Shapley ratings in brain networks

Abstract

Recent applications of network theory to brain networks as well as the expanding empirical databases of brain architecture spawn an interest in novel techniques for analyzing connectivity patterns in the brain. Treating individual brain structures as nodes in a directed graph model permits the application of graph theoretical concepts to the analysis of these structures within their large-scale connectivity networks. In this paper, we explore the application of concepts from graph and game theory toward this end. Specifically, we utilize the Shapley value principle, which assigns a rank to players in a coalition based upon their individual contributions to the collective profit of that coalition, to assess the contributions of individual brain structures to the graph derived from the global connectivity network. We report Shapley values for variations of a prefrontal network, as well as for a visual cortical network, which had both been extensively investigated previously. This analysis highlights particular nodes as strong or weak contributors to global connectivity. To understand the nature of their contribution, we compare the Shapley values obtained from these networks and appropriate controls to other previously described nodal measures of structural connectivity. We find a strong correlation between Shapley values and both betweenness centrality and connection density. Moreover, a stepwise multiple linear regression analysis indicates that approximately 79% of the variance in Shapley values obtained from random networks can be explained by betweenness centrality alone. Finally, we investigate the effects of local lesions on the Shapley ratings, showing that the present networks have an immense structural resistance to degradation. We discuss our results highlighting the use of such measures for characterizing the organization and functional role of brain networks.

A Note on Shapley Ratings in Brain Networks

“…We discuss specific features of this game in small networks and apply our model to the large-scale brain networks considered by Kötter et al (2007). Generally speaking, our results corroborate the findings of Kötter et al (2007).…”

“…We introduce an alternative coalitional game that is more intuitive from a game theoretical point of view. We use the Shapley value of this game as an alternative rating to analyze the macaque brain networks and corroborate the findings of Kötter et al (2007). Moreover, we show how missing information on the existence of certain connections can readily be incorporated into this game and the corresponding Shapley rating.…”

“…In this section we apply the Shapley rating based on the brain network game (N, v A ) to the two large-scale brain networks considered by Kötter et al (2007) and we compare the results.…”

Graphical Models of Functional MRI Data for Assessing Brain Connectivity

“…Network analysis for the compensation mechanism can take a perturbation-and-recovery approach: deleting a connection or deactivating a node, and then searching for the most efficient changes that are needed to restore the impaired connectivity. Such mechanism could be developing a new connection other than the deleted one, or increasing the functionality of the most central node of the "lesioned" network (Kötter et al, 2007).…”

“…For example, betweenness centrality (Freeman, 1977) is based on the number of shortest paths between other nodes passing through a node. It can also be assessed by the geodesic distance to other nodes, as closeness centrality (Beauchamp, 1965) does, or by deleting a node and then comparing the connectivity loss of the "impaired" network, as Shapley ratings (Kötter et al, 2007) do. Similar ideas can be applied to define the centrality of an edge or a sub-graph.…”