In-Degree & Out-Degree (Directed)

• Out-degree $\deg^{+}(v)$ counts the number of edges leaving vertex $v$, while in-degree $\deg^{-}(v)$ counts the number of edges entering $v$. These values capture different aspects of connectivity in directed graphs.
• A self-loop at vertex $v$ contributes 1 to both $\deg^{+}(v)$ and $\deg^{-}(v)$ since the edge both leaves and enters the same vertex. This differs from undirected graphs where a loop contributes 2 to the degree.
• The sum of all out-degrees equals the sum of all in-degrees, both equaling the total number of edges: $\sum_v \deg^{+}(v) = \sum_v \deg^{-}(v) = |E|$. This fundamental property reflects that every directed edge has exactly one source and one target.
• Vertices with high out-degree but low in-degree act as sources of information or influence, while vertices with high in-degree but low out-degree act as sinks. Understanding these patterns helps analyze directed networks like web pages or social media.