Exponential random graph modeling for complex brain networks.
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We illustrate the utility of ERGMs for modeling, analyzing, and simulating complex whole-brain networks with network data from normal subjects.We also provide a foundation for the selection of important local features through the implementation and assessment of three selection approaches: a traditional p-value based backward selection approach, an information criterion approach (AIC), and a graphical goodness of fit (GOF) approach.The graphical GOF approach serves as the best method given the scientific interest in being able to capture and reproduce the structure of fitted brain networks.
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Affiliation: Department of Biostatistical Sciences, Wake Forest University School of Medicine, Winston-Salem, North Carolina, United States of America. slsimpso@wfubmc.edu
ABSTRACT
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Exponential random graph models (ERGMs), also known as p* models, have been utilized extensively in the social science literature to study complex networks and how their global structure depends on underlying structural components. However, the literature on their use in biological networks (especially brain networks) has remained sparse. Descriptive models based on a specific feature of the graph (clustering coefficient, degree distribution, etc.) have dominated connectivity research in neuroscience. Corresponding generative models have been developed to reproduce one of these features. However, the complexity inherent in whole-brain network data necessitates the development and use of tools that allow the systematic exploration of several features simultaneously and how they interact to form the global network architecture. ERGMs provide a statistically principled approach to the assessment of how a set of interacting local brain network features gives rise to the global structure. We illustrate the utility of ERGMs for modeling, analyzing, and simulating complex whole-brain networks with network data from normal subjects. We also provide a foundation for the selection of important local features through the implementation and assessment of three selection approaches: a traditional p-value based backward selection approach, an information criterion approach (AIC), and a graphical goodness of fit (GOF) approach. The graphical GOF approach serves as the best method given the scientific interest in being able to capture and reproduce the structure of fitted brain networks. |
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Mentions: We implemented the model selection procedures delineated in the previous section and Appendix S1 for each of the 10 subjects using the statnet package [23] for the R statistical computing environment. The resulting models (for each approach) and their corresponding parameter estimates are displayed in Table 3. These estimates quantify the relative significance of the given metric in explaining the overall network structure; and, more specifically, they specify how much the log odds of an edge existing increases for each unit increase in the corresponding metric. For example, the final graphical GOF model for subject 10 shows that GWESP is the most important metric (other than the number of edges) in describing the structure of the subject's network given the larger absolute value of the parameter estimate. Additionally, the positiveness of the estimate associated with GWESP indicates that an edge that closes a triangle is more likely to exist than it would by chance (i.e., the network has more clustering than a random network where the probability of an edge is ) for the family of networks represented by subject 10's fitted model. As evidenced by the results in Table 3, the three model selection methods can lead to very different “best” models. The disparate final model GOF plots that can result from the three different model selection approaches are exhibited in Figures 3 and 4 (for subjects 2 and 8). Again, our aim here is not to judge the three selection methods, but to highlight the fact that they can lead to disparate final models/sets of features. These model selection approaches have been used seemingly arbitrarily in the literature; and, to our knowledge, no detailed comparisons have been performed to determine whether the approaches generally produce the same “best” model/set of features. For our purposes we recommend the graphical GOF approach as the standard and will use it in future analyses given that our main scientific interest lies in being able to capture and reproduce the structure of the fitted brain networks. With the exception of subject 8, the graphical GOF approach produces reasonably good fits for all subjects. The remaining best graphical selection model GOF plots are shown in Figures 5–12. |
View Article: PubMed Central - PubMed
Affiliation: Department of Biostatistical Sciences, Wake Forest University School of Medicine, Winston-Salem, North Carolina, United States of America. slsimpso@wfubmc.edu