A large number of systems from nature and others man-made can be described in terms of networks, composed by entities (or nodes) that interact through connections (or links). The topological and statistical properties of the nodes in a network (at microscopic level) tend to be highly heterogeneous, as can be seen by studying their degree distribution, clustering distribution and their degree-degree correlations1,2,3. On the other hand, there are also heterogeneities at mesoscopic and macroscopic levels, e.g., not all the networks have the same hierarchical structure, community structure or topology4,5,6,7,8. These heterogeneities (at microscopic, mesoscopic and macroscopic levels) have repercussions in the importance of nodes and links in the network. For example, is well known that, in highly modular networks, nodes and links that connect modules or communities, i.e., nodes with neighboring nodes in different communities are more relevant (in terms of global communications) than nodes with neighborhoods fully included in the same community and links with both extreme nodes in the same module are less relevant than links with extreme nodes in different communities, a fact that has been widely used precisely in the detection of community structures9. Thus, there exist nodes that are more important as a result of their position relative to other nodes of the network, giving us relevant information about the properties of the networks. This kind of nodes and links that have a special role in a network are called central with respect to a given role. Thus, one of the ways to address the problem of centrality define first (at least heuristically) the context in which we are talking about “centrality” and then build measures to quantify the definition of centrality used, as in the case of betweenness and closeness10. However, although there is no consensus on the concept of centrality (because as we mentioned it depends on the system under study and the context or heuristics behind the “centrality” to be measured), we can propose a definition of centrality that involves all existing definitions.

Definition. Let be a network and let be a measure quantifying a desired property. We say that a node has a μ-centrality k if μ(i) = k. So that we can talk about closeness - centrality or betweenness - centrality. The reader can find the formula for betweenness and closeness measures in the materials and methods section, equations (11) and (12).

Among the applications of measures of centrality in complex networks we have: (i) in social networks, hubs are related to the most influential people on the network11,12,13, which is of interest to understand the individual and collective social processes and the information spreading in such networks13,14, (ii) in the protein-protein interaction networks of an organism, central nodes are related to the essential genes, i.e., those genes of an organism that are critical to their survival, which is of broad interest in the research and design of drugs to combat parasitic diseases15,16,17,18, (iii) in air or urban traffic networks, the central nodes are associated with optimal points for the spread of diseases, which are of great interest to effectively prevent and control the spread of diseases, putting at these points, checkpoints and vaccination campaigns19,20,21, among many other applications22,23,24. The centrality measures are an attempt to locate these nodes and their goal is to assign a measure (or rank) to each node so that they can be sorted from highest to lowest centrality. Some of the heuristics and statistics used to define centrality are based on10:

1 How connected a node is, 2 How influential a node is in terms of its neighbourhood, 3 How easily a node can propagate an information, 4 How intermediary is a node as a connector between nodes in the network.

In this paper, we propose a new method for finding the centrality of a node in a given network, based on both the sum of the centralities of the nodes in its neighbourhood and on their dissimilarities. The neighbourhood of the node i will contribute more to its centrality in the measure in which the nodes of the neighbourhood are more dissimilar. In this work, we will consider that two nodes are dissimilar if they do not share neighbours between them (see Fig. 1).

Figure 1 In the illustrated network, green and red node are dissimilar because they do not share neighbors between them. The red node reaches the blue nodes only through the green node and therefore its contribution to the centrality of red node is greater. Full size image

Let R be a network and A its adjacency matrix, i.e., A ij = 1 if the link {i, j} is in the network and A ij = 0 otherwise. This matrix indexes .

One method known to find the centrality of the i-th node in a network25 is based on the following heuristic: “The centrality/relevance/influence of a node is proportional to the sum of centralities of its neighbours”. Mathematically, this can be written as

where c i denotes the centrality of the i-th node and 1/λ is a constant of proportionality. This leads to the eigenvectors and eigenvalues problem:

Assuming that λ = λ max = ρ(A) is the spectral radius of the adjacency matrix of the network, i.e., the largest eigenvalue, then by the Perron-Frobenius theorem26 there exists a unique nonnegative eigenvector c that satisfies the above equation, obtaining the well-known measure of eigencentrality25. Note that to know how central or influential the i-th node is in the network, we only need to know the value of the i-th entry of the vector c.

The advantages of this approach are: (i) uses local information because the centrality of a node depends explicitly on the centrality of its neighbours, (ii) uses the global information of the network through successive couplings (i.e., the centralities of the nodes in the neighbourhood of a node also depends on their neighbours and so on), involving all network nodes in the centrality of a given node, (iii) one can analyse large networks quickly, since there are a variety of numerical methods for calculating eigenvalues and eigenvectors fairly efficiently27.

The main disadvantage of this approach lies precisely in its heuristics, since it assumes that all nodes in the neighbourhood of i-th node contribute equally to its centrality, which is in general false (see Fig. 1), leading to a poor ranking, as will be discussed in the results section. Therefore, it is necessary to reformulate this heuristic.