Frontier Problems in Modeling Dynamic Social Systems

UC Irvine UC Irvine Electronic Theses and Dissertations

Title Frontier Problems in Modeling Dynamic Social Systems

Permalink https://escholarship.org/uc/item/63p464wh

Author Gibson, Charles Benjamin

Publication Date 2018

License https://creativecommons.org/licenses/by/4.0/ 4.0

Peer reviewed|Thesis/dissertation

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Frontier Problems in Modeling Dynamic Social Systems

DISSERTATION

submitted in partial satisfaction of the requirements for the degree of

DOCTOR OF PHILOSOPHY

in Sociology

Charles Benjamin Gibson

Dissertation Committee: Professor Carter T. Butts, chair, Chair Professor Katherine Faust Associate Professor Andrew Noymer

2018 c 2018 Charles Benjamin Gibson DEDICATION

To Stephanie and Joseph, and the one that comes next.

ii TABLE OF CONTENTS

Page

LIST OF FIGURES v

LIST OF TABLES vi

ACKNOWLEDGMENTS vii

CURRICULUM VITAE viii

ABSTRACT OF THE DISSERTATION x

1 Introduction 1

2 The Nature of Sexual Partnership Dissolution in the United States 4 2.1 Introduction ...... 4 2.2 Background ...... 5 2.3 Data and Methods ...... 9 2.4 Results ...... 11 2.5 Discussion ...... 19 2.6 Limitations ...... 20 2.7 Conclusion ...... 21

3 Eﬀects of Temporal Resolution Adjustments on Dynamic Sexual Contact Models 22 3.1 Introduction ...... 22 3.2 Timescale Measurement and Social Network Data ...... 24 3.3 Timescale Adjustment for Sexual Contact Network Models ...... 28 3.4 Data and Methods ...... 32 3.5 Results ...... 36 3.6 Discussion ...... 40 3.7 Limitations ...... 42 3.8 Conclusion ...... 42

4 Practical Methods for Imputing Follower Count Dynamics 44 4.1 Introduction ...... 44 4.2 Background ...... 46 4.3 Imputation for Follower Count Data ...... 50

iii 4.3.1 Evaluation Study: U.S. Public Health Accounts in the 2014 Ebola Outbreak ...... 53 4.4 Results ...... 58 4.5 Discussion ...... 66 4.6 Conclusion ...... 67

5 Conclusion 69

Bibliography 73

iv LIST OF FIGURES

Page

2.1 (a) Kaplan-Meier estimates for ethnic homophily; (b)-(c) Kaplan-Meier estimates for sex and age matching, respectively...... 13 2.2 (a) Kaplan-Meier estimate of partnership survival hazards with 95% conﬁ- dence bounds, separated by partnerships with without a concurrency and/or a concurrency memory (b) Kaplan-Meier estimates for married/cohabiting ties; (c)-(d) Kaplan-Meier estimates for tie length and primary/secondary partnerships, respectively...... 14 2.3 (a)-(f) Descriptions of concurrency resolution classes and their prevalence within the set of observed concurrencies. (g) Comparison with certain types with what would be expected given random chance...... 15 2.4 (a) Logistic regression predicting ‘transition’ concurrency; (b) Interaction between concurrency length and older tie length in predicting newer tie dissolution. 17

3.1 Illustration of ”Any” Coarsening Type ...... 27 3.2 Illustration of ”All” Coarsening Type ...... 27 3.3 Study plan for Effects of Temporal Resolution Adjustments on Dynamic Sexual Contact Models ...... 33 3.4 Effects of Model Adjustments on Average Duration of Predicted Ties . . . . . 37 3.5 Effects of Model Adjustments on Average Degree ...... 37 3.6 Effects of Model Adjustments on Predicted Average Per-Timepoint Concurrency 37 3.7 Effects of Model Adjustment on Average Predicted Per-Timepoint Network Formation Events ...... 38 3.8 Effects of Model Adjustment on Average Predicted Per-Timepoint Network Dissolution Events ...... 39 3.9 Effects of Model Adjustment on Average Predicted Forward-Reachable Vertices 39 3.10 Effects of Model Adjustment on Average Predicted Concurrency Durations . 39

4.1 Follower Count Descriptives for Ebola-responding Public Health Accounts . . 57 4.2 Kernel Density of Relative Error when Extrapolating v. Interpolating . . . . 60 4.3 Observed Follower Counts Versus Counts Predicted by Negative Binomial Regression (All Accounts Included) ...... 62 4.4 Bootstrapped Prediction Interval Width v. Regression Type ...... 64 4.5 Performance of Negative Binomial Regression Imputation Prediction Inter- vals, with Accounts Modeled Separately ...... 65 4.6 Relative Error as a Function of Proportion of Cases Missing ...... 66

v LIST OF TABLES

Page

2.1 Predicted Coeﬃcients for Cox Regression Predicting Hazard of Dissolution of Sexual Partnerships in NHSLS...... 18

4.1 Mean Absolute Relative Errors Across Methods, Interpolation v. Extrapolation 61 4.2 95% Prediction Interval Coverage across Methods ...... 64

vi ACKNOWLEDGMENTS

I would like to thank the National Science Foundation and National Institutes of Health for funding parts of this dissertation, and my graduate education.

vii CURRICULUM VITAE

Charles Benjamin Gibson

EDUCATION Doctor of Philosophy in Sociology 2018 University of California, Irvine Irvine, California Master of Arts in Demgraphic and Social Analysis 2018 University of California, Irvine Irvine, California Master of Arts in Sociology 2012 University of Alabama at Birmingham Birmingham, Alabama

RESEARCH EXPERIENCE Graduate Student Researcher 2012–2018 University of California, Irvine Irvine, California

TEACHING EXPERIENCE Teaching Assistant 2010–2012 University of Alabama at Birmingham Birmingham, Alabama

AWARDS Outstanding Research Award 2017 University of California, Irvine, Department of Sociology. Irvine, California Cliﬀord C. Clogg Award for Best Graduate Student Paper 2016 University of California, Irvine, Department of Sociology. Irvine, California Robin M. Williams Jr. Paper Award 2015 University of California, Irvine, Department of Sociology. Irvine, California Social Science Tuition Fellowship 2012-2013 University of California, Irvine. Irvine, California Ferris P. and Annie Ritchey Endowed Scholarship in Sociology 2010 Department of Sociology, UAB. Birmingham, Alabama. Dean’s Research Award 2010 College of Arts and Sciences, UAB. Birmingham, Alabama.

viii REFEREED JOURNAL PUBLICATIONS

Sutton, J., Vos, S. C., Olson, M. K.,, Woods, C. W., Cohen, E., 2017 Gibson, C. B., Phillips, N., Studts, J., Eberth, J., & Butts, C. T.. ”Lung Cancer Messages on Twitter: Content Analysis and Evaluation.” Journal of the American College of Radiology. Gibson, C. Ben and Timothy B. Mayhall. ”Comprehension Con- 2017 text and Sponsor Eﬀects in a Hospital Mental Health Study.” Sociological Methods and Research Sean M. Fitzhugh, C. Ben Gibson, Emma S. Spiro, Carter T. 2016 Butts. ”Spatio-temporal Filtering Techniques for the Detection of Disaster-related Communication” Social Science Research. Sutton, Jeannette, C. Ben Gibson, Nolan Edward Phillips, 2015 Emma S. Spiro, Cedar League, Britta Johnson, Sean M. Fitzhugh, Carter T. Butts. ”Terse Message Retransmission: A Cross-Hazard Analysis of Twitter Warning Messages.” Proceedings of the National Academy of Sciences. Sutton, Jeannette C. Ben Gibson, Emma S. Spiro, Cedar 2015 League, Sean M. Fitzhugh, Butts, Carter T. In Press. ”What it Takes to Get Passed On: Message Content, Style, and Struc- ture as Predictors of Retransmission in the Boston Marathon Bombing Response.” PLOS One. Sutton, Jeannette, Emma S. Spiro, Sean Fitzhugh, Britta John- 2014 son, C. Ben Gibson, and Carter T. Butts. ”Terse Message Am- pliﬁcation in the Boston Bombing Response.” International Journal of ISCRAM. Sutton, Jeannette, Emma S. Spiro, Britta Johnson, Sean 2014 Fitzhugh, C. Ben Gibson, and Carter T. Butts. ”Warning Tweets: Serial Transmission of Warning Messages During a Dis- aster Event.” Information, Communication and Society.

