Network Analysis

Valerie Cardenas NicolsonAssistant Adjunct Professor

Department of Radiology and p gyBiomedical Imaging

March 2, 2010

What is a network?

• Complex weblike structures– Cell is network of chemicals connected byCell is network of chemicals connected by

chemical reactions– Internet is network of routers and

computers linked by physical or wireless links

– Social network, nodes are humans and edges are social relationships

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Jefferson High Schoolsexual relationships over 18 monthssexual relationships over 18 months

63 isolated pairs 21 triads63 isolated pairs, 21 triadsVery large structure involving 52% of students with 37 steps between most distant nodes

Graph theory

• Study of complex networks• Initially focused on regular graphsInitially focused on regular graphs

– Connections are completely regular, i.e. each node is connected only to nearesteach node is connected only to nearest neighbors

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Random Graphs

• Since 1950s large-scale networks with no

Random Graphs

Since 1950s large scale networks with no apparent design principles were described as random graphs

• N nodes• Connect every pair of nodes with probability p

( 1)2

pN NE −≈

y p p y p• Approximately K edges randomly distributed

with:

( 1)2

pN NK −≈ Example:

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2 p=0.25, N=8K=7

But…

• Are real networks (such as the brain) fundamentally random?y

• Intuitively, complex systems must display some organizing principlesdisplay some organizing principles, which must be encoded in their topology– arrangement in which the nodes of the– arrangement in which the nodes of the

network are connected to each other

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How do we explore brainHow do we explore brain networks using graph theory?

• Define the network nodes• Estimate a continuous measure of association between

dnodes• Generate an association matrix, apply threshold to create

adjacency matrixadjacency matrix• Calculate network parameters • Compare network parameters to equivalent parameters of a

population of random networks

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1. Define network nodes

• EEG electrodes• MEG electrodesMEG electrodes• Anatomically defined regions

C ti l ll ti (MRI DTI DSI)– Cortical parcellation (MRI, DTI, DSI)– Individual fMRI voxels

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2. Estimate a continuous measure of association

between nodesbetween nodes• Spectral coherence between MEG sensors

Correlations in cortical thickness or MRI• Correlations in cortical thickness or MRI volume between regions (nodes)

• Connection probability between two regions• Connection probability between two regions of DTI data set

• Correlation between voxel-wise fMRI time• Correlation between voxel-wise fMRI time series

• Tract tracingTract tracingMarch 2, 2010

3. Generate association and3. Generate association and adjacency matrices

• Matrices of nodes vs. nodes• Association matrix

– Value at each (x,y) is measure of association between nodes x and y

• Adjacency matrix– Association matrix is thresholded

I di t h th d d ( ti ) i t– Indicates whether and edge (connection) exists between each pair of nodes

– Symmetrical for undirected graphsSymmetrical for undirected graphs

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Association and Adjacency

104 ROIs or nodes104 ROIs or nodes

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4. Calculate network measures

• Node degree degree distributionNode degree, degree distribution, assortativity

• Clustering coefficient• Clustering coefficient• Path length and efficiency• Connection density or cost• Hubs, centrality and robustness, y• Modularity

March 2, 2010Bullmore and Sporns, 2009

Node degree and Assortativity

• ki– number of edges connected to a node i– degree of node I

• Assortativity• Assortativity– Correlation between the degrees of

connected nodes– Positive assortativity indicates that high-

degree nodes tend to connect to each other

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Degree Distribution• Degree distribution of a graph

– Probability distribution of ki– In random graph, exponential P(k) ∼ e-αk

– WWW, power law P(k) ∼ k-α

• Existence of few major hubs (google yahoo)Existence of few major hubs (google, yahoo)– Transportation, truncated power P(k) ∼ k−α e-k/kc

• Probability of highly connected hubs greater than in a random graph but smaller than in network such as WWWgraph but smaller than in network such as WWW

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Clustering• There are “cliques” or “clusters” where every

node is connected to every other nodenode is connected to every other node• Random networks have low avg. clustering;

complex networks have high clusteringp g gLet node i have ki edges which connect it to ki other nodes. Ki is number of edges existing between ki nodes.

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Path Length• Li,j := minimal number of edges that must be

traversed to form a direct connection between two nodes i and j

• Random and complex networks have short mean path lengths (high global efficiency)path lengths (high global efficiency)

• Efficiency is inversely related to path length

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)1ln(NK

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Path length

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C=1

Degree

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Cost• Connection density or cost is the actual number of y

edges in the graph as a proportion of the total number of possible edgesE ti t f h i l t ( ) f t k• Estimator of physical cost (e.g., energy) of a network

E 10 <=<EEKmaxE

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Centrality• Centrality measures how many of the shortest paths between all

other node pairs in the network pass through it. Nodes with high centrality are crucial to efficient communication.

• Eigenvector centrality of the ith node is the ith component of the eigenvector of the adjacency matrix A associated with the largest eigenvalue

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Centrality ExampleCentrality Example

Highest closeness centrality

Highest betweennesscentralitycentrality

Highest closeness centrality

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Hubs and Robustness• Hubs are nodes with high degree or high centralityg g g y• Robustness refers either to the

• structural integrity of the network following deletions of nodes or edges

• Effects of perturbations on local or global network statesstates

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Modularity• Many complex networks consist of a number of y p

modules. Each module contains several densely interconnected nodes, and there are relatively few connections between nodes in different modulesconnections between nodes in different modules.

