A game of hide and seek in networks
AA Game of Hide and Seek in Networks ∗ Francis Bloch , Bhaskar Dutta , and Marcin Dziubi´nski Universit´e Paris 1 and Paris School of Economics48 Boulevard Jourdan75014 Paris, France [email protected] University of Warwick and Ashoka UniversityCV4 7AL Coventry, UK [email protected] Institute of Informatics, University of WarsawBanacha 2, 02-097Warsaw, Poland [email protected]
October 2019.
Abstract
We propose and study a strategic model of hiding in a network, wherethe network designer chooses the links and his position in the network facingthe seeker who inspects and disrupts the network. We characterize optimalnetworks for the hider, as well as equilibrium hiding and seeking strategieson these networks. We show that optimal networks are either equivalentto cycles or variants of a core-periphery networks where every node in theperiphery is connected to a single node in the core. ∗ Bloch’s research was supported by Agence Nationale de la Recherche though grant ANR-18-CE26-0020-01. Dziubi´nski’s work was supported by Polish National Science Centre throughgrants 2014/13/B/ST6/01807 and 2018/29/B/ST6/00174. a r X i v : . [ ec on . T H ] J a n Introduction
This paper analyses network design problem of (say) the leader of a covert organi-sation who has to construct a network connecting members of her organisation aswell as choose her own (hiding) position in the network. She faces an adversarywho can “attack” one node in the network so as to catch the leader and disrupt thenetwork. Related issues have a very long standing. According to Greek mythology,Daedalus invented the Labyrinth in order to hide the monstrous Minotaur. Tun-nels and underground chambers in Medieval castles and fortresses were built to hidetreasures or prisoners. Underground fortifications were constructed in the XXthcentury to hide weapons and combatants. In modern days, criminals and terroristsbuild covert networks in order to hide leaders, money or secret instructions.When an object or a person is being hidden, it must also be accessible forthose who need it. The Minotaur cannot be sealed off in the Labyrinth, becauseevery nine years, he receives a tribute of seven young boys and seven young girlsfrom Athens. The medieval treasures and prisoners, the weapons and combatantsof military forts also need to be recovered and freely moved. Leaders of criminaland terrorist organizations, secret plans and money must also be able to freelyand efficiently move in the network. Hence, the design of networks to hide alwaysinvolves a trade-off between security (the inviolability of the hiding place) andconnectivity (the accessibility of hidden objects and persons). We show below thatour network design problem also exhibits this trade-off, and go on to characterisean optimal network design as a function of this trade-off between security andefficiency.We construct a zero-sum game with two players, a Hider and a Seeker. Inthe first stage of the game, the Hider designs a network which is observed bothby the Hider and Seeker. In the second stage of the game, the Hider and Seekersimultaneously choose a node in the network (where the Hider hides and the Seekerseeks or attacks). The Seeker is able to observe any node that is a neighbour ofthe node attacked by her. If the Hider hides in any of the nodes observed by theSeeker, the Seeker “captures” her and obtains a penalty from the Hider. If theSeeker does not find the Hider, she is still able to disrupt part of the network, byremoving the node that she attacks. The Hider then receives a payoff which is anincreasing function of the size of the component in which she hides. The payoff inthe zero-sum two-person game thus consists of two elements: (i) a benefit (to theSeeker) of capturing the hidden object or person and (ii) a benefit (to the Hider)of using a network connecting a given number of nodes.We characterise optimal network architectures chosen by the Hider. The op-timal network can only take one of two forms: either it contains a cycle (whereall nodes are connected in a circle) or is a special core-periphery network wherehalf of the nodes form an interconnected core, and the other half are leaves, eachconnected to a single node in the core. In addition, a subset of the nodes willremain isolated. The number of isolated nodes, and the choice between the circle See Book 8 in Ovid’s Metamorphosis. If the number of nodes in the core-periphery network is odd, the architecture is slightlydifferent, with three core nodes node connected to a periphery node. Therefore, if the objective of the Hider is primarily to avoiddisruption of the network, forming a cycle will be an optimal choice for the hider.Notice however that in a cycle, every agent has two neighbors, so the probability ofdiscovery of the hidden object must be at least equal to n . In order to reduce thisprobability of discovery, while keeping the network connected, one has to allowfor the possibility that some nodes only have degree one. In the core-peripherynetwork where half of the nodes are leaves connected to one node in the core, theprobability of discovery is reduced to the minimal value for a connected graph. Inequilibrium, the Hider chooses to hide in any of the peripheral nodes, whereas theSeeker seeks in any of the core nodes. This uniform hide and seek strategy resultsin a probability of discovery equal to n , lower than in the cycle, but induces alarger disruption, as the size of the remaining component after the Seeker fails tofind the object is equal to n − n −
1. In the main characterizationTheorem, we show that no other network performs better than the cycle or thecore-periphery network. The cycle is preferred when the Hider puts more weighton avoiding disruption and the core-periphery network is preferred when the Hiderputs more weight on avoiding discovery of the hidden object.While no real network has the exact architecture of a cycle or core-peripherynetwork, our results echo some observations on the trade-off between security andefficiency in physical networks of military fortifications and human networks ofcriminals and terrorists.Following the trench warfare of World War 1, the French army built the “Mag-inot line”, a system of underground fortifications to protect the border betweenGermany and France between 1929 and 1935. The design of the undergroundtunnels struck a balance between separating blocks (where combatants could hide)and allowing for easy communication of men and materials. Figure 1 provides anexample of the underground tunnels in three of the largest fortifications of theMaginot line: the Hackenberg, Mont des Welches and Fermont “gros ouvrages”.It shows that blocks are not directly connected to each other (echoing the factthat peripheral nodes are only connected to one node in the core and not to eachother nor to a central node), while central areas (where men sleep and weapons Notice however that adding links may not change the probability of capture, if the hider onlyhides in a subset of the nodes in the cycle. Ironically, the Maginot line proved useless during the German invasion of France in May 1940,as the German army simply by-passed the line of fortifications and entered France from Belgiumand Luxembourg.
Gros ouvrage of Hackenberg Gros ouvrage of Fermont Gros ouvrage of Mont des Welches
Figure 1: Three “Gros Ouvrages” of the Maginot LineMorselli et al. (2007) illustrate the trade-off between security and efficiencyusing data on terrorist networks (Krebs (2002)’s map of the 9/11WTC terroristcells) and criminal networks (a drug-trafficking network in Canada). They arguethat terrorist networks are more likely to have longer average distances and fewerconnections with no node assuming a central position, whereas criminal networksare more clustered and exhibit a core of nodes with high centrality. In additionthey note that support nodes (which are not direct perpetrators of criminal orterrorist activities) help connect distant nodes in terrorist networks but not incriminal networks, where each support agent is attached to a single agent in thecore. These two network architectures (long lines and core-periphery with clusters)can be related to the cycle and the core-periphery network we identify in ouranalysis. Figure 2 illustrates these network architectures, by reproducing the mapof the 9/11 WTC terrorist network (Krebs (2002)) as well as the maps of twodrug-trafficking mafia groups collected by Calderoni (2012).4 he 9/11WTC terrorist network
Mapping Networks of Terrorist Cells / Krebs
46 I was amazed at how sparse the network was and howdistant many of the hijackers on the same team werefrom each other. Many pairs of team members wherebeyond the horizon of observability (Friedkin, 1983)from each other – many on the same flight were morethan 2 steps away from each other. Keeping cell mem-bers distant from each other, and from other cells,minimizes damage to the network if a cell member iscaptured or otherwise compromised. Usama binLaden even described this strategy on his infamousvideo tape which was found in a hastily deserted housein Afghanistan. In the transcript (Department ofDefense, 2001) bin Laden mentions:
Those who were trained to fly didn’t know the others. One group of people did not know the other group.
