document: the analysis of dynamic processes taking place in complex networks is a major research area with a wide range of applications in social , biological , and technological systems @xcite . 
   the spread of information in online social networks , the evolution of an epidemic outbreak in human contact networks , and the dynamics of cascading failures in the electrical grid are relevant examples of these processes . 
   while major advances have been made in this field , most modeling and analysis techniques are specifically tailored to study dynamic processes taking place in static networks . 
   however , empirical observations in social @xcite , biological @xcite , and financial networks @xcite illustrate how real - world networks are constantly evolving over time @xcite . 
   unfortunately , the effects of temporal structural variations in the dynamics of networked systems remain poorly understood .    in the context of temporal networks , we are specially interested in the interplay between the dynamics on networks ( i.e. , the dynamics of processes taking place in the network ) and the dynamics of networks ( i.e. , the temporal evolution of the network structure ) . although the dynamics on and of networks are usually studied separately , there are many cases in which the evolution of the network structure is heavily influenced by the dynamics of processes taking place in the network . 
   one of such cases is found in the context of epidemiology , since healthy individuals tend to avoid contact with infected individuals in order to protect themselves against the disease  a phenomenon called _ social distancing 
   _  @xcite . as a consequence of social distancing 
   , the structure of the network adapts to the dynamics of the epidemics taking place in the network . 
   similar adaptation mechanisms have been studied in the context of power networks  @xcite , biological and neural networks  @xcite and on - line social networks  @xcite .    despite the relevance of network adaptation mechanisms , their effects on the network dynamics are not well understood . in this research direction , we find the seminal work by gross et al .  in @xcite , where a simple adaptive rewiring mechanism was proposed in the context of epidemic models . in this model , a susceptible node can cut edges connecting him to infected neighbors and form new links to _ any _ randomly selected susceptible nodes  without structural constraint in the formation of new links . despite its simplicity 
   , this adaptation mechanism induces a complex bifurcation diagram including healthy , oscillatory , bistable , and endemic epidemic states @xcite . 
   several extensions of this work can be found in the literature  @xcite , where the authors assume homogeneous infection and recovery rates in the network . 
   another model that is specially relevant to our work is the adaptive susceptible - infected - susceptible ( asis ) model proposed in  @xcite . in this model , edges in a given contact network can be temporarily removed in order to prevent the spread of the epidemic . 
   an interesting feature of the asis model is that , in contrast with gross model , it is able to account for arbitrary contact patterns , since links are constrained to be part of a given contact graph . despite its modeling flexibility , 
   analytical results for the asis model  @xcite are based on the assumption of homogeneous contact patterns ( i.e. , the contact graph is complete ) , as well as homogeneous node and edge dynamics ( i.e. , nodes present the same infection and recovery rates , and edges share the same adaptation rates ) .    as a consequence of the lack of tools to analyze network adaptation mechanisms , 
   there is also an absence of effective methodologies for actively utilizing adaptation mechanisms for containing spreading processes . 
   although we find in the literature a few attempts in this direction , most of them rely on extensive numerical simulations  @xcite , on assuming a homogeneous contact patterns @xcite , or a homogeneous node and edge dynamics  @xcite . 
   in contrast , while controlling epidemic processes over static networks , we find a plethora of tools based on game theory  @xcite or convex optimization  @xcite .    in this paper , we study adaptation mechanisms over arbitrary contact networks . 
   in particular , we derive an explicit expression for a lower bound on the epidemic threshold of the asis model for arbitrary networks , as well as heterogeneous node and edge dynamics . in the case of homogeneous node and edge dynamics 
   , we show that the lower bound is proportional to the epidemic threshold of the standard sis model over a static network @xcite . furthermore , based on our results , 
   we propose an efficient algorithm for optimally tuning the adaptation rates of an arbitrary network in order to eradicate an epidemic outbreak in the asis model . 
   we confirm the tightness of the proposed lower bonds with several numerical simulations and compare our optimal adaptation rates with popular centrality measures . 
   in this section , we describe the adaptive susceptible - infected - susceptible ( asis ) model over _ arbitrary _ networks with _ heterogeneous _ node and edge dynamics ( heterogeneous asis model for short ) . 
