Designing virus-resistant, high-performance networks: a game-formation approach
Stojan Trajanovski, Fernando A. Kuipers, Yezekael Hayel, Eitan Altman, Piet Van Mieghem
11 Designing virus-resistant, high-performancenetworks: a game-formation approach
Stojan Trajanovski,
Member, IEEE,
Fernando A. Kuipers,
Senior Member, IEEE,
Yezekael Hayel,
SeniorMember, IEEE,
Eitan Altman,
Fellow, IEEE, and Piet Van Mieghem
Member, IEEE
Abstract —Designing an optimal network topology while balanc-ing multiple, possibly conflicting objectives like cost, performance,and resiliency to viruses is a challenging endeavor, let alone in thecase of decentralized network formation. We therefore propose agame-formation technique where each player aims to minimizeits cost in installing links, the probability of being infected by avirus and the sum of hopcounts on its shortest paths to all othernodes.In this article, we (1) determine the Nash Equilibria and thePrice of Anarchy for our novel network formation game, (2)demonstrate that the Price of Anarchy (PoA) is usually low,which suggests that (near-)optimal topologies can be formed ina decentralized way, and (3) give suggestions for practitionersfor those cases where the PoA is high and some centralizedcontrol/incentives are advisable.
Keywords — Game theory; Virus spread; Network performance;Network design; Networks of Autonomous Agents.
I. I
NTRODUCTION
Designing communication and computer networks are com-plex processes in which careful trade-offs have to be madewith respect to performance, resiliency/security and cost in-vestments. For instance, if a host in a computer networkwants to route traffic to multiple other hosts, it could directlyconnect to those other hosts, in this way increasing its expensesin installing and maintaining these connections and at thesame time also becoming more susceptible to viruses fromthose other hosts. In return, it would obtain a better andfaster performance with minimum delays, compared to whenit would have used intermediate hosts as relays. Althoughin this example, both installation costs and risk to virusesare increasing, they are linearly independent and they do notnecessarily optimize together. Indeed, reducing the number
S. Trajanovski is now with the Philips Research, Eindhoven, The Nether-lands. The research was done while the first author was with Delft Universityof Technology ([email protected]).F. A. Kuipers and P. Van Mieghem are with Delft University of Tech-nology, Faculty of Electrical Engineering, Mathematics and Computer Sci-ence, P.O. Box 5031, 2600 GA Delft, The Netherlands ( { F.A.Kuipers,P.F.A.VanMieghem } @tudelft.nl).Y. Hayel is with University of Avignon, Avignon, France([email protected]).E. Altman is with University of Cote d’Azur, INRIA, BP95, 06902 SophiaAntipolis, France. He is also member of LINCS, 23 Ave. d’Italie, 75013Paris, France and an associate member of LIA, University of Avignon, France([email protected]).This article builds on our work [1] published in Proc. of IEEE CDC 2015,Osaka, Japan. of direct connections would reduce the cost and the hostwould be less vulnerable to viruses. However, even when beingconnected to a few high-degree nodes with direct connections,the host would still be seriously imposed to a virus.In practice, hosts often are autonomous, act independentlyand do not coordinate as in P2P networks [2], peering rela-tions between Autonomous Systems [3], overlay networks [4],wireless [5], [6], [7] and mobile [8] networks, resource sharingin VoIP networks [9], social networks [10], [11] or the Inter-net [12]. Their aim is to optimize their own utility functions,which are not necessarily in accordance to the global optimum.To study global network formation under autonomous actors,the network formation game (NFG) framework [13] has beenproposed. However, resilience and notably virus protectionhave not been taken into account in that NFG context,even though their importance is undisputed. In this paper,we therefore take the NFG framework one step further byincluding performance and virus protection as key ingredients.Virus propagation will be modeled by the Susceptible-Infected-Susceptible (SIS) model [14] and we will evaluate the effect ofuncoordinated autonomous hosts versus the optimal networktopology via standard game-theoretic concepts, such as NashEquilibria and the Prices of Anarchy and Stability.Our network formation game is called the Virus Spread-Performance-Cost (VSPC) game. Each node (i.e., autonomousplayer) attempts to minimize both the cost and infection proba-bility, while still being able to route traffic to all the other nodesin a small number of hops. When the hopcount performancemetric is irrelevant, the game is driven by the cost and virus ob-jectives; a scenario we studied in [1]. That particular scenarioresulted in sparse graphs, which may not always represent real-world networks, but it helped to understand the process of virusspread better. In this paper, we generalize those results by alsoincluding the hopcount performance metric. The probabilityof the node being infected and the hopcounts to the othernodes change in a different direction, for example adding alink reduces the former, but increases the latter. Therefore,there is a tradeoff in the number of added links and howthese new links are best added. Moreover, the two metrics arelinearly independent and closed-form expressions do not exist,which makes the problem complex. Finally, the inclusion ofthe hopcount allows us to better capture realistic networks. Inparticular, our main contributions are the following: • We provide a complete characterization of the various rel-evant parameter settings and their impact on the formationof the topologies. • We show that depending on the input, the Nash Equi- a r X i v : . [ c s . G T ] O c t libria may vary from tree graphs, via graphs of differentdiameters, to complete graphs. • We demonstrate, both via theory and simulations, thatthe Price of Anarchy (PoA) is small in most of thecases, which implies that (near-)optimal topologies canbe formed in a decentralized non-cooperative manner. Wewill also identify for which scenarios the PoA may behigh. In those cases a central point of control would bedesirable to limit/steer the players’ decisions.This paper is organized as follows: The SIS-virus spreadmodel and the network-formation game model are introducedin Section II. The
Virus Spread-Performance-Cost (VSPC)game formation is analyzed in Section III. Related work ongame formation and protection against viruses is discussed inSection IV. The conclusion and directions for future work areprovided in Section V.II. M
ODELS AND PROBLEM STATEMENTS
