习, we investigate the main differences in using multiplex networks versus more general
multilayer networks when constructing frequency-based brain networks. Speciﬁcally, 我们
are concerned with the differences for estimating the algebraic connectivity λ
been related to structural, 扩散, and synchronization properties of networks. 使用
synthetic network models and real data, we show how edge-weight heterogeneity and
missing interlayer edges crucially inﬂuence the value of λ

2, which has

介绍

During the last few years, network science has undergone a conceptual revolution with the
extension of well-established techniques of network analysis to multilayer brain networks

引文: Buldú, J. M。, & Porter, 中号. A.
(2017). Frequency-based brain
网络: From a multiplex framework
to a full multilayer description. 网络
神经科学, 2(4), 418–441. https://
doi.org/10.1162/netn_a_00033

DOI:
https://doi.org/10.1162/netn_a_00033

支持信息:
https://doi.org/10.1162/netn_a_00033

已收到: 17 行进 2017
公认: 21 十月 2017

利益争夺: 作者有
声明不存在竞争利益
存在.

通讯作者:
Javier M. Buldú
javier.buldu@urjc.es

处理编辑器:
奥拉夫·斯波恩斯

版权: © 2017
麻省理工学院
在知识共享下发布
归因 4.0 国际的
(抄送 4.0) 执照

麻省理工学院出版社

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Multiplex/Multilayer Frequency-based Brain Networks

Multilayer brain network:
A brain network with more than
one layer. Layers are connected
to each other through interlayer
边缘, which link node-layers from
different layers.

Node-layer:
Within each layer, a node-layer
represents the dynamics recorded
by a given magnetometer that is
ﬁltered at a speciﬁc frequency
乐队. 因此, each magnetometer
has an associated node-layer on
each layer.

Layer:
A portion of a multilayer network
with a particular set of connections
(called “intralayer edges”) 之间
节点. In our paper, each layer
represents a speciﬁc frequency
band of a functional brain
网络.

Multiplex brain network:
A particular kind of multilayer
brain network in which interlayer
edges occur only between
counterpart nodes (哪个, 在
这篇论文, represent the same
brain region) in different layers.

(Boccaletti et al., 2014; De Domenico et al., 2013; Kivelä et al., 2014), which provide a con-
venient way to simultaneously encode different types of interactions, subsystems, 和别的
complications in networks. 最后, it has been necessary to revisit our intuitive under-
standing of both structural and dynamical properties of networks—including structural phase
transitions (Radicchi & Arenas, 2013), diffusion and other spreading processes (De Domenico,
Granell, Porter, & Arenas, 2016; Gómez et al., 2013; Salehi et al., 2015), percolation and
robustness (Buldyrev, Parshani, 保罗, 斯坦利, & Havlin, 2010; 高, Buldyrev, 斯坦利, 徐, &
Havlin, 2013), 同步 (Aguirre, Sevilla-Escoboza, Gutiérrez, Papo, & Buldú, 2014),
and others—to the new possibilities in multilayer descriptions, leading in many cases to
counterintuitive results.

The study of brain networks is currently undergoing a process of adaptation of classical
single-layer (“monolayer”) concepts and analyses to a more general multilayer description.
(For reviews, see Betzel & Bassett, 2016; De Domenico, 2017; Vaiana & Muldoon, 2018; 还
见图 1 of Betzel & Bassett, 2016 and Figures 1–2 of De Domenico, 2017 for relevant
schematics.) Some studies have considered integration of data from structural and functional
brain imaging into a multilayer network to account for both anatomical and dynamical infor-
运动. In an early study using monolayer networks, 蜂蜜, Koötter, Breakspear, and Sporns
(2007) showed that a dynamical model simulated on the anatomical network of a macaque
neocortex can successfully identify the positions of the anatomical hubs when signals are
averaged appropriately. 最近, Stam et al. (2016) analyzed how anatomical networks
support activity, leading to speciﬁc functional networks (either undirected ones or directed
那些), demonstrating that a dynamical model close to a critical transition is able to unveil
interactions between structural and functional networks.

It is also possible to combine anatomical and functional interactions of just a few nodes
instead of an entire brain network. 例如, Battiston, 尼科西亚, Chavez, and Latora (2017)
examined network motifs (IE。, overrepresented network substructures that consist of a few
节点; Milo et al., 2002) that combine anatomical connections (in one layer) and functional
relations between cortical regions (in a second layer), linking data obtained, 分别,
from diffusion-weighted magnetic resonance imaging (DW–MRI) and functional magnetic
resonance imaging (功能磁共振成像).

Efforts to combine anatomical and functional networks into a single multilayer network
face the challenge of how to normalize the weights of edges that arise from different origins.
To tackle this issue, Simas, Chavez, 罗德里格斯, and Díaz-Guilera (2015) proposed translat-
ing functional and anatomical networks into a common embedding space and then compar-
ing them in that space. They constructed functional networks (each with N nodes) 从
fMRIs of q healthy individuals. They then used the functional networks of the q = 20 在-
dividuals to construct a single functional multiplex brain network, a special type of multi-
layer network in which corresponding entities (brain regions) in different layers (个人)
can be connected to each other via interlayer edges, but other types of interlayer edges
cannot occur (Kivelä et al., 2014). They followed a similar procedure to construct an anatomi-
cal multiplex network using data obtained from DW–MRI. They then independently projected
each of the multiplex networks into a common embedding space using a series of algebraic
operations that allow one to calculate an “algebraic aggregation” of all layers into a single
层 (see Simas et al., 2015, 欲了解详情). Using such a projection, it is possible to quantify
the differences between anatomical and functional networks. Simas et al. also calculated an
“averaged aggregation” for each of the functional and anatomical multiplex networks by aver-
aging the weights of the corresponding edges over all layers. They compared the two types of

网络神经科学

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Multiplex/Multilayer Frequency-based Brain Networks

aggregation, and they were thereby able to identify certain brain regions (例如, ones related
to visual, auditory, and self-awareness processes) with signiﬁcant differences between the
functional and anatomical networks for both types of aggregation. 然而, only the alge-
braic aggregation was able to detect differences between the functional and anatomical net-
works in other regions (具体来说, the thalamus, the amygdala, the postcentral gyrus, 和
the posterior cingulate), suggesting that the averaged aggregation disregards signiﬁcant infor-
运动 (Simas et al., 2015).

One possible alternative for reducing the complexity of analysis of brain networks is to
concentrate only on “functional” (dynamical, 实际上) interactions between brain regions and
to deﬁne multilayer functional networks as a concatenation of a series of layers, 每一个
which captures the interplay between brain regions during some time window. 这种方法,
in which a layer in a multilayer network represents connection similarities over some time
window, was taken in papers such as Bassett et al. (2011, 2013) and Braun et al. (2014) 到
analyze the temporal evolution of network modules and examine dynamical reconﬁguration
and “ﬂexibility” of functional networks.

Another alternative is to construct functional multilayer networks whose layers correspond
to the well-known frequency bands at which a brain operates (Buzsáki, 2006). 作为恶魔-
strated by Brookes et al. (2016), it is possible to construct frequency-based multilayer net-
works from magnetoencephalographic (乙二醇) recordings by (我) band-pass ﬁltering the raw
MEG signals, (二) obtaining the envelope of the amplitude at each frequency band, 和 (三、)
measuring the pairwise correlations between each envelope (for whichever frequency it ac-
计数). Using such an approach, Brookes et al. (2016) constructed frequency-based multilayer
网络, in which each layer includes the interactions in a given frequency band, and showed
that the corresponding supra-adjacency matrices (which encode a linear-algebraic represen-
tation of connections in a multilayer network) convey statistically signiﬁcant differences when
comparing a control group with a group of schizophrenia sufferers.

2, λ

1, λ

最近, De Domenico, Sasai, and Arenas (2016) took the important step of analyzing
the spectral properties of matrices associated with frequency-based multiplex networks. 这
3, . . . , λ ˜N} of its eigenvalues, and it
spectrum of an ˜N × ˜N matrix consists of the set {λ
encodes valuable information about the structural properties of the corresponding network. 在
转动, these eigenvalues (as we will explain later) are related to various dynamical properties,
such as network synchronizability, robustness, and diffusion (纽曼, 2010). De Domenico,
Sasai, and Arenas (2016) compared a group of schizophrenic patients with a control group by
using fMRI data, and they found that the second-smallest eigenvalue (IE。, the algebraic connec-
tivity or “Fiedler value”; Van Mieghem, 2011) λ
2 of the combinatorial supra-Laplacian matrix1
associated with a multiplex network is a better discriminator between the two groups than what
one can obtain by studying either unﬁltered or single-band functional networks (IE。, by using
monolayer networks). De Domenico et al. also calculated centrality measures (IE。, 措施
of the importance of network components; 纽曼, 2010) on the frequency-based multi-
plex networks to demonstrate the existence of hubs that had not been classiﬁed previously
as important brain regions for functional integration. Hubs of the control group were located
in anterior cingulate, superior frontal, insula, and superior temporal cortices; 然而, 枢纽
for schizophrenic patients were distributed over frontal, 顶叶, and occipital cortices. 这些
results revealed that frequency-based multiplex networks include relevant information about

脑磁图 (乙二醇):
Brain-imaging technique that
measures the magnetic ﬁeld of the
brain using a set of magnetometers
placed on the head of an individual.

