Cost-Optimal Switching Protection Strategy in Adaptive Networks
CCost-Optimal Switching Protection Strategy in Adaptive Networks
Masaki Ogura and Victor M. Preciado
Abstract — In this paper, we study a model of network adap-tation mechanism to control spreading processes over switch-ing contact networks, called adaptive susceptible-infected-susceptible model. The edges in the network model are ran-domly removed or added depending on the risk of spreadthrough them. By analyzing the joint evolution of the spreadingdynamics “in the network” and the structural dynamics “ofthe network”, we derive conditions on the adaptation law tocontrol the dynamics of the spread in the resulting switchingnetwork. In contrast with the results in the literature, we allowthe initial topology of the network to be an arbitrary graph.Furthermore, assuming there is a cost associated to switchingedges in the network, we propose an optimization frameworkto find the cost-optimal network adaptation law, i.e., the cost-optimal edge switching probabilities. Under certain conditionson the switching costs, we show that the optimal adaptationlaw can be found using convex optimization. We illustrate ourresults with numerical simulations.
I. I
NTRODUCTION
Accurate prediction and effective control of spreadingdynamics over networks are relevant problems in epidemi-ology and public health, computer malware, or securityof cyberphysical networks. Although we find many recentadvances in the field of network epidemiology [1], thereare still many open questions to transfer this knowledge torealistic epidemiological situations. One fundamental resultin the mathematical analysis of spreading in networks is theclose connection between the eigenstructure of the contactnetwork and epidemic thresholds [2]–[4]. This result enabledthe authors in [5]–[8] to propose a convex optimizationframework to design the optimal distribution of pharmaceu-tical resources to control disease spread. This framework isspecially adapted to static network structures in which thepattern of interconnections does not change over time. Aswe argue below, this may not be the case in many practicalsituations.Social distancing is one of the most important nonphar-maceutical approaches to control disease spread over humancontact networks [9], [10]. Examples of social distancingare, for instance, isolation of patients, school closures, andavoidance of crowds. In spite of the obvious effect that suchbehavior have on the dynamics of the spread, there is a lackof studies about the role of social distancing in the spread ofdiseases over human contact networks. One of the reasons isthat social distancing induces an adaptation of the networkstructure that depends on the state of the infection. Although
The authors are with the Department of Electrical and Systems Engi-neering, University of Pennsylvania, Philadelphia, PA 19014, USA. Email: { ogura,preciado } @seas.upenn.edu This work was supported in part by the NSF under grants CNS-1302222and IIS-1447470. there are results in the literature about disease spreading overtime-varying networks (see, e.g., [11]–[13]), these works arebased on the assumption that the evolution of the network isindependent of the state of the individuals. In this paper, wepropose a tractable framework to analyze the co-evolution ofthe state-dependent network structure and the dynamics ofthe spreading process taking place on it.Most of the available studies of spreading processes overhuman networks with social distancing have been relyingon various unrealistic simplifying assumptions. The authorsin [14]–[17] propose epidemic thresholds under the so-called mixing assumption; all the individuals in a networkinteract randomly with each other. However, this assumptionis not satisfied in structured human populations. Althoughthe analysis in [18] does not rely on the mixing assumption,it relies on the quantity called a reproduction number, whosevalidity for disease spread over time-varying networks is notyet fully established [19].This paper analyzes, without the mixing assumption, thedynamics of spreading processes taking place in switchingnetworks whose structure adapt to the state of the spread. Thedisease spread is modeled by an extended version of the well-known susceptible-infected-susceptible (SIS) model, whichis called the adaptive SIS model [16]. Without the mixingassumption employed in [16], we derive conditions underwhich the network adaptation is able to protect against thespread of the disease. We furthermore use these conditionsto propose a cost-optimal adaptation policy to contain thedisease. This policy is based on the assumption that adaptingthe network structure to the state of the disease has anassociated cost. The optimal policy can be then found bysolving an optimization program. Under certain conditions,this optimization program can be effectively solved usingelements from convex optimization [20].This paper is organized as follows. In Section II, weintroduce the adaptive SIS model studied in this paper.In Section III, we analyze the exponential stability of theinfection-free equilibrium of the adaptive SIS models. Basedon our stability analysis, Sections IV and V study an cost-optimal adaptation strategy for networks of homogeneousand heterogeneous agents, respectively.
