A survey of community detection methods in multilayer networks

The research of complex networks oriented from graph theory, which is started from the “Seven Bridges of Königsberg” problem in 1736. Although naive in many respects, this approach has been extremely successful in many real-life applications. At the end of the 20th century, by employing the graph model, the famous small-world (Watts and Strogatz 1998) and scale-free (Barabási and Albert 1999) features were discovered (Newman 2018). In 2002, Girvan and Newman uncovered the community structure in networks (Girvan and Newman 2002), which opened up a continuous upsurge of relevant research. The so-called community structures are groups of nodes that are more strongly or frequently connected among themselves than with the others. Therefore, community detection is proposed to find the most reasonable partitions of a network via observed topological structures. Several conventional approaches are listed in Table 1.

In recent years, a sea of improved algorithms based on the above approaches are proposed with fruitful results. However, with the exponential growth of data scale, community detection on large volume data has encountered a serious of problems. For example, some datasets are changing with time, which brings challenges for traditional research on a static graph. In other words, traditional methods are incapable of dealing with large-scale time-varying networks.

Lancichinetti et al. (2009) raise two problems: The first one is about hierarchical structure, which means large communities are composed of small communities and in turn, large communities group together to form even larger ones. The second is overlapping communities, for example, people belong to more than one community, depending on their families, friends, professions, hobbies, etc. Nodes belong to more than one community is a pervasive scenario in networks.

Besides, the interactions among nodes are becoming more complicated than ever before. The conventional monolayer network (i.e. single-layer network) has provided plentiful cases in which a unit in a complex system is charted into a node, and the interactions between units are straightly represented as edges, no matter what type or weight of the interactions are. With the development of network modeling, we are starting to realize that the existing models cannot fully capture the details existed in some real-life problems, which even leads to incorrect descriptions of some phenomena taking place on real-world networks. Some representative problems occurred are listed as follows. By tackling the above problems with the multilayer network model (Kivelä et al. 2014; Boccaletti et al. 2014), we are able to get a more reasonable result, i.e., the community structures in multilayer networks benefit to identify functionally cohesive sub-units and reveal complex interactions and heterogeneous links.

There have been numerous attempts to address community detection problem in multilayer networks through diverse approaches, e.g., identifying communities in temporal networks by modularity-maximization (Bazzi et al. 2016), where the authors emphasize the difference between “null networks” and “null models” in modularity maximization and discuss the effect of interlayer edges on the multilayer modularity-maximization problem. De Bacco et al. (2017) propose a generative model for multilayer networks, which can be used to aggregate layers into clusters or to compress a dataset by identifying especially relevant or redundant layers. The proposed model is capable of incorporating community detection and link prediction for multilayer networks, and experimental results on both synthetic and real-world datasets shows its feasibility. Analyzing multilayer networks is of great importance because many interesting patterns cannot be obtained by analyzing single-layer networks. That’s our motivation for summarizing these approaches. The contributions of this work are: To the best of our knowledge, this is the latest work that provides a comprehensive survey on various community detection methods in multilayer networks.

The remainder is organized along the other 5 sections. In the next section, we start by presenting the multilayer network models with several real-world datasets and give brief comparisons on different definitions. Section 3 summarizes the existing community detection methods in multilayer networks and provides some metrics for quality evaluation. We introduce various applications of community detection in multilayer networks in Sect. 4, such as temporal networks, social networks, transportation systems, and biological systems. Section 5 offers concluding remarks and perspective ideas.

There are many terms for describing multilayer networks, such as multiplex network, multi-relational network, edge-colored network, node-colored network, multilevel network, multi-dimensional network, independent networks, networks of networks, temporal network and so on (Boccaletti et al. 2014; Kivelä et al. 2014). Table 2 summarizes the main notations used throughout this paper.

As we have known, a graph is a tuple \(G = (V,E)\), where V is a set of nodes and \(E \subseteq V \times V\) is the set of edges that connect pairs of nodes (Bollobás 2013). The model of multilayer networks is more complicated and there are mainly two kinds of explanations. One is summarized by Kivelä et al. (2014), described aswhere \(V_M\subseteq V\times {\mathcal {L}}_1\times {\mathcal {L}}_2\times \cdots \times {\mathcal {L}}_d,E_M \subseteq V_M\times V_M\). They employed a sequence \({\mathcal {L}}\), described aswhere \(\alpha \) is called aspect, and \({\mathcal {L}}_\alpha \) depicts an elementary layer. The layer is a product of elementary layers, which can be represented as \({\mathcal {L}}_1 \times {\mathcal {L}}_2 \times \cdots \times {\mathcal {L}}_d\). The illustration of the multilayer network is given in Fig. 6.