SOFTWARE braQCA https://cran.r-project.org/web/packages/braQCA C. Ben Gibson and Burrel Vann Jr. (2015). braQCA: Robustness assessment for QCA. CRAN.

ix ABSTRACT OF THE DISSERTATION

Frontier Problems in Modeling Dynamic Social Systems

Charles Benjamin Gibson

Doctor of Philosophy in Sociology

University of California, Irvine, 2018

Professor Carter T. Butts, chair, Chair

Social dynamics – changes in the structure, existence or occurrence of social connections (e.g. sexual, friendship) or events (e.g. conversation, armed conﬂict) – are a core interest of social sciences (Comte, 1855; Weber, 1904; Marx and Engels, 1972). One key to understanding dynamic social systems is to understand the social networks that comprise these systems (Mayhew, 1984; Moreno, 1934). Modeling networks over time is of essential interest to understanding social systems and structure, from intimate sexual partnership ties (L´evi-Strauss, 1969; Morris and Kretzschmar, 1995; Hudson, 1993; Moody, 2002) to the interactions of large-scale organizations (Blau, 1970; Childers et al., 1971; Mayhew et al., 1972). Recently, advances in computing and statistical methodology have improved our ability to model relational dynamics. One such methodology is the exponential family models for modeling social networks (i.e. temporal exponential random graph models (TERGMs), see (Krivitsky and Handcock, 2014)). This thesis aims to improve our ability to model relational dynamics through a substantive exploration of a dynamic social system, a purely predictive imputation of edge dynamics among social media accounts, and an exploration of how a dynamic model handles temporal adjustments. I thus aim to address frontier problems in relational dynamics, to improve and extend our current models by both substantive application and methodology assessment.

x Chapter 1

Introduction

1 methodology assessment.

The first chapter involves an exploration of how sexual partnerships dissolve. Current dynamic parameterizations of sexual contact networks treat dissolution homogeneously across all ties. I attempt to inform future parameterizations of TERGMs by exploring the substantive nature of partnership dissolution. Some effects I posit on dissolution heterogeneity include concurrency effects, duration effects (i.e. the length of a tie affecting its future survival), concurrency memory (edge dissolution as a function of a past concurrency), and matching effects on gender, race, and age.

The second chapter provides a method for imputing unobserved follower count dynamics for emergency management organizations’ Twitter accounts. I consider the use of three classes of simple, easily implemented method for follower imputation: polynomial functions; splines; and generalized linear models (GLMs). I evaluate the performance of each method via a case study of accounts from 236 health organizations communicating with the public during the 2014 Ebola outbreak.

The third chapter is an assessment of how well separable TEGRMs handle temporal adjustments used by researchers. I both analyze how different methods for timescale-adjusted data affect observed network statistics, and the performance of different methods for timescale- adjusted modelling of sexual contacts. I use a STERGM fit from Pavel Krivitzky’s (2012) paper on sexual contact networks.

To sum, the three chapters will ask:

• Whether and how we could improve dynamic models of sexual contact networks by systematically exploring dissolution dynamics;

• A simple, easily implemented method for imputation edge dynamics on the social media site Twitter;

2 • An evaluation of time-adjustments for STERGMs and simulated data.

In all three chapters, a blend of theoretical and methodological advancements will be applied with the purpose of improving dynamic models of social interaction in diﬀerent contexts.

3 Chapter 2

The Nature of Sexual Partnership Dissolution in the United States

2.1 Introduction

Modeling sexual contact network (SCN) dynamics helps predict the propagation of sexually transmitted infections such as HIV (Morris and Kretzschmar, 1995, 1997; Hamilton and Morris, 2015; Morris et al., 2009; Mitchell et al., 2013; Merli et al., 2015). A common way of modeling these dynamics is by using a separable parameterization – one model for the formation of sexual partnerships and another for their dissolution (Goodreau et al., 2017; Krivitsky and Handcock, 2014). In current epidemiological modeling of HIV, the primary focus of researchers modeling the underlying social dynamics of sexual contacts has been in the formation network, with little heterogeneity being considered in how sexual partnerships dissolved. Though formation of human sexual partnerships has been explored in some detail (see Krivitsky and Handcock (2014)), we aim to explore what conditions are likely responsible for the break-up of these partnerships. We draw from three literatures – formation of sexual

4 partnerships, the literature of divorce, and the dissolution of same-sex ties – to construct a series of possible eﬀects for the dissolution of sexual partnerships in the U.S. population. Using data from the National Health and Social Life Survey (NHSLS), we test several possible mechanisms of partnership dissolution in the U.S population drawn from these literatures.

2.2 Background

This review considers three sets of literatures on the nature of sexual partnerships in the United States. We draw from these literatures to construct a series of possible factors in the dissolution of sexual partnerships in the general population. The ﬁrst body of work considered is literature on partnership formation; the second is the large family literature on factors that lead to divorce; ﬁnally, we consider some work on same-sex partnerships.

According to current models of sexual partnerships (Krivitsky and Handcock, 2014), the formation of human sexual partnerships (‘edges’) depends upon if participants (‘nodes’) share some socially-relevant characteristics, like race or age; the sexual preferences of nodes involved, e.g. same-sex partnerships or different-sex partnerships; and if formation of the potential edge would result in any concurrent partnerships, i.e. the existence of more than one edge for one node. These factors have been used in prediction of sexual contact networks and simulation of networks that resemble observed data (Krivitsky and Handcock, 2014). The strong effect of demographic matching on the formation of sexual partnerships could extend into their dissolution; similarity on social characteristics could not only predict the formation of sexual partnerships, but it also could affect the stability of sexual partnerships once formed. An effect for matching on race/ethnicity, age, or education could be predictive of partnership dissolution. Matching on sex could also matter for relationship stability, as same-sex partnerships could have higher dissolution rates than different-sex partnerships (Kurdek and Schmitt, 1986).

5 In current sexual contact network models, concurrency – the existence of more than one partnership for one ego – is seen as a strong predictor of whether a partnership forms, i.e., once an ego is in a partnership, additional simultaneous partnerships are unlikely. In these models, however, no eﬀect for concurrency is parameterized for existing partnerships, and concurrency partnerships are equally likely to dissolve as partnerships without a concurrency. In general partnerships, it could be that concurrencies signal instability in current partnerships, or that concurrencies could signal high-churning partnerships for some endpoints. In either case, an eﬀect for concurrency for partnership dissolution would be predictive.

