IEEE TRANSACTIONS ON CYBERNETICS, VOL. 47, NO. 1, JANUARY 2017 165 Modeling … · 2016-12-21 · IEEE TRANSACTIONS ON CYBERNETICS, VOL. 47, NO. 1, JANUARY 2017 165 Modeling and Analysis

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 47, NO. 1, JANUARY 2017 165

Modeling and Analysis of Group Dynamicsin Alcohol-Consumption Environments

Luis Felipe Giraldo, Kevin M. Passino, Fellow, IEEE, and John D. Clapp

Abstract—High-risk drinking is considered a major concernin public health, being the third leading preventable cause ofdeath in the United States. Several studies have been conductedto understand the etiology of high-risk drinking and to designprevention strategies to reduce unhealthy alcohol-consumptionand related problems, but there are still major gaps in identifyingand investigating the key components that affect the consumptionpatterns during the drinking event. There is a need to developtools for the design of methodologies to not only identify suchdangerous patterns but also to determine how their dynamicsimpact the event. In this paper, based on current empirical evi-dence and observations of drinking events, we model a humangroup that is in an alcohol-consumption scenario as a dynamicalsystem whose behavior is driven by the interplay between theenvironment, the network of interactions between the individu-als, and their personal motivations and characteristics. We showhow this mathematical model complements empirical researchin this area by allowing us to analyze, simulate, and predict thedrinking group behaviors, to improve the methodologies for fielddata collection, and to design interventions. Through simulationsand Lyapunov stability theory, we provide a computational andmathematical analysis of the impact of the model parameterson the predicted dynamics of the drinking group at the drink-ing event level. Also, we show how the dynamical model can beinformed using data collected in situ and to generate informationthat can complement the analysis.

Index Terms—Consensus, drinking groups, Lyapunov stabilitytheory, networks, public health, social dynamics.

I. INTRODUCTION

HEAVY alcohol consumption is the cause of approx-imately 1800 deaths per year among college stu-

dents and is considered a major public health issue in theUnited States [1]. During the last 20 years, researchers havetried to understand the etiology of heavy drinking among thispopulation and design strategies to intervene to reduce heavydrinking and its consequent problems [2]. The main goal isto find the “leverage points” of the heavy drinking event,

Manuscript received June 22, 2015; revised October 13, 2015 andDecember 13, 2015; accepted December 14, 2015. Date of publicationDecember 29, 2015; date of current version December 14, 2016. This paperwas recommended by Associate Editor J. Liu.

The authors are with the Department of Electrical and ComputerEngineering, and the College of Social Work, Ohio State University,Columbus, OH 43210 USA (e-mail: [email protected];[email protected]; [email protected]).

This paper has supplementary downloadable multimedia material availableat http://ieeexplore.ieee.org provided by the authors. The supplementary filecontains implementation details of the proposed model. The total size of thefile is 113 KB.

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCYB.2015.2509444

that is, the places and times in the event that are crucial forintervention.

Several studies have been conducted to examine the behav-ior of groups that are in naturally occurring drinking eventsin situ [3]–[5]. These studies have found that the dynamics ofalcohol consumption are affected by the characteristics of theindividual (e.g., drinking motivations and history), event-levelfactors (e.g., duration of the drinking event or playing drink-ing games), and environmental factors (e.g., dancing, food, ordrink specials). Statistical tools have been employed to analyzehow these factors are correlated and to determine their signif-icance on drinking behaviors [6]. Even though these studiesare very important for better understanding high-risk drink-ing, they are very expensive, difficult to conduct, and theyhave major gaps. For example, the impact of the interactionbetween the group network and the individual motivations inthe dynamic environment has not been measured or mod-eled. The social interactions play an important role duringthe drinking activity [6]–[8], and studying them is criticalto understanding the etiology of high-risk alcohol consump-tion. There is a need to develop methodologies to not onlyidentify the most important factors that affect the alcohol-consumption patterns, but also to determine how they influencethe dynamics of the group throughout the drinking event. Thisis done here.

This need has led to an increasing interest in developingdynamical system models as tools to complement empiri-cal research that addresses not only alcohol-related problemsbut also public health issues in general [9], [10]. Models ofdynamical systems provide a way to analyze comprehensivelythe problem settings, to develop more effective interventiondesigns and evaluation methods, and to plan large-scale fieldstudies. As part of the research in public health that worksto develop strategies for the effective design of interventionsthat reduce high-risk alcohol consumption, dynamical modelshave been proposed to characterize how drinking patterns areaffected by the social interactions in large groups [11], [12].These models describe how the alcohol usage in large popu-lations changes between categories such as “heavy-drinkers,”“social-drinkers,” and “nondrinkers.” The formulation of thesemodels is closely related to the ones employed to describe thedynamics of infectious diseases that spread in a population.Also, several simulation methodologies have been designedto recreate drinking scenarios [13], [14]. However, to ourknowledge, building a model that describes how the bloodalcohol content (BAC) level in a group changes over time dur-ing the drinking event and permits analysis not only through

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166 IEEE TRANSACTIONS ON CYBERNETICS, VOL. 47, NO. 1, JANUARY 2017

Fig. 1. Conceptual diagram describing the interplay between theory, empir-ical evidence, and mathematical and computational models in the study ofthe dynamics of social systems. In that spirit, this paper uses previously col-lected field data on drinking, theoretical assumptions of group behavior, anda model of a dynamical system along with computer simulations to advanceour understanding of the etiological ecology of drinking events.

simulations but also at the mathematical level for a compre-hensive understanding of the modeled behaviors has not beenstudied yet.

Here, we construct, based on current observations andempirical data on drinking groups, a model of a system thatcharacterizes how the dynamics of the social interactions, indi-vidual characteristics, and environment translate into changesin the drinking patterns of individuals measured through theBAC level. We derive a formulation of the model and a mathe-matical analysis of the behaviors that can be characterized, andshow how this model could complement empirical research byinforming theory and testing constructs. In Fig. 1, we present aconceptual diagram of how the theoretical analysis of drinkinggroups interacts with the empirical research and mathematicaland computational system models.

