What is a network? Network Analysis

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1 What is a network? Network Analysis Valerie Cardenas Nicolson Associate Adjunct Professor Department of Radiology and Biomedical Imaging Complex weblike structures Cell 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 Jefferson High School sexual relationships over 18 months Graph theory Study of complex networks Initially focused on regular graphs Connections are completely regular, i.e. each node is connected only to nearest neighbors 63 isolated pairs, 21 triads Very large structure involving 52% of students with 37 steps between most distant nodes 1

2 Random Graphs Since 1950s large-scale networks with no apparent design principles were described as random graphs N nodes pn( N 1) E 2 Connect every pair of nodes with probability p Approximately K edges randomly distributed with: pn( N 1) K 2 Example: p=0.25, N=8 K=7 But Are real networks (such as the brain) fundamentally random? Intuitively, complex systems must display some organizing principles, which must be encoded in their topology arrangement in which the nodes of the network are connected to each other How do we explore brain networks using graph theory? Define the network nodes Estimate a continuous measure of association between nodes Generate an association matrix, apply threshold to create adjacency matrix Calculate network parameters Compare network parameters to equivalent parameters of a population of random networks March 2,

3 1. Define network nodes EEG electrodes MEG electrodes Anatomically defined regions Cortical parcellation (MRI, DTI, DSI) Individual fmri voxels 2. Estimate a continuous measure of association between nodes Spectral coherence between MEG sensors Correlations in cortical thickness or MRI volume between regions (nodes) Connection probability between two regions of DTI data set Correlation between voxel-wise fmri time series Tract tracing 3. 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 Indicates whether and edge (connection) exists between each pair of nodes Symmetrical for undirected graphs Association and Adjacency 104 ROIs or nodes March K 2,

4 1 8 3/7/ Calculate network measures Node degree, degree distribution, assortativity Clustering coefficient Path length and efficiency i Connection density or cost Hubs, centrality and robustness Modularity Node degree and Assortativity k i number of edges connected to a node i degree of node I Assortativity Correlation between the degrees of connected nodes Positive assortativity indicates that highdegree nodes tend to connect to each other Bullmore and Sporns, 2009 Degree Distribution Degree distribution of a graph Probability distribution of k i In random graph, exponential P(k) e -αk WWW, power law P(k) k -α 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 WWW random power law truncated power Clustering There are cliques or clusters where every node is connected to every other node Random networks have low avg. clustering; complex networks have high clustering Let node i have k i edges which connect it to k i other nodes. K i is number of edges existing between k i nodes. 2Ki Ci = k ( k 1) C = 1if the nearest neighborsof i are also nearest neighborsof each other i C = i C For a randomgraph, C March 2, 2010 i i i rand = p 4

5 Path Length L i,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) Efficiency is inversely related to path length L L 1 N = 2 i j rand March 2, 2010 L i, j ln N ~ K ln( 1) N, where K is number of edges Path length Clustering k i =6 K i =2 C i =2(2)/((6)(5))=0.13 Degree Examples C=1 C=0.13 Cost Connection density or cost is the actual number of edges in the graph as a proportion of the total number of possible edges Estimator of physical cost (e.g., energy) of a network 0 < K = E E max < 1 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 c ( i) = Cl c ( i) = B j, m j, m March 2, 2010 j 1 L j m i i, j σ ( i) j, m σ j, m σ : = number of σ ( i) : = number of Closeness centrality Betweenness centrality shortest paths between regions shortest paths between j and m j and m that pass through i 5

6 Centrality Example Highest closeness centrality Hubs and Robustness Hubs are nodes with high degree or high centrality Robustness refers either to the structural integrity of the network following deletions of nodes or edges Effects of perturbations ti on local l or global l network states Highest betweenness centrality Highest closeness centrality Modularity Many complex networks consist of a number of modules. Each module contains several densely interconnected nodes, and there are relatively few connections between nodes in different modules. Algorithms to assess modularity: Girvan and Newman, Community structure in social and biological networks, Proc. Natl Acad. Sci. USA 99, (2002). 5. Compare to equivalent parameters from population of random networks Lack 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 Estimate in random networks with same number of nodes and connections 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 6

7 Small Worlds Despite large size, in most networks there is a relatively short path between any two nodes Example: Six degrees of separation Stanley Milgram (1967) Path of acquaintances with typical length about six between most pairs of people in the US Small World Example Path=4 Path=3 Path=1 p is probability that pair of nodes is rewired From Guye, et al., Curr Opin Neurol 21: Why should we think about the brain as a small world network? Brain is a complex network on multiple spatial and time scales Connectivity of neurons Brain supports segregated and distributed information processing Somatosensory and visual systems segregated 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 costs Adaptive reconfiguration Small world metrics For a small world network, γ = C C λ = L L rand rand σ = γ > 1 λ >> 1 1 7

