THe notion of the disease [1] has been extended from

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1 IEEE/ACM TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING, VOL., NO., 6 Effective Network Quarantine with Minimal Restrictions on Communication Activities uanyang Zheng an Jie Wu, Fellow, IEEE Abstract This paper stuies a network structure esign problem constraine by the epiemic outbreaks. In our moel, geographic locations (noes) an their connections (eges) are moele as a ring graph. The movement of a person is represente as a flow from one location to another. A person can be infecte at a location (noe), epening on the number of infecte flows (persons) going through that location. In our paper, iseases are not limite to real human iseases; they can also refer to the general epiemic information propagations. Given esire interaction traffic from a noe to other noes in terms of flows, an a greey shortest path routing scheme that is analogous to the greey coin change, we focus on the structure esign (representing quarantine rules) that etermines the number an istribution of chors on the virtual ring network for remote connections. Our objective is to minimize the average number of routing hops, while the epiemic outbreaks are controlle uner given infection an recovery rates. We provie a systematic isomorphic structure esign on nine ifferent cases, base on three traffic istribution an three infection rate moels. Two hypercube-base structures are propose. We also provie a greey solution for constructing polymorphic structures. Our stuy reveals some intriguing theoretical results, valiate through experiments, on traeoffs between local an global infections. Our work casts new light on the effective network quarantine that places minimal restrictions on connections, i.e., maximal preservation of normal communication activities, while controlling epiemic outbreaks. Inex Terms Epiemic outbreaks, interaction, quarantine, network structure esign. INTRODUCTION Te notion of the isease [ has been extene from human iseases, such as Ebola, to general epiemic information propagations [, such as rumors in istribute networks. Controlling the sprea of a communicable isease in a population is usually one through the quarantine, where persons that have, or are suspecte to have, a communicable isease are restricte from having interactions with others. owever, a quarantine is inconvenient for communication activities, ue to its limitations on interactions. ence, there is a traeoff between the control of epiemic outbreaks an the convenience of communication activities. To unerstan this traeoff, we conuct stuies base on a virtual ring moel, which is an abstraction of the real worl through meriian an parallel. As shown in Fig. (a), meriian an parallel iscretize the Earth into geographic locations (represente by noes) on multiple rings. For each ring, noes are connecte via links (roas between nearby towns). Geographically-remote noes are connecte via chors (flights between remote cities). The geographical istance between noes can be measure by their hop istances along the ring. This paper stuies a network structure esign problem constraine by the epiemic outbreaks. In our moel, flows represent the movements of people. A person can be infecte at a location (noe), epening on the number of infecte flows (persons) going through that location. Personto-person interactions are controlle by flow routes among. Zheng an J. Wu are with the Department of Computer an Information Sciences, Temple University, Philaelphia, PA, 9. Manuscript receive May 5, 5; revise September 7, 5. (a) Meriian an parallel. Chors Links (b) Virtual ring moel. Fig.. Abstraction of the Earth with meriian an parallel. ifferent locations. The quarantine is represente as interactions with neighbors only (through a link in a -D ring, say, walking between two ajacent towns), an non-quarantine is shown as interactions with remote noes (through a chor in a -D ring, say, flying between two cities). A real-worl example is that Australia clampe own on the entry of iniviuals traveling from West Africa [3, in orer to control Ebola. Such a quarantine rule is execute by the Australian Customs an Borer Protection Service. The SIS moel [4 is use to simulate the epiemic spreaing, where flows have states of being susceptible or infecte. Flows transfer their states through a cycle of being infecte (base on a given infection rate) from susceptible, an going back to susceptible by recovery (base on a given recovery rate). We consier three infection rate moels, where the infection probability is relate to the infecte flows at a noe. The recovery rate is constant, as use in existing moels [4. We assume that each noe has fixe outgoing traffic, in terms of flows, to other noes, following a particular istribution [5. Flow interactions occur in noes along a routing path from source to estination, which is a sequence of links an chors. The routing path is etermine by

2 IEEE/ACM TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING, VOL., NO., F F 3 (a) Local infection. 9 8 Fig.. An example of the traffic flow routing F F 3 (b) Global infection. a greey-coin-change-base routing scheme [6, [7. Each noe will select a connecte relay, which is the closest to the estination, to forwar its traffic. In Fig. (a), if N wants to interact with N 5 by generating a flow F, then its path is {N, N 3, N 4 }. In general, each noe has multiple flows, incluing forwaring flows an its own traffic flows. For example, N 3 has its own esire traffic to N 4 in the flow F, while forwaring the flow F from N to N 4. Classic epiemiology controls the epiemic outbreaks by stuying the space-time patterns of iseases [8. In contrast, we stuy a network structure esign problem constraine by the epiemic outbreaks. The esigne network structure shoul minimize the average number of routing hops for noe interactions without epiemic outbreaks. The application of our work implies a quarantine rule, which can maximally preserve normal communication activities without epiemic outbreaks. Note that the effect of a given flow is not restricte over its routing path. If we esign a ring structure that has few chors, then the traffic will aggregate along local noes. In this case, susceptible flows going through locations with infecte flows are very likely to be infecte. This phenomenon is calle the local infection. For example, Fig. (a) has less chors than Fig. (b). As a result, F an F aggregate along the link