A. Modeling Group Dynamics

In the theoretical analysis of groups in social psychol-ogy, the behavior of the group is assumed to be influencedby the environment and the mutual interactions between thegroup members. This is described by Lewin, who is con-sidered the father of social psychology, through the formulaB = f (P, H), where the individual’s behavior B is a functionf of the personal characteristics and preferences P, and exter-nal influences that include the environment and other peopleH [15], [16, p. 17]. Our previous field studies on drinkingevents are consistent with this description of group dynam-ics. These studies have shown that the personal preferenceson drinking is not the only mechanism that drives the behav-ior of an individual in a drinking group. For example, using aportal design [17], we studied the group drinking behavior inbars [3]. It was observed that college students usually drinkin settings that vary in risk and protective factors from heavydrinking [6], [18]. Patrons were interviewed and breath alcoholsamples were taken upon entering and exiting the bar. Whenbar patrons are asked their intended level of intoxication whenentering a bar, their level of intoxication often failed to matchtheir previously stated intentions (e.g., to get very drunk) oncethey exited the bar. Fig. 2 shows the measured BAC using abreathalyzer when entering and exiting the bar for four differ-ent categories of the intended intoxication level. Although theaverage BAC does tend to change across categories, it is clearthat the BAC that is reported when exiting the bar does not

Fig. 2. Measured BAC on subjects before entering the bar (horizontal axis)and after they leave it (vertical axis). Each dot corresponds to the BAC sampleobtained from an individual, and the dashed line corresponds to a 45◦ linefor reference. The desired level of alcohol that the subjects reported beforeentering the bar is categorized as (a) not drinking, (b) not drinking enough toget buzzed, (c) slight buzz, and (d) drunk.

always match the intended level of intoxication. These obser-vations suggest that the behavior of individuals in a groupduring the drinking event is affected by additional basic mech-anisms. We want to propose then a model that describes thedynamics of groups in drinking environments that is consistentwith these observations and the theoretical analysis of groupdynamics.

Current studies where data are collected in situ during thedrinking events, although they are very informative, they fallshort of describing the drinking patterns throughout the drink-ing event. For example, in the study reported in [3], the BAClevel is measured only before and after the drinking event,but there is no information of how the BAC levels change inbetween. However, current technology has now made accessi-ble new types of information that helps us overcome severalof these limitations. It allows us to have a real-time moni-toring of the BAC level [19], social interactions [20], [21],and location in space [22]. Our medium-term goal then isto design and conduct new studies where observational andself-report survey data are collected along with measurementsthat quantify the dynamics of the drinking group. The con-tribution of this paper is to provide a mathematical modelthat, based on up-to-date observations and empirical data ofdrinking groups, we hypothesize captures the dynamics ofthe BAC level given the mechanisms that drive the behav-ior of the group, and can be updated once we collect datafor this purpose. In the same way conventional physics triesto explain how force translates into changes in motion, ouraim is to create a model that explains the “physics” of thedrinking event: how the influences from the individual’s per-sonal preferences (e.g., desired effect of the BAC level onhis/her body), other members of the group, and the environ-ment translate into changes in the BAC level. We do not claimthat this is a perfect model. This paper is just a step in the

GIRALDO et al.: MODELING AND ANALYSIS OF GROUP DYNAMICS IN ALCOHOL-CONSUMPTION ENVIRONMENTS 167

cyclic process depicted in Fig. 1: given current empirical evi-dence and theory of group dynamics, we propose a model thatimproves our understanding of the drinking event and allowsus to design better methodologies for the collection of newevidence. This evidence will eventually be used to evaluatethe model and make the corresponding improvements/changeson the hypotheses concerning the mechanisms that drive theevent dynamics and the construction of intervention strategies.

The proposed model follows a well-developed mathematicalframework used in engineering to study the dynamics of mul-tiple interactive agents and their stability properties [23]–[25].Under this framework, we are able to incorporate in a dynam-ical system the relationship between the BAC level and theindividuals’ personal characteristics, the environment they arein, and the influence network between the members of thedrinking group (Section II). Using computer simulations andLyapunov stability theory [26], we present a computationaland mathematical analysis of how the parameters affect themodeled dynamics (Sections III and IV), and a discussion ofthe importance of these results in the further design of inter-ventions and methodologies for the prevention of high-riskalcohol consumption (Section V).

II. CONSTRUCTION OF THE MODEL

Our goal is to model how an individual regulates his/herown BAC given the current state of the BAC and the influ-ence of his/her personal motivations, other individuals, and theenvironment the individual is in. We explain step by step howthe model is constructed, starting from the simpler case wherethe individual is only influenced by his/her own motivations,and extend it to the case where the group and the environmentaffect the dynamics of the individuals.

A. Individual Influences on Behavior

For the construction of the model, we quantify the behav-ior of an individual using his/her BAC level and its rate ofchange varying over time. Let xi(t) ≥ 0 denote the BAC levelof individual i at time t ≥ 0, and let vi(t) ∈ R denote itsrate of change. We start by assuming that there is no envi-ronmental and group pressures, and that individual i has aunique desired BAC level x∗

i . Research suggests that individ-ual factors such as drinking motives (e.g., desired outcomeand alcohol level) and drinking history (frequency of heavydrinking) influence the level of intoxication in the individualat the event-level [18], [27]. We assume therefore that x∗

i ischosen by the individual in accordance with his/her personalmotives and characteristics. The model then should describethe behavior of an individual such that he/she tends to regu-late his/her own BAC to reach the desired level. This meansthat if the actual BAC is below x∗

i , then the dynamics of theindividual (e.g., amount of alcohol consumed) should be suchthat there is an increase in his/her BAC level. On the otherhand, if the actual BAC is above x∗

i , then the dynamics ofthe individual should be such that there is a decrease his/herBAC level. An initial model that captures such dynamics isshown in the causal loop diagram in Fig. 3. In this model, itis assumed that the individual regulates his/her change rate of

Fig. 3. Causal loop diagram of the dynamics of the BAC modeled using afirst-order system. The action proportional to error corresponds to the productof the difference in BAC and a scaling factor that determines how quickly theindividual reaches the desired BAC level. In this model, it is assumed thatthere are no group and environmental pressures. The causal links that haveno sign are assumed to have positive polarity.

the BAC proportional to the difference between the desiredlevel and actual level. The constant equilibrium point of thedynamical system is x∗

i . The differential equation associatedwith the diagram in Fig. 3 is given by xi(t) = η

pi (x

∗i − x(t)),

where xi is the derivative of xi with respect to t, vi(t) = xi(t),and η

pi > 0 is the proportional action parameter, which deter-

mines how quickly the individual changes his/her BAC, andcan be seen as the commitment of individual i to reach thedesired level.