The metrics for the network in Figure 2 are shownbelow and in Table 1. We see a very long mean pathlength, 4.75, for a network of less than 20 nodes.From this metric and bin Laden’s comments above wesee that covert networks trade efficiency for secrecy. no shortcuts with shortcutsGroup SizePotential TiesActual TiesDensity 19342 54 16 % 193426619%
Geodesics = without shortcuts = with shortcuts Yet, work has to be done, plans have to be executed.How does a covert network accomplish its goals?Through the judicious use of transitory short-cuts(Watts, 1999) in the network. Meetings are held thatconnect distant parts of the network to coordinatetasks and report progress. After the coordination is
The N’Drangheta network of cocainetrafficking (operationChalonero) The N’Drangheta network of cocainetrafficking (operationStupor Mundi)
Figure 2: Three examples of terrorist and criminal networks
The related literature spans a variety of disciplines, with the earlier literaturefocusing more on the aspect of hiding and seeking. Perhaps, the first paper was byvon Neumann(1953) who discusses a zero-sum game where H chooses a cell of anexogenously given matrix, while S simultaneously chooses a column or row in thematrix. S “captures” H if the cell chosen by H lies in the row or column chosen by S . A related paper is Fischer (1993), who too analyses a similar zero-sum game,where H and S simultaneously choose vertices of an exogenously given graph. H iscaught if S chooses the same node as him or a node connected to the node chosenby him. Interestingly, the value of this “hide and seek game” on a fixed arbitrarynetwork can been computed following Fisher (1991), using fractional graph theory . . Computer scientists have also contributed to this literature. with Waniek et al.(2017) and Waniek et al. (2018) studying a related, but different problem, of hidingin a network. They consider the leader of a terrorist or criminal organization, andask the following question: How can a set of edges be added to the network in orderto reduce the leader’s measure of centrality in order to avoid detection? Wanieket al. (2017) show that, both for degree and closeness centrality, the problem isNP-complete. However, they also propose a procedure to build a new networkfrom scratch around the leader (the “captain network”) which achieves low levels (See also Theorem 1.4.1 in Scheinerman and Ullman (1997))
5f degree and closeness centrality but high values of diffusion centrality, wherediffusion centrality is measured using the independent cascade and linear thresholddiffusion models. Waniek et al. (2018) extend the analysis to betweenness centralityand to the detection of communities (rather than individuals) in the network.Notice, however, that these models are not fully strategic since S does not bestrespond to H ’s strategy.Our paper is also related to a recent strand of the economics literature analyzingnetwork design and attack and defense on networks. Baccara and Bar-Isaac (2008)study network design by an adversary (a criminal organization) taking the detectionstrategy of the defender as fixed. They highlight differences between two forms ofdetection, one which depends on the cooperation between criminals and the otherwhich does not. In both situations, they characterize the optimal network archi-tecture of the criminal network, which either consists of isolated two-player cells(with independent detection) or an asymmetric structure with one agent serving asan information hub (with cooperation-based detection). Goyal and Vigier (2014)propose an alternative model of network design where the defender designs thenetwork and chooses the distribution of defense across nodes before the attackerchooses to attack. Nodes are captured according to a Tullock contest functiongiven the resources spent by the attacker and the defender. If a node is capturedby the attacker, contagion occurs and the attacker starts attacking neighboringnodes while the defender loses his defense resources. The main message of Goyaland Vigier (2014) is that the defendant optimally forms a star and concentratesall the defenses at the hub. Dziubi´nski and Goyal (2013) analyze a related model,where the defender designs the network and chooses defense resources before theattacker attacks. As opposed to Goyal and Vigier (2014), contagion does not oc-cur and the network structure only matters through the payoffs of the two-personzero-sum game between the defender and the attacker. The objective function ofthe defender is assumed to be increasing and convex in the size of components ofthe network, reflecting the fact that the defender wants to avoid disruption in thenetwork. The analysis shows that the designer will either form a star and protectthe hub, or not protect any node and choose to form a ( k + 1)-connected networkwhen the attacker has k units, so that the attacker will not be able to disruptthe network. In the same model, Dziubi´nski and Goyal (2017) study equilibriumstrategies of the defender and attacker for any arbitrary network structure whileCerdeiro et al. (2017) consider decentralized defense decisions by the different nodesin the network.The main difference between our paper and the recent literature on design,attack and defense stems from the fact that one particular node - the location cho-sen by the Hider- has a special significance in our model. Moreover, the captureof this node has a significant impact on payoffs. In other words, our model alsoincorporates the hiding-and-seeking aspect that is missing from the recent litera-ture. Another difference comes from the timing of the game. We suppose thatthe hider and seeker simultaneously choose the nodes in which to hide and thatthey inspect, resulting in equilibria in mixed strategies as in Colonel Blotto games,whereas Goyal and Vigier (2014) and Dziubi´nski and Goyal (2013) assume thatthe defender and attacker move sequentially, allowing for pure strategy equilibria.6ote also that nodes cannot be defended in our analysis. There are two players, a
Hider ( H ) and a Seeker ( S ). The hider H constructs anetwork among n nodes and chooses a location in the network. For example, theHider may be the leader of a covert terrorist or criminal organisation, which has n − H and S ismodeled as a two-stage process, which is described below.In the first stage, H chooses a network of interactions amongst the members ofthe organisation. Formally, H chooses a graph G = (cid:104) V, E (cid:105) where V is a set of n vertices, and E is a set of undirected edges E ⊆ (cid:0) V (cid:1) . A typical edge e ∈ E will bedenoted ij , where i, j ∈ V .Both players observe the chosen network at the beginning of the second stage.After observing the network G , players H and S simultaneously choose one nodeeach. The node chosen by the hider is his (hiding) position in the network. Thenode chosen by the seeker is the node she inspects (or attacks). Let k be the nodechosen by S , and N G ( k ) = { j ∈ V | kj ∈ E } . That is, N G ( k ) is the set of allneighbours of k in G . All nodes in { k } ∪ N G ( k ) can be observed by the seeker. Ifthe chosen position of H is in { k } ∪ N G ( k ), then H is captured by S . In addition,node k is removed from the network, irrespective of whether H is captured or not.The seeker uses his choice to capture the hider and to damage the network.Payoffs depend on whether or not the hider has been captured. If caught, thehider gets payoff − β , where β ≥ f : R ≥ → R ≥ of the size of his component in the residualnetwork. We assume f to be strictly increasing with f (0) = 0. An example offunction f in line with these assumptions is the identity function, f ( x ) = x for all x ∈ R ≥ . The game is assumed to be a zero-sum game, so that the payoff to theseeker is equal to minus the payoff of the hider.Formally, given a set of nodes U ⊆ V , let G ( U ) be the set of all undirectedgraphs over U and let G = (cid:83) U ⊆ V G ( U ) be the set of all undirected graphs thatcan be formed over V or any of its subsets. A strategy for the hider is a pair( G, h ) ∈ G ( V ) × V , where G is the graph and h is the hiding place chosen by H in G . As the seeker chooses his inspected node after observing the network, a strategyfor the seeker is a function s : G ( V ) → V .Before defining the payoffs we introduce some auxiliary definitions on networks.Given a set of nodes U ⊆ V and a graph G = (cid:104) U, E (cid:105) over U , a maximal set ofnodes C ⊆ U such that any two nodes i, j ∈ C are connected in G is a component of G . The set of all components of G is denoted by C ( G ). In addition, given Two nodes i, j ∈ U are connected in G = (cid:104) U, E (cid:105) if there exists a sequence of nodes i , . . . , i l ∈ U , let C i ( G ) be the component in G containing i . Given a set of nodes U ⊆ V ,a graph G = (cid:104) U, E (cid:105) over U , and a set of nodes U (cid:48) ⊆ U , let G [ U (cid:48) ] = (cid:104) U (cid:48) , E [ U (cid:48) ] (cid:105) with E [ U (cid:48) ] = { ij ∈ E : { i, j } ⊆ U (cid:48) } be the subgraph of G induced by U (cid:48) . Given anode k ∈ V let G − k = G [ U \ { k } ] be the residual network obtained from G byremoving k and all its links from G .Given the strategy profile (( G, h ) , s ), the payoff to the hider isΠ H ( G, h, s ) = (cid:26) − β if h ∈ { s ( G ) } ∪ N G ( s ( G )) f ( | C i ( G − s ( G )) | ) otherwise. (1)The payoff to the seeker is Π S (( G, h ) , s ) = − Π H (( G, h ) , s ).The cycle network and the core-periphery networks will be important in ouranalysis. The cycle network is the unique network where every node has exactlytwo neighbors.A core-periphery network over a set V = P ∪ C of n nodes is defined as follows.There are q ≥ (cid:100) n/ (cid:101) core nodes in set C = { c , . . . , c q } and m ≤ (cid:98) n/ (cid:99) periphery nodes in set P = { p , . . . , p m } . Nodes of the core form a connected graph, whileeach periphery node, p i with 1 ≤ i ≤ m , is connected to core node c i . Nodes of thecore which are not connected to a periphery node are called orphaned . Figure 3illustrates a core-periphery network with orphaned nodes.Figure 3: A core-periphery network over 39 nodes, with 15 periphery nodes and 9orphaned core nodes.A particular class of core-periphery networks, which we call maximal , playsa crucial role in our characterization. If n is even, a core-periphery network ismaximal if and only if it has n/ If n is odd, a core-periphery network is maximal if and onlyif it has ( n − / such that i = i , i n = j , and for all k ∈ { , . . . , l } , i k − i k ∈ E . A graph is 2-connected if and only if it does not get disconnected after removing a singlenode.