from N 3 to N 4. Aggregate flows lea to a larger infection probability, an thus, local infection happens. On the other han, if we esign a ring structure that has numerous chors, then the traffic is sprea wiely through chors. An infecte flow can reach many susceptible remote flows to sprea the isease, leaing to the global infection phenomenon. For example, infecte flows from N can irectly reach more noes in Fig. (b) than Fig. (a). Since flows interacts with epiemics, the traeoff between local infection an global infection makes the structure esign problem very challenging. In this paper, we mainly focus on isomorphic structures, where each noe on the ring has the same chor istribution in terms of egree an neighbor istribution. The sensitivity experiments emonstrate that the isomorphism assumption can be relaxe, an thus they are feasible for real-worl applications. Table shows our major finings, base on three traffic istribution moels an three infection rate moels. The traffic istribution moels (inicating ifferent noe communication patterns) are shown as follows: (I) The traffic from one noe to another noe is a constant; (II) The traffic from one noe to another noe ecays slowly with respect to their geographical istance; (III) The traffic from one noe to another noe ecays quickly with respect to their geographical istance. The infection rate moels (inicating ifferent epiemic infectivities) inclue: (A) The TABLE Major results for isomorphic structures Properties of (-6) are Traffic istribution moels itemize in the text. Moel I Moel II Moel III Infection Moel A, 5, 4 rate Moel B, 6, 4 moels Moel C probability of being infecte by infecte flows is a constant; (B) The probability of being infecte is sub-linearly proportional to the traffic amount of infecte flows; (C) The probability of being infecte is linearly proportional to the traffic amount of infecte flows. For a virtual ring with n noes, the properties in Table are liste as follows: ) The global infection is the major factor for the epiemic outbreak. Epiemic outbreaks coul be controlle, if the noe egree is cappe. ) Neither local nor global infection is the major factor for the epiemic outbreak. The optimal structure remains an open problem. 3) The local infection is the major factor for the epiemic outbreak. The fully-connecte network is the optimal structure. 4) Chors shoul only connect to the geographically nearest O(ln n) noes to facilitate the communications. This is because the interaction traffic ecays exponentially with the geographical istance. 5) The Ulysses butterfly structure [9 is an asymptotically optimal structure. 6) This correspons to the most complex case, an is the focus of this paper. Two hypercube-base structures are propose an analyze. The first structure uses binary jump sizes of chors [7. The secon structure improves the first one by consiering the esire traffic istribution. We also propose a greey solution for constructing polymorphic structures, where noes have heterogenous chor istributions. This greey solution is stuie uner the esire traffic moel I an the infection rate moel A. We show the relationship between the reuce average number of routing hops (i.e., the benefit) an the increase network vulnerability to epiemics (i.e., the cost), brought by aing a chor. The chor, which has the largest benefitto-cost ratio, is iteratively ae to construct a polymorphic structure. This greey solution gives out some insights on efficiently constructing general polymorphic structures that resist epiemic outbreaks. The remainer of this paper is organize as follows. In Section, the basic moel is escribe an the problem is formulate. In Section 3, two hypercube-base isomorphic structures are stuie. In Section 4, we consier other traffic istribution an infection rate moels. In Section 5, we focus on constructing polymorphic structures. In Section 6, our network structures are evaluate. In Section 7, we conclue the paper. Supplementary material are publishe in [. MOTIVATION, MODEL, AND FORMULATION We first escribe the motivation. Then, the ring network moel, the traffic istribution moels, an the infection rate moels are escribe. Finally, the problem is formulate.

3 IEEE/ACM TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING, VOL., NO., i - j i links... Directional chors i +... Biirectional n -..., i, j Fig. 3. The virtual ring moel with biirectional links. Relay N i N j Fig. 4. A globe representing Earth with meriian an parallel.. Motivation This paper is base on the real earth. Meriians an parallels iscretize the earth into ifferent locations (cities). People move aroun ifferent locations for their lives. Epiemics (e.g., Ebola) are foun an may sprea. To avoi epiemic outbreak, people s movements are restricte by quarantine. For example, Australia clampe own on the entry of iniviuals traveling from West Africa to against Ebola [3. owever, some movements are necessary an esire, even uner the threat of epiemics. For example, foo suppliers have to move aroun to fee people. Consequently, it is challenging to fin an answer to the question of how to control these necessary movements while avoiing epiemic outbreaks. Let us consier a motivational scenario on the transportation network for foo suppliers: To fee people, foo must be transporte from some cities to some other cities by trains. These movements are necessary, even uner the threat of epiemics. To avoi epiemic outbreaks, two cities may not be irectly connecte by trains. A transportation between them is forware via some thir-party cities. The structure of the transportation network (i.e., which two cities are irectly connecte by trains) for foo suppliers is critical. It is the variable in this paper (represente by the number an istribution of chors). We want to satisfy the foo supply requirement an control the epiemics, with minimal transports. If all cities are irectly connecte by trains, then an epiemic outbreak can sprea globally. owever, if only a few cities are irectly connecte by trains, then people nee to go through multiple cities to gain access to foo supplies. People who goes through more cities also have a higher possibility of being infecte. Cities on the earth are moele by the ring network in subsection.3. Necessary people s movements (e.g., foo supplies) are escribe by given traffic istribution moels, with a routing scheme, in subsection.4. The traffic can be route irectly an inirectly (e.g., nonstop an multi-stop trains). This interacts with epiemic spreaings, which