Clustering for epidemics on networks: a geometric approach
CClustering for epidemics on networks: a geometric approach
Bastian Prasse ∗ , Karel Devriendt † and Piet Van Mieghem ∗ March 1, 2021
Abstract
Infectious diseases typically spread over a contact network with millions of individuals, whosesheer size is a tremendous challenge to analysing and controlling an epidemic outbreak. For somecontact networks, it is possible to group individuals into clusters. A high-level description of theepidemic between a few clusters is considerably simpler than on an individual level. However, tocluster individuals, most studies rely on equitable partitions, a rather restrictive structural propertyof the contact network. In this work, we focus on Susceptible-Infected-Susceptible (SIS) epidemics,and our contribution is threefold. First, we propose a geometric approach to specify all networksfor which an epidemic outbreak simplifies to the interaction of only a few clusters. Second, for thecomplete graph and any initial viral state vectors, we derive the closed-form solution of the nonlineardifferential equations of the N -Intertwined Mean-Field Approximation (NIMFA) of the SIS process.Third, by relaxing the notion of equitable partitions, we derive low-complexity approximations andbounds for epidemics on arbitrary contact networks. Our results are an important step towardsunderstanding and controlling epidemics on large networks. Modern epidemiology encompasses a broad range of spreading phenomena [27, 23, 16]. The majorityof viruses spread through a population of tremendous size, which renders individual-based modellingimpractical. However, most applications do not require to model an epidemic on individual level.Instead, a mesoscale description of the epidemic often is sufficient. For instance, suppose the outbreakof a virus is modelled on the level of neighbourhoods. Then, sophisticated lockdown measures canbe deployed which constrain neighbourhoods differently, depending on the prevalence of the virus inthe respective neighbourhood. The natural way to obtain a mesoscale description of the epidemicis clustering (or grouping) of individuals, for instance, by assigning individuals with similar age orlocation to the same cluster. Thus, all individuals in one cluster are considered indistinguishable andexchangeable. Additionally to the complexity reduction, clustering for epidemics on networks hasthe advantage that, on a mesoscale description, temporal fluctuations of the individual-based contactnetwork may average out. ∗ Faculty of Electrical Engineering, Mathematics and Computer Science, P.O Box 5031, 2600 GA Delft, The Nether-lands; email : [email protected], [email protected] † Mathematical Institute, University of Oxford, Oxford UK; email : [email protected]; Also at Alan TuringInstitute, London, UK a r X i v : . [ phy s i c s . s o c - ph ] F e b e consider a contact network with N nodes. Every node i = 1 , ..., N corresponds to an individualor a group of individuals. We focus on the Susceptible-Infected-Susceptible (SIS) epidemic process inan individual-based mean-field approximation, where every node i has a viral state v i ( t ) ∈ [0 ,
1] atevery time t . The evolution of the viral state v i ( t ) is governed by a set of N nonlinear differentialequations: Definition 1 (NIMFA [18, 49, 44]) . For every node i , the viral state v i ( t ) evolves in continuous time t ≥ as dv i ( t ) dt = − δ i v i ( t ) + (1 − v i ( t )) N (cid:88) j =1 β ij v j ( t ) , (1) where δ i > is the curing rate of node i , and β ij > is the infection rate from node j to i . If the nodes correspond to individuals, then the differential equations (1) follow from a mean-field approximation of the stochastic SIS process [49, 47], and the viral state v i ( t ) approximates theexpected value E[ X i ( t )] of the zero-one state X i ( t ) of the stochastic SIS process. For a zero-one, orBernoulli, random variable the expectation E[ X i ( t )] is equal to the probability Pr[ X i ( t ) = 1] that node i is infected at time t . In the remainder of this work, we refer to (1) as NIMFA, which stands for “ N -Intertwined Mean-Field Approximation” [49, 47]. The advantage of NIMFA is that the SIS Markovchain with 2 N states is approximated by N nonlinear differential equations. NIMFA follows from theSIS process by the approximation E[ X i ( t ) X j ( t )] ≈ E[ X i ( t )] E[ X j ( t )]. Around the epidemic threshold,the approximation of the stochastic SIS process by NIMFA might be inaccurate [49]. Furthermore, westress that NIMFA (1) assumes that the viral dynamics are Markovian and that the infection rates β ij do not depend on time t . Markovian and non-Markovian viral dynamics can be substantially different[45].The contact network, assumed to be fixed and time-invariant, corresponds to the N × N infectionrate matrix B , which is composed of the elements β ij . We denote by diag( x ) the N × N diagonalmatrix with the vector components of x ∈ R N on its diagonal. We denote the N × N curing ratematrix S = diag( δ , ..., δ N ). Then, the matrix representation of NIMFA (1) is dv ( t ) dt = − Sv ( t ) + diag ( u − v ( t )) Bv ( t ) , (2)where v ( t ) = ( v ( t ) , ..., v N ( t )) T is the viral state vector at time t , and u is the N × Homogeneous
NIMFA [49] assumes the same infection rate β and curing rate δ for all nodes, dv ( t ) dt = − δv ( t ) + β diag ( u − v ( t )) Av ( t ) , (3)where A is an N × N zero-one adjacency matrix.For NIMFA (1), the basic reproduction number R follows [42] as R = ρ ( S − B ) , (4)where ρ ( M ) denotes the spectral radius of a square matrix M . Around the epidemic threshold R ,there is a bifurcation [18]. If R ≤
1, then the all-healthy state, v i ( t ) = 0 for all nodes i , is the only2quilibrium of NIMFA (2), and it holds that v ( t ) → t → ∞ . If R >
1, then there is a secondequilibrium, the steady-state vector v ∞ , with positive components, and it holds that v ( t ) → v ∞ as t → ∞ , if v (0) (cid:54) = 0.Many papers deal with clustering of individuals into communities [7, 1, 29], where individualswithin the same community are densely connected, and there are only few links between individuals ofdifferent communities. Hence, communities are defined by structural properties of the contact graph.Most results are of the type: if the network has a certain mesoscale structure, then also the dynamicshave some structure [3, 24, 5]. In this work, we approach clustering from the other direction: wepresume structure in the dynamics and aim to find all contact networks that are compatible with thestructured dynamics.The central analysis tool in our analysis is the proper orthogonal decomposition (POD) [6] of the N × v ( t ), which is given by v ( t ) = m (cid:88) l =1 c l ( t ) y l (5)for some m ≤ N . Here, the N × agitation mode vectors y , ..., y m are orthonormal , and the scalarfunctions c l ( t ) = y Tl v ( t ) are obtained by projecting the viral state v ( t ) onto the vector y l . Since any N × v ( t ) can be written as the linear combination of N orthonormal vectors, the POD (5) isexact for any network if m = N . However, we are particularly interested in networks, for which thenumber of agitation modes m is (much) smaller than the number of nodes N . If (5) holds true, thenthe viral state vector v ( t ) is element of the m dimensional subspace V = span { y , ..., y m } (6)at any time t , where the span (the set of all linear combinations) of the vectors y , ..., y m is denotedby span { y , ..., y m } = (cid:40) m (cid:88) l =1 c l y l (cid:12)(cid:12)(cid:12) c l ∈ R (cid:41) . With the POD (5), the viral state v ( t ) can be described with less than N differential equations: denotethe right side of the NIMFA (2) by f NIMFA ( v ( t )) ∈ R N . Then, NIMFA (2) reads more compactly dv ( t ) dt = f NIMFA ( v ( t )) . (7)With the POD (5), we obtain that m (cid:88) l =1 dc l ( t ) dt y l = f NIMFA (cid:32) m (cid:88) l =1 c l ( t ) y l (cid:33) . (8)Since the vectors y , ..., y m are orthonormal, we can project (8) onto the agitation modes y l to obtainthe differential equations dc l ( t ) dt = y Tl f NIMFA (cid:32) m (cid:88) l =1 c l ( t ) y l (cid:33) , l = 1 , ..., m. (9) A set of vectors y , ..., y m is orthonormal if y Tl y k = 0 for l (cid:54) = k and y Tl y k = 1 for l = k . N to thenumber of agitation modes m . We emphasise that the POD (5) is a hybrid of linear and nonlinearanalysis: The viral state v ( t ) equals a linear combination of the agitation modes y l , which are weightedby possibly nonlinear functions c l ( t ). In [32], we have shown that the POD (5) is an accurate approx-imation for a diverse class of dynamics on networks. In this work, we study under which conditionsthe POD (5) is exact for the NIMFA epidemic model (2). (a) Path graph. 𝑣 𝑡 𝑒 𝑒 𝑦 = 𝑒 𝑦 𝑣 (b) Viral state space. Figure 1:
Proper orthogonal decomposition for a path graph. (a) : A path graph with N = 3nodes. The top, middle and bottom nodes are labelled by 1, 2 and 3, respectively. (b) : The black curvedepicts the trajectory of the viral state v ( t ) in the Euclidean space R . The shaded area illustrates theviral state set V , which equals the span of the vectors y , y , given by (12). Provided that v (0) ∈ V ,the viral state v ( t ) remains in the subspace V at every time t . Example 1.
Consider homogeneous NIMFA (3) on the path graph in Figure 1a, for which the viralstate vector v ( t ) evolves as dv ( t ) dt = − δv ( t ) + β (1 − v ( t )) v ( t ) , (10) dv ( t ) dt = − δv ( t ) + β (1 − v ( t )) ( v ( t ) + v ( t )) ,dv ( t ) dt = − δv ( t ) + β (1 − v ( t )) v ( t ) . Suppose that the initial viral states of node 1 and 3 are equal, v (0) = v (0) . Then, it holds that v ( t ) = v ( t ) at all times t due to the symmetry of the path graph. Hence, the viral state vector v ( t ) = ( v ( t ) , v ( t ) , v ( t )) T satisfies v ( t ) = c ( t ) y + c ( t ) y , (11)4 here the orthonormal vectors y , y are given by y = 1 √ , y = . (12) As illustrated by Figure 1b, the viral state v ( t ) remains in the m = 2 dimensional subspace V =span { y , y } at all times t , provided that v (0) ∈ V . On the subspace V , (9) yields that the N = 3 differential equations (10) reduce to m = 2 equations dc ( t ) dt = − δc ( t ) + √ β (cid:18) − √ c ( t ) (cid:19) c ( t ) ,dc ( t ) dt = − δc ( t ) + 2 √ β (1 − c ( t )) c ( t ) , from which the viral state v ( t ) is obtained with (11). Two conditions must hold for the set V to reduce NIMFA to m differential equations. First, theset V must be an m dimensional subspace, spanned by the basis vectors y , ..., y m . Second, if the initialviral state v (0) is element of the set V , then the viral state v ( t ) must remain in the set V at everytime t >
0. Hence, the set V must be an invariant set of NIMFA. Thus, we consider the geometricproblem: Problem 1 (Clustering in NIMFA) . For a given number of nodes N and a given number m ≤ N of agitation modes, find all N × N infection rate matrices B and the corresponding N × agitationmodes y , ..., y m , such that V = span { y , ..., y m } is an invariant set of NIMFA (2). In contrast to Example 1, for which the agitation modes y , y follow rather straightforwardly,Problem 1 considers the interdependency of arbitrary graphs and invariant sets V in full generality.If m << N , then we expect that the invariant set V , and its basis vectors y l , reflect a macroscopicstructure, or a clustering, of the contact graph. For instance, the agitation mode y in Example 1indicates that the viral states v ( t ) and v ( t ) evolve equally and nodes 1 and 3 can be assigned to thesame cluster.Furthermore, the invariant set V allows for sophisticated, low-complexity control methods for theviral state v ( t ), see [23] for a survey of control methods. More specifically, consider that an affinecontrol method is applied to NIMFA (7), dv ( t ) dt = f NIMFA ( v ( t )) + m (cid:88) l =1 g l ( t ) y l . (13)Here, the scalar function g l ( t ) is the control of the l -th agitation mode y l . If the subspace V =span { y , ..., y m } is an invariant set of NIMFA (2), then V is also an invariant set of (13). Hence, on thesubspace V , the viral state v ( t ) can be controlled with only m distinct control inputs g ( t ) , ..., g m ( t ).If the agitation mode y l corresponds to a group of nodes, such as in Example 1, then the control g l ( t )is applied to all nodes of that group. For instance, g l ( t ) could be the viral state control of individualsof a certain age group and location. 5 Related work
Clustering in NIMFA is closely related to equitable partitions [38, 46, 37]. We denote a general partitionof the node set N = { , ..., N } by π = {N , ..., N r } . Here, the cells N , ..., N r are disjoint subsets ofthe node set N , such that N = N ∪ ... ∪ N r . We adapt the definition of equitable partitions in [21, 25]as: Definition 2 (Equitable partition) . Consider a symmetric N × N infection rate matrix B and apartition π = {N , ..., N r } of the node set N = { , ..., N } . The partition π is equitable if, for all cells l, p = 1 , ..., r , the infection rates β ik satisfy (cid:88) k ∈N l β ik = (cid:88) k ∈N l β jk ∀ i, j ∈ N p . For an equitable partition π , we define the degree from cell N l to cell N p as d pl = (cid:88) k ∈N l β ik (14)for some node i ∈ N p . Definition 2 states that, for an equitable partition π , the sum of the infectionrates (14) is the same for all nodes i ∈ N p . We denote the r × r quotient matrix by B π , whose elementsare defined as ( B π ) pl = d pl . Furthermore, we define the r × u r = (1 , ..., T .As shown by Bonaccorsi et al. [5] and Ottaviano et al. [25], NIMFA (2) can be reduced to r differential equations, provided that the infection rate matrix B has an equitable partition π with r cells. For our work, we summarise the results in [5, 25] as: Theorem 1 ([5, 25]) . Consider NIMFA (2) on an N × N infection rate matrix B with an equitablepartition π = {N , ..., N r } . Assume that δ i = δ j and v i (0) = v j (0) for all nodes i, j in the same cell N l . Then, it holds that v i ( t ) = v j ( t ) at every time t > for all nodes i, j ∈ N l and all l = 1 , ..., r .Furthermore, define the r × viral state vector v π ( t ) = ( v i ( t ) , ..., v i r ( t )) T and the r × r reduced-size curing rate matrix S π = diag ( δ i , ..., δ i r ) , (15) where i l denotes an arbitrary node in the cell N l . Then, the reduced-size viral state vector v π ( t ) evolvesas dv π ( t ) dt = − S π v π ( t ) + diag ( u r − v π ( t )) B π v π ( t ) . (16)Remarkably, on both microscopic (2) and macroscopic (16) resolutions, the viral dynamics fol-low the same class of governing equation. For the Markovian Susceptible-Infectious-Susceptible (SIS)process, Simon et al. [39] proposed a lumping approach to reduce the complexity, which is an ap-proximation and merges states of the SIS Markov chain, also see the work of Ward et al. [50]. In[8], a generalised mean-field framework for Markovian SIS epidemics has been proposed, which in-cludes NIMFA as a special case. Beyond epidemics, analogous results to Theorem 1 have been proved Slightly deviating from common notation, we also refer to π as an (equitable) partition of the infection rate matrix B . on networks with equitable partitions [10, 24, 28, 36, 9]. As a directconsequence of Theorem 1, equitable partitions are related to the proper orthogonal decomposition(5): Corollary 1.