Our objective in this section is to provide optimal networks for the hider as well asto characterise the hiding and the seeking strategies on these networks. We show inour main result (Theorem 1) that these networks consist of a number of singletonnodes and a connected component which contains either a cycle or has a particular core periphery topology.Whether a cycle or a core-periphery topology is better for the hider depends onthe value of the following expression. T ( n, s ) = ( n − s − f ( n − s − − ( n − s − f ( n − s − . (2)We will show that the cycle topology is better if T ( n, s ) > β , while a core-peripherytopology is better when T ( n, s ) < β .The following lemma asserts that in an optimal network there cannot be acomponent containing just two or three nodes. The lemma will be used in theproof of the main theorem. Lemma 1.
Suppose G is an optimal network for H whose set of non-singletoncomponents is X . Then, each component C ∈ X contains at least nodes.Proof. Suppose the lemma is not true and some C ∈ X has exactly three nodes, C = { n , n , n } . Following standard arguments, C must have a non-empty in-tersection with the support of H ’s optimal hiding strategy as well as S ’s optimalseeking strategy, given G . (If not, the hider or the seeker would have profitabledeviations). Moreover, conditional on hiding in C , H is caught with probability ρ ,where ρ is the total probability with which S seeks in C . This is true because S can always search one node in C that has two neighbours, and hence observe allnodes in the component. 9et G (cid:48) be another network which coincides with G everywhere except that C is broken up into singleton components { n } , { n } , { n } . Moreover, suppose H ’shiding strategy coincides with that in G everywhere on V \ C , while H distributesthe earlier probability weight on C uniformly on the three nodes n , n , n . It isstraightforward to check that H ’s expected payoff in G (cid:48) is strictly higher than hisexpected payoff in inspecting a node in the component C , which must be equal tohis expected payoff of using a mixed strategy in G , contradicting optimality of G .A similar argument rules out an optimal network containing a component withonly two nodes. Remark 1.
An implication of this lemma is that the optimal network will either becompletely disconnected with n singletons or will contain at most n − singletons.This implication will be used throughout the proof of the theorem. At this stage, we describe the main result of the paper somewhat informally. Aformal statement appears towards the end of the section.We will construct an equilibrium that will have the following features. • The optimal network G will have a certain number of singleton nodes s (thatwill be determined) where s ≤ n − s = n . • If T ( n, s ) ≥ β and s (cid:54) = n , then G has a cycle component over n − s nodes. • If T ( n, s ) < β , n − s ≥
4, then G will have a maximal core periphery compo-nent over n − s nodes. • The hider mixes between hiding in the singleton nodes and in the connectedcomponent with probabilities that will be determined. When hiding in thesingleton nodes, he mixes uniformly across all these nodes. When hiding inthe connected component, he mixes uniformly across all the nodes when itis a cycle, mixes uniformly across the periphery nodes when it is a maximalcore-periphery network over even number of nodes, and mixes between hidingin periphery nodes, mixing uniformly across them, and the middle orphanednode, otherwise. • The seeker mixes between seeking in the singleton nodes and in the connectedcomponent. When seeking in the singleton nodes, he mixes uniformly acrossall these nodes. When seeking in the connected component, he mixes uni-formly across all the nodes when it is a cycle, mixes uniformly across thecore nodes when it is a maximal core-periphery network over even numberof nodes, and mixes between seeking in the neighbours of periphery nodes,mixing uniformly across them, and the middle orphaned node, otherwise.To get some intuition behind the result, notice that the hider faces a tradeoffbetween the cost of being caught and the value he gets in the residual network.Adding links in the network increases connectivity and hence secures a larger valueafter the the seeker’s action provided he is not caught. However, a larger numberof links also leads to higher exposure and a greater probability of being caught, asit increases the size of the neighborhoods of the nodes in which the hider can hide.10ixing the number of singleton nodes, s , the choice between a cycle and a core-periphery network is influenced by the change in f , as measured by the quantity T ( n, s ). The probability of being caught in a cycle of size n − s is 3 / ( n − s ), aseach node has exactly two neighbours, while only one node is lost from the cyclecomponent if not caught. The probability of being caught in a maximal core-periphery network (if n − s is even), on the other hand, is 2 / ( n − s ) since the hiderhides mixing uniformly across the periphery nodes; in the event of not being caught,two nodes are lost from the core periphery component since the seeker seeks mixinguniformly across the core nodes. If the change in f between n − s − n − s − T ( n, s ) > β then the marginal loss from an additionalnode being removed from a component is high, as compared to the penalty forbeing caught, and, therefore, a cycle is preferred over the core-periphery network.If the change in f is not sufficiently large, on the other hand, the marginal lossfrom an additional node being removed from a component is not sufficiently highand the hider prefers to opt for the safer, core-periphery, network.The proof of the theorem is long and we provide a brief description of thegeneral technique before giving the details.We start by constructing a feasible strategy of the seeker that, for each networkover the set of nodes V , provides a (mixed) seeking strategy on that network. Thisstrategy determines the payoffs the seeker can secure for each possible network over V . Since the game is zero-sum, minus these payoffs provide an upper bound onthe payoff the hider can get for each network. Next, for each s ∈ { , . . . , n − , n } ,we construct a network that is optimal for the hider across all possible networkswith exactly s singleton nodes. In the case of T ( n, s ) ≥ β , as well as in the case of s being even, these networks yield payoffs to the hider that meet the upper bounddetermined in the first part of the proof. In the case of T ( n, s ) < β and odd s , theupper bound from the first part of the proof is not exact. Therefore in this stepwe establish both the optimal networks for the hider and the exact upper boundon the