Even though this model captures the basic behavior of anindividual that wants to reach his/her desired BAC level, itis restrictive with respect to the dynamics that can be rep-resented. An example is the fact that the BAC trajectoriesmodeled using the system in Fig. 3 will never exhibit an over-shoot with respect to the desired BAC level, a situation thatis actually possible in drinkers who are not able to regulateaccurately the alcohol consumed during the drinking event.A model formulation that generates a richer set of trajecto-ries of the BAC level, including the possibility of modelingovershoot, is presented in Fig. 4, which is an extension ofthe one in Fig. 3 where now the dynamics of the individualare such that the difference between the desired level and theactual BAC affects the acceleration of the BAC, that is, itdirectly affects how vi changes over time. According to thediagram in Fig. 4, an individual tends to accelerate his/herBAC when the actual BAC is below the reference, and deceler-ate when it is above. Also, an individual is assumed to restrainthe BAC acceleration depending on his/her perception on howquickly the BAC level is changing. This last component ofthe model is the one that shapes the trajectory and allows thebehavior of the BAC level to have overshoot or not. The setof differential equations that represents this dynamical systemis given by

xi(t) = vi(t)

vi(t) = ηpi

(x∗

i − xi(t)) − ζivi(t) (1)

where ζi > 0. Note that an individual accelerates or decel-erates his/her BAC depending on the current value of vi(t)and where the actual BAC level is with respect to the desiredone. Parameter η

pi determines how strong the commitment of

the individual to reach the desired BAC level is [i.e., com-mitment to make x∗

i − xi(t) = 0], and parameter ζi can beseen as the strength of the opposition of individual i to quickvariations of the BAC level. Fig. S1 in the supplementary fileshows instances of BAC trajectories that can be generated fordifferent values of η

pi and ζi, given x∗

i .


Fig. 4. Causal loop diagram of second-order system that models the dynamicsof the BAC level. It is assumed that there are no group and environmentalpressures.

Although the dynamical system in Fig. 4 with differentialequation in (1) allows for modeling a wide variety of behav-iors, the assumption that the individual has a unique desiredBAC level is restrictive. It can be the case that the objectiveof an individual is to reach any BAC level within an intervalthat produces certain body reactions. For example, a personwho wants to have a “slight buzz” effect will typically reachBAC levels that are between 0.02 and 0.04 [28, Ch. 4]. Tobe able to generalize our formulation to include these casesin the model, we introduce the personal preference function.Let f p

i : R → R be a continuously differentiable function thatquantifies how preferable a given BAC is for individual i. Weassume that f p

i has a unique minimum, and without loss ofgenerality, minx f p

i (x) = 0. Lower values of f pi (xi) indicate

more preferable BAC levels for the individual. For example,if individual i wants to get drunk, then higher values of xi willcorrespond to lower values of f p

i (xi). The influence of an indi-vidual’s personal preference on his/her behavior is then givenby the negative derivative of function f p

i (x) with respect to x,since it points to the direction where f p

i (x) decreases. Thismeans that −df p

i (xi(t))/dxi indicates whether the individualshould increase or decrease his/her BAC in order to reach thedesired level. It is implicit in the system in Fig. 4 and (1)that individual i has a function f p

i (xi(t)) = (1/2)(xi(t) − x∗i )

2,where (1) can be rewritten as

xi(t) = vi(t)

vi(t) = −ηpi

df pi

dxi(xi(t)) − ζivi(t). (2)

In the specific case of the system in (1), the negative deriva-tive of the preference function with respect to xi is given by−df p

i (xi(t))/dxi = x∗i − xi(t), which corresponds to the differ-

ence in BAC part of the model. Note that in this case f pi (xi(t))

has a unique minimum at x∗i (the most desirable BAC level,

where the derivative is zero) and is symmetric around thispoint. Instead of modeling the influence of the individual onhis/her own behavior to reach the desired BAC level throughthe term η

pi (x

∗i − xi(t)), we model it in a more general way

using the negative derivative of the preference function, wherenow the goal is to reach a BAC level such that −df p

i (xi(t))/dxi

is zero.Using the concept of preference function, we can take

empirical evidence that categorizes the effect of BAC levelon an individual’s body to construct different function pro-files that describe how an individual tends to regulate his/herBAC to reach the desired effects. Fig. 5(a) shows the prefer-ence profiles for four categories of alcohol intoxication: “notenough to get buzzed,” slight buzz, “drunk,” and “no drinking.”

Fig. 5. (a) Preference functions associated with the categories of the body’sreaction to the BAC level: not enough to get buzzed, slight buzz, drunk, andno drinking. The formulation of these functions is presented in Table S1 inthe supplementary file. (b) Trajectories of the BAC level for different initialconditions and values of η

pi and ζi. In this case, the individual’s desired effect

is slight buzz, which implies that the final BAC level is within the interval[0.02, 0.04], which is where the derivative of the corresponding preferencefunction in (a) is zero. The MATLAB source code to generate these plots isavailable in Section E of the supplementary file.

The mathematical formulation of these functions is presentedin Table S1 in the supplementary file. Note that the mini-mum values of these profiles correspond to the intervals ofthe BAC level that have been reported to produce the respec-tive effects. BAC levels in the interval [0, 0.02] are associatedwith category not enough to get buzzed, interval [0.02, 0.04]with slight buzz, values greater than 0.04 with drunk, and0 with no drinking [28, Ch. 4]. The slope of the functions ischosen to be larger for values of the BAC that are before theinterval where the function is minimum than those that areafter the interval, implying that the individuals are more com-mitted to increase their BAC to reach the desired effect thanto decrease it. Fig. 5(b) shows example trajectories modeledusing (2) of individuals whose preferred effect of the BAClevel is slight buzz. All the trajectories converge to the inter-val of the BAC level [0.02, 0.04], instead of converging to asingle BAC level, due to the appropriate choice of thepreference function.

The dynamical system in (2) enables us to model a varietyof trajectories that the BAC level of an individual can have in adrinking event. It has the flexibility to model the commitmentof the individual to approach the desired levels, his/her ownperception of the change rate to restrain quick changes on hisBAC, and also through the personal preference function we


can define intervals of desired levels that are associated witheffects on the individual’s body. The dynamical system in (2)can be seen from a physics perspective. As in Newton’s secondlaw, there are “forces” acting on the individual’s acceleration.The concept of forces is not necessarily the one measured inNewtons, but it refers to influences that cause a change in theindividual’s BAC level, which in this case correspond to theinfluence of the individual to reach the desired BAC levels andthe influence that restrains the individual to quick changes inhis/her BAC according to his/her perception of the BAC rate ofchange. We will prove in Section IV-A that this last component(perception of the BAC rate of change) is necessary in themodel for the convergence of the individual’s BAC level tothe desired interval.

B. Adding Social and Environmental Influences

In the study of human group dynamics, it has been observedthat the behavior of an individual is determined not only byhis/her own personal characteristics but also by the influencesof the group and environment he/she is in [15] and [16, p. 17].This has been already observed in the particular case wherethe group is in an alcohol-consumption environment. It hasbeen shown that the social interactions influence the dynam-ics of the individuals, where the patterns of communicationbetween group members and the strength of the influence ofother members on an individual play an important role inshaping the behavior of the whole group [29], [30]. Also,data collected in situ provide evidence that environmental fac-tors have a significant effect on the dynamics of the drinkinggroups [31], [32]. For example, an environment where thereare large crowds has an impact on the behavior such thatthere is a tendency to restrain increases in the BAC. On theother hand, environments with drinking games promote higherBAC levels.

