Reducing Gang Violence through Network Influence BasedTargeting of Social Programs

Paulo Shakarian andJoseph Salmento

Network Science Center andDept. of Electrical Engineering

and Computer ScienceU.S. Military AcademyWest Point, NY 10996

paulo@shakarian.net,joseph.salmento@usma.edu

William PulleyblankNetwork Science Center and

Dept. of MathematicalSciences

U.S. Military AcademyWest Point, NY 10996

william.pulleyblank@usma.edu

John BertettoChicago Police Dept.

Chicago, IL 60653john.bertetto@chicagopolice.org

ABSTRACTIn this paper, we study a variant of the social networkmaximum influence problem and its application to intelli-gently approaching individual gang members with incentivesto leave a gang. The goal is to identify individuals whowhen influenced to leave gangs will propagate this action.We study this emerging application by exploring specificfacets of the problem that must be addressed when mod-eling this particular situation. We formulate a new influ-ence maximization variant - the “social incentive influence”(SII) problem and study it both formally and in the con-text of the law-enforcement domain. Using new techniquesfrom unconstrained submodular maximization, we developan approximation algorithm for SII and present a suite ofexperimental results - including tests on real-world policedata from Chicago.

Categories and Subject DescriptorsApplied Computing [Law, social and behavioral sci-ences]: Sociology

General TermsAlgorithms, Experimentation

Keywordscomplex networks, network diffusion

1. INTRODUCTIONViolent street gangs are a major cause of criminal activity

in the United States [2, 5]. A recent trend has been to-ward development of “smart policing” tactics to reduce theeffectiveness of these gangs. Typically, these strategies have

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focused on the allocation of law enforcement resources to in-crease arrests and deter criminal behavior. In this paper wefocus on a use of “smart policing” in a different manner: wewish to intelligently target individual gang members withincentives to leave a gang. While “pulling levers” or encour-aging dis-enrollment from a gang, is already a tactic em-ployed in cities such as Boston and Chicago, the selectionof which specific gang members to focus on is still largelyunanswered - and hence currently based on ad-hoc meth-ods. In this paper, we study this emerging application asa variant of a social network influence maximization prob-lem [14] that we refer to as the “social incentive influence”(SII) problem. We study this problem both formally andin the context of law enforcement. Then, using new tech-niques from unconstrained submodular maximization [7], wedevelop an approximation algorithm for SII and present asuite of experimental results - including tests on real-worldpolice data.

The paper is organized as follows. In Section 2 we discusssome current methods in law-enforcement used for socialprogram targeting. In Section 3 we introduce the SII prob-lem an associated technical preliminaries. This is followedby a discussion of our algorithmic approach in Section 4 andassociated experimental results in Section 5. Finally, wediscuss related work in the literature in Section 6.

2. BACKGROUNDRecent successes with so called“pulling levers”(gang mem-

bership dis-enrollment) approaches to deterring violence in-clude the Boston Gun Project and Operation CeaseFire aswell as Project Safe Neighborhoods in Chicago, and con-tinue with the on-going Group Violence Reduction Strat-egy in Chicago. Using this approach, law enforcement part-ners work with social service providers and victims advocacygroups to attempt to abate gang violence by ‘pulling’ what-ever ‘levers’ need to be applied to street gangs. The types oflevers applied, and the degree to which they are employed,depend upon the particular gang. Adjustments are madeso that the application is both customized to the specificsituation and, hopefully, more successful.

To facilitate these interactions between law enforcement,social service providers, victim’s advocacy groups, and streetgang members, two approaches are commonly employed. Inthe case of the first, law enforcement engages known street

gang members on the street as part of regular patrol activ-ities. While performed under the auspices of focused de-terrence, this type of interaction is likely to have the leastimpact. Time spent with the gang member may be lim-ited, the remaining levers (social service providers, victimsgroups, etc.) are absent so that the message conveyed to thestreet gang member is incomplete or biased toward criminalenforcement aspect. Moreover, the selection of target gangmembers is often random – the result of opportunity as thegang member is seen on the street.