A. Virus-spread model
The spread of viruses in communication and computernetworks can be described, using virus-spread epidemic mod-els [14], [15], [16]. We consider the Susceptible-Infected-Susceptible (SIS) NIMFA model [14], [17], dv i ( t ) dt = β (1 − v i ( t )) N (cid:88) j =1 a ij v j ( t ) − δv i ( t ) (1)where N is the number of network nodes and v i ( t ) is theprobability of node i being infected at time t , for all i ∈{ , , . . . , N } . If a link is present between nodes i and j , then a ij = 1 , otherwise a ij = 0 . In (1), a host with a virus caninfect its direct healthy neighbors with rate β , while an infectedhost can be cured at rate δ , after which the node becomeshealthy, but susceptible again to the virus. The probability v i ( t ) depends on the probabilities v j ( t ) of the neighbors j of node i and there is no trivial closed form expression for v i ( t ) . The model incorporates the network topology and isthus more realistic than the related population dynamic models.The model relies on the network topology, which makes itmore realistic than the related population dynamic models.The goodness of the model has been evaluated in [18]. Theprobability of a node being infected in the metastable regime,denoted by v i ∞ , where dv i ( t ) dt = 0 and v i ∞ (cid:54) = 0 , follows from(1) as [14], v i ∞ = 1 −
11 + τ (cid:80) Nj =1 a ij v j ∞ (2) where τ = βδ is called the effective infection rate. The epidemicthreshold τ c is defined as a value of τ , such that v i ∞ > if τ >τ c , and otherwise v i ∞ = 0 for all i ∈ { , , . . . , N } . The valueof v i ∞ depends of the values of all v j ∞ for all the neighbors j of i , so the network topology and the interconnectivity haveimpacts on v i ∞ s. B. Game-formation model
In our network formation game, each player i (a node inthe network) aims to minimize its own cost function J i , and the social cost J is defined as J = (cid:80) Ni =1 J i . Specifically, the optimal social cost is the smallest social cost over all possibleconnected topologies. We look for the existence, uniqueness,and characterization of ( pure ) Nash Equilibria . The Price ofAnarchy ( PoA ) and the
Price of Stability ( PoS ) are defined asthe ratio of social cost in the worst-case Nash Equilibrium (theone with highest social cost) and the optimal social cost, andthe ratio of the social cost in the best-case Nash Equilibrium(the one with lowest social cost) and the optimal social cost,respectively:
PoA = J ( worst NE )min J ,
PoS = J ( best NE )min J . (3)
PoA is an efficiency measure, illustrating how bad selfishplaying is, in comparison to the global optimum. PoS, on theother hand, reflects the best possible performance without co-ordination in comparison to the global optimum. The networkabout to be designed, is empty and every node in the network isa player. We assume the cost of building one (communication)link between two nodes is fixed. Every player i can install alink from itself to another node j . Installing a link between i and j means that both i and j can utilize it, but only one paysfor the cost, like often assumed in NFG models [12], [19],[4]. Several examples fit this scenario: (i) a friend request isinitiated by one node in a social network, but both read theposts from one another; (ii) a new road connecting two citiesis built by one city in a road network, but both utilize it; and(iii) in a hand-shake protocol in a computer network one nodeinitiates a connection used by two nodes.We consider a Virus Spread-Performance-Cost (VSPC) network formation game, where player i aims to reduce its costand the probability v i ∞ of being infected, but concurrently alsowants to improve its performance by shortening the hopcounts h ( i, j ) of the shortest paths to all the other nodes j . The costfunction of player i that unites these objectives is given by: J i = α · k i + γ N (cid:88) j =1 h ( i, j ) + v i ∞ . (4) Function J i involves the cost k i of installing all the links fromnode i , weighted by a coefficient α . The hopcounts h ( i, j ) areweighted by γ . Opposing goals meet in this game: the morelinks are installed, the shorter the paths, but the higher theprobability of being infected and the higher the cost.The social cost J for the whole network is a weighted sumover all nodes J = N (cid:88) i =1 J i = αL + γ N (cid:88) i =1 N (cid:88) j =1 h ( i, j ) + N (cid:88) i =1 v i ∞ , (5) where L denotes the number of links.III. V IRUS SPREAD - PERFORMANCE - COST (VSPC)
GAME
A. Optimal social cost, Nash Equilibria and the PoA for γ → In order to understand the effect of the virus protection, westart by setting γ to an infinitely small number (approaching A Nash Equilibrium is the state of the players’ network strategies, wherenone of the players can reduce its cost by unilaterally changing its strategy. zero ). As a result, the hopcounts are of no influence anymore,while network connectivity is still guaranteed (the hopcountbetween two disconnected nodes is assumed to be infinity).Lemma 1 limits the possible Nash Equilibria. Lemma 1.
The probability v i ∞ ( G ) of node i being infectedin the metastable state in network G does not exceed the prob-ability v i ∞ ( G + l ) of node i being infected in the metastablestate in network G + l obtained by adding a link l to G .Proof: The newly added link l = ( a, b ) is between nodes a and b . We make use of the canonical infinite form [14], v i ∞ = 1 −
11 + τ d i − τ (cid:80) Nj =1 a ij τd j − τ (cid:80) Nk =1 ajk τdk − ... . (6) After the addition of link l = ( a, b ) , the expression (6) for v i ∞ ( G + l ) has all the terms the same as in v i ∞ ( G ) , exceptthe following differences: d a → d a + 1 ; d b → d b + 1 andthe presence of the adjacency entry a ab = 0 → a ab = 1 inthe canonical representation. The last statement implies thatits contribution is a part that is the same as in v i ∞ ( G ) untilit “reaches” nodes a or b , where the expression (at a certaindepth of the canonical form) is: τ ( d a + 1) − ττ ( d b + 1) − . . . + U = τd a + U + τ (1 − τ ( d b + 1) − . . . ) , (7) where d a and d b are the degrees of a and b in G , while U is the remaining part in the canonical form. In (7), the term τ (1 − τ ( d b +1) − ... ) is positive and U increases with d a and d b . U increases with d a and d b as it is also an infinite canonicalform with any of these two variables being in the numeratoror in the denominator with a negative sign in front, in thesame way as explained in the lines above - repeating infinitelymany times. Therefore, the whole term in (7) increases, whichimplies that v i ∞ ( G + l ) > v i ∞ ( G ) also increases for eachnode i .We start by looking into the possible Nash Equilibria. Theorem 1.
