In advance I apologise for my lack of Mathematical knowledge and low quality question: I'm not trained as a mathematician, nor do I have a substantial knowledge of mathematical terminology. However, I'm eager to put my programming lines into a mathematical framework, so that's why I ended up here.
My problem is the following: Consider a semi-random network of $N$ nodes $i$ and $M$ connections, the structure of the network does not really matter here. Now I have defined the degree $k$ of a node $i$ as $$ k_i=\sum\limits_j A_{ij} $$ with $A_{ij}$ the adjacency matrix defining the connections between nodes in the network and summing over all nodes $N$.
Now I want to define hubs in the network using a hub threshold $k_h$. I actually want to count the number of hubs in my network with respect to the total number of nodes $N$ (let's call this $H$), so far I've come up with this: $$ H=\frac{1}{N}\sum\limits_i \sum\limits_{k_i>k_h} i $$ But I don't think this gives me the right answer, since say that node 10 and 11 are hub nodes, the answer would be $21/N$ instead of $2/N$. I feel like I'm very close, but I can't see it. How should I state it?
Many thanks!
I've thought about it for a while and I think I managed to come up with a solution myself using the Heaviside function: $$ H=\frac{1}{N}\sum\limits_i 1-\Theta (-k_i+k_h) $$ So a node $i$ is 'counted' as a hub node when its node degree $k_i$ is greater than threshold $k_h$. If the node degree is equal to or smaller than this threshold, the Heaviside function returns a one such that the sum becomes zero for this node.