Another model is proposed by Boccaletti et al. (2014), defined aswhere \({\mathcal {G}} = \{ G_\alpha ; \alpha \in \{1, \ldots , L\}\}\) is a family of (directed or undirected weighted or unweighted) graphs \(G_\alpha = (V_\alpha ,E_\alpha )\), which represents layers of \({\mathcal {M}}\) and \({\mathcal {C}}\) depicts the interactions between nodes of any two different layer, given byThe two models differ in the definition of “aspect”. Kivelä’s model takes into account real-life situations, e.g., a social network contains relations among various types, timeline or situations. Each relation set is regarded as an aspect, namely a classification of layers, providing a comprehensive perspective. Thus, it is more considerate. The structure with aspects concept is much richer than that of ordinary networks. Possible aspects include different types of interactions or communication channels, different subsystems, different spatial locations, different points in time, and so on (De Domenico et al. 2016). However, Boccaletti’s model is a general form that easy to understand. Specifically, the supra-adjacency matrix is a distinct tool for representation of a multilayer network, defined aswhere \(A_1, A_2,\ldots ,A_L\) are the adjacency matrix of layer \(1, 2,\ldots , L\), respectively. N is the total number of nodes, which can be calculated by \(N= \sum _{1 \le l\le L} |V^l|\). The non-diagonal block \(I_{\alpha \beta }\) represents the inter-layer edges of layer \(\alpha \) and layer \(\beta \). Thus, the interlayer edges can be represented asTake the above-mentioned 9/11 terrorists’ network for instance, the supra-adjacency matrix is represented in Fig. 7.

We also list some specific networks that can be represented by a multilayer network manner in the following.

Multiplex network is a special case of multilayer networks (Solá et al. 2013), where all layers share the same set of nodes but may have multiple types of interactions. Some works also use multi-relational networks, multidimensional networks (Berlingerio et al. 2011b) or edge-colored networks (no interlayer edges) for substitution. The underlying limitations exist in the network is node-aligned, i.e. each layer has the same nodes but merely differs in edges. Multiplex network is a special form of multilayer network, where the number of nodes in each layer is consistent, and the nodes are one-to-one correspondence. This network model simplifies the complexity of the general multilayer network form and is therefore widely used to deal with some special problems (Kanawati 2015).

Temporal networks (or time-varying networks) differ with conventional dynamic networks, which focus more on the ordinal variations of connections (Kostakos 2009). The temporal network has its own set of research models (Kempe et al. 2002; Kostakos 2009; Tang et al. 2012a), which can illustrate the dynamic characteristics of temporal networks. But it must be mentioned that the power of a multilayer network is that it can be compatible with the representation of the temporal network and can also describe the dynamic characteristics likewise. For example, the multilayer network model we have introduced (Boccaletti et al. 2014) can regard the network structure at each moment as a layer, and the arrangement between different layers is in chronological order. We collected the relationship of characters in the Game of Thrones (the first five seasons), as shown in Figs. 8 and 9.

The study of k-partite networks starts from the complete k-partite graph (i.e., a set of graph vertices decomposed into k disjoint sets such that no two graph vertices within the same set are adjacent) (Brouwer and Haemers 2012). Thus, the k-partite network is also described as node-colored networks (Kivelä et al. 2014), where the nodes are unacquainted in the same layer but have the other layers’ common neighbors with other nodes in the same layer. A sample of k-partite networks is shown in Fig. 10.

However, a special case of k-partite (i.e., \(k=2\)) networks is the bipartite network (or two-mode network), which is more accessible by us. For example, a customer’s transaction records of products can be represented by a “customer-product” network. Fig. 11 presents a classic bipartite network, namely, South women activities network (Davis et al. 2009).

Many basic metrics such as centrality, node similarity, are commonly used by community detection algorithms in monolayer networks. While in multilayer networks, the metrics need to be reformulated and adapted. Thus, in this section, some most important features of multilayer networks are introduced. Studies of structural properties include descriptors to identify the most central nodes according to various notions of importance (Battiston et al. 2014; De Domenico et al. 2015c, 2013; Halu et al. 2013; Solá et al. 2013) and quantify triadic relations such as clustering and transitivity (Battiston et al. 2014; Cozzo et al. 2015; De Domenico et al. 2013).

