Critical threshold for average weights () on networks with specified network size () and average degree ().

<p>Critical threshold for average weights () on networks with specified network size () and average degree ().</p

Illustration of representing the weighted network (a) by an unweighted network with virtual nodes (b).

<p>Illustration of representing the weighted network (a) by an unweighted network with virtual nodes (b).</p

Running time of the VN algorithm and Brandes' algorithm.

<p>(a) Networks with average degree and , 1% of the network edges are weighted with ; (b) Networks with average degree , all edges are weighted with .</p