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Exponential random graph modeling for complex brain networks.

Simpson SL, Hayasaka S, Laurienti PJ - PLoS ONE (2011)

Bottom Line: 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.

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

ABSTRACT
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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Six-node example network.The edgewise, nonedgewise, and dyadwise shared partner distributions are (ESP, , ESP)(1, 5, 1, 0, 0), (NSP, , NSP)(1, 4, 3, 0, 0), and (DSP, , DSP)(2, 9, 4, 0, 0) respectively.
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pone-0020039-g002: Six-node example network.The edgewise, nonedgewise, and dyadwise shared partner distributions are (ESP, , ESP)(1, 5, 1, 0, 0), (NSP, , NSP)(1, 4, 3, 0, 0), and (DSP, , DSP)(2, 9, 4, 0, 0) respectively.

Mentions: In order to establish the most appropriate set of explanatory metrics for each subject's brain network and provide a foundation for the development of a “best assessment” ERGM for analyzing complex brain networks, we implemented and assessed three model/metric selection methods. They include a traditional p-value based backward selection approach [30], an information criterion approach (AIC, [31]), and a graphical goodness of fit (GOF) approach [17]. The latter two techniques are used most often for metric selection in ERGMs [16], [17]; and, to our knowledge, no detailed comparisons have been performed to determine whether the approaches generally produce the same “best” model. The p-value approach is based on removing metrics that are not statistically significant. Whereas, the AIC approach selects the set of metrics that produce the estimated distribution most likely to have resulted in the observed data with a penalty for additional metrics to ensure parsimony. Alternatively, the graphical GOF method allows subjectively selecting the set of explanatory metrics that produces the model most able to capture and reproduce certain topological properties of the observed network (see Appendix S1 for more details). For each approach ERGMs were fitted to the 90-node unweighted, undirected brain networks of the 10 subjects discussed previously. The potential explanatory metrics for each of the 10 networks are listed by category in Table 2. The categories were chosen based on properties of brain networks that are regarded as important in the literature [14]. These metrics are analogous to typical brain network metrics (e.g., clustering coefficient ()) but have been developed to be statistically compatible with ERGMs. Figure 2 illustrates the calculation of the less widely used of these statistics, namely GWESP, GWNSP, and GWDSP, on a six-node example network. The distribution of the unweighted analogues of these metrics (ESP, NSP, and DSP) is given for simplicity. The weighted versions simply sum the values of the distribution giving less weight to those with more shared partners. For this example we note that the network has 1 set of connected nodes with 1 shared partners (ESP), 5 sets with 1 shared partner (ESP), 1 set with 2 shared partners (ESP), and 0 sets with 3 or 4 shared partners (ESP and ESP). Further details on the metrics are provided in Table 1 and [27]. The parameters associated with GWESP, GWDSP, GWNSP, and GWD were all assumed to be fixed and known (for reasons outlined in [17]) and set to based on preliminary analyses as this value generally led to better fitting models according to all selection methods. The three aforementioned model selection approaches are outlined in Appendix S1.


Exponential random graph modeling for complex brain networks.

Simpson SL, Hayasaka S, Laurienti PJ - PLoS ONE (2011)

Six-node example network.The edgewise, nonedgewise, and dyadwise shared partner distributions are (ESP, , ESP)(1, 5, 1, 0, 0), (NSP, , NSP)(1, 4, 3, 0, 0), and (DSP, , DSP)(2, 9, 4, 0, 0) respectively.
© Copyright Policy
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3102079&req=5

pone-0020039-g002: Six-node example network.The edgewise, nonedgewise, and dyadwise shared partner distributions are (ESP, , ESP)(1, 5, 1, 0, 0), (NSP, , NSP)(1, 4, 3, 0, 0), and (DSP, , DSP)(2, 9, 4, 0, 0) respectively.
Mentions: In order to establish the most appropriate set of explanatory metrics for each subject's brain network and provide a foundation for the development of a “best assessment” ERGM for analyzing complex brain networks, we implemented and assessed three model/metric selection methods. They include a traditional p-value based backward selection approach [30], an information criterion approach (AIC, [31]), and a graphical goodness of fit (GOF) approach [17]. The latter two techniques are used most often for metric selection in ERGMs [16], [17]; and, to our knowledge, no detailed comparisons have been performed to determine whether the approaches generally produce the same “best” model. The p-value approach is based on removing metrics that are not statistically significant. Whereas, the AIC approach selects the set of metrics that produce the estimated distribution most likely to have resulted in the observed data with a penalty for additional metrics to ensure parsimony. Alternatively, the graphical GOF method allows subjectively selecting the set of explanatory metrics that produces the model most able to capture and reproduce certain topological properties of the observed network (see Appendix S1 for more details). For each approach ERGMs were fitted to the 90-node unweighted, undirected brain networks of the 10 subjects discussed previously. The potential explanatory metrics for each of the 10 networks are listed by category in Table 2. The categories were chosen based on properties of brain networks that are regarded as important in the literature [14]. These metrics are analogous to typical brain network metrics (e.g., clustering coefficient ()) but have been developed to be statistically compatible with ERGMs. Figure 2 illustrates the calculation of the less widely used of these statistics, namely GWESP, GWNSP, and GWDSP, on a six-node example network. The distribution of the unweighted analogues of these metrics (ESP, NSP, and DSP) is given for simplicity. The weighted versions simply sum the values of the distribution giving less weight to those with more shared partners. For this example we note that the network has 1 set of connected nodes with 1 shared partners (ESP), 5 sets with 1 shared partner (ESP), 1 set with 2 shared partners (ESP), and 0 sets with 3 or 4 shared partners (ESP and ESP). Further details on the metrics are provided in Table 1 and [27]. The parameters associated with GWESP, GWDSP, GWNSP, and GWD were all assumed to be fixed and known (for reasons outlined in [17]) and set to based on preliminary analyses as this value generally led to better fitting models according to all selection methods. The three aforementioned model selection approaches are outlined in Appendix S1.

Bottom Line: 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.

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

ABSTRACT
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.

Show MeSH