If a Nash Equilibrium is reached, then theconstructed graph is a tree .Proof: If G is connected and each node can reach everyother node, then changing the strategy of node i from thecurrent one to investing in an extra link , will increase bothits cost (by , scaled by α ) and v i ∞ (by Lemma 1). Hence,unilaterally investing in an extra link is not beneficial for anode.We now assume that G is not a tree. Then, there is at leastone cycle in this graph. If a node i in that cycle changes itsstrategy from investing in a link in that cycle to not investing ,the cost is decreased by (weighted by α ) and all the othernodes in the graph are still reachable from i . Moreover, by not investing in that link, node i decreases its probability v i ∞ of being infected in the metastable state, according to The case of γ = 0 is either trivial or debatable. By neglecting thehopcounts, the optimal topology would be the (non-realistic) empty graphwith no links (cost) and no epidemic to be propagated. Moreover, infinitehopcounts will be multiplied by γ = 0 which is undefined. Lemma 1. Hence, by unilaterally changing its strategy, node i decreases its cost utility J i , which is in contradiction with aNash Equilibrium. Observation 1.
A Nash Equilibrium is achieved for both the star graph and the path graph, but not all trees are NashEquilibria.Proof:
Let us consider a star graph , where all the linksare installed by the root node as shown in Fig. 1a. (A linkis installed and paid for by the node marked with p .) Theroot node cannot unilaterally decrease its cost, because cuttingat least one of its installed links would disconnect it, whileinstalling a link from a leaf node i would increase both k i and v i ∞ (Lemma 1). Hence, the star graph is a Nash Equilibrium.Let us now assume that a path graph (Fig. 1b) is constructed,such that ( N − nodes invest in exactly one link and one ofthe leaves does not invest in installing a link. Similarly as fora star graph , none of the nodes can unilaterally decrease theircost by just installing extra links or cutting some of them. A”re-wiring” from one of the nodes by re-directing its installedlinks to another node may be in order. In such a case, if node i “re-wires” its installed link to another node, then J i wouldnot decrease. 1) If it is installed to one of the leaves, suchthat the graph is connected, we end up with an isomorphicgraph, where the position of i is the same as in the initialgraph, so J i stays the same. 2) If i ”re-wires” to one of theother nodes j (w.l.o.g., i < j ) as visualized in Fig. 1e, i wouldhave the same degree, but its “new neighbor” would have adegree instead of . The degree of j increases by to andthe degree of ( i + 1) decreases by to (node ( i + 1) willbecome terminal and ”far” from i ), while all the other degreesremain the same. Moreover, i would be equally close to anyof the nodes “behind” { , . . . , i − } , closer to the nodes “atthe end” { j + 1 , . . . , N } and equally close to the nodes in theset { i + 1 , . . . , j − } , but just in a reverse order. Based on thecanonical infinite form (6), v i ∞ would increase . Therefore,the path graph is also a Nash Equilibrium.There are also other trees that are Nash Equilibria (e.g., T (cid:48) given in Fig. 1c). Moreover, there are values of τ suchthat worst- and best-case Nash Equilibria are achieved fortrees different from star K ,N − and path P N graphs. For τ ∈ [1 . , . , tree T (cid:48) is the best-case Nash Equilibriumand has optimal social cost.However, not all the trees are Nash Equilibria. For example,the tree given in Fig. 1d. Here, whomever pays for the “central”link between a and b , can reduce its cost utility by “re-wiring”to c or d .We proceed by characterizing the worst- and best-case NashEquilibria. Theorem 2.
For sufficiently high effective infection rate τ ,the optimal social cost and the best-case Nash Equilibriumare achieved by the star graph K ,N − , while the worst-case ”Re-wiring” is a process of removing a link to node k initiated by node i and establishing a new link to another node j . The degree of node i does notchange, while the degrees of k and j are decreased and increased, respectively. v i ∞ in (6) would have bigger values by having nodes with ”biggerdegrees” as as close as possible (i.e. in fewer hops) to the node. ... ... root node pp p pp (a) Star K ,N − . ... p p p p (b) Path P N . p pp p (c) Tree T (cid:48) . a b cd p p (d) Tree T (cid:48)(cid:48) . ... p p p p pp (e) Re-wiring increases v i ∞ . Fig. 1: A link is installed by the end-node marked with p . Trees in (a), (b), and (c) are Nash Equilibria. (d) Tree T (cid:48)(cid:48) cannot bea Nash Equilibrium. Nash Equilibrium is achieved for the path graph P N , J ( K ,N − ) ≤ J ≤ J ( P N ) . Proof:
According to Theorem 1, in a Nash Equilibriumthe graph is a tree, hence it has N − links. In a general case,from a tree in which there are two nodes i and k , connectedto one another, for which d i ≥ and d k = 1 (i.e. k is a leaf),by breaking the connection between i and k and connecting k to another leaf j instead, we have: the degree of k is (remains the same); the degree of node i becomes d i − ≥ (decreased by one); and the degree of j is (increased byone). The process can be repeated until there exists a node ofdegree at least in the tree. At the end, we end up with a treewith no degree bigger than and this is a path P N . The socialcost J is increased in each step [1, Lemma 2]. In this way, theprocess converges to a path P N .In a very similar (but reverse) process, starting from anytree G , we can decrease J at each step, ending up with a star K ,N − with a maximum J ( G ) in the final step.However, what would be the optimal social cost, and theworst- and best-case Nash Equilibria highly depends on theeffective infection rate τ . Theorem 3.