   we start our exposition by considering a spreading process over a time - varying contact graph  @xmath0 , where @xmath1 is the set of nodes and @xmath2 is the time - varying set of edges . for any @xmath3 
   , @xmath4_{i , j}$ ] corresponds to the adjacency matrix of @xmath5 , and the neighborhood of node  @xmath6 at time @xmath7 is defined as @xmath8 . in the standard sis epidemic model , 
   the state of node @xmath6 at time @xmath7 is described by a bernoulli random variable @xmath9 , where node  @xmath6 is said to be _ susceptible _ if @xmath10 , and _ infected _ if @xmath11 . 
   when the contact graph evolves over time , the evolution of @xmath12 is described by a markov process with the following transition probabilities : @xmath13[.8\linewidth ] { p}(x_i(t+h ) = 1 \mid x_i(t ) = 0 ) = \beta_i\ \sum_{\mathclap{k \in { \pazocal{n}}_i(t)}}\ x_k(t)\,h + o(h ) , \end{multlined } \nonumber   \\ & { p}(x_i(t+h ) = 0 \mid x_i(t ) = 1 ) = \delta_i h + o(h ) ,   \label{eq : recovery}\end{aligned}\ ] ] where the parameters @xmath14 and @xmath15 are called the _ infection _ and _ recovery _ rates of node  @xmath6 .    in the heterogeneous asis model , 
   the epidemics takes place over a time - varying network that we model as a continuous - time stochastic graph process  @xmath16 , described below . 
   let @xmath17 be an initial connected contact graph with adjacency matrix  @xmath18_{i , j}$ ] . 
   we assume that @xmath19 is strongly connected . 
   edges in the initial graph @xmath19 appear and disappear over time according to the following markov processes : @xmath13[.9\linewidth ] { p}(a_{ij}(t+h ) = 0 \mid a_{ij}(t ) = 1)=\vspace{.1 cm } \\ 
   \phi_{ij}x_i(t ) h+ \phi_{ji}x_j(t ) h +   o(h ) , \end{multlined}\label{eq : cut } \\ & { p}(a_{ij}(t+h ) = 1 \mid a_{ij}(t ) = 0 ) = a_{ij}(0)\psi_{ij } h +   o(h ) , \label{eq : rewire}\end{aligned}\ ] ] where the parameters @xmath20 and @xmath21 are called the _ cutting _ and _ reconnecting _ rates . 
   notice that the transition rate in depends on @xmath12 and @xmath22 , inducing an adaptation mechanism of the network structure to the state of the epidemics . 
   the transition probability in   can be interpreted as a protection mechanism in which edge @xmath23 is stochastically removed from the network if either node @xmath6 or @xmath24 is infected . more specifically , because of the first summand ( respectively , the second summand ) in , whenever node @xmath6 ( respectively , node @xmath24 ) is infected , edge  @xmath25 is removed from the network according to a poisson process with rate  @xmath26 ( respectively , rate  @xmath27 ) . on the other hand , 
   the transition probability in describes a mechanism for which a ` cut ' edge @xmath25 is ` reconnected ' into the network according to a poisson process with rate @xmath28 ( see figure  [ fig : adaptive ] ) . 
   notice that we include the term @xmath29 in to guarantee that only edges present in the initial contact graph @xmath19 can be added later on by the reconnecting process . 
   in other words , we constrain the set of edges in the adaptive network to be a part of the arbitrary contact graph @xmath19 . 
   in this section , we derive a lower bound on the epidemic threshold for the heterogeneous asis model . for @xmath30 , let @xmath31 denote a poisson counter with rate  @xmath32 @xcite . in what follows , we assume all poisson counters to be stochastically independent . 
   then , from the two equations in , the evolution of the nodal states can be described by the following set of stochastic differential equations @xmath33 for all  @xmath34 . 
