Multilayer modeling and analysis of human brain networks

Abstract Understanding how the human brain is structured, and how its architecture is related to function, is of paramount importance for a variety of applications, including but not limited to new ways to prevent, deal with, and cure brain diseases, such as Alzheimer’s or Parkinson’s, and psychiatric disorders, such as schizophrenia. The recent advances in structural and functional neuroimaging, together with the increasing attitude toward interdisciplinary approaches involving computer science, mathematics, and physics, are fostering interesting results from computational neuroscience that are quite often based on the analysis of complex network representation of the human brain. In recent years, this representation experienced a theoretical and computational revolution that is breaching neuroscience, allowing us to cope with the increasing complexity of the human brain across multiple scales and in multiple dimensions and to model structural and functional connectivity from new perspectives, often combined with each other. In this work, we will review the main achievements obtained from interdisciplinary research based on magnetic resonance imaging and establish de facto, the birth of multilayer network analysis and modeling of the human brain.

The functional connectivity of the brain is usually obtained by measuring a specific type of physical signal (e.g., blood-oxygen-level dependent -i.e., BOLD -contrast as in fMRI or magnetic field as in MEG) from different regions and then comparing pairwise signals by means of some similarity measure (e.g., crosscorrelation, transfer of entropy, spectral coherence, so forth so on). If the similarity between two signals is statistically significant, a functional link is considered between the corresponding brain regions. Many studies differ in the type of signal they measure and the statistical methodology adopted to build the functional network, but they all share the approach described above.
Network modeling approaches successfully unveiled interesting features such as small-worldness -where the underlying topology is highly locally clustered and the presence of long-range connections dramatically reduce the distance between units -modular and richclub organization -where the underlying topology can be coarse-grained and described as a network of modules, with highly-connected units tending to be connected each other more frequently than random expectation. The success of network mapping increased, in parallel, the need for novel methodologies devoted to unravel the structure and the function of the brain at multiple spatial and temporal scales [3,4]. However, the lack of an appropriate mathematical framework for the representation and analysis of multivariate connectivity data forced many studies to neglect, disregard or aggregate available information, in order to cope with the high amount of underlying complexity.
More recently, researchers explored the possibility to study the human brain without necessarily either throwing out or aggregating the data deluge available nowadays. An important and promising approach is to use multilayer networks (see Refs. [5,6] for a thorough review), recently developed to provide a mathematical framework [7] to model and analyze complex data with multivariate and multi-scale information. Recent results from this research direction are exciting and provide new insights about our understanding of structure and function of the human brain.

Multilayer network representation of human brain
A multilayer network consists of several distinct classical networks, each one encoding a specific type of information about the system. In the following, we will briefly discuss different types of multilayer brain networks where layers' connectivity, measured with respect to a specific definition of similarity (e.g., crosscorrelation, spectral coherence, and so forth so on) might encode i) activity in different frequency bands; ii) time-varying activity; iii) activity with respect to different tasks; iv) structural and functional connectivity.
While standard networks can be represented by adjacency matrices, indicating the presence and the intensity of connections among system's units, multilayer networks requires higher-order matrices, i.e. tensors, to be appropriately represented [7] (see Figure 1a). In general, the components of the multilayer adjacency tensor of N nodes and L layers are indicated by M iα jβ and encode the connectivity between unit i in layer α and unit j in layer β, with i, j = 1, 2, ..., N . For instance, intra-layer connectivity in the α−th layer is given by the entries M iα jα . A standard approach is based on flattening this rank-4 tensor into a rank-2 tensor, named supra-adjacency matrix, with a block structure where diagonal blocks encode intra-layer connectivity and off-diagonal blocks encode inter-layer connectivity ( Figure 1b).
The tensorial representation of multilayer networks allows us to develop a powerful mathematical framework to extend traditional complex network analysis such as detection of modular super-units [8,9] and identification of most central units [10]. The majority of such tools is based on the analysis of how information spreads through the multilayer system (see Ref. [11] and references therein) and provides a suitable framework for the structural analysis of human brain.
While several classical network concepts have been successfully and satisfactorily extended to multilayer systems, approaches adopted to model human brain are mainly based on multiplex and interconnected multiplex topologies. In both models, the same node is usually replicated on more than one layer, where it exhibits different connectivity patterns depending on the information encoded by the layer. A multiplex topology is an edge-colored multigraph consisting of different layers that are not interconnected each other: M iα jβ = 0 for any i, j = 1, 2, ..., N and α, β = 1, 2, ..., L (with α = β), using the notation introduced before. An interconnected multiplex topology includes links across layers, although only the ones among node's replicas are allowed: M iα jβ = 0 for any i = j and α = β, whereas M iα iβ = 0 for α = β (as in Figure 1a). Other multilayer network models are possible, but they have found a few applications in neuroscience, if any.
