I did not reach a solution for the determination of upper and lower limits of the following function:
It applies to weighted unoriented network graphics, with not nodes. The formula is calculated for the node I. With $ alpha $ being a coefficient which can be chosen greater than zero. wij is the weight of the arc connection nodes I and j (with a j for all the neighbors in the network of I). The logarithm numerator represents the sum of the weights of all the arcs in the graph.
So basically we add the weights of all the arcs connected to the node I, by multiplying them before by the logarithmic term. Please note that for $ alpha> 1 $ the contribution of an arc to the final score can be negative.
Can you help me find a maximum and a minimum?
It should be expressed in terms of not and $ alpha $ which are fixed parameters. We also know the value of the maximum wij.