Introduction

Modelling social, biological and information-technology systems as complex networks has proven to be a successful approach to understand their function1,2,3,4. Among the various aspects of networks which have been investigated so far, the issue of centrality and the related problem of identifying the central elements in a network, has remained pivotal since its first introduction. The idea of centrality was initially proposed in the context of social systems, where it was assumed a relation between the location of an individual in the network and its influence and power in group processes5,6. Since then, various centrality measures have been introduced over the years to rank the nodes of a graph according to their topological importance. Centrality has found many applications in social systems6, in biology7 and in man-made spatial networks8,9,10,11.

Among the various measures of centrality, such as those based on counting the first neighbours of a node (degree centrality), or the number of shortest paths passing through a node (betweenness centrality)12,13, a particularly important class of measures are those based on the spectral properties of the graph14. Spectral centrality measures include the eigenvector centrality15,16, the alpha centrality17, Katz's centrality18, subgraph centrality19 and PageRank20 and are often associated to simple dynamics taking place over the network, such as various kinds of random walks21,22,23. As representative of the class of spectral centralities, we focus here on eigenvector centrality, which is based on the idea that the importance of a node is recursively related to the importance of the nodes pointing to it.

Results

Given an unweighted directed graph G = (V,E) with N = |V| nodes and K = |E| links, described by the N × N adjacency matrix A, the eigenvector centrality c0 of G is defined as the eigenvector of At associated to the largest eigenvalue ρ0, which in formula reads Atc0 = ρ0c015,16,17. If the graph is strongly connected, then the Perron-Frobenius theorem guarantees that c0 is unique and positive. Therefore, c0 can be normalised such that the sum of the components equals 1 and the value of the i-th component represents the centrality score of node i, i.e. the fraction of the total centrality associated to node i. In this Article we show how to change the eigenvector centrality scores of all the nodes of a graph by performing only local changes at node level. As a first step (see the Methods Section) we have proved that, given any arbitrary positive vector , c > 0 and c ≠ c0, it is always possible to assign the weights of all the links of a strongly–connected graph G and to construct a new weighted network Gω, with the same topology as G and with eigenvector centrality equal to c:

where Aω is the weighted adjacency matrix of Gω.

This is illustrated in Fig. 1 for a graph with N = 4 nodes and K = 5 links. In the original unweighted graph G, node 2 is the node with the highest eigenvector centrality, followed in order by node 3, node 4 and node 1. Now, if we have the possibility of tuning the weights of each of the five links, we can set any centrality value to the nodes of the graph. In figure we show, for instance, how to fix the weights of the five links in order to construct: i) a weighted network Gω in which all nodes have the same centrality score and ii) even a weighted network Gω in which the centrality ranking is totally reversed with respect to the ranking in G.

Figure 1
figure 1

An example of how to tune the link weights to change the node centrality scores.

The graph G with N = 4 nodes and K = 5 links shown in panel (a) is strongly-connected and has an eigenvector centrality c0 = {0.18, 0.33, 0.27, 0.22}. By ranking the nodes according to the components of c0, we obtain that node 2 is the most central one, followed in order by node 3, node 4 and node 1. We can now set the weights of the five links ω = {ω1, ω2, ω3, ω4, ω5} in such a way that equation (1) is satisfied for any given centrality vector c ≠ c0. For instance, we can get a weighted network Gω in which all nodes have the same centrality, by solving equation (1) with a centrality vector c = {1/4, 1/4, 1/4, 1/4} and ρ = 3.0. We obtain a vector of weights: ω = {α, 3, 3, 3, 3 − α} which, for 0 < α < 3, guarantees that all the link weights of the graph are positive. The resulting network Gω is shown in panel (b). As expected, we have K − N = 1 free parameter (namely α) since the graph has N = 4 nodes and K = 5 links. Instead, if we want to reverse the original node ranking we can solve the system with a centrality vector c = {0.5, 0.05, 0.2, 0.25}. Notice that, in this case, the ranking induced by c is exactly the opposite of the one induced by c0: now node 1 is the most central one, followed in order by node 4, node 3 and node 2. The solution of equation (2) gives ω = {α, 12, 15/4, 6, (3 – 10α)/5}, corresponding to a weighted network Gω with all positive weights whenever 0 < α < 3/10. The resulting network is shown in panel (c).