In the second approach, gang members are subjected to a“call-in.” The call-in sessions are prearranged meetings orga-nized by law enforcement, social service providers, and vic-tims groups during which messages of non-violence are con-veyed. The call-in is a full-spectrum effort: law enforcementmakes clear to attendees that further violence will be metwith relentless police operations and enforcement efforts, so-cial service providers offer information on how members canexit the street gang and obtain educational and vocationaltraining, and victims groups tell stories of loss in an effortto make an emotional plea for violence cessation.

Attendees for call-in sessions are typically chosen in twoways: compulsory attendance and invitation. Gang mem-bers currently on probation or parole are compelled to at-tend. The remaining attendees are invitees, and it is herewhere the selection criteria may become a bit vague. Invi-tees may be selected via some form of social network analysis(i.e. as in [5]) however such concerted efforts are not entirelyrelied upon. Often, a large amount of discretion is affordedlocal law enforcement in selecting invitees. This allows forlocal gang experts and command staff members to identifythose gang members whom they know or suspect to be in-fluential or key members of the gang and invite them to thecall-in. This can be a very successful process if the agencyhas access to these experts or knowledge of the gang’s or-ganizational structure. However, there is no guarantee thatthose persons invited are, in fact, genuinely influential in thegang or are just the “most well known to law enforcement”members of the gang. In law enforcement, where financial,personnel, and time resources are increasingly constrained,turning a more objective and analytical eye toward inviteeselection grows more important.

A key aim of gang dis-enrollment programs is to enablelaw enforcement to invite to call-ins those gang memberswhom, should the efforts to dis-enroll them be successful, aremost likely to pull additional gang members out with them.However, there is a key challenge: are influential membersalso easy to encourage to leave the gang? A recent empiricalstudy exploring non-criminal online social networks suggeststhat highly influential individuals are typically not suscep-tible [1]. However, we argue that taking both influence andsusceptibility into account are necessary: as identifying indi-viduals (or groups of individuals) that possess both qualitiesis needed for the behavior to spread.

Though, to the best of our knowledge, influence maximiza-tion techniques have not been applied to law enforcement be-fore, there is some anecdotal evidence that such approachescould bear fruit. For instance, there have been cases wheregang members thought to be “influential” successfully con-vinced others to dis-enroll from the gang. In one case, inChicago (in the summer of 20130, the district personnel (lo-cal plainclothes gang officers) knew this person to be aninfluential and as such targeted him for intervention. When

he was contacted he indicated that he had already gotten ajob, but knew several fellow gang members who could usethe offered social services and dis-enrollment opportunities.He personally contacted 20 fellow gang members, of which7 walked into the local social services facility.

In this paper we frame the problem of “pulling levers” for-mally as a variant of the social network maximum influenceproblem [14] in what we call the “social incentive influence”(SII) problem. However, there are some key nuances of the“pulling levers” strategy that we integrated into our frame-work that are not inherently considered in the maximuminfluence problem. We list these items here and address ourtechnical approach to each in the next section.

1. Duration of the diffusion process. One key differ-ence SII has from other maximum influence formula-tions is the length of time it takes for the diffusionprocess to occur. The reason for this is that gangdis-enrollment is a major life decision for an individ-ual, hence the spread of this idea will likely take time.Further complicating the matter is that there may bechanges to social network structure while the diffusionprocess occurs - based on events such as arrests, homi-cides, gang conflict and cooperation, etc.

2. Interaction with the population during the dif-fusion process. Not only does the diffusion processtake time to occur in this domain, but also the law en-forcement personnel will often make multiple attemptsto “pull levers” as the diffusion occurs.

3. Geographic locations and strength of connec-tions. Often, law enforcement data has an inherentgeospatial component. In this work, we leverage thisinformation to inform the strength of connections inthe social network - as the street gangs are also inher-ently territorial.

4. Notion of cost. Cost also becomes an importantfactor in SII as the law enforcement personnel are at-tempting to encourage a major change in the life of thegang members. Conducting a call-in session with cer-tain members costs time, money, and other resources.We also note that not all gang members will be equallysusceptible to this type of intervention - some may re-quire more or less effort to dis-enroll. Further, thereare real costs associated with encouraging dis-enrollment.For example, in North Carolina personalized lettersare created for the gang members that show the in-dividual how his association with others involved inviolence puts him or her at risk. A similar tactic isused in Chicago, where the letters are often deliveredto the homes of the gang members. This utilizes policemanpower and resources hence further increasing thecost.

