=Paper=
{{Paper
|id=Vol-2142/paper12
|storemode=property
|title=End to end influence maximization for HIV prevention
|pdfUrl=https://ceur-ws.org/Vol-2142/paper12.pdf
|volume=Vol-2142
|authors=Bryan Wilder,Laura Onasch-Vera,Juliana Hudson,Jose Luna,Nicole Wilson,Robin Petering,Darlene Woo,Milind Tambe,Eric Rice
|dblpUrl=https://dblp.org/rec/conf/ijcai/WilderOHLWPWTR18
}}
==End to end influence maximization for HIV prevention==
https://ceur-ws.org/Vol-2142/paper12.pdf
End to end influence maximization for HIV
prevention
Bryan Wilder, Laura Onasch-Vera, Juliana Hudson, Jose Luna, Nicole Wilson,
Robin Petering, Darlene Woo, Milind Tambe, Eric Rice
Center for Artificial Intelligence in Society, University of Southern California
bwilder, onaschve, jnhudson, joseluna, wilsonnj, petering, darlenew,
tambe, ericr@usc.edu
Abstract. This work aims to overcome the challenges in deploying in-
fluence maximization to support community driven interventions. In-
fluence maximization is a crucial technique used in preventative health
interventions, such as HIV prevention amongst homeless youth. Drop-
in centers for homeless youth train a subset of youth as peer leaders
who will disseminate information about HIV through their social net-
works. The challenge is to find a small set of peer leaders who will have
the greatest possible influence. While many algorithms have been pro-
posed for influence maximization, none can be feasibly deployed by a
service provider: existing algorithms require costly surveys of the entire
social network of the youth to provide input data, and high performance
computing resources to run the algorithm itself. Both requirements are
crucial bottlenecks to widespread use of influence maximization in real
world interventions.
To address the above challenges, this paper introduces the CHANGE
agent for influence maximization. CHANGE handles the end-to-end pro-
cess of influence maximization, from data collection to peer leader selec-
tion. Crucially, CHANGE only surveys a fraction of the youth to gather
network data and minimizes computational cost while providing compa-
rable performance to previously proposed algorithms. We carried out a
pilot study of CHANGE in collaboration with a drop-in center serving
homeless youth in a major U.S. city. CHANGE surveyed only 18% of
the youth to construct its social network. However, the peer leaders it
selected reached just as many youth as previously field-tested algorithms
which surveyed the entire network. This is the first real-world study of
a network sampling algorithm for influence maximization. 1
Keywords: Influence maximization · HIV prevention · social networks.
1 Introduction
This paper presents and field-tests a novel, practical agent for influence maxi-
mization, the challenge of selecting a small set of seed nodes in a social network
1
An extended version of this paper has been accepted as a full paper at AAMAS
2018.
2 Wilder et al.
who will diffuse information to many others. Such techniques have important ap-
plications ranging from preventative health [13, 1] to international development
[2].
We are particularly motivated by the challenge of preventing HIV spread
among homeless youth [11, 18, 10] (although our contributions would also assist
other public health interventions). Here, influence maximization is used to select
homeless youth who will serve as peer leaders and spread messages about HIV
prevention through their social network. Pilot studies in this domain have shown
that algorithmic approaches have great promise, substantially outperforming
status-quo heuristics [17]. However, current algorithms have a high barrier to
entry: they require a great deal of time to gather the complete social network
of the youth, expertise to select appropriate parameters, and computational
power to run the algorithms. None of these are likely available to resource-
strained service providers who will ultimately be the ones to deploy influence
maximization.
Gathering network data is particularly onerous because it requires individu-
ally surveying upwards of a hundred youth. Further, network collection is more
time intensive than simple survey methods, requiring days of time for a dedi-
cated team of social work researchers. It is not feasible for service providers with
many other responsibilities.
The other barriers are also serious impediments to wide-scale adoption of in-
fluence maximization. Service providers will not have access to the high-performance
computing resources required by previous algorithms, where high computational
cost is often incurred to find solutions robust to unknown parameters. For in-
stance, DOSIM, the state of the art algorithm for robust influence maximization
[15], takes hours of runtime on a high-performance computing system. In reality,
a deployed system would need to run in minutes on a laptop.
