The N-intertwined SIS epidemic network model

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1 Computing 0 93:47 69 DOI 0.007/s y The N-intertwined SIS epidemic network model Piet Van Mieghem Received: September 0 / Accepted: 7 September 0 / Published online: 3 October 0 The Authors 0. This article is published with open access at Springerlink.com Abstract Serious epidemics, both in cyber space as well as in our real world, are expected to occur with high probability, which justifies investigations in virus spread models in contact networks. The N-intertwined virus spread model of the SIS-type is introduced as a promising and analytically tractable model of which the steady-state behavior is fairly completely determined. Compared to the exact SIS Markov model, the N-intertwined model makes only one approximation of a mean-field kind that results in upper bounding the exact model for finite network size N and improves in accuracy with N. Wereview manypropertiestheoretically,therebyshowing,besides the flexibility to extend the model into an entire heterogeneous setting, that much insight can be gained that is hidden in the exact Markov model. Keywords Epidemics Networks Robustness Mean-field approximation Mathematics Subject Classification C50 05C8 37H0 46N0 46N30 60J8 Introduction We investigate the influence of the network topology on the spread of viruses, whose dynamics is modeled by a susceptible-infected-susceptible SIS type of process. The SIS disease model [,3,0] can be regarded as one of the simplest virus infection models, in which persons or nodes in a network are either in two states: healthy, but susceptible to infection or infected by the disease or virus and, thus, infectious to neighbors. In this article, the word virus is understood in the most general wording possible as an item transferred from the surroundings towards a node in the network. P. Van Mieghem B Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, P.O. Box 503, 600 GA, Delft, The Netherlands P.F.A.VanMieghem@tudelft.nl

2 48 P. Van Mieghem At a node, the item can be destroyed, but can also be propagated to neighbors of that node. For example, digital viruses of all kind, and generally called malware are living in cyberspace and use mainly the Internet as the transport media, while biological viruses contaminate other living beings and use contacts among their victims as their propagation network. A digital virus here can also mean a rumor, news or any kind of information that spreads over a data communications or social network. Recently, Hill et al. [7] haveconsideredhappinessofpersonsasaform of social infection and have modeled the spread of emotions over the social contact network as a SIS-type of epidemic. Our SIS, continuous-time model for the spreading of a virus in a network, called the N-intertwined virus spread model [36], is reviewed in Sect.. Itwasearlierconsid- ered by Ganesh et al. [3] andbywangetal.[40] indiscrete-time,whosepaperwas later improved in [7]afterwhichtheirdiscrete-timemodelalsoappearedinthephysics community [5]. An infected node can infect its neighbors with an infection rate β per link, but it is cured with curing rate δ per node. However, once cured and healthy, the node is again prone to the virus. Both infection and curing processes are independent. There exists a wealth of variants or refinements of the SIS model see e.g. [,0,4,,4]: there can be an incubation period, an infection rate that depends on the number of neighbors or that has a constant component as in [7], a curing process that takes a certain amount of time, and many other sophistications that we do not consider here. Whereas most of the recent contributions to epidemics in networks were made by physicists, Durrett []doubtsaboutthemathematicalrigorofmanyoftheiranalyses. AremarkablepropertyoftheSISmodelistheappearanceofaphase-transition [4,6] whentheeffectiveinfectionrateτ = β δ approaches the epidemic threshold τ c = λ,whereλ is the largest eigenvalue of the adjacency matrix A,alsocalledthe spectral radius. Below the epidemic threshold, τ<τ c, the network is virus-free in the steady-state, while for τ>τ c,thereisalwaysafractionofnodesthatremainsinfected. For the companion infection model, SIR, where R stands for recovered or removed, Kermack and McKendrick [9] haveshownthatnoepidemiccanoccurifthepopu- lation density is below a critical threshold. Recently, Youssef and Scoglio [43] have extended the N-intertwined SIS model to SIR epidemics and they have shown that the SIR-epidemic threshold also equals τ c;sir = λ.wepointheretoanotherdynamic process on a network, that bears resemblance to virus spread. The synchronization of coupled oscillators in a network features a surprisingly similar phase transition: the onset of oscillator coupling occurs [8]atacriticalcouplingstrengthg c = g 0 λ and the behavior of the phase transition around g c is mathematically similar [3]. Synchronization [9]playsaroleinsensornetworks,humanbodyheartbeat,brain,epilepsy, light emission of fire-flies, etc. The major goal of this article is to provide a consistent, theoretical overview of the steady-state in the N-intertwined virus spread model, that we consider as almost completely solved since the recent solution [3] ofthebehavioraroundtheepidemic threshold. Section explains the N-intertwined SIS virus spread model and the meanfield approximation. From Sect. 3 on, we confine to the steady-state of the N-intertwined SIS model and present general equations for the fraction y of steady-state infected nodes, from which the continued fraction expansion is derived. In addition,