Algebraic connectivity:
The second-smallest eigenvalue
(λ
(or supra-Laplacian) 矩阵.

2) of the combinatorial Laplacian

Combinatorial supra-Laplacian
矩阵:
The combinatorial Laplacian matrix
of a supra-adjacency matrix.

1 For such a matrix, λ

= 0. See the subsection Heterogeneity and Missing Interlayer Edges in Multiplex

网络 (in Results) for a detailed deﬁnition of the combinatorial supra-Laplacian matrix.

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Multiplex/Multilayer Frequency-based Brain Networks

the functional organization of brain networks that is not captured by using a classical mono-
layer approach.

Multiple recent papers have demonstrated the beneﬁts of using a multiplex description for
analyzing the functional networks of patients suffering from Alzheimer’s disease (广告). 对于前-
充足, Yu et al. (2017) constructed frequency-based multiplex networks from MEG data and
demonstrated that centrality measures calculated with layers analyzed independently are un-
able to detect signiﬁcant differences between a control group and the AD group. 然而,
when centralities are evaluated on a frequency-based multiplex network, one can ﬁnd statisti-
cally signiﬁcant differences in the hippocampus, posterior default-mode network, and occipital
地区. Guillon et al. (2017) also used frequency-based multiplex networks to differentiate be-
tween controls and people with AD. 在这种情况下, the authors proposed the use of a multi-
participation coefﬁcient (MPC) to enhance classiﬁcation of which individuals are suffering from
广告. Their MPC consists of an adaptation of the classical (IE。, monolayer) participation coef-
ﬁcient (Guimerà & Amaral, 2005) to networks composed of several layers (see also Yu et al.,
2017). A potential beneﬁt of using this MPC is that it does not depend on interlayer edge
重量, which thus do not need to be calculated. As shown in Guillon et al. (2017), using an
MPC yields better classiﬁcation accuracy and sensitivity than using only monolayer network
diagnostics.

在本文中, we investigate how to translate the dynamics of different brain regions into
a frequency-based (IE。, functional) multilayer network, in which individual layers account for
coordination within a given frequency band. We focus speciﬁcally on the consequences of an-
alyzing a multiplex network versus a more general multilayer one. The former allows interlayer
connections only between the same brain region (IE。, node-layer) in different network layers,
so coupling between oscillations in different frequency bands occurs only between the same
brain region, whereas the latter allows one to model coordination between any brain region
at any frequency band. We use resting-state MEG recordings because of their high temporal
resolution (on the order of milliseconds), which makes it possible to analyze a broad spectrum
of frequency bands (de Pasquale et al., 2010; van Diessen et al., 2015). In our case, 一套
MEG signals consist of N time series, each of which comes from a sensor that captures the
activity above a different cortical region. We then ﬁlter signals at four frequency bands (theta,
alpha, beta, 和伽玛) and construct, for each individual, a four-layer functional multilayer
network from the dynamical coordination within and between frequency bands. The existence
of interlayer edges in frequency-based functional networks relies on the phenomenon of cross-
frequency coupling (CFC), which is responsible for integrating brain activity at different spatial
and temporal scales (Canolty et al., 2006). The quantiﬁcation of CFC is a hard task, 因为
the interplay between frequency bands is very intricate (Aru et al., 2015). To address this issue,
one can set the weights of the interlayer edges to a value p, which one can estimate by using
an optimization procedure. 例如, it is possible to construct multiplex networks of two
groups of individuals with different proﬁles and determine a value of p that best distinguishes
between the two groups (看, 例如, De Domenico, Sasai, & Arenas, 2016). 然而, we adopt
a different strategy: We obtain the weights of the interlayer edges directly from time series by
calculating mutual information (MI; MacKay, 2003). We thereby capture heterogeneity (在里面
重量) of both intralayer edges and interlayer edges, and we investigate the inﬂuence of such
heterogeneity on the spectral properties of frequency-based functional networks.

Using both synthetic network models and data from laboratory experiments, we investi-
gate the effects that heterogeneity of interlayer edge weights have on the spectral properties of
both multiplex and more general multilayer networks. Speciﬁcally, we focus on the algebraic

Cross-frequency coupling (CFC):
In the brain, CFC is the mechanism
through which dynamics at a certain
frequency affect, or are affected by,
the dynamics at other frequencies.

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Multiplex/Multilayer Frequency-based Brain Networks

Combinatorial Laplacian matrix:
Given the adjacency matrix A of a
network and the degree matrix D,
which consists of the degree of the
nodes along the diagonal and 0s in
all other entries, the combinatorial
Laplacian matrix is deﬁned as
L = D − A.

connectivity λ
2, which is closely related to both structural and dynamical properties of net-
作品 (Masuda, Porter, & 兰比奥特, 2017; 纽曼, 2010; Van Mieghem, 2011). On one
手, algebraic connectivity is an indicator of modular structure in networks (Fortunato &
Hric, 2016). In the framework of multilayer networks, one can interpret the value of λ
2 和
how it changes as a function of interlayer coupling strength as a way to quantify structural
integration and segregation of different network layers (Radicchi & Arenas, 2013); also see the
discussions in DeFord and Pauls (2017). 另一方面, 1/λ
2 is proportional to the time
required to reach equilibrium in a linear diffusion process (Gómez et al., 2013). 此外,
the time tsync to reach synchronization of an ensemble of phase oscillators that are linearly
and diffusively coupled is also proportional to 1/λ
2 是
positively correlated in some situations with nonlinear coupling (Almendral & Díaz-Guilera,
2007). A recent investigation of the consequences of modifying interlayer edge weights in mul-
tiplex networks has illustrated that enhancing interlayer coupling tends to shorten the time to
achieve interlayer synchronization in a Kuramoto model of coupled oscillators (Allen-Perkins,
Albuquerque de Assis, Pastor, & Andrade, 2017).

2, and it is known that tsync and 1/λ

In the framework of functional brain networks, algebraic connectivity has been used as an
indicator of AD, such as in de Haan et al. (2012), who obtained statistically signiﬁcant differ-
ences for λ
2 of functional networks obtained from MEG in a comparison of a group of patients
suffering from AD with a control group. Phillips, McGlaughlin, Ruth, Jager, and Soldan (2015)
calculated the value of λ
2 in a group of individuals with mild cognitive impairment and AD,
although they did not report signiﬁcant differences between them. Computing λ
2 is also nec-
essary for calculating the most standard type of synchronizability parameter, which is the
ratio λN/λ
2, where λN is the largest eigenvalue of the combinatorial Laplacian matrix
(Boccaletti, Latora, Moreno, Chavez, & Hwang, 2006).
In a series of studies, the synchro-
nizability of brain networks was calculated for different frequency bands (Bassett, 迈耶-
Lindenberg, Achard, Duke, & 布莫尔, 2006), during epileptic seizures (Schindler, Bialonski,
Horstmann, Elger, & Lehnertz, 2008), and for schizophrenic individuals (七只母鸡, 韦斯,
Coppola, 温伯格, & Bassett, 2013). We aim to improve the interpretation of algebraic
connectivity for functional brain networks, and we thus investigate (我) how the fact that a con-
siderable fraction of all possible interlayer edges are not present in multiplex networks leads
to a deviation from the theoretical values expected for λ
2 和 (二) how these deviations are
related to the mean weight of the interlayer edges. We thereby scrutinize the consequences
of using a multiplex formalism, in which only CFC inside the same brain region is allowed,
instead of employing a fully multilayer approach (IE。, without any restrictions on the type of
coupling that one considers).

