Percolation Centrality: Quantifying Graph-Theoretic Impact of Nodes during Percolation in Networks Mahendra Piraveenan 1 *, Mikhail Prokopenko 2,3 , Liaquat Hossain 1 1 Centre for Complex Systems Research, Faculty of Engineering and IT, The University of Sydney, New South Wales, Australia, 2 CSIRO Information and Communications Technology Centre, Epping, New South Wales, Australia, 3 School of Physics, The University of Sydney, New South Wales, Australia Abstract A number of centrality measures are available to determine the relative importance of a node in a complex network, and betweenness is prominent among them. However, the existing centrality measures are not adequate in network percolation scenarios (such as during infection transmission in a social network of individuals, spreading of computer viruses on computer networks, or transmission of disease over a network of towns) because they do not account for the changing percolation states of individual nodes. We propose a new measure, percolation centrality, that quantifies relative impact of nodes based on their topological connectivity, as well as their percolation states. The measure can be extended to include random walk based definitions, and its computational complexity is shown to be of the same order as that of betweenness centrality. We demonstrate the usage of percolation centrality by applying it to a canonical network as well as simulated and real world scale-free and random networks. Citation: Piraveenan M, Prokopenko M, Hossain L (2013) Percolation Centrality: Quantifying Graph-Theoretic Impact of Nodes during Percolation in Networks. PLoS ONE 8(1): e53095. doi:10.1371/journal.pone.0053095 Editor: Petter Holme, Umea ˚ University, Sweden Received June 20, 2012; Accepted November 27, 2012; Published January 22, 2013 Copyright: ß 2013 Piraveenan et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: The work was not specifically funded by any grant. The University of Sydney and the Commonwealth Science and Industrial Research Organization support the positions of the authors. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. * E-mail: mahendrarajah.piraveenan@sydney.edu.au Introduction Networks are ubiquitous in today’s world. Communication networks such as world wide web, telephone networks and mobile phone networks are changing the way we live and we interact with other people. Social networks built on top of these, such as Facebook and Twitter, are redefining ways of keeping in touch. Vast airline and rail networks have given us access to the remotest parts of the world and reduced travel times by orders of magnitude. Our survival depends on the functioning of a number of biological and ecological networks. The energy needed for our domestic and industrial use is supplied by electric power networks. Indeed, the interest and awareness about networks are not only a trend in scientific research but also a social and cultural phenomenon of this age [1–5]. Percolation of a ‘contagion’ occurs in complex networks in a number of scenarios. For example, viral or bacterial infection can spread over social networks of people, known as contact networks. The spread of disease can also be considered at a higher level of abstraction, by contemplating a network of towns or population centres, connected by road, rail or air links. Computer viruses can spread over computer networks. Rumours or news about business offers and deals can also spread via social networks of people. In all of these scenarios, a ‘contagion’ spreads over the links of a complex network, altering the ‘states’ of the nodes as it spreads, either recoverably or otherwise. For example, in an epidemiological scenario, individuals go from ‘susceptible’ to ‘infected’ state as the infection spreads. The states the individual nodes can take in the above examples could be binary (such as received/not received a piece of news), discrete (susceptible/infected/recovered), or even continuous (such as the proportion of infected people in a town), as the contagion spreads. The common feature in all these scenarios is that the spread of contagion results in the change of node states in networks. Indeed, in the epidemiological domain, a few studies have successfully modelled epidemic spread as a specific example of percolation in networks [6–11]. The percolation theory is attractive because it provides connections to several well-known results from statistical physics, in terms of percolation thresholds, phase transitions, long-range connectivity, and critical phenomena in general. For instance, Newman and Watts [6] suggested using a site percolation model for disease spreading in which some fraction of the population is considered susceptible to the disease, and an initial outbreak can spread only as far as the limits of the connected cluster of susceptible individuals in which it first strikes. An epidemic can occur if the system is at or above its percolation (epidemic) threshold where the size of the largest (giant) cluster becomes comparable with the size of the entire population. Similarly, Moore and Newman [12] used a general model with two simple epidemiological parameters: (i) susceptibility, the probability that an individual exposed to a disease will contract it, and (ii) transmissibility, the probability that contact between an infected individual and a healthy but susceptible one will result in the latter contracting the disease. They pointed out that if the distribution of occupied sites or bonds is random, then the problem of when an epidemic takes place becomes equivalent to a standard percolation problem on the graph: what fraction of sites or bonds must be occupied before a ‘‘giant component’’ of connected sites forms whose size scales extensively with the total number of sites [12]. It has been noted [13] that the percolation of disease through a network depends on both the level of contagion and the structure PLOS ONE | www.plosone.org 1 January 2013 | Volume 8 | Issue 1 | e53095