3 The N-intertwined SIS epidemic network model 49 two series for y are given, whose coefficients obey recursion relations, specified in Lemmas and 3. Section3. specifies the condition for the epidemic threshold τ c = λ and the behavior of y τ around τ c,withreferencestomanyapproaches in networks to enlarge the epidemic threshold. After arguing that the transform s = τ is more natural, bounds on y s around s = E[D] are presented in Sect. 4 inspired by simulations in [4] indicatingthaty s is close to for s = E[D].Section5 discusses the viral conductance ψ of a virus spreading process in a graph, that was first proposed in [0]. We end our review by extending the N-intertwined model to a heterogeneous setting in Sect. 6:thegoverningequationandthecontinuedfractionare readily found, while the convexity Theorem 5 has interesting applications in network protection strategies [6,6]. Section 7 concludes with an outlook on open problems. TheN-intertwined SIS model AnetworkisrepresentedbyanundirectedgraphG N, L with N nodes and L links. The network topology is described by a symmetric adjacency matrix A, inwhichthe element a ij = a ji = ifthereisalinkbetweennodesi and j, otherwisea ij = 0. In the sequel, we confine ourselves and make the following simplifying assumptions. The state of a node i is specified by a Bernoulli random variable X i {0, }: X i = 0fora healthy node and X i = foraninfectednode.anodei at time t can be in one of the two states: infected, with probability v i t = Pr[X i t = ] or healthy, with probability v i t.weassumethatthecuringprocesspernodei is a Poisson process with rate δ, and that the infection rate per link is a Poisson process with rate β. All involved Poisson processes are independent. The effective infection rate is defined as τ = β δ. We assume that the initial infection state v i 0 in each node i is known. This is the general description of the simplest type of a SIS virus spread model in a network and the challenge is to determine the virus infection probability v i t for each node i in the graph G at time t. This SIS model can be expressed exactly in terms of a continuous-time Markov model with N states as shown in [36]. Unfortunately, the exponentially increasing state space with N prevents the determination of the set of {v i t} i N in realistic networks, which has triggered a spur of research to find good approximate solutions. For an overview of SIS heuristics and numerous extensions, we refer to [4,,4]. In contrast to all published SIS-type of models, the N-intertwined model, proposed and investigated in depth in [36], only makes one mean-field approximation in the exact SIS model and is applicable to all graphs.. The mean-field approximation By separately observing each node, every node i at time t in the network has two states: infected with probability v i t = Pr[X i t = ] and healthy with probability Pr[X i t = 0] = v i t.ifweapplymarkovtheorystraightaway,theinfinitesimal generator Q i t of this two-state continuous Markov chain is, Q i t = [ ] q;i q ;i q ;i q ;i

4 50 P. Van Mieghem Fig. The mean-field approximation arrow transforms the random variable q ; j into the average E [ q ; j ].Insteadofbeinginfectedattimet by the precise number of infected neighbors, the node i is now infected by the average number of infected neighbors with q ;i = δ is the curing rate and q ;i = β a ij {X j t=} where the indicator function x = iftheeventx is true else it is zero. The rate q ;i equals the sum over all infection rates of infected neighbors of node i and this rate q ;i couples or intertwines node i to the rest of the network through the appearance of the events { X j t = }.Asmentionedin[36], the total infection rate q ;i is a random variable, whereas ordinary Markov theory requires that q ; j is a real number. The random nature of q ;i is removed by an additional conditioning to all possible combinations of rates, which is equivalent to conditioning to all possible combinations of the states X j t = andtheircomplementsx j t = 0 of the neighbors of node i. Hence, the number of basic states in the Markov process dramatically increases from two states per node to all possible combinations of states for N nodes. Eventually, after conditioning each node in such a way, we end up with the exact N -state Markov chain, specified in [36]. Instead of conditioning, the mean-field approximation consists of replacing q ;i by its average E [ q ;i ],whichisarealnumberandallowsimmediateapplicationofcontinuous-time Markov theory [30]. Figure illustrates the mean-field approximation. Using E [ x ] = Pr [x],wereplaceq ;i by E [ ] N q ;i = β a ij Pr[X j t = ] =β a ij v j t which results in an effective infinitesimal generator, [ [ ] [ ] ] E q;i E q;i Q i t = δ δ Due to the dependence of E [ q ;i ] on v j t, themarkovdifferentialequation[30, 0. on p. 8] for state X i t = turnsouttobenon-linear,

