A network efficiency measure for congested networks

In this paper, we propose a network efficiency measure for congested networks, that captures demands, costs, flows, and behavior. The network efficiency/performance measure can identify which network components, that is, nodes and links, have the greatest impact in terms of their removal, due to, for example, natural disasters, structural failures, terrorist attacks, etc., and, hence, are important from both vulnerability as well as security standpoints. The new measure is applied to the Braess paradox network in which the demands are varied over the horizon and explicit formulae are derived for the importance values of the network nodes and links. This measure is applicable to such congested networks as urban transportation networks and the Internet. Introduction. – In this paper, we propose a new network efficiency measure that is appropriate for congested networks, that is, networks in which the cost associated with a link is an increasing function of the flow on the link. It is well-known that congestion is a fundamental problem in a variety of modern network systems, including urban transportation networks, electric power generation and distribution networks, as well as the Internet (cf. [1], [2], [3], [4]). Hence, an appropriate network efficiency measure for such network systems can have wide application. In addition, it can be used, as we show in this paper, to identify which nodes or links are critical or most important in a congested network in that their removal will result in a large relative efficiency drop. Furthermore, the identified important network components are those, clearly, that should be better protected or secured since their removal has a greater impact on the network system. This paper is organized as follows. We first briefly review the well-known traffic network equilibrium model, which has been applied to congested urban transportation networks and the Internet, and which is also closely related to electric power generation and distribution networks (see, e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]). We then present the new network efficiency measure for congested networks. Subsequently, we revisit the Braess Paradox [11] (see also [12]), which is as relevant to the Internet as it is to transportation networks and we apply the network measure to the case in which the demand is varied over the nonnegative real line. Explicit formulae are obtained for both the efficiency measure as well as for the importance identification of the nodes and links (along with their rankings). We conclude the paper with a summary of the results and suggestions for future research. Traffic Network Equilibrium Model. – We now recall the traffic network equilibrium model ( [1] [10]), which is widely used and applied in practice. Consider a network G with the set of directed links L with nL elements and the set of origin/destination (O/D) pairs W with nW elements. We denote the set of acyclic paths joining O/D pair w by Pw. The set of (acyclic) paths for all O/D pairs is denoted by P and there are nP paths in the network. Links are denoted by a, b, etc; paths by p, q, etc., and O/D pairs by w1, w2, etc. We assume that the demand dw is known for all w ∈ W . We denote the nonnegative flow on path p by xp and the flow on link a by fa and we group the path flows into the vector x ∈ RP + and the link flows into the vector f ∈ R nL + . The following conservation of flow equations must hold: ∑ p∈Pw xp = dw, ∀w ∈ W, (1) which means that the sum of path flows on paths connecting each O/D pair must be equal to the demand for that O/D pair. The link flows are related to the path flows, in turn,