hider’s payoff.We will use a series of lemmas to prove the theorem. We first introduce a parti-tion of the nodes into different sets that will play a crucial role in the constructionof a strategy for the seeker.Given a (possibly disconnected) network G over the set of nodes V , node i ∈ V is a singleton node if | N G ( i ) | = 0. The set of singleton nodes of G is denoted by S ( G ). Node i ∈ V is a leaf if | N G ( i ) | = 1. The set of leaves of G is denotedby L ( G ). Given node i ∈ V , let l i ( G ) = | N G ( i ) ∩ L ( G ) | denote the number ofleaf-neighbours of i .Let M ( G ) = { i ∈ V : l i ( G ) = 1 } be the set of nodes which are connected to exactly one leaf in G and let SL ( G ) = { i ∈ L ( G ) : N G ( i ) ∩ M ( G ) (cid:54) = ∅ } be the set of leaves connected to an element of M ( G ). Such leaves are called singleton leaves . Let R ( G ) = V \ ( S ( G ) ∪ SL ( G ) ∪ M ( G )) be the set of nodes in G which are neither a singleton, nor a singleton leaf, nor a neighbour of a singletonleaf. 11e now construct a strategy for the seeker which guarantees a given payoff forany network G . Take any network G over V and let s = | S ( G ) | and m = | M ( G ) | .Moreover, let GR = G [ R ( G )] be the subnetwork of G generated by the set of nodes R ( G ). In particular, when R ( G ) = ∅ , GR is the empty network with empty sets ofnodes and links. Let D ( GR ) be the set of nodes in R ( G ) that belong to two-elementsubsets of R ( G ).Consider a mixed strategy of player S , σ = ( σ , . . . , σ n ), with the followingprobabilities: σ = λ S σ S + (1 − λ S ) (cid:0) λ R σ R + (1 − λ R ) σ M (cid:1) (3)where λ R , λ S ∈ [0 , σ S i = (cid:40) s , if i ∈ S ( G ),0 , otherwise, σ M i = (cid:40) m , if i ∈ M ( G ),0 , otherwise, σ R i = l i ( GR )+1 n − s − m , if i ∈ R ( G ) \ ( L ( GR ), n − s − m , if i ∈ D ( GR ),0 , otherwise,We first show that these probabilities are well-defined. Lemma 2. σ is a feasible strategy for the seeker S .Proof. Clearly, σ S is a valid probability distribution as long as S ( G ) (cid:54) = ∅ , thatis s >
0. Similarly, σ M is a valid probability distribution as long as M ( G ) (cid:54) = ∅ ,that is m ≥
1. It is also easy to see that σ R is a valid probability distributionas long as R ( G ) (cid:54) = ∅ . To see this, notice that R ( G ) contains exactly n − s − m nodes and σ R can be obtained from a uniform distribution on R ( G ) by movingthe probability mass assigned to leaves in GR \ D ( GR ) to their neighbours. Lastly,notice that if S ( G ) (cid:54) = ∅ , then either all the non-singleton nodes in G have degree 1,in which case M ( G ) (cid:54) = ∅ , or there exists a node in G of degree 2 or more, in whichcase either M ( G ) (cid:54) = ∅ or R ( G ) (cid:54) = ∅ . Hence if S ( G ) (cid:54) = ∅ , then either σ M or σ R is a valid probability distribution. By these observations, σ is a valid probabilitydistribution as long as λ S = 1, if s = n , λ S = 0, if s = 0, λ R = 0, if R ( G ) = ∅ , and λ R = 1, if m = 0. So, the lemma is true.The idea behind the construction of strategy σ is as follows. With probability λ S , player S seeks in the set of singleton nodes, S ( G ), and with probability (1 − λ S )he seeks outside this set. Conditional on seeking outside S ( G ), with probability λ R player S seeks in the set of nodes R ( G ) and with probability (1 − λ R ) he seeksin the set SL ( G ) ∪ M ( G ). When seeking in S ( G ), S mixes uniformly across allthe singleton nodes. When seeking in SL ( G ) ∪ M ( G ), S mixes uniformly acrossall the nodes neighbouring a singleton leaf, that is all the nodes in M ( G ). Lastly,when seeking in the set of nodes R ( G ), S mixes using strategy σ R . In the nexttwo lemmas, we compute lower bounds on the probability of capture of the hiderin different regions of the network. 12 emma 3. The probability of capture of player H is at least (1 − λ S ) λ R / ( n − s − m ) , if H hides in R ( G ) \ ( S ( GR ) ∪ SL ( GR ) ∪ D ( GR )) .Proof. Take any node i ∈ R ( G ) \ ( S ( GR ) ∪ SL ( GR ) ∪ D ( GR )). Suppose, first, that i is not a leaf in GR , i.e. i ∈ R ( G ) \ L ( GR ). Then i has at least two neighboursin R ( G ) and the probability that seeker seeks at i or at one of i ’s neighbours is atleast (1 − λ S ) λ R / ( n − s − m ). Suppose, next, that i ∈ L ( GR ) \ ( SL ( GR ) ∪ D ( GR )).Then i has a neighbour j ∈ R ( G ) that has at least one more leaf neighbour in GR .Since σ j = (1 − λ S ) λ R / ( n − s − m ), the lemma is true. Lemma 4.
The probability of capture of player H is at least (1 − λ S ) λ R / ( n − s − m ) , if H hides in S ( GR ) ∪ SL ( GR ) ∪ D ( GR ) .Proof. In this case, i must have a neighbour, j , in M ( G ). For otherwise i would bea singleton node in H or a singleton leaf in H and so i would belong to S ( G ) ∪ M ( G )and not to R ( G ). Now,the probability of S putting a seeking resource in j is σ j = (1 − λ S )(1 − λ R ) (cid:18) m (cid:19) ≥ (1 − λ S ) min (cid:18) , m ( f ( n − s −
1) + β )3 m ( f ( n − s −
1) + β ) + ( n − s − m )( f ( n − s −
2) + β ) (cid:19)(cid:18) m (cid:19) = (1 − λ S ) (cid:18) f ( n − s −
1) + β )3 m ( f ( n − s −
1) + β ) + ( n − s − m )( f ( n − s −
2) + β ) (cid:19) > (1 − λ S ) (cid:18) f ( n − s −
2) + β )3 m ( f ( n − s −
1) + β ) + ( n − s − m )( f ( n − s −
2) + β ) (cid:19) = (1 − λ S ) λ R (cid:18) n − s − m (cid:19) . Thus i is caught with probability at least (1 − λ S ) λ R / ( n − s − m ).We now use these characterisations to compute lower bounds on the expectedpayoff of the seeker when the hider hides in different parts of the network. Lemma 5.
Conditional on H hiding in a node of R ( G ) and S using σ , the expectedpayoff of S is at least L R ( n, m, s ) = (1 − λ S ) (cid:18) λ R (cid:18) (cid:18) n − s − m (cid:19) β − (cid:18) − n − s − m (cid:19) f ( n − s − (cid:19) − (1 − λ R ) f ( n − s − (cid:19) − λ S f ( n − s ) (4) Proof.
Suppose that H hides in R ( G ). From lemmas 3 and 4, H is captured withprobability at least (1 − λ S ) λ R / ( n − s − m ) when S chooses σ . If not captured, onlyone node is removed when S searches in R ( G ). With probability (1 − λ S )((1 − λ R ), S searches in M ( G ) and removes two nodes. Finally, with probability λ S , S searchesin S ( G ), and does not catch H . Then, her payoff is at least − f ( n − s ) - this happensif G is connected over n − s nodes. 13imilarly, we compute a lower bound on the expected payoff of the seeker whenthe hider hides in M ( G ) or SL ( G ): Lemma 6.
Conditional on H hiding in a node of M ( G ) ∪ SL ( G ) , player S bychoosing σ obtains a payoff of at least L M ( n, m, s ) = (1 − λ S ) (cid:18) (1 − λ R ) (cid:18) (cid:18) m (cid:19) β − (cid:18) − m (cid:19) f ( n − s − (cid:19) − λ R f ( n − s − (cid:19) − λ S f ( n − s ) , Proof.