5 The N-intertwined SIS epidemic network model 5 dv i t dt = β v i t a ij v j t δv i t The governing differential equation inthen-intertwined model for a node i has the following physical interpretation: the time-derivative of the infection probability of a node i consists of two competing processes: while healthy with probability v i t,allinfectedneighbors,aneventwithprobability N a ij v j t, try to infect the node i with rate β and while infected with probability v i t,thenodei is cured at rate δ. Thisratherintuitiveexplanationhasbeendirectlyusedinformermodels such as the Kephart and White model [8]toderivethedifferentialequation,thereby implicitly making a mean-field approximation. Defining the vector V t = [ v t v t v N t ] T,thematrixrepresentation based on becomes dv t dt = β A δi V t βdiag v i t AV t where diag v i t is the diagonal matrix with elements v t,v t,...,v N t.we define the average fraction of infected nodes in the network at time t as y t = N E {X j t=} = N v j t 3 In [5,36], we show that the mean-field approximation implies that a the N-intertwined model upperbounds the exact probability v i t of infection, b the deviations between the N-intertwined and the exact model are largest for intermediate values of τ around τ c and c the random variables X j and X i are implicitly assumed to be independent. Since the latter basic assumption is increasingly good for large N, we expect that the deductions from the N-intertwined model are asymptotically for N almost exact for real-world networks. Figure shows 500 sample paths of the exact SIS process, together with the steadystate fraction y τ of infected nodes. Although the steady-state fraction is constant, the SIS infection process continues for ever, which means that an arbitrary node i moves between the healthy and infected state for v i and v i percent of the time, respectively. 3Thesteady-statefractiony of infected nodes In this section, we focus on the steady-state of the N-intertwined model, where dv v i = lim t v i t and lim i t t dt = 0. The corresponding steady-state vector is denoted by V.IntheexactSISmodel,thesteady-stateisthehealthystate,which is the only absorbing state in the Markov process. However, in networks of realistic size N, thissteady-stateisonlyreachedafteranunrealisticallylongtime[3]. The

6 5 P. Van Mieghem Fig. A time-dependent simulation of the exact SIS number of infected nodes Nyt; τ in the complete bipartite graph K 0,990 for τ = 0.5. The steady-state fraction 8 of infected nodes in the N-intertwined model gives y 0.5 = 0.6, which is the mean value of the 500 realizations of the spreading process steady-state in the N-intertwined virus spread model thus refers to the metastable state, which is reached exponentially fast and which reflects real epidemics more closely. In fact, the mean-field approximation, which transforms the linear set of N differential equations of the exact Markov chain into a set of N non-linear differential equations in, has induced the existence of the metastable state as well as the phase transition and the corresponding epidemic threshold, discussed in Sect. 3.. From, we obtain with dv i t dt = 0andv i = lim t v i t, v i = β N a ij v j β N a ij v j + δ = + τ N a ij v j 4 Beside the trivial solution v i = 0, 4 illustratesthatthereisanotherpositivesolution reflecting the metastable state in which we are interested here. For regular graphs, where each node has degree d, symmetryinthesteady-stateimpliesthatv i = v for all nodes i and it follows from 4withthedefinitionofthedegreed i = N a ij that v ;regular = y ; regular τ = τd 5 where y = N Ni= v i is the fraction of infected nodes in the steady-state as deduced from the definition 3. Besides the regular graph, [36] gives an exact

7 The N-intertwined SIS epidemic network model 53 solution of the steady-state infection behavior in the complete bipartite graph K m,n, where there are two partitions N m with m nodes and N n with n nodes [3] sothat N = m + n and L = mn, v i = mn τ τ + m n i N n 6 and v j = mn τ τ + n m j N m 7 Thus, since y = nv i +mv j n+m,weobtain mn τ { } y τ = N τ + m + τ + n 8 The complete bipartite graph, of which the star K,n is a special case, often appears as our benchmark model see e.g. [,3,33]. The nodal infection steady-state equation 4 can be solved,as proved in[36] and alternatively in [33]: Theorem For any effective spreading rate τ = β δ 0, thenon-zerosteady-state infection probability of any node i in the N-intertwined model can be expressed as a continued fraction v i = + τd i τ N a ij +τd j τ N a jk k= +τd k τ N q= a kq +τdq... 9 where d i = N a ij is the degree of node i. Consequently, the exact steady-state infection probability of any node i is bounded by 0 v i + τd i 0 The interesting feature of the continued fraction 9 is that each convergent is an upper bound for v i.another,equallyuseful,representationforv i is the Laurent series, proved in [33]: Lemma The Laurent series of the steady-state infection probability v i τ = + η m i τ m m=

