Analysis of Heuristic Techniques for Controlling Contagion

Jason Tsai, Nicholas Weller, Milind Tambe

University of Southern California, Los Angeles, CA 90089{jasontts@usc.edu, nweller@dornsife.usc.edu, tambe@usc.edu

AbstractMany strategic actions carry a ‘contagious’ component be-yond the immediate locale of the effort itself. Viral marketingand peacekeeping operations have both been observed to havea spreading effect. In this work, we use counterinsurgency asour illustrative domain. Defined as the effort to block thespread of support for an insurgency, such operations lack themanpower to defend the entire population and must focus onthe opinions of a subset of local leaders. As past researchersof security resource allocation have done, we propose usinggame theory to develop such policies and model the intercon-nected network of leaders as a graph.Unlike this past work in security games, actions in these do-mains possess a probabilistic, non-local impact. To addressthis new class of security games, recent research has usednovel heuristic oracles in a double oracle formulation to gen-erate mixed strategies. However, these heuristic oracles wereevaluated only on runtime and quality scaling with the graphsize. Given the complexity of the problem, numerous otherproblem features and metrics must be considered to betterinform practical application of such techniques. Thus, thiswork provides a thorough experimental analysis includingvariations of the contagion probability average and standarddeviation. We extend the previous analysis to also examinethe size of the action set constructed in the algorithms and thefinal mixed strategies themselves. Our results indicate thatgame instances featuring smaller graphs and low contagionprobabilities converge slowly while games with larger graphsand medium contagion probabilities converge most quickly.

IntroductionNumerous competitive situations feature actions that pro-duce a contagious effect. Word-of-mouth advertising / viralmarketing has gained significant research interest as com-panies attempt to understand what makes some products orvideos go viral (Trusov, Bucklin, and Pauwels 2009). Peace-keeping operations have also been shown to have a positiveeffect on neighboring areas (Beardsley 2011).

Counterinsurgency (COIN) is the military effort to winthe support of a local population by activities such as pro-viding security, medical supplies, and building infrastruc-ture (U.S. Dept. of the Army and U.S. Marine Corps 2007).These efforts often reach the ears of neighboring populations

Copyright c� 2012, Association for the Advancement of ArtificialIntelligence (www.aaai.org). All rights reserved.

that may then form a positive opinion about the organiza-tion/country as well. Given that counterinsurgency is, bydefinition, a competitive effort, deciding which areas to visitrequires a principled, adversary-aware approach.

Our recent work proposes the use of game theory to de-velop these resource allocation policies (Tsai, Nguyen, andTambe 2012), taking after the deployed applications nowin use by numerous security agencies (Shieh et al. 2012;Pita et al. 2011). Specifically, we model the problem as agame with two players that takes place on a graph in whichnodes represent local leaders and links between them repre-sent the influence they have upon each other. Each player’stask is then to each select a subset of the local leaders tovisit and win over. These initial leaders will then probabilis-tically spread their opinion to neighboring leaders with theprobabilities dictated by the edge weights. The goal of theinsurgents is to maximize the support of the local populacewhile the counterinsurgents seek to mitigate the insurgents’support. We then solved the game for the Nash equilibriumstrategies for each player. However, given the limitations ofknown optimal and approximate solution methods, we de-veloped heuristic techniques that provide scalable solutionsthat showed minimal quality loss in our experiments.

Given the lack of theoretical guarantees for heuristic tech-niques, the purpose of this work is to thoroughly evaluatethe performance of the proposed heuristic techniques andidentify key trends that would inform their use in practicalapplication. Our prior work evaluated runtime and solutionquality as the graph size was increased. In this work, insteadof only varying the graph size, we also evaluate the impactof changing the contagion probability on edges. In partic-ular, our prior evaluation set all edge probabilities to 0.3,while here we draw the edge probabilities from a normaldistribution and vary the average and standard deviation ofthe distribution. Furthermore, for all of the above variations,we examine not only the runtime and solution quality, butalso the support set size of the strategies generated and thenumber of actions that are generated by the iterative doubleoracle approach.

Our investigation reveals that when the graph size in-creases, all of the algorithms converge in fewer iterationsbecause the defender oracles are unable to find strong block-ing actions. Furthermore, game instances with low conta-gion probabilities prove difficult because the attacker can

AAAI Technical Report FS-12-08 Social Networks and Social Contagion

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continuously find new actions that avoid the entire set ofcurrent defender actions until a sufficient number of actionshad been added to both players’ action sets, thereby requir-ing many iterations to converge. Finally, our experimentsvarying the contagion probability distribution’s standard de-viation revealed little impact of this parameter on the perfor-mance of the algorithms tested.

