# The terminology for a node's number of in-links in weighted directed graph

Given a weighted graph $G = (V,E)$, the in-degree of a node is defined as $k_{in}(i) = \sum_{j:j \rightarrow i} A_{ji}$ where $A_{ji}$ is the weight of edge from node $j$ to $i$. My question is if there are $m$ nodes pointing to $i$, what's the terminology for $m$? Is it the number of in-links?

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The quantity you are asking about is usually called the "fan-in" of node $i$. The dual quantity which counts the number of nodes to which $i$ points is called the "fan-out". The terms are standard at least in the circuit design and computer science literature, but I can't seem to find them in graph theory texts at the moment. Searching google for something like directed graph "fan in" will reveal many papers that use the term.
I've seen papers where $m$ is called the in-degree and $k_{in}$ is called something else (such as weighted in-degree). Using in-degree and fan-in as Vel Nias suggests would be fine too. But you need to define it as there isn't universal agreement on how to interpret these names in the weighted graph context.
Thank you Brendan. I am also confused by the different definitions in differnt papers. I will take your advice to explicitly define $k_{in}$ and "fan-in" to avoid ambiguity. – zxia31 Jan 27 '13 at 19:14