   similarly , from and , the evolution of the edges can be described by the following set of stochastic differential equations : @xmath35 for all @xmath36 .    by 
   , the expectation @xmath37 $ ] obeys the differential equation @xmath38   =   -\delta_i e[x_i ] + \beta_i \ 
   \sum_{\mathclap{k\in{\pazocal{n}}_i(0)}}\   e[(1-x_i)a_{ik}x_k].\ ] ] let @xmath39 $ ] and @xmath40 $ ] . 
   then , it follows that @xmath41 where @xmath42\ ] ] contains positive higher - order terms . in what follows , 
   we derive a set of differential equations to describe the evolution of @xmath43 . from and 
   , we obtain the following equation using it rule for jump processes ( see , e.g. , @xcite ) @xmath44 therefore , @xmath45 for all @xmath46 , where @xmath47\\ & + \beta_i\ \sum_{\mathclap{k\in{\pazocal{n}}_i(0)}}\ e\bigl [ x_i(t)x_k(t)a_{ik}(t ) \\ & + ( 1-a_{ij}(t))a_{ik}(t)x_k(t)\bigr],\end{aligned}\ ] ] which contains positive higher - order terms . 
   the differential equations   and describe the joint evolution of the spreading process and the network structure .    for further analysis , it is convenient to express the differential equations   and using vectors and matrices . for this purpose , 
   let us introduce the following notation . 
   let @xmath48 and @xmath49 be , respectively , the @xmath50 identity matrix and the @xmath51-dimensional column vector of all ones . given two matrices @xmath52 and @xmath53 , their kronecker product  @xcite 
   is denoted by @xmath54 . given a sequence of matrices  @xmath55 , their direct sum , denoted by @xmath56 , is defined as the block diagonal matrix having @xmath55 as its block diagonals @xcite . if @xmath55 have the same number of columns , then the matrix obtained by stacking @xmath55 in vertical ( @xmath57 on top ) is denoted by @xmath58 . 
   based on this notation , we define the vector - variable @xmath59 , which contains the infection probabilities of all the nodes in the graph . 
   similarly , let @xmath60 and define the column vector @xmath61 . 
   define @xmath62 as the unique row - vector satisfying @xmath63 note that the length of the row vector @xmath62 and the column vector @xmath64 equals @xmath65 , where @xmath66 is the number of the edges in the initial graph @xmath19 .    using this notation 
   , we define the following matrices : @xmath67 where @xmath68 denotes the degree of node  @xmath6 in the initial graph  @xmath19 . 
   furthermore , we also define the following matrices @xmath69    stacking the set of @xmath70 differential equations in into a single vector equation , and ignoring the negative higher - order term @xmath71 , we obtain the following entry - wise vector inequality for the probabilities of infection : @xmath72 also , stacking the set of differential equations in with respect to @xmath73 , and ignoring the negative term  @xmath74 , we obtain the following entry - wise vector inequality : @xmath75 where @xmath76 . we can further stack the above inequalities with respect to the index @xmath6 to obtain the following entry - wise vector inequality : @xmath77 combining this inequality and , we obtain @xmath78/dt } \frac{d}{dt}\begin{bmatrix } p 
   q \end{bmatrix }   \leq m   \begin{bmatrix } p 
   \\ q \end{bmatrix},\ ] ] where @xmath79 is an irreducible matrix ( see appendix  [ appx : pf : irreducibility ] for the proof of irreducibility ) defined as @xmath80 therefore , the evolution of the joint vector variable  @xmath81 is upper - bounded by the linear dynamics given by the matrix @xmath79 . moreover , 
   the upper bound is tight around the origin , since both @xmath82 and @xmath83 consist of higher - order terms . from 
   , we conclude that the epidemics dies out exponentially fast in the heterogeneous asis model if @xmath84 where @xmath85 is defined as the maximum among the real parts of the eigenvalues of @xmath79 . furthermore 
   , since @xmath79 is a metzler matrix ( i.e. , has nonnegative off - diagonals ) and irreducible , there is a real eigenvalue of @xmath79 equal to  @xmath85 @xcite . 
   in the homogeneous case , where all the nodes share the same infection rate @xmath86 and recovery rate  @xmath87 , and all the edges share the same cutting rate  @xmath88 and reconnecting rate  @xmath89 , the condition   reduces to the following inequality : @xmath90 where @xmath91 is the spectral radius of the initial graph @xmath19 and @xmath92 which we call the _ effective cutting rate_. the proof of the extinction condition is given in appendix  [ appx : pf : thm : stbl : homo ] . 