Related WorkRelated work generally falls into one of two categories:(1) game-theoretic resource allocation that does not featurenetworked domains with contagion and (2) influence max-imization which has only focused on best-response gen-eration and usually features only one ‘player’. In game-theoretic resource allocation, the most relevant work pro-vides algorithms for strategy generation in games that takeplace on a network (Basilico and Gatti 2011; Jain et al. 2011;Halvorson, Conitzer, and Parr 2009). However, these do-mains do not feature a diffusion effect which results in im-mense complexity as shown by work in influence maximiza-tion.

Specifically, influence maximization studies the problemof what subset of nodes to select in a graph to maximizethe spread of influence (Chen, Wang, and Wang 2010;Leskovec et al. 2007; Kempe, Kleinberg, and Tardos 2003).The spreading occurs via known propagation dynamics,translating into a one player version of the problem we areinterested in in this work. Influence blocking maximiza-tion is a very recent area that has looked at the two-playermaximize/mitigate game we consider, but has thus far onlyfocused on best-response generation instead of equilibriumstrategies (Budak, Agrawal, and Abbadi 2011; He et al.2011). Some research exists on competitive influence max-imization where all players try to maximize their own in-fluence instead of limiting others’ (Goyal and Kearns 2012;Borodin, Filmus, and Oren 2010; Kostka, Oswald, and Wat-tenhofer 2008; Bharathi, Kempe, and Salek 2007). Further-more, these works focus on complexity results instead ofequilibrium strategy generation. Hung et al. (2011) andHoward (2010) also address the COIN problem. However,Hung et al. (2011) assume a static adversary and Howard(2010) solves for local pure strategy equilibria. These arevery restrictive assumptions that do not reflect real con-straints of the adversary.

Example Domain and Problem DefinitionWhile many domains feature a competitive scenario on anetwork with diffusion effects, we will use counterinsur-gency as an example domain to illustrate our methods.Specifically, the counterinsurgency domain includes oneplayer that is attempting to win the support of the popula-tion to their cause while the second player attempts to thwartthe first player’s efforts (Hung, Kolitz, and Ozdaglar 2011;Howard 2011; Hung 2010). Each player is assumed to pos-sess the ability to win local leaders over to their side via ac-tivities such as providing security, medical supply, or help-ing to build infrastructure. Furthermore, each leader has aknown probability of influencing other leaders to support

their affiliated player. Prior work has noted the immensesize of the problem faced by military leaders engaging incounterinsurgency. Specifically, Hung (2010) notes that inAfghanistan, a single battalion with 5-30 teams is responsi-ble for an area consisting of approximately 300-500 leaders.

The two-player interaction is modeled as a influenceblocking maximization (IBM). An IBM takes place on anundirected graph G = (V,E). One player, the attacker,will attempt to maximize the number of nodes supporting hiscause while the second player, the defender will attempt tomitigate the attacker’s influence. Vertices in the graph repre-sent local leaders that each player can affect and edges rep-resent the influence of one local leader on another. Specifi-cally, each edge e = (n,m) has an associated probabilitype that leader n will influence leader m to side with n’schosen player. Only uninfluenced nodes are eligible to beinfluenced.

Each player must decide upon a strategy to choose an ac-tion, also referred to as a seed set, where an action is a sub-set of the nodes in the graph (Sa, Sd ✓ V ). The size ofthe subset, (|Sa| = ra, |Sd| = rd), is given and modelsthe resource constraint of the player. Nodes in Sa supportthe attacker and nodes in Sd support the defender, exceptnodes in Sa \ Sd which have a 50% chance of supportingeach player. The influence then propagates synchronously,where at time step t0 only the initial nodes have been influ-enced and at t1 each edge incident to nodes in Sa [ Sd is‘activated’ probabilistically. Uninfluenced nodes incident toactivated edges become supporters of the influencing node’splayer. If a single uninfluenced node is incident to activatededges from both player’s nodes, the node has a 50% chanceof being influenced by each player in that time step. Propa-gation continues until no new nodes are influenced.

For a given pair of actions, the attacker’s payoff in an IBMis equal to the expected number of nodes that will supportthe attacker and the defender’s payoff is exactly the oppositeof the attacker’s. The natural solution concept for such azero-sum game is the Nash equilibrium mixed strategy overtheir pure strategies (subsets of nodes of size ra or rd). Theresulting mixed strategy can then be sampled each time thecounterinsurgency team is deployed. Our propagation modelimplicitly assumes that the opinions of local leaders resetsbetween deployments to reflect the difficulty of maintaininglocal support.