Communities in single-layer networks comprise a group of well-connected nodes, while in multilayer networks, communities reveal the relationships among nodes in various layers. The comparison of communities on toy examples in single-layer networks and multilayer networks is given in Fig. 12.

The problem of community detection algorithms in multilayer networks is to divide the network into a set of disjoint cohesive modules \(C_1, C_2, \ldots , C_k\) where each module \(C_k\) is comprised of a group of nodes densely connected inside and loosely connected outside the community. It can be described aswith \(\cap _{i=1}^k C_i= \phi \) for non-overlapping community detection and \(\cap _{i=1}^k C_i \ne \phi \) for overlapping scenarios. Dalibard (Dalibard 2012) gives the three requirements about community detection works as followings: Specifically, most of the existing approaches can scarcely satisfy the requirements and hold reasonable efficiency. Besides, the evaluation metrics are also varied from one to another, as introduced in the following subsection.

Quality evaluation for partition results is a complex task due to the lack of a shared and universally accepted definition of community structures. A wide variety of quality functions have been proposed to solve the community detection problem from different perspectives. Several representative metrics are analyzed and classified into three categories (Cazabet et al. 2015): Single score metrics employ quality functions associating a score to the community detection result. For example, the number of edges between partitions can be utilized to evaluate the performance of a given algorithm. Popular metrics include Modularity (Newman and Girvan 2004), Modularity density (Li et al. 2008), surprise (Aldecoa and Marín 2011), and so on. These metrics are universal but often criticized with no consensus of the several meaningful levels of partitions. The comparison with generated networks, e.g., LFR benchmark (Lancichinetti et al. 2008) is widely used to compare the partitions with the community affiliations. It is easy to recognize a good community and capable of evaluating variations in usual communities. However, sometimes the generated networks are not realistic and differ from the partitions we want. The comparison of real networks with ground truth seems to be a reasonable solution. Normalized Mutual Information (Danon et al. 2005), Precision, Recall, and \(F_\text {1-score}\) (Herlocker et al. 2004; Perry et al. 1955) are representative methods. However, these methods depend on the priori partition labels, which are probably unknown in most of real-world datasets.

Early approaches either collapse multilayer networks into a weighted single-layer network (Berlingerio et al. 2011a; Tang et al. 2012b; Taylor et al. 2016b, a), or extend the existing algorithms for each layer and then merge the partitions via consensus clustering (Tang et al. 2009; Papalexakis et al. 2013). However, these approaches have been criticized for ignoring the connections across layers, thereby resulting in low accuracy. Research to date has exhibited some novel algorithms for discovering communities in multilayer networks directly (Ma et al. 2018; Pamfil et al. 2019). Generally speaking, the strategies are mainly classified into three categories (Tagarelli et al. 2017):Flattening methods collapse the layers’ information into a single layer and then conduct the traditional monolayer algorithms for detecting communities. This strategy is very common in multiplex networks, where the multiplex network is converted into a multi-relationship network (Rocklin and Pinar 2013) or a monolayer network (Kuncheva and Montana 2015).

Aggregation methods discover the communities in each layer and then merge them by a certain aggregation mechanism, which could be useful for removing redundant information. The aggregation process requires \(2^L\) comparisons and it’s very time-consuming for a temporal network with numerous layers. Thus, “layer communities grouping” is proposed to reduce the redundant layers (Kao and Porter 2018). Dalibard proposed a parameter \(P_{C_k}^l\) for each layer \(l\in L\) to describe the probability of communities, and then aggregate the communities in each layer in terms of a correlation coefficient \(\rho _{\alpha \beta }\) between layers \(\alpha \) and \(\beta \) (Dalibard 2012). Numerical experiments on synthetic multilayer networks show that the analysis fails in aggregated networks, whereas the multilayer method can accurately identify modules across layers that originate from the same interaction process (De Domenico et al. 2015a). Thus, aggregation is not recommended.

Direct methods aim to detect the community structures directly on the multilayer network by optimizing some quality-assessment criteria without flattening (Oselio et al. 2015). For example, Pramanik et al. defined a multilayer modularity index, i.e., \(Q_M\), and combined with the improved GN and Louvain algorithms, namely GN-\(Q_M\) and Louvain-\(Q_M\), respectively (Pramanik et al. 2017).

