Functional Performance

Brain networks can be derived from anatomical or physiological observations, resulting in structural and functional networks, respectively. When interpreting brain network data sets, it is important to respect this fundamental distinction.^{7 ,13 (link)}Structural connectivity describes anatomical connections linking a set of neural elements. At the scale of the human brain, these connections generally refer to white matter projections linking cortical and subcortical regions. Structural connectivity of this kind is thought to be relatively stable on shorter time scales (seconds to minutes) but may be subject to plastic experience-dependent changes at longer time scales (hours to days). In human neuroimaging studies, structural brain connectivity is commonly measured as a set of undirected links, since the directionality of projections currently cannot be discerned.
Functional connectivity is generally derived from time series observations, and describes patterns of statistical dependence among neural elements.^{12 (link)} Time series data may be derived with a variety of techniques, including electroencephalography (EEG), magnetoencephalography (MEG), and functional magnetic resonance imaging (fMRI), and can be computed in a number of ways, including as cross-correlation, mutual information, or spectral coherence. While the presence of a statistical relationship between two neural elements is often taken as a sign of functional coupling, it must be noted that the presence of such coupling does not imply a causal relationship.^{14 (link)} Functional connectivity is highly time-dependent, often changing in a matter of tens or hundreds of milliseconds as functional connections are continually modulated by sensory stimuli and task context. Even when measured with techniques that operate with a slow sampling rate such as fMRI, functional connectivity may exhibit non-stationary fluctuations (see below).
Effective connectivity represents a third and increasinglyimportant mode of representing and analyzing brain networks.^{11 (link),15 (link)} Effective connectivity attempts to capture a network of directed causal effects between neural elements. As such it represents a generative and mechanistic model that accounts for the observed data, selected from a range of possible models using objective criteria like the model evidence. Recent developments in this area include approaches towards “network discovery”^{16 (link),17 (link)} involving the identification of graph models for effective connectivity that best explain empirical data. While effective connectivity bears much promise for the future, most current studies of brain networks are still carried out on either structural or functional connectivity data sets, and hence these two modes of connectivity will form the main focus of this review.
Within the formal framework of graph theory, a graph or network comprises a set of nodes (neural elements) and edges (their mutual connections). Structural and/or functional brain connectivity data recorded from the human brain can be processed into network form by following several steps, starting with the definition of the network's nodes and edges (Figure 1). This first step is fundamental for deriving compact and meaningful descriptions of brain networks.^{18 (link),19 (link)} Nodes are generally derived by parcellating cortical and subcortical gray matter regions according to anatomical borders or landmarks, or by defining a random parcellation into evenly spaced and sized voxel clusters. Once nodes are defined, their structural or functional couplings can be estimated, and the full set of all pairwise couplings can then be aggregated into a connection matrix. To remove inconsistent or weak interactions, connection matrices can be subjected to averaging across imaging runs or individuals, or to thresholding.
The resulting networks can be examined with the tools and methods of network science. One approach is based on graph theory and offers a particularly large set of tools for detecting, analyzing, and visualizing network architecture. A number of surveys on the application of graph theory methods in neuroscience are available.^{13 (link),20 -25 (link)} An important part of any graph-theoretical analysis is the comparison of measures obtained from empirical networks to appropriately configured populations of networks representing a “null hypothesis.” A commonly used random null model is generated by randomizing the global topology of a network while preserving local node statistics, most importantly the graph's degree sequence.
Figure 2 illustrates a selection of graph measures that are widely used in studies of human brain networks. Based on the insights they deliver, they can be classified into measures reporting on aspects of segregation, integration, and influence.^{13 (link)} Segregation (or specialization) refers to the degree to which a network's elements form separate cliques or clusters. Integration refers to the capacity of the network as a whole to become interconnected and exchange information. Influence measures report on how individual nodes or edges are embedded in the network and the extent to which they contribute to the network's structural integrity and information flow.
