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.

**Network comparison:**

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

here.

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