For low values of the effective infection rate τ ,above but sufficiently close to the epidemic threshold τ c , theoptimal social cost and the best-case Nash Equilibrium areachieved by the path graph P N , while the worst-case NashEquilibrium is achieved by the star graph K ,N − , J ( P N ) ≤ J ≤ J ( K ,N − ) . Proof:
We consider a spectral approach [20] and denote y ( τ ) = N (cid:80) i =1 v i ∞ ( τ ) the infection probability of all nodes inthe metastable state. The probabilities of a node in the graphbeing infected are non-zero and y ( τ ) > if τ > τ c = λ ,where λ is the largest eigenvalue of the adjacency matrix inthe graph [16]. For τ < λ , y ( τ ) = 0 .Lov´asz and Pelik´an [21] ordered all the trees with N nodes by the largest eigenvalues of the adjacency matrices.It turns out that, the path P N and star K ,N − are the treeswith the minimum λ ( P N ) and maximum λ ( K ,N − ) largesteigenvalues, respectively.For values τ = λ ( K ,N − ) + ε = √ N − + ε , it holds that y K ,N − ( τ ) > y T ( τ ) = 0 , where T is any tree different from K ,N − , therefore J ( K ,N − ) is the largest. For values τ = λ ( P N ) − ε =
12 cos( πN +1 ) − ε , we have y T ( τ ) > y P N ( τ ) = 0 , where T is any tree different from P N ,hence J ( P N ) is the smallest.Theorems 2 and 3 show opposite behavior depending onwhether the value τ is in the high or low regime, althoughboth revolve around the path and star graphs. For τ in theintermediate regime, different trees may give the best-/worst-case Nash Equilibrium. Corollary 1.
For both high and low effective infection rate τ ,PoS = 1 andPoA = max (cid:8) J ( P N ) J ( K ,N − ) , J ( K ,N − ) J ( P N ) (cid:9) .Proof: Based on Theorems 2 and 3, for high (low) τ ,tree K ,N − ( P N ) is both optimal in social cost and the best-case Nash Equilibrium, while P N ( K ,N − ) is the worst-caseNash Equilibrium. Based on the definitions for PoS and PoA in(3), PoS = J ( K ,N − ) J ( K ,N − ) = 1(= J ( P N ) J ( P N ) ) ; and PoA = J ( P N ) J ( K ,N − ) for large enough τ and PoA = J ( K ,N − ) J ( P N ) for τ close to theepidemic threshold τ c . Corollary 2.
For sufficiently high effective infection rate τ , inthe virus spread-cost game formation, PoA < (cid:0) τ ( α + 1) − (cid:1) , where τ ( α + 1) > . Proof:
The proof is provided in [1].The exact value of the PoA is given in Fig. 2 by makinguse of Corollary 1. It is highest ( ∼ . ) for small τ , above theepidemic threshold and it further sharply decreases reaching for a unique Nash Equilibrium. For higher τ , the PoA increasestowards its maximum around . and then it slowly decreasesapproaching .We have observed that the equilibria tree topology in whicha virus thrives is not always a star (i.e., the tree with thesmallest diameter), but that it may differ with the virusinfection rate. For most of the τ values (except maybe small τ ), a small value for the Price of Anarchy (PoA) means that atopology close to optimal can be obtained in a decentralizedmanner, even when the individual players play selfishly. B. Optimal social cost, Nash Equilibria and the PoA for γ > We start by analyzing the social cost (5). Node i is onehop away from its d i neighboring nodes, while it is atleast hops away from the other N − − d i nodes, hence = P r i ce o f A n a r c hy ( P o A ) , = 0.0 , = 0.3 , = 1.0 , = 5.0 = P r i ce o f A n a r c hy ( P o A ) (a) N = 10 . = P r i ce o f A n a r c hy ( P o A ) , = 0.0 , = 0.3 , = 1.0 , = 5.0 = P r i ce o f A n a r c hy ( P o A ) (b) N = 1000 . Fig. 2: The Price of Anarchy (PoA). The dotted lines representthe bound from Corollary 2. (cid:80) Nj =1 h ( i, j ) ≥ d i +2( N − − d i ) . Using this, for large enough τ when (cid:80) Ni =1 v i ∞ can be approximated by using truncationof Maclaurin seria [17, Lemma 1], the social cost in (5) islower bounded as J ≥ N + 2 γN ( N −
1) + ( α − γ ) L − τ N (cid:88) i =1 d i . (8) The following bound is due to Cioab˘a [22, Theorem 9], N (cid:88) i =1 d i ≤ N L + ( 1 d min − d max )( N − − LN ) , where the equality holds for regular graphs and the star graph.Based on this, d min ≥ , and d max ≤ N − , we obtain N (cid:88) i =1 d i ≤ N L + (1 − N − N − − LN )= N L + N − N ( N −
1) ( N ( N − − L ) . (9) Equality in (9) is achieved only for the star K ,N − , where d max = N − and d min = 1 , or for the complete graph K N In fact, the sum can be lower bounded [14, p. 10] by (cid:80) Ni =1 v i ∞ >N − (cid:80) Ni =1 11+( τ − d i , which is meaningful for τ > . (where L = N ( N − ). (The equality for other regular graphsis ruled out because of the inequality in (9).) Using (9) into(8) yields J ≥ N + 2 γN ( N − − N − τ + ( α − γ + 2( N − τN ( N −
1) ) L − N τL (10) Let us consider two regimes:1) If α − γ + N − τN ( N − ≥ , then the bound in (10) is anincreasing function in L , hence the optimal social costis achieved for L = N − . The bound in (10) is tightfor such L , because the bounds in (9) and (8) becomeequalities for K ,N − and any graph with a diameterat most two, respectively. Hence, J ≥ J ( K ,N − ) andequality is achieved only for the star graph K ,N − .2) If α − γ + N − τN ( N − < , then the bound in (10)increases for L < N (cid:113) τ (2 γ − α ) − N − N ( N − ) and decreases for L > N (cid:113) τ (2 γ − α ) − N − N ( N − ) . Hence, the optimal social costis achieved in one of two boundary cases: L = N − and L = (cid:0) N (cid:1) . For L = N − , similarly as in 1), weobtain that the only possibility is the star graph K ,N − ,while for L = (cid:0) N (cid:1) it is the complete graph K N . Finally, J ≥ min { J ( K ,N − ) , J ( K N ) } .It remains to compare J ( K ,N − ) and J ( K N ) : J ( K ,N − ) = N + α ( N −
1) + 2 γ ( N − − ( N − +1 τ ( N − and J ( K N ) = N + α N ( N − + γN ( N − − Nτ ( N − .Hence, J ( K N ) − J ( K ,N − ) = ( N − N − α − γ + 1 τ ( N −
1) ) . If α ≤ γ − τ ( N − , then J ( K N ) ≤ J ( K ,N − ) and theoptimal social cost is achieved for the complete graph K N . If α ≥ γ − τ ( N − , then J ( K N ) ≥ J ( K ,N − ) and the optimal social cost is achieved for the stargraph K ,N − . The last also covers case 1), because γ − τ ( N − < γ − N − τN ( N − .Now, for the optimal social cost, Theorem 4 follows. Theorem 4.