Another large literature on sexual partnerships is the family literature on divorce. Divorce has been observed to vary substantially by demographics. In one study, same-race black couples are more likely to divorce than same-race white couples (Bramlett and Mosher, 2002). Age at marriage affects the probability that the couple will get divorced, as well as the education level and religiosity of the marriage’s participants (Heaton, 2002). Marriages of longer length are less likely to dissolve, and shorter marriages are more likely to be dissolved due to compatibility, as compared to longer marriages, which tend to end due to life changes or a major event (Becker, 1991). One major event could be ‘infidelity,’ which violates typical monogamous norms in the United States (Glass and Staeheli, 2002). When an infidelity is discovered, many relationships dissolve immediately, but others dissolve after the infidelity ends or not at all. The probability of dissolution due to an infidelity differs according to how it was discovered, i.e., whether it was disclosed or discovered without the knowledge of the cheating partner (Shackelford et al., 2002). The length of the infidelity also affects the probability of divorce due to infidelity, with longer concurrencies being more likely to have emotional attachment or bonding than shorter concurrencies (Glass and Staeheli, 2002). Married partnerships that cohabited before marriage have been found to dissolve more often, though this effect has declined in recent years due to the normalization of premarital cohabitation (Teachman, 2003). The literature on divorce provides hints on what factors will affect dissolution of sexual partnerships generally. Demographics – such as age, race,

6 or education – could matter in terms of dissolution risk. The length of the partnership could affect the risk of future dissolution, as longer partnerships might have worked out compatibility issues present in earlier stages. Infidelity – which I will refer to from here forward as a ’concurrency’ – could destabilize partnerships, or otherwise signal an already- unstable partnership, resulting in higher risk of dissolution. Not all concurrencies end a marriage right away, so an effect for concurrency in the past could affect dissolution in the present, even if a concurrency does not exist in the present. Marriage itself creates a barrier to dissolution, as divorce is costly (Braver and Lamb, 2013). The presence of a marriage could then affect the probability of dissolution above and beyond an effect for length. Many of these effects, however, could exist for married couple but not the general population, greatly attenuating their effects in models of general sexual partnerships.

Finally, we draw from literature on the dissolution of same-sex partnerships. Same-sex partnerships are thought to dissolve at greater speed than heterosexual partnerships (Kurdek and Schmitt, 1986). In one study, male same-sex partnerships dissolved faster than female same-sex partnerships (Kurdek, 1998), which is thought to be in part because of the greater set of alternatives available to same-sex males due to more social mixing of males who prefer same-sex partnerships. Same-sex partnerships did not have the institutional barrier of divorce present in heterosexual partnerships, which could affect their dissolution rates. Finally, same-sex partnerships were shown to be more likely to kept secret, rendering mutual friendships unaffected by dissolution. Since secret partnerships do not affect friendships mutual to both participants, it reduces the cost of dissolution of non-secret partnerships where friends could need to take sides after the partnerships dissolve.

The literature on same-sex partnerships thus suggests competing hypotheses on the differences between same-sex partnerships and diﬀerent-sex partnerships: one theory, that same-sex partnerships dissolution rates will be higher due to institutional and social network barriers, should hold for both male and female same-sex partnerships. Another theory

7 – that more partnership alternatives create more opportunities for dissolution – suggests a higher rate of dissolution for male same-sex ties than female same-sex ties. We test both theories.

Potential Eﬀects

Demographics and demographic matching. Factors like race, age, education, and marital status could all affect the risk of relationship dissolution. Further, whether or not partners match on these social characteristics could also affect dissolution. According to the literature on divorce, whites have lower dissolution rates than blacks; higher-educated couples are more likely to persist than lower-education couples, and younger married couples are more likely to divorce. If we extend this literature to general partnerships, we will expect similar effects for a sexual partnerships in the general population. In the partnership formation literature, we see strong effects for demographic matching, where those who are closer in age, and share race and ethnicity, are more likely to form partnerships. In this study, it is an open question whether these effects will hold for dissolution as well. We expect higher dissolution rates for same-sex partnerships, which has been shown in the literature (Kurdek and Schmitt, 1986); if the same finding holds, we would expect male same-sex partnerships to have higher hazard of dissolution than heterosexual ties and same-sex female ties.

Concurrency. Concurrent partnerships are strong predictors of divorce (Glass and Staeheli, 2002) and are also strong deterrents in partnership formation (Krivitsky, 2012). When determining their effects on marriage, the duration of the concurrency matters, as well as how it was discovered (whether or not it was disclosed by the ’cheating’ partner). Some marriages end some time after the concurrency exists (Shackelford et al., 2002) . This suggests an effect for both a current concurrency and a past concurrency, as well as concurrency duration effects that could be tested on general population sexual partnerships.

8 Relationship length and institutional barriers. Longer marriages are more likely to persist than younger ones (Becker, 1991); it is an open question, however, whether this eﬀect will hold for unmarried partnerships. As relationships lengthen, the more likely they are to share assets and enter into institutional contracts like marriage, but they are also likely to accumulate baggage like repeated communicative breakdown, a concurrency, or other baggage. Marriage and cohabitation are institutional barriers to dissolution, and could aﬀect the risk that relationships with dissolve.

Each of these eﬀects will be tested in a nationally representative sample of sexual partnerships using Cox Regression. Below, I describe the methodological procedures involved in testing.

2.3 Data and Methods

This paper uses the National Health and Social Life Survey (Laumann et al., 1995), which is a retrospective life-history data collected on a national probability sample of 3332 respondents. The study was intended to be a nationally representative sample of adults aged 18-59 in the United States. The NHSLS used both face-to-face interviews as well as self-administered questionnaires to collect sexual history data. Sexual history of the respondents since age 18 is constructed from both face-to-face and self-administered sections. Respondents are asked to report sexual histories for the past year. Basic demographic information is collected both on the respondents and their partners, and start and stop times for relationships to the month, including ones begun before the study period. A tie – also referred to as an ”edge” – is thus deﬁned as a self-reported sexual partnership between an endpoint in our sample and their partner. Respondents reported start and end dates for sexual partnerships to the month level, so all models operate at the monthly timescale.

Cox Regression is used for multivariate analysis of purported eﬀects upon the expected rate

9 of dissolution in the next instant. Cox Regression predicts the expected hazard (i.e., expected rate of suﬀering an event of interest in the next instant):

h(t) = h0(t)exp(b1X1 + b2X2 + ... + bpXp) Where:

• h(t) is the expected hazard at time t

• h0(t) is the baseline hazard and represents the hazard when all of the predictors (or independent variables) X1, X2 , Xp are equal to zero

Using Cox Regression, we model the hazard of a dissolution event among observed partnerships. Dissolution is counted if the ego reported no sexual activity with their partner at least two months prior to the survey date, and no other indication of an ongoing sexual partnership was present (e.g., an ongoing cohabitation or marriage). Race/ethnicity is measured in ﬁve categories: white (71.4%), black (16%), Hispanic (9.4%), Asian (1.9%), and other (1.2%). Race/ethnicity matching means both partnership participants shared the same race/ethnicity (about 89% of partnerships shared the same race and ethnicity). The average age of egos was 36.4 years; the average age of alters was 34.5 years. Age matching is measured by ego age minus alter age (average raw age diﬀerence was 4.14 years). Relation- ship length is measured by the log + 1 of the number of months the relationship has lasted (raw mean: 95 months); raw values presented a problem when the relationship was newly formed. Concurrency is measured by a simultaneous partnership reported by the sampled unit; concurrency memory is binary variable indicating whether a concurrency existed in the past for that partnership. About 9% of all respondents report a concurrency; at any given time, 2.7% of all edges are concurrent. Same-sex partnerships share the same sex, as reported by sampled egos (about 3% of partnerships were same-sex). Marital and cohabiting

10 status is self-reported by the egos.

In all models shown, all the above measures are included in the initial Cox Regression model, but we use forward- and backward-stepwise AIC for model selection. Some variables were not significantly predictive of dissolution hazard – Cox Regression results use predictions from models estimated prior to stepwise AIC procedure, to show null effects. All other results shown use models after the stepwise procedure is applied. The best-fitting model had interaction effects between the presence of a concurrency and all other edge-level variables; interaction effects were also significant between relationship length and whether the relationship was a primary or secondary partnership. Since coefficient values of two-way and three-way interactions are difficult to understand on their own, we predict a series of outcomes in plots for clarity. All plots shown show the relationship between one variable and edge survival, with all other values held at their means.

