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# Closeness (graph theory)

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### Closeness (graph theory)

In graph theory and network analysis, centrality of a vertex measures its relative importance within a graph. Applications include how influential a person is within a social network, how important a room is within a building (space syntax), and how well-used a road is within an urban network. There are four main measures of centrality: degree, betweenness, closeness, and eigenvector. Centrality concepts were first developed in social network analysis, and many of the terms used to measure centrality reflect their sociological origin.[1]

## Definition and characterization of centrality indices

Next to the following classic centrality indices, there are dozens of other more specialized centrality indices. Despite its intuitive notion there is not yet a definition or characterization of centrality indices which captures all of them.[2] A very loose definition of a centrality index is the following:

A centrality index is a real-valued function on the nodes of a graph. It is a structural index, i.e., if $G$ and $H$ are two isomorphic graphs and $\Phi$ is the mapping from the vertex set $V\left(G\right)$ of $G$ to V(H), then the centrality of a vertex $v$ of $G$ must be the same as the centrality of $\Phi\left(v\right)$ in $H$. Conventionally, the higher the centrality index of a node, the higher its perceived centrality in the graph.[3] This definition comprises all classic centrality measures but not all measures that fulfill this definition would be accepted as centrality indices.

Borgatti and Everett summarize that centrality indices measure the position of a node along a predefined set of walks. They characterize centrality indices along four dimensions: the set of walks, whether the length or the number of these walks is considered, the position of the node on the walks (at the start=radial; in the middle=medial), and how the numbers assigned to the paths are summarized in the measure (average, median, weighted sum, ...).[2] This leads to a characterization by the way a centrality index is calculated. In a different characterization, Borgatti differentiates the centrality indices by what type of paths they consider and which type of network flow they imply.[4] The latter characterizes the centrality indices by the quality with which they predict which node is most central for a given network flow process. This characterization thus provides guidance on when to use which centrality index.

### Categorization

Reachability, Amount of flow, Vitality, Feedback; see.[5]

## Degree centrality

Main article: Degree (graph theory)

Historically first and conceptually simplest is degree centrality, which is defined as the number of links incident upon a node (i.e., the number of ties that a node has). The degree can be interpreted in terms of the immediate risk of a node for catching whatever is flowing through the network (such as a virus, or some information). In the case of a directed network (where ties have direction), we usually define two separate measures of degree centrality, namely indegree and outdegree. Accordingly, indegree is a count of the number of ties directed to the node and outdegree is the number of ties that the node directs to others. When ties are associated to some positive aspects such as friendship or collaboration, indegree is often interpreted as a form of popularity, and outdegree as gregariousness.

The degree centrality of a vertex $v$, for a given graph $G:=\left(V,E\right)$ with $|V|$ vertices and $|E|$ edges, is defined as

$C_D\left(v\right)= \text\left\{deg\right\}\left(v\right)$

Calculating degree centrality for all the nodes in a graph takes $\Theta\left(V^2\right)$ in a dense adjacency matrix representation of the graph, and for edges takes $\Theta\left(E\right)$ in a sparse matrix representation.

Sometimes the interest is in finding the centrality of a graph within a graph. The definition of centrality on the node level can be extended to the whole graph. Let $v*$ be the node with highest degree centrality in $G$. Let $X:=\left(Y,Z\right)$ be the $Y$ node connected graph that maximizes the following quantity (with $y*$ being the node with highest degree centrality in $X$):

$H= \displaystyle\left\{\sum^\left\{|Y|\right\}_\left\{j=1\right\}\left\{C_D\left(y*\right)-C_D\left(y_j\right)\right\}\right\}$

Correspondingly, the degree centrality of the graph $G$ is as follows:

$C_D\left(G\right)= \frac\left\{\displaystyle\left\{\sum^\left\{|V|\right\}_\left\{i=1\right\}\right\}\right\}\left\{H\right\}$

The value of $H$ is maximized when the graph $X$ contains one central node to which all other nodes are connected (a star graph), and in this case $H=\left(n-1\right)\left(n-2\right)$.

