Node Selection

The choice of nodes often depends on the data type. For example, if one has EEG data, network nodes may be surface sensors or hypothesized neural dipoles estimated from source localization techniques. For fMRI data, network nodes may be (i) single voxels or (ii) some aggregate of voxels chosen to represent anatomical (Achard et al. Reference Achard, Salvador, Whitcher, Suckling and Bullmore2006) or functional (Power et al. Reference Power, Cohen, Nelson, Wig, Barnes, Church, Vogel, Laumann, Miezin and Schlaggar2011) areas. In some cases, these atlases can be defined at different spatial scales (Cammoun et al. Reference Cammoun, Gigandet, Meskaldji, Thiran, Sporns, Do, Maeder, Meuli and Hagmann2012), where each scale represents a further partitioning of a region into subregions (Zalesky et al. Reference Zalesky, Fornito, Harding, Cocchi, Yücel, Pantelis and Bullmore2010; Bassett, Brown, et al. Reference Bassett, Brown, Deshpande, Carlson and Grafton2011). Thus, when defining network nodes on the basis of a parcellation, one can also study how the choice of parcellation (Wang et al. Reference Wang, Wang, Zang, Yang, Tang, Gong, Chen, Zhu and He2009; Bassett, Brown, et al. Reference Bassett, Brown, Deshpande, Carlson and Grafton2011) or parcellation scale (Zalesky et al. Reference Zalesky, Fornito, Harding, Cocchi, Yücel, Pantelis and Bullmore2010; Bassett, Brown, et al. Reference Bassett, Brown, Deshpande, Carlson and Grafton2011) affects network properties.

Edge Selection

The choice of edge also often depends on the data type. For structural brain networks, one might first perform diffusion tractography to trace white matter tracts through the brain (Sporns Reference Sporns2011). A network edge can then be defined as a weighted connection whose strength is given by the number or average fractional anisotropy of tracts connecting two nodes (Hermundstad et al. Reference Hermundstad, Bassett, Brown, Aminoff, Clewett, Freeman, Frithsen, Johnson, Tipper, Miller, Grafton and Carlson2013). For functional brain networks, one might define edges on the basis of statistical relationships between time series of regional activity, such as interregional correlation, coherence, or phase synchronization that produce undirected networks (A is connected to B with the same strength that B is connected to A). Alternatives include Granger causality that produces directed networks (connection from A to B can have a different strength than the connection from B to A).

Weighted versus Binary Networks

The methods described above produce weighted networks where the value associated with an edge represents the strength of the relationship between nodes. However, concerns regarding measurement noise have traditionally led to the application of thresholding techniques to create binary networks (Achard et al. Reference Achard, Salvador, Whitcher, Suckling and Bullmore2006), where connections between nodes either exist or do not exist. The most common thresholding technique is cumulative thresholding, where only connections whose strength is above a threshold value are retained (Bassett et al. Reference Bassett, Nelson, Mueller, Camchong and Lim2012). However, weak connections have been implicated in network function and dysfunction (Bassett et al. Reference Bassett, Nelson, Mueller, Camchong and Lim2012; Santarnecchi et al. Reference Santarnecchi, Galli, Polizzotto, Rossi and Rossi2014), and therefore caution should be exercised when thresholding techniques are employed (Ginestet et al. Reference Ginestet, Nichols, Bullmore and Simmons2011). Future work in the field should emphasize the importance of analyses and methods that incorporate the strength of connections (Rubinov and Sporns Reference Rubinov and Sporns2011).

Spatial Embedding

Finally, it is important to note that brain networks are an example of a class known as spatially embedded networks (Klimm et al. Reference Klimm, Bassett, Carlson and Mucha2014): nodes have physical locations, and internode distances can be calculated in Euclidean space. In the brain, nodes that are physically close tend to be strongly connected (Kaiser and Hilgetag Reference Kaiser and Hilgetag2006; Bassett et al. Reference Bassett, Greenfield, Meyer-Lindenberg, Weinberger, Moore and Bullmore2010), leading to an efficient info-physical structure (Kaiser and Hilgetag Reference Kaiser and Hilgetag2006; Bassett et al. Reference Bassett, Greenfield, Meyer-Lindenberg, Weinberger, Moore and Bullmore2010; Bullmore and Sporns Reference Bullmore and Sporns2012). It can be useful to take this information into account when designing null models used to compute network statistics (Expert et al. Reference Expert, Evans, Blondel and Lambiotte2011).

Network Statistics

We assume that we have an N-node graph, G, represented by an $N\times N$ adjacency matrix A, where the element $A_{ij}$ gives the strength of the connection between node i and node j.