The sample is not comparable to the United States population in terms of race, so we use Horvitz-Thompson weights for better estimates of population parameters. The weights for

each edge i are 1/πi, where πi is the inclusion probability of the sampled endpoint in the set: pi/Pi, where pi is the proportion of the sample that matches edge i’s sampled endpoint’s race, and Pi is deﬁned as the population proportion that matches the race of edge i’s sampled endpoint.

2.4 Results

In a divergence from sexual partnership formation processes, we find that mixed-race couples have no greater or less hazard of dissolution than same-race couples, and that age differences also do not significantly increase the hazard of dissolution. Partnership participants matching on race and/or age – called homophily – is a strong predictor of edge formation, suggesting

11 that a separable model for formation and dissolution, the current practice of researchers modeling sexual contact networks with exponential graph models, has a reasonable justiﬁ- cation. Once ties are formed, race and age matching do not seem to aﬀect the survival of ties.

Homophily on sex substantially increases the hazard for dissolution, but only for males (Fig 2b). Of relationships with at least one male participant, same-sex ties have nearly three times the hazard of dissolution as different-sex ties. For all ties with at least one female participant, however, same-sex (female) ties do not have significantly different hazard rates than different-sex ties. Given that 95% of new HIV infections occur among men who have sex with men (MSM), correctly modeling their dissolution rates is important for forecasting disease diffusion.

12 (a)

(b) (c)

Figure 2.1: (a) Kaplan-Meier estimates for ethnic homophily; (b)-(c) Kaplan-Meier estimates for sex and age matching, respectively.

13 (a) (b)

Figure 2.2: (a) Kaplan-Meier estimate of partnership survival hazards with 95% conﬁdence bounds, separated by partnerships with without a concurrency and/or a concurrency memory (b) Kaplan-Meier estimates for married/cohabiting ties; (c)-(d) Kaplan-Meier estimates for tie length and primary/secondary partnerships, respectively.

14 'The Affair' 'The Transition' 'The Blowup'

0.45

0.26

0.05

Proportion Proportion Proportion TIME TIME TIME

(a) (b) (c)

'The Tryouts' Same Start/Stop Unobserved

0.14 0.08 0.01 Proportion Proportion Proportion TIME TIME TIME

(d) (e) (f)

Baseline

The Affair

The Transition

The Blowup

−0.2 −0.1 0.0 0.1 0.2 Prevalence Below Baseline Prevalence Above Baseline (g)

Figure 2.3: (a)-(f) Descriptions of concurrency resolution classes and their prevalence within the set of observed concurrencies. (g) Comparison with certain types with what would be expected given random chance.

We see substantially diﬀerent survival rates for ties according to the existence of concurrency, using Cox Regression (Fig 2a). Partnerships with a concurrency have 3.6 times greater hazard of dissolution compared to non-concurrent ties. Those partnerships that begin as a

15 concurrency persist fewer months than the average for all partnerships (Fig 2c). Concurrency in the past also aﬀects dissolution in the present – those partnerships with a past concurrency have a 3.1 times greater hazard of dissolution than those with no past concurrency, including those with a current one (Fig 2a). Meaning, there is an additional hazard of having a concurrency in the past even when a concurrency is not present. With both past and present concurrencies, the hazard ratio of having both a current and past concurrency versus not is an 11-fold increase in hazard of dissolution. For modeling sexual contact networks, both the presence of concurrency and concurrency memory should be considered.

We break down concurrencies into a series of classes, according to the formation and dissolution order of the edges (Fig 3). The most common concurrency class (.45), which we obliquely call an ”affair,” occurs when a later-formed tie is dissolved before the older tie is dissolved. The second most common concurrency occurrence (.26) is a case where a concurrent tie ”transitions” – i.e. ego forms a new tie before dissolving their current one. The third most common concurrency is when two ties begin simultaneously but dissolve at different time points; we refer to this class as ‘the tryouts.’ Fourth most common are ties (.05) formed at different times but dissolve at the same time – the ‘blowup.’ Finally, a small proportion (.01) of ties begin and end at the same time point. A proportion of concurrency resolutions are unobserved (.14).

The prevalence of each class varies, but how much of this variation might be due to random chance? We construct a baseline model to compare observed concurrencies with random dissolution of partnerships. Without considering the inﬂuence of concurrency directly, the probability of simultaneous termination is much lower than ‘aﬀair’ type and ‘transition’ type, since two ties terminating at the same time is less likely by chance than one tie terminating. The probability of terminating jointly in a given time point, where p is probability of dissolution in a single time point, i.e., p = (number of dissolution events)/(tie-months-at-risk), is p2, much higher than p, which is the baseline probability of any individual tie terminat-

16 ing. When considering a time series, we account for this joint probability given that neither termination has occurred. When considering only month 1, it is p2; in the following month, both ties must still be present and we must terminate jointly, for probability (1 − p)2p2; in month 3, we must survive 2 months and terminate in the third, i.e. (1 − p)4p2, and so on.

P∞ 2(i−1) 2 Over all time points, this sums to i=1(1−p) p = p/(2−p). Even when considering this baseline, simultaneous termination is still unlikely, with the aﬀair being much more likely than expected; the transition is most below the expected baseline.

(a) (b)

Figure 2.4: (a) Logistic regression predicting ‘transition’ concurrency; (b) Interaction between concurrency length and older tie length in predicting newer tie dissolution.

The logged duration of the partnership prior to the study period greatly reduced observing the partnership’s termination during the period, suggesting that ties that have existed for longer are more stable than newer ones, mirroring the eﬀect for marriage stability. Stability of older ties seems to persist even during a concurrency event (Fig 4). Recall that the ‘transition’ type of concurrency resolution is when an ego jettisons an older tie in favor of a newer one. The eﬀect of concurrency length on a ‘transition’-type concurrency resolution decreased as

17 the length of the older tie increased. When very long-lasting (over 200 months) ties were faced with a concurrency, they typically had very low chance of dissolving regardless of the concurrency length. However, even with ties as old as 10 years, a 12-month concurrency was more likely to terminate the older tie than not. Models incorporating dissolution dynamics should 1) put an increasing weight on the persistence of ties with each additional time point it persists, 2) parameterize dissolution in such a way that penalizes newer ties in favor of older ones in the event of a concurrency, and 3) test an interaction eﬀect between older tie duration and concurrency duration.

beta exp(beta) se(beta) z Pr(>|z|) Concurrency -0.85*** 0.427 0.186 -4.574 0 Primary Partnership -0.751*** 0.472 0.133 -5.636 0 Past Concurrency 0.932*** 2.54 0.12 7.748 0 log(1 + length) -0.73*** 0.482 0.042 -17.23 0 Sex Homophily 0.096 1.101 0.234 0.411 0.681 Ego Sex 0.01 1.01 0.068 0.149 0.882 Education (Ego) 0.021 1.022 0.012 1.807 0.071 Cohab/Marital -0.58*** 0.56 0.113 -5.133 0 Ethnicity Homophily -0.023 0.978 0.09 -0.251 0.801 Primary * log(1+length) -0.245*** 0.783 0.061 -4.01 0 Sex Homophily * Sex 0.341 1.406 0.295 1.154 0.248 Conc. * Primary 0.335 1.397 0.293 1.141 0.254 Conc. * Past Conc. -1.883*** 0.152 0.517 -3.639 0 Conc. * log(1+length) 0.178** 1.194 0.061 2.908 0.004 Conc. * Sex Homophily -0.667* 0.513 0.333 -2.005 0.045 Conc. * Ego Sex -0.041 0.96 0.117 -0.349 0.727 Conc. * Cohab/Marital 0.371* 1.449 0.186 1.99 0.047 Conc. * Eth. Homophily 0.253 1.287 0.148 1.705 0.088 Conc. * Primary * log(1 + length) 0.446*** 1.562 0.103 4.322 0 Conc. * Sex Homophily * Sex 1.114** 3.047 0.426 2.613 0.009

Table 2.1: Predicted Coeﬃcients for Cox Regression Predicting Hazard of Dissolution of Sexual Partnerships in NHSLS.