8 Path Length and Clustering C(0) and L(0) are clustering coefficient and path length for regular graph. For small world, C(p)/C(0) < 1 L(p)/L(0) < 1 Empirical Examples of Small World Networks Watts and Strogatz, Nature, Vol 393: Watts and Strogatz, Nature, Vol 393: How 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 damaged nodes, model for neurodegeneration) Determine if network parameters can help diagnose or distinguish patients from controls Relate network parameters to cognition 8

9 Network efficiency and IQ van den Heuvel et al., J. Neurosci healthy subject IQ measured with WAIS-III Resting state fmri Association i was correlation between time-series i from each voxel pair (9500 voxels/nodes) Network constructed for each subject Network measures were correlated with IQ scores γ, λ and total connections k Also correlated normalized path length at each node with IQ Functional network Small world properties observed for a range of thresholds Network params vs. IQ No association between γ and IQ At higher thresholds (T=0.45, T=0.5) Negative association between IQ and λ Longer path length, lower IQ Nodes vs. IQ Path length at nodes vs. IQ Medial frontal gyrus, precuneus/posterior cingulate, bilateral inferior parietal, left superior temporal, left inferior gyrus 9

10 Conclusions Efficiency of intrinsic resting-state functional connectivity patterns is predictive of cognitive performance Short path length is crucial for efficient information processing in functional brain networks March 2011 Cortical Thickness Networks in Temporal Lobe Epilepsy Bernhardt et al., Cerebral Cortex patients with drug-resistant TLE; 63 LTLE, 59 RTLE 47 age- ad sex-matched healthy controls T1-weighted imaging at 1.5T Estimated cortical thickness in 52 ROIs Thicknesses corected for age, gender, overall mean thickness Computed r ij Pearson product moment cross-correlation across subjects in regions i and j Explored network parameters at a range of connection densities Different networks for controls, LTLE, RTLE Ensures that networks in all groups have the same number of edges or wiring cost Between group differences reflect topological organization differences; not differences in correlations Network Parameters March

11 Robustness Analysis Clinical Correlation March 2011 Reproducibility of Networks Telesford et al., Frontiers in Neuroinformatics, 2010 Methods Overview Evaluate reproducibility of measured graph metrics between two fmri runs Studied 45 healthy older adults (65-75 yrs) Participants performed an executive function task (NOT resting state fmri) Performed task two times; subjects were not removed from the scanner Pearson correlation matrix for all possible pairs of 16,000 voxels Range of thresholds applied Similar to density metric; thresholded such that the number of nodes and average number of connections at each node was consistent across subjects March

12 Results ICC (intraclass correlation coefficient) ranges from 0-1 <0.20 poor agreement fair moderate strong >0.80 almost perfect smoothed vs. unsmoothed data Their conclusions Mean graph metrics were reproducible, except for degree Voxel-wise path length and global efficiency were fairly reproducible Reproducibility lower in low degree nodes 12

13 Other reproducibility papers MEG excellent (Deuker et al. 2009) DTI excellent (Vaessen et al. 2010) CAUTION: recent Li et al shows that connection reconstruction methods leads to differences in small-world indices, identified hubs, and hemispheric asymmetries Resting state fmri (Braun et al., 2012) Makes pre-processing recommendations for most reproducible network results Weighted graphs Directional graphs Complications Bibliography Kolaczyk, Statistical Analysis of Network Data: Methods and Models, Springer Guye et al., Imaging of structural and functional connectivity: towards a unified definiton of human brain organization? Current Opinion in Neurology 2008, 21: Bassett et al., Hierarchical organization of human cortical networks in health and schizophrenia. The Journal of Neuroscience, 2009, 28(37): Van den Heuvel et al., Efficiency of functional brain networks and intellectual performance. The Journal of Neuroscience, 2009, 29(23): Bassett and Bullmore, Small-World Brain Networks. The Neuroscientist, 2006, 12(6): Bullmore and Sporns, Complex brain networks: graph theorectical analysis of structural and functional systems. Nature Reviews: Neuroscience, 2009, 10: Telesford et al., Reproducilbility of graph metrics in fmri networks. Frontiers in Neuroscience, 2010, 4:article 117. Software Brain connectivity toolbox Matlab BGL /matlab_bgl March