A. Mathematical Preliminaries
The probability of an event is denoted by P ( · ) . Theexpectation of a random variable is denoted by E [ · ] . Welet I denote the identity matrix and p the p -dimensionalvector whose entries are all one (we omit the dimension p when it is obvious from the context). A real matrix A ,or a vector as its special case, is said to be nonnegative, a r X i v : . [ c s . S I] S e p enoted by A ≥
0, if all the entries of A are nonnegative.The notations A > A ≤ A < B having the samedimensions as A , the notation A ≤ B implies A − B ≤
0. Weagain understand A < B , A ≥ B , and A > B in the obviousmanner. The Kronecker product [21] of A and B is denotedby A ⊗ B . Let A be a square matrix. The maximum real partof the eigenvalues of A is denoted by η ( A ) . We say that A isHurwitz stable if η ( A ) <
0. Also, we say that A is Metzler ifthe off-diagonal entries of A are all non-negative. We saythat A is irreducible if no similarity transformation by apermutation matrix makes A into a block upper triangularmatrix. For matrices A , . . . , A n , the direct sum (cid:76) ni = A i is defined as the block diagonal matrix having the blockdiagonals A , . . . , A n . When A , . . . , A n have the samenumber of columns, we define col ≤ i ≤ n A i = col ( A , . . . , A n ) as the block matrix obtained by stacking the matrices A , . . . , A n .A directed graph is a pair G = ( V , E ) , where V is a finiteset of nodes, and E ⊆ V × V is a set of directed edges. Unlessotherwise stated, we assume V = { , . . . , n } . A directed pathfrom i to j in G is an ordered set of nodes ( i , · · · , i (cid:96) ) suchthat i = i , ( i k , i k + ) ∈ E for k = , . . . , (cid:96) −
1, and i (cid:96) = j . Wesay that G is strongly connected if there exists a directedpath from i to j for all i , j ∈ V . The adjacency matrix of G is defined as the n × n matrix A = [ a i j ] i , j such that a i j = ( i , j ) ∈ E and a i j = G = ( V , E ) , where V is a finite set and E isa subset of unordered pairs { i , j } of the elements i , j ∈ V .The adjacency matrix of an undirected graph is defined in asimilar manner. A graph is strongly connected if and only ifits adjacency matrix is irreducible.Finally, we recall basic facts about a class of optimizationproblems called geometric programs [20]. Let x , . . . , x m denote m real positive variables. We say that a real-valuedfunction f of x = ( x , . . . , x m ) is a monomial function if thereexist c > a , . . . , a m ∈ R such that f ( x ) = cx a · · · x a m m .Also, we say that f is a posynomial function if it is a sumof monomial functions of x . Given posynomial functions f , . . . , f p and monomial functions g , . . . , g q , the optimizationproblem minimize x f ( x ) subject to f i ( x ) ≤ , i = , . . . , p , g j ( x ) = , j = , . . . , q , is called a geometric program . It is known [20] that a geo-metric program can be converted into a convex optimizationproblem.II. S USCEPTIBLE -I NFECTED -S USCEPTIBLE M ODELOVER A DAPTIVE N ETWORKS
This section introduces the model of spreading processesover adaptive networks studied in this paper and state the op-timal design problem under consideration. Each node in thenetwork can be in one of two states: susceptible or infected .The state of node i evolves over time and is represented by a binary variable x i ∈ { , } . We say that node i is susceptibleat time t if x i ( t ) =
0, and is infected at time t if x i ( t ) =