8 54 P. Van Mieghem possesses the coefficients η i = d i and η i = d i + d i a ij d j 3 and for m,thecoefficientsobeytherecursion η m+ i = N d i η m i a ij d j m + η m+ k i a ij η k j 4 Consequently, the Laurent series for the steady-state fraction of infected nodes equals y τ = + N k= { N } η m i τ m 5 m= i= 3. General relations for y Summing overalli is equivalent to right multiplication of V t by the all one vector u T because N i= v i t = u T V t.then,wefindfromthat du T V t dt = u T diag v i t β A δi V t = β u V t T AV t δu T V t Hence, we obtain a relation for y [0, ] in terms of the vector V : Ny = u T V = τ u V T AV 6 Since u T A = D T because A = A T,wecanwrite6 as y τ = τ N D T V V T AV 7 We write the degree vector D as D = u,where = diagd, d,...,d N,sothat Ny = τ u T V + V T V V T V V T AV = τ u V T V + V T A V

9 The N-intertwined SIS epidemic network model 55 Introducing the Laplacian Q = A of the graph G, thesteady-statefractionof infected nodes y is expressed as a quadratic form in terms of the Laplacian, y = τ N u V T V + V T QV After left-multiplication of the steady state version of bythevector V T diag vi k = [ v k vk ] T vk N 8 which we denote by V k T,weobtainthescalar V k T V = T v k+ j = τ V k AV V k+ T AV 9 For k = 0in9, and introducing the all one vector u = lim k 0 V k,weobtain6 again. For k = in9, the norm V = V T V = N v j obeys V T V = τ V T AV V T diag v i AV 0 When summing 9 overallk from m 0toinfinityandtaking v j < into account, the telescoping nature of the right-hand side leads to k=m V k T V = v m+ j = τ V m T AV v j When m = 0, we have that V m = u and we obtain, with the degree vector ut A = D T, τ v j v j = D T V = d j v j As shown earlier in [36], the characteristic structure ofthen-intertwined model follows more elegantly from the governing equation inthesteady-state V = τdiag v i AV 3 for finite τ such that v i <. Indeed, after left-multiplying both sides by diag vi = diag v i,wehave τ diag v i V = AV

10 56 P. Van Mieghem or where the vector V V T = [ V = AV 4 τ V v v 4 by V m T,weobtain again. v v v N v N ] T. By left-multiplication of 3. Phase transition and epidemic threshold Many authors see e.g. [3,0,8,7] mention the existence of an epidemic threshold τ c.iftheeffectivespreadingrateτ = β δ >τ c,theviruspersistsandanon-zerofraction of the nodes are infected, whereas for τ τ c,theepidemicdiesout.thefactthat the epidemic threshold occurs at τ = τ c = λ has been proved in several papers, see e.g. [7,36]. Here, we recall the fundamental lemma for the N-intertwined SIS model, proved in [36]. Lemma There exists a value τ c = λ > 0 and for τ<τ c,thereisonlythetrivial steady-state solution V = 0.BesidetheV = 0 solution, there is a second, non-zero solution for all τ>τ c.forτ = τ c + ε, itholdsthatv = αx,whereε, α > 0 are arbitrarily small constants and where x is the eigenvector belonging to the largest eigenvalue λ of the adjacency matrix A. Lemma has important practical consequences. Given a network with adjacency matrix A and an imminent infection rate β,anodalcuringrateδ>βλ can be applied in nodes in the form of anti-virus software or any other protection scheme to maintain the network virus-free. A key-point is that the security of each host depends not only on the protection strategies it chooses to adopt but also on those chosen by other hosts in the network. In a heterogeneous setting, explained in Sect. 6, the resulting gametheoretic optimum has been studied in [6], while a different optimization technique in [6] minimizestheoverallinfectioninthenetworkbydeterminingtheindividual curing rates of nodes. When the network can be modified, we possess a much larger number of ways to ban epidemics such as immunization strategies [8] and several ways to decrease the spectral radius λ of the network: quarantining using the modular form of the network [4], degree-preserving rewiring [34,38,39] that changes the assortativity and modularity, and hence, the spectral radius. The optimal strategy to remove m links from the network in order to minimize λ is proved in [37] tobe NP-hard. Consequently, several heuristics are proposed and evaluated in [37]. Lemma shows that, for all graphs, V = αx +ξ y,wherey is a vector orthogonal to x,α tends to zero as τ τ c,whileξ tends faster to zero in that limit than α. The following theorem is proved in [3]: Theorem For any graph with spectral radius λ and corresponding eigenvector x normalized such that x T x = N x j =, thesteady-statefractionofinfected