Consider NIMFA (2) on an N × N infection rate matrix B with an equitable partition π = {N , ..., N r } . Assume that δ i = δ j and v i (0) = v j (0) for all nodes i, j in the same cell N l . Then,the subspace V = span { y , ..., y m } with m = r is an invariant set, where the N × agitation modes y l are given by ( y l ) i = √ |N l | if i ∈ N l , if i (cid:54)∈ N l , and the scalar functions equal c l ( t ) = (cid:112) |N l | v πl ( t ) . In other words, Corollary 1 states that every equitable partition π yields an invariant set V , whosedimension equals the number of cells r in the partition π . Example 2 illustrates Theorem 1 andCorollary 1: N N N Figure 2:
Graph with a partition of the node set.
A graph with N = 6 nodes and the partition π = {N , N , N } , whose cells are given by N = { , , } , N = { , } and N = { } . For unit linkweights, i.e., β ij = 1 for all nodes i, j , the partition π is not equitable. If the link weights β ij satisfy(17), as in Example 2, then the partition π is equitable. Example 2.
Consider NIMFA on a graph with N = 6 nodes, whose curing rate matrix equals S =diag (cid:16) ˜ δ , ˜ δ , ˜ δ , ˜ δ , ˜ δ , ˜ δ (cid:17) for some curing rates ˜ δ , ˜ δ , ˜ δ . Furthermore, suppose that the infection rate Specifically, we believe that Theorem 1 can be generalised to the dynamics dv i ( t ) dt = − δ i v i ( t )+ (cid:80) Nj =1 β ij g ( v i ( t ) , v j ( t )),where the arbitrary function g ( v i ( t ) , v j ( t )) describes the “coupling” [41, 4, 19, 32] between node i and j . atrix B is symmetric and given by the graph in Figure 2 as B = β β β β β β β β β β β β β β β β β β β . Suppose that, for some degrees d pl > , the infection rates β ij satisfy: β = β = d ; β = β = d and β = β = d / ; β = β = β = d ; and β = d . Then, the infection rate matrix B becomes B = d d d d d / d / d d d d d / d d d d / d d d d . (17) Thus, the matrix B has the equitable partition π = {N , N , N } with the cells N = { , , } , N = { , } and N = { } . The quotient matrix equals B π = d d d d d d . For the partition π , the reduced-size viral state can be chosen as v π ( t ) = ( v ( t ) , v ( t ) , v ( t )) T . Theo-rem 1 states that the vector v π ( t ) = ( v ( t ) , v ( t ) , v ( t )) T evolves as dv π ( t ) dt = − S π v π ( t ) + diag ( u − v π ( t )) B π v π ( t ) , with the × reduced-size curing rate matrix S π = diag (cid:16) ˜ δ , ˜ δ , ˜ δ (cid:17) . Furthermore, Corollary 1 statesthat the viral state v ( t ) has the proper orthogonal decomposition v ( t ) = √ v π ( t ) y + √ v π ( t ) y + v π ( t ) y with the agitation modes y = 1 √ (cid:16) (cid:17) T ,y = 1 √ (cid:16) (cid:17) T ,y = (cid:16) (cid:17) T . But, for instance, v π ( t ) = ( v ( t ) , v ( t ) , v ( t )) T is possible as well. Exact clustering
Theorem 1 and Corollary 1 only give an incomplete answer to Problem 1: if the infection rate matrix B has an equitable partition π , then there exists an invariant set V . But are there invariant sets V ,even if the matrix B does not have an equitable partition π ?We denote the orthogonal complement of the viral state set V by V ⊥ = (cid:8) w ∈ R N | w T v = 0 , ∀ v ∈ V (cid:9) . The dimension of the set V equals m . Thus, the dimension of the orthogonal complement V ⊥ equals N − m . Since the orthogonal complement V ⊥ is a subspace, there is a set of N − m orthonormal basisvectors y m +1 , ..., y N such that V ⊥ = span { y m +1 , ..., y N } . (18)The direct sum of two subspaces S , S ⊆ R N is defined as the subspace S ⊕ S = { s + s | s ∈ S , s ∈ S } . (19)Thus, the Euclidean space is the direct sum R N = span {V} ⊕ V ⊥ of the two subspaces V , V ⊥ .We rely on four assumptions to solve Problem 1. Assumption 1.
For every viral state v ∈ V , we require that diag ( δ , ..., δ N ) v ∈ V . Suppose that the curing rates are homogeneous, i.e., δ i = δ for all nodes i . Then, Assumption 1 issatisfied, since diag ( δ , ..., δ N ) v = δv ∈ V for every viral state v ∈ V . More generally, Assumption 1states that the viral state set V is an invariant subspace of the curing rate matrix diag ( δ , ..., δ N ).Intuitively speaking, the curing rates δ , ..., δ N are “set in accordance to” the clustering given by theviral state set V , such as in Example 2. Assumption 2.
There is a viral state v ∈ V whose entries satisfy v i > for every node i = 1 , ..., N . If R > B is irreducible, then [18] there is a unique steady-state v ∞ with positivecomponents v ∞ ,i >
0. Since every viral state v converges to the steady state v ∞ , the steady state v ∞ is element of the invariant set V . Hence, Assumption 2 is always satisfied if R >
1, provided thematrix B is irreducible. Assumption 3.
The curing rates are positive and the infection rates are non-negative, i.e., δ i > and β ij ≥ for all nodes i, j . Assumption 3 is rather technical, since only non-negative curing rates and infection rates have aphysical meaning.
Assumption 4.
The infection rate matrix B is symmetric and irreducible. Assumption 4 holds if and only if the infection rate matrix B corresponds to a connected undirectedgraph [48]. Under Assumption 4, the matrix B is diagonalisable [46] as B = X Λ X T . (20)9ere, we denote the N × N diagonal matrix Λ = diag( λ , ..., λ N ) whose diagonal entries are given bythe real eigenvalues λ ≥ λ ≥ ... ≥ λ N , and the columns of the N × N matrix X = ( x , ..., x N ) aregiven by the corresponding eigenvectors x i .Lemma 1 states that the invariant set V and the orthogonal complement V ⊥ are spanned byeigenvectors of the infection rate matrix B : Lemma 1.
Suppose that Assumptions 1 and 4 hold, and consider an invariant set V = span { y , ..., y m } of NIMFA (2) and the orthogonal complement V ⊥ = span { y m +1 , ..., y N } . Then, there is some permuta-tion φ : { , ..., N } → { , ..., N } , such that V = span { x φ (1) , ..., x φ ( m ) } and V ⊥ = span { x φ ( m +1) , ..., x φ ( N ) } ,where x φ (1) , ..., x φ ( N ) denotes an orthonormal set of eigenvectors of the infection rate matrix B to theeigenvalues λ φ (1) , ..., λ φ ( N ) .Proof. Appendix AWe denote the span of the vectors x φ ( l ) of the subspace V which correspond to a non-zero eigen-value λ φ ( l ) (cid:54) = 0 as V (cid:54) =0 = span (cid:8) x φ ( l ) (cid:12)(cid:12) l = 1 , ..., m, λ φ ( l ) (cid:54) = 0 (cid:9) . Let the number of non-zero eigenvaluesbe denoted by m . Without loss of generality, we assume that, after the permutation φ , the first m eigenvalues λ φ (1) , ..., λ φ ( m ) are non-zero. Hence, the subspace V (cid:54) =0 equals V (cid:54) =0 = span (cid:8) x φ ( l ) (cid:12)(cid:12) l = 1 , ..., m (cid:9) . (21)Analogously to (21), we define the span of the vectors x φ ( l ) of the subspace V which correspond to a zero eigenvalue λ φ ( l ) = 0 as V = span (cid:8) x φ ( l ) (cid:12)(cid:12) l = 1 , ..., m, λ φ ( l ) = 0 (cid:9) = span (cid:8) x φ ( l ) (cid:12)(cid:12) l = m + 1 , ..., m (cid:9) . Thus, the subspace V is equal to the direct sum V = V (cid:54) =0 ⊕ V . (22)We emphasise that span { y , ..., y m } = span (cid:8) x φ (1) , ..., x φ ( m ) (cid:9) does not imply that y l = x φ ( k ) for some k, l . An immediate consequence of Lemma 1 is that the infection rate matrix B can be decomposedas: Lemma 2.
Suppose that Assumptions 1 and 4 hold, and consider an invariant set V = span { y , ..., y m } of NIMFA (2) and the orthogonal complement V ⊥ = span { y m +1 , ..., y N } . Then, the infection ratematrix B is decomposable as B = B V + B V ⊥ , where B V = (cid:16) y ... y m (cid:17) ˜ B V y T ... y Tm and B V ⊥ = (cid:16) y m +1 ... y N (cid:17) ˜ B V ⊥ y Tm +1 ... y TN for some m × m matrix ˜ B V and ( N − m ) × ( N − m ) matrix ˜ B V ⊥ .Proof. Appendix B. 10emma 2 shows that the sets V and V ⊥ are invariant subspaces of the matrix B . In particular,the viral state dynamics on the invariant set V are the same for all infection rate matrices B (1) , B (2) with the same submatrix B (1) V = B (2) V but different submatrices B (1) V ⊥ (cid:54) = B (2) V ⊥ . Example 3.
Suppose that Assumptions 1 and 4 hold. For some degrees d , d , d and some scalar ξ , consider the infection rate matrix B = d + ξ d − ξ d d − ξ d + ξ d d d d with the equitable partition π = {N , N } , where N = { , } and N = { } , and the quotient matrix B π = (cid:32) d d d d (cid:33) . Corollary 1 states that the subspace V = span { y , y } is an invariant set of NIMFA (2), where theagitation modes are equal to y = √ (1 , , T and y = (0 , , T . The orthogonal complement followsas V ⊥ = span { y } , where y = √ (1 , − , T . Furthermore, Lemma 2 states that the infection ratematrix can be decomposed as B = B V + B V ⊥ , where B V = (cid:16) y y (cid:17) (cid:32) d √ d √ d d (cid:33) (cid:32) y T y T (cid:33) = d d d d d d d d d and B V ⊥ = 2 ξy y T = ξ − ξ − ξ ξ
00 0 0 . The eigenvectors x φ (1) , x φ (2) are equal to a linear combination of the agitation modes y , y , and thethird eigenvector equals x φ (3) = y . Theorem 2 states our main result:
Theorem 2.
Suppose that Assumptions 1 to 4 hold. Then, any invariant set V = span { y , ..., y m } ofNIMFA (2) is equal to the direct sum V = V (cid:54) =0 ⊕ V of two subspaces V (cid:54) =0 , V . Here, the orthonormalbasis vectors y , ..., y m , where m ≤ m , of the subspace V (cid:54) =0 = span { y , ..., y m } are given by ( y l ) i = √ |N l | if i ∈ N l , if i (cid:54)∈ N l , (23) for some equitable partition π = {N , ..., N m } of the infection rate matrix B . If m = m , thenthe subspace V is empty. Otherwise, if m < m , then V = span (cid:8) x φ ( l ) (cid:12)(cid:12) l = m + 1 , ..., m (cid:9) for someeigenvectors x φ ( l ) of the infection rate matrix B belonging to the eigenvalue 0.Proof. Appendix C. 11he Euclidean space R N is always an invariant set of NIMFA. For V = R N and V = ∅ , theequitable partition π in Theorem 2 becomes trivial , i.e., π = {N , ..., N N } with exactly one node inevery cell N l . On the other hand, if there is an invariant set V of dimension m < N , then Theorem 2implies that the matrix B is equitable with m ≤ m cells.If V = ∅ , then Theorem 2 essentially reverts Corollary 1. Thus, every equitable partition π corresponds to an invariant set V , and vice versa. In other words, the macroscopic structure ofequitable partitions π and the low-rank dynamics of invariant sets V are two sides of the same coin. If V = ∅ , then the dynamics on the invariant set V = V (cid:54) =0 are given by the reduced-size NIMFA system(16) with m = m equations.If V (cid:54) = ∅ , then Theorem 2 is more general than the inversion of Corollary 1. Theorem 2 states thatinvariant set of NIMFA is equal to the direct sum V = V (cid:54) =0 ⊕ V , where the subspace V (cid:54) =0 correspondsto an equitable partition π of the infection rate matrix, and the subspace V is a subset of the kernelof the matrix B . If V (cid:54) = ∅ , then the dynamics on the invariant set V = V (cid:54) =0 ⊕ V are described by the m > m differential equations (9).The curing rates δ i satisfy Assumption 1 if there are some scalars ˜ δ , ..., ˜ δ m such that δ i = ˜ δ l forall nodes i in cell N l , where l = 1 , ..., m . However, Assumption 1 allows for more general curing rates.With Lemma 2 and Theorem 2, the infection rate matrix B can be constructed from specifying theagitation modes y l , such that V = span { y , ..., y m } is an invariant set of NIMFA (2): Example 4.