This paper presents CHANGE (CompreHensive Adaptive Network samplinG
for social influencE), a novel, end-to-end agent for influence maximization which
addresses the above barriers via a set of algorithmic contributions. CHANGE is
easy to deploy, but this simplicity is crucially enabled by a series of insights into
the social structure of homeless youth (which may be useful for other vulnerable
populations). We conducted a pilot test of CHANGE’s performance in a real
deployment by a drop-in center serving homeless youth in a major U.S. city.
CHANGE was used to plan a series of interventions designed to spread HIV
awareness among the youth. CHANGE obtained comparable influence spread to
state of the art algorithms while surveying only 18% of nodes for network data.
Overall, CHANGE offers a practical, field-tested vehicle for deployed influ-
ence maximization which drastically lowers the barrier to entry. To our knowl-
edge, this is the first real-world pilot study of a network sampling algorithm
for influence maximization and only the second ever field test of any influence
maximization algorithm.
End to end influence maximization for HIV prevention 3
2 Related Work
The influence maximization problem was introduced by Kempe et al. [7], and has
been extensively studied since then [3, 12, 4, 5]. Most work has focused on algo-
rithms which are scalable to extremely large networks, primarily in the context
of online viral marketing. Recently, HIV prevention (and preventative health
more broadly) has emerged as a new application area for influence maximiza-
tion which brings its own set of research challenges. Yadav et al. [16] proposed
HEALER, a POMDP-based algorithm for selecting influential peer leaders. Sub-
sequently, Wilder et al. [15] introduced the DOSIM algorithm which uses robust
optimization to account for uncertainty about the true probability of influence
propagation. Our approach to parameter robustness is similar to techniques used
in robust MDP planning [8], though the domains are entirely different.
Yadav et al. [17] conducted a real-world pilot study of HEALER and DOSIM,
and found that both algorithms significantly outperformed the status-quo heuris-
tic used by agencies (selecting high-degree nodes). However, neither algorithm
addresses any of the challenges described above. Both assume that the entire
social network is provided as input, which is unrealistic in practice due to the
enormous effort required. Further, only DOSIM handles uncertainty about the
probability of influence spread, and its method for doing so is extremely com-
putationally intensive (see Section 4.3).
Separate work by Wilder et al. [14] considered network data collection. They
proposed the ARISEN algorithm which samples a portion of youth in the network
to collect data from. While ARISEN can be theoretically analyzed for certain
network structures, it is not practically suitable to deployment because it relies
on querying a sequence of specific youth who may be difficult to locate (see
Section 4.2). Moreover, ARISEN does not consider either parameter uncertainty
or execution errors (the possibility that some peer leaders will not attend), both
of which we incorporate into CHANGE.
3 Problem description
Motivating domain: Our work is designed to overcome the challenges in de-
ploying influence maximization techniques to support community-driven inter-
ventions. We are specifically motivated by the challenge of raising awareness
about HIV among homeless youth. Typically, an HIV awareness intervention
will be provided by a drop in center or other organization which serves homeless
youth. Each intervention is a day-long class followed by weekly hour-long meet-
ings. Hence (as is typical in many intervention domains), the service provider
will almost never have enough resources to deliver the intervention to all of the
youth that frequent the center; instead, the intervention is usually delivered to
15-20% of the population2 . Further, limitations on space and personnel mean
that the intervention can typically be delivered to only 4-6 youth at a given
2
Note that while CHANGE directly surveys ∼18% of youth, they name others as
friends, resulting in a larger sampled graph.
4 Wilder et al.
time, so the training is broken up over a series of small sessions. These youth
are trained as peer leaders who communicate with other youth about HIV pre-
vention. This amplifies the reach of the intervention through the social network
of the homeless youth. The question is which youth will make the most effective
peer leaders, able to reach the greatest number of their peers. This is an influence
maximization problem, which we now formalize.
Influence: The youth have a social network represented as a graph G =
(V, E). Each youth is initially inactive, meaning that they have not received
information about HIV prevention. Once nodes are activated by the intervention,
they have a chance to influence their peers. We model this process through a
variant on the classical independent cascade model (ICM) which has been used
by previous work on HIV prevention and better reflects realistic time dynamics
[16, 15, 17]. The process unfolds over discrete time steps t = 1...T , where T is
a time horizon. There is a propagation probability p. When a node becomes
active, it attempts to activate each of its neighbors. Each attempt succeeds
independently with probability p. Activation attempts are made at each time
step until either the neighbor is influenced or the time horizon is reached.
