Many examples of networks exist in the world. These include structural networks (eg, the network of roads that connect cities and towns), social networks (eg, the networks of film actors that collaborate in movies), and biological networks (eg, a network of neurons connected with synapses). In each instance, we can represent the network with 2 fundamental components: nodes (the cities, actors, or neurons) and edges (the roads, films, or synapses) that connect any 2 nodes. In what follows, we will focus on binary, undirected networks. In these simplified networks, an edge either exists between 2 nodes or does not, and the presence of an edge provides no directional (ie, causal) information. In neuroscience, networks are typically divided into 2 categories: structural networks and functional networks.2
In structural networks, the edges represent physical connections between nodes. At the microscopic spatial scale, these include synaptic or gap junctional connections between individual neurons.3,4
At the macroscopic spatial scale, white matter tracts are used to infer synaptic connections between brain regions and construct structural networks.5–7
A functional network is defined by analysis of dynamic physiological activities, rather than by anatomy per se. Any time-varying physiological signal, such as electrical activity, magnetic activity, or blood oxygenation, can be the basis for inferring a functional network of brain activity. Generating the network requires recording such signals from multiple spatial locations and a metric to analyze coupling between these signals. The most commonly available clinical measure of brain activity is the routine electroencephalogram (EEG). To identify a functional network from the EEG, a coupling measure is applied to pairs of voltage signals recorded at separate EEG electrodes. Strong coupling between the activities recorded at 2 EEG electrodes defines an edge in the network, and the EEG sensors define the network nodes.
illustrates this process in a simplified case of EEG data recorded from 2 scalp electrodes. To determine whether the 2 recording sites are linked (or connected) in this 2-node network, we must determine the extent to which the EEG data recorded at the 2 electrodes match. There is no single method to optimally characterize signal coupling and many choices exist.8
Here, we demonstrate the process with a simple measure called the cross correlation. Cross correlation compares the temporal patterns contained in the 2 signals over time to see how well the voltage fluctuations align. One way to accomplish this is to compare the signals directly (the trace of signal 1 versus signal 2 at each time point) as well as indirectly: we can shift the traces in time with respect to one another to determine if the signals match more strongly at a particular lead or lag. If at some time shift the 2 signals match, then the cross correlation between the 2 signals is strong and we represent this graphically in a network by drawing an edge between the 2 electrodes (). On the other hand, if the 2 signals never match, no matter our choice of time shift, then the cross correlation between the 2 signals is weak and we leave the 2 electrodes disconnected in the network representation (). In this simple illustration, we have described a 2-node network that either contains 1 edge () or does not (). The 2 nodes represent the 2 EEG electrodes, and the edge reflects correlated activity (ie, coupling) between the voltage fluctuations recorded from the 2 electrodes. This approach can be extended to build larger networks by applying coupling measures to the voltage signals recorded from all electrode pairs, for example, in routine 10–20 configuration EEGs (). These results can then be used to create a functional connectivity network for the entire EEG montage ().
Figure 1 Example illustration of constructing functional networks from voltage signals. (A) In this simulated example, electroencephalogram (EEG) data are recorded from 2 electrodes on the scalp surface. The EEG data (black and upper gray curve) are simple sinusoids (more ...)
Having constructed a functional network, we now seek to analyze its structure. A useful, first analysis is simple visual inspection of the network (eg, visual inspection of the network in ). Even in this 10–20 configuration, the network structure becomes difficult to visualize. In fact, as the number of nodes in a network increases, so does the number of possible edges, and visual inspection becomes less useful. For a network of N nodes, the maximum possible number of edges is N(N −1)/2. When N is large (eg, in a high density EEG recording with 128 electrodes or a multielectrode array with 100 contacts) the networks typically become much too complicated for visual inspection (). To go beyond visual inspection and characterize the structure of these large networks, many measures exist.9–11
Here, we outline 3 of these measures in common use: the degree, path length, and clustering coefficient.
Figure 2 Two larger networks consisting of 200 nodes. (A,B) We arrange the nodes (gray) in the network as a ring, although this does not suggest a literal spatial sensor location in the “real” recording space. Each node represents an individual (more ...)
The degree (d) is simply the number of edges that touch a node. In , we show a 5-node network, and list the degree of each node (in this case, a value of either 2 or 4). To summarize the degree values of the entire network, we compute the average degree of all nodes, and find in this case 2.4. We note that this average value lies between the degree values we observe for each node (2 or 4), as expected. For much larger networks, the degree distribution is a useful measure that illustrates the probability of observing a node of degree d (). In many real world networks, including the film actor network and neural networks,12
the degree distribution exhibits a power law: the probability of observing a node of degree d decreases as 1 over the degree to some power ().13
In these networks, high degree nodes (which appear less frequently than the low degree nodes) can serve important functional roles in the network (ie, can act as hubs) although this is not always the case.14
Degree distributions with power law behavior are also known as scale-free because the degree distribution looks the same (just scaled by constant value) if we multiply the value of d by a constant.
Figure 3 An example 5-node network to illustrate 3 simple measures of network structure. (A) We label the 5-nodes (circles) with roman numerals and connect the nodes with edges (lines). The number in parentheses next to each node indicates its degree (d). (B) (more ...)
The path length is the minimum number of edges traversed to go from any given node to another in the network. We assume that each node is reachable from any other node, but if this is not the case, care must be taken to adjust for unreachable nodes. In the example 5-node network, the path length from node i to any other node is 1; node i can reach any other node by traversing 1 edge. Nodes ii–v can reach any other node in 1 or 2 steps. We note that many different paths exist between nodes. For example, we can travel directly from node ii to node iii, or we can pass through node i on the way to node iii. When computing the path length, by convention we always choose the shortest path between nodes. The average path length is calculated from the path length between each node and all other nodes in the network; for the 5-node network, the average path length is 1.4, a value between 1 and 2, as expected.
The final measure we consider here is the clustering coefficient. The clustering coefficient of a node is the number of connections that exist between the nearest neighbors of a node, expressed as a proportion of the maximum number of possible connections between the nearest neighbors of the node. In this context, nearest neighbor is not a spatial measure but a measure of which nodes are connected. This definition is perhaps best illustrated through the example network in . In the 5-node network, choose node ii and notice that this node has 2 neighbors (ie, 2 nodes directly connected to node ii by a single edge, nodes i and iii in ). We now examine whether an edge exists between the 2 neighbor nodes. In this example, it does (see ) so we complete a triangle or cluster in the network. In social networks, clustering is typically high; the friends (nearest neighbors) of an individual (the chosen node) also tend to be friends (ie, edges connect the nearest neighbors of the chosen node). For node i in the 5-node network, the clustering coefficient is 1/3. This result indicates that of all the possible completed triangles between the nearest neighbors of node i, only one-third exist. To complete all of the triangles between the nearest neighbors of node i would require additional connections between nodes ii and iv, nodes ii and v, nodes iii and iv, and nodes iii and v. For all other nodes, the clustering coefficient equals 1. All possible triangles between the nearest neighbors of these other nodes do exist. Often the average clustering coefficient for all nodes in a network is computed; for the 5-node network, the average clustering coefficient is 13/15.
With even these 3 simple measures in hand, we can characterize an arbitrarily large and complicated network easily and can compare networks efficiently. These, and other measures, therefore allow us to reduce the potentially overwhelming complexity of a developing brain network into a few comparable and informative measures. The next sections outline the maturation of anatomical and functional brain networks at each stage of development. We examine the application of network analysis to track the complex, but orderly, network topologies over neurodevelopment.