Double Oracle ApproachTraditionally, zero-sum games such as the one we have mod-eled are solved with a Maximin linear program. In our do-main, however, the action spaces for each player are toolarge to store in memory for anything but trivial problemsizes, necessitating an alternate approach. The one elected inTsai et. al (2012) involves an iterative approach that buildsthe action sets incrementally and is known as a double oraclealgorithm.

The double oracle algorithm features a Maximin linearprogram at the core which solves for the equilibrium strate-gies for each player given the current action set. It also pos-sesses two oracles, one for the attacker and one for the de-fender, that each produce a best-response action to the cur-

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rent opponent strategy. At each iteration, then, the Maximinlinear program solves for the current equilibrium strategiesand passes this as input to each of the oracles that each returnwith a new action to add. These new actions are added intothe current action set for each player and the Maximin linearprogram is run again. This repeats until convergence, whichoccurs when neither oracle is able to find a best-responsethat is superior to the equilibrium strategy found by the Max-imin linear program. The algorithm is outlined formally inAlgorithm 1.

Algorithm 1 DOUBLE ORACLE ALGORITHM1: Initialize D with random defender allocations.2: Initialize A with random attacker allocations.3: repeat4: (⇢d, ⇢a) = MaximinLP(D,A)5: D = D [ {DefenderOracle(⇢a)}6: A = A [ {AttackerOracle(⇢d)}7: until convergence8: return (⇢d, ⇢a)

The double oracle algorithm, when using two optimalbest-response oracles, has been shown to converge to theMaximin Nash equilibrium (McMahan, Gordon, and Blum2003). However, with heuristic oracles, the algorithm be-comes a best-response dynamics method that no longer pro-vides runtime or quality guarantees. Thus, the purpose ofthis work is to evaluate the performance of the proposedheuristic techniques under a wide range of parameters andidentify key trends that would inform their use in practicalapplication.

LSMI OracleIn studying contagion, a measure of payoff values must bespecified. Here we adopt the commonly used metric in influ-ence maximization which is the expected number of nodesthat have been influenced by each player. In our particularzero-sum game, the expected number of nodes that will beinfluenced by the attacker is the relevant measure of reward.Under the independent cascade model of propagation thatwe have used in this work, it has been shown that even de-termining the expected influence of one player is #P-Hard(Chen, Wang, and Wang 2010), necessitating approximateor heuristic techniques for larger problem sizes. A com-monly proposed method involves Monte Carlo simulationsof the propagation process for, typically, 10,000-20,000 tri-als to reliably estimate the expected influence. Theoretically,the Monte Carlo process can guarantee an arbitrarily smallerror bound with sufficient trials (Kempe, Kleinberg, andTardos 2003). To find a best-response, as each of the or-acles must do, the Monte Carlo technique can be used todetermine the payoff of all possible actions and the best canbe chosen. However, this is prohibitively slow, so an ap-proximate best-response technique was used by Budak et.al (2011) that continues to use the Monte Carlo estimationtechnique.

The LSMI oracle implements the best-response tech-nique used by Budak et. al (2011), but uses a heuristic esti-mate of the actual payoff of a given pair of actions to replace

the Monte Carlo estimation technique. The algorithm forthe heuristic estimate is given in Algorithm 2 and is basedon the idea of ignoring low probability influences. Note thatbecause L-Eval(·) is a heuristic estimation of influence prop-agation instead of an ✏-approximation as the Monte Carloscheme is, the algorithm no longer guarantees an approxi-mation bound on the best-response generated.

Algorithm 2 L-Eval(V, Sa, Sd)1: InfV alue = 02: for v 2 (V � Sa � Sd) do3: N = GetVerticesWithin✓(v) \ (Sa [ Sd)4: /* Prioritize sources by lowest hop-distance to v*/5: S =makePriorityQueue(N)6: pa = 0, pd = 07: while S 6= ; do8: s = S.poll()9: if (s 2 Sa) then

10: pa = pa + (1� pa � pd)· Prob(s, v), pd = pd11: else /* s must be in Sd */12: pd = pd + (1� pa � pd)· Prob(s, v), pa = pa13: end if14: end while15: InfV alue = InfV alue+ pa16: end for17: return InfV alue