• Algorithms to assess modularity:• Girvan and Newman, Community structure inGirvan and Newman, Community structure in

social and biological networks, Proc. NatlAcad. Sci. USA 99, 7821-7826 (2002).

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5. Compare to equivalent parameters from population of random networks

• Lack of statistical theory concerning distribution of networkLack of statistical theory concerning distribution of network metric

• How to determine if network parameters are not random?• Must build a null distribution of equivalent parameters• Must build a null distribution of equivalent parameters• Estimate in random networks with same number of nodes

and connectionsPermutation testing• Permutation testing

• Comparing network parameters from 2 populations (e.g., normal and schizophrenic)• Permutation testing• Compute difference in params for true labeling• Permute labels and compute param difference, build dist

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Small Worlds

• Despite large size, in most networks there is a relatively short path between any two y p ynodes

• Example: Six degrees of separationExample: Six degrees of separation– Stanley Milgram (1967)

Path of acquaintances with typical length about– Path of acquaintances with typical length about six between most pairs of people in the US

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Small World ExampleSmall World Example

Path=4 Path=3 Path=1p is probability that pair of nodes is rewired

From Guye et al Curr Opin Neurol 21:393 403March 2, 2010

From Guye, et al., Curr Opin Neurol 21:393-403.

Why should we think about the brain as a small world network?

• Brain is a complex network on multiple spatial andBrain is a complex network on multiple spatial and time scales– Connectivity of neurons

• Brain supports segregated and distributed• Brain supports segregated and distributed information processing– Somatosensory and visual systems segregated

Di t ib t d i ti f ti– Distributed processing, executive functions• Brain likely evolved to maximize efficiency and

minimize the costs of information processing– Small world topology is associated with high global and local

efficiency of parallel information processing, sparse connectivity between nodes, and low wiring costsAd ti fi ti

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– Adaptive reconfiguration

Small world metrics

network, worldsmall aFor

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Path Length and ClusteringC(0) and L(0)

l t iare clustering coefficient and path plength for regular graph.

For small world, C( )/C(0) 1C(p)/C(0) < 1L(p)/L(0) < 1

March 2, 2010 Watts and Strogatz, Nature, Vol 393:440-442

Empirical Examples ofEmpirical Examples of Small World Networks

Watts and Strogatz, Nature, Vol 393:440-442g

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How to use networkHow to use network analysis to study brain?

• Test for small world behavior• Model development or evolution of brain networks• Link network topology to network dynamics (structure

to function)• Explore network robustness (vulnerability to• Explore network robustness (vulnerability to

damaged nodes, model for neurodegeneration)• Determine if network parameters can help diagnose

or distinguish patients from controls• Relate network parameters to cognition

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Schizophrenia vs. ControlsBassett et al, J. Neurosci. 2008

• 203 patients with schizophrenia and related spectrum di ddisorders

• 259 healthy volunteers• T1 weighted imaging at 1 5T• T1-weighted imaging at 1.5T• Estimated gray matter volume in 104 ROIs

– Transmodal, unimodal, and multimodal

• Computed partial correlation between gray matter volumes for each possible pair; each group separateE l d t k t t f• Explored network parameters at a range of thresholds where small world properties were observed

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Normal cortical network organization

• Small world properties and hubs found– Right premotor, orbitofrontal, middle temporal,

retrosplenial, dorsolateral prefrontal, and insula• Multimodal network hierarchical: hubs had

hi h d b t l l t i t dhigh degree but low clustering; connected predominantly to nodes not otherwise connected to each otherconnected to each other

• Transmodal network had high assortativity; hubs connected to hubshubs connected to hubs

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SchizophreniaSmall orld properties different h bs• Small world properties, different hubs– Insula, thalamus, temporal pole, inferior frontal,

inferior temporal, and precentral cortexinferior temporal, and precentral cortex• Multimodal less hierarchicial• Greater multimodal connection distanceGreater multimodal connection distance• No differences in transmodal or unimodal• 23 nodes showed clustering differences;• 23 nodes showed clustering differences;

predominantly left hemisphere with increased clustering in schizophreniag p

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Multimodal network diff i hi h idifferences in schizophrenia

R d S CRed, S>CBlack, C>S

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Network efficiency and IQyvan den Heuvel et al., J. Neurosci. 2009

• 19 healthy subject• IQ measured with WAIS-III• Resting state fMRI• Association was correlation between time-series from

each voxel pair (9500 voxels/nodes)each voxel pair (9500 voxels/nodes)• Network constructed for each subject• Network measures were correlated with IQ scoresQ

– γ, λ and total connections k– Also correlated normalized path length at each node with IQ

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Functional networkFunctional network• Small world properties observed for aSmall world properties observed for a

range of thresholds

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Network params vs. IQ

• No association between γ and IQγ• At higher thresholds (T=0.45, T=0.5)

– Negative association between IQ and λNegative association between IQ and λ– Longer path length, lower IQ

Nodes vs. IQ

• Path length at nodes vs. IQPath length at nodes vs. IQ– Medial frontal gyrus, precuneus/posterior

cingulate, bilateral inferior parietal, left g , p ,superior temporal, left inferior gyrus

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Conclusions

• Efficiency of intrinsic resting-state functional connectivity patterns is y ppredictive of cognitive performance

• Short path length is crucial for efficientShort path length is crucial for efficient information processing in functional brain networksbrain networks

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Complications

• Weighted graphs• Directional graphsDirectional graphs

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