1. Inthis paper, we model the evolution of x i as a continuous-timestochastic process taking values in { , } . We also assumethat the structure of the network in which the spreadingprocess is taking place evolves over time. In particular, wemodel the network G as a continuous-time random graphprocess taking values in the set of undirected graphs with n nodes. In other words, we model the dynamics of spreadingas a stochastic process taking place over a random graphprocess. We denote by N i ( t ) the set of neighbors of node i in the graph G ( t ) , i.e., N i ( t ) = { j ∈ V : { i , j } ∈ G ( t ) } , andby A ( t ) = [ a i j ( t )] i , j the adjacency matrix of G ( t ) .The spreading models over adaptive networks studied inthis paper are formally introduced as the class of pairs ( x , G ) = ( { x i } ni = , G ) satisfying the following definition: Definition 2.1:
Let G = ( V , E ) be an undirected graphwith adjacency matrix A = [ a i j ( )] i , j . The pair ( x , G ) is saidto be an adaptive susceptible-infected-susceptible model over G ( ASIS model for short) if there exist nonnegative numbers β i , δ i , φ i , and ψ i j ( i , j = , . . . , n ) such that the followingconditions hold:a) G ( ) = G ;b) The process ( x , G ) is Markov;c) For every i , the transition probabilities of x i are given by P ( x i ( t + h ) = | x i ( t ) = ) = β i ∑ k ∈ N i ( t ) x k ( t ) h + o ( h ) , (1) P ( x i ( t + h ) = | x i ( t ) = ) = δ i h + o ( h ) , (2)where o ( h ) is a function such that lim h → o ( h ) / h = i , j , the transition probabilities of a i j are given by P ( a i j ( t + h ) = | a i j ( t ) = ) = ( φ i x i ( t ) + φ j x j ( t )) h + o ( h ) , (3) P ( a i j ( t + h ) = | a i j ( t ) = ) = a i j ( ) ψ i j h + o ( h ) . (4)e) ψ i j = ψ ji for all i and j .The constants β i , δ i , φ i , and ψ i j are respectively called infection , recovery , cutting , and rewiring rates .We can interpret the above model as follows. Item b)indicates that the future evolution of the spread, given thepresent state, does not depend on the past. The probabilitiesin c) describe how nodal states evolve. Notice that, if G ( t ) were a static network, these probabilities would coincidewith those of the NIMFA model [4] with heterogeneousinfection and recovery rates. Eqn. (1) indicates that, if node i is susceptible and its neighbor j is infected, then i becomesinfected with the instantaneous infection rate β i . Moreover,the rate is proportional to the number of infected neighbors.Eqn. (2) implies that, once node i becomes infected, it willbecome susceptible with an instantaneous recovery rate δ i .Item d) describes an adaptation mechanism of the net-work to the state of the disease. Eqn. (3) indicates that,whenever a node i is infected, the node adaptively removesedges connecting the node and its neighbors according to aPoisson process with rate φ i . This mechanism is designedto contain the spread through edges connected to infectednodes. Moreover, (4) describes a mechanism for which ‘cut’ i =1 x j x i =1 x j f i x i x j x i x j y ij Fig. 1: Adaptively switching network.edges are ‘rewired’ or added back to the network. We assumethat edge { i , j } is added to the network with a rewiringrate ψ i j . See Fig. 1 for a schematic picture of these transitionprobabilities. Finally, Item e) follows from the assumptionthat G ( t ) is undirected, although this is not an essentialrestriction and could be relaxed to account for directedcontact networks. Also, notice that we have included theterm a i j ( ) in (4) to guarantee that only those edges thatwere present at the initial time t = Remark 2.2:
A model similar to the ASIS model proposedin this paper was studied in [16], where it was assumed thatthe initial graph G is the complete graph. A major differencebetween our model and the one in [16] is the informationavailable to each node. In the model in [16], it is assumedthat nodes know the states of their neighbors. In contrast,we do not assume to have access to this knowledge in ourmodel. This difference has a direct implication in the link-breaking process. For example, in [16], an infected node doesnot break the edge between itself and its infected neighbors.On the other hand, in our model, an infected node will breakedges independent of the state of its neighbors.Once the adaptive network under consideration is de-scribed, we define the exponential stability of the infection-free equilibrium p i ( t ) = Definition 2.3:
For t ≥ p i ( t ) = P ( x i ( t ) = ) be theinfection probability of node i . We say that the infection-free equilibrium p i ( t ) ≡ ( x , G ) is exponentially stable if there exist K ≥ α > p i ( t ) ≤ Ke − α t for all i , t , and x i ( ) . We call α the decayrate .In many practical situations, there is a cost associated tothe mechanisms of cutting and rewiring edges in a network.Accordingly, we assume we have two scalar cost functions f and g , defined on [ , ∞ ) , describing the cost associated to therates of cutting and rewiring edges, respectively. The mainpurpose of this paper is to find the cost-optimal switchingstrategy, defined by the values of the cutting and rewiringrates, to drive the state of the spread towards the disease-free equilibrium at a given exponential rate. The total costof a switching strategy is given by: C = n ∑ i = f ( φ i ) + ∑ { i , j }∈ E g ( ψ i j ) . We also assume the following bounds on the rates:¯ φ ≤ φ i ≤ ¯ φ , ¯ ψ ≤ ψ i j ≤ ¯ ψ (5) for some nonnegative numbers ¯ φ , ¯ φ , ¯ ψ , and ¯ ψ . Now, we areready to state the problem investigated in this paper. Problem 2.4:
Given α >
0, find the cutting and rewiringrates φ i and ψ i j satisfying (5) such that the adaptive SISmodel is exponentially stable with decay rate α and the totalcost C is minimized.In this paper we solve Problem 2.4 under the followingreasonable assumption: Assumption 2.5: G is strongly connected. Moreover, β i > δ i >
0, and ψ i j > { i , j } ∈ E .III. S TABILITY A NALYSIS
In this section, we perform a stability analysis of the ASISmodel ( x , G ) over G . We begin by representing the modelas a set of stochastic differential equations with Poissoncounters. For γ ≥
0, we let N γ denote a Poisson counterwith rate γ . We assume that all Poisson counters appearingin this paper are stochastically independent. We will usesuperscripts for the Poisson counters to distinguish those thathas the same rates but are independent. Then, from (1) and(2), the evolution of the nodal states can be described as: dx i = − x i dN δ i + ( − x i ) ∑ k ∈ N i ( ) a ik x k dN ( k ) β i . (6)Similarly, from (3) and (4), the evolution of the edges canbe written as: da i j = − a i j ( x i dN ( j ) φ i + x j dN ( i ) φ j ) + ( − a i j ) dN ψ ij , (7)for all i and j such that { i , j } ∈ E .Using the stochastic differential equations (6) and (7), wederive an upper bounding linear model for the infection prob-abilities p i . To state the linear model, we define the followingvariables. Let us define p ( t ) ∈ R n by p = col ≤ i ≤ n p i . Also,for i = , . . . , n and j ∈ N i ( ) , define q i j ( t ) = E [ a i j ( t ) x i ( t )] and let q i = col j ∈ N i ( ) q i j and q = col ≤ i ≤ n q i . Let d i denotethe degree of node i in the initial graph G and m thenumber of the edges in G . Then, q has the dimension ∑ Ni = d i = m . We also introduce the following matrices.Define T i ∈ R × ( m ) as the unique matrix satisfying: T i q = ∑ k ∈ N i ( ) q ki . (8)Then define the matrices B = col ≤ i ≤ n ( β i T i ) , B = col ≤ i ≤ n ( β i d i ⊗ T i ) , D = (cid:76) ni = δ i , D = (cid:76) ni = ( δ i I d i ) , Φ = (cid:76) ni = ( φ i I d i ) , Ψ = (cid:76) ni = col j ∈ N i ( ) ψ i j , Ψ = (cid:76) ni = (cid:76) j ∈ N i ( ) ψ i j . Now, we can state the followingtheorem: Theorem 3.1:
Define M ∈ R ( n + m ) × ( n + m ) by M = (cid:20) − D B Ψ B − D − Φ − Ψ (cid:21) . (9)Then, for all x ( ) , . . . , x n ( ) , it holds that ddt (cid:20) pq (cid:21) ≤ M (cid:20) pq (cid:21) . (10) Since the design of infection and recovery rates have been previouslystudied in [5]–[8], we focus our attention on the design of φ i and ψ ij only(although our framework can be easily extended to include β i and δ i asadditional design variables). roof: Taking the expectations in (6) yields that ddt E [ x i ] = − δ i E [ x i ] + β i ∑ k ∈ N i ( ) E [( − x i ) a ik x k ] . Since E [( − x i ) a ik x k ] ≤ E [ a ik x k ] = q ki , from the definitionof T i in (8), we obtain d p i / dt ≤ − δ i p i + β i T i q . This impliesthat d p / dt ≤ − D p + B q , which proves the upper half blockof the inequality (10).Then, let us evaluate dq / dt . The Itˆo rule for jump pro-cesses [22] yields that d ( a i j x i ) = − a i j x i dN ( j ) φ i − a i j x i x j dN ( i ) φ j + ( − a i j ) x i dN ψ ij − a i j x i dN δ i + a i j ( − x i ) ∑ k ∈ N i ( ) a ik x k dN ( k ) β i . Taking expectations in this equation, we obtain dq i j dt = − φ i E [ a i j x i ] − φ j E [ a i j x i x j ]+ ψ i j E [( − a i j ) x i ] − δ i q i j + β i ∑ k ∈ N i ( ) E [ a i j ( − x i ) a ik x k ] . (11)Since ∑ k ∈ N i ( ) E [ a i j ( − x i ) a ik x k ] ≤ ∑ k ∈ N i ( ) E [ a ik x k ] = T i q ,we obtain dq i j / dt ≤ ψ i j p i − ( φ i + ψ i j + δ i ) q i j + β i (cid:62) d i T i q from(11). Stacking the variables q i j for all j ∈ N i ( ) yields dq i / dt ≤ col j ∈ N i ( ) ( ψ i j p i ) − ( φ i + δ i ) q i − ψ j q i + β i ( d i ⊗ T i ) q ,where ψ j = (cid:76) j ∈ N i ( ) ψ ji . This proves the lower half blockof the inequality (10) and completes the proof.From Theorem 3.1 we immediately have the followingsufficient condition for exponential stability of the infection-free equilibrium. Theorem 3.2: If M is Hurwitz stable, then the infection-free equilibrium of the adaptive SIS model is exponentiallystable with a decay rate − η ( M ) .Before closing this section, we prove the following propo-sition that plays an important role in the rest of the paper. Proposition 3.3:
The matrix M is irreducible. Proof:
Define L = (cid:20) O TJ S (cid:21) , where J = n (cid:77) i = d i , T = col ≤ i ≤ n T i , S = col ≤ i ≤ n ( d i ⊗ T i ) . (12)Since β i and ψ i j are positive by Assumption 2.5, if M i j = L i j = i and j . From this we see that,to show the irreducibility of M , it is sufficient to show that L is irreducible.In order to show that L is irreducible, we shall showthat the directed graph H , defined as the graph havingadjacency matrix L , is strongly connected. We identify thenodes 1 , . . . , n + m of H using the variables p , . . . , p n , q j ( j ∈ N ( ) ), . . . , q n j ( j ∈ N n ( ) ). Then, the upper-right block T of the matrix L shows that the graph H hasdirected edge ( p i , q ji ) for all i = , . . . , n and j ∈ N i ( ) .Similarly, from the matrices J and S , we see that H hasthe edges ( q i j , p i ) and ( q i j , q ki ) for all i = , . . . , n and j , k ∈ N i ( ) . Then, let us show that H has a directed pathfrom p i to p j for all i , j ∈ { , . . . , n } . Since G is stronglyconnected, it has a path ( i , . . . , i (cid:96) ) such that i = i and i (cid:96) = j . Therefore, from the above fact, we can see that H contains the directed path ( p i , q i , i , q i , i , . . . , q i (cid:96) , i (cid:96) − , p j ) .In the same way, we can show that H also contains thedirected path ( p i , q ji , q i j , p i ) for every { i , j } ∈ E . These twoobservations show that H is strongly connected and, hence, L is irreducible. IV. H OMOGENEOUS C ASE