The probability of capture of H is at least (1 − λ S )(1 − λ R )1 /m . If H is notcaptured, S guarantees that the component of the hider has size at most n − s − − λ S )(1 − λ R ) when the attack is in M ( G ). Furthermore, atleast one node is removed with probability (1 − λ S ) λ R when the attack is in R ( G ).Finally, the component containing H has size at most n − s when the attack is in S ( G ), and this happens with probability λ S .We now set the value of λ R in order to equalize the probability of capture ofthe hider in different regions of the network, outside singleton nodes. To this end,we assume that there exist non-singleton nodes in G , S ( G ) (cid:54) = V ,. Let ρ = ( n − s − m )( f ( n − s −
2) + β )3 m ( f ( n − s −
1) + β ) + ( n − s − m )( f ( n − s −
2) + β )= 1 − m ( f ( n − s −
1) + β )3 m ( f ( n − s −
1) + β ) + ( n − s − m )( f ( n − s −
2) + β )and λ R = (cid:40) , if R ( G ) = ∅ , ρ, otherwise. (5)Clearly ρ ∈ [0 ,
1] and λ R ∈ [0 , λ R ensures that L R ( n, m, s ) = L M ( n, m, s ), for any s ∈ { , . . . , n − } . Hence the lower bound on the payoff ofplayer S in G when H hides outside singleton nodes is L ( n, m, s ) = L R ( n, m, s ) = L M ( n, m, s ) = (1 − λ S ) A ( n, m, s ) − λ S f ( n − s ) (6)where A ( n, m, s ) = βm − (cid:0) m − m (cid:1) f ( n − s − , if R ( G ) = ∅ , (cid:16) D ( n,s ) D ( n − ,s )3 D ( n,s ) − D ( n − ,s ) (cid:17) (cid:16) β − T ( n,s )) m (3 D ( n,s ) − D ( n − ,s ))+( n − s ) D ( n − ,s ) − (cid:17) + β, otherwisewith D ( n, s ) = f ( n − s −
1) + β T ( n, s ) = ( n − s − D ( n, s ) − ( n − s − D ( n − , s ) + β In particular, the derivation above is valid for the extreme cases of m = 0 and m = ( n − s ) / A ( n, m, s ) is strictly increasing in m if T ( n, s ) > β , isstrictly decreasing in m if T ( n, s ) < β , and is constant if T ( n, s ) = β .To complete the definition of strategy σ we compute the value of the probabilityof seeking in singleton nodes, λ S . Conditional on H hiding in a node of S ( G ),using any of the strategies σ defined above, player S obtains payoff of at least L S ( n, m, s ) = λ S B ( s ) − (1 − λ S ) f (1), where B ( s ) = (cid:18) s (cid:19) β − (cid:18) − s (cid:19) f (1) , regardless of the strategy of the hider, as the probability of capture is λ S /s and, inthe case of not capturing the hider, S gets payoff − f (1). Let λ S = , if s = n , A ( n,m,s )+ f (1) A ( n,m,s )+ B ( s )+ f (1)+ f ( n − s ) , if s (cid:54) = n and A ( n, m, s ) > − f (1),0 , otherwise.To see that λ S ∈ [0 , B ( s ) > − f (1) ≥ − f ( n − s ), for any β ≥ ≤ s ≤ n − s ∈ { , . . . , n − } :(i) if A ( n, m, s ) > − f (1), then L s ( n, m, s ) = L ( n, m, s ).(ii) if A ( n, m, s ) ≤ − f (1) then L s ( n, m, s ) ≥ L ( n, m, s ).So, finally, if s ≤ n −
4, the lower bound on the payoff of player S in G is givenby Q ( n, m, s ) = (1 − λ S ) A ( n, m, s ) − λ S f ( n − s ) , Of course, if s = n , σ mixes uniformly across the singletons with λ S = 1.To summarize, the lower bound on the payoff of S in G , secured by the strategy σ , is given by Q ( n, m, s ) = B ( n ) , s = n A ( n,m,s ) B ( s ) − f (1) f ( n − s ) A ( n,m,s )+ B ( s )+ f (1)+ f ( n − s ) , if s ≤ n − A ( n, m, s ) > − f (1) ,A ( n, m, s ) , otherwise , (7)Recall that A ( n, m, s ) is increasing in m when T ( n, s ) > β , decreasing in m when T ( n, s ) < β , and constant in m when T ( n, s ) = β . This, fact, together withClaim 1 in the appendix implies that when s ≤ n − Q ( n, m, s ) is decreasing in m when T ( n, s ) < β , increasing in m when T ( n, s ) > β , and is constant in m when T ( n, s ) = β . So for all s ∈ { , . . . , n − } , Q ( n, m, s ) is minimised at m = ( n − s ) / ,when T ( n, s ) < β , and is minimised at m = 0 , when T ( n, s ) > β .
15e now turn to the construction of networks that are optimal for the hider.Fix the number of singleton nodes, s ≤ n −
4. Define a new function ¯ Q ( n, s ) asfollows¯ Q ( n, s ) = Q ( n, , s ) , if 0 ≤ s ≤ n − T ( n, s ) ≥ β , Q ( n, ( n − s ) / , s ) , if 0 ≤ s ≤ n − T ( n, s ) < β and n − s is even, Q ( n, ( n − s − / , s ) , if 0 ≤ s ≤ n − T ( n, s ) < β and n − s is odd.Consider first the case where n − s is even. Lemma 7.
Suppose H builds a network with s singleton nodes such that n − s is even. Then, an optimal strategy for H provides H with payoff − ¯ Q ( n, s ) . If T ( n, s ) < β , G is optimal if the subnetwork over n − s nodes is a maximal core-periphery network. If T ( n, s ) > β , G is optimal if the subnetwork over n − s nodesis a cycle.Proof. Fix s such that n − s is even. Let¯ A ( n, s ) = (cid:40) A ( n, ( n − s ) / , s ) , if T ( n, s ) < β , A ( n, , s ) , if T ( n, s ) ≥ β .and let κ = (cid:40) B ( s )+ f (1)¯ A ( n,s )+ B ( s )+ f ( n − s )+ f (1) if ¯ A ( n, s ) > − f (1),1 , otherwise. (8)Let H choose a network G such that :(i) G has exactly s singletons.(ii) G is a maximal core periphery on n − s nodes if T ( n, s ) < β .(iii) G is a cycle on n − s nodes if T ( n, s ) ≥ β .Moreover, suppose that the hider hides in the component of size n − s withprobability κ , mixing uniformly on the periphery nodes in the case of the componentbeing a core-periphery network, and mixing uniformly over all its nodes in thecase of the component being a cycle. Also, she hides in the singleton nodes withprobability 1 − κ , mixing uniformly on them. By similar arguments to those usedfor λ S above, κ ∈ [0 ,
1] and so the strategy is valid. If the seeker seeks in thesingleton nodes, this yields payoff of at least κf ( n − s ) − (1 − κ ) B ( s ) to the hider.Similarly, if the seeker seeks in the core-periphery component, this yields payoff ofat least − κ ¯ A ( n, s ) + (1 − κ ) f (1) to the hider. With the value of κ , above, bothvalues are equal in the case of ¯ A ( n, s ) > − f (1), and the latter is greater, otherwise.Hence, the strategy guarantees a payoff − κ ¯ A ( n, s ) + (1 − κ ) f (1) to the hider.Note that − κ ¯ A ( n, s ) + (1 − κ ) f (1) = − ¯ Q ( n, s )Recall that we have shown that ¯ Q ( n, s ) is the minimal payoff the seeker canget on any network with exactly s singleton nodes. Since the game is zero-sum, − ¯ Q ( n, s ) is the maximal payoff the hider can get on any network with exactly s singleton nodes and hence the network constructed above as well as the hidingstrategy must be optimal for the hider. 16ext, consider the case of n − s being odd. Lemma 8.