Note that the assumption that p is uniform across edges is without much loss.
As noted by He and Kempe [6], a uniform p is equivalent to each edge drawing
an individual propagation probability i.i.d. from a distribution with mean p.
This is because the following processes are analytically equivalent: (1) propagate
influence with probability p and (2) draw a propagation probability q from a
distribution with E[q] = p and then propagate influence with probability q.
Hence, our model actually subsumes any stochastic model where the propagation
probabilities are drawn from a common prior.
Interventions: At each time step t = 1...T , the algorithm selects a seed set
At containing up to K nodes. However, each seed node may or may not actually
attend the intervention. This problem is particularly acute with homeless youth
since a number of factors could prevent a given youth from attending (e.g., be-
ing arrested, running out of money for a bus ticket, etc.). Hence, we assume
that each node v has a hidden state xv ∈ {present, absent}. Each node’s state
is drawn independently from some prior distribution D. For simplicity, we will
take D to set each node to be present with probability q. However, all of our
analysis applies to arbitrary distributions. For each v ∈ At , if xv = present, then
v is activated. Nothing occurs if xv = absent. Note that an absent node can still
become activated by others, since they may still be in contact with others in
the social network. After the set At is chosen, the intervention occurs and the
hidden state of each v ∈ At is observed. We denote the set of all observations
received at time t as Ot . The algorithm may use this information to plan the
next intervention. In other words, the problem is adaptive. To model adaptivity,
we introduce the notion of a policy. A policy maps from past actions and obser-
vations to the action that should be taken next. Let A = {S ⊆ V : |S| ≤ K} be
the set of all possible actions. A history is the current sequence of actions chosen
and observations received, denoted by ψt = ((A1 , O1 ), (A2 , O2 ), ...(At , Ot )). Let
Ψ be the set of all possible histories. A policy is a mapping π : Ψ → A. Let
End to end influence maximization for HIV prevention 5
Fig. 1. Illustration of the CHANGE agent.
A(ψt ) = (A1 ...At ) be the sequence of actions taken and O(ψ) = (O1 ...Ot ) be the
corresponding observations (whether each peer leader was present or absent).
Recall that youth are trained in groups of 4-6; the policy selects a group of
youth to invite given who was trained previously. We denote the objective as
f (A(ψ)|O(ψ)). f is the expected number of nodes influenced by the seed nodes
in A(ψ) conditioned on the observations in O(ψ). We overload notation and
let f (π) = Eψ∼π [f (A(ψ)|O(ψ))] be the expected reward from running policy
π, where the expectation ranges over the hidden state x (which determines π’s
actions) as well as the influence process. Our goal is to find a policy maximizing
f (π).
Uncertainty about network structure and parameters: We consider
extensions to the core adaptive influence maximization problem which account
for the lack of information endemic in field deployments. First, we consider the
case where the structure of the network (the edges E) are unknown. To address
this challenge, we give our agent a budget of M queries to run before conducting
the intervention. Each query may target either a uniformly random node, or the
neighbor of a node already queried. When a node is queried, it reveals all of its
edges. The goal is to use the M queries to uncover a set of edges which suffice
to identify influential nodes.
We then consider an unknown propagation probability. Here, we take a robust
optimization approach and look for a policy which performs well across a range
of possible values for p. More detail on this part of the problem can be found in
Section 4.3.
4 CHANGE: a new agent for influence maximization in
the field
We now introduce the CHANGE agent for end-to-end influence maximization.
Figure 1 illustrates the three components of the agent. We start with the last
component, peer leader selection, since the other components exist to provide
the data that the peer leader selection algorithm requires. Peer leader selection
6 Wilder et al.
is performed by an adaptive greedy algorithm (Algorithm 1), which handles
the chance that some peer leaders may not attend the intervention and plans
solutions using the observations obtained so far. Algorithm 1 requires as input
a (sample of) social network and a propagation probability p. We subsequently
introduce Algorithms 2 and 3 to provide these inputs.
