I am checking correlations between arrays of values. These values are the weights of a network and some features of the nodes of the same network. I often get good correlations (sometimes > 0.6), but I always get an incredibly low p-value, usually between 1.0664e-48 and 9.6488e-181. So it's basically a 0, and this rings a few alarms in my mind, but I can't find an explanation.
How is that possible? How should I interpret these values?
EDIT: I'll try to explain myself better by simplifying the question. If I check the Pearson correlation (using a simple function in Python) between two vectors of length ~1300, and I get a good r value and an incredibly low p value, how should I interpret this?
I know what a p value is, but I never got such a low result. This is so low that it makes me wonder if there is something wrong in the analysis, so I was asking more experienced people what I could look for.