In the same way the personal preference function is usedto include in the model the influence of the individual onhis/her own behavior to reach the desired BAC levels, wedefine functions to describe how the environment and thegroup affect the behavior of individual i. First, to model theinfluence of the group on an individual’s behavior, we con-sider evidence that suggests that in a group of people thereare mutual attractions that lead the group toward consensusin drinking behavior [29], [33]. This means that, in the con-text of our model, an individual is attracted to the BAC leveland its change rate of other individuals. Since an individualmight interact with only a subset of the group and the impactof those interactions can vary in strength between people, weneed to formally define an influence network in the drinkinggroup. Assume that the drinking group has n members. Thestructure of the network is given by G = (V, E,W), whereV = {1, . . . , n} is the set of labels for each individual inthe group, and E ⊂ V × V is the set of directed links thatconnect the individuals. Link (i, j) ∈ E indicates that indi-vidual j influences individual i. It might be the case where( j, i) ∈ E does not exist, meaning that there is no influencefrom i to j. Let Ni = { j ∈ V : (i, j) ∈ E} be the set of allthe group members that have some influence on individual i.

Each link (i, j) is associated with a weight wij > 0 that cor-responds to the strength of the influence of j on i. We havethat W = {wij : i ∈ V, j ∈ Ni} is the set of all the weightsassociated with the links in E . The mathematical formulationof the social interactions in this context is in terms of attrac-tions. We say that individual i is influenced by j if there isa tendency from i to follow the BAC level and/or its rate ofchange of individual j. It is assumed that each individual isable to estimate the BAC level and its rate of change of thegroup members that influence him/her (i.e., those that are inthe set Ni). The function associated with the group influenceis formulated as

f gi (xi(t), vi(t))

= 1

2

∑

j∈Ni

wij

[(xi(t) − xj(t)

)2 + b(vi(t) − vj(t)

)2]

where b ≥ 0 is a parameter associated with the influence on therate of change of the individual’s BAC level. This function isminimized when individual i tends to follow his/her neighbors’BAC level and its rate of change. Larger values for wij implya larger attraction of individual i to follow j. The influencecomponent associated with the social pressures will be thengiven by the negative derivative of f g

i with respect to both xi

and vi

ai(t) = −df gi

dxi(xi(t)) − df g

i

dvi(vi(t))

= −∑

j∈Ni

wij[(

xi(t) − xj(t)) + b

(vi(t) − vj(t)

)]. (3)

Term −wij[(xi(t)−xj(t))+b(vi(t)−vj(t))] is a value that pointsto the direction where xj(t) and vj(t) are with respect to xi(t)and vi(t) and describes the influence of j on the behavior of i.This part of the model is the one that allows us to characterizethe drinking group as an interconnection of dynamical systems,that is, a dynamical social system.

To define the influence of the environment on the group’sdrinking behavior, we define a continuously differentiablefunction f e : R → R that is associated with the environmentthe group is in. The environment promotes BAC levels wherethe function f e takes lower values. For example, an environ-ment where there are large crowds will have a function profilewith lower values at lower BAC levels. The influence of theenvironment on individual i is given by the scaled negativederivative of this function

hi(t) = −ηei

f e(xi(t))

dxi(4)

where ηei ≥ 0 is the strength of the influence of the

environment on individual i. Equation (4) characterizes theenvironmental pressures that affect the behavior of the drinkinggroup.

Using the results in (3) and (4) along with (2), we con-struct a model that characterizes how influences of the group,environment, and individual preferences on the BAC effectstranslate into changes in a person’s BAC level. To complete themodel, we assume that the action of these three components


Fig. 6. Causal loop diagram of the final model that accounts for the effectof the individual characteristics, group, and environment on changes in theBAC level.

on behavior is additive. The complete model is illustrated inFig. 6 and defined in the set of equations

xi(t) = vi(t)

vi(t) = pi(t) + ai(t) + hi(t) + di(t) (5)

where pi(t) = −ηpi df p

i (xi(t))/dxi, and di(t) = −ζivi(t).

III. SIMULATION RESULTS

The mathematical formulation of the group dynamics in (5)allows us to characterize behaviors that have been observedin human groups involved in alcohol-consumption activities.Next, through simulations, we show the dynamics that canresult for different choices of the desired effect of the alcoholintoxication and the structure of the influence network in thedrinking group. Also, we study data from real drinking groupsthat were collected in situ. We use this information to informour model and make predictions about the behavior of thegroup given some assumptions on the parameters of the model.

A. Group Dynamics Under Different Conditions on thePersonal Preferences and Influence Networks

In the first set of simulations, we study the dynam-ics of a group of six people in three different scenarios.Individuals 1–3 have slight buzz as the preferred effect of theBAC level, meaning that these individuals prefer values of xwithin the interval [0.02, 0.04]. On the other hand, “Not drink-ing” is the preference for individuals 4–6, which means thatthey prefer having values of x that tend to zero. The profilesof the preference functions for these two subgroups are shownin Fig. 5(a). The implementation details and source code ofthe simulations that we present in this section are given inSections C and F of the supplementary file.

In the first case, the group has a network structure as shownin Fig. 7(a). Each arrow indicates the direction of the influencebetween a pair of individuals, and its thickness is proportionalto the strength of the influence [i.e., wij in (3)]. The onlyinteraction between subgroups occurs between 3 and 4. Theinfluence of 3 on 4 is stronger than the influence in any otherpair of individuals. There is no influence of 4 on 3. Fig. 7(b)shows the dynamics of the group given the initial conditionsand model parameters. Note that the individuals whose pref-erence is no drinking initially tend to have lower values ofthe BAC level even though some of them start at high values.

Fig. 7. (a) Influence network in a group where individuals 1–3 have slightbuzz as desired effect of the BAC level while individuals 4–6 prefer beingin the category no drinking. (b) Trajectories of the simulated BAC level foreach one of the individuals in the group.

People in the group whose preferred effect of the BAC levelis slight buzz tend to increase their BAC. However, the stronginfluence of 3 on 4 makes 4 change his/her behavior in a waythat his/her BAC tends to reach BAC levels in the interval thatproduces the slight buzz effect. The mutual influences of 4 on5 and 6 are not strong enough to change significantly theirbehavior.