结果

Constructing Frequency-Based Multilayer Networks

图中 1, we illustrate the process of constructing frequency-based multiplex and multi-
layer brain networks. Our starting point is a data set of MEG recordings of a group of q = 89
individuals during resting state (see Materials and Methods for details), but one can use
other experimental paradigms—including different brain-imaging techniques, such as electro-
encephalography (EEG) or fMRI—to construct multilayer networks with the same procedure.
Speciﬁcally, we record MEG activity at N cortical regions, 和 235 ≤ N ≤ 246 (看到
Supplementary Information for details; Buldú & Porter, 2017), and we then clean the data to
remove artifacts and obtain corresponding unﬁltered signals. We thereby analyze the signal
recorded by each sensor instead of carrying out a source reconstruction. We then band-pass
ﬁlter each signal to obtain four different ﬁltered time series for each brain region. We use the

网络神经科学

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Multiplex/Multilayer Frequency-based Brain Networks

数字 1. Encoding brain dynamics as a multilayer functional network. We show an illustrative
example with two frequency bands (alpha and beta). (A) We band-pass ﬁlter the MEG signals at two
频带: alpha [8–12] Hz and beta [12–30] 赫兹. We use mutual information (MI; MacKay,
2003) to quantify coordination between brain regions. This yields three different type of functional
边缘: Edge type “1” quantiﬁes coordination between different regions at the same frequency band;
edge type “2” corresponds to interlayer edges, which couple the activity of the same region at dif-
ferent frequency bands; and edge type “3” quantiﬁes cross-frequency coupling (CFC) between two
brain regions. Multiplex networks include only edges of types 1 和 2, whereas more general multi-
layer networks include all three types of edges. (乙) Schematic of the supra-adjacency matrix of a
two-layer network constructed from the data in panel A. (C) Schematic of the intralayer and interlayer
edges in the multilayer functional network.

four classical frequency bands: theta [3–8] 赫兹, alpha [8–12] 赫兹, beta [12–30] 赫兹, 和伽玛
[30–100] 赫兹. The number l of layers of the multilayer network is the number of different fre-
quencies that we examine (和 2 ≤ l ≤ 4 in our work), and the N nodes in each layer are
associated with the dynamics of the N sensors ﬁltered at a given frequency band. We number
the nodes so that nodes n, n + 氮, n + 2氮, ……, n + l N (with n ∈ {1, . . . 氮}) correspond to the
signals of the same brain region n at the l different frequency bands (IE。, layers).

Mutual information (MI):
A measure to evaluate statistical
interdependencies between a pair
变量, X and Y, 基于
comparing a joint distribution
p(X, 是) with the product p(X)p(是)
of marginal distributions.

Supra-adjacency matrix:
An adjacency matrix associated with
a multilayer network; it codiﬁes how
much different node-layers interact
with each other.

We then quantify coordination between each pair of nodes of a multilayer network, 关于-
gardless of which layers they are in, using mutual information (see Materials and Methods).
Calculating mutual information (MI) between time series of the same frequency band yields
intralayer connections between brain regions (see edge type “1” in the bottom-left plot of Fig-
ure 1A for an example), so each layer corresponds to a speciﬁc frequency band. Edges between
the signals of the same sensor at different frequency bands result in interlayer connections be-
tween layers (see edge type “2”). Such “diagonal” interlayer edges are the only type of inter-
layer edges that are allowed in multiplex networks (Kivelä et al., 2014). 最后, cross-frequency
coupling between different brain regions yields the other (“nondiagonal”) interlayer edges in
a full multilayer network (see edge type “3”). As we show in Figure 1B, we thereby obtain a
supra-adjacency matrix, where blocks along the diagonal account for intralayer connections
(layers alpha and beta in the two-layer example) and blocks off of the diagonal, marked as
P αβ
, encode the interlayer edges. Because MIij = MIji (see Equation 4 in Materials
and Methods) for time series of nodes i and j, the supra-adjacency matrix is symmetric, 所以
Pαβ = [Pβα]时间

and Pβα

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Multiplex/Multilayer Frequency-based Brain Networks

重要的, although we have chosen to use MI, there are a diversity of similarity measures
for capturing amplitude–amplitude and phase–amplitude correlations between different
频带 (see Aru et al., 2015 for a review of cross-frequency coupling measures),
and each measure has its own advantages and drawbacks. 尽管如此, as we will see, 这
same methodological implications exist no matter which speciﬁc measure one uses to evaluate
coordination between brain sites.

In our discussion, we focus on the analysis of a two-layer network with alpha and beta layers.
We will discuss the consequences of considering alternative frequency bands and numbers of
layers in the last section of the paper (see Conclusions and Discussion).

Our starting point is to compare the results from four different constructions of functional

网络:

1. Unﬁltered functional networks. We obtain these networks from the original (unﬁltered)
signals of each brain region—that is, without decomposing the signals into different fre-
quency bands—so these are monolayer networks.

2. Aggregated networks. We obtain these networks from componentwise addition of the
(Note that this is a

weights of the alpha and beta layers to form monolayer networks.
uniform aggregation.)

3. Multiplex networks. Each layer corresponds to a speciﬁc frequency band (as explained
多于), and interlayer edges are allowed only between node-layers that correspond to
the same brain region.

4. Full multilayer networks. These include the same layers as their multiplex counterparts,

but they allow all possible interlayer edges.

图中 2, we show the probability density functions (PDFs) for a group of q = 89 individ-
2, the standard deviation of the interlayer edge weights (to quantify their

uals of the values of λ
heterogeneity), and the percentage of missing interlayer edges (also see Table 1).

In column one of Figure 2, we observe that the (monolayer) unﬁltered functional network
has a similar mean and standard deviation of λ
2 as the multilayer network (also see Table 1).
然而, the aggregated networks tend to have larger values of λ
2, which makes sense, 和我们一样
construct an aggregated network by adding the weights of the two layers (alpha and beta), 和
the total “strength” of the network (IE。, the sum of all of its edge weights) is close to double
the strength of each layer. 重要的, the mean λ
2 for the multiplex networks is two orders of
magnitude smaller than the mean λ
2 of the multilayer networks. We expect this discrepancy,
because we construct a multiplex network by deleting all interlayer edges of a full multilayer
网络, except for edges (so-called “diagonal” edges) that link the same “physical” nodes (IE。,
the same brain regions). 因此, the total strength of the interlayer matrix Pαβ
is considerably
smaller in multiplex networks than in corresponding full multilayer networks. Because λ
2 是
an indicator of the amount of interconnections between communities in a network (纽曼,
2010), one expects such drastic edge removals to yield a smaller value of λ
2, as one can
construe layers as communities with a small number of edges between them (only N of the
N2 possible interlayer edges of a full multilayer network).

In columns two and three of Figure 2, we quantify the heterogeneity and the number of miss-
ing interlayer edges of the four different functional networks. In column two, we plot the PDFs
of the standard deviation of all edges (for the unﬁltered and aggregated networks) and inter-
layer edges (for the multiplex and multilayer networks). In all cases, we observe that the func-
tional networks that we construct from experimental data have non-negligible heterogeneity

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Multiplex/Multilayer Frequency-based Brain Networks

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数字 2. MEG data: from unﬁltered data to a multilayer network. Probability density functions
(PDFs) of different network characteristics of a group of q = 89 个人. (See Materials and
Methods for details.) We show the second-smallest eigenvalue λ
2 of the combinatorial Laplacian
矩阵, the standard deviation σ of the matrix elements (to quantify their heterogeneity), 和
percentage of missing edges of four different networks: (我) the functional network obtained from
the unﬁltered signals (ﬁrst row), (二) an aggregated network of the alpha and beta layers (第二
排), (三、) a multiplex network (third row), and a (四号) full multilayer network (fourth row).
In all
案例, we only consider two layers (alpha and beta). For the multiplex and multilayer networks, 笔记
that we do these computations with the combinatorial supra-Laplacian matrices. 的百分比
missing edges in the unﬁltered and aggregated networks is equal to the percentage of 0 values in
the whole matrix, but it refers only to the interlayer edges for the multiplex and multilayer networks.

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(also see Table 1). 再次, aggregated networks have values that are roughly twice as large as
those of the other kinds of networks because of the (rough) doubling of the mean strength. 在
column three, we show that there is a large percentage of missing edges. This arises from the
fact that we construe edges whose weights are smaller than those obtained with an appropri-
ate surrogate time series as not being statistically signiﬁcant, so we set their values to 0 (看
Materials and Methods for details). 这 (monolayer) aggregated networks have the smallest
mean percentage, followed by the (monolayer) unﬁltered networks, and then the two types of
multilayer networks. For the multiplex and full multilayer networks, the percentage of missing
边缘, which is larger than 40% 在这两种情况下, refers to the number of all possible interlayer
边缘. Note that missing edges are unavoidable in functional brain networks, because not all

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桌子 1. For each kind of network (see the ﬁrst column), we show the mean standard deviation
¯σ of the weights of the edges (unﬁltered and aggregated) and interlayer edges (for both multiplex
networks and full multilayer networks) and the corresponding percentage of missing edges. (See the
Supplementary Information for more details; Buldú & Porter, 2017.)

网络
Unﬁltered
Aggregated
Multiplex
(Full) Multilayer

¯σ
0.0823
0.1567
0.0618
0.0611

Missing Edges
22.75%
4.42%
52.17%
44.01%

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Multiplex/Multilayer Frequency-based Brain Networks

brain regions communicate with each other through direct connections2. 而且, 不同之处-
ent amounts of coordination between pairs of brain regions lead to functional networks with
heterogeneous weights.

Heterogeneity and Missing Interlayer Edges in Multiplex Networks

Given our prior observations, a crucial question arises: What are the consequences of hetero-
genous and missing interlayer edges, both of which are intrinsic features of brain-imaging data, 在
multiplex and full multilayer functional networks? 更具体地说, how do they affect the
value of λ
2, and what are the ensuing structural and dynamical implications of these differences?