Suppose that n − s is odd. Then, an optimal strategy for H gives hima payoff of − Q ( n, ( n − s − / , s ) . If T ( n, s ) < β , G is optimal if the subnetworkover n − s nodes is a maximal core-periphery network. If T ( n, s ) > β , G is optimalif it the subnetwork over n − s nodes is a cycle..Proof. Let ¯ A ( n, s ) = (cid:40) A ( n, ( n − s − / , s ) , if T ( n, s ) < β , A ( n, , s ) , if T ( n, s ) ≥ β .and let κ be defined as in (8). If T ( n, s ) ≥ β than choosing a cycle over n − s nodesand using the same hiding strategy as in the case of n − s being even, the hidersecures the highest possible payoff on a network with exactly s singleton nodes.Suppose that T ( n, s ) < β . Since ( n − s ) / Q ( n, s ). Recall that if T ( n, s ) < β then for any 0 ≤ s ≤ n − Q ( n, m, s ) is decreasing in m . We show below for any 0 ≤ s ≤ n −
4, the hider canattain payoff − Q ( n, ( n − s − / , s ), and that this is the maximal payoff he cansecure when n − s is odd.Suppose that the hider chooses a maximal core-periphery network (with threeorphaned nodes) over n − s nodes (c.f. Figure 5).Figure 5: A core-periphery network over 23 nodes with 3 orphaned nodes.Consider a strategy of the hider η = κ ( µ η M + (1 − µ ) η R ) + (1 − κ ) η S , where η M i = (cid:40) m , if i ∈ SL ( G ),0 , otherwise,(i.e. η M mixes uniformly on the periphery nodes of G ), η R i = (cid:40) , if i is the middle orphaned node in G ,0 , otherwise, 17 S i = (cid:40) s , if i ∈ S ( G ),0 , otherwise.(i.e. η S mixes uniformly on the singleton nodes of G ), and µ = ( n − s − f ( n − s −
2) + ( n − s − β ( n − s − f ( n − s −
1) + 2 f ( n − s −
2) + ( n − s − β . It is immediate to see that µ ∈ [0 ,
1] and so the hiding strategy is valid. If the seekerseeks in the orphaned nodes of the core-periphery component, this yields payoff ofat least κ ( µf ( n − s − − (1 − µ ) β ) + (1 − κ ) f (1) to the hider and, since the gameis zero-sum, of at most minus this value to the seeker. Similarly, if the seeker seeksin periphery nodes or their neighbours in the core-periphery component, this yieldspayoff of at least κ ( µ ( − β/ ( n − s −
3) + (1 − / ( n − s − f ( n − s − − µ ) f ( n − s − − κ ) f (1) to the hider and of at most minus this value to theseeker. With the value of µ , above, both these guarantees are equal.It is straightforward to verify that κ ( µf ( n − s − − (1 − µ ) β ) + (1 − κ ) f (1) = − κA ( n, ( n − s − / , s ) + (1 − κ ) f (1)= − Q ( n, ( n − s − / , s ) . Since Q ( n, ( n − s − / , s ) is a lower bound on the payoff that the seeker cansecure in a network with exactly s singleton nodes and at most ( n − s − / s singleton nodes and at most ( n − s − / s singleton nodes and ( n − s − / s ∈ S ∗ ( n ) singleton nodes,where S ∗ ( n ) = arg min s ∈{ ,...,n } ¯ Q ( n, s ) . Lemmas 7 and 8 have therefore proved the characterization result that we sum-marize in the following Theorem.
Theorem 1.
For any number of nodes, n ≥ , and any β ≥ there exists anequilibrium of the game, (( G, h ) , s ) such that • G has exactly s ∈ S ∗ ( n ) singleton nodes and either s ≤ n − or s = n . • If T ( n, s ) ≥ β and n − s ≥ then G has a cycle component over the remaining n − s nodes. • If T ( n, s ) < β , n − s ≥ then G has a maximal core-periphery componentover n − s nodes. The hider mixes between hiding in the singleton nodes and in the connectedcomponent. When hiding in the singleton nodes, he mixes uniformly acrossall these nodes. When hiding in the connected component, he mixes uniformlyacross all the nodes (when it is a cycle), mixes uniformly across the peripherynodes (when it is a maximal core-periphery network and n − s is even), andmixes between hiding in periphery nodes, mixing uniformly across them, andthe middle orphaned node (otherwise). • The seeker mixes between seeking in the singleton nodes and in the connectedcomponent. When seeking in the singleton nodes, he mixes uniformly acrossall these nodes. When seeking in the connected component, he mixes uni-formly across all the nodes (when it is a cycle), mixes uniformly across thecore nodes (when it is a maximal core-periphery network and n − s is even),and mixes between seeking in the neighbours of periphery nodes, mixing uni-formly across them, and the middle orphaned node (otherwise).Equilibrium payoff to the hider is − ¯ Q ( n, s ) . We have shown in the proof of Theorem 1, that the equilibrium payoff to theseeker in an optimal network with at least one singleton node is a convex combi-nation of B ( s ) which is greater than − f (1)) and − f (1) and so it is at least − f (1).Hence the payoff that the hider can secure in such a network is at most f (1). Thusif the payoff the seeker can secure in a connected component of size n , ¯ A ( n,
0) issmaller than − f (1), then the payoff the hider can secure in such a component is − ¯ A ( n, > f (1). If that inequality holds, it is optimal for the hider to choose aconnected network without singleton nodes.If, on the other hand, the cost of being caught, β , is sufficiently high then¯ A ( n, > − f (1) and the payoff the hider can secure in a connected network, − ¯ A ( n, s ≥ s = n singleton nodes.The characterization of equilibrium networks provided in Theorem 1 is notcomplete. This Theorem displays network architectures which achieve the high-est possible payoff for the hider, but does not show that these network topologiesare unique. As we prove below, if T ( n, s ) < β the connected component mustbe a maximal core-periphery network. So in this case we obtain complete char-acterization of equilibrium networks.If T ( n, s ) > β there exist network topologiesother than the cycle which are optimal. We establish necessary properties that theoptimal network topologies must possess. Theorem 2.
For any number of nodes, n ≥ , and any β ≥ , if (( G, h ) , s ) is anequilibrium of the game then • G has exactly s ∈ S ∗ ( n ) singleton nodes. • If T ( n, s ) < β , n − s ≥ , then G has a maximal core-periphery componentover n − s nodes. If T ( n, s ) > β , then G has a -connected component over n − s non-singletonnodes with at least (cid:100) ( n − s ) / (cid:101) nodes of degree and the hider never hides innodes of degree greater than in equilibrium.Proof. The fact that G must have exactly s ∈ S ∗ ( n ) singleton leaves is alreadyestablished in proof of Theorem 1. For the properties of the remaining part ofequilibrium network, we consider the cases of T ( n, s ) < β and T ( n, s ) > β sepa-rately.Suppose that T ( n, s ) < β . Suppose first that n − s is even. Since Q ( n, m, s )is decreasing in m and the maximum feasible value for m , when n − s is even, is( n − s ) / n − s non-singleton nodes in any optimal networkmust have ( n − s ) / Q ( n, s ) by mixing uniformly onthe neighbours of non-singleton leaves when seeking outside singleton nodes. This isbecause in the case of not capturing the hider, he will leave the subnetwork over n − s nodes disconnected with probability greater than 0. Hence the optimal subnetworkover n − s non-singleton nodes must be a maximal core-periphery network. Second,suppose that n − s is odd. As we showed in proof of Lemma 8, the optimal numberof singleton leaves in the subnetwork over n − s non-singleton nodes is ( n − s − / − ¯ Q ( n, s ) and so the network is not optimal.Therefore the neighbours of one of the orphaned nodes must be exactly the twoother orphaned nodes.Suppose next that T ( n, s ) > β . In this case, Q ( n, m, s ) is increasing in m andso the optimal network has no singleton leaves in the subnetwork over the n − s non-singleton nodes. Let U be the set of n − s non-singleton nodes in the networkand let F be the subnetwork over this set of nodes. As argued above, the seeker hasa seeking strategy that guarantees him a probability of capture at least 3 / ( n − s )in F . If F is not 2-connected, the seeker will leave the subnetwork disconnectedin the event of not capturing H . This gives strictly lower payoff to H than in thecycle. Hence F must be 2-connected. Hence all the nodes in F have degree at least2. Suppose that F has t < (cid:100) ( n − s ) / (cid:101) nodes of degree 2. Note that since F is 2-connected, only one node is removed if H is not captured. So, the expected