5. Profit maximization. As we consider cost, we alsomodel “benefit” in SII - which is the value of each ex-pected infectee to the diffusion process. This allowsus to adopt a profit-maximizing model (similar to theProMax problem of [15]) where we look to maximizethe expected benefit minus the cost.

3. TECHNICAL PRELIMINARIESAND ANALYSIS

Throughout this paper we assume the existence of a socialnetwork G = (V,E) where V is a set of vertices and E is aset of directed edges. We let n and m denote the cardinalityof V and E respectively. For any node v ∈ V , the set ofincoming neighbors is ηinv , and the set of outgoing neighborsis ηoutv . The cardinalities of these sets (and hence the in- andout-degrees of node v) are denoted by kinv , k

outv respectively.

For each node v, we assume a cost of marketing to thatnode denoted by cv ∈ <+. We let C denote the vectorof costs indexed by V . We let 〈c〉 denote the average cost∑v cv/n. We also assume a benefit value, b ∈ <+, which is

the associated benefit for having marketed to a given node.We assume that each node in G has an associated geolo-

cation and there exists a distance function d : V × V → <+

that meets the normal axioms: d(u, u) = 0, d(u, v) = d(v, u),and d(u,w) ≤ d(u, v) + d(v, w).

Using this distance function, we shall assume a level ofinfluence puv for each edge (u, v) ∈ E that we define usingan exponential distance-decay model [23, 16, 22] as follows:

puv = e−(d(u,v)/γ)r

where γ, r are parameters in the interval (0,∞) and e is thebase of the natural logarithm. The parameter γ is used asa scaling parameter - and we shall set it to be the averagedistance between two nodes connected with an edge. Theparameter r controls the shape of the distance decay curve,and we shall typically use r = 2. The use of the distancedecay function as a proxy for influence is the primary way weaddress the geographic nature of the law-enforcement datain this application.Diffusion Process. A key property that we utilize in ourstudy is submodularity, which we review below:

Definition 3.1 (Submodularity). Function f : 2U →<+ is submodular if for every A ⊆ B ⊆ U and u ∈ U :f(A ∪ {u})− f(A) ≥ f(B ∪ {u})− f(B).

Intuitively, the idea of submodularity represents a notionof diminishing returns: adding an element u to a set B canprovide no greater benefit than that gained by adding it toany proper subset of B.

Equivalently, function f : 2U → <+ is submodular if forevery A,B ⊆ U , f(A) + f(B) ≥ f(A ∪B) + f(A ∩B).

A set function is supermodular if its negation is submod-ular.

Next, we define a diffusion process function (dpf) whichaccepts an initial set of vertices (called the “seed set”) andreturns the expected number of infectees once the diffusionprocess completes. In this paper, we shall require this func-tion to be sub-modular and normalized. We provide a formaldefinition below.

Definition 3.2 (Diffusion Process Function). A dif-fusion process function, dpf : 2V → [0, n], is any func-tion such that: (1.) dpf(∅) = 0 and (2.) ∀V1, V2 ⊆ V :dpf(V1) + dpf(V2) ≥ dpf(V1 ∪ V2) + dpf(V1 ∩ V2).

We argue that, in general, these are reasonable restric-tions. For instance, the σ function of the independent cas-cade and linear threshold models [14], the oracle of the MIA

model [11], and the value function of the logic-programmingframework of [20] are all valid diffusion process functions.We also note that these previous studies focused only onmaximizing the number of expected infectees (subject to acardinality constraint). In this work, we disregard the cardi-nality constraint and instead seek to maximize profit, whichwe formally define below.

Definition 3.3 (Profit). Profit, pft : 2V → < is de-fined by pft(X) = b× dpf(X)−

∑v∈X cv

The SII Problem. We now have all components neces-sary to formally define the social incentive influence (SII)problem:

Definition 3.4 (SII Problem). We are given diffusionprocess function dpf, social network G = (V,E), cost vectorC and benefit value b. Find SII(dpf, G,C, b) = V ∗ ⊆ Vsuch that pft(V ∗) ≥ pft(V ′) for all V ′ ⊆ V .

Not surprisingly, the social incentive influence problem isNP-hard.

Theorem 3.1. SII is NP-hard.