are escribe by given infection rate moels in subsection.5. The problem is formulate in subsection.6. The objective is to minimize the number of routing hops for traffic flows, an the constraint is to control epiemic outbreaks.. Relate Works Our paper is closely relate to transportation network esign papers [5, [, which esign airline/train network structures to satisfy transportation emans an constraints [, [3. For example, Wieberneit [3 review the literature on esigning the freight transportation network, incluing the network structure, the freight frequency, an the freight routing path. The freight service emans are satisfie. Our research buils upon previous knowlege, but aitionally consier the constrains impose by the epiemic outbreak. This paper aims at a network structure, which brings a minimum routing for given traffics, an the constraint is that epiemics will not outbreak. This paper is also relate to peer-to-peer network structure esign problems. For example, Xu s work [5 stuies a egree-cappe network structure that achieves a minimum iameter. Our problem becomes equivalent to Xu s work, when traffics are uniformlyistribute an the epiemic is traffic-insensitive. This is because the minimum routing becomes equivalent to the minimum iameter, an the epiemic constraint becomes equivalent to the egree cap. The spreaing of epiemics has been well-stuie in the literature [4, [5, [6, [7, [8. Classic moels in epiemiology, such as SIS an SIR, are summarize in [4 with respect to ifferent network structures. These moels are known as compartmental moels, which use interactions between states (e.g., susceptible, infecte, recover) to escribe epiemics. owever, these moels o not consier the interactions between the epiemic spreaing an the traffic. Therefore, Preciao et al. [6, [7, [8, [9 introuce convex optimization techniques for the epiemic control uner ifferent traffic rates. In contrast, this paper uses the existing SIS moel to stuy the interaction between the traffic flow routing path an the epiemic spreaing through a network structure esign problem..3 The Ring Network Moel As shown in Fig. 3, our stuy is base on a virtual ring network of n noes. Since the virtual ring is an abstraction of a real-worl geographic map, each noe on the ring represents a geographic location. Noes have links with two geographical neighbors, an chors with geographicallyremote noes. The IDs of noes are from to n, where the i th noe is enote as N i. The geographical istance between N i an N j is enote as, which is measure by their hop istances along the ring (the minor part). In Fig. 3, we have,j = j an,i = n i. Both links an chors are calle eges in this paper, an are irectional. We first explore isomorphic structures, where each noe has a egree of D (both in-egree an out-egree). Polymorphic structures are consiere later in Section 5. The ring moel can represent the geographic relationships among ifferent locations. It is wiely acknowlege

4 IEEE/ACM TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING, VOL., NO., 6 4 in research on PP networks [7 an social networks (the small-worl phenomenon) [. As shown in Fig. 4, each meriian or parallel of the Earth can be abstracte as a ring. In such a case, noe communications can be ecompose on two rings. In Fig. 4, N i an N j can communicate with each other through a relay noe at the intersection of the meriian an the parallel rings (X-Y routing). More etails are presente in the supplemental material. Links in the ring represent the network backbone that maintains the network connectivity. Quarantine is represente as interactions with neighbors only (through a link in the ring, say, walking between two ajacent towns). People are always allowe to move near to their home locations. Chors stan for the normal connections. Non-quarantine is shown as the interactions with remote noes (through a chor in the ring, say, flying between two cities). A realworl example is that Australia clampe own on the entry of iniviuals traveling from West Africa [3. There is no chor from West Africa to Australia, i.e., people in West Africa are not allowe to travel to remote locations irectly..4 The Desire Traffic Distribution Moel In this subsection, we start with the esire traffic istribution moel II. Let T i,j enote the esire interaction traffic from N i to N j. T i,j can be interprete as a flow, which inclues a certain number of people moving from locations N i to N j. T l is the total interaction traffic going through N l (incluing the traffic that heas for N l, an the traffic that is forware by N l ). The mean value of T l is enote as T l. Since people are more likely to move aroun their home locations, we consier T i,j to be T i,j (monotonically ecreasing with geographical istance). For simplicity, we assume that noes have homogeneous traffic istributions. After normalization ( j T i,j = ), we have (ln n ln ): T i,j = ln n () ln n Eq. shows the esire traffic istribution moel II. We also consier two ifferent moels (moel I where T i,j, an moel III where T i,j e i,j ) in Section 4, since they represent ifferent noe communication patterns. The movement in moel I has no locality, meaning that people (e.g., businessmen) equally visit nearby an remote locations. On the other han, the movement in moel III is locality-oriente, meaning that people (e.g., farmers) only move aroun their home locations. Moel II is in the mile. Compare to existing traffic istribution moels in [, [, our moels are more simple an abstracte. They escribe the asymptotic traffic istribution with respect to the geographical istance. Flows are only allowe to be route among connecte noes (through links or chors). Flow interactions occur in noes along a routing path from source to estination, which is a sequence of links an chors. The routing path is the shortest one etermine by a greey-coin-change-base routing scheme. In this routing scheme, each noe greeily forwars the interaction traffic flow to a connecte relay that is geographically closest to the estination [6, [3. This flow routing scheme is use, since people usually make local ecisions for their movements without a global view. The routing path (representing the movement trace) from N i to n- n- 3 4 n-4 n-3 Fig. 5. An illustration of the 4-ary tree. N j is enote as P i,j, with its length being P i,j. We efine the average number of routing hops as: i,j = T i,j P i,j i,j T () i,j represents the communication convenience (a smaller means more convenient communications). Our