Based on the stability analysis presented in the previoussection, we study the optimal design problem stated inProblem 2.4. We start our analysis by assuming that the ASISmodel is homogeneous, as defined below (this restriction isrelaxed in the next section):
Definition 4.1:
We say that the adaptive SIS model is homogeneous if there exist nonnegative constants β , δ , φ ,and ψ such that β i = β , δ i = δ , φ i = φ , and ψ i j = ψ for all i and j .In the homogeneous case, the stability criterion in Theo-rem 3.2 reduces to the next simple condition. Theorem 4.2:
Assume that the adaptive SIS model ishomogeneous. Let ρ = η ( A ) . Then, the infection-free equi-librium of the adaptive SIS model is exponentially stable if δ > β ρ − φ − ψ + (cid:112) ( β ρ + φ + ψ ) − β ρφ . (13) Proof:
Assume that the model is homogeneous. Then,the matrix M defined in (9) takes the form M = (cid:20) − δ I β T ψ J β S − ( δ + φ + ψ ) I (cid:21) , where the matrices J , T , and S are defined by (12). We provethe theorem under the assumption that β ρ (cid:54) = φ . Since G is strongly connected by Assumption 2.5, A is irreducibleand therefore has a positive eigenvector v correspondingto the eigenvalue ρ (see [23]). Define the positive vec-tor w = col ≤ i ≤ n ( v i d i ) . Then, the definition of T i shows T i w = ∑ k ∈ N i ( ) w ki = ∑ k ∈ N i ( ) v k = ( Av ) i = ρ v i and therefore Tw = λ v . In the same manner, we can show Sw = ρ w . Sincewe have Jv = w , for a nonnegative number c it follows that M (cid:20) cvw (cid:21) = (cid:20) ( β ρ − c δ ) v ( c ψ + β ρ − ( δ + φ + ψ )) w (cid:21) . (14)Hence, if a real number λ satisfies the following equations: β ρ − c δ = c λ , c ψ + β ρ − ( δ + φ + ψ ) = λ , (15)then, by (14), we see that the nonnegative vector col ( cv , w ) is an eigenvector of the irreducible and Metzler matrix M corresponding to the eigenvalue λ . This implies that η ( M ) = λ (see [23, Theorem 17]). Therefore, the condition λ < λ , we solve (15) with respect to λ and obtain λ + ( δ + φ + ψ − β ρ ) λ + δ ( δ + φ + ψ ) − β ρ ( δ + ψ ) = λ = λ + , where λ + = β ρ − δ − φ − ψ + (cid:112) ( β ρ + φ + ψ ) − β ρφ . Then, the pair ( c , λ ) = ( β ρ / ( λ + + δ ) , λ + ) satisfies (15).We remark that λ + + δ is positive because of the initialassumption β ρ (cid:54) = φ . Therefore, c ≥ η ( M ) = λ + . (16)Therefore, by Theorem 3.2, the infection-free equilibrium ofthe adaptive SIS model is exponentially stable if λ + < Remark 4.3:
In the special case when the network doesnot adapt to the prevalence of infection, i.e., when φ = δ > β ρ ( A ) for the SIS models over static networks [2], [4].The following theorem provides a solution to Problem 2.4,in the homogeneous case: Theorem 4.4:
Assume that the adaptive SIS model is ho-mogeneous. Let φ and ψ be the solutions of the optimizationproblem:minimize φ , ψ n f ( φ ) + mg ( ψ ) subject to φ ≥ ( β η − δ + ) ( ψ / ( δ − α ) + ) , (17)¯ φ ≤ φ ≤ ¯ φ , ¯ ψ ≤ ψ ≤ ¯ ψ . Then, the pair ( φ , ψ ) gives the solution of Problem 2.4. Proof:
It is sufficient to show that η ( M ) ≤ − α if andonly if the (17) holds, but this easily follows from (16).V. H ETEROGENEOUS C ASE
In this section, we extend our analysis to non-homogeneous adaptive SIS models. We will show thatProblem 2.4 can be effectively solved under the followingassumption.