In L-Eval(·), V is the set of n’s local nodes and Sa/Sd arethe attacker/defender source sets. Due to the addition of n,we must recalculate the expected influence of each v 2 V .First, we determine all the nearby nodes that impact a givenv by calling GetVerticesWithin✓(v). Since only sources ex-ert influence, we intersect this set with the set of all sourcesand compile them into a priority queue ordered from lowesthop-distance to greatest. pa and pd represent the probabilitythat the attacker/defender successfully influences the givennode. From the nearest source, we aggregate the conditionalprobabilities in order. If the next nearest source is an attackersource, then pa is increased by the probability that the newsource succeeds, conditional on the failure of all closer de-fender and attacker sources. The probability that all closersources failed is exactly (1 - pa + pd). If the next nearestsource is a defender source, then a similar update is per-formed. The algorithm iterates through all impacted nodesand returns the total expected influence. For readers inter-ested in how the heuristic is incorporated into the algorithmfrom Budak et. al (2011), more detail is given in Tsai et. al(2012).

PAGERANK OraclePageRank is a popular algorithm to rank webpages (Brinand Page 1998), which we adapted due to its frequent use ininfluence maximization as a benchmark heuristic. The un-derlying idea is to give each node a rating that captures thepower it has for spreading influence that is based on its con-nectivity. For the purposes of describing PageRank, we willrefer to directed edges eu,v and ev,u for every undirectededge between u and v. For each edge eu,v , set a weightwu,v = pe/pv where pv =

Pe pe over all edges incident

to v. The rating or ‘rank’ of a node u, ⌧u =P

v wu,v · ⌧v

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Figure 1: Preliminary test, r = 10, avg. = 0.3, s.d. = 0.1

for all non-source nodes v adjacent to u. The exclusion ofsource nodes is performed because u cannot spread its influ-ence through a source node.

For the PAGERANK oracle, since the defender’s goal is tominimize the attacker’s influence, the defender oracle willfocus on nodes incident to attacker sources Na = {n|n 2V ^9en,m,m 2 Sa}. Specifically, ordering the nodes of Naby decreasing rank value, the top rd nodes will be chosen asthe best response. In the attacker’s oracle phase, the attackerwill simply choose the nodes with the highest ranks. Unlikethe LSMI oracle, the PAGERANK oracle ignores payoff es-timation entirely and generates a heuristic best-response tothe current opponent strategy based purely on graph proper-ties instead. While this results in much faster runtimes, thequality of solutions generated tend to be lower than thosefound by LSMI oracles.

ExperimentsThree types of variations were explored in this work. First,we varied the size of the graph and kept all other parametersconstant. Second, we varied the average contagion proba-bility in the graphs at three separate graph sizes. Finally, wevaried the standard deviation of the contagion probabilityin the graphs and again tested these at three separate graphsizes. All experiments featured a randomly generated scale-free graph, 10 resources per player (Sd, Sa = 10), and con-tagion probabilities on edges that were drawn from a nor-mal distribution. Scale-free graphs were chosen due to theirwidespread use as proxies for general social networks andwere generated according to the principle of ‘preferentialattachment’ as introduced by Barabasi and Albert (1999).Our particular implementation adds edges between exist-ing vertices and newly added vertices with a probability ofp = (deg(v) + 1) / (|E|+ |V |)1. 100 trials were run for everydata point shown.

Figure 1 shows a preliminary test that was conducted toprovide a benchmark for the quality results. It shows thereward for the defender when each of the four algorithmsis used as well as when no defender is present as well forgraphs of size 80, 160, and 240 and with the average con-

1http://jung.sourceforge.net/doc/api/edu/uci/ics/jung/algorithms/generators/random/BarabasiAlbertGenerator.html

(a) Runtime (b) Quality

Figure 2: Scale-up results, r = 10, avg. = 0.3, s.d. = 0.1

tagion probability set to 0.3, 0.5, and 0.7. As was done inprevious work, the reward reported is the reward achiev-able by an adversary that best-responds to our algorithm’sgenerated defender strategy by calculating the approximatebest-response via the algorithm proposed by Budak et. al(2011). The solution is guaranteed to be within 1e of optimal,making it a more consistent basis for comparison than anyheuristic best-response. This reward was chosen becausedetermining the optimal best-response for an attacker re-quired far more than 20 minutes for each best-response. Asmentioned, the graph sizes tested were limited to 260 nodesbecause for larger graphs even calculating the approximatebest-response outlined above begins to take longer than 20minutes as well.