## Closeness centrality

In connected graphs there is a natural distance metric between all pairs of nodes, defined by the length of their shortest paths. The farness of a node s is defined as the sum of its distances to all other nodes, and its closeness is defined as the inverse of the farness.[6] Thus, the more central a node is the lower its total distance to all other nodes. Closeness can be regarded as a measure of how long it will take to spread information from s to all other nodes sequentially.[7]

In the classic definition of the closeness centrality, the spread of information is modeled by the use of shortest paths. This model might not be the most realistic for all types of communication scenarios. Thus, related definitions have been discussed to measure closeness, like the random walk closeness centrality introduced by Noh and Rieger (2004). It measures the speed with which randomly walking messages reach a vertex from elsewhere in the network—a sort of random-walk version of closeness centrality.[8]

The information centrality of Stephenson and Zelen (1989) is another closeness measure, which bears some similarity to that of Noh and Rieger. In essence it measures the harmonic mean length of paths ending at a vertex i, which is smaller if i has many short paths connecting it to other vertices.[9]

Note that by definition of graph theoretic distances, the classic closeness centrality of all nodes in an unconnected graph would be 0. In a work by Dangalchev (2006) relating network vulnerability, the definition for closeness is modified such that it can be calculated more easily and can be also applied to graphs which lack connectivity:[10]

$C_C\left(v\right)=\sum_\left\{t \in V\setminus v\right\}2^\left\{-d_G\left(v,t\right)\right\}.$

Another extension to networks with disconnected components has been proposed by Opsahl (2010).[11]

## Betweenness centrality

Main article: Betweenness centrality

Betweenness is a centrality measure of a vertex within a graph (there is also edge betweenness, which is not discussed here). Betweenness centrality quantifies the number of times a node acts as a bridge along the shortest path between two other nodes. It was introduced as a measure for quantifying the control of a human on the communication between other humans in a social network by Linton Freeman.[12] In his conception, vertices that have a high probability to occur on a randomly chosen shortest path between two randomly chosen vertices have a high betweenness.

The betweenness of a vertex $v$ in a graph $G:=\left(V,E\right)$ with $V$ vertices is computed as follows:

1. For each pair of vertices (s,t), compute the shortest paths between them.
2. For each pair of vertices (s,t), determine the fraction of shortest paths that pass through the vertex in question (here, vertex v).
3. Sum this fraction over all pairs of vertices (s,t).

More compactly the betweenness can be represented as:[13]

$C_B\left(v\right)= \sum_\left\{s \neq v \neq t \in V\right\}\frac\left\{\sigma_\left\{st\right\}\left(v\right)\right\}\left\{\sigma_\left\{st\right\}\right\}$

where $\sigma_\left\{st\right\}$ is total number of shortest paths from node $s$ to node $t$ and $\sigma_\left\{st\right\}\left(v\right)$ is the number of those paths that pass through $v$. The betweenness may be normalised by dividing through the number of pairs of vertices not including v, which for directed graphs is $\left(n-1\right)\left(n-2\right)$ and for undirected graphs is $\left(n-1\right)\left(n-2\right)/2$. For example, in an undirected star graph, the center vertex (which is contained in every possible shortest path) would have a betweenness of $\left(n-1\right)\left(n-2\right)/2$ (1, if normalised) while the leaves (which are contained in no shortest paths) would have a betweenness of 0.

From a calculation aspect, both betweenness and closeness centralities of all vertices in a graph involve calculating the shortest paths between all pairs of vertices on a graph, which requires $\Theta\left(V^3\right)$ time with the Floyd–Warshall algorithm. However, on sparse graphs, Johnson's algorithm may be more efficient, taking $O\left(V^2 \log V + V E\right)$ time. In the case of unweighted graphs the calculations can be done with Brandes' algorithm[13] which takes $O\left(V E\right)$ time. Normally, these algorithms assume that graphs are undirected and connected with the allowance of loops and multiple edges. When specifically dealing with network graphs, oftentimes graphs are without loops or multiple edges to maintain simple relationships (where edges represent connections between two people or vertices). In this case, using Brandes' algorithm will divide final centrality scores by 2 to account for each shortest path being counted twice.[13]

## Eigenvector centrality

Eigenvector centrality is a measure of the influence of a node in a network. It assigns relative scores to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes. Google's PageRank is a variant of the Eigenvector centrality measure.[14] Another closely related centrality measure is Katz centrality.