11 The N-intertwined SIS epidemic network model 57 nodes y obeys y τ = N x j λ N N x 3 τc τ + O τc τ j 5 when τ approaches the epidemic threshold τ c from above. Since the eigenvectors x, x,...,x N belonging to the eigenvalues λ λ λ N of the adjacency matrix A span the N-dimensional vector space, we can write the steady-state infection probability vector V τ as a linear combination of the eigenvectors of A, V τ = γ k τ x k 6 k= where the coefficient γ k τ = x T k V τ is the scalar product of V τ and the eigenvector x k and where the eigenvector x k obeys the normalization x T k x k =. Physically, 6mapsthedynamicsV τ of the process onto the eigenstructure of the network, where γ k τ determines the importance of the process in a certain eigendirection of the graph. The definition y τ = N ut V τ shows that y τ = N γ k τ u T x k 7 k= Substitution of 6 into6 yields y τ = τ N k= λ k γ k τ u T x k γ k τ 8 For irregular graphs, generally, γ m τ = x T m V τ = 0form > andalleigenvalues and eigenvectors in 8playarole.Moreover,γ m τ can be negative, as well as λ m,while N k= λ k = 0see[3, p.30].thelargerthespectralgapλ λ and the smaller λ N,themorey is determined by the dominant k = termin8, and the more its viral behavior approaches that of a regular graph. Graphs with large spectral gap possess strong topological robustness [3], in the sense that it is difficult to tear that network apart. Theorem suggests, for all k N, theexistenceofthepowerseries γ k τ = c j k τc τ j 9

12 58 P. Van Mieghem where c k = 0for k N, c = λ N and x j 3 all other coefficients c j k can be determined in a recursive way as specified in the following lemma, which is proved in [33]: Lemma 3 Defining X m, l, k = x m q x l q x k q the coefficients c j m in 9 obey, for m > and j >, therecursion q= c j m = c j m λ λ m { c λ + λ m X m, m, } c λ λ m λ λ m while, for j = and m >, k=;k =m j n= l= k= λ + λ k c j k X m, k, c j n l c n k λ k X m, l, k X m,, c m = λ λ m λ X,, and c m = 0. Form =, thereholdsthatc = j >, thecoefficientsc j satisfy the recursion c j = λ X,, n= l= k= λ + λ k c j k X,, k k= j c j+ n l c n k λ k X, l, k λ N x 3 j and for The radius of convergence of the Laurent series andoftheseries9 is,in general, unknown and still an open problem. From the definition 7, we obtain the series expansion of y τ around τc τ as { } y τ = c j k u T x k τc N k= τ j 30

13 The N-intertwined SIS epidemic network model 59 Fig. 3 The steady-state fraction y of infected nodes versus s = /τ and vs. τ in the inset for a regular graph with degree d = andastar.bothgraphshaven = 00 nodes and almost the same average degree valid for τ λ.similarly,fromtheeigenvectorexpansion6, all steady-state infection probabilities v i τ are expanded as v i τ = { N } c j kx k i τc k= τ j Finally, Sahneh and Scoglio [] haveextendedthe N-intertwined model to three states: besides the susceptible and infected state, an alert state is introduced, that takes the behavior into account when a node realizes that it is surrounded by infected neighbors. The three-state extension is shown to exhibit two distinct epidemic thresholds. Between the two thresholds, the infection spreads at first, but eventually dies out due to increased alertness in the network. 3.3 Transform s = τ Figure 3 illustrates the typical behavior of y s versus s = τ and y τ versus τ in the insert for a regular and irregular graph. Theoretically though debatable, one might argue that the effective curing rate s = β δ = τ is more natural than the effective infection rate τ = β δ,becauseataylorexpansionofy s around s = 0exists,while the corresponding one 5 fory τ is a Laurent series in τ around τ.the Taylor expansion of y s around s c = λ = τc follows directly from 30.