   we remark that , in the special case when the network does not adapt to the prevalence of infection , i.e. , when @xmath93 , we have that @xmath94 and , therefore , the condition in   is identical to the extinction condition  @xmath95 corresponding to the homogeneous networked sis model over a static network  @xcite . 
   it is worth comparing the condition in with the epidemic threshold @xmath96 given in @xcite for the case in which @xmath19 is the complete graph : @xmath97 where @xmath98 is the link - creating rate , @xmath99 is the link - breaking rate , and @xmath100 is a positive and `` slowly varying '' function depending on the metastable long - time average of the number of infected nodes ( for details , see @xcite ) . 
   we first notice that our lower bound in can be checked directly from the parameters of the model , namely , the adjacency matrix of the initial graph @xmath19 and the relevant rates of the model . 
   this is in contrast with the threshold in  , since it depends on the metastable average of the number of infected nodes and , thus , can only be computed via numerical simulations . 
   we also remark that the lower bound on the epidemic threshold in and the epidemic threshold in both exhibit affine dependence on the effective link - breaking rates  @xmath101 and  @xmath102 , respectively . 
   finally , we see that the recovery rate @xmath103 appears in different places in the two conditions , namely , inside the expression of @xmath101 in and inside the function  @xmath100 in . 
   however , the consequences of this difference are not obvious , since @xmath100 is defined via the metastable state and , therefore , does not allow an analytical investigation .    [ cols="^,^,^,^ " , ]     once the cost functions are selected , we must solve the problem of finding the optimal tuning investment to achieve a desired exponential decay rate in the probabilities of infection . from the inequality in , 
   the infection probabilities @xmath104 , @xmath105 , @xmath106 decay exponentially at a rate ( at least ) @xmath107 if @xmath108 since @xmath79 is an irreducible and metzler matrix , we can use perron - frobenius theory  @xcite to prove that is satisfied if there exists an entry - wise positive vector  @xmath109 satisfying the following entry - wise vector inequality ( see @xcite for more details ) : @xmath110 therefore , problem  [ prb : ] can be reduced to the following equivalent optimization problem : @xmath111 as we show in appendix  [ appx : gp ] , we can equivalently transform this optimization problem into a geometric program  @xcite , which can be efficiently solved using standard optimization software . 
   the computational complexity of solving the resulting geometric program is @xmath112 , where @xmath70 is the number of nodes and @xmath66 is the number of edges in the initial network @xmath19 .    in the rest of this section , 
   we compute the optimal tuning profiles for three different graphs and compare our results with several network centralities . in our simulations , we consider the following three graphs with @xmath113 nodes : _ 1 _ ) an erds - rnyi graph with @xmath114 edges , _ 2 _ ) a barabsi - albert random graph with @xmath115 edges , and _ 3 _ ) a social subgraph ( obtained from facebook ) with @xmath116 edges . for simplicity in our simulations , we assume that all nodes share the same recovery rate  @xmath117 and infection rate @xmath118 , where @xmath91 denotes the spectral radius of each initial graph . since @xmath119 , the extinction condition indicates that the infection process does not necessarily die out without adaptation , i.e. , when @xmath120 . 
   the rest of parameters in our simulations are set as follows : we let @xmath121 , @xmath122 , and @xmath123 . also , the parameters in the cost functions are @xmath124 and @xmath125 . 
   the desired exponential decay rate is chosen to be @xmath126 .    using this set of parameters , we solve the optimization problem   following the procedure described in appendix  [ appx : gp ] . 
   our numerical results are illustrated in various figures included in table  [ table : ] . 