For sufficiently high τ , the optimal social cost isachieved for the star K ,N − if α ≥ γ − τ ( N − , and for thecomplete graph K N , otherwise. We proceed with characterization of the Nash Equilibria andthe Price of Anarchy for sufficiently high τ . In the VSPCgame, Nash Equilibria topologies can be complex, while thestar and the complete graph can appear as extreme cases: • The complete graph K N is a Nash Equilibrium, if andonly if α ≤ γ − τ ( N − . Since new links cannot be added,changing the strategy for a node i means deleting k of itslinks ( ≤ k ≤ N − ). The corresponding change wouldincrease the cost J i of i by k ( γ − α ) − τ ( N − − k ) + τ ( N − = k ( γ − α ) − kτ ( N − N − − k ) ≥ kτ ( N − (1 − N − − k ) ≥ . Hence, node i has no interest to deviatefrom its current strategy. On the other hand, if α > γ − τ ( N − and node i changes its strategy by cutting ( N − links (all except one - to keep its connectivity), the changein J i is equal to ( N − γ − α − τ ( N − ) < , whichwill reduce its cost. • The star graph K ,N − is a Nash Equilibrium, if and onlyif α ≥ γ − τ ( N − . The root node cannot delete a link,because this would make its cost infinity. If i is a leaf,for some k ≥ , changing its strategy means: (i) adding k links, then the hopcounts to these nodes are reducedfrom to , hence the contribution from the hopcounts ischanged by − kγ ; or (ii) deleting the link installed by him(if any) and installing ( k +1) links, where k +1 < N − .In (ii), the hopcount to the root node is increased from to , the hopcount to ( k + 1) links is decreased from 2to 1, and the hopcounts to the other (cid:0) ( N − − ( k + 1) (cid:1) nodes are increased from to . The change in the sumof hopcounts is: − ( k +1) γ +1 · γ + (cid:0) ( N − − ( k +1) (cid:1) γ = − kγ + (cid:0) ( N − − ( k + 1) (cid:1) γ ≥ − kγ , hence the changeof the hopcount is again at least − kγ . Thus, the changein J i is at least kα − kγ − τ ( k +1) + τ = k ( α − γ + τ ( k +1) ) ≥ k ( α − γ + τ ( N − ) ≥ . On the other hand,if α < γ − τ ( N − , the change in J i by adding ( N − links from one leaf to all the other leaves in K ,N − , is ( N − α − γ ) − τ ( N − + τ = ( N − α − γ + τ ( N − ) < , i.e. it is not a Nash Equilibrium.The above two points resolves the conditions for two specificgraphs, but they do not cover all the possibilities for theNash Equilibria and the Price of Anarchy, which may varyon different intervals and a case analysis, as provided in thefollowing, is required. We will consider the case α < γ − τ and the case α > γ − τ . Case α < γ − τ . Now, γ > τ . A Nash Equilibrium is achieved only forgraphs with a diameter at most - an argument used in the laterpoints (b) and (c). The proof is by contradiction. Let us assumenode i is at least hops away from another node. Clearly, d i ≤ ( N − and if i installs a link from i to j , the difference in J i is at least α − γ + τd i ( d i +1) ≤ − τ + τd i ( d i +1) < . Hence, i reduces its cost and the graph is not a Nash Equilibrium. Weconsider three sub-intervals (a), (b) and (c): (a) If α < γ − τ , adding a link from i will change J i by atleast α − γ + τd i ( d i +1) < − τ + τd i ( d i +1) ≤ . Therefore, thecomplete graph K N is the only Nash Equilibrium. Because, α < γ − τ ≤ γ − τ ( N − for N ≥ and, according toTheorem 4, it also has optimal social cost. Finally, PoA = PoS = 1 . (b) If γ − τ ≤ α ≤ γ − τ ( N − and we assume, bycontradiction, that there is a Nash Equilibrium different from K N , we have the following: • If there is a link in the graph, installed by node i suchthat its deletion increases the sum of hopcounts from i byonly , then J i is increased by: γ − α − τd i ( d i − > .On the other hand, adding a link would change J i to: α − γ + τd i ( d i +1) > . The last two inequalities imply, < α − γ + τd i ( d i +1) < − τd i ( d i − + τd i ( d i +1) = − τ ( d i − d i ( d i +1) < , which is a contradiction. Hence, there is no other Nash Equilibrium different from K N and PoA = PoS = 1 . • If deleting any of the links installed by i would increasethe sum of hopcounts by at least ; by link deletion, thedifference in J i is at least γ − α − τd i ( d i − and wehave γ − α − τd i ( d i − ≥ γ + τ ( N − − τd i ( d i − ≥ τ − τd i ( d i − + τ ( N − ≥ τ ( N − > . We proceedby considering the properties of the possible Nash Equi-libria in particular sub-intervals: − τ ( k − k ≤ α − γ < − τk ( k +1) for k ∈ { , , . . . , (cid:98) (cid:113) N − − (cid:99)} . By linkaddition, the difference in J i is α − γ + τd i ( d i +1) and anecessary condition for a Nash Equilibrium is d i < k .On the other hand, k ≤ (cid:113) N − − ≤ √ N − ,hence d i < √ N − . Therefore, we have less than √ N − nodes that are on a distance from a node i .Each of these nodes is directly connected by less than √ N − − nodes different from i . Hence, there lessthan √ N − √ N − √ N − −
1) = N − nodesthat are at most hops from i , hence at least one nodethat is more than hops away from i , a contradictionto the general claim (before (a))! Hence, K N is the onlyNash Equilibrium and PoA = PoS = 1 . (c) If γ − τ ( N − ≤ α < γ − τ , then K ,N − is a NashEquilibrium. Graphs that are of diameter at most are alsocandidates for a Nash Equilibrium.Because the diameter of the graph is not bigger than , (8)becomes an equality J = N + 2 γN ( N −