In the second case, individuals 5 and 6 influence each other,as shown in Fig. 8(a). Now, the influence of 4 on 5 and 6is stronger than the one in the first case in Fig. 7(a). Thetrajectories of the BAC through time in Fig. 8(b) show thatindividuals 4–6 are affected by the influence of individual 3on 4, where their BAC tends to increase even though theirpersonal preference is no drinking.

B. Estimation of Group Dynamics From Field Data

The information in [3] contains field data from 1024 peoplesurveyed at 30 different bars. Data collected from each subjectinclude the BAC measurements before entering the bar andafter leaving it, duration time in the bar, whether the subjectis alone or not, the amount of money available to spend onfood, and the amount of money available to spend on alcohol.The subjects reported the level of alcohol intoxication thatthey desired to reach during the drinking activity by choosingone of the following categories: not drinking, not drinking


Fig. 8. (a) Influence network in a group where individuals 1–3 have slightbuzz as preferred effect of the BAC level, and individuals 4–6 prefer nodrinking. The difference of this network with respect to the one in Fig. 7(a)is that now individuals 5 and 6 influence each other, and the influence of 4on 5 and 6 is stronger. (b) Trajectories of the simulated BAC for each one ofthe individuals in the group where the influence network is given in (a).

enough to get buzzed, slight buzz, and drunk. In addition tothis information, there is observational data that report howcrowded the bar was, and whether the subjects were exposed todrinking games and alcohol and food specials. Despite of thelarge amount of data collected from drinking events at differentplaces, the only available information about the individual’sdrinking patterns is the BAC measured at the entrance andexit of the bar. We show in this section how we can have atentative estimation of the trajectory of the BAC level duringthe drinking event given the available field data and someassumptions on the dynamics of the group.

First, we define the personal preference and environmentfunctions. Since the subjects reported the desired effect ofthe BAC level that they wanted to reach during the drinkingevent, we use the preference profiles as shown in Fig. 5(a). Theenvironment function is constructed as the linear combinationof influences that either protect against or favor increases inthe alcohol level. The amount of money available to spendon food, whether food specials are offered or not, and howcrowded the bar was, is information used to construct the com-ponent of the environment profile that promotes lower alcohollevels. On the other hand, the presence of drinking games, theamount of money to spend on alcohol, and alcohol specialsrepresent information useful to construct the component of the

environment profile that promotes higher BAC. The perceptionof the BAC rate of change ζi in (5) is assumed to be the samefor all the individuals.

The only information available about the social interaction iswhether the subject is alone or not. Since there are no reportedobservations of the social interactions between subjects, weassume in our model that there exists a nonobserved socialcomponent that allows the subject to achieve the reported finalBAC level, given the preferred effect of the BAC level, envi-ronment profile, initial BAC level, and the duration time of thedrinking activity. Assuming that parameter b = 0, for thosethat reported that were not alone during the drinking event (3)can be rewritten as

ai(t) = −wi(xi(t) − xi(t)) (6)

where wi = ∑j∈Ni

wij is the total influence strength actingon i, and xi = (1/wi)

∑j∈Ni

wijxj is the weighted average ofthe BAC level of those that influence individual i. Here, weassume that xi points to the reported final BAC level. Parameterwi > 0 is computed to be the smallest scalar such that the tra-jectory of the BAC level generated using (3) along with (6)and the previous assumptions reaches the measured final BAClevel during the reported time of the drinking activity and start-ing at the measured initial BAC level. Low values of wi do notimply that there was not social interactions during the drinkingevent. It implies that, under the assumptions on the parame-ters of the model and the reported information, the unobservedsocial pressures are not significantly stronger than other influ-ences to modify the drinking patterns of the individual. Thedetails of the implementation, including the construction ofthe environment profile, can be found in Section D of thesupplementary file.

Fig. 9 shows the results obtained using our model follow-ing the methodology and assumptions described above on thedata collected in situ from individuals that reported drunkas desired effect of BAC level to reach during the drinkingevent. Using the corresponding preference function as shownin Fig. 5(a), the environment function constructed from theinformation in the field data, the initial and final BAC, andthe time spent in the bar, we show in Fig. 9(a) the estimatedtrajectories followed by ten randomly selected subjects duringthe drinking activity (solid lines), and the estimated strengthof the social influence on the subjects (thickness of the lines),which is proportional to wi in (6). Note that the thickest linecorresponds to the trajectory of a subject that maintains arelatively low level of alcohol with respect to his/her pref-erence of getting drunk. This person reported that he/she didnot get food, the bar was not crowded, there were not foodspecials, and there were alcohol specials and drinking games.Under the assumptions on the parameters of the model and thereported information, the dynamics estimated from the modelsuggest that there were strong social pressures on the subjectthat influenced his/her behavior.

Also, we show in Fig. 9(b) a plot of the BAC when the sub-jects exited the bar versus the BAC before they entered it. Thesize of the markers is proportional to the estimated strengthof the social influence on the individuals. Note that individu-als who maintain lower levels of the BAC tend to have larger


Fig. 9. (a) Estimated trajectories of the BAC provided by the model forten randomly selected subjects that reported drunk as the preferred level ofalcohol intoxication, where symbol “x” marks the BAC measured when thesubject exited the bar. (b) Samples of the BAC level when entering the bar(horizontal axis) and after leaving it (vertical axis). The dashed line is a 45◦line for reference. The thickness of the lines in (a) and the size of the markersin (b) are proportional to the strength of the social influence acting on thesubjects.

social pressures acting on them than those that have larger lev-els of the BAC. These results are consistent with the fact thatthis set of subjects had drunk as the desired level of intoxi-cation to reach while they were in the bar, hence additionalpressures on the behavior are required to maintain lower valuesof the BAC. The sample pointed by the arrow corresponds toan individual that was not exposed to drinking games and hadlittle money to spend in alcohol. In this case, the results givenby the model suggest that the social pressures were not signif-icantly larger than the other influences to change the dynamicsof the individual.

IV. MATHEMATICAL ANALYSIS OF THE GROUP DYNAMICS

In the previous section, we provided computer simulationsto show the dynamics that the proposed model is able todescribe and that are consistent with the observations of groupsthat are in a alcohol-consumption environment for a given setof parameters and initial conditions. Now, through a mathemat-ical analysis of the model, we study how the model parametersaffect all possible trajectories of the BAC level that can bedescribed by the system in (5) given any initial condition.This analysis is based on Lyapunov stability theory [26], which

equips us with mathematical tools to derive statements on thedynamics of the group given specific parameter settings. First,we study the case when the influence strength of the socialinteractions and the environment is zero and the behavior ofthe individual is driven mainly by his/her desire to reach aspecific effect of the BAC level. Theorem 1 shows that eachindividual reaches the desired effect in the drinking activitywhen there is no influence of other members of the group andthe environment. Then, through Theorem 2, we show how thedynamics of the group are affected when there are social inter-actions and also environmental pressures on the group. We alsodiscuss the implications of these mathematical results.

