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Braess like Paradox in a Small World Network

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.