ArticlePDF AvailableLiterature Review

Complex brain networks: Graph theoretical analysis of structural and functional systems



Recent developments in the quantitative analysis of complex networks, based largely on graph theory, have been rapidly translated to studies of brain network organization. The brain's structural and functional systems have features of complex networks--such as small-world topology, highly connected hubs and modularity--both at the whole-brain scale of human neuroimaging and at a cellular scale in non-human animals. In this article, we review studies investigating complex brain networks in diverse experimental modalities (including structural and functional MRI, diffusion tensor imaging, magnetoencephalography and electroencephalography in humans) and provide an accessible introduction to the basic principles of graph theory. We also highlight some of the technical challenges and key questions to be addressed by future developments in this rapidly moving field.
We have known since the nineteenth century that the
neuronal elements of the brain constitute a formidably
complicated structural network1,2. Since the twentieth
century it has also been widely appreciated that this
anatomical substrate supports the dynamic emergence
of coherent physiological activity, such as phase-locked
high-frequency electromagnetic oscillations, that can
span the multiple spatially distinct brain regions that
make up a functional network3,4. Such networks are
thought to provide the physiological basis for informa-
tion processing and mental representations5–9. In this
article, we focus on graph theoretical approaches to the
analysis of complex networks that could provide a pow-
erful new way of quantifying the brain’s structural and
functional systems (BOX 1).
Since the mid 1990s, developments in our under-
standing of the physics of complex systems10–12 have led
to the rise of network science13 as a transdisciplinary
effort to characterize network structure and function.
In this body of literature, complexity arises in the macro-
scopic behaviour of a system of interacting elements that
combines statistical randomness with regularity14. The
ascendancy of network science has been driven by the
growing realization that the behaviour of complex sys-
tems — be they societies, cells or brains — is shaped
by interactions among their constituent elements. The
increasing availability and tractability of large, high-
quality data sets on a wide range of complex systems15–17
has led to a fundamental insight: substantively different
complex systems often share certain key organizational
principles, and these can be quantitatively characterized
by the same parameters (BOX 2). In other words, many
complex systems show remarkably similar macroscopic
behaviour despite profound differences in the micro-
scopic details of the elements of each system or their
mechanisms of interaction.
One example of an apparently ubiquitous macro-
scopic behaviour in complex systems is the small-world
phenomenon18 (BOX 3). Recently, small-world architec-
tures have been found in several empirical studies of
structural and functional brain networks19–22 in humans
and other animals, and over a wide range of scales in
space and time; small-worldness is naturally therefore
a key topic for this Review. However, discovering that
brain networks are small-world networks is only the
first step towards a comprehensive understanding of
how these networks are structurally organized and how
they generate complex dynamics. In network science,
methodological advances allow us to quantify other
topological properties of complex systems — such as
modularity23, hierarchy24, centrality25 and the distribu-
tion of network hubs26,27 — many of which have already
been measured in brain networks. There have also been
significant efforts to model the development or evolu-
tion of complex networks28, to link network topology to
network dynamics, and to explore network robustness
*University of Cambridge,
Behavioural & Clinical
Neurosciences Institute,
Department of Psychiatry,
Addenbrooke’s Hospital,
Cambridge, CB2 2QQ, UK.
Clinical Unit Cambridge,
Addenbrooke’s Hospital,
Cambridge, CB2 2QQ, UK.
§Department of Psychological
and Brain Sciences, Indiana
University, Bloomington,
Indiana 47405, USA.
Correspondence to E.B.
Published online
4 February 2009
Graph theory
A branch of mathematics that
deals with the formal
description and analysis of
graphs. A graph is defined
simply as a set of nodes
(vertices) linked by connections
(edges), and may be directed
or undirected. When describing
a real-world system, a graph
provides an abstract
representation of the system’s
elements and their interactions.
Complex brain networks: graph
theoretical analysis of structural and
functional systems
Ed Bullmore* and Olaf Sporns§
Abstract | Recent developments in the quantitative analysis of complex networks, based
largely on graph theory, have been rapidly translated to studies of brain network organization.
The brain’s structural and functional systems have features of complex networks — such as
small-world topology, highly connected hubs and modularity — both at the whole-brain
scale of human neuroimaging and at a cellular scale in non-human animals. In this article, we
review studies investigating complex brain networks in diverse experimental modalities
(including structural and functional MRI, diffusion tensor imaging, magnetoencephalogra-
phy and electroencephalography in humans) and provide an accessible introduction to the
basic principles of graph theory. We also highlight some of the technical challenges and key
questions to be addressed by future developments in this rapidly moving field.
MARCH 2009
© 2009 Macmillan Publishers Limited. All rights reserved
Nature Reviews | Neuroscience
Temporal pole
Inferior temporal
Histological or
imaging data
Anatomical parcellation Recording sites
Graph theoretical analysis
Time series data
Structural brain network Functional brain network
Complex network
An informal description of a
network with certain
topological features, such as
high clustering,
small-worldness, the presence
of high-degree nodes or hubs,
assortativity, modularity or
hierarchy, that are not typical
of random graphs or regular
lattices. Most real-life networks
are complex by this definition,
and analysis of complex
networks therefore forms an
important methodological tool
for systems biology.
Adjacency matrix
An adjacency matrix indicates
the number of edges between
each pair of nodes in a graph.
For most brain networks, the
adjacency matrix is specified
as binary — that is, each
element is either 1 (if there is
an edge between nodes) or 0
(if there is no edge). For
undirected graphs the
adjacency matrix is
Box 1 | Structural and functional brain networks
Structural and functional brain networks can be explored using graph theory through the following four steps (see the figure):
•Define the network nodes. These could be defined as electroencephalography or multielectrode-array electrodes, or as
anatomically defined regions of histological, MRI or diffusion tensor imaging data.
•Estimate a continuous measure of association between nodes. This could be the spectral coherence or Granger causality
measures between two magnetoencephalography sensors, or the connection probability between two regions of an
individual diffusion tensor imaging data set, or the inter-regional correlations in cortical thickness or volume MRI
measurements estimated in groups of subjects.
•Generate an association matrix by compiling all pairwise associations between nodes and (usually) apply a threshold to
each element of this matrix to produce a binary adjacency matrix or undirected graph.
•Calculate the network parameters of interest in this graphical model of a brain network and compare them to the
equivalent parameters of a population of random networks.
Each step entails choices that can influence the final results and must be carefully informed by the experimental question.
At step 1, parcellation schemes can use prior anatomical criteria or be informed by the functional connectivity profiles of
different regions. Several such parcellation schemes may be available and can affect network measures147. In most magneto-
encephalography and electroencephalography studies, network nodes are equivalent to individual electrodes or sensors,
but networks could also be based on reconstructed anatomical sources. However, some reconstruction algorithms will
determine the brain location of each source by minimizing the covariance between sensors, which has major effects on the
configuration of functional networks. At step 2, a range of different coupling metrics can be estimated, including measures
of both functional and effective connectivity. A crucial issue at step 3 is the choice of threshold used to generate an
adjacency matrix from the association matrix: different thresholds will generate graphs of different sparsity or connection
density, and so network properties are often explored over a range of plausible thresholds. Finally, at step 4 a large number of
network parameters can be quantified (BOX 2). These must be compared with the (null) distribution of equivalent parameters
estimated in random networks containing the same number of nodes and connections. Statistical testing of network
parameters may best be conducted by permutation- or resampling-based methods of non-parametric inference given the
lack of statistical theory concerning the distribution of most network metrics.
Most graph theoretical network studies to date have used symmetrical measures of statistical association or functional
connectivity — such as correlations, coherence and mutual information — to construct undirected graphs. This approach
could be generalized to consider asymmetrical measures of causal association or effective connectivity — such as Granger
causal148,149 or dynamic causal66 model coefficients — to construct directed graphs. It is also possible to avoid the
thresholding step (step 3) by analysing weighted graphs that contain more information than the simpler unweighted and
undirected graphs that have been the focus of attention to date. Structural brain network image is reproduced from
REF. 59. Functional brain network image is reproduced, with permission, from REF. 70 © (2006) Society for Neuroscience.
MARCH 2009
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Nature Reviews | Neuroscience
Connector hub
Provincial hub
Community 1
Community 2
and vulnerability — topics that are likely to become
increasingly relevant in relation to neuroscience.
