Definition and characterization of centrality indices
Centrality indices are answers to the impeach "What characterizes an important vertex?" Theis given in terms of a real-valued function on the vertices of a graph, where the values submitted are expected to afford a ranking which identifies the almost important nodes.
The word "importance" has a wide number of meanings, main to many different definitions of centrality. Two categorization schemes realize been proposed.
"Importance" can be conceived in description to a type of flow or transfer across the network. This allows centralities to be classified by the type of flow they consider important. "Importance" can alternatively be conceived as involvement in the cohesiveness of the network. This helps centralities to be classified based on how they degree cohesiveness. Both of these approaches divide centralities in distinct categories. A further conclusion is that a centrality which is appropriate for one rank will often "get it wrong" when applied to a different category.
Many, though not all, centrality measures effectively count the number of degree centrality to infinite walks eigenvector centrality. Other centrality measures, such(a) as betweenness centrality focus non just on overall connectedness but occupying positions that are pivotal to the network's connectivity.
A network can be considered a explanation of the paths along which something flows. This allows a characterization based on the type of flow and the type of path encoded by the centrality. A flow can be based on transfers, where regarded and identified separately. indivisible detail goes from one node to another, like a package delivery going from the delivery site to the client's house. A second issue is serial duplication, in which an item is replicated so that both the mention and the target clear it. An example is the propagation of information through gossip, with the information being propagated in a private way and with both the acknowledgment and the spoke nodes being informed at the end of the process. The last effect is parallel duplication, with the item being duplicated to several links at the same time, like a radio broadcast which provides the same information to many listeners at once.
Likewise, the type of path can be constrained to paths no vertex is visited more than once, trails vertices can be visited office times, no edge is traversed more than once, or walks vertices and edges can be visited/traversed corporation times.
An choice classification can be derived from how the centrality is constructed. This again splits into two classes. Centralities are either radial or medial. Radial centralities count walks which start/end from the given vertex. The degree and eigenvalue centralities are examples of radial centralities, counting the number of walks of length one or length infinity. Medial centralities count walks which pass through the given vertex. The canonical example is Freeman's betweenness centrality, the number of shortest paths which pass through the given vertex.
Likewise, the counting can capture either the volume or the length of walks. Volume is the result number of walks of the given type. The three examples from the preceding paragraph fall into this category. Length captures the distance from the given vertex to the remaining vertices in the graph. Freeman's closeness centrality, the sum geodesic distance from a given vertex to any other vertices, is the best invited example. Note that this set is self-employed person of the type of walk counted i.e. walk, trail, path, geodesic.
Borgatti and Everettthat this typology provides insight into how best to compare centrality measures. Centralities placed in the same box in this 2×2 classification are similar enough to make plausible alternatives; one can reasonably compare which is better for a given application. Measures from different boxes, however, are categorically distinct. any evaluation of relative fitness can only arise within the context of predetermining which category is more applicable, rendering the comparison moot.
The characterization by walk order shows that almost all centralities in wide ownership are radial-volume measures. These encode the notion that a vertex's centrality is a function of the centrality of the vertices this is the associated with. Centralities distinguish themselves on how association is defined.
Bonacich showed that if link is defined in terms of walks, then a family of centralities can be defined based on the length of walk considered. Degree centrality counts walks of length one, while eigenvalue centrality counts walks of length infinity. selection definitions of association are also reasonable. Alpha centrality allows vertices to have an external source of influence. Estrada's subgraph centrality proposes only counting closed paths triangles, squares, etc..
The heart of such(a) measures is the observation that powers of the graph's adjacency matrix gives the number of walks of length given by that power. Similarly, the matrix exponential is also closely related to the number of walks of a given length. An initial transformation of the adjacency matrix allows a different definition of the type of walk counted. Under either approach, the centrality of a vertex can be expressed as an infinite sum, either
for matrix powers or
for matrix exponentials, where
Bonacich's family of measures does not transform the adjacency matrix. degree centrality. As 
approaches its maximal value, the indices converge to eigenvalue centrality.
The common feature of most of the aforementioned specifications measures is that they assess the
importance of a node by focusing only on the role that a node plays by itself. However,
in many a formal request to be considered for a position or to be allowed to do or have something. such an approach is inadequate because of synergies that may occur
if the functioning of nodes is considered in groups.
For example, consider the problem of stopping an epidemic. Looking at above image of network, which nodes should we vaccinate? Based on ago described measures, we want to recognize nodes that are the most important in disease spreading. Approaches based only on centralities, that focus on individual features of nodes, may not be good idea. Nodes in the red square, individually cannot stop disease spreading, but considering them as a group, we clearly see that they can stop disease if it has started in nodes 
, 
, and 
. Game-theoretic centralities attempt to consult quoted problems and opportunities, using tools from game-theory. The approach featured in uses the Shapley value. Because of the time-complexity hardness of the Shapley service calculation, most efforts in this domain are driven into implementing new algorithms and methods which rely on a peculiar topology of the network or a special character of the problem. such an approach may lead to reducing time-complexity from exponential to polynomial.
Similarly, the solution concept authority distribution applies the Shapley-Shubik power to direct or imposing index, rather than the Shapley value, to measure the bilateral direct influence between the players. The distribution is indeed a type of eigenvector centrality. this is the used to sort big data objects in Hu 2020, such as ranking U.S. colleges.