Consider NIMFA (2) on a network of N = 5 nodes and the subspaces V (cid:54) =0 = span { y , y } , V = span { y } , where the agitation modes equal y = 1 √ (cid:16) (cid:17) T ,y = 1 √ (cid:16) (cid:17) T ,y = 1 √ (cid:16) − (cid:17) T . Furthermore, let y , y be two vectors, with y T y = 0 and y T y = y T y = 1 , that are orthogonal to theagitation modes y , y , y . With Lemma 2, define the infection rate matrix as B = (cid:16) y y (cid:17) ˜ B V (cid:54) =0 (cid:32) y T y T (cid:33) + (cid:16) y y (cid:17) ˜ B V ⊥ (cid:32) y T y T (cid:33) , where the symmetric × matrices ˜ B V (cid:54) =0 , ˜ B V ⊥ are chosen such that the matrix B is irreducible and con-tains only non-negative elements. Furthermore, consider the curing rate matrix S = diag(˜ δ , ˜ δ , ˜ δ , ˜ δ , ˜ δ ) for some curing rates ˜ δ , ˜ δ , ˜ δ > . Then, Assumptions 1 to 4 are satisfied, and Theorem 2 states thatthe subspace V = V (cid:54) =0 ⊕ V is an invariant set of NIMFA (2). (An alternative choice for the curingrate matrix is S = diag(˜ δ , ˜ δ , ˜ δ , ˜ δ , ˜ δ ) , which also satisfies Assumption 1.) In [33], we derived the solution of the NIMFA model (2) around the epidemic threshold R = 1.More precisely, under mild assumptions, we derived the approximation v apx ( t ) = c ( t ) v ∞ with anexplicit, closed-form expression for the scalar function c ( t ). If the initial viral state satisfies (cid:107) v (0) (cid:107) ≤ ˜ σ ( R − for some constant ˜ σ as R ↓
1, then it holds that (cid:107) v ( t ) − v apx ( t ) (cid:107) ≤ σ ( R − at everytime t for some constant σ as R ↓
1. Hence, the viral state v ( t ) converges to the approximation v apx ( t )12niformly in time t . Remarkably, since v apx = c ( t ) v ∞ , the viral state v ( t ) lies in the one-dimensionalsubspace V = span { v ∞ } when R ↓
1, for an arbitrarily large and heterogeneous contact network.Figure 3 illustrates the uniform convergence result in [33, Theorem 3]. 𝑣 𝑡 𝑣 ∞ 𝑣 apx 𝑡 width ≤ 𝜎 𝑅 − radius ≤ 𝜎 𝑅 − 𝑣 𝑒 𝑒 Figure 3:
Viral dynamics around the epidemic threshold R = 1 . An illustration of the uniformconvergence result in [33, Theorem 3] for a network with N = 2 nodes. The black curve shows thetrajectory of the 2 × v ( t ) as time t evolves. The blue line shows the steady state v ∞ . The red curve depicts the trajectory closed-form approximation v apx ( t ) = c ( t ) v ∞ , which is inthe subspace span { v ∞ } at every time t . If the initial viral state v (0) is positive and in the disk ofradius ˜ σ ( R − for some constant ˜ σ , then the approximation error (cid:107) v ( t ) − v apx ( t ) (cid:107) is bounded by σ ( R − for some constant σ at every time t as R ↓ v ( t ) converges to the one-dimensional dynamics v apx ( t )as R ↓ Are there networks for which the approximation v apx ( t ) is exact, for any basic reproductionnumber R > ? The infection rate matrix B is regular if N (cid:88) k =1 β ik = N (cid:88) k =1 β jk (24)for all nodes i, j . From Theorem 2, we obtain: Corollary 2.
Suppose that Assumptions 1 to 4 hold and consider that R > . Then, there is an m = 1 dimensional invariant set V = span { y } of NIMFA (2) if and only if V = ∅ , the agitationmode equals either y = v ∞ / (cid:107) v ∞ (cid:107) or y = − v ∞ / (cid:107) v ∞ (cid:107) and the infection rate matrix B is regular.Furthermore, the approximation v apx ( t ) = c ( t ) v ∞ is exact if and only if the matrix B is regular and v (0) = c (0) v ∞ for some scalar c (0) .Proof. Appendix D 13 .1 Decomposition of the viral dynamics
Suppose the infection rate matrix B has an equitable partition π and the infection rates β ij are thesame between all nodes i, j in any two cells . Then, we can decompose the dynamics of the viral state v ( t ) as: Theorem 3.
Consider NIMFA (2) on a symmetric N × N infection rate matrix B with an equitablepartition π = {N , ..., N r } . Furthermore, suppose that the curing rates δ i are the same for all nodes i in any cell N l , and that the infection rates β ij are the same for all nodes i in any cell N l and allnodes j in any cell N p . Denote the subspace V (cid:54) =0 = span { y , ..., y r } , with the basis vectors y l definedin (23), and denote the kernel of the matrix B by ker( B ) = span { y r +1 , ..., y N } . At every time t ≥ ,consider the viral state decomposition v ( t ) = ˜ v ( t ) + v ker ( t ) , where the projection of the viral state v ( t ) on the subspace V (cid:54) =0 equals ˜ v ( t ) = r (cid:88) l =1 (cid:0) y Tl v ( t ) (cid:1) y l , and the projection of the viral state v ( t ) on the kernel ker( B ) equals v ker ( t ) = N (cid:88) l = r +1 (cid:0) y Tl v ( t ) (cid:1) y l . Furthermore, denote the r × reduced-size projection ˜ v π ( t ) = (cid:0) ˜ v πi ( t ) , ..., ˜ v πi r ( t ) (cid:1) T , where i l denotesan arbitrary node in cell N l . Then, the reduced-size projection ˜ v π ( t ) evolves, independently of theprojection v ker ( t ) , as d ˜ v π ( t ) dt = − S π ˜ v π ( t ) + diag ( u r − ˜ v π ( t )) B π ˜ v π ( t ) (25) with the quotient matrix B π and the matrix S π given by (15), and the projection v ker ( t ) obeys dv ker ( t ) dt = − ( S + diag ( B ˜ v ( t ))) v ker ( t ) . (26) Proof.
Appendix E.In Theorem 3, the set V is equal to the kernel ker( B ), which is equivalent to V ⊥ = ∅ and assumingthe same infection rates β ij between all nodes i, j in any two cells. In contrast to Theorem 1, we donot consider that the initial state satisfies v i (0) = v j (0) for all nodes i, j in the same cell N l .With the definition of the agitation mode y l in (23), the viral state average in cell N l follows fromthe projection of the viral state v ( t ) on the vector y l as1 |N l | (cid:88) i ∈N l v i ( t ) = 1 (cid:112) |N l | y Tl v ( t ) If the matrix B is decomposable as B = B V + B V ⊥ as in Lemma 2, then the infection rates β ij are the same betweenall nodes i, j in any two cells if and only if B V ⊥ = 0. l = 1 , ..., r . Furthermore, the subspace V (cid:54) =0 is spanned by the vectors y , ..., y r . Hence,the dynamics of the projection ˜ v ( t ) on the subspace V (cid:54) =0 describes the evolution of viral state averagesof every cell N l , which is described by r differential equations (25) on the quotient graph B π . Sincethe steady state v ∞ ,i of every node i in the same cell N l is the same [5, 25], it holds that v ∞ ∈ V (cid:54) =0 ,which implies that v ker ( t ) → t → ∞ . Furthermore, from Theorem 1 it follows that, if v ker (0) = 0,then v ker ( t ) = 0 at every time t . Thus, the evolution of the projection v ker ( t ) describes convergence ofthe viral states v i ( t ) to the respective cell-averages. By (25), Theorem 3 implies that the viral statecell-averages evolve independently of the dynamics on the kernel ker( B ) . Schaub et al. [36] obtainedan analogous result for linear dynamics on networks.If we can derive the closed-form expression for the projection ˜ v ( t ) by solving (25), then the dynamics v ker ( t ) follow by the linear time- varying system (26). Furthermore, the reduced-size steady state v π ∞ = (cid:16) ˜ v π ∞ ,i , ..., ˜ v π ∞ ,i r (cid:17) T is an equilibrium of (25). Thus, if ˜ v ( t ) = v ∞ , then the dynamics of theprojection v ker ( t ) obey the linear time- invariant (LTI) system dv ker ( t ) dt = − ( S + diag ( Bv ∞ )) v ker ( t ) . Thus, the affine subspace (cid:8) v ∞ + v ker (cid:12)(cid:12) v ker ∈ ker( B ) (cid:9) is an invariant set of NIMFA, on which the viraldynamics are linear.Loosely speaking, Theorem 3 shows that a crucial challenge for solving NIMFA on graphs withequitable partitions is the dynamics of the projection ˜ v ( t ), since solving the set of nonlinear equations(25) seems more difficult than solving the linear time-varying system (26) for a given ˜ v ( t ). Fora complete graph, the solution ˜ v ( t ) to set of nonlinear equations (25) is one-dimensional and can bestated in closed form [43]. Thus, we obtain the solution of NIMFA on the complete graph, for arbitrary initial viral states v (0), as: Theorem 4.
Consider NIMFA (2) on the complete graph, whose infection rates equal β ij = β for allnodes i, j = 1 , ..., N . Suppose the curing rates satisfy δ i = δ for all nodes i . Then, for any initial viralstate v (0) ∈ [0 , N , the solution of NIMFA (2) equals v ( t ) = c ( t ) v ∞ + c ( t ) v ker (0) , where the steady-state vector equals v ∞ = (cid:16) − δβN (cid:17) u , and the N × vector v ker (0) is given by v ker (0) = (cid:18) I − N uu T (cid:19) v (0) . The scalar function c ( t ) equals c ( t ) = 12 (cid:16) (cid:16) w t + Υ (0) (cid:17)(cid:17) (27) with the viral slope w = βN − δ and the constant Υ (0) = arctanh (cid:18) v T ∞ v (0) (cid:107) v ∞ (cid:107) − (cid:19) , and the scalar function c ( t ) equals c ( t ) = Υ (0) e − Φ t sech (cid:16) w t + Υ (0) (cid:17) (28)15 ith the constant Φ = βN v ∞ ,i / δ , for an arbitrary node i , and the constant Υ (0) = v T ker (0) v (0) (cid:107) v ker (0) (cid:107) cosh (Υ (0)) . (29) Proof.
Appendix F. . . . t V i r a l S t a t e v ( t ) v ( t ) v ( t ) (a) Viral state v ( t ) versus time t . − . . . . t V i r a l S t a t e v ker , ( t ) v ker , ( t ) v ker , ( t ) ˜ v i ( t ) (b) Projections ˜ v ( t ) and v ker ( t ) versus time t . Figure 4:
Closed-form solution of NIMFA on the complete graph.
The solution of NIMFA (1)for a complete graph with N = 3 nodes and homogeneous spreading rates. As stated by Theorem 3,the viral state satisfies v ( t ) = ˜ v ( t ) + v ker ( t ), where ˜ v ( t ) and v ker ( t ) denote the projection of the viralstate v ( t ) on the subspace V (cid:54) =0 and the kernel ker( B ), respectively. (a) : The viral state v i ( t ) versustime t for every node i . (b) : The projections ˜ v ( t ) and v ker ( t ), which follow from Theorem 4 as˜ v i ( t ) = c ( t ) v ∞ ,i and v ker ,i ( t ) = c ( t ) ( y ) i for all nodes i , where the scalar functions c ( t ) and c ( t )are given by the closed-form expressions (27) and (28), respectively. Since the steady state v ∞ ,i is thesame for every node i in the complete graph, it holds that ˜ v i ( t ) = ˜ v j ( t ) for all nodes i, j .Figure 4 illustrates the closed-form solution of NIMFA for complete graphs, as given by Theorem 4.As shown by Figure 4, even though the viral state average ˜ v ( t ) is monotonically increasing, the viralstate v ( t ) = ˜ v ( t ) + v ker , ( t ) is decreasing until t ≈
1, which is due to the dynamics of the projection v ker ( t ) on the kernel ker( B ). 16 Approximate clustering
As shown by Theorem 2, equitable partitions and low-dimensional viral state dynamics in NIMFAare equivalent. Many networks possess some macroscopic structure, which may resemble an equitablepartition, but which is not precisely an equitable partition.