In this article, we describe and discuss the expanding
interface between systems neuroscience and the physics
of complex networks. We review the existing empirical
data on topological and dynamical properties of struc-
tural and functional brain networks, and ask what this
literature tells us about how structural networks shape
functional brain dynamics. Space limitations prevent us
from providing coverage of all animal models, in vitro
Box 2 | Network measures
A network is defined in graph theory as a set of nodes or vertices and the edges or lines between them. Graph topology
can be quantitatively described by a wide variety of measures, some of which are discussed here. It is not yet established
which measures are most appropriate for the analysis of brain networks. The figure shows a schematic diagram of a brain
network drawn as a directed (left) and an undirected (right) graph; both structural and functional networks can be either
directed or undirected (BOX 1).
Node degree, degree distribution and assortativity
The degree of a node is the number of connections that link it to the rest of the network — this is the most fundamental
network measure and most other measures are ultimately linked to node degree. The degrees of all the network’s nodes
form a degree distribution15. In random networks all connections are equally probable, resulting in a Gaussian and
symmetrically centred degree distribution. Complex networks generally have non-Gaussian degree distributions, often
with a long tail towards high degrees. The degree distributions of scale-free networks follow a power law90. Assortativity is
the correlation between the degrees of connected nodes. Positive assortativity indicates that high-degree nodes tend to
connect to each other.
Clustering coefficient and motifs
If the nearest neighbours of a node are also directly connected to each other they form a cluster. The clustering coefficient
quantifies the number of connections that exist between the nearest neighbours of a node as a proportion of the
maximum number of possible connections18. Random networks have low average clustering whereas complex networks
have high clustering (associated with high local efficiency of information transfer and robustness). Interactions between
neighbouring nodes can also be quantified by counting the occurrence of small motifs of interconnected nodes150. The
distribution of different motif classes in a network provides information about the types of local interactions that the
network can support48.
Path length and efficiency
Path length is the minimum number of edges that must be traversed to go from one node to another. Random and
complex networks have short mean path lengths (high global efficiency of parallel information transfer) whereas regular
lattices have long mean path lengths. Efficiency is inversely related to path length but is numerically easier to use to
estimate topological distances between elements of disconnected graphs.
Connection density or cost
Connection density is the actual number of edges in the graph as a proportion of the total number of possible edges and is
the simplest estimator of the physical cost — for example, the energy or other resource requirements — of a network.
Hubs, centrality and robustness
Hubs are nodes with high degree, or high centrality. The centrality of a node measures how many of the shortest paths
between all other node pairs in the network pass through it. A node with high centrality is thus crucial to efficient
communication151. The importance of an
individual node to network efficiency can be
assessed by deleting it and estimating the
efficiency of the ‘lesioned’ network.
Robustness refers either to the structural
integrity of the network following deletion of
nodes or edges or to the effects of
perturbations on local or global network
Many complex networks consist of a number
of modules. There are various algorithms that
estimate the modularity of a network, many of
them based on hierarchical clustering23. Each
module contains several densely
interconnected nodes, and there are relatively
few connections between nodes in different
modules. Hubs can therefore be described in
terms of their roles in this community
structure27. Provincial hubs are connected
mainly to nodes in their own modules,
whereas connector hubs are connected to
nodes in other modules.
MARCH 2009
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In the brain, connectivity can
be described as structural,
functional or effective.
Structural connectivity denotes
a network of anatomical links,
functional connectivity denotes
the symmetrical statistical
association or dependency
between elements of the
system, and effective
connectivity denotes directed
or causal relationships
between elements.
A neuronal network composed
of specific cell types and
synaptic connections, often
arranged in a modular
architecture and capable of
generating functional outputs.
The complete description of
the structural connections
between elements of a nervous
preparations and human studies that have contributed to
this endeavour: we necessarily focus on what we consider
to be some representative examples of graph theoretical
research in brain networks, with an emphasis on studies
of the human brain. We thus consider the implications of
complex brain networks for an understanding of neuro-
psychiatric disorders and conclude with some general
remarks about the evolution of scale-invariant topology
in brain networks and some key future challenges for this
emerging field.
Structural brain networks
Topological and physical properties of structural net-
works. The anatomical configuration of brain networks,
ranging from inter-neuronal connectivity to inter-regional
connectivity, has long been a focus of empirical neuro-
science. Network analysis, and in particular graph theory
(BOX 4), offers new ways to quantitatively characterize
anatomical patterns. According to graph theory, struc-
tural brain networks can be described as graphs that are
composed of nodes (vertices) denoting neural elements
(neurons or brain regions) that are linked by edges rep-
resenting physical connections (synapses or axonal
projections). Although graph theory emphasizes topo-
logical connectivity patterns, the topological and physical
distances between elements in brain networks are often
intricately related29. Neurons and brain regions that are
spatially close have a relatively high probability of being
connected, whereas connections between spatially remote
neurons or brain regions are less likely30–32. Longer axonal
projections are more expensive in terms of their material
and energy costs33. It has been suggested that the spatial
layout of neurons or brain regions34,35 is economically
arranged to minimize axonal volume. Thus, conserva-
tion of wiring costs is likely to be an important selection
pressure on the evolution of brain networks.
Mapping structural networks in animal models. Tracing
individual neuronal processes has been a long-standing
technological challenge, and few structural brain net-
works have been mapped at cellular resolution. Serial
reconstruction of electron microscopy sections allowed
the complete connection matrix of the nematode
Caenorhabditis elegans to be visualized36. Currently,
this is the only nervous system to have been compre-
hensively mapped at a cellular level, and it was the first
to be described as a small-world network18. In larger
brains, combinations of physiological and anatomical
techniques have allowed patterns in neuronal connec-
tions to be identified, leading to the identification of
neuronal microcircuits37 and the formulation of proba-
bilistic connection rules30. Small-world properties have
been demonstrated in biologically accurate models of
cellular networks in the reticular formation of the verte-
brate brainstem38. Reconstruction of cellular networks in
the mammalian neocortex from multielectrode activity
recordings has revealed several highly nonrandom fea-
tures of connectivity39, including a tendency for synaptic
connections to be reciprocal and clustered. A promising
approach for mapping connectivity involves the stochas-
tic expression of several fluorescent proteins40, and this
may ultimately deliver a complete map of the cellular
interconnections of an entire brain41.
Histological dissection and staining, degeneration
methods and axonal tracing have been used to map cer-
ebral white matter connections. The pathways identi-
fied by these methodologies have formed the basis for
the systematic collation of species-specific anatomical
connection matrices, including those for the macaque
visual cortex42 and the cat thalamocortical system43.
Network analyses of such data sets demonstrated high
clustering of functionally related areas with short average
path lengths44–46, hallmarks of a small-world architecture.
Clusters identified by network analysis map on to known
functional subdivisions of the cortex43. Long-distance
cortical projections facilitate short-path communica-
tion despite increasing axonal volume47. The cortical
networks of several mammalian species also consistently
demonstrated an overabundance of motif classes associ-
ated with network modularity and functionally diverse
Mapping structural networks in the human brain.
Several attempts have been made to map the struc-
tural networks of the human brain, also known as the
human connectome50, at the scale of brain regions. One
Box 3 | Random, scale-free and small-world networks
In random graphs each pair of nodes has an equal probability, p, of being connected152.
Large random graphs have Gaussian degree distributions (BOX 2). It is now known that
most graphs describing real-world networks significantly deviate from the simple
random-graph model.
Some networks (including the Internet and the World Wide Web) have degree
distributions in the form of a power law: that is, the probability that a node has degree k
is given as Prob(k) ~ kλ. In biological systems, the degree exponent λ often ranges
between 2 and 3, and the very gradual (‘heavy-tail’) power law decay of the degree
distribution implies that the network lacks a characteristic scale — hence such
networks are called ‘scale-free’ networks. Barabási and Albert90 demonstrated that
scale-free networks can originate from a process by which each node that is added to
the network as it grows connects preferentially to other nodes that already have high
degree. Scale-free networks are unlikely if the attachment of connections is subject to
physical constraints or associated with a cost15. Therefore, physically embedded
networks, in which nodes have limited capacity for making connections, often do not
have pure power law degree distributions but may instead demonstrate exponentially
truncated power law degree distributions, which are associated with a lower
probability of very high degree nodes.
