A new betweenness centrality measure based on an algorithm for ranking the nodes of a network

We propose and discuss a new centrality index for urban street patterns represented as networks in geographical space. This centrality measure, that we call ranking-betweenness centrality, combines the idea behind the random-walk betweenness centrality measure and the idea of ranking the nodes of a network produced by an adapted PageRank algorithm. We initially use a PageRank algorithm in which we are able to transform some information of the network that we want to analyze into numerical values. Numerical values summarizing the information are associated to each of the nodes by means of a data matrix. After running the adapted PageRank algorithm, a ranking of the nodes is obtained, according to their importance in the network. This classification is the starting point for applying an algorithm based on the random-walk betweenness centrality. A detailed example of a real urban street network is discussed in order to understand the process to evaluate the ranking-betweenness centrality proposed, performing some comparisons with other classical centrality measures.

New glimpses on convex infinite optimization duality

Given a convex optimization problem (P) in a locally convex topological vector space X with an arbitrary number of constraints, we consider three possible dual problems of (P), namely, the usual Lagrangian dual (D), the perturbational dual (Q), and the surrogate dual (Δ), the last one recently introduced in a previous paper of the authors (Goberna et al., J Convex Anal 21(4), 2014). As shown by simple examples, these dual problems may be all different. This paper provides conditions ensuring that inf(P)=max(D), inf(P)=max(Q), and inf(P)=max(Δ) (dual equality and existence of dual optimal solutions) in terms of the so-called closedness regarding to a set. Sufficient conditions guaranteeing min(P)=sup(Q) (dual equality and existence of primal optimal solutions) are also provided, for the nominal problems and also for their perturbational relatives. The particular cases of convex semi-infinite optimization problems (in which either the number of constraints or the dimension of X, but not both, is finite) and linear infinite optimization problems are analyzed. Finally, some applications to the feasibility of convex inequality systems are described.