# Generalized Epidemic Mean-Field Model for Spreading Processes over Multi-Layer Complex Networks

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4 4 Network Layers Fig. 3. Network layers describe the different types of interactions among agents in GEMF. The vertical dotted lines emphasize that all graphs have same nodes but the edges are different. our model. A representation of the network layering structure is depicted in Fig. 3. V. AGENT-LEVEL DECRPTON OF THE MARKOV PREADNG PROCE The network state () follows a continuous-time Markov process. Knowing that the network is in state () at time, what is the network state ( + ) at time +? n a network of interacting agents, this question can be very complicated. nstead, a more natural approach is to describe the agent state (+ ) given the network state () at time. The spreading process is fully described if the probability to record a transition from compartment to compartment for agent, conditioned on the network state (), is known for all possible values of,, and. n this section, we focus on deducing an expression for Pr[ ( + ) = () = ()], which will be used later on to develop the GEMF model. The challenge in deducing an expression for Pr[ (+ ) = () = ()] is that there are too many possibilities for the dependence of the transition of the individual agent on the network state. Here are a few examples: the transition happens completely independently from the states of other agents, the transition happens if the number of other agents in compartment are more than the number of agents in compartment, the transition happens if agents 1 and 2 are both in compartment and the rate of the transition is the logarithm of the number of agents in compartment. All of these examples are legitimate so far. However, we need to specify the transition possibilities properly to develop a coherent and consistent epidemic spreading model. A. Epidemic preading Process Modeling The N-ntertwined model [12] gives very good insights in how to properly define the transition possibilities to describe an epidemic spreading process. n the model, there are two transitions. The curing process, which is basically the transition from infected state to susceptible, occurs independently from the states of other agents. nstead, the infection process, which refers to transition from susceptible state to infected state, happens through a different mechanism. A susceptible agent is in contact with some other agents. During the time interval ( + ], the susceptible agent receives the infection from its infected neighbor with some probability. The process of receiving the infection from one infected neighbor is independent from the process of receiving the infection from another neighbor. All the infected neighbors compete to infect the susceptible agent. The susceptible agent becomes infected when one of the neighbors succeeds to transmit the infection. The transitions in the epidemic model are very similar to the transitions in most existing epidemic models. As a result, we impose a similar structure of independent competing processes to the generalized spreading model. Assumption 1: A transition for agent is the result of several statistically independent competing processes: the process for agent that happens independently of the states of other agents, and the process for agent because of interaction with agent 6=, for each {1 }\{}. According to Assumption (1), the interaction of agent with agent 6= is statistically independent of its interaction with agent { }. Define the auxiliary counting process () () corresponding to the interaction of agent with agent. For convenience of notations, let () () correspond to the transition for agent occurring independently from the states of other agents. According to Assumption 1, conditioned on the network state, these counting processes are statistically independent. The transition occurs in the time interval ( + ] if any of these counting processes record an event. Therefore, Pr[ ( + ) = () = ()] can be written as Pr[ ( + ) = () = ()] = Pr[ {1} s.t. () (+ ) () () 6= 0 ()] (4) Each of the counting processes () () is a Poisson process with the rate () (), to be determined. Therefore, Pr[ () (+ ) () () 6= 0 ()] = () () +( ) (5) The sum of independent Poisson processes is also a Poisson process with aggregate rate equal to the sum of the individual rates (see Th in [16]). Therefore, Pr[ ( + ) = () = ()] = X () ()+( ) (6) =1 The remaining part of this section is dedicated to properly find () (). 1) Nodal Transition: As discussed earlier, a nodal transition occurs independently from the states of other agents. For example, in the model, the curing process is the result of a nodal transition. The curing process happens with rate