Is it possible to reduce the number ofNIMFA equations, if the network has an “almost” equitable partition?
For two N × x, y , x ≥ y denotes that x i ≥ y i for all entries i = 1 , ..., N . Theorem 5 showsthat NIMFA (2) on any network can be bounded by increasing or decreasing the spreading rates β ij , δ i : Theorem 5.
Consider two NIMFA systems with respective positive curing rates δ i and ˜ δ i , non-negativeinfection rates β ij and ˜ β ij , and viral states v i ( t ) and ˜ v i ( t ) . Suppose that the initial viral state v i (0) , ˜ v i (0) are in [0 , for all nodes i and that the matrices B and ˜ B , with elements β ij and ˜ β ij , respectively, areirreducible. Then, if ˜ δ i ≤ δ i and ˜ β ij ≥ β ij for all nodes i, j , ˜ v (0) ≥ v (0) implies that ˜ v ( t ) ≥ v ( t ) atevery time t .Proof. Appendix G.We emphasise that Theorem 5 does not assume symmetric infection rate matrices B , ˜ B . Buildingupon Theorem 5, we aim to bound the viral state v ( t ) of any network at every time t by the viralstate of networks with equitable partitions. In the following, we consider a partition π = {N , ..., N r } of the node set N = { , ..., N } of an arbitrary network. We stress that π can be any , not necessarilyequitable, partition. We define the minimum d min ,pl of the sum of infection rates from cell N l to N p as d min ,pl = min i ∈N p (cid:88) k ∈N l β ik (30)and the maximum d max ,pl as d max ,pl = max i ∈N p (cid:88) k ∈N l β ik . (31)Furthermore, we denote the r × r matrices B min and B max , whose elements are given by d min ,pl and d max ,pl , respectively. Analogously, we define the minimum δ min ,l of the curing rates in cell N l as δ min ,l = min i ∈N l δ i and the maximum δ max ,l as δ max ,l = max i ∈N l δ i . (32)We combine Theorem 1 and Theorem 5 to obtain: Theorem 6.
Suppose that the Assumptions 3 and 4 hold. At every time t , consider the r × reduced-size lower bound v lb ,l ( t ) and r × upper bound v ub ,l ( t ) , which evolve as dv lb ( t ) dt = − diag ( δ max , , ..., δ max ,r ) v lb ( t ) + diag ( u r − v lb ( t )) B min v lb ( t ) (33)17 nd dv ub ( t ) dt = − diag ( δ min , , ..., δ min ,r ) v ub ( t ) + diag ( u r − v ub ( t )) B max v ub ( t ) . Then, if the initial states satisfy v lb ,l (0) ≤ v i (0) ≤ v ub ,l (0) for all nodes i in any cell N l , the viral state v i ( t ) of all nodes i in any cell N l is bounded by v lb ,l ( t ) ≤ v i ( t ) ≤ v ub ,l ( t ) ∀ t ≥ . (34) Proof.
Appendix H.Theorem 6 states that the N × v ( t ) on any network is bounded by the r × v lb ( t ), v ub ( t ) on networks with equitable partitions and r cells. Reducing the N -dimensionalviral state dynamics to r -dimensional dynamics comes at the cost of an approximate description bythe bounds in (34). If the partition π is equitable, then it holds that d min ,pl = d max ,pl , and the boundsin Theorem 6 can be replaced by the exact statement in Theorem 1.Similarly to the lower bound and upper bound of the degrees in (30) and (31), respectively, wedefine the average degree from cell N l to N p for any partition π as¯ d pl = 1 (cid:12)(cid:12) N p (cid:12)(cid:12) (cid:88) i ∈N p (cid:88) k ∈N l β ik . Then, we define the r × r reduced-size infection rate matrix ¯ B , which consists of the elements ¯ d pl .Furthermore, we define the average curing rate of any cell N l as¯ δ l = 1 (cid:12)(cid:12) N l (cid:12)(cid:12) (cid:88) i ∈N l δ i . Then, we approximate the viral state by v i ( t ) ≈ ¯ v l ( t ) for all nodes i in any cell N l . Here, the r × v ( t ) evolves as d ¯ v ( t ) dt = − diag (cid:0) ¯ δ , ..., ¯ δ r (cid:1) ¯ v ( t ) + diag ( u r − ¯ v ( t )) ¯ B ¯ v ( t ) , (35)and, for all cells N l , the initial state equals¯ v l (0) = 1 (cid:12)(cid:12) N l (cid:12)(cid:12) (cid:88) i ∈N l v l (0) . If the matrix B has an equitable partition π and the rates δ i , β ij are the same between all nodes i, j in any two cells as in Theorem 3, then the approximation ¯ v ( t ) coincides with the projection ˜ v ( t ) ofthe viral state v ( t ) on the subspace V (cid:54) =0 .To illustrate the accuracy of the bounds in Theorem 6 and the reduced-size viral state ¯ v ( t ) fornetworks without equitable partitions, we consider the Stochastic Blockmodel (SBM), originally intro-duced by Holland et al. [14]. We consider a network with N = 1000 nodes and a partition π with r = 5 cells N , ..., N . The cells are of size |N | = 400, |N | = 250, |N | = 200, |N | = 100 and |N | = 50. With a probability of 0 .
7, there are no links between two cells N p , N l , i.e., β ij = β ji = 0for all nodes i ∈ N p and j ∈ N l . Otherwise, with a probability of 0 .
3, we denote the mean of the links18etween the cells N p , N l by ¯ β pl = ¯ β lp , which is set to a uniform random number in [0 . , . β ij = β ji for all nodes i ∈ N p and j ∈ N l is set to a random number [ ¯ β pl , ¯ β pl (1 + σ rel )],where we vary the relative variance σ rel for different scenarios in the numerical evaluation. If σ rel = 0,then the partition π is equitable. The larger the variance σ rel , the “less equitable” the partition π .For every node i , the curing rate δ i is set to a uniform random number in [1 , σ rel ], and the initialviral state v i (0) is set to a uniform random number in [0 . , . σ rel )]. Hence, if the variance σ rel = 0, then it holds that v lb ,l ( t ) = v lb ,l ( t ) = v i ( t ) for every node i in any cell N l . Lastly, the curingrates are decreased to δ i ← cδ i , where the scalar c is chosen such that the basic reproduction number(4) equals R = 3. To obtain the viral state v ( t ), we discretise NIMFA (1) with a sufficiently smallsampling time, see [26, 31, 20] for a detailed analysis of the resulting discrete-time NIMFA model.Figure 5 illustrates the accuracy of the bounds v lb ,l ( t ), v lb ,l ( t ) in Theorem 6 and the approximationaccuracy of ¯ v ( t ) in (35) for the largest cell N and the smallest cell N . For both σ rel = 0 .
25 and σ rel = 0 .
5, the approximation ¯ v l ( t ) is close to the exact average viral state in cell N l , v avg ,l ( t ) = 1 (cid:12)(cid:12) N l (cid:12)(cid:12) (cid:88) i ∈N l v l ( t ) . The accuracy of the bounds v lb ,l ( t ), v lb ,l ( t ) on any viral state v i ( t ) in cell N l decreases when thevariance σ rel is increased. Nonetheless, the bounds v lb ,l ( t ), v lb ,l ( t ) are reasonably accurate for both σ rel = 0 .
25 and σ rel = 0 . Approximating the viral state dynamics by m < N equations requires the specification of a partition π of the nodes. In some cases, this partition is given a priori , as in the experiments in Figure 5,where the node partition π was chosen corresponding to the SBM blocks. In contrast, for real-worldnetworks, it is more challenging to determine an appropriate clustering and, hence, to obtain anaccurate description of the viral state dynamics by m < N equations.We consider a two-step approach to reduce NIMFA to m = r < N equations. First, we obtain apartition π of the nodes by the Bethe spectral clustering algorithm [34], which makes use of the BetheHessian H ± = ( d avg − I ± d avg B + D , with the average degree d avg and the degree matrix D =diag( d , ..., d N ). When the matrix B has an (approximate) SBM structure, the negative eigenvaluesof H ± have corresponding eigenvectors which are (approximately) piecewise constant on the blocks of B . The spectral clustering algorithm partitions the nodes of B based on a k -means clustering of thenegative eigenvector entries of H ± . Second, we evaluate the accuracy of reduced-size viral state ¯ v ( t )in (35) by the deviation of the prevalence, (cid:15) avg = n (cid:88) k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N (cid:88) i =1 v i ( k ∆ t ) − N r (cid:88) l =1 |N l | ¯ v l ( k ∆ t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (36)Here, ∆ t denotes the sampling time, k is the discrete time, and the number of observations n is chosensuch that the viral state v ( n ∆ t ) practically converged to the steady state v ∞ .We applied the Bethe clustering algorithm to three real-world networks, which were accessedthrough [17]: the American football network [12] with N = 115 nodes and L = 613 links, for which r = 10 clusters were detected; the primary school contact network (day 1) [40] with N = 236 nodes19 .
05 0 . . . . . t V i r a l S t a t e v avg ,l ( t ) v ub ,l ( t )¯ v l ( t ) v lb ,l ( t ) (a) Cell N and relative variance σ rel = 0 . .
05 0 . . . . . t V i r a l S t a t e v avg ,l ( t ) v ub ,l ( t )¯ v l ( t ) v lb ,l ( t ) (b) Cell N and relative variance σ rel = 0 . .
05 0 . . . . . . t V i r a l S t a t e v avg ,l ( t ) v ub ,l ( t )¯ v l ( t ) v lb ,l ( t ) (c) Cell N and relative variance σ rel = 0 . .
05 0 . . . . . t V i r a l S t a t e v avg ,l ( t ) v ub ,l ( t )¯ v l ( t ) v lb ,l ( t ) (d) Cell N and relative variance σ rel = 0 . Figure 5:
Low-dimensional approximation of the viral state dynamics.
For a stochasticblockmodel network with N = 1000 nodes and r = 5 cells, the accuracy of the approximation ¯ v l ( t )and the tightness of the bounds v lb ,l ( t ), v lb ,l ( t ) are depicted. The reduced-size viral states ¯ v ( t ), v lb ( t )and v lb ( t ) are equal to the linear combination of m = r = 5 agitation modes y l , each of whichcorresponds to one cell. The first and second row correspond to the relative variance σ rel = 0 . σ rel = 0 .
5, respectively. The left column corresponds to the largest cell N , the right columncorresponds to the smallest cell N . The viral state v i ( t ) of every node i in the respective cell N l iswithin the shaded grey area.and L = 5899 links, resulting in r = 8 clusters; and the train bombing network [13] with N = 64nodes, L = 243 links and r = 3 identified clusters. For all networks, we considered homogeneousspreading rates β ij , δ i , which were set such that the basic reproduction number equals R = 3. The20nitial viral state was set to v i (∆ t ) = 1 /N for every node i . To evaluate the accuracy of the Betheclustering approach, we additionally considered a collection of random partitions, which are obtainedby randomly permuting the nodes in the partition π of the Bethe clustering. Bethe Random22 . . E rr o r ǫ a v g (a) American football. Bethe Random758085 E rr o r ǫ a v g (b) Primary school. Bethe Random150200250 E rr o r ǫ a v g (c) Train bombing. Figure 6:
Low-dimensional approximation of epidemics on real-world networks.
The error (cid:15) avg of the reduced-size viral state ¯ v ( t ), in (35), for partitions obtained by Bethe clustering and randompartitions.Figure 6 shows that, for the football and the school network which have a clear community struc-ture, the Bethe spectral clustering approach results in significantly more accurate low-dimensionalviral dynamics ¯ v ( t ) than for random partitions. For the train network, which does not possess a clearcommunity structure, there is a smaller advantage of Bethe clustering. Thus, our results indicate thatif the network has an underlying community structure, then spectral clustering may be used to findan accurate low-dimensional approximation of the viral state dynamics.Furthermore, for any partition π of the nodes, there are low-dimensional bounds v lb ,l ( t ), v ub ,l ( t )of the viral state dynamics, as stated by Theorem 6. We define the errors (cid:15) ub and (cid:15) lb of the bounds v ub ,l ( t ) and v lb ,l ( t ) analogously to (36). Figure 7 demonstrates that the partition of the nodes bythe Bethe clustering algorithm results in significantly more accurate lower bounds v lb ,l ( t ) than thoseobtained from random partitions, and somewhat more accurate upper bounds v ub ,l ( t ). In this work, we focussed on reducing NIMFA on a network with N nodes to only m << N differentialequations. We believe that the geometric clustering approach outlined in this work can be appliedto other dynamics on networks, particularly to general epidemic models [35, 30] and the class ofdynamics in [41, 4, 19, 32]. Our contribution is composed of three parts. In the first part, we showedthat the viral dynamics evolve on an m -dimensional subspace V if and only if the contact networkhas an equitable partition with m ≤ m cells. Thus, low-dimensional viral state dynamics and themacroscopic structure of equitable partitions are equivalent.In the second part, we focussed on equitable partitions π with the same spreading rates β ij and δ i forall nodes i, j in the same cell N l . We considered the decomposition of the viral state v ( t ) = v ker ( t )+˜ v ( t )into two parts: the term ˜ v ( t ) describes the viral state average in every cell N l ; and the term v ker ( t )equals the projection of the viral state v ( t ) onto the kernel of the infection rate matrix B . By showing21 ethe Random550600650700 E rr o r ǫ ub (a) American football. Bethe Random600620640660 E rr o r ǫ ub (b) Primary school. Bethe Random8009001 , E rr o r ǫ ub (c) Train bombing. Bethe Random600700800 E rr o r ǫ l b (d) American football. Bethe Random650652654656 E rr o r ǫ l b (e) Primary school. Bethe Random470472474 E rr o r ǫ l b (f) Train bombing. Figure 7:
Low-dimensional bounds of epidemics on real-world networks.