### Using the adjacency matrix to find eigenvector centrality

For a given graph $G:=\left(V,E\right)$ with $|V|$ number of vertices let $A = \left(a_\left\{v,t\right\}\right)$ be the adjacency matrix, i.e. $a_\left\{v,t\right\} = 1$ if vertex $v$ is linked to vertex $t$, and $a_\left\{v,t\right\} = 0$ otherwise. The centrality score of vertex $v$ can be defined as:

$x_v = \frac\left\{1\right\}\left\{\lambda\right\} \sum_\left\{t \in M\left(v\right)\right\}x_t = \frac\left\{1\right\}\left\{\lambda\right\} \sum_\left\{t \in G\right\} a_\left\{v,t\right\}x_t$

where $M\left(v\right)$ is a set of the neighbors of $v$ and $\lambda$ is a constant. With a small rearrangement this can be rewritten in vector notation as the eigenvector equation

$\mathbf\left\{Ax\right\} = \left\{\lambda\right\}\mathbf\left\{x\right\}$

In general, there will be many different eigenvalues $\lambda$ for which an eigenvector solution exists. However, the additional requirement that all the entries in the eigenvector be positive implies (by the Perron–Frobenius theorem) that only the greatest eigenvalue results in the desired centrality measure.[15] The $v^\left\{th\right\}$ component of the related eigenvector then gives the centrality score of the vertex $v$ in the network. Power iteration is one of many eigenvalue algorithms that may be used to find this dominant eigenvector.[14] Furthermore, this can be generalized so that the entries in A can be real numbers representing connection strengths, as in a stochastic matrix.

## Katz centrality and PageRank

Main article: Katz centrality

Katz centrality [16] is a generalization of degree centrality. Degree centrality measures the number of direct neighbors, and Katz centrality measures the number of all nodes that can be connected through a path, while the contributions of distant nodes are penalized. Mathematically, it is defined as $x_i = \sum_\left\{k=1\right\}^\left\{\infin\right\}\sum_\left\{j=1\right\}^N \alpha^k \left(A^k\right)_\left\{ij\right\}$ where $\alpha$ is an attenuation factor in $\left(0,1\right)$.

Katz centrality can be viewed as a variant of eigenvector centrality. Another form of Katz centrality is $x_i = \alpha \sum_\left\{j =1\right\}^N a_\left\{ij\right\}\left(x_j+1\right).$ Compared to the expression of eigenvector centrality, $x_j$ is replaced by $x_j+1$.

It is shown that [17] the principal eigenvector (associated with the largest eigenvalue of $A$, the adjacency matrix) is the limit of Katz centrality as $\alpha$ approaches $1/\lambda$ from below.

PageRank satisfies the following equation $x_i = \alpha \sum_\left\{j \right\} a_\left\{ji\right\}\frac\left\{x_j\right\}\left\{L\left(j\right)\right\} + \frac\left\{1-\alpha\right\}\left\{N\right\},$ where $L\left(j\right) = \sum_\left\{j\right\} a_\left\{ij\right\}$ is the number of neighbors of node $j$ (or number of outbound links in a directed graph). Compared to eigenvector centrality and Katz centrality, one major difference is the scaling factor $L\left(j\right)$. Another difference between PageRank and eigenvector centrality is that the PageRank vector is a left hand eigenvector (note the factor $a_\left\{ji\right\}$ has indices reversed).[18]

## Centralization

The centralization of any network is a measure of how central its most central node is in relation to how central all the other nodes are.[19] The general definition of centralization for non-weighted networks was proposed by Linton Freeman (1979). Centralization measures then (a) calculate the sum in differences in centrality between the most central node in a network and all other nodes; and (b) divide this quantity by the theoretically largest such sum of differences in any network of the same degree.[19] Thus, every centrality measure can have its own centralization measure. Defined formally, if $C_x\left(p_i\right)$ is any centrality measure of point $i$, if $C_x\left(p_*\right)$ is the largest such measure in the network, and if $max \sum_\left\{i=1\right\}^\left\{N\right\} C_x\left(p_*\right)-C_x\left(p_i\right)$ is the largest sum of differences in point centrality $C_x$ for any graph of with the same number of nodes, then the centralization of the network is:[19]

$C_x=\frac\left\{\sum_\left\{i=1\right\}^\left\{N\right\} C_x\left(p_*\right)-C_x\left(p_i\right)\right\}\left\{max \sum_\left\{i=1\right\}^\left\{N\right\} C_x\left(p_*\right)-C_x\left(p_i\right)\right\}$

## Extensions

Empirical and theoretical research have extended the concept of centrality in the context of static networks to dynamic centrality[20] in the context of time-dependent and temporal networks.[21][22][23]

For generalizations to weighted networks, see Opsahl et al. (2010).[24]

The concept of centrality was extended to a group level as well. For example, Group Betweenness centrality shows the proportion of geodesics connecting pairs of non-group members that pass through the group.[25][26]