With the aim of answering the above questions, we perform a series of numerical com-
putations in which we compare the theoretical values of λ
2 in multiplex and full multilayer
networks from homogeneous interlayer-edge distributions with ones from networks with het-
erogeneous and missing interlayer edges.

and Mβ

We start with a two-layer multiplex network, whose layers alpha and beta have Nα = Nβ
nodes and Lα = Lβ intralayer edges, 分别. We number the nodes of the alpha layer
from k = 1 to k = Nα and the nodes of the beta layer from m = Nα + 1 to m = Nαβ = Nα + Nβ.
The matrices Mα
are the corresponding adjacency matrices for each layer. We then
introduce lc connector edges (IE。, diagonal interlayer edges) between each node k of the alpha
layer to its corresponding node m = k + Nα of the beta layer to construct a multiplex network.
We suppose that intralayer edges have weight wintra
both belonging
either to the alpha layer or to the beta layer), and we set the weights of the interlayer edges
km = pkm (IE。, for k ∈ α and m ∈ β), where pkm are the elements of a vector (西德:2)p of the
to winter
weights of the interlayer connections. Under these conditions, we obtain a supra-adjacency
matrix Mαβ
) and two off-diagonal blocks
and Pβα
(Pαβ

that consists of two diagonal blocks (Mα
, where Pαβ = [Pβα]T = (西德:2)p I, 在哪里 (西德:2)p is a row vector). 那是,
(西德:2)

我(西德:4)j(西德:4) = 1 (IE。, for i(西德:4)

and Mβ

and j(西德:4)

(西德:3)

Mαβ =

Mα
(西德:2)p I
(西德:2)p I Mβ

(1)

where I is an identity matrix. Following the same procedure, one can also extend the deﬁ-
nition of a supra-adjacency matrix to include an arbitrary number of layers. 例如, 如果
one considers layers for each of the theta, alpha, beta, and gamma bands (see Materials and
方法), one obtains a supra-adjacency matrix Mθαβγ

The combinatorial supra-Laplacian matrix Lαβ

of the multiplex network is

(西德:2)

Lαβ =

Lα + (西德:2)p I
-(西德:2)p I

-(西德:2)p I
Lβ + (西德:2)p I

(西德:3)

where the layer combinatorial Laplacians Lα,β

have coordinates

⎧
⎪⎨

⎪⎩

Lα,β

ij =

si ,
−1 ,
0 ,

if i = j
if i and j are adjacent
否则

(2)

(3)

and si = ∑i(西德:5)=j wij is the weighted degree (IE。, total weight of incident edges) of node i.

2 一般来说, networks that are constructed from pairwise time-series similarities have nonzero edge weights in
全部 (or almost all) intralayer adjacency entries (Bassett & 斯波恩斯, 2017). 然而, in functional brain networks,
the deletion of entries that are not statistically signiﬁcant can lead to a non-negligible number missing edges, 作为
is the case with our data.

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Multiplex/Multilayer Frequency-based Brain Networks

数字 3. Consequences of heterogenous and missing interlayer edges in a multiplex network.
2 of the combinatorial supra-Laplacian matrix Lαβ
Algebraic connectivity λ
. Each of the two layers
is a G(氮, pcon) Erd ˝os–Rényi network with N = 250 nodes and a connection probability of pcon =
0.25. Each circle corresponds to a mean over 100 realizations. (A) We quantify the heterogeneity
of the interlayer edges with the standard deviation σ of their weights: (我) σ = 0 (black circles),
(二) σ ≈ 0.407 (red circles), (三、) σ ≈ 0.523 (green circles), 和 (四号) σ ≈ 0.581 (blue circles). Lines
correspond to analytical solutions for the case σ = 0 (IE。, a homogenous multiplex network). 这
(Lα + Lβ) (Radicchi & Arenas, 2013;
= 2p, and the solid line is λ
λ
dashed line is λ
2,agg
2
2, the difference of λ
Sahneh et al., 2015). In the inset, we show Δλ
2 between the homogeneous
multiplex network and a multiplex network with heterogeneous interlayer edges. (In the inset, 这
solid vertical line indicates the mean value (西德:7)p∗(西德:9) of the transition point.) (乙) Algebraic connectivity
λ
2 of the multiplex networks as a function of the number of missing interlayer edges. In the inset,
2 between the homogeneous multiplex network (IE。, σ = 0 和
we show Δλ
all possible interlayer edges) and the multiplex networks with missing interlayer edges. As before,
the solid vertical line indicates the mean value (西德:7)p∗(西德:9) of the transition point.

2, the difference of λ

= 1
2

图中 3, we show the consequences of heterogeneity in the distribution of the interlayer-
edge weights of the multiplex network Mαβ
. 在这个例子中, the multiplex network has an
interlayer connectivity matrix Pαβ = (西德:2)p I, where I is an Nα × Nα (equivalently, Nβ × Nβ, 作为
Nα = Nβ in this example) identity matrix and (西德:2)p = p(西德:2)h is a row vector that controls the weights
(西德:2)h encodes the
of interlayer edges, where p modulates the vector’s magnitude and the vector
(西德:2)h to follow a uniform distribution
heterogeneity of the interlayer edges. We set the elements of
min, hmax]. These elements have a mean value of ¯h = 1 and a standard de-
over the interval [H
viation of σ. We set ¯h = 1 and construct networks with interlayer-edge-weight heterogeneities
that range from σ = 0 (blue circles) to σ ≈ 0.581 (green circles). We then examine the interplay
between the weights of the interlayer edges and the heterogeneity by increasing the value of
p. Note that σ = 0 corresponds to what we call a homogeneous multiplex network, 哪个
has uniformly-weighted interlayer edges (IE。, (西德:2)p = p for all interlayer edges). We obtain the
results in Figure 3 by calculating a mean over 100 realizations of two-layer networks with the
G(氮, pcon) Erd ˝os–Rényi (ER) 模型 (with pcon = 0.25) in each layer and N = Nα = Nβ = 250
节点 (纽曼, 2010).

As explained in Radicchi and Arenas (2013), modifying the weight parameter p of the in-
terlayer edges has important consequences for the value of λ
2 for homogeneous multiplex
网络. The existence of two regimes of qualitatively distinct dynamics, separated at a tran-
sition point p∗
, was discussed in Radicchi and Arenas (2013) (and in various subsequent
文件): When p (西德:10) p∗
= 2p; 为了
p (西德:11) p∗
2 approximates that of the (uniformly) aggregated network

2 follows the linear relation λ

, the algebraic connectivity λ

, 然而, the value of λ

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Multiplex/Multilayer Frequency-based Brain Networks

(Lα + Lβ)]. 图中 3, the dashed and solid lines indicate the theoretical
= 1
λ
[IE。, λ
2
2,agg
2
predictions of λ
2 in the homogeneous case for small (dashed) 和大 (solid) values of p. 我们
obtain a mean value of p∗ ≈ 2.870 by locating the intersection between the eigenvalues λ
2
and λ

3 of the supra-Laplacian matrix Lαβ

(Sahneh, Scoglio, & Van Mieghem, 2015).

2. 此外, values of p near p∗

In Figure 3A, we illustrate that the heterogeneity of the (weights of the) interlayer edges
leads to non-negligible differences in λ
have the maximum
discrepancy between heterogeneous cases (colored circles) and the homogeneous case (黑色的
circles). In studies of multiplex networks, it is simplest to take interlayer edge weights to be
homogeneous (IE。, given by a constant value p), especially when it is not clear how to estimate
their values Kivelä et al. (2014). When possible, 然而, it is desirable to be more sophisti-
cated, and it is possible to quantify the coupling strength between different frequency bands
in various ways when studying frequency-based functional brain networks (Aru et al., 2015).
The choice of incorporating versus ignoring heterogeneity of inter-frequency edge weights then
leads to unavoidable differences in the estimation of λ
2, especially near the transition point
p∗
. It is also important to think about the different interlayer edge weights that one obtains

from different measurements of coupling strength (see Materials and Methods).

Discrepancies in the values of λ

2 are typically even larger when some interlayer edges are
丢失的.
图中 3(乙), we show the effects of removing some percentage of the interlayer
edges in a “holey” multiplex network (IE。, a multiplex network with an interlayer connectivity
matrix Pαβ
whose elements are either 1 或者 0 along the diagonal and are 0 in all other entries).
We observe that increasing the number of missing interlayer edges causes the multiplex net-
works to have drastically reduced values of λ
2 always grows
with a slope that is smaller than 2p; for p (西德:11) p∗
2 never reaches the
value of λ
again have the largest
discrepancies with respect to the homogeneous case.