Strong positive interaction effects exist between 1) concurrency and being a cohabiting/married partnership, 2) concurrency and relationship duration, and 3) concurrency and being a primary partnership (effect sizes and direction may be difficult to interpret alone, given that three-way interactions are used). Signals that the partnership is a trusted, long-term partnership increases the effect of a concurrency on the partnership’s dissolution hazard. Those

18 partnerships with shorter length and no marital/cohab status are more likely to be destabi- lized by a concurrency, even accounting for the partnership being a ‘primary’ partnership.

2.5 Discussion

Here, we show that substantial heterogeneity exists in the dissolution of sexual partnerships. Factors that matter substantially in the formation of partnerships, specifically race mixing, age difference, and same-sex partnerships, did not result in different hazard rates of dissolution. The observation of high similarity in partnerships in regards to race and age is not due to increased dissolution; i.e., more same-race human partnerships exist not because they are more likely to dissolve, but because they typically do not form in the first place. Once formed, different-race or different-age partnerships have no more hazard of dissolving than other types. Heterogeneity in same-sex partnership dissolution rates disappears with controls. The increased dissolution hazard of same-sex partnerships without controls was especially affected when interacted with the edge concurrency variable.

Though many factors that matter for formation did not apply to dissolution, heterogeneity remains in dissolution hazards. Not only did concurrent partnerships have a higher dissolution hazard, the existence of a concurrency in the past also mattered. Models should take into account the past history of concurrencies to more accurately represent dissolution dynamics. In addition, the hazard of sexual partnership dissolution decreased the longer the partnership lasted. Institutional barriers – specified using a variable for a cohabiting or married edge – had a small effect on dissolution after controls were entered in the model. More important for dissolution was the perception of the partnership being the ‘primary’ partnership of the ego. This effect was much larger than the institutional barrier, suggesting that institutional barriers are a small deterrent from dissolution when taking into account relationship length, concurrency, and other factors.

19 Strong positive interaction eﬀects between 1) concurrency and being a cohabiting/married partnership, 2) concurrency and relationship duration, and 3) concurrency and being a primary partnership shows that concurrency more powerfully increases the hazard of dissolution when partnerships are in a more committed partnership. This aligns with family literature that has found that long-term partnerships are more likely to dissolve by a concurrency than by other means (Glass and Staeheli, 2002; Becker, 1991). Since marital/cohab status and relationship length are both independently related to dissolution hazard beyond whether or not the relationship is considered a primary partnership, it suggests that the more elements of a trusted partnership exist, the more a concurrency aﬀects the partnership.

2.6 Limitations

The National Health and Social Life Survey is a nationally-representative sample of sexual partnerships. While it is often used for modeling of sexual partnerships, it does come with limitations: partnerships are aggregated to the monthly level, which aggregates all partnerships for the same ego that existed in the same month as having simultaneously existed. This could overestimate concurrency; however, this is likely to only have minimal eﬀects on observed concurrency, since sexual partnerships churn slowly relative to the monthly scale (see Section 4 of this thesis). Second, the number of partnerships reported by men are higher than those reported by women, likely caused by a small number of over-reporting males (Morris, 1993). Updating the collection to both collect instantaneous network data (to measure instantaneous concurrency rates) and observation of gender diﬀerences in sexual contact reporting (to determine if women are under-reporting, or men are over-reporting) would be fruitful.

20 2.7 Conclusion

Substantial heterogeneity exists in sexual partnership dissolution in the United States. Dif- ferences in tie formation order, duration of ties, and past histories of concurrency all have eﬀects upon how partnerships are resolved.

Dissolution heterogeneity in same-sex partnerships disappears when controlling for other factors in our model. Race/ethnicity mixing, as well as age diﬀerences – both major determinants of relationship formation – do not appear to impact dissolution hazards. In addition, concurrent sexual partnerships seem to resolve by one partnership ending, rather than both ending simultaneously.

The results above suggest a few eﬀects that could be useful in future estimation – concurrency, formation order of concurrent partnerships, partnership duration, and institutional barriers. Though dissolution is treated homogeneously in current model estimations, there is substantial heterogeneity in the nature of sexual contact concurrency.

21 Chapter 3

Eﬀects of Temporal Resolution Adjustments on Dynamic Sexual Contact Models

3.1 Introduction

Modeling and simulating sexual contact dynamics help predict the propagation of sexually transmitted infections such as HIV, both by using the cross-sectional network structure and how that structure changes over time (Krivitsky, 2012; Krivitsky and Handcock, 2014). The speed of the formation and dissolution of sexual contacts, as well as the rates in which concurrent partnerships exist in a network, could affect the speed and breadth of the diffusion of STI: higher rates of concurrent partnerships may diffuse HIV wider than in networks with low concurrency (Morris and Kretzschmar, 1995, 1997), and fast-churning networks, such as partnerships among sex workers, may increase the spread of bacterial infections like syphilis (Mitchell et al., 2013; Merli et al., 2015). Fundamentally important to modeling STI diffusion

22 is to accurately simulate the underlying dynamics of how quickly networks churn, which are often used in diﬀusion predictions (Goodreau et al., 2017; Hamilton and Morris, 2015; Morris et al., 2009).

When modeling networks, the choice of temporal resolution – the timescale between network changes – is an important theoretical consideration. Researchers make choices both in the timescale of the network model, and the timescale in which data is collected. Dynamic network data is often collected by either following a panel of units forward over time (e.g. McKusick et al. (1990)), or by extracting retrospective life history (RLH) data from sampled units (e.g. Laumann et al. (1995)). When collecting data in any of these collection methods, decisions are made on the unit of time to track changes in network ties. When following a panel over multiple time points, researchers are restricted by survey resources and informant load, limiting the number of times units can be sampled; in retrospective life history data, human informants are limited in their ability to recall exact dates of tie changes. These limitations require researchers to make a choice of collection timescale, within the bounds of their resource constraints. Network data are thus ”coarsened” by design – collecting data on network dynamics at every instant is not feasible in most contexts. The choice of ”coarsening” the timescale of data collection could have substantial consequences for the observed statistics of the network (Butts, 2009; Moody, 2002).

A second consideration is that the timescale of collection could differ from the timescale researchers would prefer to model. For example, if we want to use sexual contact network dynamics to predict disease diffusion, a full month between observation points could mask many short-duration sexual partnerships, as many of these network ties last for less than a month. To make disease diffusion predictions, we might miss ties relevant for its spread. In these cases, a model ”correction” is necessary – an adjustment of the model to make predictions at a different timescale than it was measured. Still, these corrections might miss underlying network dynamics.

23 Here, we assess model timescale adjustments on simulated data characteristics of a often- modeled type of network that is important for disease diﬀusion: sexual contact networks (SCNs). We also observe the eﬀects of timescale adjustments on observed sexual contact networks.

3.2 Timescale Measurement and Social Network Data

Different kinds of social dynamics operate on different timescales. Conversations may last, on average, no more than ten minutes, but sexual partnerships persist more than ten years on average (Krivitsky, 2012). When tasked with understanding different kinds of social dynamics, the choice of timescale matters when building models to predict them. When modeling finely-scaled social connections (e.g. turn-taking on a radio network, (Butts, 2008)), a relational event modeling framework is appropriate; for types of social connections that persist for years (e.g. friendship, kinship), a TERGM framework is often used (e.g. Krivitsky, 2012; Goodreau et al., 2017; Fitzhugh and DeCostanza, 2017).