goal is to esign a structure that can minimize without epiemic outbreaks. Then, we have the following observation: Proposition. If the network structure is isomorphic an all the noes in the network have out-egrees of D, then the average traffic receive by a noe is. The average amount of traffic going through each ege is /D. Proof: The interaction traffic from N i to N j is T i,j, with a path length P i,j. Each noe on the path P i,j, except for the source noe N i, receives this traffic. Therefore, P i,j noes receive a traffic of T i,j, an the total traffic is T i,j P i,j = i,j T i,j T i,j P i,j i,j i,j T = n (3) i,j i,j Since the esire interaction traffic of each noe is a specifie unit, we have i,j T i,j = n. Therefore, the average traffic receive by a noe is. Note that the traffic of a noe is share by D eges, an thus the average amount of traffic going through each ege is /D. Proposition means that longer routing paths bring heavier forwaring traffic on each noe. If the ring network has fewer chors, then people nee to go through more locations to reach their estinations. Meanwhile, we have: Theorem. In any network structures, if all the noes have out-egrees no more than D, then Ω(log D n). Proof: Since the out-egree of each noe is boune by D, a noe can reach at most D P noes through a path no longer than P. When the network structure is a D-ary tree, the root noe can reach the most noes with a limite path length. The lower boun of is that each noe can be regare as a root of a D-ary tree. Let us look into a specifie noe, N, as the root noe. The neighbors of N in the ring are put into the higher layers of the tree, an the remote noes are assigne into the lower layers of the tree, as shown in Fig. 5. This is because N has more interaction traffic to nearby noes than remote noes. In Fig. 5, noes N, N, N n, N n are place into the secon layer, while the interaction traffic between the root an these noes are T, = T,n = ln n an T, = T,n = ln n. Then, we have the following inequality: i = T n,i P,i i T =,i i= log D n [ x ln n x= n T,i P,i = T,i P,i (D x +...+D )/ (D x +...+D )/ i= y y (4)

5 IEEE/ACM TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING, VOL., NO., 6 5 This is because the tree has O(log D n) layers. For the x th layer, the tree contains D x noes. Then, we have (D x +...+D )/ (D x +...+D )/ Therefore, Eq. 4 can be rewritten as y y = ln Dx D D x > ln D (5) D > [log D n x ln D log D n ln D = ln n ln n log D n (6) x= where log D n = ln n ln D. ence, we have Ω(log D n)..5 The Infection Rate Moel We start with the infection rate moel B, which is a simplification of the classic SIS moel [4. The eigenvalue approach [4 is not use because of complexity. In our moel, the infection rate can be traffic-sensitive. We highlight that a esire traffic flow with ifferent routing paths have ifferent impacts on epiemic spreaings [4. This paper uniquely connect the traffic flow routing an the epiemic spreaing through the network structure esign. Flows have states of being susceptible or infecte. A susceptible flow represents a group of moving people who o not have the isease, but can be infecte. A infecte flow represents a group of moving people, who have the isease an can sprea the isease at the locations they go through. Infecte flows can go back into the susceptible state upon recovery, an can be reinfecte. For isomorphic structures, Theorem shows that the average traffic per ege is /D. In the infection rate moel B, let us consier a noe with D incoming eges, each of which loas a traffic of /D. If the flows on one incoming ege are infecte, then they woul bring an infection rate (the infection probability per time unit) of λ /D, where λ is a coefficient (λ > ). The square root epicts the ecreasing hazar infection of subsequent interactions [5. Other sub-linear functions can also be use here. Section 4 further consiers a constant rate moel of λ as moel A, an a linear rate moel of λ/d as moel C. While moel A is traffic-insensitive, moel C is traffic-sensitive. Moel B is in the mile. Base on the existing literature [4, λ an r are fixe. Let f(t) an g(t) (or simply f an g) enote the average fractions of eges that loa infecte an susceptible flows at time t, respectively. Each noe has, on average, Df incoming eges that loa infecte flows. ence, the infection probability for susceptible outgoing flows per time unit is: ( λ D )Df λ Df = µf (7) In Eq. 7, we efine µ = λ D for presentation simplicity. We assume λ /D. Otherwise, the epiemic outbreak cannot be controlle. ence, a total fraction, µf g, of all flows are infecte uring a time unit. Let r enote the recovery rate of infecte flows. Therefore, we have: g f = rf µfg an = µfg rf (8) t t Base on the existing literature [4, the solution to Eq. 8 is: (µ r)e (µ r)t f(t) = f() µ r + µf()e (µ r)t (9) Notation N i n D TABLE Notations in this paper Description Noe with an ID of i in the ring. Total number of noes in the ring. In-/Out-egrees of all noes (only use for isomorphic structures). Number of chors for half a ring. We have = D/. Geographical istance between N i an N j. T i,j Desire interaction traffic from N i to N j. P i,j Routing path from N i to N j. P i,j enotes the path length. T l Total interaction traffic going through N l. Average number of routing hops. f (g) Fraction of eges that loa infecte (susceptible) flows. λ, µ Coefficients for infection rate moels. Recovery rate of the infecte flows. Fraction of noes with in-egree D (only use for polymorphic structures). Mean value of the corresponing variable. r Q(D) If µ < r, f(t) ecreases exponentially, meaning that the epiemic outbreak is controlle. ence, we have: r µ < r or D < () λ A larger egree D generally brings a smaller, since more chors can shrink the network iameter. Eq. means that short routing paths can contribute to the control of epiemic outbreaks. This is because flows visit fewer locations, when is smaller. owever, the pattern of Eq. is very complex: If logarithmically ecreases with D, the constraint of controlling epiemic outbreaks is equivalent to a egree limitation. For example, if = ln n/ ln D, then D increases monotonically with D >.78, i.e., D shoul be smaller than a threshol. Our goal becomes minimizing with a limite egree. On the other han, if exponentially ecreases with D, then a larger D is better, which is counterintuitive. For example, if = n/ D, then D ecreases monotonically with D >.443. The above phenomena shows the traeoff between local an global infections. If D is small, then the traffic will aggregate along local noes ( is large). In this case, susceptible flows going through locations with infecte flows are very likely to be infecte, resulting in a local infection phenomenon. On the other han, if D is large, then the traffic is sprea wiely through chors. At this time, an infecte flow can reach many susceptible remote flows to sprea the isease, leaing to the global infection phenomenon. The challenge is that cannot be irectly erive as a function of D. This is because networks with the same D can have a ifferent ue to ifferent network structures..6 Problem Formulation This paper stuies a network structure esign problem constraine by the epiemic outbreaks. The network structure refers to the number an istribution of chors on the ring. Our objective is to minimize the average number of routing hops (minimize ), an the constraint is that epiemics will not outbreak. The variable of our problem is the number

