Optimizing algebraic connectivity by edge rewiring

Robustness in complex networks is an ongoing research effort that seeks to improve the connectivity of networks against attacks and failures. Among other measures, algebraic connectivity has been used to characterize processes such as damped oscillation of liquids in connected pipes. Similar characterizations include the number of edges necessary to disconnect a network: the larger the algebraic connectivity, the larger the number of edges required to disconnect a network and hence, the more robust a network. In this paper, we answer the question, ''Which edge can we rewire to have the largest increase in algebraic connectivity?''. Furthermore, we extend the rewiring of a single edge to rewiring multiple edges to realize the maximal increase in algebraic connectivity. The answer to this question above can provide insights to decision makers within various domains such as communication and transportation networks, who seek an efficient solution to optimize the connectivity and thus increase the robustness of their networks. Most importantly, our analytical and numerical results not only provide insights to the number of edges to rewire, but also the location in the network where these edges would effectuate the maximal increase in algebraic connectivity and therefore, enable a maximal increase in robustness. Complex networks including biological, power grid, Internet, and transportation networks can range from hundreds to millions of vertices, and their characterization provides mechanisms to enhance their performance and realize their impact to our standard of living [1–3]. For example, the US freight network (a classic example of a transportation network), transports over $200 billion worth of products every year. The ailing transportation infrastructure is unable to sustain the economical needs of the US and a predicted increase in population (an estimated 70 million additional individuals by 2035) and trade only exacerbates the situation. To further substantiate this, in 2009 Americans spent 4.2 billion hours sitting in traffic: a cost of $115 billion dollars [4]. From a post analysis of the transportation infrastructure report for 2011, the deployment of new roadways contributes to an overall solution to alleviate traffic congestion. For this example, our results provide insights to select endpoints for roadways to increase connectivity and robustness of the transportation network. These results can apply similarly to power grid utility companies such as AEP that propose an increase of 19,000 miles to the national transmission line network, in an effort to upgrade the aging power grid infrastructure [5]. Algebraic connectivity is a …