Epidemiological models tasked with predicting the spread of disease are often informed by underlying sexual contact dynamics (Goodreau et al., 2017; Sullivan et al., 2012; Beyrer et al., 2012). Using information from the dynamic structure of social connections, disease diffusion models can be better informed about the diffusion potential through a social network. The choice of simulation of timescale differs according to both the nature of the type of network and the nature of the disease being predicted. When predicting the spread of the flu, the type of social interaction may be fleeting and brief (e.g. conversation, interaction in a social space), which requires very short timescales in its simulation. For HIV specifically, simulation of networks of ongoing sexual partnerships are used in predictions of HIV magnitude and spread (Goodreau et al., 2017; Hamilton and Morris, 2015; Morris et al., 2009). While these sexual partnerships are on average long-lasting, large numbers of short-term partnerships

24 (such as brothels) can do more to spread highly-infectious STI than long-term partnerships; alternatively, longer-term partnerships might be more important for the spread of HIV than shorter-term partnerships. Even within sexual contact networks (SCNs) being used in the prediction of disease, the predicted disease matters for how ﬁnely-scaled we aim to model the underlying SCN dynamics. Methods for adjustment of time scaling for simulated data is thus important for adapting network simulations for diﬀerent types of epidemiological predictions.

Adding to the need for adjusted-timescale models for social dynamics is the discrepancy between the speed with which network ties are measured versus how quickly the ties change. Two common ways to collect social network data are panel follow-up studies and panel retrospective life history (RLH) data. The first method requires follow-up with a panel across two or more time points; the second method requires a survey of respondents’ past sexual histories measured at one time point. Follow-up panel data measures a network state for respondents at the time of interview; RLH data gets time-resolved data according to the memory of respondents surveyed at one time point. The choice of timescale between observed network states depends on both the resources available to researchers, the rates of which the observed ties churn, and the speed with which the predicted phenomena operates. For example, a study observing kinship ties’ effects on labor market outcomes (e.g. Nguyen and Nordman, 2017) can afford to wait years between observations, as both the tie type and outcome change relatively slowly. Other types of network ties – such as sexual partnerships – churn relatively quickly and must be measured at more refined time points. In addition, disease diffusion among sexual partnerships can happen quickly, depending on the disease; sometimes, one sexual event is enough to spread disease. Quick measurement of sexual partnership churn is thus essential to predict the spread of certain diseases.

Another consideration is the constraint of resources upon the ability to measure networks at reﬁned timescales. Measuring the exact start and stop time of a sexual partnership to the

25 minute likely puts too much strain on respondent memory to get accurate measurements. For sexual partnerships, it is conventional to ask the start and stop times of a partnership at the monthly timescale (e.g. Laumann et al., 1995), as it produces relatively accurate ranges of past partnerships compared to a weekly or daily timescale. If conducting a panel study with multiple observations, administering a survey costs time and money for both researchers and respondents, and network slices observed at the daily or weekly timescale may not be practical in most circumstances. In many cases, our ability to model networks at desired timescales is limited by our ability to collect the data over time.

Such ‘coarsening’ may have a substantial impact on the observed state of the network during collection. If ties are reported at the monthly level, a decision is made whether to count the tie as having existed for the entire month, or having existed up to the month reported. We refer to these coarsening methods as the ”any” method – any tie reported during the interval is treated as having existed for the entire time period, or the ”all” method, where only ties that have existed for the entire time point are treated as having existed at all during the time point. Figure 3.1 shows an illustration of counting two ties that existed for a partial time point, but are treated as existing for the entire time point: an increase in duration of both ties, and an observation of a concurrency that did not in fact exist. The longer the interval, the more this will occur: an increase in duration of ties with an increase in concurrency rates, both being important elements of sexual partnership networks that predict HIV diﬀusion. Alternatively, Figure 3.2 shows the ”all” coarsening type, where ties are only counted as having existed for a time interval if they existed for the entire interval. In this case, durations are under-estimated relative to an instantaneous measurement, and a concurrency is not counted that existed. When observing this for sexual contact networks, the magnitude of these biases have not been measured, despite data being coarsened in measurement. Here, we attempt to measure the impact of coarsening methods on our observation of network properties.

26 Actual Observed

TIME TIME

(a) (b)

Figure 3.1: Illustration of ”Any” Coarsening Type

Actual Observed

TIME TIME

(a) (b)

Figure 3.2: Illustration of ”All” Coarsening Type

27 Parameters from a STERGM or DNR have a specific interpretation that only applies to the timescale of the fitted data. If data were measured at a monthly timescale, how might we make predictions from this model of network change at a daily scale? Here, we test two different methods of adjustment: naive adjustment of nominal time units and duration correction .

3.3 Timescale Adjustment for Sexual Contact Network

Models

STERGMs extend ERGMs for a dynamic model in discrete time (Krivitsky, 2012; Krivitsky and Handcock, 2014). When modeling social networks using ERGM, the model ﬁts one model to a single, cross-sectional network. STERGMs use several discrete observations of a single network, and has a separable structure: one model that underlies the formation network, and another that underlies the dissolution network. To build a STERGM, two ERGM formulas are speciﬁed in the same model – one that corresponds to how network ties form, and another than corresponds to how existing ties persist.

Given a set of ties yt in a network, a formation network Y+ is the set of ties formed in an ERGM with formation parameters θ+ and formation statistics g+(y+,X):

exp{θ+ · g+(y+, X)} Pr(Y+ = y+Yt; θ+) = , y+ ∈ Y+(yt) (3.1) c(θ+, X, Y+(Yt))

where c(θ+,X, Y+(Y t)) is a normalizing constant equal to the summation over the space of possible formation networks on n nodes.

For the same set of relations yt, a dissolution network Y− is simultaneously generated from

28 a q-vector of dissolution parameters θ− and statistics g−(y−, X):

exp{θ− · g−(y−, X)} Pr(Y− = y−|Yt; θ−) = , y− ∈ Y−(yt) (3.2) c(θ−, X, Y−(Yt))

The resulting network is built by applying the changes in Y+ and Y− to yt:

Yt+1 = Y+ − (Yt − Y−) (3.3)

Both θ− and θ+ conventionally include intercepts, but with diﬀerent interpretations: the formation intercept represents a tendency for nodes to form ties; the dissolution intercept is the baseline logged odds that any tie will dissolve any a given time point, which is equal to 1/(average length of ties). In typical parameterization for sexual contact networks, the dissolution ERGM is conventionally modeled using an intercept alone.

Researchers have adjusted the timescale of simulated data by adjusting the dissolution rate – which controls the length of partnerships measured in discrete model time steps – to apply to diﬀerent timescales (Goodreau et al., 2012). If data were collected at the monthly timescale, changing the dissolution rate to a daily scale would involve dividing the risk of dissolution by 30 (days in a month). No changes are made to formation network dynamics. Formation parameters – mixing terms, formation propensity, degree terms, and others – are all kept identical to the unadjusted timescale in the conventional adjustment. However, problems may arise from this adjustment method: reducing the odds of dissolution at each time step while keeping the formation propensity the same could increase average degree; if already-formed ties are less likely to dissolve without a similar adjustment for decreased

29 formation propensity. This could lead to misleading network simulations. To our knowledge, no systematic evaluation of simulations has been undertaken to compare networks between adjusted and unadjusted model simulations.

This paper evaluates this timescale adjustment to determine whether the resulting simulated networks reasonably match the underlying STERGM-generated network at target timescales. If data simulated from these models result in networks that do not match what we would expect with observed data, other adjustments may need to be made.

When adjusting models to apply to different timescales, some consideration is needed about what exactly is desired from the adjustment. When simulating from timescale-adjusted model, does the researcher want simulations to aggregate tie dynamics in the time interval, or instantaneous draws from the network between intervals (i.e. at each time interval observation, do we want to observe all social activity within the interval or only a cross-sectional snapshot of the network at the interval)? Since the goals can differ according to context, it is useful to define what these goals are, and what we intend to recover with timescale adjustment.