Originally described in social networks153, the ‘small-world’ property combines high
levels of local clustering among nodes of a network (to form families or cliques) and
short paths that globally link all nodes of the network. This means that all nodes of a
large system are linked through relatively few intermediate steps, despite the fact that
most nodes maintain only a few direct connections — mostly within a clique of
neighbours. Small-world organization is intermediate between that of random
networks, the short overall path length of which is associated with a low level of local
clustering, and that of regular networks or lattices, the high-level of clustering of which
is accompanied by a long path length18. A convenient single-number summary of
small-worldness is thus the ratio of the clustering coefficient to the path length after
both metrics have been standardized by comparing their values to those in equivalent
random networks154. Evidence for small-world attributes has been reported in a wide
range of studies of genetic, signalling, communications, computational and neural
networks. These studies indicate that virtually all networks found in natural and
technological systems have non-random/non-regular or small-world architectures
and that the ways in which these networks deviate from randomness reflect their
specific functionality.
MARCH 2009
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Diffusion tensor imaging
(DTI). An MRI technique that
takes advantage of the
restricted diffusion of water
through myelinated nerve
fibres in the brain to map the
anatomical connectivity
between brain areas.
Diffusion spectrum imaging
An MRI technique that is
similar to DTI, but with the
added capability of resolving
multiple directions of diffusion
in each voxel of white matter.
This allows multiple groups of
fibres at each location,
including intersecting fibre
pathways, to be mapped.
Cortical parcellation
A division of the continuous
cortical sheet into discrete
areas or regions; Brodmann’s
division of the cortex into areas
defined by their
cytoarchitectonic criteria is the
most famous but not the only
parcellation scheme.
Recordings of epileptiform
electrical activity at specific
sites in the cortex following
topical application of a
pro-convulsive drug to a distant
cortical site; rapid propagation
of electrical activity from
stimulation to recording sites
implies that the sites are
anatomically connected.
Functional MRI
(fMRI). The detection of
changes in regional brain
activity through their effects on
blood flow and blood
oxygenation (which, in turn,
affect magnetic susceptibility
and tissue contrast in magnetic
resonance images).
(EEG). A technique used to
measure neural activity by
monitoring electrical signals
from the brain, usually through
scalp electrodes. EEG has good
temporal resolution but
relatively poor spatial
(MEG). A method of measuring
brain activity by detecting
minute perturbations in the
extracranial magnetic field that
are generated by the electrical
activity of neuronal
study derived structural connection patterns from cross-
correlations in cortical thickness or volume across
individual brains, which might indirectly indicate the
presence of corticocortical pathways51,52. Graph analysis
revealed small-world attributes and the existence of local
communities of brain regions. A more detailed analysis53
of the modularity or community structure of connection
data sets derived from cortical thickness correlations
revealed significant overlap between anatomical network
modules and functional systems in the cortex.
Human brain structural networks have also been
mapped using diffusion imaging and tractography. A map
of 70–90 cortical and basal brain grey matter areas was
constructed using diffusion tensor imaging (DTI) and ana-
lysed using methods from graph theory54,55. The network
exhibited high clustering and short path length, and con-
tained motif classes similar to those identified from tract-
tracing data48. Several areas, including the precuneus, the
insula, the superior parietal cortex and the superior frontal
cortex, were found to have high ‘betweenness centrality’
and thus to constitute putative hubs. Another study that
mapped connections between 78 cortical regions using
DTI also identified several hub regions, including the
precuneus and the superior frontal gyrus56.
Due to limitations in the model that is used to infer
fibre bundle orientation, DTI has difficulty detecting
crossing fibre bundles. Diffusion spectrum imaging can ov er-
come this limitation by reconstructing multiple diffusion
directions in each voxel57 and was used to build cortical
connection matrices between 500–4,000 homogeneously
distributed regions of interest58. Again, network analyses
revealed the small-world architecture of the cortical net-
work. A more extensive analysis of 998 region-of-interest
networks obtained from 5 participants59 identified
structural modules interconnected by highly central hub
regions. When considering multiple network measures
(including node degree, connection strength and central-
ity), a particular set of brain regions located predomi-
nantly in the posterior medial cortex, including portions
of the posterior cingulate and the precuneus, was highly
and densely interconnected, forming a structural core59.
Although they differ in terms of their experimen-
tal methodology and cortical parcellation, most of these
studies reveal highly clustered large-scale cortical
networks, with most pathways existing between areas
that are spatially close and functionally related. These
clusters or modules are interlinked by specialized hub
regions, ensuring that overall path lengths across the
network are short. Most studies identified hubs among
parietal and prefrontal regions, providing a potential
explanation for their well-documented activation
by many cognitive functions. Particularly notable is
the prominent structural role of the precuneus55,56,59,
a region that is homologous to the highly connected
posteromedial cortex in the macaque60. The precuneus
is involved in self-referential processing, imagery and
memory61, and its deactivation is associated with anaes-
thetic-induced loss of consciousness62. An intriguing
hypothesis suggests that these functional aspects can
be explained on the basis of its high centrality in the
cortical network.
Functional brain networks
Although analysing structural networks helps us to
understand the fundamental architecture of inter-
regional connections, we must also consider functional
networks directly to elucidate how this architecture sup-
ports neurophysiological dynamics. Despite considerable
heterogeneity in the methodological approaches, there is
an encouraging degree of convergence between studies
of functional brain networks. The first such study used a
set of neuronographic measurements of the propagation of
epileptiform activity following localized applications
of strychnine to the macaque cortex63. This demon-
strated a pattern of functional connections between
cortical areas that was consistent with a small-world
network. As we discuss below, these findings have been
extended by studies based on functional MRI (fMRI),
electroencephalography (EEG), magnetoencephalography
(MEG) or multielectrode array (MEA) data.
Although such studies based on graph theory are
the focus of this Review, we note that other methods to
investigate brain functional systems have recently been
developed, including mathematical models of effec-
tive connectivity between regions. Effective connectiv-
ity models, such as structural equation modelling64,65,
dynamic causal modelling66 or Granger causality67,
involve estimating the causal influence that each ele-
ment of a system exerts on the behaviour of other
elements. Thus, measures of effective connectivity
Box 4 | Origins of graph theory
Graph theory is rooted in the physical world155. In 1736, Euler showed that it was impossible to traverse the city of
Königsberg’s seven bridges across the river Pregel exactly once and return to the starting point. To prove this conjecture,
Euler represented the problem as a graph, and his original publication156 is generally taken to be the origin of a new
branch of mathematics called graph theory. In the middle of the nineteenth century, the analysis of electrical circuits and
the exploration of chemical isomers led to the discovery of additional graph theoretic concepts149. Today, graph theory
pervades many areas of science.
Significant progress in graph theory has come from the study of social networks157. One prominent experiment153
tracked paths of acquaintanceship across a large social network and found that even very large networks could be
traversed, on average, in a small number of steps. Although this ‘small-world’ phenomenon quickly captured the public
imagination, its origins remained obscure until its association with specific types of connectivity was demonstrated
(BOX 3). The dual discoveries of small-world18 and scale-free90 networks launched the modern era of graph theory, which
now extends into biology and neuroscience. Neural systems have long been described as sets of discrete elements linked
by connections. Nonetheless, graph theory has essentially only been applied to neuroscience in the past 10 years.
MARCH 2009
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Multielectrode array
(MEA). A technique for
simultaneously measuring the
electrical activity of local
neuronal populations or single
neurons, usually in tissue slices
or cell cultures in vitro.
Association matrix
A matrix that represents the
strength of the association
between each pair of nodes in
a graph. Association between
nodes can be quantified by
many continuously variable
metrics, such as correlation or
mutual information. Either
functional or effective
connectivity measures can be
used to construct an
association matrix.
Blood oxygen
level-dependent (BOLD)
Changes in magnetic
susceptibility and MRI tissue
contrast that are indirectly
indicative of underlying
changes in spontaneous or
experimentally controlled brain
Default-mode network
A set of brain regions, including
medial frontal and posterior
cingulate areas of the cortex,
that are consistently
deactivated during the
performance of diverse
cognitive tasks.
between multiple regions can be used to generate a
directed graph, which can then be topologically described
using graph theory. However, the functional network
studies reviewed below have all been based on undi-
rected graphs, derived from simpler measures of func-
tional connectivity or symmetrical statistical association
between brain regions. The key point is that, in principle,
graph theory could be applied to an association matrix of
either functional or effective connectivity measures, to
generate either undirected or directed graphs, respec-
tively, although all neuroscientific studies to date have in
fact been based on measures of functional connectivity.
