Complex network analysis, describes important properties of complex systems by quantifying topologies of their respective network representations. Complex network analysis has its origins in the mathematical study of networks, known as graph theory. However, unlike classical graph theory, the analysis primarily deals with real-life networks that are large and complex—neither uniformly random nor ordered.
Brain connectivity datasets comprise networks of brain regions connected by anatomical tracts or by functional associations. Brain networks are invariably complex, share a number of common features with networks from other biological and physical systems, and may hence be characterized using complex network methods.
Construction of brain networks:
A network is a mathematical representation of a real-world complex system and is defined by a collection of nodes (vertices) and links (edges) between pairs of nodes. Nodes in large-scale brain networks usually represent brain regions, while links represent 1> anatomical, 2> functional, or 3> effective connections depending on the dataset. The details of the connections are as follows -
1> Anatomical connections typically correspond to white matter tracts between pairs of brain regions.
2> Functional connections correspond to magnitudes of temporal correlations in activity and may occur between pairs of anatomically unconnected regions.
3> Effective connections represent direct or indirect causal influences of one region on another and may be estimated from observed perturbations.
Binary links denote the presence or absence of connections, while weighted links also contain information about connection strengths. Weights in anatomical networks may represent the size, density, or coherence of anatomical tracts, while weights in functional and effective networks may represent respective magnitudes of co relational or causal interactions.
Links may be differentiated by the presence or absence of directionality. Anatomical and effective connections may conceptually be represented with directed links.
Measures of brain networks:
1> Functional segregation::
Functional segregation in the brain is the ability for specialized processing to occur within densely interconnected groups of brain regions. Measures of segregation primarily quantify the presence of such groups, known as clusters or modules, within the network.
Measures of segregation have straightforward interpretations in anatomical and functional networks. The presence of clusters in anatomical networks suggests the potential for functional segregation in these networks, while the presence of clusters in functional networks suggests an organization of statistical dependencies indicative of segregated neural processing.
Simple measures of segregation are based on the number of triangles in the network, with a high number of triangles implying segregation. Locally, the fraction of triangles around an individual node is known as the clustering coefficient and is equivalent to the fraction of the node's neighbours that are also neighbours of each other. The triangles in the anatomical networks indicate groups of specialized functional areas, such as the visual and somatomotor.
2> Functional integration::
Functional integration in the brain is the ability to rapidly combine specialized information from distributed brain regions. Measures of integration characterize this concept by estimating the ease with which brain regions communicate and are commonly based on the concept of a path. Lengths of paths consequently estimate the potential for functional integration between brain regions, with shorter paths implying stronger potential for integration. On the other hand, functional connectivity data, by its nature, already contain such information for all pairs of brain regions. Paths in functional networks represent sequences of statistical associations and may not correspond to information flow on anatomical connections. Consequently, network measures based on functional paths are less straightforward to interpret.
The average shortest path length between all pairs of nodes in the network is known as the characteristic path length of the network and is the most commonly used measure of functional integration.
3> Small-world brain connectivity::
Small-world networks are formally defined as networks that are significantly more clustered than random networks, yet have approximately the same characteristic path length as random networks .More generally, small-world networks should be simultaneously highly segregated and integrated.
Anatomical and effective networks in are simultaneously highly segregated and integrated, and consequently have small-world topologies. In comparison, the functional network is also highly segregated but has a lower global efficiency, and therefore weaker small-world attributes.
4> Centrality ::
Important brain regions (hubs) often interact with many other regions, facilitate functional integration, and play a key role in network resilience to insult. There are many measures of centrality.
The degree is one of the most common measures of centrality. The degree has a straightforward neurobiological interpretation: nodes with a high degree are interacting, structurally or functionally, with many other nodes in the network. The degree may be a sensitive measure of centrality in anatomical networks with non homogeneous degree distributions. In modular anatomical networks, degree-based measures of within-module and between-module connectivity may be useful for heuristically classifying nodes into distinct functional groups.
Nodes with a high within-module degree but with a low participation coefficient(known as provincial hubs)are hence likely to play an important part in the facilitation of modular segregation. On the other hand, nodes with a high participation coefﬁcient (known as connector hubs) are likely to facilitate global intermodular integration.
Many measures of centrality are based on the idea that central nodes participate in many short paths within a network and consequently act as important controls of information ﬂow. Bridging nodes that connect disparate parts of the network often have a high betweenness centrality . The notion of betweenness centrality is naturally extended to links and could therefore also be used to detect important anatomical or functional connections. The calculation of betweenness centrality has been made signiﬁcantly more efﬁcient with the recent development of faster algorithms.
Measures of centrality may have different interpretations in anatomical and functional networks. For instance, anatomically central nodes often facilitate integration, and consequently enable functional links between anatomically unconnected regions. Such links in turn make central nodes less prominent and so reduce the sensitivity of centrality measures in functional networks. In addition path-based measures of centrality in functional networks are subject to the same interpretational caveats as path-based measures of integration.
Differences in density between anatomical and functional networks make global comparisons between these networks less straightforward. Functional networks are likely to be denser than anatomical networks, as they will typically contain numerous connections between anatomically unconnected regions. These differences in density are likely to become more pronounced in larger, more highly resolved networks, as anatomical connectivity in such networks becomes increasingly sparse, while functional connectivity remains comparatively dense. Notably, comparisons between anatomical and functional modular structure remain meaningful despite differences in density.
Other factors that may affect comparisons of network topology include degree and weight distributions. The signiﬁcance of such comparisons will become more obvious with increased knowledge about causal relationships between brain regions, as mediated by direct anatomical connections. The development of a detailed anatomical map of the human brain is an important step in this direction.
Complex network analysis has emerged as an important tool for characterization of anatomical and functional brain connectivity. It's a description of a collection of measures that quantify local and global properties of complex brain networks. For further details click