Let us imagine that we have a network process unfolding in continuous time, and let y be a realization of that process. Define C(y, t) to be a t-coarsening of y, i.e. a representation of y (which is a series of spells) into a time series whose slices reflect successive periods of duration t. (Obviously, we can imagine multiple such coarsening functions, leading to different coarsenings.) Likewise, M(z) be a discrete-time network model fit to time series z, and let g be a function defined on network time series. We reasonably expect such a model to recover properties of the fitted data:

Eg(C(Y, t)) = Eg(M(C(y, t))) (3.4)

30 for y drawn from Y . That is, recovering the expected properties of the distribution of t- coarsened Y realizations by feeding M a t-coarsened realization of Y . Now, let a t-adjustment of M, A(M, t), be a modiﬁcation of M that translates it from the timescale of its input data to timescale t. Similarly, we reasonably expect such an adjustment to recover network properties of timescale-adjusted data:

Eg(C(Y, t0)) = Eg(A(M(C(y, t)), t0)) (3.5)

where, again, y is drawn from Y . On the left, we have the expectation of some given function g applied to t0-coarsened realizations from Y . On the right, we again have the expectation of g, but now the function is being applied to draws from the t0-adjusted model, M, that was originally fit to some t-coarsened draw y from Y . (Algorithmically: (1) draw y from Y ; (2) t-coarsen y to get the initial time series data; (3) fit M to the coarsened data; (4) adjust M to timescale t0; and (5) take an infinite number of draws from the adjusted model, calculate g on each, and take the mean.) We intend to find the timescale correction that will reproduce the properties of the coarsened data (or, more exactly, coarsened draws from the generating process).

The above is a reasonable goal for an adjusted network model – to recover the properties of adjusted data fed to the model. Another goal could be to simulate precisely the network state at each adjusted time point:

Eg(Y, t0) = Eg(A(M(y, t), t0)) (3.6)

31 This differs from Equation 3.5 in that, instead of reproducing data that resembles timescale- adjusted data, it instead resembles instantaneously-drawn cross-sectional slices of the network over arbitrary timescales. Obviously, such a goal would be difficult in practice for refined timescales using RLH data. RLH measures coarsened data by design, and so simulating instantaneous draws from a network process at refined timescales is perhaps intractable without directly-measured, instantaneously-drawn network observations (i.e., follow-up panel studies). RLH data are the only observations available For nationally-representative sexual contact data – panel follow-up studies could be a good alternative for future work, which, by design, measure the network instantaneously at each interview time.

Since our analysis is restricted to RLH data, we compare simulated network properties from adjusted models with that of timescale-adjusted data from the original data consistent with Equation 3.5. Further research should explore the possibility of reproducing multiple cross- sectional draws at diﬀerent time scales, when data is measured only at one time scale.

3.4 Data and Methods

This study is split into two broad parts: observing the effect of timescale adjustment on observed statistics of sexual contact network data; second, the ability of adjusted models to reproduce statistics from the adjusted data. We first use Krivitzky et al’s model (2012) fitted to observed data at the monthly timescale; we use this model to simulate a sexual partnership network, and then adjust the data to different timescales, e.g. ”coarsened” to yearly timescale or ”refined” to daily timescale. Changes in network statistics (e.g. degree, concurrency) are observed. Second, we use the same model from Krivitzky’s (2012) parameterization, and adjust parameter coefficients consistent with three different adjustment methods (see below). We then assess the ability of model timescale adjustments to reproduce networks consistent with data adjustments (Equation 3.5). Data and methods involved in each collection of

32 Simulate Timepoints Calculate Data from Adjusted Adjustment Statistics Adjustment Model

Simulate STERGM Dissolution Calculate Data from Adjusted Model Adjustment Statistics Adjustment Model

Formation Edges+ Simulate Calculate Dissolution Data from Adjusted Statistics Adjustment Model

Calculate Refne Data Statistics

Calculate Simulate All Time Points STERGM Statistics Data Model from Model

Coarsen Data Calculate Any Time Point Statistics Study Plan Figure 3.3: Study plan for Eﬀects of Temporal Resolution Adjustments on Dynamic Sexual Contact Models analyses are described below.

We use existing models with parameters drawn from Krivitzky et. al. (2012) for our evaluation. The STERGM was ﬁtted to National Health and Social Life Survey (NHSLS) – a retrospective life-history data collected on a national probability sample of 3332 respondents (Laumann et al., 1995). The data was intended to be a nationally representative sample of adults aged 18-59 in the United States. Researchers used both face-to-face interviews as well as self-administered questionnaires to collect sexual history data. Sexual history of the respondents since age 18 is constructed from both the face-to-face and self-administered sections. Respondents are asked to report sexual histories for the past year. Basic demographic information is collected both on the respondents and their partners, and start and stop times for relationships to the month, including ones begun before the study period. The result is

33 3576 total ties.

The formation model includes the following terms: an intercept (offset), actor activity by sex and race, homophily on sex, race/ethnicity (split by homophily on black, Hispanic, white, and other), age effects (normalized age, square root of normalized age, and age difference), and an effect for being an older-male and younger-female; the dissolution model has only an intercept: all ties have an equal chance of dissolving at every time step, set to the logged mean duration of ties in the set.

Adjustments to the time scaling in the past have adjusted only the dissolution parameter by scaling the mean duration (measured in time steps) to be equivalent to clock time at the requisite scaling. Where t0 equals the level of reﬁnement (1/2, 1/3, 1/4, ...1/R) or coarsening (2, 3, 4...C) and d¯ is deﬁned as the average duration of ties.

1 − (1/d¯)(1/t0) θ0− = log( ) (3.7) 0 (1/d¯)(1/t0)

This adjustment has been used to adjust from monthly to daily timescaling for sexual contact networks (Goodreau et al., 2012; Carnegie et al., 2015; Goodreau et al., 2017). We assess this adjustment method, as well as two others. The ﬁrst alternative method is to keep all parameters identical, but to scale the number of timepoints simulated to match the clock time of requisite scaling. For example, to adjust from monthly scaling to daily timescale, an identical model is simulated 30 times longer. Since STERGM simulates an equilibrium distribution of ties, this adjustment will preserve statistics that converge at the original timescale (such as degree and concurrency) but will not preserve average duration of ties.

34 A second alternative is scale the formation edges term along with the dissolution edges term:

1 − (1/N)(1/t0) θ0+ = log( ) (3.8) 0 (1/N)(1/t0) where t0 equals the level of refinement (1/2, 1/3, 1/4, ...1/R) or coarsening (2, 3, 4...C) and N is defined as the population size of the network. When scaling only the dissolution parameter to account for average duration, you adjust the speed of dissolution without scaling the speed of formation. When dissolution is slowed down, as in refinement adjustments, the speed of formation stays constant, with a slowdown in their dissolution. This could increase the number of edges formed in the same clock time in the refined model as in the unadjusted model. The formation edges parameter scaling adjusts the formation speed analogously to the dissolution parameter adjustment. Intuitively, this adjusts all formation model parameters to scale with adjustments to the arbitrary 1/N baseline provided by the above STERGM parameterization.

Data simulated from the above adjustments will be compared with temporally-adjusted data simulated from the unadjusted model. In our simulations, we use two types of coarsening methods: in the first, a partnership is counted as existing for a month if the partnership exists for any duration in the month (any time point); second, a partnership is counted only if it exists for the full duration of the month (all time points). We expect substantial differences in observed network statistics coarsening type. One expectation is that with an ‘any time point’ method, concurrency will be much more common in data simulated from fitted coefficients. In data adjusted with the ‘all time points’ coarsening method, ties are removed from analysis, thus reducing concurrency and degree. Each coarsening type thus has its own potential challenges, but with unclear magnitude. Here, we observe the effect of coarsening type on the observed statistics. Adjusting simulated data to reflect more finely-

35 grained timescales involves duplicating each time point to the target scale. For example, for an adjustment from the monthly to weekly timescale, we duplicate each month four times to reﬂect the number of weeks in a month. To adjust from monthly to daily, we duplicate each monthly observation thirty times.