Degree

The degree of a node can be computed as

(1)$\begin{array}{}{k}_i=\sum _j{A}_{ij}\mathrm{.}\end{array}$

Nodes identified as having an unusually high degree when compared to other nodes in the network are often referred to as “hubs.” The distribution of node degrees can be an important distinguishing feature between different networks: random networks display a Poisson distribution (Newman Reference Newman2010), whereas brain networks display a heavy-tailed distribution (Achard et al. Reference Achard, Salvador, Whitcher, Suckling and Bullmore2006), indicating the presence of hubs (van den Heuvel and Sporns Reference van den Heuvel and Sporns2013).

Path Length

To measure the distance between two nodes in topological space, we compute the shortest path length, $d_{ij}$, as the minimal number of edges that must be traversed to walk from node i to node j along edges in the graph. Note that the shortest path length computed in network space is mathematically unrelated to the physical distance between two nodes. We can define the characteristic path length,

(2)$\begin{array}{}L=\frac{1}{N(N-1)}\sum _{ij\mathrm{,}i\ne j}{d}_{ij}\mathrm{,}\end{array}$

as the average shortest path length.

Clustering Coefficient

The local clustering coefficient of a node, $c_i$, can be defined as the fraction of a node’s neighbors that are also connected to one another. For a binary network, we define the global clustering coefficient to be the mean local clustering coefficient values over all N nodes in the graph:

(3)$\begin{array}{}C=\frac{1}{N}\sum _i{c}_i\mathrm{.}\end{array}$

For weighted networks, one can define related diagnostics that account for the local clustering of edge weights.

Efficiency

The global efficiency of a network is defined as

(4)$\begin{array}{}{E}_{\mathrm{glob}}=\frac{1}{N(N-1)}\sum _{ij\mathrm{,}i\ne j}\frac{1}{d_{ij}}\mathrm{.}\end{array}$

Because this formulation uses information regarding the shortest paths through the network (Latora and Marchiori Reference Latora and Marchiori2001), efficiency is often interpreted as a measure of how efficiently the network transmits information (although, for caveats on this interpretation in the context of brain networks, see Rubinov and Bassett [Reference Rubinov and Bassett2011]).

The local efficiency of a node in a network is given by the average global efficiency of that node’s local subgraph, and therefore the local efficiency of the entire network is given by

(5)$\begin{array}{}{E}_{\mathrm{loc}}=\frac{1}{N}\sum _i{E}_{\mathrm{glob}}\left(G_i\right)\mathrm{,}\end{array}$

where $G_i$ is the local subgraph of node i consisting of the nearest neighbors of node i and their edges to other nodes in the subgraph.

Modularity

An interesting feature of many complex networks is their community structure: the presence of subsets of nodes that are more densely connected among themselves than they are to the rest of the network (Porter, Onnela, and Mucha Reference Porter, Onnela and Mucha2009). A popular method to detect communities in networks is to maximize the following modularity quality function (Newman Reference Newman2006):

(6)$\begin{array}{}Q=\frac{1}{2m}\sum _{ij}(A_{ij}-\gamma \frac{k_i{k}_j}{2m}){\delta }_{C_i\mathrm{,}{C}_j}\mathrm{,}\end{array}$

where $m=(1/2m){\sum }_{ij}{A}_{ij}$, γ is a structural resolution parameter (often set to unity), ${\delta }_{C_i\mathrm{,}{C}_j}=1$ if nodes i and j are in the same community, and ${\delta }_{C_i\mathrm{,}{C}_j}=0$ otherwise. The modularity is computed over the landscape of all possible partitions of the network into communities and is maximal for the partition with the greatest community structure as quantified by the modularity quality index Q. Unfortunately, maximizing Q is an NP-complete problem, and therefore heuristic algorithms are used such as the locally greedy Louvain algorithm (Blondel et al. Reference Blondel, Guillaume, Lambiotte and Lefebvre2008).

Brain Organization

Network statistics form a toolbox to characterize the organization of human brain networks, offering insight into human cognition (Medaglia et al. Reference Medaglia, Lynall and Bassett2015). Here, we review three basic features observed consistently in human brain networks: small-world structure, community structure, and rich-club structure.

Multilayer Networks

To address this limitation, we turn to the new mathematical formalism of multilayer networks (Kivelä et al. Reference Kivelä, Arenas, Barthelemy, Gleeson, Moreno and Porter2014): representations in which each layer constitutes a different network, for example, constructed from a different subject, patient group, experimental condition, cognitive task, or time point. One can also combine multiple aspects, thus creating a network with more than one multilayer dimension: for example, a multilayer network in which one dimension is subject and a second dimension is time. A node can exist in all layers or in a subset of layers and is linked throughout the layers by an edge representing the node’s identity. Multiple types of edges can also link nodes within and between layers representing different types of relationships between network elements.

