The Braess' paradox for pendent twins

Abstract

The Kemeny's constant κ(G) of a connected undirected graph G can be interpreted as the expected transit time between two randomly chosen vertices for the Markov chain associated with G. In certain cases, inserting a new edge into G has the counter-intuitive effect of increasing the value of κ(G). In the current work we identify a large class of graphs exhibiting this "paradoxical" behavior -namely, those graphs having a pair of twin pendant vertices. We also investigate the occurrence of this phenomenon in random graphs, showing that almost all connected planar graphs are paradoxical. To establish these results, we make use of a connection between the Kemeny's constant and the resistance distance of graphs.

Families of graphs with twin pendent paths and the Braess edge

“…Kirkland and Zeng [13] provides a particular family of trees, with a vertex adjacent to two pendent vertices (such two vertices are called twin pendent vertices), such that inserting an edge between the twin pendent vertices causes Kemeny's constant to increase. Furthermore, Ciardo [8] extends the result to all connected graphs with twin pendent vertices. Unlike the works [13] and [8], Hu and Kirkland [11] establishes equivalent conditions for complete multipartite graphs and complete split graphs to have every non-edge as a Braess edge.…”

“…Furthermore, Ciardo [8] extends the result to all connected graphs with twin pendent vertices. Unlike the works [13] and [8], Hu and Kirkland [11] establishes equivalent conditions for complete multipartite graphs and complete split graphs to have every non-edge as a Braess edge.…”

On Kemeny's constant for trees with fixed order and diameter

“…. defined by (8) satisfies the requirements of Proposition 6.2. The result follows since, clearly, those requirements cannot be satisfied by two distinct sequences.…”

“…As a consequence, one can control the long-run diffusion rate of the information flow by performing modifications on the network that lead to the desired change in the value of Kemeny's constant. Particularly interesting in this regard, is the phenomenon known as the Braess' paradox for graphs, which occurs when adding a new connection in the network has the counter-intuitive effect of increasing the value of Kemeny's constant instead of decreasing it [8][9][10].…”

“…Step (II) was performed via MATLAB, by applying multivariate regression on a sample of 100 randomly generated broom-stars, and it produced the candidate expression (8). To show that this candidate is correct (step (III)), we only need to check that it satisfies the recursive relation of Proposition 6.2.…”