payoff of H (and hence S ) only depends on the probability of capture. Consider any strategy η of H and let T be its support on U . Let σ (cid:48) T be a mixed strategy of the seeker20hat mixes uniformly on N F [ T ]. Let σ T = λ σ (cid:48) T + (1 − λ ) σ S be a strategy of theseeker that mixes uniformly on the singleton nodes with probability 1 − λ and uses σ (cid:48) T with probability λ , where λ is such that the lower bound on the expected payoffto the seeker when the hider hides in T is equal to the lower bound on the expectedpayoff to the seeker when the hider hides in singleton nodes. Notice that the lowerbound on the expected payoff to the seeker from using σ (cid:48) T when the hider hides in T is strictly higher than 3 / ( n − s ). For if T contains a node of degree at least 3 thenthe seeker captures the hider with probability strictly greater than 3 / | N F [ T ] | ≥ / ( n − s ), and if T does not contain a node of degree 3 then | N F [ T ] | ≤ | T | < n − s and the seeker captures the hider with probability 3 / | N F [ T ] | > / ( n − s ). Hencethere exists p T > ¯ Q ( n, s ) such that the expected payoff to the seeker from using σ T against any strategy of the hider, η , with support T on U is at least p T . Taking ε = min T ⊆ U ( ¯ Q ( n, s ) − p T ) shows that F cannot be optimal forv H . Notice also thatif the support of H ’s strategy in a network with 2-connected component F containsnodes of degree greater then 3 then strategy σ guarantees the seeker payoff strictlygreater than ¯ Q ( n, s ). Therefore, in equilibrium, the hider never hides in nodes ofdegree greater than 2 in the 2-connected component of an optimal network.We next provide examples of topologies of the connected component other thanthe cycle in equilibrium networks for the case of T ( n, s ) > β . Suppose that n − s =3 t where t ≥ U be the set of nodes of the component. Supposethat the nodes in U are connected, forming a cycle, and let T ⊆ U , | T | = t , bea subset of the nodes such that any two nodes in T are separated by two nodesfrom U \ T . Any network obtained from from the cycle by adding links betweenthe nodes in U \ T is optimal (an example is presented in Figure 6). Both playersmixing uniformly on U is an equilibrium on any such network.Figure 6: An optimal component for n − s = 12.Theorems 1 and 2 provide a characterization of optimal networks for the hiderin terms of the quantity T ( n, s ). As this expression is not transparent, we providesufficient conditions on the utility function f ( · ) which guarantee that the connectedcomponent of an optimal network is a maximal core-periphery network. Given graph G = (cid:104) V, E (cid:105) and a set of nodes U ⊆ V , set N G [ U ] = U ∪ { v ∈ V : uv ∈ E for some u ∈ U } is the closed neighbourhood of U in G . heorem 3. Suppose that either(i) f is concave, or(ii) f is convex and for all x ≥ f ( x + 1) < xx − f ( x ) Then, for all n ≥ , and any β ≥ , G is an equilibrium network if and only if G has s ∈ S ∗ ( n ) singleton nodes and a maximal core-periphery component over n − s nodes. In addition, if f is linear then S ∗ ( n ) = { , , n } .Proof. Notice that T ( n, s ) = ( n − s − f ( n − s − − f ( n − s − T ( n, s + 1) = ( n − s − f ( n − s − − f ( n − s − T ( n, s + 1) − T ( n, s ) = − ( n − s − f ( n − s − − ∆ f ( n − s − − ( n − s − f ( n − s − . where ∆ f ( x ) = f ( x + 1) − f ( x ) is the first-order (forward) difference of f at x and∆ f ( x ) = ∆ f ( x + 1) − ∆ f ( x ) is the second-order (forward) difference of f at x .Hence, if f is concave, then ∆ f ( n − s − ≤
0, and so T ( n, s + 1) − T ( n, s ) ≥ s ≤ n − T ( n, n −
4) = f (3) − f (2) which is negative if f is concave and strictlyincreasing. Thus for all n ≥ s ≤ n − T ( n, s ) < ≤ β .From Theorems 1 and 2, G is an equilibrium network if and only if its connectedcomponent is a maximal core-periphery network over n − s nodes.If f is convex then ∆ f ( n − s − ≥ T ( n, s + 1) − T ( n, s ) ≤
0, for all s ≤ n −
4. Thus T ( n, s ) is decreasing in s on [0 , n − n ≥ f ( x + 1) < x/ ( x − f ( x ) for all x ≥ Then T ( n,
0) = ( n − f ( n − − ( n − f ( n − < T ( n, s ) ≤ T ( n, < β, for all s ∈ [0 , n − . Again, by Theorems 1 and 2, G is an equilibrium network if and only if itsconnected component is a maximal core-periphery network over n − s nodes.Next, note that if n ≤
5, then Lemma 1 shows that s ∗ ≤
1. Suppose that f islinear and that n ≥
6. We show in the Appendix (Lemma 10) that if n ≥
6, then Q ( n, ( n − s ) / , s ) is minimised either at s = 0 or at s = 1 or at s = n . This showsthat s ∗ ∈ { , , n } and completes the proof of the theorem. Remark 2.
The theorem establishes a full characterization of equilibrium networkswhen f is concave or convex but growing slowly. An example of a family of strictly increasing convex functions that satisfy this property arethe functions f ( x ) = x γ / ( x + 1) γ − with γ > Conclusions
We proposed and studied a strategic model network design and hiding in the net-work facing a hostile authority that attempts to disrupt the network and capturethe hider. We characterized optimal networks for the hider as well as optimal hid-ing and seeking strategies in these networks. Our results suggests that the hiderchooses networks that allow him to be anonymous and peripheral in the network.We also developed a technique for solving such models in the setup of zero-sumgames.There are at least two avenues for future research. Firstly, different forms ofbenefits from the network could be considered. For example, the utility of the hidercould dependent not only on the size of his component but also on his distance tothe nodes in the component. Given our results, we conjecture that this wouldmake the core periphery components with better connected core more attractive.But answering this problem precisely requires formal analysis. Secondly, the seekercould be endowed with more than one seeking unit and the units could be usedeither simultaneously or sequentially. Our initial investigation suggests that solvingsuch an extension might be an ambitious task.23 eferences
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Nature Human Behaviour , 2:139–147, 2018.24 ppendixA Proofs
Lemma 9. If n − s is odd and T ( n, s ) < β , the hider obtains a higher expectedpayoff in a core-periphery network with ( n − s − / singleton leaves than in acore-periphery network with ( n − s − / singleton leaves.Proof. In a core-periphery network with ( n − s − / R ( G ) consist of exactly one node and this node is connected to at least two nodesin M ( G ). It cannot be connected to one node in M ( G ), because in this case itsneighbour would have two leaf-neighbours and could not be a member of M ( G ).Let ˜ σ = λ σ S + (1 − λ ) σ M , where σ M and σ S are the mixed strategies of theseeker, defined earlier in the proof, λ = (cid:40) X ( n,s )+ f (1) B ( s )+ X ( n,s )+ f (1)+ f ( n − s ) , if X ( n, s ) > − f (1),0 , otherwise,and X ( n, s ) = 2 βn − s − − (cid:18) − n − s − (cid:19) f ( n − s − . Using this strategy, with probability λ , S mixes uniformly on the nodes in M ( G )and with probability (1 − λ ), S mixes uniformly on the singleton nodes of G . Thepayoff to S conditional on H hiding in a singleton node is at least λB ( s ) − (1 − λ ) f (1)and the payoff to S conditional on H hiding outside singleton nodes is at least(1 − λ ) X ( n, s ) − λf ( n − s ). It is easy to verify that the value of λ is such that boththese payoffs are equal (in the case of X ( n, s ) > − f (1)) or the latter is higher, forany value of λ . Therefore the payoff to S from using ˜ σ against any strategy of H is at least Y ( n, s ) = (cid:40) B ( s ) X ( n,s ) − f (1) f ( n − s ) B ( s )+ X ( n,s )+ f (1)+ f ( n − s ) , if X ( n, s ) > − f (1), X ( n, s ) , otherwise,and so the upper bound on the payoff to the hider on any network with s singletonnodes and ( n − s − / − Y ( n, s ). To see that − Q ( n, ( n − s − /s, s ) > − Y ( n, s ) notice that X ( n, s ) − A ( n, ( n − s − / , s ) =2( f ( n − s − − f ( n − s − f ( n − s −
2) + β )( n − s − n − s − f ( n − s − n − s −
3) + 2 f ( n − s −
2) + β ( n − s − > X ( n, s ) > A ( n, ( n − s − / , s ).Next consider the following Claim: Claim 1.