6 IEEE/ACM TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING, VOL., NO., 6 6 chors chors n/ n - -(n/) -(n/4).... +(n/4) Fig. 6. An illustration of the binary-cut network structure (only the chors of N are shown). +(n/) n/ an istribution of chors (how noes connect each other on the ring; as a result, the egree D is a variable). The istribution of the esire traffic (T i,j ) is given. The routing scheme (given the number an istribution of chors, which path oes T i,j route from N i to N j ) is given. The epiemic outbreak constraint ( D < r λ for infection rate moel B) is given. In general, more chors brings a smaller, but the network may be more vulnerable to epiemic outbreaks ( D may be larger, since the egree D is larger). Networks with the same number of chors can have a ifferent ue to ifferent chor istributions. Our problem stans for an effective network quarantine with minimal restrictions on communication activities. The ring network is an abstraction of the real-worl geographic map, as shown in Fig.. The quarantine rule is represente by the number an istribution of chors. Two unconnecte noes (no chor) mean that they are isolate from each other in the quarantine, in orer to control epiemic outbreaks. A real-worl example is that Australia clampe own on the entry of iniviuals traveling from West Africa [3. Australia an West Africa are represente as two noes on the ring, while there is no chor connecting these two noes. Such a quarantine rule is execute by the Australian Customs an Borer Protection Service. The average number of routing hops represents the convenience level of the communication activities. A smaller means that fewer restrictions are put on the communication activities. The epiemics in this paper inclue human iseases, as well as epiemic information propagations in istribute systems (more etaile escriptions can be foun in [). In the next section, we esign isomorphic structures, uner the esire traffic istribution moel II an the infection rate moel B. In Section 4, we iscuss other traffic istribution moels an infection rate moels. The reason for the isomorphic structures is that networks with heterogenous egrees are more vulnerable to epiemics [6. We also show a greey solution for constructing polymorphic structures in Section 5. Finally, all the notations are shown in Table. 3 ISOMORPIC STRUCTURE DESIGN In this section, we esign two hypercube-base isomorphic structures that correspon to property (6) in Table. Since the structure is isomorphic, we only escribe the chor istribution of noe N for clear presentation. A chor from the noe N to N i (i< n ) inicates that N has a jump size of i. The out-egree of N is D, incluing two links an D chors. We use symmetric esigns for each sie of the ring, meaning that = D chors are assigne for the left-sie (or right-sie) half ring of N. The core iea of our structure esign is to fin a goo traeoff between the egree an the average number of routing hops, while the greey-coinchange-base routing is well-supporte. All of these factors can be foun in hypercube structures [5, [7 because of their structural symmetries. Epiemic outbreaks are reveale by the traeoff between the local an global infections, which are, in turn, controlle by the egree. First, we iscuss the hypercube-base binary-cut structure, which is shown in Fig. 6. In this network structure, the jump sizes are {+ n/, + n/,..., + n/, +} for the right-sie half ring an { n/, n/,..., n/, } for the left-sie half ring. For example, if n = 6 an = (D = 6), N woul have chors to noes {N, N 4, N, N 4 } an links to noes {N, N 5 }. We have assume < log n such that noes in the range ( n/, + n/ ) are archive through stepby-step links (jump sizes of ). Then, the bouns of for the binary-cut structure are: Theorem. In the binary-cut structure, the bouns of are O( 3 n 4 ln n ) an Ω( n ln n ). The proof of Theorem is attache in the supplemental material. Asymptotically, D monotonically ecreases with D, meaning that a larger egree is better. This structure can support at most = log n i i n n i i, the jump sizes are set to be {+ n/, + 4 n/, + 8 n/,..., + n/, +} an { n/, 4 n/, 8 n/,..., n/, }. ere, the esire traffic from noe N to the nearest n/ noes is the same as its traffic to the remaining noes. The bouns of for the binary-traffic-cut structure are: chors for half a ring. The jump sizes for half a ring are {+, +, +4..., + n/ }. owever, the binary-cut structure fails to consier the esire traffic istribution for optimizing. While each noe has more traffic to its geographically-nearby noes, we shoul provie more jumps to geographically-nearby noes an fewer jumps to geographically-remote noes. Following this intuition, we can improve the binary-cut network structure to the binary-traffic-cut network structure. Instea of jumping to the mile noe of each interval, we jump to the noe that forwars half of its esire traffic. Consiering that we have n Theorem 3. In the binary-traffic-cut structure, the upper an lower bouns of are O( n ln n + ln nn ) an Ω( n ln n + ln n n ), respectively. The proof of Theorem 3 is attache in the supplemental material. The binary-traffic-cut structure can support at most O(ln ln n) chors. owever, even if is very small, the n first part of ln n ominates the bouns of. For this structure, D monotonically increases with D, meaning that we nee to control the egree to avoi epiemic outbreaks: Corollary. For a binary-traffic-cut network structure without epiemic outbreaks, we have O( n ln n + ln nn ) an Ω( n ln n + ln n n ), where (r ln n)/(λ n).