2 for the aggregated network. 最后, values of p close to p∗

, 然而, the value of λ

2. When p (西德:10) p∗

, the value of λ

Mutilplex Networks Versus Full Multilayer Networks

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We now identify the qualitative differences (and some of their consequences) between mul-
tiplex networks and full multilayer networks. The latter have an interlayer connectivity matrix
Pαβ = pC, where C has elements cij that account for the weight between each pair {我, j} 的
nodes i ∈ α and j ∈ β, and the parameter p allows one to modulate the mean weight of the
interlayer edges. In Figure 4A, we connect the same layers as in our previous example, but now
we use a homogeneous interlayer connectivity matrix Pαβ
with weights pij = p (IE。, cij = 1
for all i and j). Increasing p from 0 leads to the transition point p∗ ≈ 0.013, which we obtain
3 of the combinatorial supra-Laplacian Lαβ
from the intersection of the eigenvalues λ
2 and λ
.
We observe for p < p∗ = 2p(cid:7)c(cid:9) (see the dashed line in Figure 4A), where (cid:7)c(cid:9) = 250 is the mean weighted degree of the matrix C, as demonstrated in Radicchi and Arenas (2013). Interestingly, after the transition point p = p∗ , one can describe the value of λ 2,min(α,β) is the value of the smaller λ 2 of the two isolated layers. As we show in the inset of Figure 4A, when we introduce a certain amount of heterogeneity into the interlayer connectivity matrix, we observe slight differences from homogeneous case. Speciﬁcally, for heterogeneous cases, these differences are larger for values of p above the mean (cid:7)p∗(cid:9) of the transition point. 2 follows the linear function λ 2 by the function λ that the value of λ 2,min(α,β) + λ (pC), where λ = λ 2 2 2 f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 As we saw in our analysis of multiplex networks, deleting interlayer edges from a full multilayer network increases the discrepancy in λ 2 values between networks with homoge- neous interlayer edges and networks with heterogeneous ones. As we show in Figure 4B, the Network Neuroscience 428 Multiplex/Multilayer Frequency-based Brain Networks 2 for a homogeneous interlayer connectivity matrix Pαβ Figure 4. Edge heterogeneity and missing interlayer edges in a full multilayer network. Alge- 2 of the combinatorial supra-Laplacian matrix Lαβ (black circles) as a function braic connectivity λ of the weight parameter p of the interlayer edges. Each of the two layers is a G(N, pcon) Erd ˝os– Rényi network with N = 250 nodes and a connection probability of pcon = 0.25. Each circle corresponds to a mean over 100 realizations of such multilayer networks. (A) Algebraic connec- =pC, where each of the elements of tivity λ C is cij = 1. The dashed line, given by λ = 2p(cid:7)c(cid:9), corresponds to the analytical solution for a homo- geneous matrix Pαβ , where C has a mean weighted degree of (cid:7)c(cid:9) = 250. The solid line is given = λ 2,min(α,β) is the value of the smaller λ by λ 2 of the two isolated layers. The inset shows the differences Δλ 2 between a multilayer network with homogeneous inter- layer connectivity matrix Pαβ and a series of multilayer networks with increasing heterogeneity (quantiﬁed by the standard deviation σ) in Pαβ . Differences with the homogeneous case increase after the mean (cid:7)p∗(cid:9) of the transition point (vertical solid line). (B) Algebraic connectivity λ 2 of the full multilayer networks as a function of the percentage of missing interlayer edges. (See the values of the percentages in the legend.) In the inset, we show the increment of λ 2 versus the number of missing interlayer edges. 2,min(α,β) + λ (pC), where λ 2 2 2 value of λ 2 decreases with the number of missing edges, which one should expect, because having a smaller number of interlayer edges implies that one needs larger edge weights to Interestingly, the deviation from the the- maintain the same amount of interlayer coupling. oretical predictions is signiﬁcant even for p < (cid:7)p∗(cid:9). Additionally, for p > (西德:7)p∗(西德:9), we ob-
serve (as expected) a change in the slope of λ
2, but the theoretical predictions given by
(pC) (solid line) begin to fail, leading to a discrepancy that increases
λ

= λ

2,min(A,β) + λ

with the percentage of missing edges.

Because full multilayer networks have up to N2 interlayer edges, whereas multiplex net-
works can have only N of them, the former tend to have interlayer connectivity matrices with
larger strengths SP = ∑ij pij. 图中 5, we show the algebraic connectivity λ
2 of the com-
binatorial supra-Laplacian matrices Lαβ
for a series of networks with identical layers and in-
terlayer strength SP, but with a different number of interlayer edges, ranging from a multiplex
网络 (with N interlayer edges) to a full multilayer network with no nonzero entries (IE。,
with N2 interlayer edges). The dashed lines correspond to the analytical solutions for the full
= 2p(西德:7)C(西德:9), 和 (西德:7)C(西德:9) = 250) and the multiplex network
multilayer network (black dashed line; λ
2
= 2p) when p < p∗ (red dashed line; λ . The solid curves are the corresponding theoretical solutions when p > (西德:7)p∗(西德:9) for the full multilayer network (black line; λ
(pC))
(Lα + Lβ)). We observe the effect that
and the multiplex network (red curve; λ
adding interlayer edges to multiplex networks has on the value of λ
2 and the associated tran-
sition from a multiplex network to a full multilayer architecture.
有趣的是, the different
numbers of interlayer edges in the two types of networks leads to a difference in the position

2,min(A,β) + λ

= 1
2

= λ

2,agg

网络神经科学

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Multiplex/Multilayer Frequency-based Brain Networks

数字 5. Transition from a multiplex network to a full multilayer network. Algebraic connectivity
2 of the combinatorial supra-Laplacian matrix Lαβ
λ
for different values of the percentage of edges in
the interlayer connectivity matrix Pαβ
. Departing from a multiplex network, we add interlayer edges
uniformly at random and calculate the percentage of existing edges in Pαβ
. Each circle corresponds
to a mean over 100 realizations. We set all active edges of the interlayer connectivity matrix Pαβ
to pij = p. The dashed lines correspond to the analytical solutions for the full multilayer network
(black dashed line; λ
= 2p(西德:7)C(西德:9), 和 (西德:7)C(西德:9) = 250) and the multiplex network (red dashed line;
= 2p) for p < (cid:7)p∗(cid:9). The solid curves are the theoretical solutions when p > (西德:7)p∗(西德:9) for the full
λ
2
(pC)) and the multiplex network (red curve;
multilayer network (black line; λ
2 versus SP/(Sα + Sβ), where SP, Sα, and Sβ are,
λ
2,agg
分别, the strength of the interlayer connectivity matrix Pαβ
, the strength of the alpha layer,
and the strength of the beta layer. Note that differences between the values of λ
2 of the multilayer
structures increase signiﬁcantly for SP/(Sα + Sβ) (西德:2) 0.1.

(Lα + Lβ)). In the inset, we show λ

2,min(A,β) + λ

= 1
2

= λ

的 (西德:7)p∗(西德:9) (which one can infer by looking at the change of slope of λ
orders of magnitude.

2) that can reach several

The inset of Figure 5 illustrates the same results normalized by the total strength of the inter-
layer connectivity matrices. 那是, we show SP/(Sα + Sβ), where Sα = ∑ Mα
ij and Sβ =
∑ Mβ
ij, 分别, are the strengths of layers alpha and beta. This allows us to compare net-
works with the same value of SP, regardless of whether they are close to a multiplex archi-
tecture or to a fully multilayer architecture with no nonzero entries. For p < (cid:7)p∗(cid:9), we obtain It is only for p > (西德:7)p∗(西德:9) that the particular
similar values of λ
structure of the interlayer connectivity matrix begins to play a role in the value of λ
2. 我们
observe that differences start to arise at SP/(Sα + Sβ) ≈ 0.1, which is a relatively small value.

2 for all network architectures.

Although we used ER intralayer networks in our above calculations, we obtain similar
results for other network models. 尤其, the results are qualitatively the same when we
construct the intralayer networks using a Barabási–Albert (BA) 模型 (巴拉巴斯 & 阿尔伯特, 1999).
See the Supplementary Information for details (Buldú & Porter, 2017).

The Meaning of λ2 in Experimental Data

Now that we have analyzed the effects of edge-weight heterogeneity and the number of missing
interlayer edges, let’s revisit the multiplex and multilayer networks that we constructed from the
MEG recordings. As we saw in Figure 2, both edge-weight heterogeneity and missing interlayer

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Multiplex/Multilayer Frequency-based Brain Networks

2 of the combinatorial supra-Laplacian matrix Lαβ

数字 6. Edge heterogeneity and missing interlayer edges in frequency-based functional brain
网络. Algebraic connectivity λ
as a function
of the mean strength of the interlayer connectivity matrix Pαβ= pC, where C encodes the weights
of the interlayer edges, for four different types of networks: (我) homogeneous multiplex networks
(黑色的), (二) heterogeneous multiplex networks (红色的), (三、) homogeneous full multilayer networks
(orange), 和 (四号) heterogeneous full multilayer networks (蓝色的). Each circle is a mean over 89 印迪-
个人. The blue and black dashed lines, 分别, are the values of λ
2 for the unﬁltered and
= 2p(西德:7)C(西德:9), 在哪里 (西德:7)C(西德:9) is the mean of the
aggregated networks. The red dashed line corresponds to λ
weighted degree. The red solid line is given by λ
2,min(A,β) 是个
2
value of the smaller λ
2 of the two isolated layers.