Assumption 5.1:
1) The values of ψ i j are given for every { i , j } ∈ E ;2) There exist constants r > ¯ φ and s such that the func-tion F : [ r − ¯ φ , r − ¯ φ ] → R : x (cid:55)→ s + f ( r − x ) is a posyn-omial function.In order to state the main result of this section, we willneed the next proposition. Proposition 5.2:
Let ¯ δ = max i δ i and define ˜ δ i = ¯ δ − δ i .Similarly, let ¯ ψ = max i , j ψ i j and define ˜ ψ i j = ¯ ψ − ψ i j .Let ˜ φ , . . . , ˜ φ n be real numbers. Define the matrices˜ D = (cid:76) ni = ˜ δ i , ˜ D = (cid:76) ni = ( ˜ δ i I d i ) , ˜ Φ = (cid:76) ni = ( ˜ φ i I d i ) , and˜ Ψ = (cid:76) ni = (cid:76) j ∈ N i ( ) ˜ ψ i j . Define the nonnegative matrix˜ M = (cid:20) ˜ D + ¯ ψ I + rI B Ψ B + ˜ D + ˜ Φ + ˜ Ψ (cid:21) . Then, for a given α >
0, the following statements areequivalent: • There exist φ , . . . , φ n ∈ [ ¯ φ , ¯ φ ] such that η ( M ) ≤ − α . • There exist ˜ φ , . . . , ˜ φ n ∈ [ r − ¯ φ , r − ¯ φ ] such that η ( ˜ M ) + α ≤ ψ + ¯ δ + r . Moreover, between { φ i } ni = and { ˜ φ i } ni = , there is a one-to-onecorrespondence given by the equation φ i = r − ˜ φ i . (18) Proof:
Assume that there exist φ , . . . , φ n ∈ [ ¯ φ , ¯ φ ] sat-isfying η ( M ) ≤ − α . Define ˜ φ i by (18). Then we see that˜ M = M + ( ¯ δ + r + ¯ ψ ) I . This implies η ( ˜ M ) + α ≤ ¯ δ + r + ¯ ψ .We also have ˜ φ i ∈ [ r − ¯ φ , r − ¯ φ ] . The other direction can beshown in the same way and, hence, its proof is omitted.Using Proposition 5.2, we can reduce Problem 2.4 to ageometric program under Assumption 5.1, as stated in thefollowing theorem: Theorem 5.3:
Let ˜ φ , . . . , ˜ φ n be the solutions to thefollowing geometric program:minimize ˜ φ i , v n ∑ i = F ( ˜ φ i ) (19a)subject to (cid:0) ˜ M + α I (cid:1) v ≤ (cid:0) ¯ δ + r + ¯ ψ (cid:1) v , (19b) v > , (19c) r − ¯ φ ≤ ˜ φ i ≤ r − ¯ φ . (19d)Then, { φ i } ni = , defined in (18) solve Problem 2.4. Proof:
By Proposition 5.2, Problem 2.4 is equivalentto the optimization problemminimize ˜ φ i n ∑ i = f ( r − ˜ φ i ) subject to η ( ˜ M + α I ) ≤ ¯ δ + r + ¯ ψ , (20) r − ¯ φ ≤ ˜ φ i ≤ r − ¯ φ , after the change of variables (18). Minimizing the objectivefunction in this problem is equivalent to minimizing the onein (19) by the definition of F , whose constant term s can beignored in the optimization. Then, since ˜ M + α I is irreducibleby Proposition 3.3, we can replace the constraint (20) into(19b) and (19c) in the same way as in [8] using Perron-Frobenius lemma. Also, by a similar argument as in [8], wecan show that (19) is a geometric program. This is because F is a posynomial and each entry of the matrix ˜ M + α I isa posynomial in the variables ˜ φ , . . . , ˜ φ n . The details areomitted. Remark 5.4:
When ψ i j are also design variables, theabove argument reduces Problem 2.4 to a signomial program,which are (in general) hard to solve [20].VI. N UMERICAL R ESULTS