7 IEEE/ACM TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING, VOL., NO., 6 7 If (r ln n)/(λ n)<, epiemic outbreak is inevitable for this structure. In terms of minimizing, the binarytraffic-cut structure outperforms the binary-cut structure, since it consiers the esire traffic istribution. owever, in the view of minimizing without epiemic outbreaks, it is har to analytically compare these two structures, since they have ifferent vulnerabilities to epiemic outbreaks. 4 OTER TRAFFIC DISTRIBUTION AND INFECTION RATE MODELS FOR ISOMORPIC STRUCTURES In this section, we stuy isomorphic structures with ifferent traffic istribution an infection rate moels. 4. Other Traffic Distribution Moels In previous sections, we only stuy the esire traffic istribution moel II, where T i,j /. In this subsection, two ifferent traffic istribution moels (moel I an moel III) are iscusse. These three moels present ifferent noe communication patterns. Compare to moel II (i.e., the most general case), moels I an III are two more extreme moels that have simpler properties as follows. Let us start with the esire traffic istribution moel I, where we have a uniform istribution of T i,j = n. In this case, the binary-cut structure is equivalent to the binarytraffic-cut structure. For each jump size on one sie of the ring, half of the flows to noes on that sie will take it. For example, if n = 6 an =, flows from N to {N 4, N 5, N 6, N 7 } inclue a jump size of +4, flows from N to {N, N 3, N 6, N 7 } inclue a jump size of +, an flows from N to {N, N 3, N 5, N 7 } inclue a jump size of +. Therefore, the average path length through chors is Θ(). For each noe in the interval of (, n ), it is achieve through links (jump sizes of ), the average path length of which is Θ( n ). Therefore, we have Θ( + n ), where Θ(log n) is the best choice. If D < r /λ, we have Θ(log n) for the binary-cut structure. owever, better structures may exist. For the same number of chors, the Ulysses butterfly network structure [9 achieves a lower network iameter than oes the hypercube structure. The esire traffic istribution moel III has an exponential traffic istribution of T i,j e i,j. Note that, the istribution in moel III ecays exponentially, an thus the traffic to remote noes can be ignore (people s movements are extremely locality-oriente). Assuming N is the source, then each noe can be achieve in, at most, steps (by links), an at least step (by a chor). Then, we have the following inequation: n n ln n e i i < n e ln n ln n n e i i () Eq. means that the nearest ln n noes on one sie of the ring are more important, while the traffic from N to the remaining noes on that sie can be ignore. Even if all the other noes can only be achieve by links, their influences on are much smaller than the nearest ln n noes. Therefore, all the chors are only necessary to connect the nearest O(ln n) noes, regarless of the remaining remote noes. The insight is that we nee smaller jump sizes for a faster traffic ecay, where remote connections are useless. This result is state as the property 4 in Table. 4. Other Infection Rate Moels In previous sections, we use the infection rate moel B, where µ = λ D. The reason behin the sub-linearity of moel B is that people are more likely to be infecte by the initial interactions with infectors than the subsequent interactions with infectors. In this subsection, two ifferent infection rate moels (moel A an moel C) are iscusse. These three infection rate moels present ifferent epiemic infectivities. Compare to moel B (i.e., the most general case), moels A an C are two more extreme moels that have simpler properties as follows. First, we consier the infection rate moel A, where the infection rate is a constant, i.e., traffic-insensitive. In this case, Eq. 7 can be rewritten as (let µ = λd for consistency): ( λ) Df λdf = µf () Eqs. 8 an 9 remain the same, while Eq. changes to: D < r λ (3) Eq. 3 means that the constraint of controlle outbreaks is equal to the egree limitation. The insight is that the global infection ominates, an thus we nee to restrict the connections among ifferent locations. This result is state as the property in Table. Uner the traffic istribution moel I an the infection rate moel A, our problem is reuce to minimizing with a limite egree. As shown in [9, Ulysses butterfly network structure is asymptotically optimal for this problem, where we have Ω(log r/λ n). This result is state as property 5 in Table. As for the infection rate moel C, we consier that the infection rate is traffic-sensitive. For a given location, each incoming ege that loas infecte flows woul inepenently bring an infection probability that is linearly proportional to the traffic on that ege. Then, Eq. 7 can be rewritten as (let µ = λ for presentation consistency): ( λ D )Df λf = µf (4) Eqs. 8 an 9 remain the same, while Eq. changes to: < r λ (5) Eq. 5 means that the network structure with a smaller can resist epiemics better, an thus the fully-connecte network is the best choice. This is because the local infection becomes the major factor for epiemic outbreaks. It implies that the communication restrictions lea to negative effects, uner the infection rate moel C. In this case, one noe shoul connect to as many noes as possible, in orer to mitigate the traffic aggregation on local links. This result is state as property 3 in Table. 5 POLYMORPIC STRUCTURE DESIGN In this section, we stuy polymorphic structures, uner the esire traffic moel I an the infection rate moel A. Other moels are not explore ue to their complexities.

8 IEEE/ACM TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING, VOL., NO., 6 8 Algorithm Greey Construction Input: Output: A virtual ring network that has no chor; A polymorphic network structure; : while D D < r λ o : for each pair of unconnecte noes o 3: Calculate the corresponing an [ D D, if a chor is ae for this pair of noes; 4: A the chor with the highest / [ D D ; 5: Remove the last ae chor to guarantee D D < r λ ; 6: return the current network structure; Algorithm Greey Dismantlement Input: Output: A fully-connecte virtual ring network; A polymorphic network structure; : while D D > r λ o : for each pair of noes connecte by a chor o 3: Calculate the corresponing an [ D D, if a chor is remove for this pair of noes; 4: Remove the chor with the lowest / [ D D ; 5: return the current network structure; 5. Contributions of an Aitional Chor This subsection stuies how reuces, if an aitional chor is ae into the current ring network (uner the esire traffic moel I). Let enote the amount of reuce. Let us start with a virtual ring network that has no chor, an then consier aing a chor from N i x to N i with a jump size of x. Then, noes {N i n/,..., N j,..., N i x } can benefit from this chor. For N j specifically, its routing paths of flows to noes {N i,..., N j+n/ } are shortene by x. Uner the esire traffic moel I, we have: n x = (x ) ( n x j) (n x) x (6) j= Eq. 6 is maximize, when we use a chor with the jump size of x = n 6. The insight behin Eq. 6 is the traeoff between () the number of routing paths that can benefit from the chor an () the save lengths of routing paths brought by the chor. ere, we o not further explore the contribution of an aitional chor on a ring with multiple chors. owever, note that the brought by an aitional chor can be calculate in polynomial time. This is because we have O(n ) pairs of noes, while the routing path for each pair of noes can be etermine within O(n). 