1) + ( α − γ ) L − τ (cid:80) Ni =1 1 d i for sufficiently large τ . Applying the condition of(c) leads to J ( worst NE ) < N + 2 γN ( N − − τ L − τ N (cid:88) i =1 d i = N + 2 γN ( N − − τ N (cid:88) i =1 ( d i d i ) ≤ N + 2 γN ( N − − N τ = N (1 + 2 γ ( N − − τ ) (11) due to the fact that d i + d i ≥ (equivalent to ( d i − d i − ≥ ). Equality holds (only) in the last line of (11) if d i = 1 or d i = 2 for all i (e.g., the ring C N or the path P N graphs),otherwise a strict inequality in the second part also holds.Finally, knowing that the optimal social cost is attained by thecomplete graph K N and the condition inequality condition in(c) for J ( K N ) : PoA = J ( worst NE ) J ( K N ) < γ ( N − − τ γ ( N − − τ − τ ( N − = (1+ γ ( N − − τ − τ ( N − ) − ( + τ − τ ( N − )1+ γ ( N − − τ − τ ( N − ≤ (1+ γ ( N − − τ − τ ( N − )1+ γ ( N − − τ − τ ( N − = for each N ≥ ; because + τ − τ ( N − > . This bound is approached, for instance,when α and γ are large and bigger than τ : K N is the socialoptimum and K ,N is the worst-case Nash Equilibrium andthe bounds in (11) and the inequality for PoA are closely ap-proached. If α < γ − τ ( N − N − , K N is a Nash Equilibriumand PoS = 1 , otherwise PoS > . Case α > γ − τ . We first consider the links, whose deletion leaves the graphconnected. For any node i , we focus on the links installed by i . Let l = ( i, j ) be one such link and the number of all nodes q that use l as a link for the shortest paths from j to q is z .According to Schoone et al. [23, Theorem 2.1., case k = 1 ],all the distances from i to the other nodes are increased by atmost d , where d is the diameter in the original graph. In aNash Equilibrium, dzγ − α − τd i ( d i − > for any possiblevalue of d i ≥ , i.e. we obtain dzγ − α − τ > . Hence, z > α + τ dγ and then the number of such links to node j isnot bigger than dγNα + τ . Taking into account all possible nodes,the number of links whose deletion does not disconnect thegraph is not bigger than dγN α + τ . On the other hand there areat most ( N − links such that a removal of any of thoselinks disconnects the graph. Indeed, a connected graph has aspanning tree T and a link removed from T disconnects thegraph, while a removal of a link that is not in T leaves thegraph connected. Therefore, L ≤ N − dγN α + τ (12) If two nodes i and j are a hops apart from each other, addinga link from i to j would reduce the hopcounts from i to allthe nodes in the “second half” along the previous path to j byat least half of their lengths, by a − , a − , . . . , for a evenor by a − , a − , . . . , for a odd. Hence, the total reductionin the sum of shortest paths from i is (cid:80) a i =1 (2 i −
1) = a or (cid:80) a − i =1 (2 i ) = a − for a even or odd, respectively. Assuminga Nash Equilibrium and i is a starting node on the diameter,considering the change in cost J i , the following inequalitywould hold for any d : α − d − γ + τd i ( d i +1) > . Using d i ≥ , and the absolute maximum for d being N − , wearrive at d < min { (cid:114) γ ( α + 12 τ ) , N } (13) Each node i has at least one neighbor and all the others are nomore than d hops apart, hence: (cid:80) Nj =1 h ( i, j ) ≤ N − d .Applying the arithmetic-harmonic mean inequality leads to τ (cid:80) Ni =1 1 d i ≥ τ N (cid:80) Ni =1 d i = N Lτ . We proceed by upper bound-ing J in (8), J ≤ N + αL + γN (1 + ( N − d ) − N Lτ (14) Applying (12) modifies (14), into J ≤ N + α ( N −
1) + γN + N (cid:0) ( N −
1) + 2 αNα + τ (cid:1) γd − N τ ( N − dγN α + τ ) (15) We distinguish two sub-cases, (a) and (b): (a) If α ≥ γ − τ ( N − , then the optimal social cost (anda Nash Equilibrium) is achieved for the star graph K ,N − , hence PoS=1. Now, using (15) for PoA, PoA ≤ N + αL + γN (1 + ( N − d ) − N Lτ N + α ( N −
1) + 2 γ ( N − − ( N − +1 τ ( N − (16) and applying (12), PoA ≤ N + α ( N −
1) + Nγ + Nγ ( N − αNα + τ ) d − N τ ( N − dγN α + 12 τ ) N + α ( N −
1) + 2 γ ( N − − ( N − +1 τ ( N − (17) • γ · d is not infinitesimally small. For sufficiently large N , after some algebraic transformation, division by N inboth the numerator and denominator of (17) and applying(12), PoA ≤ O (cid:0) ( 12 + αα + τ ) (cid:114) γ ( α + 12 τ ) (cid:1) (18) • γ · d is infinitesimally small. According to (12), L = O ( N ) . Now, (16) yields PoA ≤ O (cid:16) N + αL − N Lτ N + α ( N − − ( N − +1 τ ( N − (cid:17) (19) (b) If γ − τ ( N − ≥ α ≥ γ − τ , then the optimal socialcost is achieved for the complete graph K N , and using (15),PoA ≤ N + αL + γN (1+( N − d ) − N Lτ N + α N ( N − + γN ( N − − Nτ ( N − . Now, • γ · d is not infinitesimally small. Using the bound for L in(12), for sufficiently high enough N , (17) is transformedinto PoA ≤ O (cid:16) √ γ +4 γ ( α + τ ) α +2 γ (1+ αα + τ ) (cid:17) and we havea constant value for PoA. • γ · d is infinitesimally small. Then α is small and PoAhas a value close to .Based on these results, we present Theorem 5. Theorem 5.