Full size image

As shown in the example, given a graph G, by controlling the weights of all the links, it is always possible to set any arbitrary vector c as the eigenvector centrality of the graph. However, tuning the weights of all the K links of a given network is practically unfeasible, especially in large systems. Fortunately, this is not necessary, either. In fact, in the case of Fig. 1, a weighted graph with all nodes having the same centrality score can also be obtained by changing the weights of only four links, while leaving unchanged the weight of the link from node 1 to node 2. More in general, it can be proved that the eigenvector centrality of the whole network can be controlled by appropriately tuning the weights of just N of the K links. The only constraint is that the N links must belong to a subset such that, for every node i ∈ V , there is a link pointing to i (see Methods Section). This is illustrated in Fig. 2 for three real social networks. In each of the three cases, it is possible to set any arbitrary eigenvector centrality by changing only the weights of the red arcs, while keeping unchanged (and equal to 1) the weights of all remaining arcs, shown in yellow. The nodes from which the links in E′ originate are also coloured in red and are referred to as a controlling set of the network (see Methods Section). What is striking is that, in each of the three networks, the set E′ can be chosen in such a way that all the links in E′ originate from a relatively small subset of nodes. For instance, the controlling set reported for the student government network of the University of Ljubljana contains only two nodes. This is also a minimum controlling set, since the graph does not admit another controlling set with a smaller number of nodes. This finding indicates that only two members of the student government, namely node 2 and node 8, can in principle set the centrality of all the other members by concurrently modifying the weights of some of their links. It is in fact reasonable to assume that the weight of the directed link from i to j, representing in this case the social credit (in terms of reputation, esteem or leadership acknowledgement) given by individual i to individual j, can be strengthen or decreased only by i. Consequently, nodes 2 and 8 can modify at their will the weights of their out-going links and, If these changes are opportunely coordinated, they can largely alter the actual roles of all the other individuals. Analogously, only five monks can control the centrality of the Sampson's monk network, while only 4 members of the Zachary's karate club network can set the eigenvector centrality of the remaining 30 members.

Figure 2
figure 2

Minimum controlling sets in three real social networks.

The graph in panel (a), with N = 11 vertices and K = 41 arcs, shows who asks who for an opinion among the members of the student government of the University of Ljubljana in 199224. The minimum controlling set is made by the two nodes marked in red, namely node 2 and node 8. These two nodes are linked to each other and point to all the remaining nodes in the graph. Therefore, nodes 2 and 8, by cooperatively modifying the weights of their red links, can set any arbitrary eigenvector centrality to the entire system. The graph in panel (b) has N = 18 nodes and K = 55 arcs and describes the social relations between the monks of an isolated contemporary American monastery, as recorded by Sampson in 196925. Here, the minimum controlling set contains five nodes, shown in red. In this case, the subset of links E′ (red links), does not contain links pointing to node 5, so that the red nodes can control the centrality of all the network nodes, except node 5. Finally, the Zachary's karate club network shown in panel (c) has N = 34 nodes and K = 78 undirected edges and describes the social network of friendships among the members of a US university karate club in 197026. In this network, the minimum controlling set contains node 1, the instructor Mr. Hi, node 34, the club administrator Mr. John A and also nodes 7 and 26. Notice that just two nodes, namely 1 and 34, can control the centrality of 95% of the graph nodes.

Full size image

A question of practical interest is to investigate the size of the minimum controlling set in various complex networks. When is small with respect to N, then the centrality of the network is easy to control. Conversely, when the number of nodes in the minimum controlling set is large, the network G is more robust with respect to centrality manipulations. We have used two greedy algorithms to compute approximations of minimum controlling sets in various real systems (see Methods Section). In Table 1 we report the best approximation for , i.e. the size of the smallest controlling set produced by either of the two algorithms in networks whose sizes range from hundreds to millions of nodes. In the majority of the cases we have found unexpectedly small controlling sets, containing only up to 10 – 20% of the nodes of the network. For instance, in the graph of Jazz musicians, there exists a controlling set made by just 16 of the 198 musicians. These 16 individuals alone can, in principle, decide to set the popularity of all the other musicians, enhancing the centrality of some of the nodes and decreasing the centrality of others, just by playing more or less often with some of their first neighbours. Among all the networks we have considered, the one with the smallest controlling set is the Wikipedia talk communication network, a graph with 2,394,385 nodes in which just 2% of nodes are able to alter the centrality of the entire system. The quantities in parenthesis indicate that for this network a set of just 1% of the nodes can control the centrality of 99% of the nodes.