The errors of thelow-dimensional bounds v lb ,l ( t ) and v ub ,l ( t ), stated by Theorem 6, for partitions obtained by Betheclustering and random partitions. The subplots in the first and second row show the errors (cid:15) ub and (cid:15) lb of the upper bound v ub ,l ( t ) and the lower bound v ub ,l ( t ), respectively.that the term ˜ v ( t ) evolves independently from the projection v ker ( t ) and the projection v ker ( t ) obeysa linear time-varying system, we derived the solution of the NIMFA differential equations on thecomplete graph for arbitrary initial conditions v (0).Strictly speaking, most contact networks do not have an equitable partition, and an exact reductionof the number of NIMFA equations is not possible. In the third part, we considered arbitrary contactnetworks with a (not necessarily equitable) partition of the nodes into m cells. For any partition ofthe nodes, we derived bounds and approximations of the NIMFA epidemics with only m differentialequations. The “more equitable” the partition, the more accurate the approximation. Thus, finding(almost) equitable partitions is crucial for reducing an epidemic outbreak in a large population to theinteraction of only few groups of individuals. Acknowledgements
We are grateful to Massimo Achterberg for helpful discussions on this material.22 eferences [1] E. Abbe, “Community detection and stochastic block models: recent developments,”
The Journalof Machine Learning Research , vol. 18, no. 1, pp. 6446–6531, 2017.[2] M. Abramowitz and I. A. Stegun,
Handbook of mathematical functions: with formulas, graphs,and mathematical tables . Courier Corporation, 1965, vol. 55.[3] A. Arenas, A. Diaz-Guilera, and C. J. P´erez-Vicente, “Synchronization reveals topological scalesin complex networks,”
Physical Review Letters , vol. 96, no. 11, p. 114102, 2006.[4] B. Barzel and A.-L. Barab´asi, “Universality in network dynamics,”
Nature Physics , vol. 9, no. 10,p. 673, 2013.[5] S. Bonaccorsi, S. Ottaviano, D. Mugnolo, and F. D. Pellegrini, “Epidemic outbreaks in networkswith equitable or almost-equitable partitions,”
SIAM Journal on Applied Mathematics , vol. 75,no. 6, pp. 2421–2443, 2015.[6] S. L. Brunton and J. N. Kutz,
Data-driven science and engineering: Machine learning, dynamicalsystems, and control . Cambridge University Press, 2019.[7] A. Clauset, C. Moore, and M. E. Newman, “Hierarchical structure and the prediction of missinglinks in networks,”
Nature , vol. 453, no. 7191, pp. 98–101, 2008.[8] K. Devriendt and P. Van Mieghem, “Unified mean-field framework for Susceptible-Infected-Susceptible epidemics on networks, based on graph partitioning and the isoperimetric inequality,”
Physical Review E , vol. 96, no. 5, p. 052314, 2017.[9] K. Devriendt and R. Lambiotte, “Nonlinear network dynamics with consensus–dissensus bifurca-tion,”
Journal of Nonlinear Science , vol. 31, no. 1, pp. 1–34, 2021.[10] M. Egerstedt, S. Martini, M. Cao, K. Camlibel, and A. Bicchi, “Interacting with networks:How does structure relate to controllability in single-leader, consensus networks?”
IEEE ControlSystems Magazine , vol. 32, no. 4, pp. 66–73, 2012.[11] S. Friedberg, A. Insel, and L. Spence,
Linear Algebra . Prentice Hall, 1989.[12] M. Girvan and M. E. Newman, “Community structure in social and biological networks,”
Pro-ceedings of the National Academy of Sciences , vol. 99, no. 12, pp. 7821–7826, 2002.[13] B. Hayes, “Connecting the dots,”
American Scientist , vol. 94, no. 5, pp. 400–404, 2006.[14] P. W. Holland, K. B. Laskey, and S. Leinhardt, “Stochastic blockmodels: First steps,”
SocialNetworks , vol. 5, no. 2, pp. 109–137, 1983.[15] E. Kamke, “Zur Theorie der Systeme gew¨ohnlicher Differentialgleichungen. II.”
Acta Mathemat-ica , vol. 58, pp. 57–85, 1932.[16] I. Z. Kiss, J. C. Miller, and P. L. Simon, “Mathematics of epidemics on networks,”
Cham:Springer , vol. 598, 2017. 2317] J. Kunegis, “Konect: the Koblenz network collection,” in
Proceedings of the 22nd InternationalConference on World Wide Web . ACM, 2013, pp. 1343–1350.[18] A. Lajmanovich and J. A. Yorke, “A deterministic model for gonorrhea in a nonhomogeneouspopulation,”
Mathematical Biosciences , vol. 28, no. 3-4, pp. 221–236, 1976.[19] E. Laurence, N. Doyon, L. J. Dub´e, and P. Desrosiers, “Spectral dimension reduction of complexdynamical networks,”
Physical Review X , vol. 9, no. 1, p. 011042, 2019.[20] F. Liu, S. Cui, X. Li, and M. Buss, “On the stability of the endemic equilibrium of a discrete-timenetworked epidemic model,” arXiv preprint arXiv:2001.07451 , 2020.[21] D. Mugnolo,
Semigroup methods for evolution equations on networks . Springer, 2014, vol. 20,no. 4.[22] M. M¨uller, “ ¨Uber das Fundamentaltheorem in der Theorie der gew¨ohnlichen Differentialgleichun-gen,”
Mathematische Zeitschrift , vol. 26, no. 1, pp. 619–645, 1927.[23] C. Nowzari, V. M. Preciado, and G. J. Pappas, “Analysis and control of epidemics: A survey ofspreading processes on complex networks,”
IEEE Control Systems Magazine , vol. 36, no. 1, pp.26–46, 2016.[24] N. O’Clery, Y. Yuan, G.-B. Stan, and M. Barahona, “Observability and coarse graining of con-sensus dynamics through the external equitable partition,”
Physical Review E , vol. 88, no. 4, p.042805, 2013.[25] S. Ottaviano, F. De Pellegrini, S. Bonaccorsi, and P. Van Mieghem, “Optimal curing policyfor epidemic spreading over a community network with heterogeneous population,”
Journal ofComplex Networks , vol. 6, no. 5, pp. 800–829, 2018.[26] P. E. Par´e, J. Liu, C. L. Beck, B. E. Kirwan, and T. Ba¸sar, “Analysis, estimation, and validationof discrete-time epidemic processes,”
IEEE Transactions on Control Systems Technology , 2018.[27] R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, “Epidemic processes incomplex networks,”
Reviews of Modern Physics , vol. 87, no. 3, pp. 925—-979, 2015.[28] L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchro-nization and isolated desynchronization in complex networks with symmetries,”
Nature Commu-nications , vol. 5, no. 1, pp. 1–8, 2014.[29] T. P. Peixoto, “Hierarchical block structures and high-resolution model selection in large net-works,”
Physical Review X , vol. 4, no. 1, p. 011047, 2014.[30] B. Prasse and P. Van Mieghem, “Network reconstruction and prediction of epidemic outbreaks forgeneral group-based compartmental epidemic models,”
IEEE Transactions on Network Scienceand Engineering , 2020.[31] ——, “The viral state dynamics of the discrete-time NIMFA epidemic model,”
IEEE Transactionson Network Science and Engineering , 2019. 2432] ——, “Predicting dynamics on networks hardly depends on the topology,” arXiv preprintarXiv:2005.14575 , 2020.[33] ——, “Time-dependent solution of the NIMFA equations around the epidemic threshold,”
Journalof Mathematical Biology , vol. 81, no. 6, pp. 1299–1355, 2020.[34] A. Saade, F. Krzakala, and L. Zdeborov´a, “Spectral clustering of graphs with the BetheHessian,” in
Advances in Neural Information Processing Systems , Z. Ghahramani, M. Welling,C. Cortes, N. Lawrence, and K. Q. Weinberger, Eds., vol. 27. Curran Associates, Inc.,2014. [Online]. Available: http://papers.nips.cc/paper/5520-spectral-clustering-of-graphs-with-the-bethehessian.pdf[35] F. D. Sahneh, C. Scoglio, and P. Van Mieghem, “Generalized epidemic mean-field model forspreading processes over multilayer complex networks,”
IEEE/ACM Transactions on Networking(TON) , vol. 21, no. 5, pp. 1609–1620, 2013.[36] M. T. Schaub, N. O’Clery, Y. N. Billeh, J.-C. Delvenne, R. Lambiotte, and M. Barahona, “Graphpartitions and cluster synchronization in networks of oscillators,”
Chaos: An InterdisciplinaryJournal of Nonlinear Science , vol. 26, no. 9, p. 094821, 2016.[37] M. T. Schaub and L. Peel, “Hierarchical community structure in networks,” arXiv preprintarXiv:2009.07196 , 2020.[38] A. J. Schwenk, “Computing the characteristic polynomial of a graph,” in
Graphs and Combina-torics . Springer, 1974, pp. 153–172.[39] P. L. Simon, M. Taylor, and I. Z. Kiss, “Exact epidemic models on graphs using graph-automorphism driven lumping,”
Journal of Mathematical Biology , vol. 62, no. 4, pp. 479–508,2011.[40] J. Stehl´e, N. Voirin, A. Barrat, C. Cattuto, L. Isella, J. Pinton, M. Quaggiotto, W. Van denBroeck, C. R´egis, B. Lina, and P. Vanhems, “High-resolution measurements of face-to-facecontact patterns in a primary school,”
PLOS ONE , vol. 6, no. 8, p. e23176, 08 2011. [Online].Available: http://dx.doi.org/10.1371/journal.pone.0023176[41] M. Timme, “Revealing network connectivity from response dynamics,”
Physical Review Letters ,vol. 98, no. 22, p. 224101, 2007.[42] P. Van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemicequilibria for compartmental models of disease transmission,”
Mathematical Biosciences , vol.180, no. 1-2, pp. 29–48, 2002.[43] P. Van Mieghem, “SIS epidemics with time-dependent rates describing ageing of informationspread and mutation of pathogens,”
Delft University of Technology , vol. 1, no. 15, 2014.[44] P. Van Mieghem and J. Omic, “In-homogeneous virus spread in networks,” arXiv preprintarXiv:1306.2588 , 2014. 2545] P. Van Mieghem and R. Van de Bovenkamp, “Non-Markovian infection spread dramatically altersthe susceptible-infected-susceptible epidemic threshold in networks,”
Physical Review Letters , vol.110, no. 10, p. 108701, 2013.[46] P. Van Mieghem,
Graph spectra for complex networks . Cambridge University Press, 2010.[47] ——, “The N-Intertwined SIS epidemic network model,”
Computing , vol. 93, no. 2-4, pp. 147–169,2011.[48] ——,
Performance Analysis of Complex Networks and Systems . Cambridge University Press,2014.[49] P. Van Mieghem, J. Omic, and R. Kooij, “Virus spread in networks,”
IEEE/ACM Transactionson Networking , vol. 17, no. 1, pp. 1–14, 2009.[50] J. A. Ward and M. L´opez-Garc´ıa, “Exact analysis of summary statistics for continuous-timediscrete-state Markov processes on networks using graph-automorphism lumping,”
Applied Net-work Science , vol. 4, no. 1, p. 108, 2019.
A Proof of Lemma 1
Let w denote a vector in the orthogonal complement V ⊥ of the invariant set V . Hence, it must holdthat w T v ( t ) = 0 for every time t ≥ v (0) ∈ V , which is equivalent to both w T v (0) = 0 and d ( w T v ( t )) dt = 0 ∀ v ( t ) ∈ V , w ∈ V ⊥ . (37)We replace the notation v ( t ) ∈ V by v ∈ V . Then, we obtain from the NIMFA equations (2) that (37)is equivalent to w T ( − Sv + diag( u − v ) Bv ) = 0 ∀ v ∈ V , w ∈ V ⊥ . Under Assumption 1, it holds that Sv ∈ V . Hence, the vector w ∈ V ⊥ is orthogonal to the vector Sv ,which yields that w T diag( u − v ) Bv = 0 . Since diag( u ) is the identity matrix, we obtain that w T Bv = w T diag( v ) Bv. (38)Since the invariant set V is a subspace of R N , v ∈ V implies that γv ∈ V for any scalar γ ∈ R . Forthe vector γv , where we consider γ >
0, it follows from (38) that γw T Bv = γ w T diag( v ) Bv, which is equivalent to w T Bv = γw T diag( v ) Bv. γ > w T diag( v ) Bv = γw T diag( v ) Bv, which implies that w T diag( v ) Bv = 0 . (39)Then, from (38), it follows that w T Bv = 0for all vectors w ∈ V ⊥ , v ∈ V . The vector Bv is orthogonal to all vectors w ∈ V ⊥ , only if Bv ∈ V .Thus, the set V is an invariant subspace [11] of the infection rate matrix B . The sets of vectors y , ..., y m and y m +1 , ..., y N span the invariant set V and the orthogonal complement V ⊥ , respectively,see (6) and (18). Thus, we can express the symmetric matrix B as B = (cid:16) y ... y N (cid:17) (cid:32) M M M (cid:33) y T ... y TN (40)for some m × m symmetric matrix M and some ( N − m ) × ( N − m ) symmetric matrix M . The m × ( N − m ) matrix M describes the mapping from the subspace V ⊥ to the subspace V . Since thematrix B is symmetric, it holds that M = 0, and (40) becomes B = (cid:16) y ... y N (cid:17) (cid:32) M M (cid:33) y T ... y TN . Furthermore, since the matrix B is diagonalisable as (20), the matrices M and M are diagonal-isable [11, Exercise 24, Section 5.4]. Thus, there is some orthogonal m × m matrix C and someorthogonal ( N − m ) × ( N − m ) matrix C such that B = (cid:16) y ... y N (cid:17) (cid:32) C C (cid:33) (cid:32) Λ
00 Λ (cid:33) (cid:32) C T C T (cid:33) y T ... y TN . (41)where the m × m diagonal matrix Λ and the ( N − m ) × ( N − m ) diagonal matrix Λ contain theeigenvalues of B . In contrast to the N × N matrix Λ in (20), the diagonal entries of the matrices Λ and Λ may not be ordered with respect to their magnitude. Hence, there is some permutation φ : { , ..., N } → { , ..., N } of the eigenvalues λ , ..., λ N such thatΛ = diag (cid:0) λ φ (1) , ..., λ φ ( m ) (cid:1) and Λ = diag (cid:0) λ φ ( m +1) , ..., λ φ ( N ) (cid:1) .