A. No Social and Environmental Pressures

In this analysis, we use the concept of asymptotic stabilityto show that when there are no social and environmental pres-sures, the individuals achieve their personal preferences in thelong term. We start by assuming that f p

i (personal preferencefunction of individual i) is a strictly convex function that has aunique minimum at x∗

i , for i = 1, . . . , n. Without loss of gen-erality, f p

i (x∗i ) = 0 for every i ∈ {1, . . . , n}. Let xi = xi −x∗

i bethe relative BAC level of individual i with respect to his/herdesired BAC level. Assume that there is no influence fromthe other members of the group and the environment on theindividual. Then, the dynamics of i can be written as

˙xi = vi

vi = −ηpi

df pi

dxi

(xi + x∗

i

) − ζivi, i = 1, . . . , n. (7)

The following theorem shows that an individual modulateshis/her own behavior such that his/her BAC level reaches thedesired one when there is no pressures from the social interac-tions and the environment the group is in, and that the conceptof perception of the BAC rate of change affecting the BAClevel has to be included in the model to guarantee convergenceof the trajectories of the BAC level.

Theorem 1: Consider the dynamics of the individuals in thegroup characterized in (7). The point xi = 0 and vi = 0 is aglobally asymptotically stable equilibrium.

Proof: Let Vi(xi, vi) = ηpi f p

i (xi + x∗i ) + (1/2)|vi|2. From the

assumptions that f pi (x∗

i ) = 0 and that f pi is strictly convex and

has a unique minimum at x∗i , we have that f p

i (xi + x∗i ) > 0

for all xi = 0. Since |vi|2 > 0 for all vi = 0, it implies thatVi(xi, vi) = 0 only when xi = 0 and vi = 0, and Vi(xi, vi) > 0otherwise. We use then Vi as our Lyapunov function candidate.Its derivative with respect to time is

Vi(xi, vi) = ηpi

df pi

dxi

(xi + x∗

i

) ˙xi + 1

2vivi

= −ζi|vi|2 ≤ 0

where Vi(xi, vi) is negative semidefinite. However, from (7),we can see that if vi(t) = 0 for a given t, variable vi(t) isnot zero unless xi(t) = 0. This is consistent with LaSalle’sinvariance principle [26, Th. 4.4]. Let S = {x, v : V(x, v) = 0}.From (7), we have that no solution can stay identically in S


other than xi = 0 and vi = 0. Therefore, from LaSalle’s invari-ance principle, the solution xi = 0 and vi = 0 is globallyasymptotically stable.

Remark 1: Since there is no effect of the environment onthe group and there is no coupling between its members, theforce that drives the individual’s behavior is his/her own pref-erences. Hence, the dynamics of an individual in the drinkinggroup modeled in (7) are the result of an optimization processdriven by the cost function f p

i .Remark 2: The fact that ζi is greater than zero guarantees

that the individual approaches the point xi = 0 and vi = 0as t tends to infinity. The term −ζivi in (7) causes a gradualdeceleration of the trajectories of xi as xi approaches zero, i.e.,when xi approaches x∗

i .

B. Influence of the Environment and Social Interactionson the Group Dynamics

To examine the influence of the environmental and socialpressures on the dynamics of the drinking group, we now thinkof the group as a dynamical system that is a perturbation ofthe “nominal” system given in (7), where the perturbationsare given by the components of the force associated with theenvironment and social interactions in (3) and (4). In this case,we study the dynamics of the system through the concept ofuniform ultimate boundedness [26, Ch. 4.8], which allows usto characterize how the resultant dynamics of the BAC levelin the group members deviate from their desired level in termsof the model parameters.

Assume that f e (environment function) is a strictly con-vex function and has a unique minimum xe ∈ R, and that itsderivative is Lipschitz continuous, that is, for any z, y ∈ R

there exists a constant Me > 0 such that |(df e/dx)(z) −(df e/dx)(y)| ≤ Me|z−y|. Also, as in Section IV-A, we assumethat f p

i is strictly convex and has a unique minimum at x∗i

with f pi (x∗

i ) = 0, and that its derivative is also Lipschitz con-tinuous with constant Mp

i > 0, for every i = 1, . . . , n. Letx = [x1, . . . , xn]�, x∗ = [x∗

1, . . . , x∗n]�, x = [x1, . . . , xn]�, and

v = [v1, . . . , vn]�. Also, let

Fp(x) =[

ηp1

df p1

dx(x1), . . . , η

pn

df pn

dx(xn)

]�

Fe(x) =[ηe

1df e

dx(x1), . . . , η

en

df e

dx(xn)

]�.

We define matrix L such that its entry at the ij position isgiven by

[L]ij =⎧⎨

⎩

0 if i = j and j /∈ Ni

−wij if i = j and j ∈ Ni∑l∈Ni

wil if i = j

where wij > 0 is the strength of the influence of j on i definedin (3). This matrix corresponds to the Laplacian of the graph Gthat defines the influence network in the drinking group. Then,the model of the drinking group in (5) can be written as

˙x = v

v = −L(x + x∗) − bLv

− Fp(x + x∗) − Fe

(x + x∗) − Dv (8)

where D is a n × n diagonal matrix defined as D =diag(ζ1, . . . , ζn). The following theorem shows how the socialand environmental pressures affect the behavior of each indi-vidual in a way that it deviates from their personal preferences.

Theorem 2: Let the social pressures in the drinking groupcharacterized in (8) be such that the strength of the attractionssatisfy

b∑

j∈Ni

wij + 2ζi > b∑

j∈�i

wji, i = 1, . . . , n (9)

where �i = { j ∈ V : i ∈ Nj} is the set of all members of thegroup that are influenced by individual i. Then, the trajectoriesof the social system in (8) are uniformly ultimately boundedwith ultimate bound γ given by

γ = β maxi

⎡

⎣ 1

δi√

2n

∑

j∈Ni

wij

∣∣∣x∗

i − x∗j

∣∣∣ + ηe

i Me∣∣xe − x∗

i

∣∣

⎤

⎦ (10)

where δi = max [cθi,∑

j∈�iwji + ηe

i Me], and constants β > 0,Me > 0, and θi ∈ (0, 1) for i = 1, . . . , n. Constant c is theminimum eigenvalue of the matrix D + b/2(L + L�).

Proof: Let Lo be a n×n matrix whose ijth entry is given by

[Lo]ij =⎧⎨

⎩

0 if i = j and j /∈ �i

−wji if i = j and j ∈ �i∑l∈�i

wli if i = j.