   each figure is a scatter plot where each point corresponds to a particular edge @xmath46 ; the ordinate of each point corresponding to its optimal cutting rate @xmath26 , and the abscissa of each point corresponds to a particular edge - centrality measure . in these figures , we use three different edge - centrality measures : a ) the product of the degrees of nodes @xmath6 and @xmath24 ( left column ) , b ) the product of the eigenvector - centralities of nodes @xmath6 and @xmath24 ( center column ) , and c ) the betweenness centralities of edge @xmath25 ( right column ) .    in our numerical results , 
   we observe how both degree - based and eigenvector - based edge - centralities are good measures for determining the amount of investment in tuning cutting rates . 
   in contrast , betweenness centrality does not show a significant dependency on the optimal cutting rates . 
   in particular , for the synthetic networks in rows 1 ) and 2 ) in table [ table : ] , we observe an almost piecewise affine relationship with the centrality measures in columns a ) and b ) . in particular , in these subplots 
   we observe how edges of low centrality require no investment , while for higher - centrality edges , the tuning investment tends to increase affinely as the centrality of the edge increases  as expected . 
   for the real social network in row c ) , the relationship between investment and centralities is still strong  although not as clear as in synthetic networks . in the scatter plot corresponding to the relationship between the optimal investment and the eigenvector - based centrality in the real social network ( lower center figure in table [ table : ] ) , we observe a collection of several stratified parallel lines . 
   we conjecture that each line corresponds to a different community inside the social network ; in other words , the relationship between centrality and optimal investment is almost affine inside each community . 
   we have studied an _ adaptive _ susceptible - infected - susceptible ( asis ) model with heterogeneous node and edge dynamics and arbitrary network topologies . 
   we have derived an explicit expression for a lower bound on the epidemic threshold of this model in terms of the maximum real eigenvalue of a matrix that depends explicitly on the network topology and the parameters of the model . for networks with homogeneous node and edge dynamics , the lower 
   bound turns out to be a constant multiple of the epidemic threshold in the standard sis model over static networks ( in particular , the inverse of the spectral radius ) . 
   furthermore , based on our results , we have proposed an optimization framework to find the cost - optimal adaptation rates in order to eradicate the epidemics . 
   we have confirmed the accuracy of our theoretical results with several numerical simulations and compare cost - optimal adaptation rates with popular centrality measures in various networks . 
   43ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] 
   + 12$12  & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1038/30918 [ * * ,   ( ) ] @noop _ _ 
   ( ,  ) @noop _ _  ( ,  ) link:\doibase 10.1007/s00779 - 005 - 0046 - 3 [ * * ,   ( ) ] @noop * * ( ) in  link:\doibase 1102.0629v1 [ _ _ ]  ( )  pp . 
   link:\doibase 10.1093/bib / bbp057 [ * * , ( ) ] link:\doibase 10.1038/nature02555 [ * * ,   ( ) ] link:\doibase 10.1038/nbt.1522 [ * * ,   ( ) ] link:\doibase    10.1140/epjb / e20020151 [ * * ,   ( ) ] link:\doibase 10.1016/j.physrep.2012.03.001 [ * * ,   ( ) ] link:\doibase 10.3201/eid1201.051371 [ * * ,   ( ) ] link:\doibase 10.1098/rsif.2010.0142 [ * * ,   ( ) ] link:\doibase 10.1209/epl / i2004 - 10533 - 6 [ * * ,   ( ) ] link:\doibase 10.1038/304158a0 [ * * ,   ( ) ] link:\doibase 10.1161/01.atv.0000069625.11230.96 [ * * ,   ( ) ] link:\doibase 10.1186/s40649 - 015 - 0023 - 6 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.96.208701 [ * * ,   ( ) ] link:\doibase 10.1098/rsif.2007.1229 [ * * ,   ( ) ] link:\doibase 