One of the first graph structure studies was performed by Weiss and Jacobson and published in 1955. Since then, several types of algorithms have been studied and implemented, using different types of rules.
As for the node importance problem, the first thing to think of in the case of a community detection problem is the definition of the metric or objective function that will quantify how good the graph partitions are. The most common definition of community states that a community is a set of nodes with more infra-community connections (more edges between nodes in the same community) than inter-community connections (edges between nodes in two different communities). But even with that definition, there are several possible ways to achieve a satisfactory partitioning.
Many algorithms have been proposed, using different metrics. For instance, hierarchical clustering uses some rules to create a dendrogram, creating a hierarchy of clusters...