Proof. We show NP-hardness by reducing SIMPLE MAXCUT [13] to SII. The SIMPLE MAX CUT problem takes asinput a graph G = (V,E) and returns sets V1, V2 ⊆ V suchthat |{(u, v) ∈ E : u ∈ V1, v ∈ V2}| is maximized. The fol-lowing construction can be performed in polynomial time.Let dpf(X) =

∑v∈X fv(X) where fv(X) = 1 if v ∈ X and

|ηinv ∩ X| otherwise. Note that dpf(∅) = 0 and, becauseeach fv is submodular, dpf is submodular. For each v setcv = 1 and set b = 1. Then pft(X) =

∑v∈V \X : ηinv ∩X| =

|{(u, v) ∈ E|u ∈ X, v ∈ V \X}|. Hence pft becomes equiva-lent to the objective function for SIMPLE MAX CUT.

However, note that our profit function pft is submodular.

Proposition 3.1. pft is submodular.

Proof. It is well known that subtracting a supermodularfunction from a submodular function yields a submodularfunction. Since dpf is submodular (and b is positive) and thesum of costs is supermodular, the proposition follows.

One-Step Diffusion. As stated earlier, two key challengesin this domain are the duration of the diffusion process andthe effect of the law-enforcement personnel interacting dur-ing the diffusion process. This has led us to model the diffu-sion process as a “one-step” influence model where we onlyconsider the immediate effect of the diffusion process onetime step in the future. Our envisioned use case is thatthe law-enforcement analysts will use the most current dataavailable to make a decision as to which gang members toreach-out to based on this model. Attempts will be madeto influence those individuals, after which changes to thesocial network (both resulting the outreach and other ex-ternal factors) will be incorporated before repeating the cy-cle. Because we expect the time for diffusion to generallytake longer, the repetition of the cycle will generally oc-cur after about one time period. We formally define thefollowing“one-step” influence model.

Definition 3.5. The one-step diffusion model, σ1 :2V → <+ is defined as follows:

σ1(V ′) =∑u∈V

(1−

∏u∈ηinv ∩V ′

(1− puv))

Note that we also assume that a node v is infected by anode u independently of which others of its incoming neigh-bors were previously infected. An easy proof shows that σ1

is a valid diffusion process function.

Proposition 3.2. σ1 is a valid diffusion process.

Proof. Clearly, σ1(∅) = 0 by inspection. Next, we showthat the quantity

∏u∈ηinv ∩V ′(1−puv) is supermodular. Sup-

pose, by way of contradiction, that it is not, then we havefor V ′ and nodes q, r /∈ V ′ the following for each v ∈ V :∏u∈ηinv ∩(V ′∪{q,r})

(1− puv)−∏

u∈ηinv ∩(V ′∪{r})

(1− puv) <

∏u∈ηinv ∩(V ′∪{q})

(1− puv)−∏

u∈ηinv ∩V ′

(1− puv)

Let us assume that q, r are both neighbors of v (the othercases cause both sides to be equal). This gives us the fol-lowing:

(1− prv)( ∏u∈ηinv ∩(V ′∪{q})

(1− puv)−∏

u∈ηinv ∩V ′

(1− puv))<

∏u∈ηinv ∩(V ′∪{q})

(1− puv)−∏

u∈ηinv ∩V ′

(1− puv)

As∏u∈ηinv ∩(V ′∪{q})(1−puv) ≤

∏u∈ηinv ∩V ′(1−puv), we have

1 < 1 − prv which is clearly a contradiction. Note that thesupermodularity of this quantity implies the submodularityof 1−

∏u∈ηinv ∩V ′(1−puv). The rest of the statement follows

from the fact that σ1 is a positive linear combination ofsubmodular functions.

We also note, that with many diffusion processes func-tions, the calculation of their outcome may yield an individ-ual probability of activation for each node. In this case, wecan identify a specific benefit for each node. The frameworkcan be easily adapted for such a case.

4. APPROACHWhile the submodularity of the pft function is encourag-

ing, we note that because marketing to each node incurs anassociated cost, it is possible to experience a loss by mar-keting to additional nodes. For instance, if we market to anadditional individual who provides us no increase in the dif-fusion process leads to a loss. This is not considered in pre-vious diffusion models designed to maximize the expectednumber of infectees. Hence, the greedy approximation of[17] no longer provides us an approximation guarantee. Ourcase can instead be viewed as an “unconstrained” submodu-alr function. Recently, a deterministic approximation algo-rithm was introduced in [7] that requires only a linear num-ber of evaluations of the function. We recall their algorithmhere (adapted for SII).