27e define the N × m matrix E V and the N × ( N − m ) matrix E V ⊥ as E V = (cid:16) y ... y m (cid:17) C and E V ⊥ = (cid:16) y N − m ... y N (cid:17) C . Since the matrices C and C are nonsingular, the columns of the matrices E V and E V ⊥ span thesubspaces V and V ⊥ , respectively. We obtain that B = (cid:16) E V E V ⊥ (cid:17) diag (cid:0) λ φ (1) , ..., λ φ ( N ) (cid:1) (cid:32) E T V E T V ⊥ (cid:33) . Thus, the matrices E V , E V ⊥ are equal to E V = (cid:16) x φ (1) ... x φ ( m ) (cid:17) (42)and E V ⊥ = (cid:16) x φ ( N − m ) ... x φ ( N ) (cid:17) , where the columns x φ (1) , ..., x φ ( N ) are eigenvectors to the eigenvalues λ φ (1) , ..., λ φ ( N ) of the matrix B ,which completes the proof. B Proof of Lemma 2
From (41), it follows that B = (cid:16) y ... y m (cid:17) C Λ C T y T ... y Tm + (cid:16) y m +1 ... y N (cid:17) C Λ C T y Tm +1 ... y TN . We complete the proof by identifying the m × m matrix ˜ B V = C Λ C T and the ( N − m ) × ( N − m )matrix ˜ B V ⊥ = C Λ C T . C Proof of Theorem 2
The proof of Theorem 2 is based on four lemmas. First, Lemma 3 relates the product diag( w ) v to thesubspaces V (cid:54) =0 and V ⊥ : Lemma 3.
For all vectors v ∈ V (cid:54) =0 and w ∈ V ⊥ , it holds that diag( w ) v ∈ V ⊥ .Proof. Since w T diag( v ) = ( w v , ..., w N v N ) = v T diag( w ), we obtain from (39) that v T diag( w ) Bv = 0 . v T B diag( w ) v = 0 . (43)The invariant set V is given by the span of some orthogonal vectors y , ..., y m . By Lemma 1, it holdsthat V = span { x φ (1) , ..., x φ ( m ) } , where x φ ( l ) is an eigenvector of the matrix B to the eigenvalue λ φ ( l ) for some permutation φ . Thus, every vector v ∈ V can be written as v = (cid:16) x φ (1) ... x φ ( m ) (cid:17) z (44)for some m × z = ( z , ..., z m ) T , and the subspace V equals V = (cid:110)(cid:16) x φ (1) ... x φ ( m ) (cid:17) z (cid:12)(cid:12) z ∈ R m (cid:111) . With (44), we can rewrite (43) as z T Λ x Tφ (1) ... x Tφ ( m ) diag( w ) (cid:16) x φ (1) ... x φ ( m ) (cid:17) z = 0 , (45)with the m × m diagonal matrix Λ = diag( λ φ (1) , ..., λ φ ( m ) ). The quadratic form (45) equals zero forall vectors cz ∈ R m if and only ifΛ x Tφ (1) ... x Tφ ( m ) diag( w ) (cid:16) x φ (1) ... x φ ( m ) (cid:17) = 0 , which implies, with (44), that Λ x Tφ (1) ... x Tφ ( m ) diag( w ) v = 0for all vectors v ∈ V . Componentwise, we obtain that λ φ ( l ) x Tφ ( l ) diag( w ) v = 0 (46)for all rows l = 1 , ..., m and all vectors v ∈ V . Equation (46) is satisfied if and only if λ φ ( l ) =0 or x Tφ ( l ) diag( w ) v = 0 for all rows l = 1 , ..., m . The subspace V contains the vectors x φ ( l ) forwhich λ φ ( l ) = 0, and the subspace V ⊥ contains the vectors x φ ( m +1) , ..., x φ ( N ) which are orthogonal tothe vectors x φ (1) , ..., x φ ( m ) . Thus, the vector diag( w ) v must be element of the subspaces V or V ⊥ , orthe vector diag( w ) v must be equal to the sum of two vectors in the subspaces V and V . Hence, withthe direct sum (19), we can reformulate (46) asdiag( w ) v ∈ V ⊥ ⊕ V (47)for all vectors v ∈ V . We define the N × m matrix E V (cid:54) =0 as E V (cid:54) =0 = (cid:16) x φ (1) ... x φ ( m ) (cid:17) N × ( m − m ) matrix E V as E V = (cid:16) x φ ( m +1) ... x φ ( m ) (cid:17) . Thus, the definition of the matrix E V in (42) implies that E V = (cid:16) E V (cid:54) =0 E V (cid:17) , and the matrix diag( w )can be written as diag( w ) = (cid:16) E V (cid:54) =0 E V E V ⊥ (cid:17) M M M M M M M M M E T V (cid:54) =0 E T V E T V ⊥ for some matrices M ij , where i, j = 1 , ,
3, whose dimensions follow from the dimension of the matri-ces E V (cid:54) =0 , E V and E V ⊥ . The matrices M and M describe the mapping of the matrix diag( w ) fromthe subspaces V (cid:54) =0 and V , respectively, to the subspace V (cid:54) =0 . From (47), we obtain that M = 0 and M = 0. Furthermore, since the matrix diag( w ) is symmetric, it holds that M = M T = 0. Hence,to satisfy (47), the matrix diag( w ) must be equal todiag( w ) = (cid:16) E V (cid:54) =0 E V E V ⊥ (cid:17) M M M M M M E T V (cid:54) =0 E T V E T V ⊥ , which implies for all vectors v ∈ V (cid:54) =0 that diag( w ) v ∈ V ⊥ .Lemma 3 states that for all vectors v ∈ V (cid:54) =0 and w ∈ V ⊥ , there must be some vector ˜ w ∈ V ⊥ suchthat diag( w ) v = ˜ w. (48)We aim to find all subspaces V (cid:54) =0 and V ⊥ whose elements v and w, ˜ w , respectively, satisfy (48). FromLemma 1 it follows that a basis of the N − m dimensional subspace V ⊥ is given by the columns of thematrix E V ⊥ = (cid:0) x φ ( m +1) (cid:1) ... (cid:0) x φ ( N ) (cid:1) ... . . . ... (cid:0) x φ ( m +1) (cid:1) N ... (cid:0) x φ ( N ) (cid:1) N . (49)For every matrix, the column rank equals the row rank. Since the columns of the matrix E V ⊥ arelinearly independent, there are N − m linearly independent rows of the matrix E V ⊥ . Without loss ofgenerality , we assume that the first N − m rows of the matrix E V ⊥ are linearly independent. Hence,the first N − m rows span the Euclidean space R N − m ,span (cid:0) x φ ( m +1) (cid:1) ... (cid:0) x φ ( N ) (cid:1) , (cid:0) x φ ( m +1) (cid:1) ... (cid:0) x φ ( N ) (cid:1) , ..., (cid:0) x φ ( m +1) (cid:1) N − m ... (cid:0) x φ ( N ) (cid:1) N − m = R N − m . (50) Otherwise, consider a permutation of the rows, which is equivalent to a relabelling of the nodes. all vectors w ∈ V ⊥ and v ∈ V (cid:54) =0 , there is a vector ˜ w ∈ V ⊥ whose first N − m entries satisfy(48), i.e., ˜ w i = w i v i , i = 1 , ..., N − m. (51)The last m entries of the vector ˜ w ∈ V ⊥ are determined by the first ( N − m ) entries of the vector w , asshown by Lemma 4. (Lemma 4 is not a novel contribution, but we include Lemma 4 for completeness.) Lemma 4.
Suppose that that the first N − m rows of the matrix E V ⊥ are linearly independent. Then,there are some ( N − m ) × vectors χ N − m , ..., χ N such that the last m entries of any vector w ∈ V ⊥ follow from the first ( N − m ) entries as w i = χ Ti w ... w N − m , i = N − m + 1 , ..., N. Proof.
With the definition of the matrix E V ⊥ in (49), every vector w ∈ V ⊥ can be written as w = (cid:16) x φ ( m +1) ... x φ ( N ) (cid:17) z m +1 ... z N (52)for some scalars z m +1 , ..., z N ∈ R . Thus, the first N − m entries of the vector w follow as w ... w N − m = M z m +1 ... z N , (53)where the ( N − m ) × ( N − m ) matrix M equals the first N − m rows of the matrix E V ⊥ , M = (cid:0) x φ ( m +1) (cid:1) ... (cid:0) x φ ( N ) (cid:1) ... . . . ... (cid:0) x φ ( m +1) (cid:1) N − m ... (cid:0) x φ ( N ) (cid:1) N − m . By assumption, the first N − m rows of the matrix E V ⊥ are linearly independent. Hence, the matrix M is nonsingular, and the scalars z m +1 , ..., z N follow from (53) as z m +1 ... z N = M − w ... w N − m . Thus, we obtain the last m entries of the vector w with (52) as w N − m +1 ... w N = (cid:0) x φ ( m +1) (cid:1) N − m +1 ... (cid:0) x φ ( N ) (cid:1) N − m +1 ... . . . ... (cid:0) x φ ( m +1) (cid:1) N ... (cid:0) x φ ( N ) (cid:1) N z m +1 ... z N = (cid:0) x φ ( m +1) (cid:1) N − m +1 ... (cid:0) x φ ( N ) (cid:1) N − m +1 ... . . . ... (cid:0) x φ ( m +1) (cid:1) N ... (cid:0) x φ ( N ) (cid:1) N M − w ... w N − m .
31o complete the proof, we define the vectors χ N − m +1 , ..., χ N as χ TN − m +1 ... χ TN = (cid:0) x φ ( m +1) (cid:1) N − m +1 ... (cid:0) x φ ( N ) (cid:1) N − m +1 ... . . . ... (cid:0) x φ ( m +1) (cid:1) N ... (cid:0) x φ ( N ) (cid:1) N M − . We combine Lemma 4 and (51), which yields for the last ( N − m ) entries of the vector ˜ w ∈ V ⊥ that ˜ w i = N − m (cid:88) j =1 χ ij ˜ w j = N − m (cid:88) j =1 χ ij w j v j , where i = N − m + 1 , ..., N . Furthermore, (48) states that ˜ w i = v i w i . Thus, it must hold that w i v i = N − m (cid:88) j =1 χ ij w j v j for the entries i = N − m + 1 , ..., N . Since the vector w is element of the subspace V ⊥ , we applyLemma 4 again and obtain that N − m (cid:88) j =1 χ ij w j v i = N − m (cid:88) j =1 χ ij w j v j . Thus, for all entries i = N − m + 1 , ..., N , it must hold that N − m (cid:88) j =1 χ ij w j ( v i − v j ) = 0 (54)for all vectors w ∈ V ⊥ and v ∈ V (cid:54) =0 . Since the first N − m rows of the matrix E V ⊥ are linearlyindependent, see (50), it follows that (54) must be satisfied for all scalars w , ..., w N − m in R . Hence,for all vectors v ∈ V (cid:54) =0 , it must hold that χ ij ( v i − v j ) = 0 for all indices j = 1 , ..., N − m , which isequivalent to χ ij = 0 or v j = v i . Thus, the non-zero entries of the vectors χ i indicate which nodes j have the same viral state as node i . Example 5.
