  Braess \cite{1} has been studied about a traffic flow on a diamond type
network and found that introducing new edges to the networks always does not
achieve the efficiency. Some researchers studied the Braess' paradox in similar
type networks by introducing various types of cost functions. But whether such
paradox occurs or not is not scarcely studied in complex networks. In this
article, I analytically and numerically study whether Braess like paradox
occurs or not on Dorogovtsev-Mendes network\cite{2}, which is a sort of small
world networks. The cost function needed to go along an edge is postulated to
be equally identified with the length between two nodes, independently of an
amount of traffic on the edge. It is also assumed the it takes a certain cost
$c$ to pass through the center node in Dorogovtsev-Mendes network. If $c$ is
small, then bypasses have the function to provide short cuts. As result of
numerical and theoretical analyses, while I find that any Braess' like paradox
will not occur when the network size becomes infinite, I can show that a
paradoxical phenomenon appears at finite size of network.

Physics and Society;Social and Information Networks