14 60 P. Van Mieghem 4Behaviorofy s around s = E[D] In [5], we have shown, for any graph, that y for τ E[D]. Moreover, simulations in [5] indicatethatthemaximumvarianceforthen-intertwined model is reached for τ E[D]. Via extensive simulations, Youssef et al. [4] haveobservedthaty around s = E[D]. Inthissection, severallemmas, provedin[33], explain and support these simulations. Lemma 4 For any graph, it holds that y s + E [D] for s = E [D] λ 3 and equality is only possible for the regular graph. Lemma 5 For any graph, it holds that E [D] y τ τ 4 + V T QV N 3 and equality is only possible for the regular graph for which V T QV = 0. Lemma 5 shows for the regular graph that the tangent line through the origin τ = 0 lies above y ;regular and only touches y ;regular at the point τ = E[D] = d. For any other graph, the slope is not larger than E[D] 4 + V T QV N and, after transforming s = τ in 3, we find y s + V T QV NE[D] that complements 3. The correction for s = E [D] 33 E[D] λ in 3andthecorrection V T QV NE[D] in 33 arepositiveandsmall,butneverthelessshowthaty s can be larger than at s = E[D],asnumericallyfoundin[4]. Lemma 6 For any graph, it holds that y τ τ r τ = N where the lower bound obeys, for any τ, v j d j λ v j 34 r τ N 4Nλ and where N k = u T A k udenotesthenumberofwalksoflengthk.

15 The N-intertwined SIS epidemic network model 6 We will now estimate the value ξ for which r ξ = N 4Nλ in irregular graphs where Var[D] > 0. Provided that there exists a value of τ = ξ for which v j ξ = d j λ for each j N,thatmaximizesr τ, wehavethat y ξ = N d j = L λ λ N = E [D] < λ Lemma 6 then states that y ξ ξ N 4Nλ or from which λ L N ξ N 4Nλ ξ 4L E [D] = N E [D] + Var [D] < E [D] In conclusion, there may exist a value ξ such that τ c <ξ< E[D] for which y ξ = E[D] λ <. We end by deducing, approximately though, another type of lower bound. Concavity of y τ for τ τ c = λ similarly leads to y qτ c + q mτ c q y mτ c.forsufficientlylargem, thelaurentseries up to first order leads to y mτ c mτ c N because the second order term of O Choosing qτ c + q mτ c = E[D] y E [D] > mτ c d j provides us with E[D] τ c m τ c mτ c N λ E [D] m is positive due to η i > 0in3. d j λ E [ ] D m Ignoring the integer nature of m, the maximizer of the right-hand side occurs at m = λ E [ D ], resulting in

16 6 P. Van Mieghem Notice that λ E[D] [ ] 4λ E D λ E[D] y E [D] 4λ E [ ] D λ E[D] < 4 λ. E[D] In summary, both last lower bound arguments illustrate, together with the upper bounds in Lemmas 4 and 5 that, for values of τ approaching E[D],thesteady-statefraction of infected nodes y τ is close to in any graph, in agreement with simulations [4]. 5Theviralconductance The viral conductance ψ of a virus spreading process in a graph was first proposed in [0] as a new graph metric and then elaborated in more detail in[4]. The viral conductance ψ is defined as ψ = λ 0 y s ds 35 where s = τ and λ is the spectral radius i.e. the largest eigenvalue of the adjacency matrix A of the graph and equal to s c = τ c.belowtheepidemicthreshold τ c,thenetworkisvirus-freeinthesteady-state.hence,v i τ = 0forτ<τ c,and equivalently, v i s = 0fors > τ c = λ.sincethefunctiony τ versus τ is not integrable over all τ,kooijetal.[0]haveproposedtoconsidery τ versus s = τ see Fig. 3. In most published work so far, network G was considered to be more robust against virus spread than network G if the epidemic threshold τ c G >τ c G. For example, in Fig. 3,theregulargraphwiththesamenumberN of nodes and nearly the same number L of links possesses a higher epidemic threshold than the star, and, thus, according to the above robustness criterion, the regular graph is more robust against virus propagation than the star. However, when the effective infection rate τ> = τ c Gregular,weobservefromFig.3 that the percentage of infected nodes in the star is smaller than in the regular graph. The extent of the virus-free region is one aspect of the network s resilience against viruses, but once that barrier, the epidemic threshold, is crossed, the virus may conduct differently in networks with high and low epidemic threshold. This observation has led Kooij et al. [0] topropose theviral conductance as an additional metric. The viral conductance ψ is a graph metric that represents the overall conductance of the virus for all possible effective infection rates τ: whenψ is high for the graph G, theviruscanspreadeasilying. Thus,insteadofgradinggraphsonlybasedon their epidemic threshold τ c from virus vulnerable, where τ c is small high spectral radius λ to virus robust where τ c is large, the viral conductance complements this classification with an average infection notion because