Consider a network of N = 5 nodes with an invariant set V of dimension m = 3 .Furthermore, consider that V = ∅ , which implies with (22) that V = V (cid:54) =0 . Thus, there are N − m = 2 vectors χ , χ . Suppose that the vectors χ , χ are equal to χ = ( χ , T and χ = (0 , χ ) T , where χ , χ (cid:54) = 0 . Then, (54) implies that v = v and v = v for every viral state v ∈ V . Hence, thesubspace V = span { y , y , y } is given by the basis vectors y = 1 √ , y = 1 √ , y = . or l = 1 , , , the eigenvector x φ ( l ) of the infection rate matrix B equals a linear combination of thebasis vectors y , y , y . From (54), we can determine disjoint subsets N , N , ... of the set of all nodes N = { , ..., N } asfollows: If two nodes i, j are element of the same subset N l ⊆ N , then the viral states are equal, v i = v j , for every viral state v ∈ V (cid:54) =0 . If a subset contains only one node, N l = { i } , then the viralstate can be arbitrary v i ∈ R , independently of the viral state v j of other nodes j (cid:54) = i . Every subsetdefines a basis vector y l of the subspace V (cid:54) =0 as( y l ) i = √ |N l | if i ∈ N l , i (cid:54)∈ N l . (55)Then, the subspace V (cid:54) =0 equals the span of the vectors y l of all subsets N l . Since the dimension of thesubspace V (cid:54) =0 is m , there must be m subsets N , ..., N m . Every node i is element of at most onesubset N l . Hence, the vectors y l , y ˜ l are orthogonal for l (cid:54) = ˜ l .Furthermore, some nodes i might not be element of any subset N , ..., N m , which would implythat ( y l ) i = 0 for all basis vectors y l of V (cid:54) =0 . We define the subset N m +1 , whose elements are thenodes i that are not in any other subset N , ..., N m . As shown by Lemma 5, the set N m +1 is empty: Lemma 5.
Under Assumptions 1 to 4, it holds that N m +1 = ∅ .Proof. Under Assumption 2, there is a viral state vector v ∈ V with positive entries. The positiveviral state vector v satisfies v = m (cid:88) l =1 z l y l + m (cid:88) l = m +1 z l y l (56)for some scalars z , ..., z m ∈ R . We denote the projection of the viral state v onto the subspace V as v ker = m (cid:88) l = m +1 z l y l Every basis vector y l of the subspace V (cid:54) =0 satisfies ( y l ) i = 0 for all nodes i ∈ N m +1 . Thus, we obtainwith (56) that ( v ker ) i = v i > i ∈ N m +1 . Any vector ˜ v ∈ V (cid:54) =0 is orthogonal to the vector v ker ∈ V . Hence, it holdsthat N (cid:88) i =1 (˜ v ) i ( v ker ) i = 0 . We split the sum m (cid:88) l =1 (cid:88) i ∈N l (˜ v ) i ( v ker ) i + (cid:88) i ∈N m (˜ v ) i ( v ker ) i = 0 . v ) i = 0 for all nodes i ∈ N m +1 , we obtain that m (cid:88) l =1 (cid:88) i ∈N l (˜ v ) i ( v ker ) i = 0 ∀ ˜ v ∈ V (cid:54) =0 . (58)Furthermore, we define the N × u a with the entries( u a ) i = i (cid:54)∈ N m +1 , i ∈ N m +1 . From the definition of the basis vectors y l in (55), it follows that the vector u a equals u a = m (cid:88) l =1 (cid:112) |N l | y l . Thus, vector u a is element of V (cid:54) =0 . Since the vector v ker is in the kernel of the matrix B , it holds that Bv ker = 0, which implies that u Ta Bv ker = 0 . (59)We decompose the vector v ker as v ker = v ker ,a + v ker ,b , where the first addend equals( v ker ,a ) i = ( v ker ) i if i (cid:54)∈ N m +1 , i ∈ N m +1 , and the second addend equals ( v ker ,b ) i = i (cid:54)∈ N m +1 ( v ker ) i if i ∈ N m +1 . (60)Then, (59) becomes u Ta Bv ker ,a + u Ta Bv ker ,b = 0 . Since u a ∈ V (cid:54) =0 and V (cid:54) =0 is an invariant subspace of the matrix B , it holds that Bu a ∈ V (cid:54) =0 . Thus,(58) implies that u Ta Bv ker ,a = 0, and we obtain that u Ta Bv ker ,b = 0 , which is equivalent to m (cid:88) l =1 (cid:88) i ∈N l N (cid:88) j =1 β ij ( v ker ,b ) j = 0 . With the definition of the vector v ker ,b in (60), we obtain that m (cid:88) l =1 (cid:88) i ∈N l (cid:88) j ∈N m β ij ( v ker ) j = 0 . (61)As stated by (57), the entries ( v ker ) j are positive for all nodes j ∈ N m +1 . Furthermore, the infectionrates β ij are non-negative under Assumption 3. Hence, (61) is satisfied only if β ij = 0 for all nodes j ∈ N m +1 and i ∈ N l for all subsets l = 1 , ..., m . In other words, the nodes in N m +1 are notconnected to any nodes in N , ..., N m , which contradicts the irreducibility of the matrix B underAssumption 4. Hence, it must hold that N m +1 = ∅ .34ince N m +1 = ∅ , it holds that N ∪ ... ∪ N m = N . Hence, the disjoint subsets N , ..., N m definea partition of the set of all nodes N = { , ..., N } . To complete the proof of Theorem 2, we must showthat the subsets N , ..., N m are an equitable partition of the infection rate matrix B . Hence, we mustshow that the sum of the infection rates β ij , (cid:88) j ∈N l β ij , (62)is the same for all nodes i ∈ N p and all cells l, p = 1 , ..., m . Lemma 1 states that V (cid:54) =0 = span { y , ..., y m } = span (cid:8) x φ (1) , ..., x φ ( m ) (cid:9) . Thus, there must be some nonsingular m × m matrix H such that (cid:16) x φ (1) ... x φ ( m ) (cid:17) = (cid:16) y ... y m (cid:17) H. (63)Since the set eigenvectors x i and the set of vectors y l are orthonormal, the matrix H is orthogonal .The eigendecomposition of the matrix B reads B = (cid:16) x φ (1) ... x φ ( m ) (cid:17) diag (cid:0) λ φ (1) , ..., λ φ ( m ) (cid:1) x Tφ (1) ... x Tφ ( m ) + (cid:16) x φ ( m +1) ... x φ ( m ) (cid:17) diag (cid:0) λ φ ( m +1) , ..., λ φ ( m ) (cid:1) x Tφ ( m +1) ... x Tφ ( m ) + (cid:16) x φ ( m +1) ... x φ ( N ) (cid:17) diag (cid:0) λ φ ( m +1) , ..., λ φ ( N ) (cid:1) x Tφ ( m +1) ... x Tφ ( N ) . With (63), and since the eigenvalues λ φ ( l ) = 0 for l = m + 1 , ..., m , we obtain that B = (cid:16) y ... y m (cid:17) H diag (cid:0) λ φ (1) , ..., λ φ ( m ) (cid:1) H T y T ... y Tm (64)+ (cid:16) x φ ( m +1) ... x φ ( N ) (cid:17) diag (cid:0) λ φ ( m +1) , ..., λ φ ( N ) (cid:1) x Tφ ( m +1) ... x Tφ ( N ) . Consider two nodes i ∈ N p and a subset N l for some l = 1 , ..., m . Since( y l ) j = √ |N l | if j ∈ N l , j (cid:54)∈ N l , Since x Ti x j = 1 if i = j and x Ti x j = 0 if i (cid:54) = j and analogously for the vectors y i , y j , it follows from x Ti x j = y Ti H T Hy j that the matrix H is orthogonal.
35e can express the sum (62) as (cid:88) j ∈N l β ij = (cid:112) |N l | (cid:16) β i ... β iN (cid:17) y l . Thus, with the N × e i , it holds that (cid:88) j ∈N l β ij = (cid:112) |N l | e Ti By l . From the orthogonality of the vectors y , ..., y m and from x Tφ ( k ) y l = 0 for k = m + 1 , ..., N , we obtainwith (64) that (cid:88) j ∈N l β ij = (cid:112) |N l | e Ti (cid:16) y ... y m (cid:17) H diag (cid:0) λ φ (1) , ..., λ φ ( m ) (cid:1) H T e m × ,l , (65)where the l -th entry of the m × e m × ,l equals one, and the other entries of e m × ,l equalzero. Since node i is element of exactly one subset N p , it holds that e Ti (cid:16) y ... y m (cid:17) = 1 (cid:112) |N p | ˜ e Tm × ,p . Then, (65) becomes (cid:88) j ∈N l β ij = d il , where d il = (cid:112) |N l | (cid:112) |N p | e Tm × ,p H diag (cid:0) λ φ (1) , ..., λ φ ( m ) (cid:1) H T e m × ,l is the same for all nodes i ∈ N p , which completes the proof. D Proof of Corollary 2
Since R >
1, the viral state v ( t ) converges to a positive steady state v ∞ as t → ∞ . Thus, the steadystate v ∞ must be element of the m = 1 dimensional invariant set V = span { y } , which implies that v ∞ = ˜ cy for some scalar c . Hence, the unit-length agitation mode equals either y = v ∞ / (cid:107) v ∞ (cid:107) or y = − v ∞ / (cid:107) v ∞ (cid:107) . Without loss of generality assume that y = v ∞ / (cid:107) v ∞ (cid:107) . Then, under Assumption 4,the matrix B is connected, which implies that By (cid:54) = 0 since the vector y is positive. Thus, thesubspace V must be empty.To prove Corollary 2, we must show two directions. “If ” direction : Suppose the infection ratematrix B is regular. Then, the viral state v ∞ ,i is the same for all nodes i , and v (0) ∈ V implies that v i (0) = v j (0) for all nodes i, j . Since the matrix B is regular and the initial viral state v i (0) is thesame for every node i , the approximation v apx ( t ) = c ( t ) v ∞ is exact [43, 33]. Since v ( t ) = c ( t ) v ∞ atevery time t , the invariant set V = span { y } is indeed a one-dimensional invariant set of NIMFA. “Only if ” direction : Suppose the one-dimensional subspace V = span { y } is an invariant set ofNIMFA. Then, Theorem 2 yields that the infection rate matrix B has the equitable partition π = {N } ,36here the cell N = { , ..., N } contains all nodes. Thus, (14) yields, that there exists some degree d which satisfies d = (cid:88) k ∈N β ik = N (cid:88) k =1 β ik for all nodes i . Thus, we obtain with definition (24) that the matrix B is regular. E Proof of Theorem 3
By assumption, the infection rates β i,j are the same for all nodes i in any cell N l and all nodes j inany cell N p . Thus, with the definition of the vectors y , ..., y r in (23), the symmetric infection ratematrix equals B = (cid:16) y ... y r (cid:17) ˜ B V (cid:54) =0 y T ... y Tr (66)for some symmetric r × r matrix ˜ B V (cid:54) =0 . Since the kernel ker( B ) is the orthogonal complement of thesubspace V (cid:54) =0 , it holds that R N = V (cid:54) =0 ⊕ ker( B ). Thus, any viral state vector v ( t ) ∈ [0 , N can bedecomposed as v ( t ) = ˜ v ( t ) + v ker ( t ), where ˜ v ( t ) ∈ V (cid:54) =0 and v ker ( t ) ∈ ker( B ). With the decomposition v ( t ) = ˜ v ( t ) + v ker ( t ), NIMFA (2) becomes dv ( t ) dt = − S (˜ v ( t ) + v ker ( t )) + diag ( u − ˜ v ( t ) − v ker ( t )) B (˜ v ( t ) + v ker ( t ))= − S ˜ v ( t ) − Sv ker ( t ) + diag ( u − ˜ v ( t ) − v ker ( t )) B ˜ v ( t ) , where the second equality follows from Bv ker ( t ) = 0. Further rearrangement yields that dv ( t ) dt = ( B − S ) ˜ v ( t ) − diag (˜ v ( t )) B ˜ v ( t ) − Sv ker ( t ) − diag ( v ker ( t )) B ˜ v ( t ) . (67)We decompose the derivative dv ( t ) /dt into two addends, by making use of two lemmas: Lemma 6.
Suppose that the assumptions in Theorem 3 hold true. Then, if ˜ v ∈ V (cid:54) =0 , the vector B ˜ v − S ˜ v − diag (˜ v ) B ˜ v (68) is element of V (cid:54) =0 .Proof. We consider the three addends of the vector (68) separately. First, (66) shows that the addend B ˜ v is element of V (cid:54) =0 if ˜ v ∈ V (cid:54) =0 . Second, we consider the addend S ˜ v . By assumption, the curing rates δ i are the same for all nodes i in the same cell N l . Thus, we obtain from the definition of the agitationmodes y l in (23) that Sy l = δ i y l (69)37or l = 1 , ..., r , where i denotes an arbitrary node in cell N l . Since the agitation modes y , ..., y r spanthe subspace V (cid:54) =0 , (69) implies that S ˜ v if ˜ v ∈ V (cid:54) =0 .Third, we consider the addend diag (˜ v ) B ˜ v . Since ˜ v ∈ V (cid:54) =0 , it holds that˜ v = r (cid:88) l =1 (cid:0) y Tl ˜ v (cid:1) y l . Similarly, since B ˜ v ∈ V (cid:54) =0 , it holds that B ˜ v = r (cid:88) l =1 (cid:0) y Tl B ˜ v (cid:1) y l . (70)Thus, we obtain that diag (˜ v ) B ˜ v = r (cid:88) l =1 r (cid:88) p =1 (cid:0) y Tl ˜ v (cid:1) (cid:0) y Tp B ˜ v (cid:1) diag( y l ) y p . (71)From the definition of the vectors y l in (23) it follows thatdiag ( y l ) y p = y l if l = p, l (cid:54) = p, where the N × y l = (cid:0) ( y l ) , ..., ( y l ) N (cid:1) T denotes Hadamard product of the vector y l with itself.Thus, (71) becomes diag (˜ v ) B ˜ v = r (cid:88) l =1 (cid:0) y Tl ˜ v (cid:1) (cid:0) y Tl B ˜ v (cid:1) y l . (72)With (23), the Hadamard product y l equals( y l ) i = |N l | if i ∈ N l , i (cid:54)∈ N l , which implies that ( y l ) = y l / (cid:112) |N l | and yields with (72) thatdiag (˜ v ) B ˜ v = r (cid:88) l =1 (cid:0) y Tl ˜ v (cid:1) (cid:0) y Tl B ˜ v (cid:1)(cid:112) |N l | y l . Thus, the vector diag (˜ v ) B ˜ v is a linear combination of the vectors y , ..., y r , which implies thatdiag (˜ v ) B ˜ v V (cid:54) =0 . Hence, we have shown that all three addends of the vector (68) are in V (cid:54) =0 , whichcompletes the proof. Lemma 7.
