7.1 C
entrality
A large volume of research on networks has been devoted to the concept ofcen- trality. This research addresses the question, “Which are the most important or central nodes in a network?” There are many possible definitions of impor- tance and there are correspondingly many centrality measures for networks. In the following several sections we describe some of the most widely used such measures.
7.1.1 Degree centrality
Perhaps the simplest centrality measure for a node in a network is just its degree, the number of edges connected to it (see Section 6.10). Degree is sometimes calleddegree centralityin the social networks literature, to emphasize its use as a centrality measure. In directed networks, nodes have both an in-degree and an out-degree, and both may be useful as measures of centrality in the appropriate circumstances.
Although degree centrality is a simple centrality measure, it can be very illuminating. In a social network, for instance, it seems reasonable to suppose that individuals who have many friends or acquaintances might have more influence, more access to information, or more prestige than those who have fewer. A non-social network example is the use of citation counts in the evalu- ation of scientific papers. The number of citations a paper receives from other papers, which is its in-degree in the directed citation network, gives a quan- titative measure of how influential the paper has been and is widely used for judging the impact of scientific research.
7.1.2 Eigenvector centrality
Useful though it is, degree is quite a crude measure of centrality. In effect, it awards a node one “centrality point” for every neighbor it has. But not all neighbors are necessarily equivalent. In many circumstances a node’s impor- tance in a network is increased by having connections to other nodes thatare themselves important. For instance, you might have only one friend in the world, but if that friend is the president of the United States then you yourself may be an important person. Thus centrality is not only about how many people you know but also who you know.
Eigenvector centralityis an extension of degree centrality that takes this factor into account. Instead of just awarding one point for every network neighbor a node has, eigenvector centrality awards a number of points proportional to the centrality scores of the neighbors. This might sound tautological—in order to
work out the score of every node, I need to know the score of every node. But in fact it is straightforward to calculate the scores with just a little work.
Consider an undirected network ofn nodes. The eigenvector centralityxi
of node i is defined to be proportional to the sum of the centralities of i’s neighbors, so that
xiκ−1 Õ
nodesjthat are neighbors ofi
xj, (7.1)
where we have called the constant of proportionalityκ−1for reasons that will become clear. For the moment we will leave the value ofκarbitrary—we will choose a value shortly.
With eigenvector centrality defined in this way, a node can achieve high centrality either by having a lot of neighbors with modest centrality, or by having a few neighbors with high centrality (or both). This seems natural: you can be influential either by knowing a lot of people, or by knowing just a few people if those few are themselves influential.
An alternative way to write Eq. (7.1) is to make use of the adjacency matrix:
xiκ−1 Õn
j1
Ai jxj. (7.2)
Note that the sum is now over all nodes j—the factor ofAi j ensures that only the terms for nodes that are neighbors oficontribute to the sum. This formula can also be written in matrix notation asxκ−1Ax, or equivalently
Axκx, (7.3)
wherexis the vector with elements equal to the centrality scoresxi. In other words,xis an eigenvector of the adjacency matrix.
This doesn’t completely fix the centrality scores, however, since there aren different eigenvectors of then×nadjacency matrix. Which eigenvector should we use? Assuming we want our centrality scores to be non-negative, there is only one choice: x must be the leading eigenvector of the adjacency matrix, i.e., the eigenvector corresponding to the largest (most positive) eigenvalue.
We can say this with certainty because of thePerron–Frobenius theorem, one of the most famous and fundamental results in linear algebra, which states that for a matrix with all elements non-negative, like the adjacency matrix, there is only one eigenvector that also has all elements non-negative, and that is the leading eigenvector. Every other eigenvector must have at least one negative
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entrality
element.1,2
So this is the definition of the eigenvector centrality, as first proposed by Bonacich in 1987 [73]: the centralityxiof nodeiis theith element of the leading eigenvector of the adjacency matrix.
This also fixes the value of the constantκ—it must be equal to the largest eigenvalue. The centrality scores themselves are still arbitrary to within a multiplicative constant: if we multiply all elements ofxby any constant, Eq. (7.3) is unaffected. In most cases this doesn’t matter much. Usually the purpose of centrality scores is to pick out the most important nodes in a network or to rank nodes from most to least important, so it is only the relative scores of different nodes that matter, not the absolute numbers. If we wish, however, we can normalize the centralities by, for instance, requiring that they sum ton(which ensures that average centrality stays constant as the network gets larger).