For positive, unconstrained submodular maximization, [7]proves that SII-Approx provides a result that is at least 1/3 ofoptimal. However, we note that pft can potentially providea negative solution. However, we can leverage their resultsto provide the following approximation guarantee:

Corollary 4.1. Given VALG as returned by SII-Approxfor an instance of SII and optimal soluition VOPT we havethe following relationship:

pft(VOPT )

3− n

3(〈c〉 − b) ≤ pft(VALG)

Algorithm 1 SII-Approx

INPUT: Social network G = (V,E), cost vector C, benefitb, distance function d.OUTPUT: Approximation V ′ to SII.

1: V ′ = ∅, V ′′ = V2: for v ∈ V do3: a = pft(V ′ ∪ {v})− pft(V ′)4: b = pft(V ′′ \ {v})− pft(V ′′)5: if a ≥ b then6: V ′ = V ′ ∪ {v}7: else8: V ′′ = V ′′ \ {v}9: end if

10: end for11: return V ′.

Proof. In the proof of Theorem I.1 of [7], the authorsshow that pft(VOPT ) ≤ 3pft(VALG) − pft(∅) − pft(V ). Wenote that, by definition, pft(∅) = 0 and pft(V ) = bn−〈c〉n =−n(c− b), which gives the result.

Note that if 〈c〉 ≤ b then we recover the 1/3 approximationof [7].

5. EXPERIMENTAL RESULTSAll experiments were run on a computer equipped with an

Intel X5677 Xeon Processor operating at 3.46 GHz with a 12MB Cache and 288 GB of physical memory under the RedHat Enterprise Linux version 6.1 operating system. Onlyone core was used for experiments. Our implementation ofSII-Approx was written in Python 2.7 using the NetworkXlibrary1.

Police Dataset. We used a dataset consisting of arrestrecords of individuals from March 2010 - March 2013 in asingle police district in Chicago. This data set included ar-rest location and relationships among the individuals. Fromthis data, we were able to construct a social network (“ar-rest network”) consisting of 1836 nodes and 2531 edges. Forsome experiments we also generated simulated networks tosupplement our analysis.

Comparison with Optimal Solution. Our first test wasto evaluate SII-Approx compared to an optimal solution foundby enumeration. We did this by sampling the police datasetand by generating simulated networks. We generated 20connected samples from the overall police network rangingin size from 11 to 20 nodes. We defined cost cv = 1 for allv ∈ V and we set benefit b = 1. The worst approximationratio obtained in these tests was 0.70 - more than doublethe theoretical bound of 1/3 in this case. The average-casebound was better still at 0.92. Additionally, we also studiedthe behavior of SII-Approx on several standard generatedgraph types including complete graphs of size 20 and 25,Erdos-Reyni (E-R) random graphs, preferential-attachmentgenerated scale-free graphs (SF), and the “Florentine Fami-lies”(FF) network [6]. In all of these tests, we never achievedan approximation ratio lower than 0.80. The results are

1http://networkx.github.io/

Dataset Num. Avg. Min. Max. Avg. Std.Sams. Size Appx. Appx. Appx. Dev.

Police 20 13.55 0.70 1.00 0.92 0.09Compl. 2 22.5 1.00 1.00 1.00 0.00

E-R 9 20 0.84 1.00 0.93 0.05SF 27 20 0.80 1.00 0.98 0.06FF 1 15 1.00 1.00 1.00 0.00

Table 1: Empirically Determined ApproximationRatios for SII-Approx (when compared to the opti-mal solution)

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10 15 20

Spp

ed

up

(lo

g sc

ale

)

Number of Nodes

Figure 1: Network size vs. speedup obtained bySII-Approx over exact approach.

shown in Table 5.

