+ An alternative way of viewing the clustering coefficient—rather than as a measure of local clustering—is that it counts the number of closed triangles within each node’s local neighborhood.
+ Rather than just counting triangles, we could consider more complex structures, such as cycles of a particular length. We could also characterize nodes by counts of how often these different motifs occur in their ego graph.

Closed triangles, ego graphs, and motifs

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]:

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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

+ Node degree simply measures how many neighbors a node has, but this is not
necessarily sufficient to measure the importance of a node in a graph.
+ We define a node’s eigenvector centrality e(u) via a recurrence relation in which the node’s centrality is proportional to the average centrality of its neighbors.

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