An entropy maximization problem in shortest path routing networks

Abstract

In the context of an IP network, we investigate an interesting case of the inverse shortest path problem using the concept of network centrality. For a given network, the centrality distribution associated with the links of a network can be determined based on the number of shortest paths passing through each link. An entropy measure for this distribution is defined, and we then forumulate the inverse shortest problem in terms of maximizing this entropy. We then obtain a centrality distribution that is as broadly distributed as possible subject to the topology constraints. An appropriate change in the weight of a link alters the number of shortest paths that pass through it, thereby modifying the centrality distribution. The idea is to obtain a centrality distribution that maximizes the entropy. This problem is shown to be NP-hard, and a heuristic approach is proposed. An application to handling link failure scenarios in Open Shortest Path First routing is discussed.

A Survey of Information Entropy Metrics for Complex Networks

“…A number of authors use information entropy as a network or graph metric [ 15 , 16 , 18 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 31 , 34 , 35 , 37 , 38 , 39 , 40 , 41 , 50 , 53 ]. However, a distinction should be made: while all of these works measure the entropy of the network, some use graph entropy to indirectly measure the centrality of the nodes (see Section 4.2 ).…”

“…In fact, probability distribution definitions while using information functionals are used in several graph entropy metrics. For example, information functionals are based on edge or node betweenness centrality [ 24 , 25 , 34 , 50 , 53 ] distances to a given vertex [ 28 ], degree, degree power or probability distribution of degrees [ 31 , 41 ], paths or paths’ length [ 16 , 35 ], and closeness or eigenvector centrality [ 53 ].…”

“…Chellappan et al [ 24 , 25 , 34 ] used edge betweenness centrality instead of the node betweennnes. They defined , where the double-bullet notation indicates all pairs of nodes.…”

“…The authors posed that high edge betweenness entropy is indicative of a high diversity of paths in tactical communication networks [ 24 ]. They further proposed the use of entropy maximization and betweenness entropy in order to make communications routing decentralized [ 25 ] and handle single edge failures [ 34 ]. Similarly, Zhang et al [ 50 ] used edge betweenness centrality in order to analyze transportation networks in Xiamen.…”