Multiple transitions of the susceptible-infected-susceptible epidemic model on complex networks Ang´ elica S. Mata 1 and Silvio C. Ferreira 1 1 Departamento de F´ ısica, Universidade Federal de Vi¸cosa, 36570-000, Vi¸cosa, MG, Brazil The epidemic threshold of the susceptible-infected-susceptible (SIS) dynamics on random networks having a power law degree distribution with exponent γ> 3 has been investigated using different mean-field approaches, which predict different outcomes. We performed extensive simulations in the quasistationary state for a comparison with these mean-field theories. We observed concomitant multiple transitions in individual networks presenting large gaps in the degree distribution and the obtained multiple epidemic thresholds are well described by different mean-field theories. We observed that the transitions involving thresholds which vanishes at the thermodynamic limit involve localized states, in which a vanishing fraction of the network effectively contribute to epidemic activity, whereas an endemic state, with a finite density of infected vertices, occurs at a finite threshold. The multiple transitions are related to the activations of distinct sub-domains of the network, which are not directly connected. PACS numbers: 89.75.Hc, 05.70.Jk, 05.10.Gg, 64.60.an I. INTRODUCTION Phase transitions involving equilibrium and non- equilibrium processes on complex networks have begun drawing an increasing interest soon after the boom of network science at late 90’s [1]. Percolation [2], epi- demic spreading [3], and spin systems [4] are only a few examples of breakthrough in the investigation of criti- cal phenomena in complex networks. Absorbing state phase transitions [5] have become a paradigmatic issue in the interplay between nonequilibrium systems and complex networks [6–10], being the epidemic spreading a prominent example where high complexity emerges from very simple dynamical rules on heterogeneous sub- strates [3, 11–15]. The existence or absence of finite epidemic thresholds involving an endemic phase of the susceptible-infected- susceptible (SIS) model on scale-free (SF) networks with a degree distribution P (k) ∼ k −γ , where γ is the degree exponent, has been target of a recent and intense inves- tigation [11–17]. In the SIS epidemic model, individuals can only be in one of two states: infected or suscepti- ble. Infected individuals become spontaneously healthy at rate 1 (this choice fixes the time scale), while the sus- ceptible ones are infected at rate λn i , where n i is the number of infected contacts of a vertex i. Distinct theoretical approaches for the SIS model were devised to determine an epidemic threshold λ c separating an absorbing, disease-free state from an active phase [11– 19]. The quenched mean-field (QMF) theory [18] explic- itly includes the entire structure of the network through its adjacency matrix while the heterogeneous mean-field (HMF) theory [3] performs a coarse-graining of the net- work grouping vertices accordingly their degrees. The HMF theory predicts a vanishing threshold for the SIS model for the range 2 <γ ≤ 3 while a finite threshold is expected for γ> 3. Conversely, the QMF theory states a threshold inversely proportional to the largest eigen- value of the adjacency matrix, implying that the thresh- old vanishes for any value of γ [11]. However, Goltsev et al. [12] proposed that QMF theory predicts the thresh- old for an endemic phase, in which a finite fraction of the network is infected, only if the principal eigenvec- tor of adjacency matrix is delocalized. In the case of a localized principal eigenvector, that usually happens for large random networks with γ> 3 [20], the epidemic threshold is associated to the eigenvalue of the first delo- calized eigenvector. For γ< 3, there exists a consensus for SIS thresholds: both HMF and QMF are equivalent and accurate for γ< 2.5 while QMF works better for 2.5 <γ< 3 [13, 19]. Lee et al. [15] proposed that for a range λ QMF c <λ< λ c with a nonzero λ c , the hubs in a random network be- come infected generating epidemic activity in their neigh- borhoods but high-degree vertices produce independent active domains only when they are not directly con- nected. These independent domains were classified as rare-regions, in which activity can last for very long times (increasing exponentially with the domain size [21]), gen- erating Griffiths phases (GPs) [21, 22]. The sizes of these active domains increase for increasing λ leading to the overlap among them and, finally, to an endemic phase for λ>λ c . However, on networks where almost all hubs are directly connected, it is possible to sustain an en- demic state even in the limit λ → 0 due to the mutual reinfection of connected hubs. Inspired in the appeal- ing arguments of Lee et al. [15], Bogu˜ n´ a, Castellano and Pastor-Satorras (BCPS) [14] proposed a semi-analytical approach taking into account a long-range reinfection mechanism and found a vanishing epidemic threshold for γ> 3. They compared their theoretical predictions with simulations starting from a single infected vertex and a diverging epidemic lifespan was used as a criterion to determine the thresholds. However, the applicability of BCPS theory to determine a phase transition involving an endemic phase has been debated [23]. In this work, we performed extensive simulations and arXiv:1403.6670v3 [physics.soc-ph] 9 Jan 2015