Table 1 Number of nodes N, average degree 〈k〉 and the relative size of the mimimum controlling set found in 35 different real world networks. The values of reported are expressed as percentage of the network size N. The algorithms used to find approximations of minimum controlling sets, mark as controllers also nodes not controlling other nodes, simply because, at a certain iteration of the greedy procedure, they have remained with no out-going links. Therefore, we also report in parenthesis the relative size of the effective minimum controlling set and the percentage of the controlled nodes. The notation x% → y%, indicates that x% of the nodes is able to control the centrality of y% of the network. We report also, for each network, the relative size of the controlling set in randomized versions which preserve the original degree sequence. We have considered averages over 100 different randomizations. In the last two columns we report, respectively, the ratio α between the total degree of the nodes in the minimum controlling set and the number of nodes in the network and the percentage p quantifying how many of the top-100 nodes with the highest degree belong to the minimum controlling set. From top to bottom, the networks are divided into six classes, respectively World-Wide-Web, collaboration/communication, citation, spatial, words and socio–economical networks
Full size table

For each real network G, we have also computed the typical size of the minimum controlling set in its randomised counterpart (see the fifth column in Table 1). In particular, we have considered a randomisation which preserves the degree sequence of the original graph. In most of the cases , relevant exceptions being some spatial man-made networks, such as power grids, road networks and electronic circuits and also the patents citation network. This fact suggests that, in the absence of other limitations, such as strong spatial/geographic constraints11, the structure of real networks has probably evolved to favour the control of spectral centrality by a small group of nodes. To better compare the controllability of networks with different sizes, we report in Fig. 3 the ratio as a function of the number of graph nodes N. The smallest values of the ratio are found for collaboration/communication systems, WWW and socio-economical networks. The five most controllable networks are respectively Wiki-talk, Internet at the AS level, movie actors, the Stanford World Wide Web and the collaboration network of researchers in astrophysics. These are all networks in which single nodes can tune, at their will, the weights of their out-going links. A scientist can decide whether to weaken or strengthen the connections to some of the collaborators. The administrators of an Internet Autonomous System can control the routing of traffic through neighbouring ASs, by modifying peering agreements28. And, similarly, the owner of a Web page can change the weights of hyperlinks, for instance by assigning them different sizes, colour, shapes and positions in the Web page.

Figure 3
figure 3

Relative size of the minimum controlling set in various real systems.

We report, as a function of N, the ratio between the sizes of the minimum controlling set in real networks and in their respective randomized versions (we have considered averages over 100 different realizations). Different symbols and colors refer to the six network classes considered in Table 1. The observed ratio is lower than 1 in most of the cases, with the smallest values corresponding to collaboration/communication systems, WWW and socio-economical networks. The ratio is equal to 1 in three cases. The networks with ratio larger than 1, with the exception of one socio-economical system (namely Epinions), are all spatially constrained systems: three electronic circuits, the US power grid and three road networks.