The multilayer framework is particularly appealing in the context of understanding statistically significant similarities and differences in network structure between experimental scenarios or across time, a traditionally difficult enterprise (Muldoon, Bridgeford, and Bassett Reference Muldoon, Bridgeford and Bassett2016). For example, a perennial question lies in how brain structure constrains brain function (Sporns Reference Sporns2013). To address this question, one might acquire a multimodal data set of structural and functional images and wish to build a multilayer network with two aspects: one layer containing the structural networks from a group of subjects and the other aspect containing the functional networks from a group of subjects. Edges representing nodal identity then link nodes between the two aspects and many layers to ensure that a given region can be identified consistently. A second interesting scenario in the context of disease (Bassett and Bullmore Reference Bassett and Bullmore2009; Stam Reference Stam2014) lies in building a multilayer network in which one aspect contains brain networks from healthy individuals and a second aspect contains brain networks from a patient cohort. Finally in the context of cognitive neuroscience (Sporns Reference Sporns2014; Medaglia et al. Reference Medaglia, Lynall and Bassett2015), one might wish to construct a multilayer network in which each aspect represents a functional brain network constructed during a different task.

The ability to combine different aspects (structure and function, healthy and disease groups, or time windows) within the same overall multilayer network framework allows one to develop network statistics that operate on a single object, using interaspect links to quantify the interactions between layers. This approach directly addresses the inherent dependencies between networks, offering a more principled framework for statistical inference.

Temporal Networks

The first use of multilayer networks in neuroscience addressed the temporal evolution of brain dynamics (Bassett, Wymbs, et al. Reference Bassett, Wymbs, Porter, Mucha, Carlson and Grafton2011) in which each layer represented a functional network estimated from a single time window of fMRI data and nodes were linked across sequential layers to indicate identity. This specific instantiation of the framework is referred to as a temporal network (Holme Reference Holme2012). This framework has proved particularly powerful in characterizing reconfiguration of human brain networks during learning (Bassett, Wymbs, et al. Reference Bassett, Wymbs, Porter, Mucha, Carlson and Grafton2011, Reference Bassett, Porter, Wymbs, Grafton, Carlson and Mucha2013; Bassett et al. Reference Bassett, Yang, Wymbs and Grafton2015; Mantzaris et al. Reference Mantzaris, Bassett, Wymbs, Estrada, Porter, Mucha, Grafton and Higham2013), where the multilayer formalism enables a statistically robust identification of dynamic changes in community structure that explain individual differences in learning ability (Bassett, Wymbs, et al. Reference Bassett, Wymbs, Porter, Mucha, Carlson and Grafton2011, Reference Bassett, Porter, Wymbs, Grafton, Carlson and Mucha2013; Bassett et al. Reference Bassett, Yang, Wymbs and Grafton2015). The tools have similarly proved useful in understanding lexical processing (Doron, Bassett, and Gazzaniga Reference Doron, Bassett and Gazzaniga2012), working memory (Braun et al. Reference Braun, Schäfer, Walter, Erk, Romanczuk-Seiferth, Haddad, Schweiger, Grimm, Heinz, Tost, Meyer-Lindenberg and Bassett2015), and motor behavior (Wymbs et al. Reference Wymbs, Bassett, Mucha, Porter and Grafton2012), and they complement a growing interest in dynamic functional connectivity in both rest (Hutchison et al. Reference Hutchison, Womelsdorf, Allen, Bandettini, Calhoun, Corbetta, Penna, Duyn, Glover, Gonzalez-Castillo, Handwerker, Keilholz, Kiviniemi, Leopold, de Pasquale, Sporns, Walter and Chang2013) and task states.

Studies describing, predicting, and manipulating brain dynamics in these temporal networks compose the relatively new field of dynamic network neuroscience (Bassett et al. Reference Bassett, Yang, Wymbs and Grafton2015). By modeling brain networks as evolving, interconnected systems, dynamic network neuroscience seeks to understand the complex nature of neurophysiological phenomena that underlie cognition. However, to achieve this goal, current network statistics must be modified to take into account the multidimensional aspects implicit in multilayer networks, a growing area of innovation (Kivelä et al. Reference Kivelä, Arenas, Barthelemy, Gleeson, Moreno and Porter2014). In addition to extending traditional statistics, there is a need to develop new statistics formalized specifically with the features of temporal networks in mind. Examples include network flexibility (Bassett, Wymbs, et al. Reference Bassett, Wymbs, Porter, Mucha, Carlson and Grafton2011) and stationarity (Palla et al. Reference Palla, Derényi, Farkas and Vicsek2005), which each quantify temporal changes in community structure. Finally, important challenges remain in defining appropriate statistical approaches to infer nonrandom structure in temporal networks (Bassett, Wymbs, et al. Reference Bassett, Wymbs, Porter, Mucha, Carlson and Grafton2011; Bassett, Porter, et al. Reference Bassett, Porter, Wymbs, Grafton, Carlson and Mucha2013), as well as tools to use these approaches to discriminate different roles that brain regions may play in network dynamics (Bassett, Wymbs, et al. Reference Bassett, Wymbs, Rombach, Porter, Mucha and Grafton2013).