Mapping functional networks using fMRI. The first
graph theoretical study of fMRI data measured the
partial correlations of resting-state blood oxygen level-
dependent (BOLD) signals between 90 cortical and sub-
cortical regions and reported small-world properties of
the resulting whole-brain networks68. Almost simulta-
neously, another study reported small-world properties
of functional networks derived from a set of activated
voxels in fMRI data; this voxel-level network was also
reported to have a scale-free degree distribution69.
Subsequently, small-world properties, with parameter
values similar to those previously reported in topologi-
cal studies of cat and macaque anatomical connectivity
matrices50, were confirmed in a low-frequency (0.03–
0.06 Hz) whole-brain network derived from wavelet
correlations between regional mean time series70. The
high-degree nodes or hubs of this network were mostly
regions of multimodal association cortex, and the degree
distribution was an exponentially truncated power law70.
Other studies have explored the community structure of
fMRI networks using a hierarchical cluster analysis68,71,72
and shown that functionally and/or anatomically related
brain regions are more densely interconnected, with rela-
tively few connections between functional clusters, again
echoing prior work on anatomical connectivity matrices.
The high density of connections between functionally
related regions increases the clustering coefficient of
the graph, whereas the long-range connections between
different modules or clusters, even though they are rela-
tively few in number, keep the path length low. Thus, the
small-world architecture of a brain functional network
is closely related to its modularity72.
There are several other metrics for quantifying small-
world architecture in brain functional networks. Studies
in statistical physics73,74 have shown that path length is
inversely related to the global efficiency of a network for
the transfer of information between nodes by multiple
parallel paths, and that global efficiency is easier to esti-
mate than path length when studying sparse networks.
Furthermore, the clustering coefficient can be regarded as
a measure of the local efficiency of information transfer,
or of the robustness of the network to deletion of individ-
ual nodes. The structural network of the macaque brain
was found to have high global and local efficiency and to
be sparsely connected73,74. Thus, the macaque cortex has
economical small-world’ properties: it has high global
efficiency of parallel information transfer and high local
fault tolerance for relatively low connection density.
These concepts were translated to the analysis of
resting-state fMRI data acquired from young and eld-
erly adults75, using the wavelet correlation (a measure
of the association between time series in a specific fre-
quency band) to estimate the functional connectivity
between regional BOLD time series endogenously oscil-
lating in the frequency interval 0.06–0.1 Hz (for a more
detailed review of the rationale for wavelet analysis in
fMRI, see REF. 76). In younger adults, functional brain
networks demonstrated small-world properties over a
broad range of connection densities or ‘costs’. Relatively
sparse networks were associated with maximum cost
efficiency. The older age group also showed evidence of
small-world properties, but had significantly reduced
cost efficiency: they had to be relatively over-connected
to provide the efficiency of parallel information trans-
fer seen in a younger brain network. The suggestion
that aging is associated with changes in the economi-
cal small-world properties of brain functional networks
converges with studies that show differences in atten-
tional and default-mode networks between children and
young adults77. Normal processes of brain maturation
and senescence might thus be reflected in quantifiable
changes in functional network topology.
Mapping functional networks using electrophysiologi-
cal techniques. When comparing the results of fMRI
studies to results obtained using electrophysiological
techniques (EEG, MEG or MEA), many aspects of the
data clearly differ. fMRI has good spatial resolution (on
the order of millimetres) but poor temporal resolution
(on the order of seconds), restricting the measurable
bandwidth to approximately 0.001–0.5 Hz, and fMRI
measures activation-related haemodynamics rather than
neuronal activity per se. All electrophysiological meth-
ods measure neuronal activity more directly and have
better temporal resolution, with bandwidths typically of
1–100 Hz, but they often have worse spatial resolution
(on the order of millimetres or centimetres for MEG and
EEG) or less complete anatomical coverage (in the case
of MEA) than fMRI. Another point of difference is that
the nodes of a network derived from fMRI data will be
anatomically localized regions or voxels of the image,
whereas the nodes of a network derived from MEG or
EEG data could be the surface sensors or recording elec-
trodes. However, we can compare the topologies of net-
works derived from these different data sets by invoking
a general operating principle of complex network analy-
sis: microscopically distinct systems can be informatively
compared in terms of their macroscopic organization
using graph theory.
Functional connectivity between pairs of electrodes
has been estimated using a measure of generalized syn-
chronization78, and then thresholded to generate func-
tional networks, in several studies of EEG or MEG data
sets. This has shown that small-world topology is repre-
sented at many frequency intervals79 and can be related
to cognitive performance and normal aging80. An alter-
native approach used the wavelet correlation to estimate
frequency-dependent functional connectivity between
MEG sensors, again revealing that many topological and
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© 2009 Macmillan Publishers Limited. All rights reserved
Metastable dynamics
Transitions between marginally
stable network states; these
transitions can occur
spontaneously or as a result of
weak external perturbations.
Resting state
A cognitive state in which a
subject is quietly awake and
alert but does not engage in or
attend to a specific cognitive
or behavioural task.
dynamic properties of brain functional networks were
conserved across frequencies81. This timescale invari-
ance of wavelet correlations and the brain functional
network parameters derived from them is a theoretically
predictable corollary of the long-range autocorrelations
of neurophysiological time series82–84.
Conservation of functional network properties. There
have been fewer studies of functional networks in non-
human species. In anaesthetized rats, fMRI has been
used to demonstrate small-world and modular prop-
erties of whole-brain functional networks85. A recent
study considered functional coupling between pairs of
cortical neurons, measured by multiunit electrodes, in
the visual cortex of the anaesthetized cat86. Most of the
properties of these neuronal-interaction networks could
be accounted for by the pairwise correlations between
electrodes87,88, and the networks had small-world prop-
erties with some highly connected (hub) neurons. The
small-world organization of functional networks at a cel-
lular level has also been described on the basis of MEA
recordings from in vitro cultures of cortical networks89.
These results indicate that small-worldness might be a
conserved property of brain functional networks over
different species and spatial scales.
Preliminary evidence suggests that other topologi-
cal properties of brain functional networks, such as the
degree distribution, might also be conserved over spatial
and temporal scales. Many large networks exhibit scale-
free power law degree distributions90 indicative of the
existence of highly connected nodes (BOX 3). Some stud-
ies of functional brain networks carried out at high spatial
resolution (single voxels in fMRI) have provided evidence
of scale-free organization69,91. However, in other studies of
functional networks derived from fMRI and MEG data
sets over a wide range of frequency intervals70,81, the
empirical degree distribution conforms to an exponen-
tially truncated power law, implying that the probabil-
ity of a highly connected node or hub is greater than
in an equivalent random network but less than would
be expected in a scale-free network. Truncated power
law degree distributions have also been reported from
analysis of anatomical networks derived from structural
MRI data in humans55, and from analysis of functional
networks derived from MEA data in cats86. These dis-
tributions are typical of physically embedded networks,
such as the global air-transportation network, in which
the maximum degree is limited by physical considera-
tions such as the finite capacity of any node to receive
connections27. The reasons for the differences between
these findings and reports of scale-free properties are
currently unknown; however, it is notable that pure
power law scaling of the degree distributions of human
brain functional networks has only been reported by
voxel-level analysis, whereas exponentially truncated
power laws have been reported by region-level analysis.
The form of the degree distribution could be affected
by spatial interpolation, and other pre-processing steps
applied before the construction of functional networks
and the standardization of such methodological pro-
cedures will be important in elucidating the impact of
anatomical resolution on the extent to which power
law scaling of the degree distribution is truncated at
high degree.
Structure–function relations in brain networks
How do functional brain networks emerge from struc-
tural brain connectivity? Structural maps indicate that
each neural node maintains a specific pattern of structural
connections with other nodes. Different nodes often have
different functionalities, such as specific response prefer-
ences to sensory stimuli. From a network perspective, the
functionality of an individual neural node is partly deter-
mined by the pattern of its interconnections with other
nodes in the network92: nodes with similar connection
patterns tend to exhibit similar functionality36. Although
functional properties are expressed locally, they are the
result of the action of the entire network as an integrated
system. Structural connectivity places constraints on
which functional interactions occur in the network.