6 6 a) 1 b) 26 1 According to (11), the probability of remaining in the previous state is Pr[ ( + ) = () = ()] = l,51 l,36 l,32 1 X 6= Pr[ ( + ) = () = ()] (13) Combining (2), (11), and (13) leads to 4 4 l,43 Fig. 6. Transition rate graphs in GEMF: a) nodal transition rate graph; nodes represent compartments, directed link ( ) represent nodal transition weighted by the transition rate 0, and b) edge-based transition graph of network layer G ; directed link ( ) represent the edge-based transition weighted by the transition rate 0 in network layer. The inducer compartment of layer is. C. Agent-Level Markov Description of the preading Process n ection V-A, we developed the expressions for the nodal transition and edge-based transitions. ubstituting (7) and (8) into (6) yields Pr[ ( + ) = () = ()] = + X ()+( ) (11) =1 for = {1 } and 6=, where (), X 1 { ()= } (12) =1 is the number of neighbors of agent in G that are in the corresponding influencer compartment. Equation (11) provides an agent-level description of the Markov process. t can directly be used for Monte Carlo numerical simulation of the spreading process. V. GEMF: GENERALZED EPDEMC MEAN-FELD MODEL The objective of this section is to derive the time evolution of the state occupancy probabilities. A. Exact Markov Differential Equation n the previous section, the spreading model was described and the corresponding Markov process was derived in (11). The evolution of the state occupancy probabilities, associated to a Markov process, follows a set of differential equations known as the Kolmogorov differential equations. The derivation of the Kolmogorov differential equation of a Markov process is fairly standard (see, [16], [29]) when the transition rates between the states of the Markov process are known. However, the challenge here is that the network states are the actual Markov states and instead of the transition rates between the network states, we have the agent-level description of the transitions in (11). n this section, we derive the differential equations directly from (11). [ ( + ) () = ()] = 1 + P =1 1 (). ( 1) + P =1 ( 1) () P =1 () (+1) + P + =1 + ( ) (+1) (). + P =1 () (14) where, X 6=, X 6= (15) and ( ) is a function of higher order terms of satisfying the condition ( ) =0 (16) where is the all ones vector with appropriate dimensions. Define the generalized transition matrices R and R with the elements ( ),, (17) Accordingtodefinitions (15) and (17), the matrices and are actually the Laplacian matrices of transition rate graphs, definedinectionv-b. Using (14) and the definition (17), [ ( + ) ()] is obtained as [ ( + ) ()] = () X () () =1 + ()+( ) (18) where () is defined in (12). Computing the expected value of each side of (18), we get [ ( + )] = [ ()] X [ () ()] =1 +[ ()] + ( ) (19) where ( ) =[( )] and the formula for iterative expectation (see [30]) rule [[ ]] = [] hasbeenusedto