2
2,min(A,β) + λ

(pC), where λ

= λ

edges occur in our experimental data, and it is thus desirable to investigate how close the
experimental networks that we are analyzing are to a transition point p∗
and how this proximity
(or lack thereof ) inﬂuences the expected value of λ

图中 6, we show the values of λ

2 that we obtain for four different network reconstruc-
tions based on the MEG data: a homogeneous multiplex network (black circles), a hetero-
geneous multiplex network (red circles), a homogeneous full multilayer network (blue circles),
and a heterogeneous full multilayer network (cyan circles). Each network has two layers—one
for the alpha band and one for the beta band—and each point corresponds to the mean over
the group of 89 个人. We obtain the heterogeneous versions of the multiplex and multi-
layer networks by calculating MI values between the brain regions in the frequency bands, 作为
we described previously (also see Materials and Methods). 第二, we construct the homo-
geneous versions of both the multiplex and multilayer networks by assigning the same weight
(西德:7)C(西德:9) to all interlayer edges, 在哪里 (西德:7)C(西德:9) is the mean of the weights of the interlayer edges in their
heterogeneous counterparts. 注意, 在这种情况下, the homogeneous multiplex and mul-
tilayer networks do not correspond to real functional networks; 反而, they are reference
networks that we use to quantify the consequences on λ
2 of the intrinsic edge heterogeneity
and missing edges in real functional networks.

To assess how close the real networks are to the transition point p∗

, we multiply the values
of the interlayer edges by a parameter p, which we increase from p = 0 to a value above 1. 我们
then calculate the strength SP of the corresponding interlayer connectivity matrix Pαβ
to allow a
comparison between the multiplex network and the multilayer networks, and we plot the value
of λ
2 versus SP. The red dashed line in Figure 6 corresponds to the theoretical predictions for
p < p∗ = 2p(cid:7)c(cid:9), where (cid:7)c(cid:9) is the mean weighted degree of the nodes in the interlayer (i.e., λ connectivity matrix Pαβ ). The black and blue dashed lines are, respectively, the value of λ 2 2 Network Neuroscience 431 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / / t e d u n e n a r t i c e - p d l f / / / / / 2 4 4 1 8 1 0 9 2 2 3 8 n e n _ a _ 0 0 0 3 3 p d t . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Multiplex/Multilayer Frequency-based Brain Networks 2 2 λ 2,agg = 1 2 = 2p(cid:7)c(cid:9) to λ (Lα + Lβ)) and the value of λ for the aggregated network divided by 2 (i.e., λ 2 for the unﬁltered (monolayer) functional network. The vertical solid line corresponds to the case p = 1 for the mean of the heterogeneous full multilayers networks (i.e., the networks that we obtain by taking into account all interlayer correlations without modifying their weights). Interestingly, this network occupies the region in which the evolution of λ 2 with respect to (pC), which suggests that the p changes slope from λ frequency-based multilayer networks that we are analyzing are close to the transition point p∗ 2 is most inﬂuenced by the effects of edge heterogeneity and missing interlayer edges. It is also worth noting that the four network representations have rather different values of algebraic connectivity (and hence, we expect, have different spectral properties more generally), except when p (cid:10) p∗ (see the inset of Figure 6). Although it looks from the plot that the algebraic connectivities of the heterogeneous multilayer networks (cyan circles) and those of the homogeneous multiplex networks (black circles) may converge to the same value for large SP, they eventually cross each other when we further increase SP (not shown). . As we have seen, it is near this point where the value of λ 2,min(α,β) + λ = λ 2 2 In Figure 7, we show (top panel) the PDF of the values of p∗ for the multilayer networks of each of the 89 individuals and (bottom panel) the percentage of deviation of λ 2 with respect 2,agg of the aggregated network. We observe that the peak of the PDF for p∗ to the value λ is near p = 1. That is, the multilayer networks that we constructed from empirical data are close to the transition point. Figure 7 also illustrates discrepancies in the values of λ 2 between the aggregated and multilayer networks, indicating that it is necessary to differentiate between the two cases when interpreting the value of λ 2, especially when comparing results from different studies. Finally, we investigate how the combination of layers from different frequency bands affects the value of λ 2. In our analysis thus far, we have focused on a two-layer network that we constructed using the alpha and beta frequency bands, because they are often associated with l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / / t e d u n e n a r t i c e - p d l f / / / / / 2 4 4 1 8 1 0 9 2 2 3 8 n e n _ a _ 0 0 0 3 3 p d t . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Figure 7. MEG of full multilayer networks and the transition point p∗ point p∗ corresponds to p∗ = 1. (Bottom) Percentage of deviation of λ = 100(λ 2,agg, where λ gregated network: % Δλ of the full multilayer network. Each circle represents one of the 89 individuals. . (Top) PDF of the transition for two-layer networks (of the alpha and beta layers) of the 89 individuals. The dashed line 2 with respect to the value of the ag- 2 denotes the algebraic connectivity )/λ − λ 2,agg 2 2 Network Neuroscience 432 Multiplex/Multilayer Frequency-based Brain Networks 2 of all possible combinations of two-layer networks versus λ Figure 8. Combining different frequency bands into multilayer networks. We show the algebraic connectivity λ 2,θαβγ of the four-layer full multilayer networks. We construct the layers from the theta (θ), alpha (α), beta (β), and gamma (γ) frequency bands. The labels at the top of each plot correspond to the associated frequency bands. Each circle corresponds to one of the 89 individuals. brain activity during the resting state. Nevertheless, because the signal has been ﬁltered into four different frequency bands, there are eight possible combinations of two layers. In Figure 8, we show the relation between λ 2 for all possible combinations of two-layer networks versus that for the full four-layer multilayer network. Observe the strong correlation of the two-layer networks that include the gamma layer (especially the one that consists of the theta and gamma layers) with the full multilayer network. One can explain such a correlation by inspecting the In Table 2, we separate the intralayer and interlayer total strength of each frequency band. strengths to facilitate interpretation of the results. We observe that the gamma band is the least-active layer, as it is the one with the smallest intralayer strength. Nevertheless, it has the largest interlayer strength (i.e., the sum of the weights of its interlayer edges with all other layers is the largest), so it is the layer that appears to interact most strongly with the other layers. Because (as we have seen) the full multilayer functional networks are close to the mean (cid:7)p∗(cid:9) of the transition point, the weight of the intralayer connections has a strong inﬂuence on the value of λ 2. Therefore, the two-layer theta–gamma network, which includes the layers with the largest interlayer strengths, is the one with the strongest correlation of λ 2 with the full multilayer network (see Figure 8C). CONCLUSIONS AND DISCUSSION Using network analysis as a tool for analyzing brain-imaging data, and (more speciﬁcally) im- plementing and studying a multilayer description of brain activity, has both advantages and drawbacks that must be investigated carefully. As we have discussed, it is possible to en- code such imaging data either as a multiplex network or as a more general type of multilayer network, but different choices lead to different results, which one must then interpret from a neuroscientiﬁc perspective. In our paper, we have analyzed the implications of such a choice on spectral information—and speciﬁcally on the algebraic connectivity λ 2, which has been Network Neuroscience 433 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . t / / e d u n e n a r t i c e - p d l f / / / / / 2 4 4 1 8 1 0 9 2 2 3 8 n e n _ a _ 0 0 0 3 3 p d . t f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Multiplex/Multilayer Frequency-based Brain Networks related to structural, diffusion, and synchronization properties of networks (Almendral & Díaz- Guilera, 2007; Gómez et al., 2013; Masuda et al., 2017; Newman, 2010; Radicchi & Arenas, 2013; Van Mieghem, 2011). We have seen how heterogeneity (of the weights) of the interlayer edges of multiplex networks leads to deviations of the theoretical predictions that one obtains when all interlayer edges have an equal weight p, and we observed that the deviation is even larger when interlayer edges are missing. The importance of these results, which entail large differences in qualitative dynamics, is underscored by the fact that both edge heterogeneity and missing interlayer edges are common features of brain-imaging data. It is also important to understand that although a multiplex description of brain networks is an important and useful approach to integrate multivariate information, and it is often an extremely natural approach—such as for integrating anatomical and functional networks, examining the temporal evolution of a functional network, and so on—it is just an initial step towards developing increasingly complete models to better analyze the spatial and temporal complexity of brain networks (Papo, Zanin, & Buldú, 2014). In this quest, a natural step— although certainly not the ﬁnal step—is to use a full multilayer description for subsequent analyses of frequency-based brain networks.3 In this type of network, it is more biologically plausible to represent brain activity as a full multilayer network than as a multiplex network. The reasons are twofold: (i) a brain region does not necessarily coordinate with itself at different frequencies; and (ii) there may also exist CFCs between different brain regions. Consequently, interlayer connections do not necessarily follow the multiplex paradigm, because some of the interlayer edges probably should not be present, whereas others should be included to ac- count for interlayer coupling between different nodes (i.e., different brain regions), thereby necessitating a fully general multilayer approach.4 Nevertheless, the use of frequency-based multiplex networks has been an effective strategy in previous work for successfully distin- guishing between healthy individuals and individuals who are suffering from schizophrenia (De Domenico, Sasai, and Arenas, 2016), so there are times when a multiplex approach works well. As we have illustrated with our computations, because the number of interlayer edges scales with N in multiplex networks but with N2 in the multilayer networks that we examined, there are important quantitative differences in the values of the algebraic connectivity in the two types of networks. Nevertheless, the effects of edge-weight heterogeneity and missing inter- layer edges on full multilayer networks are qualitatively similar to those in multiplex networks. Speciﬁcally, the analytical predictions that we obtained with homogeneous multilayer net- works deviate from those of heterogeneous cases, showing an important discrepancy for val- ues of the coupling-strength parameter p that are larger than the transition point p∗ . The effect of missing interlayer edges is particularly dramatic for p > p∗