We have defined the eigenvector centrality here for the case of an undirected network. In theory it can be calculated for directed networks too, but it works best for the undirected case. In the directed case other complications arise.
First of all, a directed network has an adjacency matrix that is, in general, asymmetric (see Section 6.4). This means it has two sets of eigenvectors, the left eigenvectors and the right eigenvectors, and hence two leading eigenvectors.
Which of the two should we use to define the centrality? In most cases the
1Technically, this result is only true for connected networks, i.e., networks with only one component. If a network has more than one component then there is one eigenvector with non- negative elements for each component. This doesn’t pose a practical problem though: one can simply split the network into its components and calculate the eigenvector centrality separately for each one, which again guarantees that there is only one vector with all elements non-negative.
2A detailed discussion and proof of the Perron–Frobenius can be found, for example, in the books by Meyer [331] and Strang [440]. The basic intuition behind it is simple though. Suppose we take a random vectorx(0)and multiply it repeatedly by a symmetric matrixAthat has all elements non-negative. Aftertmultiplications we get a vectorx(t)Atx(0). Now let us writex(0)as a linear combinationx(0)Í
iciviof the eigenvectorsviofA, for some appropriate choice of constantsci. Then
x(t)Atx(0)AtÕ
i
civiÕ
i
ciκtiviκt1Õ
i ci
κi κ1
t vi,
whereκiare the eigenvalues ofAandκ1is the eigenvalue of largest magnitude. Since|κi/κ1|<1 for alli,1, all terms in the sum other than the first decay exponentially astbecomes large, and hence in the limitt→ ∞we getx(t)/κt1→c1v1. In other words, the limiting vector is simply proportional to the leading eigenvector of the matrix.
But now suppose we choose our initial vectorx(0)to have only non-negative elements. Since all elements of the adjacency matrix are also non-negative, multiplication byAcan never introduce any negative elements into the vector andx(t)must have all elements non-negative for all values oft. Thus the leading eigenvector ofAmust also have all elements non-negative. As a corollary, this also implies thatκ1must be positive, sinceAxκ1xhas no solutions for negativeκ1if both Aandxhave only non-negative elements.
correct answer is to use the right eigenvector. The reason is that centrality in directed networks is usually bestowed by other nodes that point towards you, rather than by you pointing to others. On the World Wide Web, for instance, it is a good indication of the importance of a web page that it is pointed to by many other important web pages. On the other hand, the fact that a page might itself point to important pages is neither here nor there. Anyone can set up a page that points to a thousand others, but that does not make the page important.3 Similar considerations apply also to citation networks and other directed networks. Thus the correct definition of eigenvector centrality for a node i in a directed network makes it proportional to the centralities of the nodes that point to it:
xi κ−1Õ
j
Ai jxj, (7.4)
which givesAxκxin matrix notation, wherexis the right leading eigenvector.
A
B
Figure 7.1: A portion of a directed net- work.Node A in this network has only outgoing edges and hence will have eigenvector centrality zero. Node B has outgoing edges and one ingoing edge, but the ingoing one originates at A, and hence node B will also have centrality zero.
However, there are still problems with this definition. Consider Fig. 7.1. Node A in this figure is connected to the rest of the network, but has only outgoing edges and no ingoing ones. Such a node will always have eigenvector centrality zero because all terms in the sum in Eq. (7.4) are zero. This might not seem to be a problem:
perhaps a node that no one points toshouldhave centrality zero. But then consider node B. Node B has one ingoing edge, but that edge originates at node A, and hence B also has centrality zero—all terms in the sum in Eq. (7.4) are again zero. Taking this argument further, we see that a node may be pointed to by others that themselves are pointed to by many more, and so on through many generations, but if the trail ends at a node or nodes that have in-degree zero, it is all for nothing—the final value of the centrality will still be zero.
In mathematical terms, only nodes that are in a strongly con- nected component of two or more nodes, or the out-component of such a strongly connected component, can have non-zero eigen- vector centrality.4 In many cases, however, it is appropriate for nodes with high in-degree to have high centrality even if they are not in a strongly-connected component or its out-component. Web pages with many links, for instance, can reasonably be considered important even if they are not
3Arguably, this is not entirely true. Web pages that point to many others are often directories of one sort or another and can be useful as starting points for web surfing. This, however, is a different kind of importance from that highlighted by the eigenvector centrality and a different, complementary centrality measure is needed to quantify it. See Section 7.1.5.