Runtime Evaluation. We evaluated runtime in two ways:(1) we compared the runtime of SII-Approx with an exactenumeration based computation and (2) we studied how SII-Approx scaled with network size. Both results are depictedin Figures 1-2. For the comparison with the exact computa-tion, we studied the effect of runtime on our 20 samples fromthe police dataset. We studied at the speedup provided bySII-Approx (defined as the runtime for the exact approachdivided by the runtime for SII-Approx on the same input) asa function of network size. We found a significant speedupin all cases and that the speedup increased exponentiallywith network size (R2 = 0.96) - which is clearly due to theexponential runtime of the enumeration approach.

Runtime also scaled monotonically with the size of thenetwork (quadratic fit, R2 = 0.97). Hence, for the size ofthe datasets used by the Chicago police department (order103 nodes), this is a viable approach with the current im-plementation. However, we think further improvement inruntime for the heuristic is possible with further practicalmodifications.

Cost Model Evaluation. One of the more useful char-acteristics of our framework is the ability to consider nodecosts. We studied two variants. First, we studied the case inwhich all nodes have the same cost, considering several dif-ferent values. Second, we set the cost to be proportional toa network centrality measure. In both cases, we also variedthe value of the benefit. We used the entire police datasetin these trials. The results for both sets of trials are shown

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Figure 2: Network size vs. runtime for the heuristicalgorithm.

in Figures 3-4.We examined a fixed/uniform cost model where all nodes

we assigned the same value. We examined cost values from0.25 to 2.0 in intervals of 0.25 and compared them to uniformbenefit values in the same range. In general, there was alinear relationship between the benefit and profit for all fixedcost models examined (R2 values ranged from 0.97 to 0.99).Further, as expected, decreased cost led to increased profit.

For our centrality-based cost trials, we studied degree cen-trality (number of adjacent edges), closeness centrality (see[24]), eigenvector centrality (see [4]), shell number (based onshell-decomposition, see [19]), and clustering coefficient (see[24]). Cost was set to be proportional to these values foreach node. We also normalized the cost so that the averagewould be 1.0 in each case. Just as with the fixed-cost tri-als, we compared the profit for various benefit values in therange [0.25, 2.0] in intervals of 0.25.

For centrality-based cost models, we also observed a lin-ear relationship between benefit and profit (R2 values ap-proaching 1.0). In examining the difference among central-ity measures, we found the most expensive centralities weredegree and shell-number followed by closeness. As these canbe considered radial measures of centrality, meaning theymeasure centrality in terms of the number of paths thatoriginate from a given node, then this result should be ex-pected. Clustering coefficient was less expensive than thesemeasures, which again was as expected as this measure isless dependent upon the number of adjacent nodes and moredependent upon the neighborhood. Perhaps most interest-ing was that eigenvector centrality was the “least expensive”cost model. We believe that this is due to the wide distribu-tion of values assigned by this measure which ranged from1.66 × 10−43 to 166.45 (compared to degree, which rangedfrom 0.36 to 6.17).

Heuristic for Improved Solution Quality. SII-Approx,as presented in this paper, does not take into account theorder in which the vertices are selected. We found thatif vertices are ordered by descending degree by the node’sClusterRank [8] provided an improvement to the profit asreturned by the algorithm when compared to our randombaseline (average over 10 runs) for the case of uniform cost(∀v ∈ V, cv = 1) and various settings for benefit. The resultsare depicted in Figure 5. In [8] nodes of high ClusterRankwere shown to encourage diffusion under the SIR model -which is related to the one-step process of this paper. The

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Pro

fit

Benefit

Fixed (.25)

Fixed (.5)

Fixed (.75)

Fixed (1)

Fixed (1.25)

Fixed (1.5)

Fixed (1.75)

Fixed (2)

Figure 3: Profit obtained from SII-Approx for the po-lice dataset for fixed/uniform cost models with var-ious benefit settings.

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Figure 4: Profit obtained from SII-Approx for thepolice dataset for centrality-based cost models withvarious benefit settings.

ClusterRank of node v is defined as follows:

crv = 10Cv∑

u∈ηoutv

(1 + koutu )

Where Cv is the clustering coefficient for node v. This is par-ticularly helpful as the computation of these measures reliesonly on local information and can be calculated quickly. Ad-ditionally, we examined ordering by degree, clustering coeffi-cient, closeness centrality, shell number, and weighted degreecentrality (for each v ∈ V the quantity

∑u∈V pvu). While

all of these measures showed some improvement over therandom baseline, they were out performed by ClusterRankfor all benefit values above 0.5. We are currently examiningthe performance of centrality-based ordering heuristics on avariety of inputs for the algorithm.

