Efficiency and Betweenness Centrality of Graphs and some Applications

The distance $d_{G}(i,j)$ between any two vertices $i$ and $j$ in a graph $G$ is the minimum number of edges in a path between $i$ and $j$. If there is no path connecting $i$ and $j$, then $d_G(i,j)=infty$. In 2001, Latora and Marchiori introduced the measure of efficiency between vertices in a graph. The efficiency between two vertices $i$ and $j$ is defined to be $in_{i,j}=frac{1}{d_G(i,j)}$ for all $ineq j$. The textit{global efficiency} of a graph is the average efficiency over all $ineq j$. The {it power of a graph} $G^m$ is defined to be $V(G^m)=V(G)$ and $E(G^m)={(u,v)|d_G(u,v)le m}$. In this paper we determine the global efficiency for path power graphs $P_n^m$, cycle power graphs $C_n^m$, complete multipartite graphs $K_{m,n}$, star and subdivided star graphs, and the Cartesian products $K_{n}times P_{m}^{t}$, $K_{n}times C_{m}^{t}$, $K_{m}times K_{n}$, and $P_{m}times P_{n}$. The concept of global efficiency has been applied to optimization of transportation systems and brain connectivity. We show that star-like networks have a high level of efficiency. We apply these ideas to an analysis of the Metropolitan Atlanta Rapid Transit Authority (MARTA) Subway system, and show this network is 82% as efficient as a network where there is a direct line between every pair of stations. From BOLD fMRI scans we are able to partition the brain with consistency in terms of functionality and physical location. We also find that football players who suffer the largest number of high-energy impacts experience the largest drop in efficiency over a season. Latora and Marchiori also presented two local properties. The textit{local efficiency} $E_{loc}=frac{1}{n}sumlimits_{iin V(G)}E_{glob}left(G_{i}right) $ is the average of the global efficiencies over the subgraphs $G_{i}$, the subgraph induced by the neighbors of $i$. The clustering coefficient of a graph $G$ is defined to be $CC(G)=frac{1}{n}sumlimits_{i}C_{i}$ where $C_{i}=|E(G_i)|/binom{|V(G_i)|}{2}$ is a degree of completeness of $G_{i}$. In this paper, we compare and contrast the two quantities, local efficiency and clustering coefficient. Betweenness centrality is a measure of the importance of a vertex to the optimal paths in a graph. Betweenness centrality of a vertex is defined as $bc(v)=sum_{x,y}frac{sigma_{xy}(v)}{sigma_{xy}}$ where $sigma_{xy}$ is the number of unique paths of shortest length between vertices $x$ and $y$. $sigma_{xy}(v)$ is the number of optimal paths that include the vertex $v$. In this paper, we examined betweenness centrality for vertices in $C_n^m$. We also include results for subdivided star graphs and $C_3$ star graphs. A graph is said to have unique betweenness centrality if $bc(v_i)=bc(v_j)$ implies $i=j$: the betweenness centrality function is injective over the vertices of $G$. We describe the betweenness centrality for vertices in ladder graphs, $P_2times P_n$. An appended ladder graph $U_n$ is $P_2times P_n$ with a pendant vertex attached to an tbl endtbr. We conjecture that the infinite family of appended graphs has unique betweenness centrality