Introduction

Network biology and related approaches are fundamental tools used for simplification, reconstruction, analysis, and comprehension of complex behavior in biological systems. Finding the most relevant elements in a network is crucial in network analysis. In various fields, this concept is discussed under different terms such as influential users, influential spreaders, authorities, hubs, key players, spread nodes, essentiality, lethality, and centrality. Although the centrality concept, which has its origin in social sciences, has been adopted into biology [1], there is no consensus among biologists regarding its definition and applications. Indeed, the difference between biological and other types of networks has prompted extensive research efforts aimed at developing more appropriate and reliable methods for the generalization of the centrality concept that could enable its applications in biology. For instance, systems biologists interpreted the centrality, essentiality, and lethality equivalents based on the centrality–lethality rule [2]. According to such interpretations, essential genes that show correlations with human disease genes are suitable candidates for biomarkers and drug targets. Many efforts performed for improving the existing methods and developing new insights of centrality usage, e.g., using percolation [3], the game theory [4], and perturbation [5] to prepare an effective new centrality index.

There are applications for network analysis on different platforms. Majority of these tools compute classical and common centralities, with a maximum of less than 20 computable measures. We categorized these tools into five groups. Group 1 consisted of the tools that are in the form of a standalone software such as CentiBin, which calculates 16 indices [6]. Group 2 consisted of plug-ins, packages, and toolboxes such as igraph, CentiLib, and CentiScaPe, which calculate 13, 14, and 8 indices, respectively [7–9]. Group 3 consisted of libraries and/or frameworks that can be used via programming languages such as Java and C/C++ (S1 File). The implementation codes proposed by developers such as LeaderRank [10] formed group 4. Algorithms provided by the author(s) such as the motif-based centrality [11] formed group 5. In addition to these tools, some centralities, such as the neighborhood functional centrality, have been coined without any proposed computational approaches [12]. The lack of user-friendly and comprehensive software products for computing some centralities have limited their use. However, comparing the capability and efficiency of all measures is not practical. Gathering available studies as an applicable, especially interactive, website could undoubtedly be a good solution for the mentioned problems. In addition, this not only results in saving researchers’ time but also can be an efficient tool for the creation of new ideas. According our knowledge, there is no complete and comprehensive collection of all current measures in either application that calculates the greater number of measures.

Methods and Implementation

This comprehensive encyclopedia mainly covers fundamental centrality definitions and computational algorithms in the field of network biology. Nearly all papers related to centrality were carefully reviewed for this survey. To make the materials accessible, a web page of pertinent materials, tools (by which the given measures are calculable), and relevant references was created. At present, the encyclopedia contains 113 entries.

Applications

Recently, many approaches and models have been developed to computationally identify the essential genes according to two main ideas. 1) Essential genes have a common feature in all organisms and they are evolutionary conserved [14]. This feature is based on functional importance, gene duplicability, and mRNA expression level [15]. 2) The network approach (graph theory) can promote an innovative way of thinking in biology [16, 17]. According to this method, after reconstruction of biological networks by high-throughput data, different network analysis methods have been employed to investigate the behavior of such complex large networks. Topological analysis of biological networks using mathematical approaches computes centrality indices and finds essential nodes. The concept of essential genes (or their products; essential proteins) has been widely used in various fields of biological sciences. These genes are vital to maintain cellular life and their deletion will result in lethality or infertility [18].

Centrality indices are a local network property that ranks the graph nodes. The higher the rank, the more important the node is in the network and may play key roles in controlling cellular responses [19].

As a simple illustration, we downloaded the protein-protein interaction network of Escherichia coli K12 from STRING (Search Tool for the Retrieval of Interacting Genes/Proteins) [20] version 9.1. Only interactions with an experimental score were selected. The loops and parallel edges were removed and the maximum connected component that contains 3138 vertexes (proteins) and 20309 edges (interactions) was selected. The essential proteins of E. coli were obtained from the Online GEne Essentiality (OGEE) database [21]. Thirty four centrality indices were calculated for given the network by CentiServe. The percentage of essential proteins in the top 100 ranked proteins by various centrality indices are displayed in Fig 1. The best indices were Laplacian centrality, Barycenter centrality, and 3 variations of closeness centrality, which determined about 71–74 essential proteins among top 100 ranked proteins. Previously Schoch and Brandes showed that closeness centrality is an appropriate measure for predicting essential nodes in biological networks [22]. In our study, in addition to confirming previous results, we concluded that the variants of closeness centrality index led to the same results.

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Fig 1. The percentage of essential proteins in the top 100 ranked proteins by various centrality indices.

https://doi.org/10.1371/journal.pone.0143111.g001

Each centrality measure provides an ordered list of proteins (S3 File). For combining these ordered lists into an overall top 20 proteins list, which most likely would be more essential than any individual list by itself, we implemented the rank aggregation method in the RankAggreg R package [23]. For this purpose we applied the RankAggreg function by a Genetic Algorithm and the Spearman distance parameters to 34 rows of ordered lists. After adequate iterations, the optimal ordered list that converged all ordered lists with a minimum objective function score of 53.08 was: groL (b4143), dnaK (b0014), rplB (b3317), rpsD (b3296), rpsB (b0169), rplD (b3319), rplA (b3984), rpsE (b3303), rpsC (b3314), htpG (b0473), tufB (b3980), tufA (b3339), fusA (b3340), rpsG (b3341), rpsI (b3230), rplC (b3320), rplM (b3231), rplQ (b3294), rpoC (b3988) and ybcV (b0558). Except rplA (b3984) and htpG (b0473), all of these genes were marked as essential ones previously.

The result of this analysis emphasized that the centrality is an anticipative topological measurement for predicting essential nodes in biological networks, because although about 20 percent of proteins in the protein interaction network of E. coli were marked as essential genes, the centrality index predicted about 74% of these proteins ranked in the top 100. Moreover, the aggregation method is a promising way for predicting more reliable and accurate essential proteins by unifying all ordered protein lists extracted from each centrality index. Discussing the biological significance of the RankAggreg result is beyond the scope of this article.

Conclusions

Centrality and essentiality are important and ongoing topics in the areas of social and biological networks. The centrality concept and related algorithms have found wide spread use in identifying essential nodes in a distinct network. With respect to the expansion of networks in all scientific fields, much efforts have been done in this regard. Since these methods have not met the needs, continuing efforts for optimizing indices and creating more efficient indices are expected. Determining the prominent nodes in a network completely depends on our importance definition and the measure we use for a specific purpose. Therefore, selecting an appropriate measure is critical. Additionally, uninformed use of measures without considering the essence of the network and the desired goal could result in misleading results, especially in biology [22]. A query in the resources suggests that the measures of centrality are not limited to the well-known ones. Although several new measures have recently been developed, the majority of them remained unknown. A number of these measures were defined initially based on biology notions, whereas others are optimizations of previous theories. Our goal while developing this encyclopedia and web application tool was to first provide all available centrality values, definitions, calculating methods, algorithms, and related software products in one place. This encyclopedia of centrality with its computational facilities was created to help researchers to make better choices of measures by reviewing the previous works, come up with new ideas, and avoid repetition. The second goal was to develop an application, which calculates more centrality measures in comparison to other tools. Access to all measures will not only help scientists to develop and optimize indices, but also provide an opportunity to calculate more centrality values for meaningful analysis of biological networks.

The site's materials will be kept updated and new contents will be added. The number of computable measures will be increased to improve the capability of the website. Adding practical applications of each index could significantly contribute the functionality of the site.

Supporting Information