pseudo clique with at least connectivity x and maximum weight of the nodes
Your problem is NP hard, because with $x=1$ (real clique) and the same weight for each node you get the  maximum clique problem, which is NP complete. You can try to find a reduction in order to use a SAT solver, or just enumerate the $2^N$ subgraphs.

Furthermore, unless P = NP, there can be no polynomial time algorithm that approximates the maximum clique to within a factor better than $O(n^{1 − \varepsilon})$, for any $\varepsilon &gt; 0$. Wikipédia

We can't even find a good probabilistic algorithm, unless NP is in BPP. 

If your $x$ is fixed and $&lt;1$, look at the following article:

An Efficient Algorithm for Solving Pseudo Clique Enumeration Problem.  Takeaki Uno.  Algorithmica, January 2010, Volume 56, Issue 1, pp 3-16.

It considers a variant of the problem with no weights, but you can substitute each vertex by a clique of size square root of its weight, if the weights are not too big, and then apply their techniques.