4.1 Adaptive greedy planning
Algorithm 1 Adaptive greedy
1: for t = 1...T do
2: At = ∅
3: for k = 1...K do //greedily select seeds for action t
4: v = arg maxv∈V ∆(At ∪ {v}|ψt−1 ) − ∆(At |ψt−1 )
5: At = At ∪ {v}
6: execute At and observe Ot
7: ψt = ψt−1 + (At , Ot ) //add action/observation to history
Given as input the graph G and propagation probability p, finding the opti-
n
mal policy
is a difficult planning problem. There are 2 possible hidden states
n
and K possible actions. While it is possible to formulate the problem as a
POMDP, these exponentially large state and action spaces place even small in-
stances beyond the reach of off-the-self solvers. Hence, we exploit the structure of
the problem to formulate a scalable greedy algorithm which obtains (provably)
near-optimal solutions.
Pseudocode for adaptive greedy, our online planning algorithm, can be found
in Algorithm 1. Algorithm 1 selects the action at each step which maximizes the
expected gain in influence spread, conditioned on the observations received so
far. Then, it waits until this action has been executed, observes which peer lead-
ers attended the intervention, and greedily plans the next step. Formally, let
∆(At |ψt−1 ) = f (A(ψt−1 ) ∪ At |O(ψt−1 )) − f (A(ψt−1 )|O(ψt−1 )) denote the ex-
pected marginal gain to selecting At at time t. The greedy policy is to select
At = arg max|A|≤K ∆(A|ψt−1 ) (the outer loop of Algorithm 1). However, com-
puting the maximizing action is itself computationally intractable (as there are
n
K possible choices). Hence, Algorithm 1 uses an additional greedy inner loop
which greedily selects the elements of At one at a time (lines 3-5). Note that ∆
can be computed by averaging over random simulations over both the hidden
state (which nodes are present/absent) as well as how influence spreads via the
ICM.
We prove the following theorem, which shows that greedy planning is suffi-
cient to obtain a guaranteed approximation ratio:
Theorem 1. Let πG be Algorithm
1’s greedy policy and π∗ be an optimal policy.
e−1
It holds that f (πG ) ≥ 2e−1 f (π∗ ).
End to end influence maximization for HIV prevention 7
A proof may be found in the supplemental material. We use the adaptive
submodularity framework of Golovin and Krause, which generalizes the classical
notion of a submodular set function to adaptive policies. Their framework does
not straightforwardly apply to our problem since our algorithm selects a sequence
of actions, not a set. The order in which actions are selected matters since peer
leaders who are selected earlier will have more time to influence others. We show
that our problem can be reformulated as maximizing an adaptive submodular set
function subject to a more complex set of constraints (a partition matroid). This
is the first approximation guarantee for adaptive influence maximization under
execution errors, which is a well-known challenge in domains such as ours [15,
17].
4.2 Network collection
Algorithm 2 Network sampling
1: input: vertex set V , budget M
2: E = ∅ //set of edges observed
3: S = ∅ //set of nodes surveyed
4: for i = 1... M
2
do
5: Sample v uniformly at random from V \ S
6: S = S ∪ {v}
7: E = E ∪ {(v, u) : u ∈ N (v)}
8: Sample u uniformly at random from N (v) \ S
9: E = E ∪ {(u, w) : w ∈ N (u)}
10: S = S ∪ {u}
11: return E
The adaptive greedy algorithm assumes that the graph G is fully specified.
However, in order for an intervention to deployed in practice, the social network
needs to be laboriously gathered by interviewing the entire population of home-
less youth (potentially hundreds of youth in total). This is not practical for a
service provider to carry out on their own. We present an approach (Algorithm
2) which randomly samples a small number of youth to survey. Our procedure is
easy for a service provider to implement in the field without much computational
assistance. This simplicity is enabled by underlying insights about the structure
of homeless youth social networks, which may assist with intervention design in
other vulnerable populations.
We assume that the service provider has the ability to survey up to M youth.
Each youth, when surveyed, reveals all of their edges. Algorithm 2 chooses M 2
nodes uniformly at random from the population to survey (line 5). For each
surveyed node, it choses a uniformly random neighbor to survey as well (line 8).
Lastly, it returns the graph consisting of the reported edges. The intuition for
why this procedure succeeds is that it leverages the friendship paradox : a phe-
nomena where a random node’s neighbor has more friends, on average, than the
8 Wilder et al.
node itself. Essentially, high-degree nodes are overrepresented when we sample
a random neighbor instead of a uniformly random node. Thus, Algorithm 2 is
disproportionately likely to find central nodes in the network who will reveal
many edges and may be good potential seeds.