7 7 find [ ( + )]. Movingthe[ ()] term in (19) to the left side and dividing both sides by yields [ ( + )] [ ()] = X [ ()] [ () ()] + =1 Letting 0 in (20), we obtain [ ()] = [ ()] 1 ( ) (20) X [ () ()] (21) Using (12), the term [ () ()] in (21) can be written as X [ () ()] = [( ) ()] (22) =1 The term [( ) ()] is actually embedded in [ () ()]. Therefore, the evolution of [ ()] depends on [ () ()] term, which is the joint state of pairs of nodes. This means that the marginal information about the compartmental occupancy probabilities is not enough to fully describe the time evolutions of the probabilities. f we continue to derive the evolution law for [ () ()], it turns out that the time derivative of [ () ()] depends on terms of the form [ () () ()], which are the joint states of triplets. This dependency of the evolution of expectation of -node groups on expectation of ( +1)-node groups continues until reaches to =. As a result, any system describing the evolution of the expected value of the joint state of any group of nodes is not a closed system. When =, the expectation of the joint state of all nodes [ 1 () ()], which according to definition (3) is actually the expectation of the network state, satisfies a differential equation of the form [] = Q [] (23) where Q R is the infinitesimal generator (see [16], [29]) of the underlying Markov process. The Kolmogorov differential equation (23) is, which we refer to as the exact Markov model, is derived explicitly in the Appendix. The exact Markov equation (23) fully describes the system. However, the above differential equation has states. Therefore for large values of, it is neither analytically nor computationally tractable. n the following section, it is shown that through a mean-field type approximation, a differential equation with states can be derived. B. GEMF: Generalized Epidemic Mean-Field Model One way to reduce the state space size is to use closure approximation techniques. As explained in the previous section, expectations of order depend on expectation of order +1. The goal of closure techniques is to approximate the expectations of order +1, and express them in terms of expectations of order less or equal to. nthisway,a new set of differential equations is obtained which is closed =1 and has the state space size, which is polynomially growing by. The simplest approximation is the mean-field type approximation [31]. n first order mean-field models [12], the states of nodes are assumed to be independent random variables. t is also possible to consider higher order mean-field approximations. Cator and Van Mieghem [32] used a second order mean-field approximation and found more accurate performance of the model. Another approach is called the moment closure technique, where the joint states of triplets are assumed to have a specific distribution (usually normal or lognormal) [11], [31]. n this way the joint expectation of triplets are expressed in terms of expectations of pairs. Taylor et al. [31] have conducted a comparison of the performances of different approximations. n this paper, we use a fist order mean-field type approximation. Using a first order mean-field type approximation, the joint expected values are approximated in terms of marginal expected values. pecifically, the term [( ) ()] in (22) is approximated by [( ) ()] ' ( [( ) ])[ ()] (24) This approximation assumes independence between the random variables. Using the approximation (24), we can describe the time evolution of the expected values through a set of ordinary differential equations with states. Denote by (), the expected value of at time, i.e., (), [ ()] (25) ubstituting [ () ()] = P =1 ( ()) () in (21), from (22) and (24), yields () = () X X =1 =1 ( ()) () (26) Arranging the terms in (26) specifies our generalized epidemic spreading model GEMF: X X = ( ) = {1 } =1 =1 (27) Having initially ( 0 )=1, the sum of the probabilities is guaranteed to be 1 at any time. The reason is that from (27) does not change over time because = = ( ) X X ( ) =1 =1 X X ( )( ) =1 =1 = 0 (28) The last conclusion is for the fact that =0and =0, since indeed and are the graph Laplacians for which is the eigenvector corresponding to a zero eigenvalue. GEMF has a systematic procedure to develop different spreading mean-field models. For any specific spreading scenario, the compartment set, the network layers, and their

12 12 APPENDX DERVATON OF EXACT MARKOV EQUATON n this section, we explicitly derive the expression for Q in (23). The idea is to derive the expression for [( + )] as a function of [()]. For this end, first we find the expression for the conditional expectation [(+ ) ()]. Then, the expression for [( + )] is found by averaging out the conditional. For small values of, wecanassume that only one transition happens at each time step, i.e., starting at network state at time, the network state can only go to a new state at time + for which only the state of a single node has been changed. Given the network state () =, state () = of each agent are can be determined and we have = 1 (A.1) ince only at most one single node can make a transition, the conditional expected value of the network state at time + is and Q is such that its th column is col(q )= X =1 1 X ( 1 { = } ) =1 (A.6) By letting 0 in (A.5), the time evolution of [] can be fully described by [] = Q [] (A.7) where Q is defined as X Q = Q + Q (A.8) The differential equation (A.7) is referred to as the exact Markov equation. =1 [( + ) () = ]= X 1 =1 [ ( + ) () = ] (A.2) where from (18), the expression for [ ( + ) () = ] is [ ( + ) () = ]= X X 1 {= } =1 =1 + ()+( ) (A.3) By averaging over all of the possible network states, the expected value of the network state at time + is found [( + )] X = [( + ) () = ]Pr[() = ] =1 X = ( X =1 1 =1 [ ( + ) () = ] )Pr[() = ] (A.4) ubstituting for [ (+ ) () = ] from (A.3), [(+ )] is deduced to be [( + )] = Q [()] X Q [()] + [()] + ( ) =1 (A.5) where Q = X =1

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