17 The N-intertwined SIS epidemic network model 63 y = λ λ 0 y s ds < so that ψ = y τc < τ c = λ.graphswithsmallepidemicthresholdmaypossessa small average fraction of infected nodes y so that the viral conductance can be equal to graphs with large epidemic threshold and large y. Using both expansions 5 and30 intothe definition35 of ψ yields, subject to the condition that the radius of convergence of both series is at least λ, where the remainder is { ψ = λ N N d k= k R = N m= j k= u T x λ N x 3 j } λ 8 + R { N } η m k + c m k u T λ m+ x k m+ m + Subject to the above convergence condition, the expansion R j can be numerically computed up to any desired accuracy when the adjacency matrix A is given, from which the eigenvectors x, x,...,x N belonging to the eigenvalues λ λ λ N can be computed. Several bounds for the viral conductance are derived in [33] of which we only recall here a few. A simple upper bound, deduced from the convexity of y s in s [0,λ,is ψ λ More accurate lower and upper bounds are ψ λ { Z + ζ Z + min s [0,λ ] y s { λ 3 3λ ζ + 3ζ }} 36 and ψ λ { Z + ζ Z + max s [0,λ ] y s {λ 3 3λ ζ + 3ζ }} where Z = N N x j N x 3 j d min < andζ = Z [ λ E D ] Z. Other types of bounds are λ ψ<ln + {E [D] } d min

18 64 P. Van Mieghem The largest possible ratio λ d min viral conductance is bounded by N = O occurs in the star, which illustrates that the E [D] log N. Likely,thestarpossessesthelargest viral conductance among all graphs with N nodes and L links. It is further conjectured in [4] thattheregulargraphattainsthelowestviralconductanceallgraphswithn nodes and L links. This conjecture has only partially been proved so far in [33] tobe true for the class of complete bipartite graphs, but not yet for all graphs. 6HeterogeneousN-intertwined model The homogenous N-intertwined model, where the infection and curing rate is the same for each link and node in the network, has been extended to a heterogeneous setting in [35], where an infected node i can infect its neighbors with an infection rate β i,but it is cured with curing rate δ i. Heterogeneity rather than homogeneity abounds in real networks. For example, in data communications networks, the transmission capacity, age, performance, installed software, security level and other properties of networked computers are generally different. Social and biological networks are very diverse: a population often consists of a mix of weak and strong, or old and young species or of completely different types of species. The network topologies for transport by airplane, car, train, ship are different. Many more examples can be added illustrating that homogeneous networks are the exception rather than the rule. This diversity in the nodes and links of real networks will thus likely affect the spreading pattern of viruses. In previous Sections, only a homogeneous virus spread was investigated, where all infection rates β i = β and all curing rates δ i = δ were the same for each node. We believe that the extension to a full heterogeneous setting is, perhaps, the best SIS model that we can achieve. The governing differential equation isstraightforwardlygeneralizedto dv i t dt = β j a ij v j t v i t β j a ij v j t + δ i 37 while the corresponding matrix equation is dv t dt = Adiag β j V t diag vi t Adiag β j V t + C 38 where diagv i t is the diagonal matrix with elements v t,v t,...,v N t and the curing rate vector is C = δ,δ,...,δ N.WenotethatA diagβ i is, in general and opposed to the homogeneous setting, not symmetric anymore, unless A and diagβ i commute, in which case the eigenvalue λ i Adiag β i = λ i A β i and both β i and λ i A have a same eigenvector x i.

