In mathematics graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, or lines. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another; see Graph (discrete mathematics) for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics.
Resting state functional magnetic resonance imaging data were acquired using multi echo imaging, echo-planar imaging pre-operatively from five participants each with a right temporal-parietal-occipital glioblastoma. Complex networks analysis was initiated by parcellating the brain into anatomically regions amongst which connections were identified by retaining the most significant correlations between the respective wavelet decomposed time-series.
Key characteristics of complex networks described in healthy controls were preserved in these patients, including ubiquitous small world organization. An exponentially truncated power law fit to the degree distribution predicted findings of general network robustness to injury but with a core of hubs exhibiting disproportionate vulnerability. Tumours produced a consistent reduction in local and long-range connectivity with distinct patterns of connection loss depending on lesion location.
Connectome analysis is a feasible and novel approach to brain mapping in individual patients with brain tumours. Applications to pre-surgical planning include identifying regions critical to network function that should be preserved and visualising connections at risk from tumour resection. In the future one could use such data to model functional plasticity and recovery of cognitive deficits 1).