<p>Several methods were suggested.</p>

<p>One simple yet effective method was suggested in <a href="https://arxiv.org/abs/0803.0476" rel="nofollow noreferrer">Fast unfolding of communities in large networks</a> (Blondel et al., 2008). It supports weighted networks. Quoting from the abstract:</p>

<blockquote>
  <p>We propose a simple method to extract the community structure of large
  networks. Our method is a heuristic method that is based on modularity
  optimization. It is shown to outperform all other known community
  detection method in terms of computation time. Moreover, the quality
  of the communities detected is very good, as measured by the so-called
  modularity.</p>
</blockquote>

<p>Quoting from the paper:</p>

<blockquote>
  <p>We now introduce our algorithm that finds high modularity partitions
  of large networks in short time and that unfolds a complete
  hierarchical community structure for the network, thereby giving
  access to different resolutions of community detection.</p>
</blockquote>

<p>So it supposed to work well for complete graph, but you should better check it.</p>

<p>A C++ implementation is available <a href="https://sites.google.com/site/findcommunities/" rel="nofollow noreferrer">here</a> (now maintained <a href="https://sourceforge.net/projects/louvain/" rel="nofollow noreferrer">here</a>).</p>

<p>Your other idea - using weight-threshold - may prove as a good pre-processing step, especially for algorithms which won't partition complete graphs. I believe it is best to set it to some percentile (e.g. to the median) of the weights.</p>
Community Detection in complete and weighted networks
<p>I do have a complete network graph where every vertex is connected with each other and they only differ in form of their different weights. A example network would be: a trade network, where every country is connected with each other somehow and only differ in form of different trading volumina.</p>

<p>Now the question is how I could perform a community detection in that form of network. The usual suspects (algorithm) are only able to perform in either unweighted or incomplete networks well. The main problem is that the geodesic is everywhere the same. </p>

<p>Two option came into my mind:</p>

<ol>
<li>Cut the network into smaller pieces by cutting them at a certain "weight-threshold-level"</li>
<li>Or use a hierarchical cluster algorithm to turn the whole network into a blockmodel. But I think the problem "no variance in geodesic terms" will remain.</li>
</ol>