For sufficiently high τ in the VSPC game, thePoA depends on the parameters α , γ and τ ,1) if α ≥ γ − τ ( N − , then PoS=1 and if • γd is not small, thenPoA ≤ O (cid:0) ( + αα + τ ) (cid:113) γ ( α + τ ) (cid:1) . • γ · d is small, then PoA is given by Corollary 1.2) if γ − τ ( N − ≥ α ≥ γ − τ and • γd is not small, thenPoA ≤ O (cid:16) √ γ +4 γ ( α + τ ) α +2 γ (1 + αα + τ ) (cid:17) . • γ · d (and γ ) is small, then α is also small and we havea constant value for PoA close to .3) if γ − τ > α ≥ γ − τ ( N − , then Nash Equilibria graphshave diameters at most two and PoS ≤ PoA < .4) if γ − τ ( N − > α , then K N is the only Nash Equilibriumand PoA = PoS = 1 . Theorem 5 and Corollaries 1 and 2 are compatible for small γ . Significantly larger compared to the coefficients α , γ and τ . , P r i ce o f A n a r c hy ( P o A ) virus spread (big = )virus spread (moderate = )virus spread (small = )non virus spread (a) Average Price of Anarchy (PoA) γ = 5 . , A v e r age nu m be r o f li n ks i n N a s h E qu ili b r i u m virus spread (big = )virus spread (moderate = )virus spread (small = )non virus spread (b) Average number of links γ = 5 . , A v e r a g e S ho r t e s t P a t h i n N a s h E qu ili b r i u m virus spread (big = )virus spread (moderate = )virus spread (small = )non virus spread (c) Average hopcount γ = 5 . , P r i ce o f A n a r c hy ( P o A ) virus spread (big = )virus spread (moderate = )virus spread (small = )non virus spread (d) Average Price of Anarchy (PoA) γ = 1 . , A v e r a g e nu m b e r o f li nk s i n N a s h E qu ili b r i u m virus spread (big = )virus spread (moderate = )virus spread (small = )non virus spread (e) Average number of links γ = 1 . , A v e r a g e S ho r t e s t P a t h i n N a s h E qu ili b r i u m virus spread (big = )virus spread (moderate = )virus spread (small = )non virus spread (f) Average hopcount γ = 1 . , P r i ce o f A n a r c hy ( P o A ) virus spread (big = )virus spread (moderate = )virus spread (small = )non virus spread (g) Average Price of Anarchy (PoA) γ = 0 . . , A v e r a g e nu m b e r o f li nk s i n N a s h E qu ili b r i u m virus spread (big = )virus spread (moderate = )virus spread (small = )non virus spread (h) Average number of links γ = 0 . . , A v e r a g e S ho r t e s t P a t h i n N a s h E qu ili b r i u m virus spread (big = )virus spread (moderate = )virus spread (small = )non virus spread (i) Average hopcount γ = 0 . . Fig. 3: Simulation results of the heuristic algorithm for the obtained networks in a Nash Equilibrium. The three regimes big,moderate and small τ are represented with values . , . and , respectively. The number of nodes is N = 10 . C. Computational aspects and simulations
Since, (i) for small τ below the epidemic threshold τ c , v i ∞ = 0 and (ii) for τ = ∞ , v i ∞ = 1 for all i ∈ { , , . . . , N } ,the problem of finding a best response in the VSPC gameincludes the best-response problem described in [12], which isNP-hard. We therefore use a best-response heuristic algorithm,as in [4]. The steps of the algorithm are the following:1) We start with an initial random graph G = G .2) Time t is slotted and the first time slot is t = 1 . 3) Each node takes only two actions at each time slot t .We fix the order of actions from node to node N . Thepossible actions for each node are: dropping a link (D);adding a link (A); or doing nothing (N).4) We denote by G it the graph at time t before the action ofnode i .5) Starting from node , each node i first computes themaximum reduction of its cost J i induced by droppinga link (D) from graph G it , or takes action (N) if no reduction could be realized. Taking the obtained graph,node i computes the maximum reduction of its cost J i induced by adding a link (A), or takes action (N) if noreduction could be realized.6) After the decision of node i at time t , the graph becomes G i +1 t . After the decision of node N , the algorithm movesto time t + 1 (i.e., to graph G t +1 ).7) An equilibrium is reached at time t when all the nodestake the action (N) or the algorithm stops after a certainnumber of iterations t max is reached.In Fig. 3, results are given for the Price of Anarchy (PoA),the average number of links and the average hopcount as afunction of installation cost α for different effective infectionrates τ (namely big, moderate and small τ with values . , . and , respectively that well represents the 3 regimes)and different weights (costs) for the hopcounts γ in a graphwith N = 10 nodes. We also display some typical outcomesof the algorithm for different values of α and τ and threedifferent values of γ in Fig. 4, Fig. 5 and Fig. 6 (see thevisualizations at the end of the paper, after the bibliography).For all the metrics shown in Fig. 3, there is an interestingbehavior for the curve with “no virus,” in the sense that itfollows the same shape as the curves where the virus is present,but is often shifted/delayed from them. This is due to the”enhancing” effect from the virus spread on the installationcost contribution.For small values of α , due to the resulting cheap installationcost, and for non-negligible performance values γ , the NashEquilibrium is a very dense graph, often the complete graph K N , for all τ (Fig. 4). This reflects case 4) in Theorem 5,although the interval for α , where K N is the only NashEquilibrium would shrink (and may vanish) for small γ (Fig. 5and Fig. 6). In the latter case, the corresponding PoAs inFig. 