10.1007/s10867 - 008 - 9060 - 9 [ * * ,   ( ) ] link:\doibase    10.1103/physreve.82.036116 [ * * ,   ( ) ] link:\doibase    10.1103/physreve.83.026102 [ * * ,   ( ) ] link:\doibase 10.1088/1742 - 5468/2012/08/p08018 [ * * ,   ( ) ] link:\doibase 10.1007/s00285 - 012 - 0555 - 4 [ * * ,   ( ) ] link:\doibase 10.1103/physreve.90.022801 [ * * , ( ) ] link:\doibase    10.1103/physreve.88.042802 [ * * ,   ( ) ] link:\doibase 10.1103/physreve.92.030801 [ * * ,   ( ) ] link:\doibase 10.1103/physreve.88.042801 [ * * ,   ( ) ] link:\doibase 10.1186/1471 - 2458 - 12 - 679 [ * * ,   ( ) ] link:\doibase 10.1103/physreve.85.036108 [ * * ,   ( ) ] link:\doibase    10.1109/tcns.2015.2426755 [ * * ,   ( ) ] link:\doibase 10.1016/j.jcss.2006.02.003 [ * * ,   ( ) ] link:\doibase 10.1109/mcs.2015.2495000 [ * * ,   ( ) ] link:\doibase    10.1109/tcns.2014.2310911 [ * * ,   ( ) ] link:\doibase 10.1109/tnet.2008.925623 [ * * ,   ( ) ] @noop _ _ 
   ( ,  ) @noop _ _  ( ,  ) @noop _ _  ( ,  ) link:\doibase 10.1103/physreve.87.062816 [ * * ,   ( ) ] link:\doibase 10.1007/s11081 - 007 - 9001 - 7 [ * * , ( ) ] in  link:\doibase 10.1109/cdc.2013.6761078 [ _ _ ]  ( )  pp . @noop _ _  ( ,  ) ( ) 
   we show that the matrix @xmath79 defined in is irreducible , that is , there is no similarity transformation that transforms @xmath79 into a block upper - triangular matrix . for this purpose , 
   define @xmath127 where @xmath128 , @xmath129 , and @xmath130 . since the rates @xmath131 and @xmath28 are positive , if @xmath132 , then @xmath133 for all distinct @xmath6 and @xmath24 . 
   therefore , to prove the irreducibility of @xmath79 , it is sufficient to show that @xmath134 is irreducible .    in order to show that @xmath134 is irreducible , we shall show that the directed graph @xmath135 on the nodes @xmath136 , defined as the graph having adjacency matrix  @xmath137 , is strongly connected . 
   we identify the nodes @xmath138 , @xmath105 , @xmath139 and variables @xmath104 , @xmath105 , @xmath106 , @xmath140 ( @xmath141 ) , @xmath105 , @xmath142 ( @xmath143 ) . 
   then , the upper - right block  @xmath144 of the matrix @xmath134 indicates that the graph @xmath135 contains the directed edge  @xmath145 for all @xmath146 and @xmath147 . 
   similarly , from the matrices @xmath148 and  @xmath149 in @xmath79 , we see that @xmath135 contains the directed edges  @xmath150 and @xmath151 for all @xmath146 and @xmath152 .    using these observations , let us first show that @xmath135 has a directed path from @xmath153 to @xmath154 for all @xmath155 . since @xmath19 is strongly connected , it has a path @xmath156 such that @xmath157 and @xmath158 . 
   therefore , from the above observations , we see that @xmath135 contains the directed path @xmath159 . in the same way 
   , we can show that @xmath135 also contains the directed path @xmath160 for every @xmath161 . 
   these two types of directed paths in @xmath135 guarantees that @xmath135 is strongly connected . hence the matrix  @xmath134 is irreducible , as desired . 
   in the homogeneous case , the matrix @xmath79 takes the form @xmath162 in what follows , we show that @xmath163 if and only if holds true . 
   since @xmath19 is strongly connected by assumption , its adjacency matrix  @xmath164 is irreducible and therefore has an entry - wise positive eigenvector  @xmath109 corresponding to the eigenvalue  @xmath91 ( see  @xcite ) . define the positive vector @xmath165 . 
   then , the definition of  @xmath62 in shows @xmath166 and therefore @xmath167 . in the same manner , we can show that @xmath168 . 
   moreover , it is straightforward to check that @xmath169 
   . therefore , for a real number @xmath170 , it follows that @xmath171 hence , if a real number @xmath172 satisfies the following equations : @xmath173 then @xmath174 is an eigenvector of @xmath79 . since @xmath79 is irreducible ( shown in appendix  [ appx : pf : irreducibility ] ) , by perron - frobenius theory  @xcite , if @xmath175 then @xmath176 ( see ( * ? ? ? * theorem  17 ) ) . 