Suppose that the assumptions in Theorem 3 hold true. Then, if ˜ v ∈ V (cid:54) =0 and v ker ∈ ker( B ) ,the vector Sv ker + diag ( v ker ) B ˜ v (73) is element of ker( B ) . roof. The kernel ker( B ) is the orthogonal complement of the subspace V (cid:54) =0 . Thus, the vector (73) iselement of ker( B ) if Sv ker is orthogonal to every basis vector y , ..., y r of the subspace V (cid:54) =0 . We showseparately that both addends of the vector (73) are orthogonal to every vector y , ..., y r . First, forany l = 1 , ..., r , we obtain for the first addend in (73) that y Tl Sv ker = ( Sy l ) T v ker , since the matrix S is symmetric. With (69), we obtain for an arbitrary node i ∈ N l that y Tl Sv ker = δ i y Tl v ker = 0 . Thus, the addend Sv ker is element of ker( B ).Second, for any l = 1 , ..., r , we obtain for the second addend in (73) with (70) that y Tl diag ( v ker ) B ˜ v = r (cid:88) q =1 (cid:0) y Tl B ˜ v (cid:1) y Tl diag ( v ker ) y q = r (cid:88) q =1 (cid:0) y Tq B ˜ v (cid:1) v T ker diag ( y l ) y q . Analogous steps as in the proof of Lemma 6 yield that y Tl diag ( v ker ) B ˜ v = (cid:0) y Tl B ˜ v (cid:1)(cid:112) |N l | v T ker y l . Thus, by the orthogonality of the vectors v ker and y l , y Tl diag ( v ker ) B ˜ v = 0 , which completes the proof.With Lemma 6 and Lemma 7, we obtain from (67) that dv ( t ) dt = d ˜ v ( t ) dt + dv ker ( t ) dt , where d ˜ v ( t ) dt = − S ˜ v ( t ) + diag ( u − ˜ v ( t )) B ˜ v ( t )and dv ker ( t ) dt = − Sv ker ( t ) − diag ( v ker ( t )) B ˜ v ( t ) , which completes the proof, sincediag ( v ker ( t )) B ˜ v ( t ) = diag ( B ˜ v ( t )) v ker ( t ) . Proof of Theorem 4
Since the spreading rates are homogeneous, β ij = β and δ i = δ , the infection rate matrix equals B = βuu T , (74)and the curing rate matrix equals S = δI. (75)Thus, with r = 1 cell N = { , ..., N } , Theorem 3 yields that the viral state v ( t ) can be decomposedas v ( t ) = ˜ v ( t ) + v ker ( t ). We prove Theorem 4 in two steps. First, we show that the projection ˜ v ( t )equals c ( t ) v ∞ at every time t . Second, we prove that the projection v ker ( t ) equals c ( t ) y at everytime t . F.1 Projection on the subspace V (cid:54) =0 With the reduced-size curing rate matrix S π = δ and the quotient matrix B π = N β , Theorem 1 yieldsthat the projection on the subspace V (cid:54) =0 satisfies ˜ v ( t ) = v π ( t ) u . The evolution (16) of the reduced-size,scalar viral state v π ( t ) becomes dv π ( t ) dt = − δv π ( t ) + (1 − v π ( t )) N βv π ( t ) , whose solution equals [43, 33] v π ( t ) = v π ∞ (cid:16) (cid:16) w t + Υ (0) (cid:17)(cid:17) with the reduced-size steady state v π ∞ = 1 − δβN , the viral slope w = βN − δ and the constantΥ (0) = artanh (cid:18) v (0) v ∞ − (cid:19) . Thus, the projection ˜ v ( t ) = v π ( t ) u is equal to c ( t ) v ∞ at every time t . F.2 Projection on the kernel ker( B ) With (74) and (75), Theorem 3 yields that the projection v ker ( t ) obeys dv ker ( t ) dt = − (cid:0) δI + β diag (cid:0) uu T ˜ v ( t ) (cid:1)(cid:1) v ker ( t ) . Since ˜ v ( t ) = c ( t ) v ∞ and v ∞ = v ∞ ,i u for an arbitrary node i , we obtain that dv ker ( t ) dt = − ( δI + βN c ( t ) v ∞ ,i I ) v ker ( t ) . From the function c ( t ) given in (27), it follows that dv ker ( t ) dt = − δv ker ( t ) − βN v ∞ ,i (cid:16) (cid:16) w t + Υ (0) (cid:17)(cid:17) v ker ( t ) . (76)40or any initial condition v ker (0) ∈ ker( B ), the right side of (76) is element of the one-dimensionalsubspace span { v ker (0) } . Thus, the projection v ker ( t ) obeys v ker ( t ) = c ( t ) v ker (0). We solve (76) intwo steps. First, we compute the initial condition v ker (0). Since v (0) = v ker (0) + c (0) v ∞ , the initialcondition v ker (0) is obtained as v ker (0) = v (0) − c (0) v ∞ = v (0) − v T ∞ v (0) (cid:107) v ∞ (cid:107) v ∞ . Since v ∞ = v ∞ ,i u , it follows that v ker (0) = v (0) − N u T v (0) u, which simplifies to v ker (0) = (cid:18) I − N uu T (cid:19) v (0) . Second, using v ker ( t ) = c ( t ) v ker (0), we project (76) on the initial condition v ker (0) to obtain that thescalar function c ( t ) obeys the linear differential equation dc ( t ) dt = − δc ( t ) − βN v ∞ ,i (cid:16) (cid:16) w t + Υ (0) (cid:17)(cid:17) c ( t )and hence, with the constant Φ = βN v ∞ ,i / δ ,log ( c ( t )) = − (cid:90) t (cid:18) Φ + 12 βN v ∞ ,i tanh (cid:16) w ξ + Υ (0) (cid:17)(cid:19) dξ. The integral of the hyperbolic tangent equals to the logarithm of the hyperbolic cosine [2], (cid:90) tanh ( ξ ) dξ = log (cosh( ξ )) , which yields that log ( c ( t )) = − Φ t − βN v ∞ ,i w log (cid:16) cosh (cid:16) w t + Υ (0) (cid:17)(cid:17) + K (0) (77)for some constant K (0). With the definition of the viral slope w in Subsection F.1 and v ∞ ,i = 1 − δβN ,we obtain that βN v ∞ ,i w = βN (1 − δβN ) βN − δ = 1 . Thus, (77) becomes log ( c ( t )) = − Φ t + log (cid:18) cosh (cid:16) w t + Υ (0) (cid:17) − (cid:19) + K (0) , and we obtain, with the hyperbolic secant sech( x ) = cosh( x ) − , that c ( t ) = Υ (0) e − Φ t sech (cid:16) w t + Υ (0) (cid:17) . (78)with the constant Υ (0) = exp( K (0)). At the initial time t = 0, (78) becomes c (0) = Υ (0) sech (Υ (0)) , and it holds that c (0) = v T ker (0) v (0) (cid:107) v ker (0) (cid:107) . Thus, with sech( x ) = cosh( x ) − , we obtain the constant Υ (0) as (29), which completes the proof.41 Proof of Theorem 5
The viral state ˜ v i ( t ) evolves as d ˜ v i ( t ) dt = ˜ f NIMFA ,i (˜ v ( t )) , where we define, for every node i ,˜ f NIMFA ,i (˜ v ( t )) = − ˜ δ i ˜ v i ( t ) + (1 − ˜ v i ( t )) N (cid:88) j =1 ˜ β ij ˜ v j ( t ) . (79)Since ˜ β ij ≥ β ij and ˜ δ i ≤ δ i for all nodes i , we obtain an upper bound on NIMFA (1) as dv i ( t ) dt ≤ − ˜ δ i v i ( t ) + (1 − v i ( t )) N (cid:88) j =1 ˜ β ij v j ( t )= ˜ f NIMFA ,i ( v ( t )) . Since dv i ( t ) /dt ≤ ˜ f NIMFA ,i ( v ( t )), we can apply the Kamke-M¨uller condition [15, 22], see also [16]: If v ≤ ˜ v and v i = ˜ v i implies that ˜ f NIMFA ,i ( v ) ≤ ˜ f NIMFA ,i (˜ v ) for all nodes i , then v (0) ≤ ˜ v (0) implies that v ( t ) ≤ ˜ v ( t ) at every time t ≥ v ≤ ˜ v and v i = ˜ v i implies that ˜ f NIMFA ,i ( v ) ≤ ˜ f NIMFA ,i (˜ v ). From(79), we obtain that˜ f NIMFA ,i ( v ) − ˜ f NIMFA ,i (˜ v ) = − ˜ δ i ( v i − ˜ v i ) + (1 − v i ) N (cid:88) j =1 ˜ β ij v j − (1 − ˜ v i ) N (cid:88) j =1 ˜ β ij ˜ v j . From v i = ˜ v i , it follows that˜ f NIMFA ,i ( v ) − ˜ f NIMFA ,i (˜ v ) = (1 − v i ) N (cid:88) j =1 ˜ β ij v j − (1 − v i ) N (cid:88) j =1 ˜ β ij ˜ v j , which yields that ˜ f NIMFA ,i ( v ) − ˜ f NIMFA ,i (˜ v ) = N (cid:88) j =1 ˜ β ij ( v j − v i v j − ˜ v j + v i ˜ v j )= N (cid:88) j =1 ˜ β ij (1 − v i ) ( v j − ˜ v j ) . Since ( v j − ˜ v j ) ≤
0, we obtain that ˜ f NIMFA ,i ( v ) ≤ ˜ f NIMFA ,i (˜ v ), which completes the proof. H Proof of Theorem 6
Here, we prove that v i ( t ) ≥ v lb ,l ( t ) for all nodes i in any cell N l . The proof of v i ( t ) ≤ v ub ,l ( t ) followsanalogously. First, we define the curing rates ˜ δ max ,i by˜ δ max ,i = δ max ,l for all nodes i in any cell N p . Thus, (32) implies that ˜ δ max ,i ≥ δ i for all nodes i = 1 , ..., N .42 emma 8. For all nodes i, j , there are infection rates ˜ β ij , which satisfy ˜ β ij ≤ β ij and (cid:88) j ∈N l ˜ β ij = d min ,pl (80) for all nodes i in any cell N p and all cells N l .Proof. With the definition of the lower bound d min ,pl in (30), we obtain that (80) is satisfied if (cid:88) j ∈N l ˜ β ij = min i ∈N p (cid:88) k ∈N l β ik . (81)Denote the difference of the infection rates by ε ij = β ij − ˜ β ij . Thus, ˜ β ij ≤ β ij and ˜ β ij ≥ ≤ ε ij ≤ β ij . We obtain from (81) that the differences ε ij must satisfy (cid:88) j ∈N l β ij − (cid:88) j ∈N l ε ij = min i ∈N p (cid:88) k ∈N l β ik , which yields that (cid:88) j ∈N l ε ij = (cid:88) j ∈N l β ij − min i ∈N p (cid:88) k ∈N l β ik . (82)To complete the proof, we must show that there exist some ε ij ∈ [0 , β ij ] that solve (82). Since (cid:88) j ∈N l β ij ≥ min i ∈N p (cid:88) k ∈N l β ik and β ij ≥
0, the right side of (82) is some value in [0 , (cid:80) j ∈N l β ij ]. Since the feasible values of theinfection rate differences ε ij are in the interval [0 , β ij ], the left side of (82) may attain an arbitraryvalue in [0 , (cid:80) j ∈N l β ij ]. Thus, there are some infection rate differences ε ij ∈ [0 , β ij ] that solve (82),which completes the proof.Lemma 8 states the existence of an N × N matrix ˜ B min whose elements ˜ β min ,ij satisfy ˜ β ij ≤ β ij and (80). Thus, π is an equitable partition of the matrix ˜ B min . We define the N × v lb ( t )as d ˜ v lb ( t ) dt = − diag (cid:16) ˜ δ max , , ..., ˜ δ max ,N (cid:17) ˜ v lb ( t ) + diag ( u − ˜ v lb ( t )) ˜ B min ˜ v lb ( t ) (83)with the initial viral state ˜ v lb ,i (0) = min j ∈N p v j (0)for all nodes i in any cell N p . Since ˜ v lb ,i (0) ≤ v i (0), ˜ δ max ,i ≥ δ i and ˜ β min ,ij ≤ β ij for all nodes i, j ,Theorem 5 yields that ˜ v lb ,i ( t ) ≤ v i ( t ) for every node i at every time t . Furthermore, Theorem 1 yieldsthat the N -dimensional dynamics of the viral state ˜ v lb ( t ) in (83) can be reduced to the r -dimensionaldynamics of the reduced-size viral state v lb ( tt