It is worth remarking that one should be cautious in the choice of the network model to adopt for the analysis of human brain, if one is interested in exploiting the tensorial algebra developed to naturally extend the majority of classical network descriptors to the multilayer realm [7]. In fact, when interconnectivity is absent, analysis based on the multilayer adjacency tensor provides the same results of classical analysis of each layer separately.
In the following we will consider applications involving interconnected multiplex networks and we will refer to them as multiplex, for sake of simplicity.

Frequency-based decomposition
Frequency-based decomposition is an approach that provides a multilayer functional representation of human brain. In the case of fMRI, signals are filtered and components between 0.01 and 0.1 Hz are usually kept [12,13,14] (see Ref. [15] for a review). The choice of the frequency band might have deep impact on the functional representation of the brain. In fact, standard methodologies do not distinguish the contributions coming from different frequency bands, considering only one specific range. The resulting network provides a functional map of the brain and allows to identify special regions which act as hubs, i.e. units either with larger connectivity than others or with strategic functions which maximize the information flow through them [16,17,18,19] (see Ref. [15] for a review). It is worth remarking that while the concept of information flow is well defined for structural networks, it requires to be careful in the case of functional ones. In fact, functional representations encode statistically significant correlations between brain regions and, strictly speaking, the concept of information flowing through such (often non-physical) links is poorly defined. Here, the interpretation of some network descriptors in terms of information flow is given to better elucidate the meaning of the descriptor in a structural context, rather than to characterize physical information dynamics.
Given their functional importance, hubs mediate interactions among other regions and might favor the brain's integrated operation. They are generally identified by centrality descriptors [20] and they are of particular interest in many applications [21,22,23,24]. Recent studies have shown that the importance of each region is subjected to dramatic changes depending on the frequency cuts [25] and that hubs might be very different when functional connectivity is measured in different frequency bands [26]. These results, together with previous findings concerning the importance of topological information measured from components above 0.1 Hz [27,28,29], suggest that a novel framework for modeling and analysis of human brain functional connectivity is required.
The new framework must be able to consider functional information from different frequency bands, simultaneously: in practice, for each band it is sufficient to build a functional network and then to analyze the resulting system as a whole. Multilayer networks provide the mathematical background [7] for this purpose. In this new framework, each region of the brain is mapped into a network node and replicated across all layers, encoding frequency bands, where they are connected with other nodes by means of functional links -corresponding to significant correlations in a specific frequency band. The methodology is summarized in the top panels of Fig. 2, while the result of the procedure applied to a real human brain is visualized in Fig. 3.
Nodes are interconnected with their replicas -also known as state nodes -across layers and the weight of these links is, in general, a free parameter which must be estimated from the data or by maximizing a specific cost function [30]. For each unit, the set of state nodes constitutes a physical node corresponding to a specific brain region. State nodes of a single physical node are interconnected categorically, i.e., they build a clique.
The first question to answer is to which extent such an enriched representation of functional connectivity is more valuable than other aggregated (or less rich) representations. The answer has been recently given in Ref. [30], where it has been shown that each functional layer -in a range between 0.01 Hz and 0.25 Hz, in steps of 0.02 Hz -provides unique information and should be neither aggregated with other layers nor neglected. The result is based on the analysis of structural reducibility [31], a modern technique grounded on information entropy.
The irreducibility of the multilayer functional representation of human brain raises the necessity for multilayer analysis of the underlying architecture and a few first results have been recently reported about the identification of hubs. In other contexts, it has been shown that hubs in a multilayer network might be dramatically different from hubs in each layer of the system [32]. An intuitive example is given in the following. Let us consider a two-layer system where a certain node is in the periphery of both networks, and let us consider that such a node is the only one in common to the two layers. It is clear that this node is crucial for the exchange of information between the two layers and, as a consequence, it will be most central with respect to this criterion. In a classical analysis, where the layers are considered separately, the node is still peripheral and it would be the less central [1] .
The multilayer analysis of brain's regions centrality reveals that hubs are, in general, different from the hubs identified by standard methodologies based on single-layer network analysis. The most surprising finding is that such hubs can be used to distinguish, with high accuracy and sensitivity (above 80% in both cases), the brain of a schizophrenic patient from a healthy brain in resting state [30], thus improving our understanding of schizophrenia and opening the door to the analysis of other brain disorders within the same framework.