   this , in particular , shows that @xmath163 if and only if there exist @xmath175 and @xmath177 such that holds .    the two equations in have two pairs of solutions @xmath178 and @xmath179 such that @xmath180 , and @xmath181 . 
   therefore , we need to show @xmath182 if and only if holds true . to see this 
   , we notice that @xmath183 and @xmath184 are the solutions of the quadratic equation @xmath185 following from . since @xmath186 
   , we have @xmath187 if and only if the constant term @xmath188 of the quadratic equation is positive , which is indeed equivalent to . 
   this completes the proof of the extinction condition stated in  . 
   we first give a brief review of geometric programs  @xcite . 
   let @xmath189 , @xmath105 , @xmath190 denote positive variables and define @xmath191 . 
   we say that a real function  @xmath192 is a _ monomial _ if there exist @xmath193 and @xmath194 such that @xmath195 . 
   also , we say that a function  @xmath196 is a _ posynomial _ if it is a sum of monomials of  @xmath197 ( we point the readers to  @xcite for more details ) . given a collection of posynomials @xmath198 , @xmath105 , @xmath199 and monomials @xmath200 
   , @xmath105 , @xmath201 , the optimization problem @xmath202 is called a _ geometric program_. a constraint of the form @xmath203 with @xmath196 being a posynomial is called a posynomial constraint . 
   although geometric programs are not convex , they can be efficiently converted into equivalent convex optimization problems  @xcite .    in the following , we rewrite the optimization problem   into a geometric program using the new variables @xmath204 and @xmath205 . 
   by , these variables should satisfy the constraints @xmath206 also , using these variables , the cost function @xmath207 can be written as @xmath208 , where @xmath209 are posynomials . in order to rewrite the constraint  , 
   we first define the matrices @xmath210 we also introduce the positive constants @xmath211 , @xmath212 , @xmath213 . now , 
   adding @xmath214 to both sides of , we equivalently obtain @xmath215 where @xmath216 is given by      summarizing , we have shown that the optimization problem   is equivalent to the following optimization problem with ( entry - wise ) positive variables : @xmath218 in this optimization problem , the objective function is a posynomial in the variables @xmath219 , @xmath220 , and @xmath221 . 
   also , the box constraints , , and can be written as posynomial constraints  @xcite . 
   finally , since each entry of the matrix @xmath222 is a posynomial in the variables @xmath219 , @xmath220 , and  @xmath221 , the vector - constraint yields @xmath139 posynomial constraints . 
   therefore , the optimization problem is a geometric program , as desired . 
   furthremore , a standard estimate on the computational complexity of solving geometric program ( see , e.g. , ( * ? ? ? 
   * proposition  3 ) ) shows that the computational complexity of solving the optimization problem in is given by @xmath112 . 
   finally we remark that @xmath222 contains both the terms  @xmath28 and @xmath223 so that we can not use @xmath28 as the decision variable in the geometric program   due to the positivity constraint on decision variables . by this reason , we can not design the reconnecting rates @xmath28 under the framework presented in this paper . 
   this issue is left as an open problem .
summary: in this paper , we study the dynamics of epidemic processes taking place in adaptive networks of arbitrary topology . 
   we focus our study on the adaptive susceptible - infected - susceptible ( asis ) model , where healthy individuals are allowed to temporarily cut edges connecting them to infected nodes in order to prevent the spread of the infection . in this paper 
   , we derive a closed - form expression for a lower bound on the epidemic threshold of the asis model in arbitrary networks with heterogeneous node and edge dynamics . for networks with homogeneous node and edge dynamics , we show that the resulting lower bound is proportional to the epidemic threshold of the standard sis model over static networks , with a proportionality constant that depends on the adaptation rates . 
   furthermore , based on our results , we propose an efficient algorithm to optimally tune the adaptation rates in order to eradicate epidemic outbreaks in arbitrary networks . 
   we confirm the tightness of the proposed lower bounds with several numerical simulations and compare our optimal adaptation rates with popular centrality measures .