This corresponds to the column Laplacian matrix of the graphthat defines the structure of the influence network G. Let

V(x, v) = 1

2x�(

L + Lo)x +n∑

i=1

ηpi f p

i

(xi + x∗

i

) + 1

2v�v. (11)

From the assumption that f pi is strictly convex and has

a unique minimum at x∗i and f p

i (x∗i ) = 0 for all i =

1, . . . , n, we have that∑n

i=1 ηpi f p

i (xi + x∗i ) is positive def-

inite with respect to x. Also, v�v is positive definite withrespect v. From [34, Lemma 5], we have that x�(L + Lo)x =∑n

i=1∑

j∈Niwij

∣∣xj − xi∣∣2 is positive semidefinite with respect

to x. Hence, V(x, v) = 0 only when x = 0 and v = 0, andV(x, v) > 0 otherwise. We select then V as our Lyapunovfunction candidate. The derivative of V with respect to time is

V(x, v) = −v�(D + bL)v + v�(Lox − Lx∗)

−n∑

i=1

v�i ηe

idf e

dx

(xi + x∗

i

)

= −v�(D + bL)v

−n∑

i=1

v�i

⎡

⎣∑

j∈Ni

wij

(x∗

i − x∗j

)

+∑

j∈�i

wji(xi − xj

) + ηei

df e

dx

(xi + x∗

i

)⎤

⎦.

(12)

In (12), term v�(D + bL)v is positive definite with respect tov if and only if the eigenvalues of the matrix D+b/2(L+L�)


are positive. From Gershgorin’s circle theorem [35, Th. 7.2.1],every eigenvalue λ of this matrix is in the region

∣∣∣∣∣∣λ − b

∑

j∈Ni

wij − ζi

∣∣∣∣∣∣≤ b

2

⎡

⎣∑

j∈Ni

wij +∑

j∈�i

wji

⎤

⎦.

From the assumption (9), we have that all the eigenvalues ofthis matrix are positive. Let c = λmin[D+b/2(L+L�)] be theminimum eigenvalue of the matrix D + b/2(L + L�). Then,using the fact that v�(D + bL)v = v�(D + b/2(L + L�)v,and from Rayleigh quotient [36, Th. 10.13], we have that−v�(D + bL)v ≤ −c

∑ni=1 |vi|2. Also, since (df e/dx) is

assumed to be Lipschitz continuous, we know that thereexists a constant Me > 0 such that |(df e/dx)(xi + xe) −(df e/dx)(xe)| = |(df e/dx)(xi + xe)| ≤ Me|xi − x∗

ei|, wherex∗

ei = xe − x∗i . Using this result in (12), we obtain

V(x, v) ≤n∑

i=1

⎡

⎣−c|vi|2 + |vi|⎛

⎝∑

j∈Ni

wij

∣∣∣x∗i − x∗

j

∣∣∣

+∑

j∈�i

wji∣∣xi − xj

∣∣ + ηei Me

∣∣xi − x∗ei

∣∣

⎞

⎠

⎤

⎦.

(13)

For a constant θi ∈ (0, 1), we have that −c|vi|2 = −c(1 −θi)|vi|2−cθi|vi|2. Then, we can rewrite the inequality in (13) as

V(x, v) ≤n∑

i=1

−c(1 − θi)|vi|2 for all

cθi|vi| −⎛

⎝∑

j∈�i

wji + ηei Me

⎞

⎠|xi| −∑

j∈�i

wji∣∣xj

∣∣

≥∑

j∈Ni

wij

∣∣∣x∗i − x∗

j

∣∣∣ + ηei Me

∣∣xe − x∗i

∣∣, i = 1, . . . , n.

(14)

Equation (14) can be written in a compact yet more conser-vative way

V(x, v) ≤n∑

i=1

−c(1 − θi)|vi|2

for all∥∥∥[x, v

]�∥∥∥ ≥ μ (15)

where

μ = maxi

⎡

⎣ 1

δi√

2n

∑

j∈Ni

wij

∣∣∣x∗i − x∗

j

∣∣∣ + ηei Me

∣∣xe − x∗i

∣∣

⎤

⎦ (16)

and

δi = max

⎡

⎣cθi ,∑

j∈�i

wji + ηei Me

⎤

⎦.

This expression was obtained using the norm inequality[26, p. 648]

n∑

i=1

|xi| + |vi| =∥∥∥∥[x�, v�]�∥∥

∥∥1

≤ √2n

∥∥∥∥[x�, v�]�∥∥

∥∥2.

Equation (15) indicates that V is negative semidefinite forall ‖[x�, v�]�‖2 ≥ μ. However, similar to the proof ofTheorem 1, from (8) we know that if v(t) = 0 for a given t,variable v(t) is not zero unless additional conditions on bothx(t) and v(t) are satisfied. Hence, V is negative as long as thebound in (15) is satisfied. Thus, the trajectories of the socialsystem in (8) are uniformly ultimately bounded. From now on,unless it is indicated, the vector norm ‖ · ‖ corresponds to theL2 vector norm.

To compute the ultimate bound, we need to find strictlyincreasing functions α1 and α2, with α1(0) = 0 and α2(0) = 0,such that

α1

(∥∥∥∥[x�, v�]�∥∥∥∥

)≤ V(x, v) ≤ α2

(∥∥∥∥[x�, v�]�∥∥∥∥

).

According to [26, Th. 4.18], the ultimate bound will be givenby γ = α−1

1 (α2(μ)). From (11), we know that V(x, v) satisfies

V(x, v) ≥ 1

2λmin

[L + Lo]‖x‖2 + 1

2‖v‖2

≥ β1

[‖x‖2 + ‖v‖2

]= α1

(∥∥∥∥[x�, v�]�∥∥

∥∥

)(17)

where β1 = min{(1/2)λmin[L + Lo], (1/2)}.To find the upper bound of V(x, v), we use the assumption of

Lipschitz continuity on (df pi /dx) for i = 1, . . . , n. Here, from

[37, Proposition A.24], we have that there exists a constantMp

i > 0 such that f pi satisfies f p

i (xi + x∗i ) ≤ Mp

i |xi|2/2. Usingthis property, we obtain

V(x, v) ≤ 1

2λmax

[L + Lo]‖x‖2 +

n∑

i=1

ηpi Mp

i

2|xi|2 + 1

2‖v‖2.

(18)

Equation (18) then can be bounded as

V(x, v) ≤n∑

i=1

β2i

[|xi|2 + |vi|2

]

≤ β2

∥∥∥∥[x�, v�]�∥

∥∥∥

2

= α2

(∥∥∥∥[x�, v�]�∥

∥∥∥

)(19)

where

β2i = max

[1

2λmax

[L + Lo],

ηpi Mp

i

2,

1

2

]

and

β2 = maxi=1,...,n

β2i.