5. Epiemics in Polymorphic Structures This subsection introuces an avance epiemic moel [6 for polymorphic structures uner the infection rate moel A. To capture the structural heterogeneity, let Q(D) enote the fraction of noes with in-egree D, an let f D (t) enote the fraction of incoming eges that loa infecte flows in locations with in-egree D at time t. For the moel A of the constant infection probability, we have: f D (t) t = λd[ f D (t)θ(f(t)) rf D (t) (7) Eq. 7 is similar to Eq. 8, which is for isomorphic structures. The fraction of outgoing eges that loa susceptible flows in locations with in-egree D is [ f D (t). Θ(f(t)) is the total fraction of eges that loa infecte flows. Therefore, the first term, λd[ f D (t)θ(f(t)), inicates the fraction of eges that loa new infecte flows at locations with in-egree D. The last term, rf D (t), shows the recovery. Existing work in [6 shows the following result: D D + D < r D λ (8) D is average noe egree, an D D represents the noe egree variance. Eq. 8 epicts the prerequisite of controlle outbreaks in polymorphic structures, uner the infection rate moel A. In contrast, the constraint for isomorphic structures are shown in Eq. 3. The insight of Eq. 8 is that a larger egree variance also brings a more vulnerable network. Uner the infection rate moel A, both the average egree an the egree variance etermines the network resistance to epiemics. D D inicates the network vulnerability to epiemics (the larger, the more vulnerable). 5.3 Constructing Polymorphic Structures This subsection escribes two greey algorithms for polymorphic structures. Algorithm starts with a ring network that has no chors, an then iteratively as chors with the consieration of () that inicates the reuce average number of routing hops (the benefit of that chor) an () [ D D that represents the increase network vulnerability (the cost of that chor). Algorithm iteratively as the chor with the highest ratio of to [ D D (the benefitto-cost ratio). In the event of a tie, a ranom one is picke. A chor with a large means that it can effectively minimize the number of average hops. A chor with a small [ D D inicates that aing this chor only slightly increases the network vulnerability to epiemics. In contrast, Algorithm starts with a fully-connecte ring network, an then iteratively removes the chor with the lowest ratio of to [ D D. Algorithms an are extenable to heterogeneous traffic istribution an epiemic spreaing moels [7, [8, since they construct polymorphic network structures base on the benefit-to-cost ratio of each chor. owever, both Algorithms an are suboptimal. They may be trappe into local optima ue to the greey nature. The optimal solution remains to be explore. 6 EXPERIMENTS This section conucts experiments to evaluate the propose structures. Coes are publishe in [9. 6. Experiments on Ring Networks Settings. In this subsection, we focus on the propose isomorphic structures in a ring network. The ring network has n = 7 noes, with the number of chors ranging from to 3. The infection coefficient λ is., while the recovery rate r varies in ifferent settings, in orer to observe the epiemic outbreak point. % of the total eges

9 IEEE/ACM TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING, VOL., NO., µ< r λ µ< r λ µ r λ (a) T i,j, µ = λ D, r = (b) T i,j = n, µ = λ D, r = (c) T i,j, µ = λd, r =. Fig. 7. The evaluation results. The top line figures show the relationship between an, an the bottom line figures show the relationship between an the infection percentage of noes after, time units. (a) The evaluation results uner the esire traffic moel II an the infection rate moel B, with a recovery rate r =.5. (b) The evaluation results uner the esire traffic moel I an the infection rate moel B, with a recovery rate r =.3. (c) The evaluation results uner the esire traffic moel II an the infection rate moel A, with a recovery rate r =.. are initialize as being infecte, while the percentage of infectors is checke again after, time units. The time perio of, time units is long enough to guarantee that the infection percentage becomes steay. We o not consier the other initializations on the percentage of infectors, since [4 has shown that the initialize percentage of infectors has a very limite influence on the eventual epiemic outbreak (or not). ere, the esire traffic moel III is not consiere, since all chors ten to connect to the nearest noes. The infection rate moel C is not consiere, since it results in a fully-connecte network as an optimal solution. Therefore, we focus on the esire traffic moel I (T i,j = n ) an moel II (T i,j ). As for the infection rate moels, we only pay attention to moel A (µ = λd) an moel B (µ = λ D). Three aitional isomorphic structures are use for comparison. The first one is the uniform-cut structure, where the chors uniformly cut the half-ring. In this structure, the jump sizes are {+, + n, + n ( )n,..., + } for half a ring. The secon one is the structure, where each noe connects to the nearest D geographical neighbors. The thir one is the structure [3, which was propose for the connections among servers in ata center networks. Testing Different Structures. The experimental results are shown in Fig. 7. The top-line figures show the relationship between an, while the bottom-line figures show the relationship between an the percentage of infecte flows after, time units. The three subfigures in Fig. 7 escribe three ifferent settings of traffic istribution moels an infection rate moels. The top lines of Figs. 7(a) an 7(c) are the same, since they have the same traffic moel. In Fig. 7(b), the performance of the binary-cut an binarytraffic-cut structures are the same, ue to the uniform traffic moel. In the top line figures, the ashe boxes show the theoretical interval of controlle epiemic outbreaks (i.e., the theoretical constraint of µ < r λ ). The real infection percentages, after, time units (i.e., epiemic outbreaks or not), are shown in the bottom line of Fig. 7. It can be seen that the uniform-cut an the structures are useless, since they cannot control epiemic outbreaks. They have very large average numbers of routing hops. Although has a ecent average number of routing hops, it cannot control epiemic outbreaks. For the esire traffic moel II an the infection rate moel B in Fig. 7(a), the binary-cut structure controls outbreaks with a large, an the binary-traffic-cut structure controls outbreaks with a small. This is consistent with our theoretical results. The binary-cut structure can achieve the smallest without outbreaks. For the esire traffic moel I an the infection rate moel B in Fig. 7(b), the performance of the two propose structures are the same, an they can control epiemic outbreaks when is large enough. The esire traffic moel II an the infection rate moel A in Fig. 7(c) show that the epiemic outbreaks only epen on the egree. These results verify that the prerequisite of controlle epiemic outbreaks (i.e., µ < r λ ) is accurate. Sensitivity Experiments. One step further, here we also stuy the sensitivities of the propose structures, since the assumption of isomorphic structures may be too strong for real-worl applications. We want to check whether this assumption can be relaxe or not. ence, we ranomly rewire a fraction of chors for the propose structures to check their sensitivities. If a chor is rewire, it will reconnect to a pair of noes that are ranomly selecte. For the binary-cut an binary-traffic-cut structures, the resulting structures with % rewire chors are enote as binarycut* an binary-traffic-cut*, respectively. Then, the resulting