These studies provide evidence for and support the hypothesis that functional layers do not act as independent entities, suggesting the existence of mechanisms for integration and segregation of brain activity [1] Here, information exchange can be modeled by bits diffusing through the system either along random walks [33] or shortest paths [34,35] between two endpoints .   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 within and across different frequency bands. Very recently, a mechanistic model for this process has been proposed [38]. The authors have compared the performance of two models: in the model A, each brain region generates oscillations in a single frequency; in model B, each brain region can generate oscillations in multiple frequency bands. The model B, named multi-frequency model, does not take into account cross-frequency interactions but it still outperforms single-frequency model in reproducing empirical MEG data [38].
In a more recent work, MEG recordings during resting states in subjects affected by the Alzheimer's disease have been used to build a multilayer network where layers represent functional connectivity in different frequency bands (2-4 Hz, 4-8 Hz, 8-10.5 Hz, 10.5-13 Hz, 13-20 Hz, 20-30 Hz, 30-45 Hz). The study has provided evidence that regional connectivity in unhealthy subjects was abnormally distributed across frequency bands -a feature with no counterpart in healthy individuals -revealing an abnormal loss of inter-frequency centrality in memory-related association areas. The proposed methodology has led to high classification accuracy (78.4%) and sensitivity (91.1%) of subjects, confirming the superior performance of multilayer analysis as compared to more traditional approaches [39].
All the results briefly described in this section support and reinforce the possibility to adopt multilayer techniques as a potential non-invasive biomarkers for neurodegenerative diseases and mental disorders.
Time-varying network model and task-based decomposition Instead of building functional layers in the frequency domain, it might be desirable to consider the brain activity in the time domain, because temporal changes and their mapping might be biologically meaningful. It is worth noting that historically this was, in fact, the first multilayer approach to the analysis of brain networks, even when a formal theory for this type of structure was not yet available [40].
Usually, the measured BOLD activity is divided into a series of time windows named snapshots, -which can be overlapping or not -and a pairwise measure of correlation between regions of interest is calculated to build a functional network for each snapshot. However, it is fundamental to remark that this processing phase is far from providing a rigorous and well established method to build functional networks from this type of data [41]. In practice, overlapping and non-overlapping windows are not statistically independent [42,43], their length is a free parameter and their choice requires careful inspection of the data [44,45] to avoid mapping spurious connectivity fluctuations.
The resulting network is a multilayer graph where each layer corresponds to a functional snapshot of brain activity. This approach has the advantage of building a static backbone of the underlying functional dynamic of human brain that can be used, for instance, to better understand how it operates during specific tasks or on the onset of an epileptic seizure. In this regard, multilayer networks describing how functional connectivity changes across time provide a richer framework than traditional approaches [46]. In this framework, state nodes are interconnected only with their subsequent replicas, like in a chain. This methodology, summarized in the bottom panels of Fig. 2, has opened the door to several studies and triggered the development of novel theoretical measures to identify the most influent brain regions during learning [47] and how they cluster together in functional modules [48], to cite some of them.
The multilayer model for time-varying networks can be used to explore the role of functional fluctuations while in resting state or performing specifying activities (see Ref. [3] for an up-to-date review), where in the latter case one defines a task-based representation of brain activity [48,49]. This type of decomposition is of particular interest because it is possible to map the reconfiguration of brain regions' correlated activity between different tasks or during a learning process [50,51].
Besides the variety of its applications, very recently, this novel framework has been used to better characterize high-level language processing in humans by using fMRI data from 22 human subjects, asked to perform a language comprehension task. While it is known that the activity of left frontal, temporal, and parietal cortices is very correlated -constituting a functional system -when an individual is performing a naturalistic language comprehension task or she is resting, it is still poorly understood how they become part of such an integrated functional system. By identifying functional modules within the multilayer framework, involving the generalization of classical modularity maximization to the multilayer domain [8], it has been shown that a stable core of mutually co-activating brain regions emerges mainly in the left hemisphere, whereas a periphery of brain regions is developed in the right hemisphere, while co-activating with different regions at different times. One might ask if it is required to use such a complicated computational tool for this purpose. While it is possible to perform communityor any other network descriptor -analysis in each layer separately, only by performing multilayer analysis it is possible to account for the continuity of communities -or centrality, influence, clusters, and so forth so on -over time, a key advantage that has no counterpart in other single-layer or aggregated approaches. This result, heavily based on the multilayer analysis of functional brain connectivity, suggests the existence of trade-off between a region's specialization and its capacity for flexible network reconfiguration [52] and highlights the power of this novel analytical framework to improve our understanding of brain's functional dynamics.