From (17) and (19), we can obtain an expression for theultimate bound

γ = α−11 (α2(μ)) =

√α2(μ)

β1=

√β2

β1μ = βμ

where μ is defined in (16), and β = √β2/β1.

Remark 3: Theorem 2 indicates that the social interactionsand the environment add dissipative elements in the dynamicsthat make the individuals deviate from their personal prefer-ences. Note that the ultimate bound in (10) mainly dependson the social influence and environment parameters.


Remark 4: If the ultimate bound γ defined in (10) is zero,it means that every individual follows his/her personal prefer-ences. There are two cases when this situation happens. First,when there is no social interaction and no influence of theenvironment (i.e., wij = 0, η

pi = for all i ∈ V , and j ∈ Ni),

which is the case described in Theorem 1. The other case iswhen x∗

i = x∗j , and x∗

i = xe, for all i ∈ V , j ∈ Ni). It meansthat, given the initial conditions of the variables x and v, inthe long term all the individuals will reach their desired BAClevel even though there are some force components that pres-sure the individuals to have the same BAC and its rate ofchange during the drinking activity.

Remark 5: The ultimate bound in (10) tends to zero asthe strength of the social interactions and/or the differencebetween the desired positions of the individuals decrease.

Remark 6: In Theorem 2, Lipschitz continuity in the gra-dient of the personal preference and environment functions isa reasonable assumption that implies some convenient prop-erties for the functions that allow us to obtain results that areeasily interpretable.

Remark 7: Note that there are no assumptions on the con-nectivity of the structure of the influence network G. The onlyrelated assumption is the one in (9), which guarantees that, inthe long term, the rate of change in BAC level of the individu-als v will be zero. There are special cases when this assumptioncan be satisfied: when the network G is balanced, that is,∑

j∈Niwij = ∑

j∈�iwji for all i ∈ V (i.e., the total strength

of the influence of individuals in the group on i equals thetotal strength of the influence of individual i on others); andwhen there are no attractions on the rate of change in BAC inthe group, that is, b = 0.

The results given in Theorem 1 suggest that there must existan influence acting on the individual’s behavior that dependson his/her perception on how quickly his/her own BAC levelis changing and modulates the dynamics of the BAC accord-ingly. Also, the ultimate bound provided in Theorem 2 showsthat the compound action of the environment (with strengthproportional to ηe

i ) and social influences (with strength pro-portional to the wij) can have a significant impact on theBAC level trajectories at the individual level, as expected. It isimportant to note that the design of interventions has focusedmainly on affecting the conditions in the environment throughindividual incentives to prevent problems like heavy alcoholconsumption. However, as shown in the ultimate bound fromTheorem 2, an individual’s drinking behavior can be impactedby having either a large influence coming from the environ-ment, or small social influences that add up together. Thisobservation suggests that intervention designs at the sociallevel could lead to promising outcomes in terms of preventionof unhealthy drinking behaviors. For example, a recent studyhas shown that using social pressures instead of individualincentives to increase physical activity levels in a communityprovided significantly better results [10, Ch. 4].

V. CONCLUSION

Methodologically, this paper illustrates how field dataand computational and mathematical modeling complement

each other. In our original field studies, we did not have theability to directly measure or model the influence of groups ondrinking behavior. The combination of these approaches rep-resents a way to maximize the data collected in large studies.Furthermore, measuring dynamical processes using traditionalsocial science methods is often not possible or extremelydifficult [38]. As part of the cyclic process in Fig. 1, ourmodel along with the mathematical analysis and simulationspresented above will help us refine our future field studies,especially as they relate to the interplay between individual,group, and environment and the relationship to alcohol intox-ication. Also, as recent technological advances improve ourability to collect real-time data, we will better inform theempirical specification of the proposed model [5].

Theoretically, through the model presented above and itsanalysis through stability theory and simulations, we hope toinform our understanding of group dynamics as they relateto drinking behavior. Traditional social psychological mod-els [15] have given us the foundation from which to buildmore sophisticated complex dynamical models. Understandinggroup influence in the context of environment, network rela-tionships, and individual preference—taking into account forthe influence of psychoactive substances—affords a richer eti-ological understanding of real world phenomena like drinkingbehavior. In turn, understanding the influences at differ-ent levels, how environment can moderate personal in-groupinfluence for example, may ultimately help guide applied pre-ventive solutions to problems like heavy alcohol consumptionand the problems that flow from that consumption.

On the modeling side of this paper, our next steps includedeveloping models that incorporate additional informationabout the individual, such as gender or weight. In this way,the model can be coupled with Wegner’s equations of alco-hol content [39] to characterize the trajectories of the BACgiven the number and type of drinks per time unit. Onthe empirical side of this paper, we are designing a real-time data collection process at drinking events that includesmeasurements of variables at the individual, group, and envi-ronmental level [19], [20], [22]. Our aim is to study thesedata using tools from both statistics and dynamical systemtheory [26]. We plan to do system identification [40] tofind the parameters that allow the model to have the clos-est approximation to the measured behaviors. Depending onthe results, we will validate and improve our hypotheseson the mechanisms that drive behavior during the drink-ing event. We hope that this subsequent round of modeland field validations will have contributed to our understand-ing sufficient to engage in a series of interventions at theevent level.

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Luis Felipe Giraldo is currently pursuing the Ph.D.degree in electrical and computer engineering withOhio State University, Columbus, OH, USA.

His current research interests include modelingand analysis of linear and nonlinear dynamical sys-tems, pattern recognition, and signal processing.

Kevin M. Passino (S’79–M’90–SM’96–F’04)received the Ph.D. degree in electrical engineeringfrom the University of Notre Dame, Notre Dame,IN, USA, in 1989.

He is a Professor of Electrical and ComputerEngineering and the Director of the HumanitarianEngineering Center with Ohio State University,Columbus, OH, USA. He has published a bookentitled Humanitarian Engineering: CreatingTechnologies that Help People, Edition 2(Columbus: Bede, 2015).

John D. Clapp received the Ph.D. degree in socialwork from Ohio State University, Columbus, OH,USA, in 1995.

He is a Professor and the Associate Dean ofResearch and Faculty Development with the Collegeof Social Work, Ohio State University, where healso serves as the Director of the Higher EducationCenter for Alcohol and Drug Misuse Preventionand Recovery. He is the Editor-in-Chief of theInternational Journal of Alcohol and Drug Research.

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IEEE TRANSACTIONS ON CYBERNETICS, VOL. 47, NO. 1, JANUARY 2017 165 Modeling … · 2016-12-21 · IEEE TRANSACTIONS ON CYBERNETICS, VOL. 47, NO. 1, JANUARY 2017 165 Modeling and Analysis