There has been plenty of endeavors made in the last decades, however, research for multilayer networks is still in infancy (Tagarelli et al. 2017), and it is promising to extract communities without collapsing multilayer networks. Several representative methods are introduced in the following subsections.

In the last decade, a plethora of approaches have been proposed to address the community detection problem with enormous network data. We list several representative methods (from 2009 to 2019), as shown in Table 4.

Table 4 shows that most of the presented methods are holding relatively high complexity, where GN-\(Q_M\), Louvain-\(Q_M\) and LART methods are based on multiplex modularity maximization and unfavorable on general multilayer networks. Moreover, the majority are designed for multiplex networks, which require the nodes in each layer should be aligned. As we have introduced in the previous subsection, some improved version of classic monolayer algorithms, e.g., GenLouvain has been regarded as a benchmark and is really worth expecting for general multilayer networks. We can anticipate four prospective directions, i.e., random walk-based method, tensor decomposition, nonnegative matrix factorization, and modularity optimization will receive increasing attention over time.

In addition to the above-mentioned directions, quite a part of algorithms focus on overlapping community detection (Liu et al. 2018) and local community detection (Interdonato et al. 2017; Jeub et al. 2015; Li et al. 2019). On the one hand, with the increasing of network scale, global computation becomes time-consuming, which promoting local community detection into our view. Liu et al. (2017) proposed an improved multi-objective evolutionary approach for community detection in multilayer networks. Aiming at solving the local community detection problem, they employ a string-based representative scheme and genetic operation and local search. However, the algorithm adapts the strategy of conducting the Louvain algorithm (Blondel et al. 2008) on each layer and then merges the partitions, which seems to deviate from the multilayer community concept. More than that, comparisons with other competitors are not provided. On the other hand, overlapping communities are also ubiquitous in multilayer networks (De Domenico et al. 2016), i.e., some nodes are attached to multiple partitions simultaneously (Chen et al. 2016). Kao and Porter (2018) proposed a method based on computing pairwise similarities between layers and then executing community detection for grouping structurally similar layers in multiplex networks. The algorithm is verified in both synthetic and empirical multiplex networks. As most of the compared algorithms are designed for multiplex networks, there’s still a great deal of works to do in community detection in the general multilayer networks.

In brief, the research on community detection for multilayer networks is just in its infancy. At the time of this writing, there is still no standard algorithms for general multilayer networks and quite a few problems remain to be solved, such as the optimization of algorithm process to avoid time-consuming procedures, the extending of algorithms for applying in general form of multilayer networks, the simplification of mathematical model, and so on.

Community detection in temporal networks, i.e., temporal community detection, is required to find how communities emerge, grow, combine, and decay in an evolving process (Kawadia and Sreenivasan 2012). A common approach to detect temporal communities is to obtain communities independently in each snapshot by utilizing static methods and then map the partitions between two snapshots together as many as possible. Obviously, it fails to achieve the goal of revealing the evolving process because such methods do not adequately use partitions found in past snapshots to inform the identification for the optimal partition on the current snapshot (Jiao et al. 2017). Thus, as the foremost step of modeling, the traditional graph model is incapable of presenting inter-connections in a temporal network.

Multilayer network model is commonly employed in the study of time-varying networks (or temporal networks, multi-slice networks), in which the time snapshots are modeled as layers and the layers are ordered by a certain sequence. However, there’s a foundational question: Across how many layers must a community persist in order for layer aggregation to benefit detection? To solve this problem, a layer aggregation approach (De Domenico et al. 2014) is proposed to reduce data size or as a data filter to benefit network-analysis outcomes. Since Mucha et al. (2010) introduced the multiplex modularity optimization method, numerous attempts were made in this field (Drugan et al. 2011; Nguyen et al. 2011; Li and Garcia-Luna-Aceves 2013), which opened up a upsurge in unveiling the communities in time-varying networks. Taylor et al. (2017) proposed the random matrix theory and found layer aggregation to significantly influence detectability. The detectability limitation is described as the ability of network structure to form a community, i.e., if the community structure is too weak, it cannot be found upon inspection of the network (Lancichinetti and Fortunato 2011). When the aggregative network corresponds to the summation of the adjacency matrices encoding the network layers, aggregation always improves detectability. The research is beneficial to understand the contraction of network layers and analyze pairwise-interaction data to obtain sparse network representations. The application of layer aggregation can be used for anomaly detection in network data, e.g., in cybersecurity, detecting harmful events such as attacks, intrusions, and fraud.