Our analysis of experimental imaging data conﬁrms our results with synthetic networks. 这
small number of interlayer edges in a multiplex network leaves the network in a region in which
layers behave as if they are structurally independent, and a deeper analysis reveals that it is
necessary to increase the weights of the interlayer edges from the experimental data by several
orders of magnitude to reach the transition point p∗
. 最后, using a multiplex repre-
sentation leads to a network with structurally independent layers, unless one uses extremely

3 It is also desirable, 例如, to examine this type of network in a time-resolved manner (ideally using

continuous time), to incorporate spatial constraints, 等等.

4 Analogous critiques are also relevant for research, including our own, on multilayer analysis of time-
dependent brain networks (看, 例如, Bassett et al., 2011), for which one can envision incorporating time delays
in coordination between brain regions.

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Multiplex/Multilayer Frequency-based Brain Networks

桌子 2. Percentage of strength of each layer in a four-layer multilayer network from MEG data.
The ﬁrst column indicates the layer. The second column indicates the percentage of strength of
all intralayer edges in the full four-layer multilayer network that come from that layer; 那是, 为了
a layer l ∈ {我, A, β, C}, the percentage is given by 100 × (西德:12)Ml(西德:12)1
1 是个
(西德:12)Mθαβγ(西德:12)1
entrywise 1-norm of M, corresponding to the sum of all elements of the matrix. The third column
gives the percentage of the strength of the interlayer edges. 那是, for a layer l(西德:4)
, the percentage
是 100 × 2 ∑l(西德:5)=l(西德:4) (西德:12)Pl(西德:4) 我 (西德:12)1
, where l(西德:4) ∈ {我, A, β, C}. Each percentage in the table is a mean over the 89
(西德:12)Mθαβγ(西德:12)1
个人.
Layer
theta (我)
alpha (A)
beta (β)
gamma (C)

Interlayer Strength
9.53%
7.49%
9.34%
10.47%

Intralayer Strength
12.69%
31.43%
12.18%
6.85%

, where the operator (西德:12)中号(西德:12)

large weights (IE。, p (西德:11) 1) for the interlayer edges. 然而, the weights of the interlayer edges
cannot be increased artiﬁcially without limits, because eventually they lose their biophysical
meaning in attempts to compare them to intralayer edge weights.

有趣的是, our results changed drastically when we used a full multilayer description, 在
which a brain region whose dynamics are in a given frequency band can now couple with
other brain regions in a different frequency band. When we considered all possible interlayer
边缘, the majority of the 89 individuals in the experiment are close to the transition point p∗
,
even though the percentage of weight in the interlayer edges is always about 10% of that of
the intralayer ones (见表 2). As we showed in our analysis of synthetic data, it is around
this point that edge-weight heterogeneity and missing interlayer edges begin to yield important
discrepancies from the theoretical results of homogeneous multilayer networks (see Figure 4A).

It is important to note that the percentage of total weight of interlayer edges (relative to the
total weight of all edges) depends strongly on how one measures coordination between brain
地区. When one uses diagnostics other than mutual information, one calculates different
values for the percentages, which can yield multilayer networks either above or below the
transition point p∗
. 因此, no matter what measure one uses, it is mandatory to ﬁrst calculate
the percentage of total edge weight that arises from interlayer edges and to interpret the value
of λ

2 with respect to the value of p∗

The fact that, in frequency-based multilayer networks, the weights of interlayer edges
strongly inﬂuence the value of λ
2 despite being much smaller than the weights of intralayer
边缘 (见表 2) highlights the importance of adequately evaluating CFCs, which tradition-
ally have been disregarded when representing brain activity in terms of functional networks.
As has been discussed prominently in neuroscience, including in critiques of connectomics
(Kopell, Gritton, Whittington, & 克莱默, 2014), the dynamics matter, and investigations of
dynamics must include incorporation and analysis of coupling between different frequency
乐队. The methodology that is used to quantify interactions between brain regions at differ-
ent frequencies leads to different values of λ
2; and there are also other important dynamical
问题, such as phase–amplitude correlations, that we have not investigated in this paper. (看
Bastos & 锄, 2016 for a review of how common reference, volume conduction, 领域
传播, or common input affect the quantiﬁcation of coordinated activity between brain re-
gions.) These caveats notwithstanding, our analysis illustrates a viable approach for examining
the effects of such phenomena on λ
2 (and hence on spectral structure) in multilayer functional
网络, and we expect that similar qualitative phenomena will arise in both multiplex and

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Multiplex/Multilayer Frequency-based Brain Networks

full multilayer networks that are constructed using other choices (例如, different measures than
MI) from the ones that we chose in order to provide a concrete illustration.

We initially studied two-layer networks of the alpha and beta frequency bands, 因为 (我)
it is known that these bands incorporate a large portion of the power spectrum of brain activity
during resting state and (二) these frequency bands typically exhibit stronger synchronization
between brain regions (例如, see Brookes et al., 2016) than the others. The alpha and beta
bands are thus the most commonly studied frequency bands in resting-state studies, 和我们
followed this tradition. 尽管如此, in our particular case, our comparison between two-
layer and four-layer networks illustrates that the gamma band is the one that most inﬂuences
the spectral properties of the full four-layer multilayer network (见图 8). As we showed
表中 2, the theta and gamma layers are the ones whose interlayer edges give the largest
contributions, thereby leading to the strongest correlation between a two-layer network (和
theta and gamma layers) and the complete (four-layer) multilayer network. This fact high-
lights the importance of the well-known phase–amplitude correlations between the theta and
gamma frequency bands (Aru et al., 2015; Canolty et al., 2006), as the former acts as a carrier
of fast-amplitude ﬂuctuations in the latter. 最后, theta–gamma coupling may be of
fundamental importance for understanding the multilayer nature of functional brain networks.

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It also worth mentioning that the facts that we are constructing frequency-based multilayer
networks from (我) data at the sensor level and (二) using MI as a measure to quantify mutual
interdependency between brain regions leads to unavoidable errors in the quantiﬁcation of
edge weights, due to signal mixing and spurious correlations from common sources. 然而,
迄今为止, there does not exist an error-free methodology to construct functional brain networks,
and other alternatives (such as source reconstruction or using different measures to evaluate
interdependency between brain regions) have their own drawbacks. For a detailed discussion,
see the subsection Methodological Considerations (in Materials and Methods).

最后, although our calculations with experimental data used resting-state MEG record-
英格斯, we expect to observe similar behavior in frequency-based multilayer networks of different
origins—whether obtained from any of a large variety of different cognitive or motor tasks, 和
different brain-imaging techniques, or even if they come from a completely different system
(such as functional climate networks; Tsonis & Roebber, 2004). 然而, despite the gener-
ality of our results, it is important to examine richer models of frequency-based brain network,
such as ones that include spatial constraints and temporal evolution.

材料和方法

Data Acquisition

The data were made available by the Human Connectome Project (HCP); 参见http://万维网.
humanconnectome.org/ and Larson-Prior et al. (2013) 欲了解详情. The experimental data con-
sist of magnetoencephalographic (乙二醇) recordings of a group of 89 个人, during rest-
ing state, for a period of approximately 2 min. During the scan, subjects were supine and
maintained ﬁxation on a projected red crosshair on a dark background. Brain activity was
scanned on a whole-head MAGNES 3600 (4D Neuroimaging, 圣地亚哥, CA, 美国) 系统
housed in a magnetically-shielded room, and it included up to 248 magnetometer channels.
The root-mean-squared noise of the magnetometers is about 5 fT/sqrt (赫兹) on average in the
white-noise range (多于 2 赫兹). The data were recorded at a sampling rate of fs ≈ 508.63 赫兹.
Five current coils attached to a subject, in combination with structural-imaging data and head-
surface tracings, were used to localize the brain in geometric relation to the magnetometers

网络神经科学

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Multiplex/Multilayer Frequency-based Brain Networks

and to monitor and partially correct for head movement during MEG acquisition. Artifacts, 坏的
channels, and bad segments were identiﬁed and removed from the MEG recordings, 哪个
were processed with a pipeline based on independent component analysis to identify and
clean environmental and subject artifacts (Larson-Prior et al., 2013). After this process, 这
number N of channels considered for each individual was in the range 235 ≤ N ≤ 246
(with a mean of 243.42), because some channels were used as references and others were
disregarded.