An important measure of segregation is the clustering coefficient of a given node, essentially measuring the density of connections among a node's topological neighbors. If these neighbors are densely interconnected they can be said to form a cluster or clique, and they are likely to share specialized information. The average of clustering coefficients over all nodes is the clustering coefficient of the network, often used as a global metric of the network's level of segregation. Another aspect of connectivity within local (ie, topologically connected) sets of network nodes is provided by the analysis of network motifs, constituting subgraphs or “building blocks” of the network as a whole.^{26 (link)} Every network can be uniquely decomposed into a set of motifs of a given size, and the distribution of different motifs can reveal which subgraphs occur more frequently than expected, relative to an appropriate null model.
Measures of integration are generally based on the concept of communication paths and their path lengths. A path is any unique sequence of edges that connects two nodes with one another, and its length is given by the number of steps (in a binary graph) or the sum of the edge lengths (in a weighted graph). The length of the shortest path between each pair of nodes corresponds to their distance (also often referred to as the “shortest path length”), and the global average of all distances across the entire network is called the network's characteristic path length. Closely related to this measure is the global network efficiency, which is computed as the average of the inverse of all distances.^{27 (link)} One can see easily that the global efficiency of a fully connected network would be maximal (equal to one) while the global efficiency of a completely disconnected network would be minimal (equal to zero). Short path lengths promote functional integration since they allow communication with few intermediate steps, and thus minimize effects of noise or signal degradation.
Measures of influence attempt to quantify the “importance” of a given node or edge for the structural integrity or functional performance of a network. The simplest index of influence is the node degree, and in many (but not all) cases the degree of a node will be highly correlated with other more complex influence measures. Many of these measures capture the “centrality” of network elements, for example expressed as the number of short communication paths that travel through each node or edge.²⁸ This measure of “betweenness centrality” is related to communication processes, but is also often found to be highly correlated with the related measure of “closeness,” quantifying the proximity of each node to the rest of the network. Another class of influence measures is based on the effect of node or edge deletion on short communication paths or network dynamics. For example, vulnerability measures the decrease (or, in some cases, the increase) in global efficiency due to the deletion of a single node or edge.²⁹ The most central or influential nodes in a network are often referred to as “hubs,” but it should be noted that there is no unique way of detecting these hubs with graph theory tools. Instead, a conjunction of multiple influence measures (eg, degree, betweenness, vulnerability) should be used when attempting to identify hub nodes.^{30 (link)}While measures of segregation, integration, and influence can express structural characteristics of a network from different perspectives, recent developments in characterizing network communities or modules can potentially unify these different perspectives into a more coherent account of how a given network can be decomposed into modules (segregation), how these modules are interconnected (integration), and which nodes or edges are important for linking modules together (influence). Community detection is an extremely active field in network science.³¹ A number of new community detection techniques have found applications in the analysis of structural and functional brain networks. One of the most commonly- used community detection algorithms is based on Newman's Q-metric³² coupled with an efficient optimization approach.³³ Another approach called Infomap^{34 (link)} identifies communities on the basis of a model of a diffusive random walk, essentially utilizing the fact that a modular network restricts diffusion between communities. In contrast, the Q-metric essentially captures the difference between the actually encountered within-module density of connections compared with what is expected based on a corresponding random model, given a particular partitioning of the network into modules. Since combinatorics makes it impractical to examine all possible module partitions, an optimization algorithm is needed to identify the single partition for which the Q-metric is maximized.
Several methodological issues have arisen in recent years that impact the way community detection is carried out in brain networks, particularly in networks describing functional connectivity (Figure 3). The first issue concerns the widespread practice of thresholding functional networks to retain only a small percentage (often less than 10%) of the strongest functional connections. In addition, the remaining connections are then set to unit strength, resulting in a greatly sparsified binary network which is then subjected to standard graph analysis. Since the appropriate value of the threshold is a free and completely undetermined parameter, most practitioners vary the threshold across a broad range and then compute and compare graph metrics for the resulting networks. The practice of thresholding functional networks has two immediate consequences, a much sparser topology which then tends to result in more and more separate clusters or modules, and a topology that discards all (even strong) negative correlations. While the status of negative correlations in resting fMRI remains controversial,^{35 (link)-38 (link)} it could be argued that the presence of an anticorrelation between two nodes does contribute information about their community membership. Building on this idea, variants of the Q-metric and other related measures that take into account the full weight distribution of a network have been proposed.^{39 (link)} These new metrics can also be applied to functional networks regardless of their density (including fully connected networks), thus eliminating the need for thresholding entirely.
