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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.

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Thank you Vel. I googled it and this is the exact word I am looking for. "fan−in" describes the situation vividly. – zxia31 Jan 27 '13 at 19:09

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.

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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

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