While brain activity during a single task can be studied by means of a temporal network, it has been recently shown that the networks corresponding to different tasks can be used to encode the layers of a multitask multilayer topology [53]. At variance with the temporal networks described above, where replicated nodes are interconnected across layers following the arrow of time (i.e., node i in layer corresponding to snapshot τ is linked to node i in layer(s) corresponding to τ > τ ), interconnectivity in multitask networks is categorical (i.e., node i in layer α is linked to all of its replicas in layers β = α). Results from this research direction indicate that several inter-region temporal patterns observed at rest are preserved during different tasks, suggesting the existence of a primary intrinsic functional network architecture -similar to the one observed in resting state -that is enriched by a secondary task-dependent functional connectivity [53].

Structural and functional decomposition
Understanding the interplay between brain structure, function and dynamics is a longstanding challenge [54,55,56,57,58,2]. The novel multilayer framework provides a unique opportunity to study, simultaneously, structural and functional information and, in fact, it has been recently used for this purpose [59,60].
The first study concerns motifs, specific subgraphs of reduced size (generally 3 or 4 nodes) that play a fundamental role for the stability of the underlying system and several functions [61]. The significance of a motif is usually estimated by its occurrence with respect to a null model of the network. While the relationship between structural and functional brain motifs has been studied in the past [62], in Ref. [59] the authors have exploited the recent mathematical advances in network analysis to investigate multiplex motifs [63].
In their setup, each multiplex network consists of two layers: one reflecting anatomical connectivity -inferred from Diffusion Magnetic Resonance Imagingand one encoding functional relationships -inferred from functional Magnetic Resonance Imaging -among the brain regions of healthy subjects. In this context, multiplex motifs are potentially more informative than their single-layer (either structural or functional) counterparts taken separately, because a larger number of configurations, accounting for both layers simultaneously, is considered. The results indicate that when a physical connection between different brain regions coexists with a non-trivial positive correlation in their activities, the corresponding motif is statistically significant, i.e. it occurs more frequently than random expectation. As a consequence, this works provides further quantitative support to the hypothesis that functional connectivity is non-trivially constrained by brain architecture.
In the same spirit, another study explored the relationship between structure and function the Macaque cortical network [60]. In this case, the functional layer has been derived from simulated neural activity, whereas structural information is provided by anatomical connectivity. From the study of multiplex clustering, involving triangles of nodes on the two layers, the authors have investigated the emergence of functional connections that have no structural counterpart and the dependence of the multiplex network on the neural dynamical regime.

Conclusion and Outlook
Increasing evidences show that our understanding of human brain cannot prescind from using more complex multi-scale and multilayer models than a decade ago. The new models have to account for the hierarchical organization of the brain in both spatial and temporal dimensions, as well as its functional organization changes across temporal and frequency domains, while interplaying with the underlying structure. The recent advances in network science led to the development of a powerful mathematical framework for multilayer networks [7], topologies able to account for the simultaneous existence of different types of relationships between system's units and their variation over time [5,6,11].
The present epoch is mature enough for multilayer analysis of human brain, to investigate the functional role of brain regions in different domains. While the field is still in its infancy, intense research activity is ongoing, based on advanced mathematical models to represent structural and functional connectivity, their evolution over time and their interdependence. Network science is just coming out of the multilayer revolution, which triggered hundreds of applications in all disciplines, from life sciences to humanities, in a few years. While there are still many theoretical challenges to tackle, such as the definition of appropriate null models to compare against the connectivity of empirical multilayer systems [64,65,66,67], the outcome of such a revolution already provides several computational tools to identify key units in multilayer systems [68,7,34,35,69,32], determine their organization in modules [8,9,70,71], reduce connectivity into simpler architectures [31] and discover the hierarchical organization of layers [72] .   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 The application of some of these tools to the analysis of human brain provided exciting novel insights about its structure and function. Nevertheless, from a methodological point of view it is still a challenge to define a physical meaning for inter-layer connectivity, beyond purely mathematical or computational arguments. For instance, in multiplex networks representing multimodal connectivity or structural-functional relationships, as well as in time-varying networks, nodes are replicated across different layers are linked with each other: in the former all the replicas are interconnected, whereas in the latter only replicas corresponding to subsequent temporal snapshots are connected in order to preserve the underlying causal structure of the data. However, the weight to assign to these inter-layer connections is a free parameter, as in the case of frequency-based decompositions (in this last case, a partial solution might be given by the analysis of cross-frequency correlations).