Coordination Between Brain Regions

To estimate coordination between brain regions, we ﬁrst apply a band-pass ﬁlter to the prepro-
cessed signals to obtain, for each of the N sensors, a set of four different time series, 每一个
which corresponds to a speciﬁc frequency band: theta [3–8] 赫兹, alpha [8–12] 赫兹, beta [12–30]
赫兹, 和伽玛 [30–100] 赫兹. We thereby obtain 4N time series of t = 149646 points for each
of the q = 89 个人. We then order the ﬁltered signals according to their corresponding
频带, so the time series Xs
with s ∈ {1, . . . , 氮} corresponds to the theta band,
Xs
with s ∈ {2氮 + 1, . . . , 3氮}
corresponds to the beta band, and Xs
with s ∈ {3氮 + 1, . . . , 4氮} corresponds to the gamma
乐队. Each of the 4N time series corresponds to a node-layer of a frequency-based network.
We calculate mutual information (MI) between the time series Xi
of a pair of nodes i
and j with the formula

with s ∈ {氮 + 1, . . . , 2氮} corresponds to the alpha band, Xs

and X j

(西德:8)

(西德:9)

MIij = ∑
你,v

puv ln

puv
pu pv

(4)

√

and X j

where pu is the probability that Xi = xu, the quantity pv is the probability that X j = xv, 和
puv is the joint probability that Xi = xu and X j = xv at the same time point. We set the
number of bins of the PDFs to be u = v = 5 × (西德:13)(
t/10)(西德:15), where t the is the number of time
points and (西德:13)y(西德:15) corresponds to the nearest integer function of the real number y (where we
round up for .5). When Xi
are independent variables, puv = pu pv, and the resulting
value of MIij is 0. When Xi = X j
for each time series, MIij achieves its maximum value. 笔记
that MIij = MIji, so we obtain an undirected edge between the two time series, and we are
disregarding causality. Calculating MIij allows one to detect coordinated activity even for time
series that include different frequency bands. See Bastos and Schoffelen (2016) and Pereda,
Quiroga, and Bhattacharya (2005) for a review of different measures for quantifying coordina-
tion between brain regions and a discussion of their advantages and pitfalls. Note that MIi(i+N)
measures the coordination between two different frequency bands in the same brain region i.
After calculating MIij using Equation 4, we generate surrogates (which we subsequently use
as a threshold for mutual information) by using a block-permutation procedure (Canolty et al.,
2006): We simultaneously cut each time series into blocks of 1018 点 (关于 2 s), 和
we permute the resulting blocks uniformly at random. This procedure preserves lower fre-
quencies and time-series features (such as nonstationarity and nonlinearity) below the chosen
temporal scale. (See the Supplementary Information (Buldú & Porter, 2017) for further discus-
sion and a detailed investigation of the inﬂuence of block length on the surrogates.) We then
evaluate the mutual information between each surrogate time series to obtain MIrand.

Frequency-Based Multilayer Networks

We construct a frequency-based multilayer network for each individual from the matrix that
encodes the MI of each pair of sensors for the four different frequencies bands. Each layer
includes nodes with the same frequency band, yielding four different layers: theta (我), alpha
(A), beta (β), 和伽玛 (C). We use MIrand from the surrogate times series as a threshold

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Multiplex/Multilayer Frequency-based Brain Networks

and construct a weighted supra-adjacency matrix W with elements Wij = MIij − MIrand
如果
MIij > MIrand
and Wij = 0 否则. We thereby account only for edges with statistically-
signiﬁcant edges. (图中 2, one can see what fractions of edges are 0 in each case.) 最后,
we apply a linear normalization to W to obtain

Mij =

Wij − w
wmax − w

min

where w
min and wmax are, 分别, the largest and smallest entries of W. This ensures that
Mij ∈ [0, 1] for each individual, thereby facilitating comparisons between them. The weighted
supra-adjacency matrix M includes some number of 0 entries, which account for interac-
tions that we deem to not be statistically signiﬁcant. It has four blocks along the diagonal that
encode interactions within each layer (IE。, the same frequency band for different brain regions),
and it has off-diagonal blocks that quantify coordination between different frequencies. 看
Figure 1B for a schematic.

Methodological Considerations

An important issue is the applicability of our results to functional brain networks that are con-
structed using different approaches, such as by using source reconstruction or with different
ways of evaluating coordination between brain regions.
In our study, we constructed func-
tional brain networks from magnetoencephalographic recordings, which are measured at the
sensor level, instead of using the actual magnetic ﬁeld that is generated in each brain region
(IE。, at a source). (One can infer such a ﬁeld using source-reconstruction methods.) Working
with time series at the sensor level necessarily entails signal mixing (锄 & 总的, 2009),
whereas the use of source reconstruction has what is known as the “inverse problem” (of in-
ferring the actual magnetic ﬁeld that is created by the brain regions), and it thus requires the
introduction of a priori assumptions in the model that one uses for source reconstruction. 这
issue has led to a diversity of algorithms to obtain source-reconstructed time series and, 在
many cases, different algorithms yield qualitatively different time series at the source level
(看, 例如, 锄 & 总的, 2009; Belardinelli, Ortiz, 巴恩斯, Noppeney, & Preissl, 2012).
最后, there is an open debate about what constitutes the most appropriate method-
ology to construct functional networks using source reconstruction (崇拜 & 崇拜, 2012; van
Diessen et al., 2015).

Regardless of how one constructs a functional brain network (and whether one uses sensors
or sources), our analysis has the same qualitative implications. Speciﬁcally, estimating the
value of λ
2 from matrices associated with function brain networks is affected strongly by edge-
weight heterogeneity and missing interlayer edges. One also observes the same qualitative
differences between a multiplex construction and a full multilayer one, as this phenomenon
depends on the number of interlayer edges and not on speciﬁc details of the construction of
functional brain networks.

Similar reasoning applies to the construction of frequency-based brain networks using
different ways of quantifying coordination between brain regions. We used MI because it is
able to successfully capture both linear and nonlinear correlations, it is an adequate method
for quantifying interdependencies between signals that are split into frequency bands (大卫,
Cosmelli, & 弗里斯顿, 2004), and it has been used widely in the past to construct functional
brain networks (看, 例如, Bassett et al., 2009; Becker et al., 2012; Chai, Walther, Beck, &
Fei-Fei, 2009; Deuker et al., 2009; Jin, Seol, Kim, & 钟, 2011). On the down side, MI
cannot avoid zero-lag correlations that originate from common sources. (This ﬂaw also occurs

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in many other commonly-employed similarity measures, such as partial correlation, coher-
恩斯, and phase-locking value (Bastos & 锄, 2016).) The systematic deletion of zero-
lag correlations entails a very strict assumption, but because not all zero-lag correlations are
due to common sources or volume conduction (Christodoulakis et al., 2013; Porz, Kiel, &
Lehnertz, 2014; Vicente, Gollo, Mirasso, Fischer, & Pipa, 2008), there does not exist a con-
sensus on how to eradicate them (看, 例如, 布鲁克斯, 伍尔里奇, & 巴恩斯, 2012; 斯塔姆, 诺尔特,
& Daffertshofer, 2007). It is thus important for future work to conduct a systematic comparison
between the most widespread coordination measures and their consequences on the spectral
properties of functional networks. As we stated previously, the effect that edge-weight hetero-
geneity and missing interlayer edges have on functional multiplex and multilayer networks is
a general phenomenon, as it does not rely on the speciﬁc similarity measure that one employs
to evaluate coordination between brain regions, as long as there is sufﬁcient heterogeneity
in the weights of the interlayer edges. As shown in Brookes et al. (2016), such heterogeneity
is expected in frequency-based functional brain networks, given the complicated interactions
between different brain regions at different frequencies.

作者贡献

Javier M. Buldú: 概念化; 数据管理; 形式分析; 方法; Writing —
original draft; Writing — revisions. Mason Alexander Porter: 概念化; Methodol-
奥吉; Writing — original draft; Writing — revisions.

致谢

We thank John Allen for fruitful conversations. We also thank the referees for their helpful
suggestions.

J.M.B. acknowledges ﬁnancial support from Spanish MINECO (project FIS2013-41057) 和
from Salvador de Madariaga Program (PRX15/00107), which allowed him to visit University
of Oxford in summer 2016.

资金信息

Javier M. Buldú, Ministerio de Economía y Competitividad (http://dx.doi.org/10.13039/
Javier M. Buldú,
501100003329), Award IDs:
Ministerio de Economía y Competitividad (http://dx.doi.org/10.13039/501100003329), 奖
ID: PRX15/00107.

FIS2013-41057 and FIS2017-84151-P.

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