Uniform Edge Betweenness Centrality

On the number of geodesics of Petersen graph $GP(n,2)$

In any network, the interconnection of nodes by means of geodesics and the number of geodesics existing between nodes are important. There exists a class of centrality measures based on the number of geodesics passing through a vertex. Betweenness centrality indicates the betweenness of a vertex or how often a vertex appears on geodesics between other vertices. It has wide applications in the analysis of networks. Consider $GP(n,k)$. For each $n$ and $k \,(n > 2k)$, the generalized Petersen graph $GP ( n , k )$ is a trivalent graph with vertex set $\{ u_ i ,\, v_ i \,|\, 0 \leq i \leq n - 1 \}…

图的介数中心

Betweenness centers of graphs

本文研究图的介数中心与外围，给出其位于单一图块的充分条件，证明任意图可作为某图的介数中心子图，并证明树的介数中心含于路径，且枚举了至多20阶的树。

Constructions of betweenness-uniform graphs from trees

Betweenness centrality is a measure of the importance of a vertex x inside a network based on the fraction of shortest paths passing through x. We study a blow-up construction that has been shown to produce graphs with uniform distribution of betweenness. We disprove the conjecture about this procedure's universality by showing that trees with a diameter at least three cannot be transformed into betweenness-uniform by the blow-up construction. It remains open to characterize graphs for which the blow-up construction can produce betweenness-uniform graphs.

Paths and flows for centrality measures in networks

We consider the number of paths that must pass through a subset $X$ of vertices of a network $N$ in a maximum sequence of arc-disjoint paths connecting two vertices $y$ and $z$. We show that when $X$ is a singleton, that number equals the difference between the maximum flow value from $y$ to $z$ in $N$ and the maximum flow value from $y$ to $z$ in the network obtained by $N$ setting to zero the capacities of arcs incident to $X$. That fact theoretically justifies the common identification of those two concepts in network literature. We also show that the same equality does not hold when $|X|\g…

About the Discriminant Power of the Subgraph Centrality and Other Centrality Measures About the Discriminant Power of the Subgraph Centrality and Other Centrality Measures(Working paper)

The discriminant power of centrality indices for the degree, eigenvector, closeness, betweenness and subgraph centrality is analyzed. It is defined by the number of graphs for which the standard deviation of the centrality of its nodes is zero. On the basis of empirical analysis it is concluded that the subgraph centrality displays better discriminant power than the rest of centralities. We also propose some new conjectures about the types of graphs for which the subgraph centrality does not discriminate among nonequivalent nodes.