In the next future, we expect more complex structural and dynamical models able to account for several types of information, simultaneously. Such models will incorporate multivariate information from different domains, e.g., space, time and frequency, and across different scales, from cellular level to entire brain regions, with the ultimate goal to shed light on how the interplay between structure and dynamics is related to brain diseases and give rise to cognition.

Competing interests
The author declares that he has no competing interests.
All the referees provided valuable suggestions that improved the quality of this review. We have taken into account all the recommendations, and a more detailed reply to each point is attached at the end of this letter.
Summarizing the changes:  Extended Background section, to better explain structural and functional brain networks;  Extended second section, to provide more details about salient features of different types of multilayer networks, with special focus to the ones of interest for neuroscience applications;  Extended the section on frequency decomposition, to cover one study appeared while the manuscript was under review;  Extended the time-varying network section, to include a more appropriate review of taskbased decomposition;  Modified the Conclusion section and extensively extended it to include outlooks.  Added Figure 3, upon suggestion of several colleagues. The figure intends to show a practical visualization of an empirical multilayer functional brain from different perspectives.

Reviewer #1
This review deals with the important topic of multi-layer network methods for neuroscience. It is well-written, in general, and discusses many of the important papers sitting at the confluence of these two fields.
Thank you for your positive assessment of our manuscript.
I have little to add in terms of writing style and content, though I have a few suggestions for how the scope could be broadened and some of the discussions made to be more balanced. Throughout the review, the author discusses functional brain networks in terms of information transfer and transmission. I think that it is important to note that functional brain networks do not transfer "information" nor is "information" capable of flowing over their links or between layers (functional networks represent brainwide correlation patterns that arise as a consequence of some dynamics constrained by an underlying physical network (i.e. a structural or anatomical brain network). This is a point that is often overlooked, partly because the measures used to diagnose or infer information transfer are generally agnostic as to whether they are applied to a functional or structural network, e.g. one can compute betweenness centrality measures on both classes of networks, but the concept of shortest path structure in a network whose links are based on correlations or coherence estimates is vague. In short, I would suggest revising some of the statements on information transfer and functional networks (e.g. on p.2 when discussing information flow over multi-frequency networks).
We agree with the referee on this point and we have added the following text to avoid confusion in the reader: It is worth remarking that while the concept of information flow is well defined for structural networks, it requires to be careful in the case of functional ones. In fact, functional representations encode statistically significant correlations between brain regions and, strictly speaking, the concept of information flowing through such (often non-physical) links is poorly defined. Here, the interpretation of some network descriptors in terms of information flow is given to better elucidate the meaning of the descriptor in a structural context, rather than to characterize physical information dynamics.
Another point that, in my opinion, should be noted, is the lack of rigorous methods for estimating time-varying functional connectivity. The author presents this topic as though constructing such networks is as simple as specifying a window length, calculating a correlation, and then aggregating the "snapshots." In general, the problem is more difficult -for overlapping (and even non-overlapping) windows successive windows are not independent of one another [3,1], the length of the window is a free parameter and needs to be chosen carefully [4,2], whether to taper the window is an open question [5], and so on. The authors should at least touch on this idea and note that while the multi-layer framework may be prepared to deal with time-varying networks, the process of estimating the connectivity over time is not yet resolved.
We thank the referee for this valuable comment. We completely agree with the referee and, in the time-varying network section (where overlapping windows are mentioned for the first time), we have added a brief discussion about this point and we have cited the suggested references.
It is also the case that the author focuses primarily on networks estimated from MRI data. While it is true that most network-based analysis in the neurosciences is relegated to these kinds of data, it would maximize the readibility and relevance of the article if the author could also include some discussion on how these approaches could influence, say, cellular connectomics (this is where the author could speculate a bit).
We agree with the referee, but or choice has been mainly driven by the fact that this review will be part of a thematic series on (f)MRI. Nevertheless, we have partially accounted for this comment in the Conclusion and Outlook section.
Finally, it would be good to touch on null models. The use of rewiring models for "static" networks is well-documented, but te appropriate models for time-varying or multi-frequency multi-layer networks are not as well understood. It would be nice to offer some discussion of the matter.