10 IEEE/ACM TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING, VOL., NO., * ** * ** * ** * ** * ** 5 5 * ** * ** * ** 5 5 (a) T i,j, µ = λ D, r = * ** * ** 5 5 (b) T i,j = n, µ = λ D, r = * ** * ** 5 5 (c) T i,j, µ = λd, r =. Fig. 8. The sensitivity experiments. The settings are the same as those in Fig. 7. The binary-cut* an binary-traffic-cut* enote binary-cut an binary-traffic-cut structures with % ranomly rewire chors, respectively. Similarly, The binary-cut** an binary-traffic-cut** enote binary-cut an binary-traffic-cut structures with 5% ranomly rewire chors, respectively. Fig. 9. The geographic map of the real ata-riven experiment. structures with 5% rewire chors are enote as binarycut** an binary-traffic-cut**, respectively. The sensitivity experiment results are shown in Fig. 8. It can be seen that, rewire chors have a very limite impact, in terms of both the average number of routing hops () an the infection percentage. The network traffic moel an infection rate moel also o not significantly change the result. The average number of routing hops () slightly increases, since the hypercube-base structures are greatly fault-tolerant with respect to greey-coin-change-base routings. The variance of the infection percentage is also limite, since the isomorphic structure is not estructe too much. Overall, these two structures are not sensitive to a small portion of ranomly rewire chors, i.e., the isomorphism assumption can be relaxe for real-worl applications. 6. Real Data-Driven Experiments Settings. This subsection conucts real ata-riven experiments to verify the applicability of our approach. We use a real airline ataset of OpenFlights [3, which contains 59,36 routes between 3,9 airports on 53 airlines spanning the globe, as shown in Fig. 9 (airports are marke as re points). Airports correspon to noes in our moel. We iscretize the Earth through meriians an parallels (, points). Each airport is roune to its nearest geographic point. The existing airline routes imply the information on the esire traffic istribution. In other wors, the esire traffic istribution moel is given, where each flight stans for a unit traffic. If we use the existing airline routes as chors, then the average number of routing hops () is one, but this structure fails to control epiemic outbreaks. To apply our ring moel, X-Y routing scheme is use for noe communications. The routing has two stages: the first stage routes along the meriian ring of the source noe, an the secon stage routes along the parallel ring of the estination noe. An example of such a routing has been shown in Fig. 4. The number of routing hops is the sum of these two stages on two ifferent rings. We use the real Ebola ataset in three countries (Guinea, Liberia, an Sierra Leone) reporte by Wor ealth Organization [3 to moel epiemic spreaings. Table 3 shows the ata statistics, in terms of the number of recovers an eaths. The number of recovers are significantly smaller than the number of eaths in Guinea, approximately equal to the number of eaths in Liberia, an significantly larger than the number of eaths in Sierra Leone. The fractions of recovers an eaths are use as the recover an infection rates, respectively. ere, the infection rate is trafficinsensitive (infection rate moel A). The other settings are the same as those in the previous subsection. Experimental results. The results for the real ata-riven experiments are shown in Fig.. Fig. (a) shows the relationship between an. A larger noe egree leas to a smaller average number of routing hops. The binarytraffic-cut outperforms other structures, when is large. It

11 IEEE/ACM TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING, VOL., NO., (a) Number of routing hops (b) Epiemics in Guinea (c) Epiemics in Liberia () Epiemics in Sierra Leone. Fig.. Evaluation results for real ata-riven experiments. TABLE 3 Ebola statistics reporte by Wor ealth Organization. Dataset Recover Death Total Guinea (/8/5),68,536 3,84 Liberia (5/9/5) 5,86 4,86,666 Sierra Leone (/7/5),67 3,955 4, ACKNOWLEDGMENTS This work is supporte in part by NSF grants CNS 4986, CNS 4693, CNS 4697, CNS 43967, CNS 3774, ECCS 346, ECCS 89, an CNS has an of about 4, when = 8. In contrast, has the largest among all the structures. Another notable point is that has the smallest when is small (when = ). Fig. (b), Fig. (c), an Fig. () shows the eventual infection percentages uner the ataset of Guinea, Liberia, an Sierra Leone, respectively. A larger always leas to a larger infection percentage. Note that, in our structure esign, each airport can have four routes to nearby airports (on meriian an parallel), an routes to remote airports ( for meriian-remote airport an for parallel-remote airport). This is because the infection rate is not traffic-sensitive, an thus, we shoul cap the noe egree to avoi epiemic outbreaks. Fig. (b) has the highest infection percentages, since the ataset of Guinea has the highest eath-to-recover ratio, an the infection rate is larger than the recovery rate. In Fig. (b), uniform-cut an can only control epiemic outbreaks when = with 7. controls epiemic outbreaks when = 3 with 6. Binary-cut an binary-trafficcut control epiemic outbreaks when = 5 with 5. In Fig. (c), uniform-cut an control epiemic outbreaks when = 4 with 6. controls epiemic outbreaks when = 6 with 7. Binary-cut an binarytraffic-cut control epiemics with 4. Fig. () shows similar results with Fig. (c), since Sierra Leone has the lowest eath-to-recover ratio. Real ata-riven experiments confirm the real-worl applicability of our approach. 7 CONCLUSION This paper stuies network structures with minimum traffic flow routings, while controling epiemic outbreaks through regulating the number an istribution of chors on a ring network. The objective is to esign a structure (quarantine rules) that can minimize the average number of routing hops, while the epiemic outbreaks are controlle. For isomorphic structures, we provie a systematic structure esign on nine ifferent cases. Two hypercube-base structures are explore. Sensitivity experiments emonstrate that the isomorphism assumption can be relaxe. For polymorphic structures, we provie a greey solution. Our work casts new light on the effective network quarantine that places minimal restrictions on communication activities. REFERENCES [ C. Lagorio, M. Dickison, F. Vazquez, L. A. Braunstein, P. A. Macri, M. V. Migueles, S. avlin, an. E. 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