3g and Fig. 2 have comparable shapes and the obtainedtopologies are trees (see Fig. 3h), although not necessarily stargraphs (see Fig. 6).Because of the higher installation cost (higher α ) in intervals3) and 2) from Theorem 5, the Nash Equilibria topologiesare sparse: (i) in particular non-tree networks for γ = 5 (constant average number of links and sum of hopcounts) andmostly trees for γ equal to or . . Consequently, the PoAlinearly decreases with α on this interval. For the interval 1) inTheorem 5 (high installation cost), the PoA increases with α (and τ ), reaching a local maximum and then have a differentbehavior for larger α . Namely, it decreases towards for small γ , while it is unpredictable for higher γ (the left column,subfigures (a), (d) and (g), in Fig. 3). But most importantlythe effect of the epidemics part is noticeable and higher τ introduces inefficiency, which is reflected by a high PoA. Itis also important to note that for comparable α and γ , thealgorithm displays somewhat fluctuating behavior in terms ofPoA (middle row of Fig. 3) due to the heuristic nature of thealgorithm.Being able to detect the intervals with high PoA, as wehave done in this section, means that for those intervalssome coordination/incentives of and for the players is needed.Since the PoS is generally low, in the best case with NashEquilibrium a small amount of coordination likely suffices. For the intervals where the PoA is low, the selfish behavior of theplayers might still lead to (near)-optimal topologies withoutany coordination. IV. R ELATED WORK
Virus spread in networks has been thoroughly exploredduring the last decades [14], [15], [16], [17], [24]. Theseworks involve studies ranging from virus-spread propagation,the computation of the number of infected hosts [14] to theepidemic threshold [15] in various epidemic models on net-works. There is a large body of literature on game formation,that mostly minimizes a cost utility based on hopcount andthe cost for installing links [12], [3], [19], [4], [25], [26], [27].Fabrikant et al. [12] have studied the case, where a node’sutility is a weighted sum of the installed links and the sumof hopcounts from each node in an undirected graph. Thefollow up work by Albers et al. [19] resolved some openquestions from [12]. Chun et al. [4] have conducted extensivesimulations on the same type of game formation. A gameformation problem involving hopcounts and costs, applied toP2P networks has been considered by Moscibroda et al. [25].Meirom et al. [28], [29] have provided dynamic and dataanalyses (apart from their static analysis) in an NFG settingwith heterogenous players and robustness objectives. Nahir etal. [27] have considered similar NFG problems in directedgraphs. A coalition and bilateral agreements between playersin NFG and game-theory, in general, have been consideredin [30], [8], [31], [3]. In order to evaluate “the goodness” ofthe equilibria, the prices of anarchy and stability [26], [32]have been used.In this work, we have considered virus protection aspectstogether with cost and the length of shortest paths. In thissense, our work extends (with virus spread) and generalizesthe related work [12], [19], [27]. However, to the best ofour knowledge, network formation games concerning virusspread and protection both with or without the performanceaspects have not been considered in the NFG framework,although security games [33], [34], [35], [36], [37], [38]have been used in modeling the virus spread suppression andnetwork immunization. Performance aspects, represented bythe hopcounts are linearly independent from the resilienceto virus spread– the two metrics do not possess closed-formexpressions– making the NFG problem challenging, apart fromthe novelty. V. C
ONCLUSION
We have considered a novel network formation game (NFG),called the virus spread-performance-cost (VSPC) game, forcommunication networks in which the aspects link installationcosts, virus infection probability, and performance in termsof the number of hops needed to reach other nodes in thenetwork, all need to be balanced. We have characterized theNash Equilibria and the Price of Anarchy (PoA) for variouscases. In most of the cases, the PoA is not high, often close to , which implies that the decisions of non-cooperative playerswould lead, in a decentralized way, to an optimal topology. When the aspect of the shortest hopcounts is not important,we have found that only trees (but not all) could be NashEquilibria. In that case, surprisingly, a path graph is the worst-and the star graph is the best-case Nash Equilibrium for bigvirus infection rate τ , while it is the opposite for small τ . Forintermediate values of τ , other trees are optimal. The PoA isthe highest for values of τ just above the epidemic threshold.However, the PoA is generally small and close to , does notdepend on the number of players, and is inversely proportionalto τ and the installation cost α .When the hopcounts do matter, the Nash Equilibria mightbe formed by more complex topologies. The PoA highlydepends on τ , the installation links and hopcount costs α and γ , respectively, as shown by both theory and simulation.Although the PoA is small for most of the cases, for someintervals of those parameters, the PoA could be high. Hence,a central control and regulatory mechanism should be in placein such cases. Being able to detect those intervals, as we havedone, helps in the design of optimal, efficient, virus-free andcheap overlay, P2P or wireless networks by limiting the non-cooperative freedom of the hosts’ decisions.There are several possibilities for follow-up work, such as astudy on mixed Equilibria, player coalitions, inhomogeneouscosts, or time-varying networks.R EFERENCES[1] S. Trajanovski, F. A. Kuipers, Y. Hayel, E. Altman, and P. VanMieghem, “Designing virus-resistant networks: a game-formation ap-proach,” in
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