For any finite directed multigraph, the branching ratio of a node's input tree exists and equals the Perron eigenvalue of the subgraph of all nodes that can reach it.
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Branching Ratios of Input Trees for Directed Multigraphs
For any finite directed multigraph, the branching ratio of a node's input tree exists and equals the Perron eigenvalue of the subgraph of all nodes that can reach it.