NETWORK CLASSIFICATION WITH APPLICATIONS TO BRAIN CONNECTOMICS

Abstract

While statistical analysis of a single network has received a lot of attention in recent years, with a focus on social networks, analysis of a sample of networks presents its own challenges which require a different set of analytic tools. Here we study the problem of classification of networks with labeled nodes, motivated by applications in neuroimaging. Brain networks are constructed from imaging data to represent functional connectivity between regions of the brain, and previous work has shown the potential of such networks to distinguish between various brain disorders, giving rise to a network classification problem. Existing approaches tend to either treat all edge weights as a long vector, ignoring the network structure, or focus on graph topology as represented by summary measures while ignoring the edge weights. Our goal is to design a classification method that uses both the individual edge information and the network structure of the data in a computationally efficient way, and that can produce a parsimonious and interpretable representation of differences in brain connectivity patterns between classes. We propose a graph classification method that uses edge weights as predictors but incorporates the network nature of the data via penalties that promote sparsity in the number of nodes, in addition to the usual sparsity penalties that encourage selection of edges. We implement the method via efficient convex optimization and provide a detailed analysis of data from two fMRI studies of schizophrenia.

Keywords: fMRI data; graph classification; high-dimensional data; variable selection.

Fig 1:

Regions of interest (ROIs) defined by Power et al. (2011), colored by brain systems, and the total number of nodes in each system.

Fig 2:

Brain network from one of the subjects, showing the value of the Fisher z-transformed correlations between the nodes, with the 264 nodes grouped into 14 brain systems.

Fig 3:

Expected adjacency matrices for each class. There are 50 active nodes $\mathcal{G}$ on communities 4 and 7, and edge weights on 25% of the edges within $\mathcal{G}\times \mathcal{G}$ have been altered for the second class of networks (Y = 1).

Fig 4:

Variable selection performance of different methods in terms of edge AUC (top) and node AUC (bottom) as a function of the fraction of differentiating edges in the subgraph induced by the active node set $\mathcal{G}$. As the proportion of active edges increases, methods that use network structure improve their performance when only a subset of the nodes is active.

Fig 5:

Classification error of different methods as a function of the fraction of differentiating edges in the subgraph induced by the active node set $\mathcal{G}$. Our method is more accurate when only a subset of the nodes is active.

Fig 6:

Cross-validated results for the two data sets. Classification accuracy (left), fraction of zero edge coefficients (middle), and fraction of inactive nodes (right).

Fig 7:

Fitted coefficients for COBRE and UMich datasets, with tuning parameters selected by the “one standard error rule”. Positive coefficients corresponds to higher edge weights for schizophrenic patients.

Fig 8:

Cross-validated accuracy and number of nodes selected as a function of the number of edges used.

Fig 9:

Nodes shown in green are endpoints of edges selected by stability selection shown in Table 2. Node shown in purple are nodes not selected by any of the sparse solutions within one standard error of the most accurate solution.