4For the left eigenvector it would be the in-component.
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entrality
in a strongly connected component. Recall also that acyclic networks, such as citation networks, have no strongly connected components of more than one node (see Section 6.12.1), so all nodes will have centrality zero, making the eigenvector centrality completely useless for acyclic networks.
There are a number of variants of eigenvector centrality that address these problems. In the next two sections we discuss two of them: Katz centrality and PageRank.
7.1.3 Katz centrality
One solution to the issues of the previous section is the following: we simply give each node a small amount of centrality “for free,” regardless of its position in the network or the centrality of its neighbors. In other words, we define
xiαÕ
j
Ai jxj+β, (7.5)
whereαandβare positive constants. The first term is the normal eigenvector centrality term in which the centralities of the nodes pointing toiare summed, and the second term is the “free” part, the constant extra amount that all nodes receive. By adding this second term, we ensure that even nodes with zero in-degree still get centralityβ, and once they have non-zero centrality they can pass it along to the other nodes they point to. This means that any node that is pointed to by many others will have a high centrality, even if it is not in a strongly connected component or out-component.
In matrix terms, Eq. (7.5) can be written
xαAx+β1, (7.6)
where1is the uniform vector(1,1,1, . . .). Rearranging forx, we then find that xβ(I−αA)−11. As we have said, we normally don’t care about the absolute magnitude of centrality scores, only about the relative scores of different nodes, so the overall multiplierβis unimportant. For convenience we usually setβ1, giving
x(I−αA)−11. (7.7)
This centrality measure was first proposed by Katz in 1953 [258] and we will refer to it as theKatz centrality.
The definition of the Katz centrality contains the parameterα, which gov- erns the balance between the eigenvector centrality term in Eq. (7.5) and the constant term. If we wish to make use of the Katz centrality we must first choose a value for this constant. In doing so it is important to understand
thatαcannot be arbitrarily large. If we letα→0, then only the constant term survives in Eq. (7.5) and all nodes have the same centralityβ(which we have set to 1). As we increaseαfrom zero the centralities increase and eventually there comes a point at which they diverge. This happens when(I−αA)−1diverges in Eq. (7.7), i.e., when det(I−αA)passes through zero. Rewriting this condition as
det(α−1I−A)0, (7.8)
we see that it is simply the characteristic equation whose rootsα−1 are equal to the eigenvalues of the adjacency matrix.5 Asαincreases, the determinant first crosses zero whenα−1κ1, the largest (most positive) eigenvalue ofA, or alternatively when α 1/κ1. Thus, we should choose a value ofαless than this if we wish the expression for the centrality to converge.6
Beyond this, however, there is little guidance to be had as to the value that αshould take. Most researchers have employed values close to the maximum of 1/κ1, which places the maximum amount of weight on the eigenvector term and the smallest amount on the constant term. This returns a centrality that is numerically quite close to the ordinary eigenvector centrality, but gives small non-zero values to nodes that are not in strongly connected components of size two or more or their out-components.7
The Katz centrality provides a solution to the problems encountered with ordinary eigenvector centrality in directed networks. However, there is no reason in principle why one cannot use Katz centrality in undirected networks as well, and there are times when this might be worthwhile. The idea of adding a constant term to the centrality so that each node gets some weight just by virtue of existing is a natural one. It allows a node that has many neighbors to have high centrality regardless of whether those neighbors themselves have high centrality, and this could be useful in some applications.
5The determinant of a matrix is equal to the product of the eigenvalues of the matrix. The matrixxI−Ahas eigenvaluesx−κiwhereκiare the eigenvalues ofA, and hence its determinant is det(xI−A)(x−κ1)(x−κ2). . .(x−κn), which is a degree-npolynomial inxwith zeros at xκ1, κ2, . . . Hence the solutions of det(xI−A)0 give the eigenvalues ofA.
6Formally one recovers finite values again when one moves past 1/κ1 to higherα, but in practice these values are meaningless. The method returns good results only forα <1/κ1.