As can be seen, all of the algorithms provide at least a30-40% improvement in reward obtained as opposed to hav-ing no defender present across all of the cases tested. Sincethis was intended as a preliminary justification for the algo-rithms, we will provide more in-depth analysis of the solu-tion quality of the algorithms in the following subsections.

Graph size scale-upThe first set of experiments explored the impact of scalingup the size of the graph alone. Specifically, the four al-gorithms (all combinations of the LSMI and PAGERANKoracles) were run on randomly generated scale-free graphswith 80-260 nodes in increments of 20, with 10 resourcesand contagion probabilities drawn from a normal distribu-tion N (0.3, 0.1). Graph sizes were limited to 260 nodesbecause the adversary best-response technique used to de-termine the defender’s reward became too cumbersome forlarger graphs.

Figure 2a shows the impact on runtime as the graph sizeis scaled up. As can be seen, the solution technique thatfeatures two LSMI oracles (DOLL) requires the longest runtime at 40-50 seconds for all of the game sizes tested. Inter-estingly, there did not appear to be a consistent increase inruntime as was observed in the other 3 algorithms (each ofwhich had at least one PAGERANK oracle). The other 3 al-gorithms were much faster, all requiring less than 30 secondswith a consistent trend as the graph size increases. DOPL re-quires more time than DOLP because of the fact that the de-fender PAGERANK oracle explicitly adapts to the attacker’sstrategy (only uses nodes adjacent to attacker nodes), whilethe attacker PAGERANK oracle does not. Previous work

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(a) Action set size (b) Support set size

Figure 3: Scale-up results, r = 10, avg. = 0.3, s.d. = 0.1

explored scaling to larger graphs with more resources, butsince this is not the focus of our work, we refer the inter-ested reader to Tsai et. al (2012).

Figure 2b shows the impact on solution quality as thegraph size is scaled up. Unsurprisingly, as the size of thegraph increases, it becomes increasingly difficult for the de-fender to block the adversary’s influence spread and the de-fender receives a correspondingly lower reward. As wasseen in previous work, we also observe a large differencebetween algorithms that use a LSMI oracle for the defenderas opposed to a PAGERANK oracle for the defender, with thelatter providing much lower rewards. This is expected, dueto the higher sophistication of the LSMI defender oracle aswas noted in previous work.

Figure 3a shows the final number of actions in the de-fender’s action set as the size of the graph is increased. Theaction set is defined as the number of actions available to thedefender in the CoreLP phase of the double oracle algorithmand is exactly the number of new best-responses that havebeen found by the defender oracle. In the worst case, thiswould include all possible actions in the game, but as canbe seen is generally far smaller, making the problem muchmore tractable. The attacker’s action set size was always ex-tremely similar if not identical to the defender’s action setsize.

Figure 3b shows a similar metric and features the numberof actions in the support set of the final defender strategy.The support set is the set of actions that have non-zero prob-ability in the final mixed strategy. Again, the final attackersupport set size was always extremely similar if not identicalto the defender’s.

As can be seen, both the action set and the support setsizes are much larger with the DOLL algorithm than for anyof the other algorithms. This is due to the sophistication ofthe LSMI oracles as opposed to the PAGERANK oracle. ThePAGERANK oracles converge extremely quickly to a smallset of actions and often do not generate new actions in re-sponse to new adversary strategies. This is especially truefor the PAGERANK attacker oracle, since the defender ora-cle actually chooses nodes directly adjacent to the attacker.Thus, even when only one PAGERANK oracle is used, thealgorithm overall converges quickly. The DOLL algorithmis iterating many more times than algorithms featuring thePAGERANK oracle, leading to the previous runtime resultwith DOLL being far slower than the other algorithms.

(a) Runtime (b) Quality

Figure 4: Contagion probability average results, s.d. = 0.1

Furthermore, the trends seen in both Figure 3a and b showthe size of the final action set and support set decreasing asthe graph size is increased. This is due to the fact that asthe graph grows larger, very few actions are useful for thedefender to use to defend against the spread of the attacker’sinfluence. For the attacker, randomization becomes less es-sential for the same reason. Thus, both players converge toa very small set of actions for the final mixed strategy.

Contagion probability: AverageTo explore the impact of changing the contagion probabil-ities on the four algorithms, we tested three different con-tagion probability averages for three separate graph sizes.Specifically, we ran all four algorithms with the contagionprobabilities drawn from normal distributions N (0.3, 0.1),N (0.5, 0.1), and N (0.7, 0.1). The graph sizes tested were80, 160, and 240 node random scale-free graphs with 10 re-sources allowed per player. We measured the same 4 metricsas in the previous section: runtime, solution quality, actionset size, and support set size.