Structural and functional connectivity in cellular net-
works undergo dynamic changes. The degree to which
synaptic connectivity is modified in the adult brain is
highly debated. Although some evidence suggests that
mammalian cellular networks are continually remod-
elled93, other evidence indicates that most synaptic
spines are stable94. Changes in neuronal connectivity
necessitate homeostatic mechanisms to ensure func-
tional stability95. Multielectrode recording data suggest
that cellular functional networks exhibit transient syn-
chronization96 and metastable dynamics97. These changes
occur within seconds, and it seems possible that even
more rapid transitions and network reconfigurations
may take place33. These observations of relatively slow
structural modifications accompanied by faster changes
in functional linkages pose major unresolved questions
regarding functional stability in neural circuits.
It is currently unknown whether large-scale cortical
networks in the adult brain undergo structural modi-
fications on fast timescales. Most of the changes that
have been observed were associated with aging, disease
progression or experience-dependent plasticity. By con-
trast, patterns of functional connectivity between brain
regions undergo spontaneous fluctuations and are
highly responsive to perturbations, such as those that
are induced by sensory input or cognitive tasks, on a
timescale of hundreds of milliseconds. These rapid recon-
figurations do not affect the stability of global topologi-
cal characteristics81,98. On longer timescales of seconds
to minutes, correlations between spontaneous fluctua-
tions in brain activity99–101 form functional networks that
are particularly robust. For example, a set of posterior
medial, anterior medial and lateral parietal brain regions
comprises the default mode network102,103.
The persistence of functional networks associated
with the brain’s resting state provides an opportunity
to investigate how much of the pattern of functional
connections is determined by underlying structural
networks. Observations from a single cortical slice104,
structural imaging of fibre bundles linking components
of the default network105, and direct comparisons of
structural and functional connectivity in the same cohort
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© 2009 Macmillan Publishers Limited. All rights reserved
Nature Reviews | Neuroscience
Functional brain network
Mutual information Cross-correlation
Structural brain network
Coupling matrix and neural dynamics
tract tracing
time series
Simulated BOLD
time series
of participants63,106,107 suggest that structural connections
are highly predictive of functional connections. Indirect
interactions can account for additional functional link-
ages. Such indirect connections can lead to discrepan-
cies between structural and functional connectivity;
however, current evidence suggests that topological
parameters are generally conserved between structural
and functional networks.
Computational models offer a complementary
method to investigate structure–function relations in
brain networks. Several studies have used empirically
derived structural brain networks to specify the wiring
between neural units, which then results in spatially
patterened dynamic interactions (FIG. 1). The covari-
ance structure of the endogenous activity generated by
these neural units yields a functional network that can
be generated by computer simulations or in some cases
analytically108. This strategy has been used to determine
the effects of structural topology on functional networks
and dynamics. Simulation studies of large-scale corti-
cal networks demonstrated the emergence of complex
spatio temporal structure in neural correlations at mul-
tiple timescales109, as well as realistic patterns of mod-
elled BOLD resting-state correlations that depend on
the topology109 and time delays110 in the structural cou-
pling matrix. The topology of structural and functional
networks was identical when functional connectivity
was estimated from long time samples, but functional
networks estimated on shorter time samples or at higher
frequencies were less strongly constrained by the struc-
tural wiring diagram109. Likewise it has been shown
that the modularity of structural networks can deter-
mine the hierarchical organization of functional net-
works111,112 and may be important for generating diverse
and persistent dynamic patterns113. In a computational
model of phase synchronization between coupled neu-
rons, patterns of local versus global synchronization
were found to depend strongly on the balance between
high clustering and short path length in the connectiv-
ity of the network114. Computational studies have also
begun to determine the effects of functional activity on
structural topology. It has been shown that an initially
random wiring diagram can evolve through synaptic
plasticity to a functional state characterized by a small-
world topology of the most strongly connected nodes
and by self-organized critical dynamics115.
These findings indicate that the brain’s structural
and functional networks are intimately related and
share common topological features, such as modules
and hubs (FIG. 2). Although most studies provide sup-
port for the idea that structural networks determine
some aspects of functional networks, especially at low
frequencies or over long time periods, it is less clear how
the structural topology both supports the emergence of
fast and flexibly reconfigured functional networks and
is itself remodelled by function-related plasticity on a
slower timescale.
Clinical and translational aspects
Since the work of pioneers such as Wernicke, Meynert
and Dejerine, it has been appreciated that many neuro-
logical and psychiatric disorders can be described as dys-
connectivity syndromes116. The emergence of symptoms
or functional impairment in these disorders can be theo-
retically related to the disruption or abnormal integration
of spatially distributed brain regions that would normally
constitute a large-scale network subserving function.
Using network properties as diagnostic markers. One
application of complex-network theory in this context is
to provide new measures to quantify differences between
patient groups and appropriate comparison groups.
Several studies have reported that the parameters of
Figure 1 | Computational modelling of structural and functional brain networks.
Computational models have been used to demonstrate how dynamic patterns arise as a
result of interactions between anatomically connected neural units. Shown is how such
a model is generated and used. A structural brain network derived from anatomical data
serves as a matrix of coupling coefficients that link neuronal nodes, the activities of which
unfold through time. This time evolution is governed by physiologically motivated dynamic
equations. In the example shown, the surface of the macaque cortex was subdivided into
47 areas (nodes) and a structural brain network linking these nodes was compiled from
anatomical tract-tracing data. The dynamic equations were derived from a model of large
neuronal populations, the parameters of which were set to physiological values109. Data
from computer simulations then yield functional brain networks. Such networks are
derived from measures of association between the simulated time series — for example, an
information theoretic measure such as the mutual information (computed on
voltage–time data) or cross-correlations in neural activity that are computed from
simulated blood oxygen level-dependent (BOLD) data. These matrices can then be
thresholded to yield binary networks from which network measures can be derived. The
fact that both structural and functional networks are completely specified in the model
facilitates their comparative analysis. The structural brain network panel is reproduced,
with permission, from REF. 109 (2007) National Academy of Sciences. The rest of the
figure is modified, with permission, from REF. 158 (2009) Academic Press.
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© 2009 Macmillan Publishers Limited. All rights reserved
Nature Reviews | Neuroscience
Cellular functional network
Whole-brain structural network
Module 1 Module 2Hub node
brain networks derived from fMRI, EEG or structural
MRI data are altered in patients with schizophrenia or
Alzheimer’s disease (AD).
In an fMRI study of AD, clustering was significantly
reduced at a global level (in whole-brain networks
operating at frequencies below 0.1 Hz) and at a local
level (in both hippocampi), and global clustering was
able to discriminate AD patients from age-matched
comparison subjects with high specificity and sensi-
tivity, implying that the loss of small-world network
properties might provide a clinically useful diagnostic
marker117. In a comparable EEG study, path length in
beta band (15–35 Hz) functional networks was signifi-
cantly increased in patients with AD. Importantly, the
variability of cognitive function across both control and
AD groups was negatively correlated with path length,
providing direct evidence that functional-network
topology can be related to variation in cognitive per-
formance118. A third MEG study of resting-state func-
tional networks confirmed a degradation of small-world
attributes in patients with AD and suggested that this
effect is due to disease-related changes at highly con-
nected network hubs119. However, in a fourth study that
used between-subject covariation in regional measures
of cortical thickness to infer anatomical networks from
a large structural MRI data set, global clustering was
significantly increased in patients with AD and there
were abnormalities in the topological configuration
of crucial, high-centrality nodes in regions of the
multimodal association cortex120.
Schizophrenia has been investigated by comparable
approaches. The economical small-world properties of
low-frequency functional networks derived from fMRI
data were shown to be impaired in patients with schizo-
phrenia121. Studies that used EEG to measure synchro-
nization likelihood or nonlinear interactions between
cortical nodes found that the clustering and path-length
parameters of functional networks were closer to their
values in random graphs for patients with schizophrenia
than for comparison subjects122,123. Anatomical networks
derived from inter-regional covariation of grey mat-
ter density in structural MRI data showed differences
in the hierarchy and assortativity of multimodal and
transmodal cortical subnetworks in healthy volunteers,
suggesting that these major divisions of the cortex may
have developed according to different growth rules or
evolved to meet different selection criteria. Topological
abnormalities in people with schizophrenia included
a reduced hierarchy of multimodal cortex (FIG. 3). The
schizophrenic group’s networks also exhibited relatively
long physical distances between connected regions,
compatible with inefficient axonal wiring124.