4.3 Parameter robustness
Algorithm 3 Robust parameter selection
1: input: parameter values p1 ...pL
2: for i = 1...L do
3: for j = 1...L do
4: g(pi , pj ) = value obtained by Algorithm 1 using pi evaluated under pj
g(p ,p )
5: return arg maxi=1...L minj=1...L g(pji ,pjj )
A further complication is that the adaptive greedy algorithm assumes that
the propagation probability p is known, in order to calculate the marginal gain ∆.
However, p is never known precisely in practice; each intervention takes months
to deploy so we are unlikely to observe the many repeated cascades needed to for
learning-based approaches. Previous work has attempted to resolve this dilemma
via robust influence maximization [6, 3, 9, 15] which finds a seed set which per-
forms well in the worst case over an uncertainty set of possible parameters.
However, the only previous work which addresses robust influence maximiza-
tion in an adaptive domain is the DOSIM algorithm. DOSIM requires hours or
even days of runtime on a high-performance computing cluster because it needs
to brute force over a grid of possible parameter settings. Such computational
expense is far beyond the capabilities of the average service provider, motivat-
ing the development of lightweight but effective heuristics for robust influence
maximization.
Algorithm 3 gives the heuristic used by CHANGE. It searches for a good
nominal value of the parameter p, which (when given to Algorithm 1) will
result in high performance no matter what the true value of p actually is.
We first discretize the interval [0, 1] into L points p1 ...pL . Let g(pi , pj ) de-
note the expected influence obtained when we run adaptive greedy planning
based on propagation probability pi , but the true parameter is pj . We then find
g(p ,p ) g(p ,p )
p∗ = arg maxi=1...L min j = 1...L g(pji ,pjj ) . Here, g(pji ,pjj ) , is the ratio of the value
based on planning with parameter pi to the value that could have been obtained
if we new the true parameter pj . p∗ is the parameter which maximizes the worst-
case value of this ratio. Notably, this procedure requires only L2 runs of adaptive
3
greedy; we take L = 10 in practice. By contrast, DOSIM requires O n runs
of a greedy algorithm to achieve approximation error . This quickly reaches
thousands (or tens of thousands) of runs even for moderately sized networks
and requires high-performance computing resources.
End to end influence maximization for HIV prevention 9
5 Pilot study procedure
Table 1. Number of youth recruited, trained, and retained for follow-up in each study.
CHANGE refers to the study conducted in this work to test the CHANGE agent. The
other columns are taken from Yadav et al. [17], who conducted pilot tests of HEALER
and DOSIM.
CHANGE HEALER DOSIM
Youth recruited 64 62 56
Queried for links 18.75% 100% 100%
PL trained 15.6% 17.7% 17.85%
Retained 54.7% 73% 73%
The major contribution of this work is carrying out a pilot study which
tests the CHANGE agent in a field deployment at a real drop in center serving
homeless youth in a major U.S. city. Here, we outline the procedure followed for
the pilot study. There were two studies, the feasibility study and the intervention
study. In the feasibility study, we just tested the first component of CHANGE
(network data collection) to validate that it works in practice to gather high-
quality data. In the intervention study, we carried out actual interventions with
homeless youth at the center. This step used all three steps of the CHANGE
agent: we gathered the network, found a robust set of parameters, and then
carried out interventions.
For each of the studies, we enrolled (respectively) 72 and 64 youth. Each
youth was paid $20 to enroll in the study (all monetary incentives were the same
as prior studies [17]). We ran CHANGE’s data collection mechanism, randomly
sampling a subset of youth to query for ties. Each youth who enrolled was also
asked to complete a baseline survey. As part of this survey, we also gathered
the full network consisting of ties from all of the youth. We emphasize that
this data was collected just for analysis. We did not use the full network to plan
interventions, and we would not expect an agency to conduct this step in a regular
deployment.
In the intervention study, trained social workers delivered the Have You
Heard intervention, previously published in the public health literature [11].
