Research Article

Quantification of Spatial Parameters in 3D Cellular Constructs Using Graph Theory

Table 1

Description of graph metrics.

Number of nodesThe number of vertices in the graph, in this case cell nuclei as defined by image segmentation.
Number of edgesThe number of links between the vertices of the graph, in this case the probability of cellular interaction based on distance.
Average degreeThe number of neighbors/connections that a given node has averaged over all the nodes in the graph.
Clustering coefficient (C)Is defined as , where is the number of neighbors of node and is the number of existing links between s neighbors. measures the ratio of existing links to possible links for each node.
Path length or hop count:The shortest distance between two nodes reflective of the weight of each link.
Number of nodesThe number of vertices in the graph, in this case cell nuclei as defined by image segmentation.
Eccentricity and closenessThe maximum and average of the shortest lengths, respectively. The average of these two features provides global measure of average eccentricity and average closeness.
DiameterDefined as the maximum eccentricity, greatest distance between two nodes in the graph.
Central pointsDefined as nodes that have an eccentricity equal to the average eccentricity.
Hop plot valueReflects the size of a neighborhood between any two nodes with a “hop”, a single edge.
Hop plot exponentSlope of the hop plot values as a function of in log-log scale.
Effective diameterDefined as , where and are the number of nodes and edges, and is the hop plot exponent.
Giant connected componentDefined as the largest set of nodes that are reachable from each other.
Giant connected ratioThe ratio of the size of the giant connected component to the number of nodes in the graph.
Isolated nodeA node with no edges and therefore a degree of 0.
End nodeA node with one edge.