Eigenvector centrality measures a node’s importance recursively: a node is more central when it is connected to neighbors that are themselves highly central. Unlike node degree, which only counts neighbors, eigenvector centrality accounts for the centralities of those neighbors.

Saint Olaf College

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Includes:
 + Node degree.
 + Node centrality.
 + The clustering coefficient.
 + Closed triangles, ego graphs, and motifs.



Node-level statistics and features


The degree of a node $$u \in V$$, denoted $$d(u)$$, is the most straightforward node-level statistic: it simply counts the number of edges incident to that node. Despite its simplicity, node degree is often one of the most informative features used in traditional machine learning models applied to node-level tasks.

Node degree

The clustering coefficient measures the proportion of closed triangles in a node’s local neighborhood. A popular local variant of the clustering coefficient is computed as follows [Watts and Strogatz, 1998]:

Clustering Coefficient

A node’s ego graph is its local neighborhood viewed as a subgraph. The clustering coefficient summarizes closed triangles in this neighborhood, while motif counts characterize it more broadly by recording how often structures such as triangles or cycles of a specified length occur.

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