7In fact, the Katz centrality becomes formally equal to the eigenvector centrality in the limitα→ 1/κ1. Moreover, it is equivalent to degree centrality in the limitα→0. So the Katz centrality includes both these other measures as special cases and interpolates between them for intermediate values ofα. See Exercise 7.3 for more details.
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entrality
7.1.4 PageRank
The Katz centrality of the previous section has one potentially undesirable feature. If a node with high Katz centrality has edges pointing to many others then all of those others also get high centrality. A high-centrality node pointing to one million others gives all one million of them high centrality. One could argue that this is not always appropriate. In many cases it means less if a node is only one among many that are pointed to. The centrality gained by virtue of receiving an edge from a prestigious node is diluted by being shared with so many others. For instance, websites like Amazonor eBaylink to the web pages of thousands of manufacturers and sellers; if I’m selling something on Amazon it might link to me. Amazon is an important website, and would have high centrality by any sensible measure, but should I therefore be considered important by association? Most people would say not: I am only one of many that Amazon links to and its contribution to the centrality of my page will get diluted as a result.
We can allow for this by defining a variant of the Katz centrality in which the centrality I derive from my network neighbors is proportional to their centrality divided by their out-degree. Then nodes that point to many others pass only a small amount of centrality on to each of those others, even if their own centrality is high.
In mathematical terms this centrality is defined by xiαÕ
j
Ai j
xj
koutj +β. (7.9)
This gives problems, however, if there are nodes in the network with out- degreekoutj 0. For such nodes the corresponding term in the sum in Eq. (7.9) is indeterminate—it is equal to zero divided by zero (becauseAi jis always zero ifjhas no outgoing edges). This problem is easily fixed however. It is clear that nodes with no out-going edges should contribute zero to the centrality of any other node, which we can contrive by artificially setting koutj 1 for all such nodes. (In fact, we could set koutj to any non-zero value and the calculation would give the same answer.)
In matrix terms, Eq. (7.9) is then
xαAD−1x+β1, (7.10)
with1being again the vector(1,1,1, . . .)andDbeing the diagonal matrix with elementsDii max(kouti ,1). Rearranging, we find thatx β(I−αAD−1)−11, and thus, as before,βplays the role only of an unimportant overall multiplier
for the centrality. Conventionally, we setβ1, giving
x(I−αAD−1)−11. (7.11) This centrality measure is commonly known asPageRank, which is a name given it by the Google web search corporation. Google uses PageRank as a central part of their web ranking technology for web searches, which estimates the importance of web pages and hence allows the search engine to list the most important pages first [82]. PageRank works for the Web precisely because having links to your page from important other pages is a good indication that your page may be important too. But the added ingredient of dividing by the out-degrees of pages ensures that pages that simply point to an enormous number of others do not pass much centrality on to any of them, so that, for instance, network hubs like Amazon do not have a disproportionate influence on the rankings. PageRank also finds applications in other areas besides web search—see Gleich [205] for a review.
As with the Katz centrality, the formula for PageRank, Eq. (7.11), contains a free parameterα, whose value must be chosen somehow before the algorithm can be used. By analogy with Eq. (7.8) and the argument that follows it, we can see that the value ofαshould be less than the inverse of the largest eigenvalue of AD−1. For an undirected network this largest eigenvalue turns out to be one,8 and thusαshould be less than one. There is no equivalent result for a directed network, the leading eigenvalue differing from one network to another, although it is usually still roughly of order one.
The Google search engine uses a value of α 0.85 in its calculations, although it’s not clear that there is any rigorous theory behind this choice.
More likely it is just a shrewd guess based on experimentation to find out what works.
One could imagine a version of the PageRank equation (7.9) that did not have the additive constant termβin it at all:
xi αÕ
j
Ai j
xj
kj, (7.12)
which is similar to the original eigenvector centrality introduced back in Sec- tion 7.1.2, but now with the extra division by kj. Particularly for undirected
8This is straightforward to show. The corresponding (right) eigenvector is(k1,k2,k3, . . .), wherekiis the degree of theith node. It is easily confirmed that this is indeed an eigenvalue of AD−1with eigenvalue 1. Moreover, since this vector has all elements non-negative it must be the leading eigenvector and 1 must the the largest (most positive) eigenvalue by the Perron–Frobenius theorem discussed in Section 7.1.2—see footnote 2 on page 161.