19 The N-intertwined SIS epidemic network model The steady-state The metastable steady-state follows from 38 as Adiag β i V diag v i Adiag β i V + C = 0 where V = lim t V t.wedefinethevector and write the stead-state equation as or w = Adiag β i V + C 39 w C = diag v i w I diag v i w = C Ignoring extreme virus spread conditions the absence of curing δ i = 0 and an infinitely strong infection rate β i, then the infection probabilities v i cannot be one such that the matrix I diag v i = diag v i is invertible. Hence, w = diag Invoking the definition 39ofw, weobtain v i C vi Adiag β i V = diag C v i δ i = diag V 40 v i that generalizes 4. The i-th row of 40 yieldsthenodalsteadystateequation, a ij β j v j = v i δ i 4 v i Let Ṽ = diagβ i V and the effective spreading rate for node i,τ i = β i δ i,thenwe arrive at Q τ i v i Ṽ = 0 4

20 66 P. Van Mieghem where the symmetric matrix Q q i = diag q i A 43 = diag q i d i + Q can be interpreted as a generalized Laplacian,becauseQ d i = Q = A,where = diagd i.theobservationthatthenon-linearsetofsteady-stateequationscanbe written in terms of the generalized Laplacian Q q i is fortunate, because the powerful theory of the normal Laplacian Q applies. Many properties of the generalized Laplacian Q q i are given in [35] thatenabledtoprovethreeimportanttheorems. The first theorem is Theorem 3 The critical threshold is determined by vectors τ c = τ c,τ c,...,τ Nc that obey λ max R =, whereλ max R is the largest eigenvalue of the symmetric matrix R = diag τi Adiag τi whose corresponding eigenvector has positive components if the graph G is connected. Several bounds for λ max R are derived and λ max R for the complete graph K N is solved exactly. The generalization of Theorem is Theorem 4 The non-zero steady-state infection probability of any node i in the N-intertwined model can be expressed as a continued fraction 44 v i = + γ i δ i δi N β j a ij + γ j δ δ Nk= β k a jk j j + γ k δ δ k k Nq= a qk βq where the total infection rate of node i, incurred by all neighbors towards node i, is γ i = a ij β j = j neighbori β j 46 Consequently, the exact steady-state infection probability of any node i is bounded by 0 v i + γ i δ i Perhaps the most important theorem proved in [35] is All eigenvalues of the Laplacian Q = A in a connected graph are positive, except for the smallest one that is zero. Hence, Q is positive semi-definite. Much more properties of the Laplacian Q are found e.g. in [5,9,3].

21 The N-intertwined SIS epidemic network model 67 Theorem 5 Given that all curing rates δ j for j = i Nareconstantand independent from the infection rates β j,thenon-zerosteady-stateinfectionprobability v i δ,...,δ i,...,δ N > 0 is strict convex in δ i,whileallothernon-zero steady-state infection probabilities v j δ,...,δ i,...,δ N > 0 are concave in δ i. AdirectconsequenceofTheorem5 to the homogeneous setting is that y s is convex for s [0,λ or y τ is concave for τ>τ c. 7Conclusion The N-intertwined SIS network model has been introduced and many derived results in the steady-state have been reviewed. While extensions of the model are certainly expected in the future, we believe that the steady-state theory of the homogeneous N-intertwined SIS network model is almost entirely established. The time-dependent theory, on the other hand, needs much more efforts towards maturity. Although the N-intertwined model is not exact, the only a mean-field approximation has enabled analytic computations as presented here that are, to the best of our knowledge, not possible with any other SIS model that is more accurate than the N-intertwined model. An open problem is to determine of the overall accuracy of the N-intertwined model with respect to the exact SIS Markov process for any value of τ in a broad class of interesting networks. So far, numerical simulations [] point to a promisingly good accuracy that improves with N. Anewlyenvisioneddirectionisthecouplingofthevirusspreadprocesswiththe underlying topology. In other words, the presented model has assumed that the adjacency matrix A is fixed and is not changed by the virus spread process. Hence, nodes can only protect themselves against the virus by increasing their curing rate δ. While taking medicine or vaccination is one measure in the fight against the virus, a more natural one is to avoid contact with infected people. The latter assumes that the adjacency matrix A is changed by the process and the knowledge that a node s neighbors is are infected. The precise description of the coupling between virus spread process and topology as well as the solution of the far more complex set of differential equations stand on the agenda of future research. Acknowledgements We are very grateful to Caterina Scoglio, Mina Youssef, Faryad Darabi Sahneh, Cong Li and Rob Kooij for many comments on an early version of the manuscript. This research was supported by Next Generation Infrastructures Bsik and the EU FP7 project ResumeNet project No Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original authors and source are credited. References. Anderson RM, May RM 99 Infectious diseases of humans: dynamics and control. Oxford University Press, Oxford. Asavathiratham C 000 The influence model: a tractable representation for the dynamics of networked markov chains. Ph.D thesis, Massachusetts Institute of Technology, Cambridge

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