Thus, there is convergent evidence from methodo-
logically disparate studies that both AD and schizo-
phrenia are associated with abnormal topological
organization of structural and functional brain net-
works. However, there are also inconsistencies between
existing studies — for example, clustering is reportedly
increased in the structural networks120 but decreased in
the functional networks of patients with AD117 — which
might be attributable to the clinical heterogeneity of the
Figure 2 | Cellular and whole-brain networks demonstrate consistent topological
features. The top panel shows a cellular functional network constructed from multi-
electrode-array recordings made in the anaesthetized cat; each node (represented by a
circle) corresponds approximately to one neuron and the connections represent high
functional connectivity between neurons86. The different coloured nodes constitute
separate clusters or modules. The plots in each circle illustrate cellular responses to
stimuli of different orientations, and the circle size corresponds to the degree (number of
functional connections) of each node. The bottom panel shows a whole-brain structural
network constructed from histological data on the macaque cortex; each node
corresponds to a brain area and the connections represent axonal projections between
areas49. The network has two main modules, shown here with yellow and grey circles
corresponding to mostly dorsal and ventral visual regions, respectively. Both networks
exhibit the small-world attributes of high clustering and short path length (see BOX 3);
both have an exponentially truncated power law degree distribution (see BOX 2),
associated with the existence of high-degree ‘hubs’ (V4 in the anatomical network); and
both have a community structure characterized by sparse connectivity between modules
(each module is enclosed by stippled lines) and linked by hubs (nodes circled in red).
AITv, anterior inferotemporal ventral area; CITd, central inferotemporal dorsal area;
CITv, central inferotemporal ventral area; DP, dorsal preluneate area; FEF, frontal eye
field; FST, floor of superior temporal area; LIP, lateral intraparietal area; MT, middle
temporal area; PIP, posterior intraparietal area; PITd, posterior inferotemporal dorsal
area; PITv, posterior inferotemporal ventral area; PO, parieto-occipital area; TF, area TF;
TH, area TH ;V1–4, visual cortical areas 1–4; VIP, ventral intraparietal area; VOT, ventral
occipitotemporal area; VP, ventral posterior area. The top panel is reproduced, with
permission, from REF. 86 (2008) Oxford University Press. The bottom panel is
reproduced from REF. 49.
MARCH 2009
© 2009 Macmillan Publishers Limited. All rights reserved
Nature Reviews | Neuroscience
a Healthy volunteers b People with schizophrenia
22 44
19 36
20 19'
High clustering
Low clustering
A measure of the tendency for
nodes to be connected to
other nodes of the same or
similar degree.
A quantifiable biological
marker of the genetic risk for a
neuropsychiatric disorder.
patient groups as well as to the differences in imag-
ing and analytic methods. Some of these differences
may perhaps be resolved by studies combining network
measurements on structural and functional neuroim-
aging data acquired on the same patients. It also seems
likely that evidence for network abnormalities in other
neuropsychiatric disorders and conditions (such as epi-
lepsy125127, attention-deficit hyperactivity disorder128 or
spinal cord injury129) will accumulate as the disorders
are increasingly investigated from this perspective.
Understanding the pathogenesis and treatment of brain
disorders from a network perspective. Many psychiatric
disorders are highly heritable and are likely to repre-
sent the clinical outcome of aberrations in the forma-
tion of large-scale networks in utero or during early
post natal life. Measures of network topology may be
worth investigating as intermediate phenotypes, or
endophenotypes, that indicate the genetic risk for a
neuro psychiatric disorder; however, network metrics
have not yet been adopted for this purpose. A study of
healthy twin pairs has shown that classical small-world
metrics on brain functional networks derived from EEG
data have high heritability130, a necessary prerequisite for
their candidacy as disease endophenotypes. Another
study of graph theoretical measures of anatomical net-
works derived from inter-regional correlations in corti-
cal-thickness MRI measurements on a sample of normal
twins, singletons and singleton siblings of twins showed
that genetically determined frontoparietal networks had
small-world properties131. Network metrics are arguably
more attractive as intermediate phenotypes than local
measures of brain (dis)organization, because computa-
tional models of network development are often avail-
able to test mechanistic hypotheses for how an observed
profile of anatomical or functional dysconnectivity in a
mature network might have been generated by earlier
developmental abnormalities24,132.
Another example of how empirical and computational
approaches can be usefully combined is provided by stud-
ies that have ‘lesioned’ anatomical or functional network
models — for example, by deleting nodes or connections
— to explore how acute and focal damage could affect the
overall performance of brain networks70,133,134. Networks
can be lesioned by random deletion of nodes or edges,
or by targeted attack on the highest-degree nodes in the
network. The vulnerability of the network to damage is
assessed by comparing its topological or dynamical behav-
iour after the lesioning to its intact behaviour. Different
network topologies confer different vulnerabilities to the
effects of random or targeted attack. For example, scale-
free networks are robust to random error but highly
vulnerable to deletion of the network hubs. Brain func-
tional networks with an exponentially truncated power
law degree distribution were found to be less vulnerable
to attack than scale-free networks70. In an anatomically
informed computational model, deletion of hub nodes
produced widespread disruptions of functional connec-
tivity53,134 that were consistent with effects reported in
focal human brain lesions135,136. Computational lesioning
of network models was also used to explore the func-
tional consequences of a gradual and precisely specified
disease process: the elimination of long-range projections
and the sprouting of short-range connections in a model
of epileptogenesis in the rat dentate gyrus137. The topol-
ogy of the normal or non-epileptic dentate gyrus became
relatively over-connected and dynamically hyperexcitable
as a result of cellular changes previously described in rela-
tion to temporal lobe epilepsy. Other studies of models
of temporal lobe epilepsy have shown loss of small-world
topology in cellular networks during hypersynchronized
bursting138 and have shown that variation of small-world
topological and synaptic properties of a computational
model can cause transitions between normal, bursting
and seizing behaviours139.
It is also conceivable that network analysis can be
used to further our understanding of the therapeutic
effects of pharmacological or psychological therapies.
Figure 3 | Disease-related disorganization of brain anatomical networks derived
from structural MRI data. In both parts, the nodes (circles) represent cortical regions
and the connections represent high correlation in grey matter density between nodes.
The nodes are arranged vertically by degree and are separated horizontally for clarity of
representation. The numbers indicate approximate Brodmann area, and the prime
symbols () denote left-sided regions. The clustering coefficient of each node, a measure
of its local connectivity, is indicated by its size: nodes with high clustering are larger.
a | The brain anatomical network of the healthy volunteers has a hierarchical
organization characterized by low clustering of high-degree nodes24. b | The equivalent
network constructed from MRI data on people with schizophrenia shows loss of this
hierarchical organization — high-degree nodes are more often highly clustered. Figure is
reproduced, with permission, from REF. 124 (2008) Society for Neuroscience.
MARCH 2009
© 2009 Macmillan Publishers Limited. All rights reserved
Dopaminergic drugs can modulate measures of functional
connectivity in animal and human fMRI140,141 and MEG
recordings142, suggesting that drug effects might be quan-
tifiable in terms of altered functional network topology.
This has been confirmed directly in a study which demon-
strated that a dopamine D2 receptor antagonist impaired
the economical small-world properties of human brain
fMRI networks75. Future work might include efforts
using graph theoretical measures to quantify how thera-
peutically effective treatments remediate topologically
sub-optimal network configurations in patients.
Conclusions and prospects
It is clear that certain aspects of the organization of com-
plex brain networks are highly conserved over different
scales and types of measurement, across different species
and for functional and anatomical networks. The arche-
typal brain network has a short path length (associated
with high global efficiency of information transfer), high
clustering (associated with robustness to random error),
a degree distribution compatible with the existence of
hubs, and a modular community structure. Furthermore,
anatomical networks are sparsely connected, especially
between nodes in different modules, and the ‘wiring
length’ (the physical distance that connections span) is
close to minimal. This profile of topological and geomet-
ric properties is typical not just of brain networks but
also of many other complex networks, including trans-
port systems and intracellular signalling pathways17,73.
Why might this be so?