On account of the critical role of transportation system in modern society, the study on traffic dynamics has become one of the most successful applications of complex network theroy. However, the vast majority of researches treat transportation networks as an isolated system, which is inconsistent with the fact that many complex networks are interrelated in a nontrivial way (Du et al. 2016). Analogously, the transportation system has a variety of traffic manners, such as bus, subway, tram, high-speed train, airline, ship, etc, hence a comprehensive study should cover many of such manners. Early researches of traffic networks mainly focus on a single traffic way and ignore the interactions between their counterparts (Calimente 2012; Chen et al. 2014). Du et al. (2016) utilized a two-layered traffic network to study the distribution-based strategy and improved the generating rate of passengers using a particle swarm optimization algorithm. The multilayer network model utilized in this work is an idealized transportation system, in which each layer has a different topology and supports different traveling speeds. The passengers are allowed to travel along the path of minimal traveling time and with the additional cost they can transfer from one layer to another to avoid congestion. The research indicates that a degree centrality-based strategy is not overly beneficial in enhancing the performance of the system. However, starting from such a strategy and reassigning transfer costs using a particle swarm optimization algorithm improve the capacity and several other properties of the system at a reasonable computational cost. The research is rewarding to the selection of traffic manners and exemplifies how multilayer network models are applied in the urban transportation system.

Inspired by the complex network theory and the multilayer network representation, Hong and Liang (2016) analyze the Chinese airline transportation system with the multilayer network framework, in which each layer is defined by a commercial airline (company) and the weights of links are set by the number of flights, the number of seats and the geographical distance between pairs of airports, respectively. By calculating the clustering coefficient, average shortest path length, and assortativity coefficient of the airports, the research has shown that the Chinese airline is of considerably higher value of a maximal degree and betweenness than the other top airlines. Ding et al. (2018) proposed a method for measurements in areas of Kuala Lumpur (i.e., the national capital of Malaysia) to detect communities. The multilayer network model employed in their research contains the railway layer and urban street layer, which mainly focuses on detecting the changing structures of a rail network and mining in urban network communities. The experimental results suggest that rail network growth triggers structural and community changes, i.e., when an upper-layer rail network grows from a simple tree-like network to a more intricate form, the network diameter and average shortest path length decrease dramatically. The growth of the network allows the remainder of the network to be easily visited, which provides suggestive patterns for city development. Yildirimoglu and Kim (2018) analyzed the urban traffic network by combing bus lines, passenger trajectories, and vehicle trajectories together and formed a three-layered network. By applying the Louvain algorithm (Blondel et al. 2008) independently on the three layers, they found that aggerated patterns can shape geographically well-connected communities in the urban traffic network. The spatial structure is quite alike for the bus and passenger layers, which benefits transit authority in making location decisions. The research is beneficial from a planning perspective that sub-regional borders designate the influential areas around local centers, shopping districts, school zones, etc., and cities can develop policies in order to improve the accessibility to them and enhance network performance.

Another hot-point of community detection research in multilayer networks is social network analysis (Alhajj and Rokne 2014). Social networks have been studied fairly extensively over the last couple of decades, mainly in the general context of analyzing interactions between people in order to determine important structural patterns in such interactions. With the utilization of plentiful data resources from online social media such as Facebook, Twitter, and Flickr, there’s an increasing tendency in discovering community structures in such time-varying social networks (Alimadadi et al. 2019; Rozario et al. 2019; Zhou et al. 2007, 2016). The emergence of online social networks has altered millions of web users’ behavior so that their interactions with each other produce huge amounts of data on various activities. Facebook and Twitter, as the top-two popular social media in our daily life, have been widely employed for social network analysis in recent years (Alimadadi et al. 2019; Türker and Sulak 2018).