The social workers conducted a day-long class with the selected youth, covering
HIV awareness and prevention, and training the youth as peer leaders to com-
municate with other youth at the agency. Peer leaders were paid $60. Three sets
of peer leaders were selected by CHANGE, with approximately 4 peer leaders in
each set. This matches the size of trainings used in previous influence maximiza-
tion pilot studies [17]. Table 1 reports specific values on the number of youth
enrolled, queried for edges, and trained as peer leaders for our pilot test as well
as pilot tests of previous algorithms. One month after the start of the study, we
conducted a follow up survey with all of the youth who initially enrolled. Some
10 Wilder et al.
youth were lost to follow up (see Table 1). We asked the youth whether they
had received information about HIV prevention from a peer who was part of the
study. Youth were paid $20 to respond to the follow up survey. We emphasize
that all aspects of the intervention study (the training materials for peer lead-
ers, survey instruments, etc.) are identical to Yadav et al. [17], so our results are
directly comparable.
6 Pilot study results
We focus here on results from the intervention study, which tested the entirety
of the CHANGE agent. In this study, we recruited a population of 64 homeless
youth from a drop-in center. Table 1 gives the total number of youth recruited
for different activities, as well as the corresponding figures for previous pilot
tests of the HEALER and DOSIM algorithms by Yadav et al. [17]. We gathered
the full social network from all 64 youth, and in parallel ran Algorithm 2 with a
budget of M = 12 youth to collect a sampled network (querying 18.75% of youth
in total for links). Only the sampled network was used to plan interventions; the
full network was gathered only for analysis. We then ran the CHANGE pol-
icy for three steps, training 10 total peer leaders (15.6% of the network). This
percentage is comparable to previous studies (HEALER and DOSIM trained
approximately 17% of the network each). However, HEALER and DOSIM used
the entire network to plan their intervention, compared to the 18.75% of sam-
pled youth used by CHANGE. At one month, we conducted a follow-up survey
to assess whether youth received information about HIV prevention from the
peer leaders. 54.7% of youth were retained in the follow-up survey, which is a
somewhat lower percentage than in previous studies. Nevertheless, we obtain a
population of 34 youth who provided follow-up data.
6.1 Influence spread results
Fig. 2. Percentage of youth who were not peer leaders reached by each algorithm in
its respective real-world pilot test.
We now present our core result: the number of youth who received a message
about HIV prevention. We examine the percentage of youth in the follow-up
End to end influence maximization for HIV prevention 11
group who were not peer leaders (and hence eligible to become influenced) who
reported receiving information. Figure 2 shows this percentage for our pilot
study of CHANGE as well as the percentages reported by Yadav et al. [17]
in their pilot studies of the state of the art algorithms HEALER and DOSIM.
CHANGE reached 80% of non-peer leaders compared to approximately 70%
for each of HEALER and DOSIM. Thus, CHANGE was able to reach just as
many youth while gathering data from only 18.75% of the network. The 10%
difference between CHANGE and HEALER/DOSIM could be attributable to
random variation; we do not claim that CHANGE is actually more effective than
algorithms which gather the entire network. Nevertheless, this result provides
empirical evidence that CHANGE can perform comparably to existing state of
the art influence maximization agents while drastically reducing the amount of
data required.
7 Discussion and conclusion
This paper presents the CHANGE agent for influence maximization, a multia-
gent problem with many applications in preventative health and other domains.
CHANGE addresses major barriers to the deployment of influence maximiza-
tion by service providers through a series of algorithmic contributions, backed
by simulation results on real-world networks. We then conducted a real-world
pilot study of CHANGE with a drop-in center serving homeless youth, the first
such pilot study of sampling-based influence maximization and only the second
study testing any influence maximization agent in the real world. CHANGE
obtained comparable influence spread to previously field tested algorithms, but
surveyed only 18% of youth to obtain network data. CHANGE has empirical
promise in delivering high-quality influence maximization solutions in a manner
which can be feasibly implemented by a service provider.
While the algorithms underlying CHANGE are easy to implement, they draw
on a series of insights into the social behavior of homeless youth. One lesson
learned from our study is that, to be successful in the field, algorithms must be
designed with their target population and setting in mind. CHANGE both nav-
igates challenges specific to homeless youth (e.g., the difficulty of locating youth
to query for edges or serve as peer leaders) and leverages properties of their so-
cial network (the friendship paradox). Our experience shows that accounting for
both challenges and opportunities in the target population is crucial to produce
a practically deployable algorithm.
Acknowledgments: This research was supported by MURI Grant W911NF-
11-1- 0332, the California HIV/AIDS Research Program, and a NSF Graduate
Research Fellowship.
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