The function ϕ ( Z ) = (cid:40) B ( s ) Z − f (1) f ( n − s ) Z + B ( s )+ f ( n − s )+ f (1) , if Z > − f (1) , Z, otherwise,is strictly increasing in Z . roof. Notice that ϕ ( − f (1)) = − f (1) when Z = − f (1). Moreover, ϕ is increasingin Z if Z < − f (1). Let Z > − f (1). Taking the derivative of ϕ with respect to Z we get ϕ (cid:48) ( Z ) = ( B ( s ) + f (1))( B ( s ) + f ( n − s ))( Z + B ( s ) + f ( n − s ) + f (1)) and it is immediate to see that ϕ (cid:48) ( Z ) > ϕ increases in Z when B ( s ) > − f (1)and B ( s ) ≥ − f ( n − s ). Notice that B ( s ) = ( β + f (1)) /s − f (1) > − f (1) forany β ≥ s >
0. Also f ( n − s ) ≥ f (1) for all s ∈ [0 , n − ϕ , above, ϕ ( Z ) increases when Z increases.Claim 1, together with X ( n, s ) > A ( n, ( n − s − / , s ), implies that Y ( n, s ) >Q ( n, ( n − s − / , s ), completing the proof of the Lemma. Lemma 10.
Let λ > and let f ( x ) = λx , for all x ∈ R ≥ . For any natural n ≥ , t ∈ { , } and any s ∈ { t + 1 , . . . , n } , Q ( n, ( n − s ) / , s ) > min( Q ( n, , n ) , Q ( n, ( n − t ) / , t )) Proof.
Let f ( x ) = λx , with λ >
0, and let (cid:101) β = β/λ . Let (cid:101) A ( n, s ) = A ( n, ( n − s ) / , s ) = λ (cid:32) (cid:32) (cid:101) β − n − s (cid:33) + 4 − ( n − s ) (cid:33) , for 0 ≤ s ≤ n − B ( s ) = λ (cid:32) (cid:101) β + 1 s − (cid:33) , and (cid:101) Q ( n, s ) = Q ( n, ( n − s ) / , s ) = (cid:101) A ( n, s ) , if (cid:101) A ( n, s ) ≤ − λ or s = 0, AB ( n, s ) , if 1 ≤ s ≤ n − (cid:101) A ( n, s ) > − λB ( n ) , otherwise,with AB ( n, s ) = (1 − ρ ) (cid:101) A ( n, s ) − ρλ ( n − s ) (9)where ρ solves (1 − ρ ) (cid:101) A ( n, s ) + ρλ ( s − n ) = ρB ( s ) − (1 − ρ ) λ. (10)Solving (10) we get ρ = s (2( (cid:101) β − − ( n − s )( n − s − s (2( (cid:101) β − − ( n − s )( n − s − n − s )( s ( n − s −
1) + (cid:101) β + 1) . Notice that 2( (cid:101) β − − ( n − s )( n − s − > (cid:101) A ( n, s ) > − λ , and( n − s )( s ( n − s −
1) + (cid:101) β + 1) > s ≤ n −
1. Thus if (cid:101) A ( n, s ) > − λ then ρ ∈ (0 , B ( s ) > − λ , for all s >
0, so if ρ ∈ (0 ,
1) then AB ( n, s ) > − λ . Moreover, (cid:101) A ( n, s ) is increasing in s on [0 , n −
2] and it is equal to β at s = n −
2. By the26bservations above, if (cid:101) A ( n, ≤ − λ then ¯ Q ( n,
0) = (cid:101) A ( n, < (cid:101) A ( n,
1) = (cid:101) Q ( n, ≤− λ < ¯ Q ( n, s ), for all s ∈ { , . . . , n } , and the claim of the lemma holds.For the remaining part of the proof suppose that (cid:101) A ( n, > − λ . This implies2( (cid:101) β − > ( n − n −
6) and, consequently, (cid:101) β > n ≥
6. We will show that (cid:101) Q ( n, s ) is either decreasing or first increasing and then decreasing on [0 , n − s = inf { s ∈ [0 , n −
2) : (cid:101) A ( n, s ) ≥ − λ } . Since (cid:101) A ( n, s ) is increasing in s andequal to β ≥ s = n − s is well defined. On [0 , ˜ s ], (cid:101) Q ( n, s ) = (cid:101) A ( n, s ) and, as we argued above, (cid:101) Q ( n, s ) is increasing. Consider theinterval [˜ s, n − B ( s ) > − λ ≥ − λ ( n − s ), for all 0 < s ≤ n − (cid:101) A ( n, ˜ s ) = − λ so AB ( n, ˜ s ) = − λ . In addition, AB ( n, n ) = B ( n ). We will showthat AB ( n, s ) is either decreasing or first increasing and then decreasing on [0 , n ].Inserting ρ into (9) we get AB ( n, s ) = ( n ( (cid:101) β + 1) − n ( s ( β/λ −
1) + 2( (cid:101) β + 1)) + s ( (cid:101) β −
3) + 6 s (cid:101) β − (cid:101) β + 1)( (cid:101) β − s (4 s − (cid:101) β + 5) − n (4 s + (cid:101) β + 1) . Taking the derivative of AB ( n, s ) with respect to s we get ∂AB ( n, s ) ∂s = ( (cid:101) β + 1) W ( s )( s (4 s − (cid:101) β + 5) − n (4 s + (cid:101) β + 1)) , where W ( s ) = Xs − Y s + (cid:32) n + (cid:101) β − (cid:33) Y − (cid:32) (cid:101) β − (cid:33) ( n − (cid:101) β + 1) , with X = 4 n − (cid:101) β −
15 and Y = 4 n + n ( (cid:101) β − − (cid:101) β − ∂AB/∂s is the same as the sign of W ( s ). Notice that W ( n ) = − (cid:101) β − n + (cid:101) β − <
0, as n ≥ (cid:101) β >
2. When
X >
0, then W ( s ) isan (cid:83) -shaped parabola and, since W ( n ) ≤
0, either W is negative or W is firstpositive and the negative on [0 , n ]. Thus in this case AB is either increasing or firstincreasing and then decreasing on [0 , n ]. Similar observation holds when X = 0.Suppose that X <
0. In this case W ( s ) is an (cid:84) -shaped parabola and it has amaximum at s ∗ = Y /X . Suppose that s ∗ ∈ (0 , n − X <
Y < n ≥ X < β > W ( s ∗ ) = − Y s ∗ + (cid:32) n + (cid:101) β − (cid:33) Y − (cid:32) (cid:101) β − (cid:33) ( n − (cid:101) β + 1)= (cid:32) n − s ∗ + (cid:101) β − (cid:33) Y − (cid:32) (cid:101) β − (cid:33) ( n − (cid:101) β + 1) < . Thus W is either negative or first positive then negative on [0 , n ], for any natural n ≥
5. Hence
ABQ is either decreasing or first increasing and then decreasing on[0 , n ], for any natural n ≥ (cid:101) A ( n, > − λ then AB ( n, s ) is either decreasing orfirst increasing and then decreasing in s on [0 , n ] and AB ( n, n ) = B ( n ). Hence, bythe definition of (cid:101) Q ( n, sn, s