The second issue relates to the optimization of the module partition given a cost or quality metric like Newman's Q. Studies of various real-world networks have shown that identifying the single optimal partition can not only be computationally difficult, but that many real networks can be partitioned at near-optimal levels in a number of different or “degenerate” ways;⁴⁰ Aggregating these degenerate solutions can provide additional information about the robustness with which a given node pair is affiliated with the same or a different module. This idea has been developed further into a quantitative approach called “consensus clustering.” ⁴¹ Consensus clustering has not yet been widely applied to brain networks,^{39 (link),42} but it may soon become a useful tool since it provides information about the strength with which individual neural elements affiliate with their “home community.” An attractive hypothesis is that elements with generally weak affiliation are good candidates to assume functional roles as hub nodes that crosslink diverse communities.
The next three sections of the article will review our current knowledge about the network architecture of structural brain networks, how structural networks relate to functional networks in both rest and task conditions, and what we can learn by applying network approaches to clinical problems.

To measure performance, in Figs. 2I, 3 B and D and 4 G and H, we computed the circular average distance (72 ) of the estimate μ_T from the true HD ϕ_T at the end of a simulation of length T = 20 from P = 5′000 simulated trajectories by $m_1=\frac{1}{P}\sum _{k=1}^{P}exp\left(i\left({\mu }_T^{(k)}-{\varphi }_T^{(k)}\right)\right)$ . The absolute value of the imaginary-valued circular average, 0 ≤ |m₁|≤1, is unitless and denotes an empirical accuracy (or “inference accuracy”) and thus measures how well the estimate μ_T matches the true HD ϕ_T. Here, a value of 1 denotes an exact match. The reported inference accuracy is related to the circular variance via Var_circ = 1 − |m₁|. In SI Appendix, Fig. S5, we provide histograms with samples μ_T − ϕ_T with different numerical values of |m₁| to provide some intuition for the spread of estimates for a given value of the performance measure.
We estimated performance through such averages for a range of HD observation information rates in Figs. 2I, 3B and 4G. This information rate is a simulation time-step size-independent quantity, which measures the Fisher information that HD observations provide about true HD per unit time. For individual HD observations of duration dt, Eq. 6, this Fisher information approaches I_zt(ϕ_t)→(κ_zdt)²/2 with dt → 0 (31 , Theorem 2]. Per unit time, we observe 1/dt independent observations, leading to a total Fisher information (or information rate) of γ_z = κ_z²dt/2. As in simulations, γ_z needs to remain constant with changing Δt to avoid increasing the amount of provided information, the HD observation reliability κ_z needs to change with the size of simulation time-step size Δt. To keep our plots independent of this time-step size, we thus plot performance as a function of the HD observation information rate rather than κ_z. For the inset of Fig. 3B, and for Figs. 3 D and F, we additionally performed a grid search over the fixed-point κ^* (Fig. 3B, inset) or both the fixed-point κ^* and of the decay speed β (Figs. 3 D and F). For each setting of κ^* and β, we assessed the performance by computing an average over this performance for a range of HD observation information rates, weighted by how likely each observation reliability is assumed to be a priori. The latter was specified by a log-normal prior, p(γ_z)=Lognormal(γ_z|μ_γz, σ_γz²), favoring intermediate reliability levels. We chose μ_γz = 0.5 and σ_γz² = 1 for the prior parameters, but our results did not strongly depend on this parameter choice. The performance loss shown in Fig. 3D also relied on such a weighted average across information rates γ_z for a particle filter benchmark (PF, SI for details). The loss itself was then defined as $1-\frac{\text{Performance}}{\text{Performance PF}}$ .