Iterative Application. We envision real-world police useof SII-Approx to occur in an iterative manner. One way thiscould be done is as follows: we initially consider a uniformcost model and identify initial nodes to seed. Then, wecalculate the diffusion process function based on that seedset in a manner that yields the probability pv of each nodev being activated. Then, for the next iteration, we removefrom the network all previously seeded nodes (or whicheversubset dis-enrolled from the gang) and set the cost for eachnode v to be 1 − pv. The intuition is that it will be lessexpensive to seed nodes that already obtained influence fromother members departing the gang. The process of re-settingthe cost function and social network prior to re-running SII-

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Figure 5: Improvement to profit as returned by SII-Approx when the vertices are ordered by Cluster-Rank.

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Figure 6: Iterative applications of SII-Approx – Num-ber of nodes targetted at each iteration.

Approx is then performed continually.We applied SII-Approx iteratively five times to the police

dataset in the manner described above and studied the sizeof the set targeted as well as the resulting profit (see Fig-ures 6-7). We observed, under the assumption that all pre-viously seeded members left the gang, that the profit gaineddecreased monotonically with the number of iterations whilethe number of targeted vertices increased slightly in the sec-ond iteration, followed by a steep decrease and then con-verged to zero. The success of the second iteration indi-cated that viral marketing may be successful in encouragingneighboring individuals to dis-enroll. However, beyond thesecond step, there is limited profit to further marketing fordis-enrollment. We note that at this point, if successful, ap-proximately half of the gang members are dis-enrolled, whichwould be a significant reduction. Further, we also note thattopological changes to the network may become more sig-nificant after the second (and possibly even after the first)round of dis-enrollment.

6. RELATED WORKThe maximum influence problem was introduced in [14]

and later studied in work such as [9, 11, 15, 21]. We referthe reader to the book [10] for a summary of recent workin this area. However, to our knowledge, no other work ad-dresses all the challenges presented here for the SII problemsimultaneously. For instance, [11] presents a model wherethe diffusion is restricted to shortest paths - which is a sim-ilar restriction to our one-step model, but does not considerthe idea of profit. Likewise, [15] considers the idea of profit,

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Figure 7: Iterative applications of SII-Approx – Profitobtained at each iteration.

but only applies it to the linear threshold model - whichrelies on the number of an individuals neighbors reachinga certain threshold. Another issue is that in most of thesemodels, the diffusion process function is difficult to compute- for instance the dpf for the independent cascade model isshown to be #P -hard in [11]. As a result, in most otherpieces of work the diffusion process is approximated usingsimulation, which is as expensive operation. (Most law en-forcement agencies we work with have limited computationalpower). One notable exception regarding this issue are de-terministic models such as that described in [12, 21]. Wenote that in our previous work we have looked at utilizingthis model in a law-enforcement setting [18, 3]. However theresults of that work were primarily used to describe charac-teristics of the gangs and not to make operational decisions.This work did not study the operational issues associatedwith encouraging gang dis-enrollment as considered in thispaper.

7. CONCLUSIONIn this paper we introduced the “social incentive influ-

ence” (SII) problem, a variant of the maximum influenceproblem, designed to help law-enforcement personnel iden-tify members of street gangs that they can encourage todis-enroll. We studied this problem both formally and ex-perimentally in the context of the law-enforcement domain.Utilizing techniques from unconstrained submodular maxi-mization, we developed a heuristic technique to help policebetter identify sets of influential individuals to target withdis-enrollment incentives. We implemented our approachand performed an experimental evaluation. Our next stepis to integrate this method with our GANG/ORCA analysissoftware [18] that is currently in use by the Chicago Police.In doing so, we can then work to better understand how SIIis employed in practice - allowing us to identify componentsof this framework that can be adjusted for improved resultsin a real-world setting.

8. ACKNOWLEDGMENTSThe authors are supported by the Army Research Office

(project 2GDATXR042) and the U.S. Air Force. The opin-ions in this paper are those of the authors and do not nec-essarily reflect the opinions of the funders, the U.S. MilitaryAcademy, the U.S. Army, the U.S. Air Force, or the U.S.Department of Defense.

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