A parsimonious hypothesis is that many spatially
embedded complex networks have evolved to optimize
the same set of competitive selection criteria — high effi-
ciency of information transfer between nodes at low con-
nection cost — or to achieve an optimal balance between
functional segregation and integration that yields high
complexity dynamics14. If wiring cost was exclusively
prioritized the network would be close to a regular lat-
tice, whereas if efficiency was the only selection criterion
the network would be random. The existence of a few
long-range anatomical connections can deliver benefits
in terms of efficiency and could arguably account for
the evolution of economical small-world properties in
brain networks at all scales20,51. This hypothesis needs to
be more directly explored and tested, perhaps using evo-
lutionary algorithms in computational models of brain
network selection48.
A key issue for the future will be to consolidate our
understanding of how functional networks interact with
their structural substrates. At low frequencies, or over long
time periods, there are reasons to expect that functional
networks should be highly isomorphic with underlying
structural networks84,109. But clearly function can be adap-
tive over much shorter timescales than structure. We need
to understand more about the non-stationarity or metast-
ability143 of brain functional networks. How does func-
tional network topology change over time? Do functional
networks exist in a dynamically critical state at some or all
frequency intervals144–146? What constraints on the itiner-
ancy of network dynamics are imposed anatomically and
how does the long-term history of functional activity in a
network feed back on the development and remodelling
of the anatomical connections between nodes?
A related question concerns how the parameters of
complex brain networks relate to cognitive and behav-
ioural functions. One can make an intuitively reason-
able claim that high clustering favours locally specialized
processing whereas short path length favours globally
distributed processing; but the empirical evidence is cur-
rently almost non-existent. This will probably be a key
focus of future work that might be combined with fur-
ther studies of clinical disorders or cohorts at different
stages of normal development.
The emerging field of complex brain networks raises
a number of interesting questions and provides some of
the first quantitative insights into general topological
principles of brain network organization. The funda-
mental growth in the statistical mechanics of complex
networks, and the power and elegance of graph theo-
retical analysis, suggests that this approach will play an
increasingly important part in our efforts to comprehend
the physics of the brains connectome.
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E.B. was supported by a Human Brain Project grant from the
National Institute of Mental Health and the National Institute
of Biomedical Imaging & Bioengineering. The Behavioural &
Clinical Neurosciences Institute in the University of Cambridge
is supported by the Wellcome Trust and the Medical Research
Coun cil (UK). O.S. w as s uppo rted by the JS McD onne ll
Competing interests statement
The authors declare competing financial interests: see web
version for details.
Ed Bullmore’s homepage:
Olaf Sporns’ homepage:
Brain Connectivity Toolbox: http://www.brain-
BrainwaveR Toolbox:
MARCH 2009
© 2009 Macmillan Publishers Limited. All rights reserved
... The topological properties of the human brain's anatomical networks can be analyzed quantitatively using graph theory [19,20], which is a branch of mathematics and permeates all scientific disciplines. The graph-theoretical approaches are centrally important to understanding the architecture, development, and evolution of brain networks [21] and are widely used in the study of psychiatric disorders such as schizophrenia [20] and depression [22]. ...
... The topological properties of the human brain's anatomical networks can be analyzed quantitatively using graph theory [19,20], which is a branch of mathematics and permeates all scientific disciplines. The graph-theoretical approaches are centrally important to understanding the architecture, development, and evolution of brain networks [21] and are widely used in the study of psychiatric disorders such as schizophrenia [20] and depression [22]. Most of the previous structural neuroimaging studies of PD applied traditional voxel-based analyses [12,13,15,[23][24][25][26], which could not find imbalanced interactions between the brain regions or extensive pathological changes on a large-scale level. ...
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Panic disorder (PD) is an anxiety disorder that impairs life quality and social function and is associated with distributed brain regions. However, the alteration of the structural network remains unclear in PD patients. This study explored the specific characteristics of the structural brain network in patients with PD by graph theory analysis of diffusion tensor images (DTI). A total of 81 PD patients and 48 matched healthy controls were recruited for this study. The structural networks were constructed, and the network topological properties for individuals were estimated. At the global level, the network efficiency was higher, while the shortest path length and clustering coefficient were lower in the PD group compared to the healthy control (HC) group. At the nodal level, the PD group showed a widespread higher nodal efficiency and lower average shortest path length in the prefrontal, sensorimotor, limbic, insula, and cerebellum regions. Overall, the current results showed that the alteration of information processing in the fear network might play a role in the pathophysiology of PD.
... Hub, having high number of edges connected to it, but low clustering. [22,23,24] ...
Mental health is an intricate branch of medicine that involve the various brain circuits including frontal, temporal and occipital lobes, may also involve the structure and functional unit of brain. Many psychiatrist treat on the basis of subjective experience rather than implying the pathophysiology of the disease process. So in this article, we highlighted the concept of various brain circuits, role of neurochemicals and its involvement in various neuro-psychiatric illness like schizophrenia, Obsessive compulsive disorder, depression etc. Analogous to our human brain is Deep Neural Network (DNN) and Machine learning, based on "Graph Theory"(Artificial Neural Network), play a crucial role in working of Artificial Intelligence (AI) techniques. Various neuroimaging techniques like fMRI, is described in detail. Importance of AI in early diagnosis, individual treatment, counselling and research of various illness and AI based applications has been narrated.
... Neuroimaging approaches were used to examine the underlying neurobiological mechanisms of IGD, and previous studies reported that IGD was associated with systemlevel alterations between the brain regions rather than functional impairment of isolated regions (Song et al., 2020). With the advent of connectomics, it is currently feasible to shift the view from an isolated regional perspective toward a system-level perspective (i.e., a network perspective) based on the integration of various forms of anatomical/functional data to assess the connectivity of networks in brain diseases including IGD (Bullmore and Sporns, 2009;Weinstein et al., 2017). A wide range of measures can be computed to assess the topological properties of the underlying brain networks (Rubinov and Sporns, 2010). ...
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Although recent evidence suggests that dysfunctional brain organization is associated with internet gaming disorder (IGD), the neuroanatomical alterations related to IGD remain unclear. In this diffusion tensor imaging (DTI) study, we aimed to examine alterations in white matter (WM) structural connectomes and their association with IGD characteristics in 47 young men with IGD and in 34 well-matched healthy controls. Two approaches [namely, network-based statistics (NBS) and graph theoretical measures] were applied to assess differences in the specific topological features of the networks and to identify the potential changes in the topological properties, respectively. Furthermore, we explored the association between the alterations and the severity of internet addiction. An NBS analysis revealed widespread alterations of the cortico-limbic-striatal structural connectivity networks in young people with IGD: (1) an increased subnet1 comprising the insula and the regions responsible for visual, auditory, and sensorimotor functions and (2) two decreased subnet2 and subnet3 comprising the insula, striatum, and limbic regions. Additional correlation analysis showed a significant positive association between the mean fractional anisotropy- (FA-) weighted connectivity strength of subnet1 and internet addiction test (IAT) scores in the IGD group. The present study extends our knowledge of the neuroanatomical correlates in IGD and highlights the role of the cortico-limbic-striatal network in understanding the neurobiological mechanisms underlying this disorder.
... Functional connectivity, distinguished from anatomical connectivity, represents the relationship between dynamic neuronal activities 80 . The measure of modularity, representing the community structure employed in assemblies of functionally distinct clusters, is usually identified in graph-theoretic methods 91 . Neurons or active electrodes can be regarded as a subset of nodes, and the links correspond to the relation between the nodes (Fig. 7). ...
A bidirectional in vitro brain-computer interface (BCI) directly connects isolated brain cells with the surrounding environment, reads neural signals and inputs modulatory instructions. As a noninvasive BCI, it has clear advantages in understanding and exploiting advanced brain function due to the simplified structure and high controllability of ex vivo neural networks. However, the core of ex vivo BCIs, microelectrode arrays (MEAs), urgently need improvements in the strength of signal detection, precision of neural modulation and biocompatibility. Notably, nanomaterial-based MEAs cater to all the requirements by converging the multilevel neural signals and simultaneously applying stimuli at an excellent spatiotemporal resolution, as well as supporting long-term cultivation of neurons. This is enabled by the advantageous electrochemical characteristics of nanomaterials, such as their active atomic reactivity and outstanding charge conduction efficiency, improving the performance of MEAs. Here, we review the fabrication of nanomaterial-based MEAs applied to bidirectional in vitro BCIs from an interdisciplinary perspective. We also consider the decoding and coding of neural activity through the interface and highlight the various usages of MEAs coupled with the dissociated neural cultures to benefit future developments of BCIs.