We illustrate our results with a numerical example. Let G be the graph of a social network of n =
247 nodes and m =
940 edges. The adjacency matrix of the graph has spectralradius ρ = .
53. We assume that all nodes have identicalrecovery rate δ = . β = δ / ( . ρ ) = . × − . Since δ / β = ( . ) ρ > ρ , Theorem 4.2 doesnot guarantee the stability of the infection-free equilibriumwhen φ =
0, i.e., when the network does not adapt.Let us design the cost-optimal cutting rates so that thespread stabilizes towards the disease-free equilibrium in the egrees C u tt i n g r a t e s ¯ φ Fig. 2: Cost-optimal cutting rates for stabilizationadaptive network. We assume ¯ φ =
0, ¯ φ = β , and ψ i j = β anduse the following cost function in our numerical simulations: f ( x ) = ( r − x ) − − ( r − ¯ φ ) − ( r − ¯ φ ) − − ( r − ¯ φ ) − . We have chosen this function since it is increasing andpresents diminishing returns. Also, we have normalized it,so that f ( ¯ φ ) = f ( ¯ φ ) =
1, and fixed r = φ . Let thedesired exponential decay rate be α = .
005 and solve thegeometric program in Theorem 4.4 to obtain the optimalcutting rates φ i . Fig. 2 shows a scatter plot for the optimalrates, φ i , versus the degrees of the nodes for all i ∈ V . Theresulting switching policy suggests that, in general, nodeswith a larger degree should have higher cutting rates (ascould be naturally expected). However, the relationship be-tween the optimal cutting rates and the degrees is not trivial.Alternatively, we have also studied the relationship betweencutting rates and other network centrality measures and K -scores (though we omit these figures for space limitations).Our simulations do not show any trivial dependency betweencutting rates and any of the measures considered.VII. C ONCLUSION
In this paper, we have studied the dynamics of spreadingprocesses taking place in networks that adapt their structuredepending on the state of the dynamics. Our model isbased on a collection of stochastic differential equationswith Poisson jumps that model the joint evolution of thestates of the process taking place in the network, as wellas the evolution of the network structure. To illustrate ourframework, we have focused our attention in a popular modelof spreading dynamics, the SIS model, and study it dynamicsover adaptive, switched networks. For this particular model,we have derived conditions for the dynamics of the spreadto converge towards the disease-free equilibrium. Using thisstability result, we have then formulated an optimizationprogram to find the cost-optimal adaptive strategy to achievestability. We have also showed that this optimization programcan be efficiently solved using geometric programming. Anumerical example was included to illustrate our results. Aninteresting future work is to fully investigate the difference of information structures in our model and the one in [16]addressed in Remark 2.2.R
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