Another class of centrality measures takes a geometric approach to identifying important vertices, relying on geodesic paths between pairs of vertices. Notably, geodesic distances are not metric— they do not obey the triangle inequality—which means applying our (Euclidean) intuition may provide incorrect interpretations of the results. In many cases, the most central vertices under these measur...

The central vertices in complex networks are of particular interest because they might play the role of organizational hubs. Here, we consider three different geometric centrality measures, excentricity, status, and centroid value, that were originally used in the context of resource placement problems. We show that these quantities lead to useful descriptions of the centers of biological netwo...

Betweenness centrality is a metric that seeks to quantify a sense of the importance of a vertex in a network graph in terms of its ‘control’ on the distribution of information along geodesic paths throughout that network. This quantity however does not capture how different vertices participate together in such control. In order to allow for the uncovering of finer details in this regard, we in...

Betweenness centrality is a measure based on the overall amount of shortest paths passing through given vertex. A graph betweenness-uniform if all its vertices have same betweenness centrality. We study properties graphs. In particular, we show that every connected either cycle or 3-connected graph. Also, uniform graphs high maximal degree small diameter.

Financial market has been investigated from many perspectives. The recent emerging financial network methods model the market as a network. The edge between two vertices, or two equities are modeled as the correlation coefficient of the return on the prices of two equities in a period. A common question one can pose is that how can we determine the importance of an equity in the market. From a ...

Processing large graphs is an emerging and increasingly important computation in a variety of application domains, from social networking to genomics and marketing. One of the important and computationally challenging structural graph metrics is node betweenness centrality, a measure of influence of a node in the graph. The best known algorithm for computing exact betweenness centrality runs in...

Betweenness is a well-known centrality measure that ranks the nodes of a network according to their participation in shortest paths. Because exact computations are prohibitive in large networks, several approximation algorithms have been proposed. Besides that, recent years have seen the publication of dynamic algorithms for efficient recomputation of betweenness in networks that change over ti...

Let G be a connected graph, suppose that v is a vertex of G, and denote the subgraph formed from G by deleting vertex v by G \ v. Denote the algebraic connectivities of G and G \ v by α(G) and α(G \ v), respectively. In this paper, we consider the functions φ(v) = α(G) − α(G \ v) and κ(v) = α(G\v) α(G) , provide attainable upper and lower bounds on both functions, and characterise the equality ...