Full size image

In order to characterize the properties of the controlling set, we report in the last two columns of Table 1 the link redundancy α of the minimum controlling set (see Methods for details) and the percentage p, representing how many of the 100 nodes with the highest degree are contained in the minimum controlling set. In most of the cases, the value of α found is close to 1, indicating that every node in the graph is controlled, on average, by a relatively small number of nodes (generally no more than two or three) or, equivalently, that there is small overlap among the sets of nodes controlled by two different controlling nodes. Notice also that the values of p reported in the rightmost column of Table 1 range from 1% up to 95%. Although nodes with larger degree have in general a higher probability to be included in a controlling set, not all the nodes with the top highest degree always belong to the minimum controlling set. For instance, in the graph of movie actors, 90 of the top 100 nodes with the highest degree belong to the minimum controlling set. Conversely, there are cases, such as that of the collaboration network of researchers in astrophysics, where only an extremely small number of nodes with the highest degree actually belong to the minimum controlling set. We have therefore computed the degree distribution of the minimum controlling set for each of the real networks in Table 1 and compared it with the degree distribution of the corresponding network. Four typical cases (movie actors (a), Notre Dame Web(b), astrophysics coauthorship network (c) and Berkley-Stanford Web (d)) are shown in the four top panels of Fig. 4, where we report both the normalized degree distributions of the original network (red circles) and that of corresponding minimum controlling set (greeen squares). As shown in figure, the degree distribution of the minimum controlling can vary from that of the original network in many respects. For instance, the minimum controlling set of the networks in panel a) and b) contains a lower percentage of low-degree nodes and a relatively higher percentage of intermediate and high degree nodes than the original network. Conversely, the minimum controlling set of networks in panel c) and d) exibit a higher abundance of low-degree nodes and a lower percentage of nodes with intermediate degree. These results suggest that it is in general not possible to predict the composition of the minimum controlling set from the degree distribution of the original network. Analogously, we have verified that the presence and nature of degree-degree correlations have no direct influence on the features of the minimum controlling set. Particularly striking is the case of the networks reported in panel b) and d). These two networks correspond to the same kind of technological system, namely World-Wide-Web networks. The networks are both scalefree with the same value of the degree distribution exponent γ = 2.3 and they have the same disassortative degree-degree correlations. As shown in the figure, also the degree distributions of the two corresponding minimum controlling set are scale-free. However, while for the Notre Dame Web we extract an exponent γMCS = 2.3 equal to that of the original network, in the case of Berkley-Stanford Web we get an exponent γMCS = 1.9. Also the average degree of the nodes in the minimum controlling set is different in the two cases. In the Notre Dame Web network the average degree of the controlling set is around 12.89, i.e. much larger than the average degree of the original network (〈k〉 = 6.85), while in the case of Berkley-Stanford Web, the opposite happens, i.e. the the average degree of the controlling set is equal to 15.05 and is smaller than the average degree of the original network (〈k〉 = 22.2). Finally, in panel e) of Fig. 4 we report, for each network, the ratio between the average degree of the minimum controlling set and that of the original graph, showing that in one third of the cases the ratio is smaller than 1.

Figure 4
figure 4

Degree of the nodes in the minimum controlling set.

The top four panels report the degree distribution of the graph (red circles) and the degree distribution of the minimum controlling set (green squares) for the graph of movie actors (a), Notre Dame Web (b), the astrophysics coauthorship network (c) and Berkley-Stanford Web (d). The degree distribution of the minimum controlling set is, in general, different from that of the original graph. For instance the minimum controlling sets of the two graphs of the Web (panel b) and d)) have different degree distributions, even if these two networks correspond to the same kind of system, have the same degree distribution (p(k) ∼ k−γ with γ ≃ 2.3) and the same disassortative degree-degree correlations (knn(k) ∼ k−ν with ν ≃ 0.3). In panel (e) we report for each network the ratio between the average degree of the minimum controlling set and that of the original graph.

Full size image

Discussion

In this work, we have shown how a small number of entities, working cooperatively, can set any arbitrary eigenvector centrality for all the nodes of a real complex network. It is straightforward to extend our results to other spectral centralities, such as α-centrality and Katz's centrality. Similar arguments can also be applied, with some limitations, to PageRank: in this case, the inverse centrality problem has solutions only for some particular choices of c. Such findings suggest that rankings obtained from centrality measures can be easily controlled and even distorted by a small group of cooperating nodes. The high controllability of real networks potentially has large social and commercial impact, given that centrality measures are nowadays extensively used to identify key actors, to rank Web pages and also to assess the value of a scientific research.

Methods

Solution to the inverse centrality problem

The set of N linear equations with K variable weights, ω1,…, ωK, in equation (1) can be rewritten as a system of N linear equations with K variables:

where now B is a N × K matrix of real numbers and ω ≡ {ω1,…, ωK}. Notice that the linear system in equation (2) has solutions since the rank of B is N < K (all the equations are separated and each of the variables, ω1,…, ωK, appears in one equation only) and the in-degree of all nodes is positive by definition. Hence, there always exists such that equation (1) is satisfied. It is convenient to rewrite equation (2) in a form that emphasises the dependence of matrix B from c. We choose to label the arcs as follows: denotes the l-th arc entering node i, where is the in–degree of node i. Likewise, Si,l is the source of arc (i, l), while ωi,l is the corresponding weight. Using this notation, the i-th component of equation (2) can be written as:

