# Statistical Properties of SIS Processes with Heterogeneous Nodal Recovery Rates in Networks
## 1 Introduction  
The modeling and analysis of epidemic-like dynamical processes in complex networks have gained much attention among researchers over the past few decades [1; 2; 3; 4; 5; 6]. From the view point of dynamics, two basic models are commonly employed to describe epidemic spread in networks. They are named the susceptible-infectious-susceptible (SIS) model and the susceptible-infectious-recovery (SIR) model. It is convenient to characterize the change in dynamics in populations by means of the notation named the system state, which can be defined as the collection of all nodal infection states as well as the network structure. Many efforts have been made to investigate how the system parameters impact the system state. Specifically, it is of great interest to investigate how many infected nodes there when the system state changes over a long time period. It is also intriguing to gain insights into the relations between the number of infected nodes and the system parameters such as the infection rate and the network structure. [PERSON] et al. [2] reviewed related works about the epidemic spread in systems with static or dynamical single-layer network structures and provided insights into the impact of network structure on the steady-state system state. [PERSON] [3] provided a thorough overview of multilayer networks, which denotes the structure of a network of networks, and reported how epidemic spread is affected by the structure of a multilayer network. Usually, it is possible to determine the epidemic threshold by the eigenstructure of the topology of a finite-size network [2; 3]. It is usually assumed that the infection process and the recovery process that constitutes the epidemic processes are Markovian processes, as studied in most previous works. [PERSON] et al. [7] and [PERSON] et al. [8] investigated non-Markovian SIS and SIR epidemic processes innetworks, respectively. Furthermore, [PERSON] et al. [9] extended the results for the Markovian case to non-Markovian epidemics in networks with time-changing topology. From the view point of methodology, several theoretical frameworks have been proposed to investigate the dynamical statistical properties of the processes and obtain physical insights into the impact of the network's structure on the dynamics. Typical methodologies and frameworks include the mean-field approximation [2; 10; 11], message passing [12], master equation [13] and Markov theory [14]. Markov theory is capable of exactly capturing the time-evolution change of each system state but is limited to small-size networks because of its computational complexity [12]. The mean-field approximation framework appears in various forms with respect to the abstraction of the system. One merit of the mean-field approximation framework is that it is possible to strike a balance between the simplicity and accuracy of the theory. A common underlying assumption made with the basic SIS model or the SIR model is that all nodal infection rates are equal and that the same applies to the nodal recovery rates. This simplicity of the model makes it possible to theoretically and experimentally study the epidemic process and obtain several intriguing insights, such as the relationship between the network's structure and the epidemic threshold, in a convenient manner. However, a real-world epidemic process usually shows heterogeneous nodal recovery and nodal infection rates, which makes it difficult to propose a theory suited to any general scenario that would enable us to obtain physical insights into its general dynamics. Compared with studies on the basic SIS processes in complex networks, studies on SIS processes with heterogeneous nodal recovery and infection rates are still limited. In this work, we focus on the SIS process with heterogeneous recovery rates in a finite-size network that was previously used to characterize some real-world scenarios of interest, such as the impact of a mixture of mitigation methods on interest flooding attacks in named data networking (NDN) [15].