Figure 4a shows the results pertaining to runtime. Thex-axis is divided into three sets of three bars each. Eachset represents one setting for the contagion probability av-erage (0.3, 0.5, 0.7) while each bar represents the runtimeresult for one algorithm. At averages of 0.5 and 0.7, con-sistent trends can be seen, with larger graphs taking longerand higher probabilities leading to longer runtimes for al-gorithms with LSMI oracles. This is because LSMI ora-cles speed up heuristic estimation by calculating only highprobability influences, but when contagion probabilities arehigher, this leads to many more nodes that must be processedby the algorithm.

For the case of 0.3, however, the trend is not consistentfor the DOLL algorithm. Experiments suggest that with lowcontagion probabilities, two LSMI oracles continually findnew best-responses to each other’s strategies. This occursbecause at low contagion probabilities, different parts of thegraph interact minimally and the attacker is able to move to‘new’ nodes and entirely avoid the defender, resulting in acat-and-mouse game that requires many more iterations toconverge than when a PAGERANK oracle is used.

Figure 4b shows the reward for the defender using thesame approximate best-response technique described previ-ously. Unsurprisingly, larger graphs lead to lower rewardfor the defender because it is harder to defend. Higher con-

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(a) Action set size (b) Support set size

Figure 5: Contagion probability average results, s.d. = 0.1

Figure 6: Contagion probability s.d. results, avg. = 0.3

tagion probabilities also result in lower defender rewards forthe same reason.

As we noticed in the scale-up experiments, larger graphslead to fewer actions in the action set as well as the finalsupport set, as shown in Figures 5a and b. As mentioned, atthe lowest contagion probability tested (0.3), the action andsupport set sizes are very large for DOLL, causing very highruntimes due to the many iterations required to generate theobserved action sets.

Contagion probability: Standard deviationNext we tested variations of the standard deviation of thenormal distribution that the contagion probabilities on edgesare drawn from. Specifically, we ran all four algorithms withthe contagion probabilities drawn from normal distributionsN (0.3, 0.0), N (0.3, 0.05), N (0.3, 0.1), and N (0.3, 0.15).These results, however, did not show statistically significantdifferences in the results when the standard deviation waschanged under the particular parameter settings we tested.We only show the runtime results in Figure 6 to support thisclaim, but the quality, action set size, and support set sizeresults all looked similarly homogenous across the differentstandard deviations tested.

ConclusionPrevious research in game-theoretic contagion blocking pro-posed heuristic algorithms to scale to realistic domain sizes.In this work, we perform a thorough investigation of the at-tributes of these algorithms and the solutions they generateby systematically varying the graph size, contagion proba-

bility distribution’s average, and contagion probability dis-tribution’s standard deviation. Our experiments reveal thatwhen the graph size increases, all of the algorithms con-verge in fewer iterations because the defender becomes un-able to find strong blocking actions. Interestingly, low con-tagion probabilities provided difficult problem instances be-cause the attacker could continuously find new actions thatavoided the entire set of current defender actions until a suf-ficient number of actions had been added to both players’action sets, thereby requiring many iterations to converge.For the LSMI oracle, which increases in complexity as con-tagion probabilities are increased, this results in low conta-gion probability problems requiring high runtimes (due tomore iterations), high contagion probability problems alsorequiring high runtimes (due to more time required per iter-ation), and a relatively easy middle region.

While these findings serve as a much more thorough in-vestigation of the behavior of these heuristic algorithms, ad-ditional avenues for future analysis clearly exist. For ex-ample, only scale-free graphs were used in this work. Whilethis is accepted as a good approximation for general-purposesocial networks, many other social network structures havebeen observed in the literature. Also, sociologists have notedthe importance of structural factors in the social networkssuch as the clustering coefficient, skew of degree distribu-tion, degree correlation, and path lengths on the contagionbehavior. In future work we will analyze the algorithms ongraphs across the spectrum of these metrics and design newheuristics tailored for specific parts of the parameter spacethat remain challenging for existing methods.

AcknowledgmentsThis research was supported by the United States Depart-ment of Homeland Security through the National Center forRisk and Economic Analysis of Terrorism Events (CRE-ATE) under award number 2010-ST-061-RE0001. Anyopinions, findings, conclusions or recommendations hereinare those of the authors and do not reflect views of the UnitedStates Department of Homeland Security, or the Universityof Southern California, or CREATE.

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