By direct computation, one positive solution of eqauation (3) is given by

where i = 1…N and by continuity there are infinite many solutions such that ωi,l are all positive. In particular, if for node i we have , then the i-th equation of equation (3) has a unique solution, while if , there are always infinitely many solutions depending on parameters. Summing up, eqauation (2) has only one solution if all the node in-degrees are equal to one, while there are, in general, infinitely many solutions depending on K – N parameters. Notice that ρ can be different from ρ0, meaning that it is also possible to set the value of the largest eigenvalue of the weighted graph.

Tuning a subset of the graph links

Here, we show that it is not necessary to fix the weights of all the graph links in order to get an arbitrary centrality vector c > 0. In fact, given a subset of links containing at least one incoming link for each node, it is sufficient to assign some positive weights to each , while keeping constant , for instance all equal to 1, such that the resulting weighted graph has eigenvector centrality equal to c. Without loss of generality we can assume that the first incoming links of each node i belong to E′, so that the components of equation (3) can be written as:

Therefore, since ci > 0 for each 1 ≤ i ≤ N, then there is a ρ0 > 0 such that for every ρ > ρ0

and hence, by a similar continuity argument as above, we can ensure that there are infinitely many positive solutions to equation (5).

Finding minimum controlling sets

A controlling set of graph G is any set of nodes such that:

This means that, for each node j in the graph, at least one of the two following conditions holds: a) j ∈ C, or b) j is pointed by at least one node in C. We use |C| to denote the size of the controlling set, i.e. the number of nodes contained in C. Finding the minimum controlling set C* of a graph G, i.e. a controlling set having minimal size, is equivalent to computing the so-called domination number of G. The domination number problem is a well known NP-hard problem in graph theory27. Therefore, the size of the minimum controlling set can be determined exactly only for small N graphs as those in Fig. 2. To investigate larger graphs we have used two greedy algorithms. The first algorithm, called Top–Down Controller Search (TDCS), works as follows. We initially set Gt = 0 = G. We select the node i0 with the maximum out-degree in Gt = 0 and mark it as controlling node (or controller). Then, all the nodes in the out-neighbourhood of i0 are marked as controlled and are removed from Gt = 0, together with i0 itself. In this way, we obtain a new graph Gt = 1 and we store the controlling node i0, together with the list of nodes controlled by i0. Notice that, removing a generic node j from Gt = 0, also implies that Gt = 1 does not contain any of the links pointing to j or originating from it. The same procedure is iteratively applied to Gt = 1, Gt = 2 and so on, until all the nodes of G are either marked as controller or as controlled nodes. The algorithm produces a set , with , which is a controlling set of G by construction. The second algorithm is called Bottom–Up Controller Search (BUCS) and it works as follows. We set Gt = 0 = G and consider the set M(0) containing all the nodes in Gt = 0 with minimum in–degree. For each node i ∈ M(0), we consider the set of nodes pointing to i and select from this set the node mi with the maximal out–degree. This node is marked as controller. Then we obtain a new graph Gt = 1 by removing from Gt = 0 all the controlling nodes mi for all i ∈ M(0), together with all the nodes, marked as controlled, pointed by them. The same procedure is iteratively applied to Gt = 1, Gt = 2 and so on, until all the nodes of G are either marked as controller or as controlled nodes. If a graph Gt contains isolated nodes, these are marked as controller and removed from Gt. The algorithm finally produces a set which is a controlling set of G by construction. We have verified that the controlling sets obtained by both TDCS and BUCS for each of the networks considered are much smaller than those obtained by randomly selecting the controlling nodes. Moreover, the set of controlling nodes found by TDCS is in general different from that obtained on the same network by BUCS. Also the sizes of the two controlling sets obtained by the two algorithms are different.

Link redundancy of a controlling set

Given a controlling set , we define the link redundancy α of as the ratio between the sum of the degrees of the nodes in the controlling set and the number N of nodes in the network:

By definition 1 ≤ α ≤ 2K/N. In particular we get the minimal redundancy α = 1 when every graph node is controlled by just one node of the controlling set, while α = 2K/N when the controlling set contains all nodes in the graph.