The analysis of social networks is usually accompanied by various applications such as information propagation, internal trades analysis, influential spreaders identification, and so on. Diffusion processes, like the propagation of information or the spreading of diseases, are fundamental phenomena occurring in social networks. While the study of diffusion processes in single networks has received a great deal of interest from various disciplines for over a decade, diffusion on multilayer networks is still a young and promising research area, presenting many challenging research issues (Salehi et al. 2015). Numerous attempts have been made to uncover the community structures in international trades, typically represented as bipartite networks in which connections can be established between countries and industries (Alves et al. 2019). Biondo et al. (2017) present a multilayer network model with contagion dynamics, which is able to simulate the spreading of information and the transactions phase of a typical financial market. In their two-layered network framework, the first layer comprises the trading decisions of investors, and the second layer is constructed of the information dynamics, which is fruitfully beneficial to explain the aggregate behavior of markets. Basaras et al. (2017) proposed an effective method to detect influential spreaders in multilayer networks based on the underlying community structures. The experimental evaluation shows that the proposed method outperforms the major competitors proposed so far for either single-layer or multilayer networks.

The above-mentioned applications mainly focus on a local structure or some certain context-based community detection for the case of large volume and various dynamic changes of networks. Thus, it is of great significance in designing some smart algorithms to mine the valuable information among plentiful social network resources.

Biological systems, from a cell to the human brain, are intrinsically complex (Ma’ayan 2017). Multilayer networks, described by an intricate network of relationships across multiple scales, are most widely employed in representing such systems. The majority of the biological processes are constituted by a group of proteins that are connected densely (Cui et al. 2012). The protein-protein interaction (PPI) network contains the communications among the protein groups that communicate with each other closely, which can be used to predict the complexity of the function of normal proteins (Srihari et al. 2017). In general, there are two typical protein communities: protein complexes and protein functional modules. Protein complexes are sets of proteins that interact with each other to execute a single multimolecular mechanism. Protein functional modules are sets of proteins that participate in a particular biological process, and interact with each other at different time and places (Spirin and Mirny 2003). Recently, several studies are highlighting how simple networks, i.e., obtained by aggregating or neglecting temporal or categorical descriptions of biological data, are not able to account for the richness of information characterizing biological systems (De Domenico 2018). Chen et al. (2018a) proposed an MLPCD algorithm by integrating Gene Expression Data (GED) and a parallel solution of MLPCD using cloud computing technology. They reconstructed the weighted protein-protein interaction (WPPI) network by combining PPI network and related GED, and then defined simplified modularity as the ratio of in-degrees and out-degrees of proteins in a community. By utilizing an improved Louvain algorithm (Blondel et al. 2008), they have achieved the goal of detecting protein complexes and protein function modules.

Although non-overlapping communities are more commonly studied in network neuroscience, a model of community structure that allows for overlapping networks offers a more realistic presentation of brain network organization (Wu et al. 2011). Taking overlapping communities into consideration, Zhang et al. (2018a) propose a central edge selection (CES) based community detection algorithms for PPI networks. Experimental results on three benchmark networks and two PPI networks indicate the excellent performance of the proposed CES algorithm. Kurmukov et al. (2017) propose a framework to compare both overlapping and non-overlapping community structures of brain networks within the machine learning settings. The performance of the proposed framework is verified in the task of classifying Alzheimer’s disease, mild cognitive impairment, and healthy participants. Pan et al. (2018) present an aggregation approach to detect communities in multilayer biological networks, which first constructs a consensus graph form multiple networks and then applies traditional algorithms to detect communities. Inspired by the fact of few shared edges existed among different networks, they merge the weights of edges from different layers and cut off the nodes with low weights. The approach is simple but limited by the application scenarios.

Another notable direction of biological research is about human brain networks. Cantini et al. (2015) propose a multi-network-based strategy to integrate different layers of genomic information and use them in a coordinated way to identify diving cancer genes. The multi-networks they focus, combine transcription factor co-targeting, microRNA, cotargeting, protein-protein interaction, and gene co-expression networks. The combination of different layers benefits extracting from the multi-networks indications on the regulatory pattern and functional role of both the already known and the new candidate diver genes. Sanchez-Rodriguez et al. (2019) introduce an approach for the detection of a modular organization by considering the temporal scales of the information flow over large-scale brain graphs, and several organizational patterns existing in the brain anatomical and functional networks are found. The structures may coexist together, in a dynamical way that is given by the temporal scales of the activity they produce, guaranteeing functional independence and coordination.

In brief, discovering the underlying patterns in biological networks is experiencing a blossom. With the development of network science, multi-biological networks provide plentiful data resources than ever before, which requires us to dedicate more to this promising field.