... To reveal the interpersonal neural coupling, we used wavelet transform coherence (WTC, Grinsted et al. 2004) to assess the IBS for LF and FF pairs. Considering that leader emergence is characterized by higher IBS for LF pairs than that for FF pairs (Jiang et al. 2015) and that groups with higher creative performance have stronger IBS (Lu et al. 2019, 2020), we hypothesized that (III) the IBS increment for LF would be higher than that for FF in condition E. Finally, we adopted the graph theoretical approach (Bullmore and Sporns 2009) to explore the characteristics of the hyperbrain network consisting of intra-and interbrain synchronization across group members. The graph-based approach is a powerful way of quantifying brain systems to analyze complex brain networks that served as physiological basis of information transfer and mental representations (Strogatz 2001). ...
This study aimed to investigate how the ways leaders arise (appointed vs. emergent) affect the leader–follower interaction during creative group communication. Hyperscanning technique was adopted to reveal the underlying interpersonal neural correlates using functional near-infrared spectroscopy. Participants were assigned into 3-person groups to complete a creative problem-solving task. These groups were randomly split into conditions of appointed (condition A) and emergent (condition E) leaders. Creative group outcomes were better in condition E, accompanied by more frequent perspective-taking behaviors between leaders and followers. The interpersonal brain synchronization (IBS) increment for leader–follower pairs was significantly higher at the right angular gyrus (rAG), between the rAG and the right supramarginal gyrus (rSMG), and between the right middle temporal gyrus and the right motor cortex in condition E and positively correlated with perspective-taking behaviors between leaders and followers. The graph-based analysis showed higher nodal betweenness of the rAG and the rSMG in condition E. These results indicated the neural coupling of brain regions involved in mentalizing, semantic processing and motor imagery may underlie the dynamic information transmission between leaders and followers during creative group communication.
Introduction Neurodegenerative diseases can be considered as ‘disconnection syndromes’, in which a communication breakdown prompts cognitive or motor dysfunction. Mathematical models applied to functional resting-state MRI allow for the organization of the brain into nodes and edges, which interact to form the functional brain connectome. Areas covered The authors discuss the recent applications of functional connectomics to neurodegenerative diseases, from preclinical diagnosis, to follow up along with the progressive changes in network organization, to the prediction of the progressive spread of neurodegeneration, to stratification of patients into prognostic groups, and to record responses to treatment. The authors searched PubMed using the terms ‘neurodegenerative diseases’ AND ‘fMRI’ AND ‘functional connectome’ OR ‘functional connectivity’ AND ‘connectomics’ OR ‘graph metrics’ OR ‘graph analysis’. The time range covered the past 20 years. Expert opinion Considering the great pathological and phenotypical heterogeneity of neurodegenerative diseases, the identification of a common framework to diagnose, monitor and elaborate prognostic models is challenging. Graph analysis can describe the complexity of brain architectural rearrangements supporting the network-based hypothesis as a unifying pathogenetic mechanism. Although a multidisciplinary team is needed to overcome the limit of methodologic complexity in clinical application, advanced methodologies are valuable tools to better characterize the functional disconnection in neurodegeneration.
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Chronic pain is one of the most common symptoms of temporomandibular disorders (TMD). Although its pathophysiology is still a challenge, TMD has been associated with changes in central nervous system activity related to pain modulatory capacity. This study was conducted to examine the cortical activity of patients with temporomandibular disorders and chronic pain of myofascial origin using quantitative electroencephalography (qEEG). Individuals with TMD and chronic pain and healthy controls were evaluated using qEEG in four consecutive conditions, all with closed eyes: 1) initial resting condition; 2) non-painful motor imagery task of hand movement; 3) painful motor imagery task of clenching the teeth; 4) final resting condition. Participants with TMD and chronic pain overall presented decreased alpha power density during baseline at rest, and non-painful and painful motor imagery tasks when compared to healthy controls. Furthermore, functional brain connectivity was distinct between groups, with TMD and chronic pain showing lower small-world values for the delta (all conditions), theta (eyes closed, painful and non-painful motor imagery task), and alpha bands (painful motor imagery task), and an increase in the beta band (all conditions). These results suggest that TMD and related chronic pain is associated with maladaptive plasticity in the brain, which may correspond to a reduced ability to modify brain activity during different mental tasks, including painful and non-painful imagery. These changes can be detected by qEEG, a method which may be very important because of its characteristics of good temporal resolution and the possibility to be performed in naturalistic setups.
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Small-world properties have been demonstrated for many complex networks. Here, we applied the discrete wavelet transform to functional magnetic resonance imaging (fMRI) time series, acquired from healthy volunteers in the resting state, to estimate frequency-dependent correlation matrices characterizing functional connectivity between 90 cortical and subcortical regions. After thresholding the wavelet correlation matrices to create undirected graphs of brain functional networks, we found a small-world topology of sparse connections most salient in the low-frequency interval 0.03– 0.06 Hz. Global mean path length (2.49) was approximately equivalent to a comparable random network, whereas clustering (0.53) was two times greater; similar parameters have been reported for the network of anatomical connections in the macaque cortex. The human functional network was dominated by a neocortical core of highly connected hubs and had an exponentially truncated power law degree distribution. Hubs included recently evolved regions of the heteromodal association cortex, with long-distance connections to other regions, and more cliquishly connected regions of the unimodal association and primary cortices; paralimbic and limbic regions were topologically more peripheral. The network was more resilient to targeted attack on its hubs than a comparable scale-free network, but about equally resilient to random error. We conclude that correlated, low-frequency oscillations in human fMRI data have a small-world architecture that probably reflects underlying anatomical connectiv-ity of the cortex. Because the major hubs of this network are critical for cognition, its slow dynamics could provide a physiological substrate for segregated and distributed information processing.
Many real networks in nature and society share two generic properties: they are scale-free and they display a high degree of clustering. We show that these two features are the consequence of a hierarchical organization, implying that small groups of nodes organize in a hierarchical manner into increasingly large groups, while maintaining a scale-free topology. In hierarchical networks, the degree of clustering characterizing the different groups follows a strict scaling law, which can be used to identify the presence of a hierarchical organization in real networks. We find that several real networks, such as the Worldwideweb, actor network, the Internet at the domain level, and the semantic web obey this scaling law, indicating that hierarchy is a fundamental characteristic of many complex systems.
In recent years, many new cortical areas have been identified in the macaque monkey. The number of identified connections between areas has increased even more dramatically. We report here on (1) a summary of the layout of cortical areas associated with vision and with other modalities, (2) a computerized database for storing and representing large amounts of information on connectivity patterns, and (3) the application of these data to the analysis of hierarchical organization of the cerebral cortex. Our analysis concentrates on the visual system, which includes 25 neocortical areas that are predominantly or exclusively visual in function, plus an additional 7 areas that we regard as visual-association areas on the basis of their extensive visual inputs. A total of 305 connections among these 32 visual and visual-association areas have been reported. This represents 31% of the possible number of pathways it each area were connected with all others. The actual degree of connectivity is likely to be closer to 40%. The great majority of pathways involve reciprocal connections between areas. There are also extensive connections with cortical areas outside the visual system proper, including the somatosensory cortex, as well as neocortical, transitional, and archicortical regions in the temporal and frontal lobes. In the somatosensory/motor system, there are 62 identified pathways linking 13 cortical areas, suggesting an overall connectivity of about 40%. Based on the laminar patterns of connections between areas, we propose a hierarchy of visual areas and of somato sensory/motor areas that is more comprehensive than those suggested in other recent studies. The current version of the visual hierarchy includes 10 levels of cortical processing. Altogether, it contains 14 levels if one includes the retina and lateral geniculate nucleus at the bottom as well as the entorhinal cortex and hippocampus at the top. Within this hierarchy, there are multiple, intertwined processing streams, which, at a low level, are related to the compartmental organization of areas V1 and V2 and, at a high level, are related to the distinction between processing centers in the temporal and parietal lobes. However, there are some pathways and relationships (about 10% of the total) whose descriptions do not fit cleanly into this hierarchical scheme for one reason or another. In most instances, though, it is unclear whether these represent genuine exceptions to a strict hierarchy rather than inaccuracies or uncertainties in the reported assignment.