We reﬁne original, monthly-scaled data into intervals of 2 (twice a month), 3 (three times a month), and 6 (six times per month); we coarsen data into intervals of 2 (bimonthly), 3 (quarterly), 6 (twice a year), and 12 (yearly). For each adjustment, we simulate 100 temporal networks over 10 years of clock time. We burn in all networks to 400 months clock time before assessing statistics. Total, we simulate 3500 temporal networks, which amounts to 35,000 years of simulated sexual contact network data.

Our chief benchmark for model adjustment is if simulated data from adjusted models will recover statistics calculated from adjusted data. If statistics from simulated data from adjusted models predict statistics that match adjusted data, we consider that a successful model adjustment.

3.5 Results

Figure 3.4 shows the relative ability of model timescale adjustments in reproducing average durations seen in timescale-adjusted data. Since the timepoints adjustment (left) does not at all adjust for duration, ties wildly vary in duration across timescales. When the duration- only adjustment is applied (center), the durations match targets up until 12-month intervals; the combined duration and formation edges adjustment (right) also retains target durations. This shows that the model duration adjustment is eﬀective at scaling average durations to target values.

Figure 3.5 shows the ability of model timescale adjustments to preserve average degree. The

36 Formation Edges + Timepoints Adjustment Duration Adjustment Duration Adjustment 150 150 150

Timepoints Adjustment Duration Adjustment Formation Edges + Duration Adj. Refined Data Refined Data Refined Data Coarsened Data, Any Coarsened Data, Any Coarsened Data, Any Coarsened Data, All Coarsened Data, All Coarsened Data, All 100 100 100 Average Duration Average Duration Average Duration Average 50 50 50

0 Refine Coarsen 0 Refine Coarsen 0 Refine Coarsen

6 3 2 2 3 6 12 6 3 2 2 3 6 12 6 3 2 2 3 6 12 Interval Adjustment (Months) Interval Adjustment (Months) Interval Adjustment (Months)

Figure 3.4: Eﬀects of Model Adjustments on Average Duration of Predicted Ties

Formation Edges + Timepoints Adjustment Duration Adjustment Duration Adjustment 1.4 1.4 1.4 Timepoints Adjustment Duration Adjustment Formation Edges + Duration Adj. Refined Data Refined Data Refined Data Coarsened Data, Any Coarsened Data, Any Coarsened Data, Any 1.2 Coarsened Data, All 1.2 Coarsened Data, All 1.2 Coarsened Data, All 1.0 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 Average Degree Average Degree Average Degree Average 0.4 0.4 0.4 0.2 0.2 0.2

0.0 Refine Coarsen 0.0 Refine Coarsen 0.0 Refine Coarsen

6 3 2 2 3 6 12 6 3 2 2 3 6 12 6 3 2 2 3 6 12 Interval Adjustment (Months) Interval Adjustment (Months) Interval Adjustment (Months)

Figure 3.5: Eﬀects of Model Adjustments on Average Degree

Formation Edges + Timepoints Adjustment Duration Adjustment Duration Adjustment

0.15 Timepoints Adjustment 0.15 Duration Adjustment 0.15 Formation Edges + Duration Adj. Refined Data Refined Data Refined Data Coarsened Data, Any Coarsened Data, Any Coarsened Data, Any Coarsened Data, All Coarsened Data, All Coarsened Data, All 0.10 0.10 0.10 0.05 0.05 0.05 Instantaneous Concurrency Instantaneous Concurrency Instantaneous Concurrency 0.00 0.00 0.00

Refine Coarsen Refine Coarsen Refine Coarsen −0.05 −0.05 −0.05 6 3 2 2 3 6 12 6 3 2 2 3 6 12 6 3 2 2 3 6 12 Interval Adjustment (Months) Interval Adjustment (Months) Interval Adjustment (Months)

Figure 3.6: Eﬀects of Model Adjustments on Predicted Average Per-Timepoint Concurrency

37 Formation Edges + Timepoints Adjustment Duration Adjustment Duration Adjustment 40 40 40 Timepoints Adjustment Duration Adjustment Formation Edges + Duration Adj. Refined Data Refined Data Refined Data Coarsened Data, Any Coarsened Data, Any Coarsened Data, Any Coarsened Data, All Coarsened Data, All Coarsened Data, All 30 30 30 20 20 20 10 10 10 Average Formation Events Average Formation Events Average Formation Events Average 0 0 0

−10 Refine Coarsen −10 Refine Coarsen −10 Refine Coarsen

6 3 2 2 3 6 12 6 3 2 2 3 6 12 6 3 2 2 3 6 12 Interval Adjustment (Months) Interval Adjustment (Months) Interval Adjustment (Months)

Figure 3.7: Effects of Model Adjustment on Average Predicted Per-Timepoint Network For- mation Events timepoints adjustment preserves degree, but the duration-only adjustment increasingly biases degree at increasing levels of adjustment. However, when adding an edges adjustment, predicted degree matches target values. Similar cross-sectional network statistics are preserved in timepoints adjustment as well as duration-plus-formation edges adjustment, while duration statistics are preserved in both duration-only adjustment and formation edges- plus-duration adjustments. In general, adding a formation edges adjustment successfully reproduces target statistics associated with the ”all time points” data adjustment. We sus- pect that the change in degree that is seen when scaling only the dissolution parameter is due to the adjusted speed of dissolution without a scaling of the speed of formation. For example, a refined model in this adjustment type leaves the speed of formation unadjusted, but with a slowdown in their dissolution. This in turn increases the number of edges formed in the same clock time in the refined model as in the unadjusted model, because more nodes are isolates and thus have high propensity to form ties in the next time step. This explains Figure 3.5, which shows increasing degree with refinement and decreasing degree with coarsening when only adjusting the model persistence term. Similar effects can be found with concurrency and forward-reachable vertices: the more refinement, the higher the connectivity of the graph. With a equally-scaled model formation edges term, this effect is moderated.

38 Formation Edges + Timepoints Adjustment Duration Adjustment Duration Adjustment 40 40 40 Timepoints Adjustment Duration Adjustment Formation Edges + Duration Adj. Refined Data Refined Data Refined Data Coarsened Data, Any Coarsened Data, Any Coarsened Data, Any Coarsened Data, All Coarsened Data, All Coarsened Data, All 30 30 30 20 20 20 10 10 10 Average Dissolution Events Average Dissolution Events Average Dissolution Events Average 0 0 0

−10 Refine Coarsen −10 Refine Coarsen −10 Refine Coarsen

6 3 2 2 3 6 12 6 3 2 2 3 6 12 6 3 2 2 3 6 12 Interval Adjustment (Months) Interval Adjustment (Months) Interval Adjustment (Months)

Figure 3.8: Eﬀects of Model Adjustment on Average Predicted Per-Timepoint Network Dis- solution Events

Formation Edges + Timepoints Adjustment Duration Adjustment Duration Adjustment 8 8 8 Timepoints Adjustment Duration Adjustment Formation Edges + Duration Adj. Refined Data Refined Data Refined Data Coarsened Data, Any Coarsened Data, Any Coarsened Data, Any Coarsened Data, All Coarsened Data, All Coarsened Data, All 6 6 6 4 4 4 Logged FRVs Logged FRVs Logged FRVs 2 2 2

0 Refine Coarsen 0 Refine Coarsen 0 Refine Coarsen

6 3 2 2 3 6 12 6 3 2 2 3 6 12 6 3 2 2 3 6 12 Interval Adjustment (Months) Interval Adjustment (Months) Interval Adjustment (Months)

Figure 3.9: Eﬀects of Model Adjustment on Average Predicted Forward-Reachable Vertices

Formation Edges + Timepoints Adjustment Duration Adjustment Duration Adjustment

120 Timepoints Adjustment 120 Duration Adjustment 120 Formation Edges + Duration Adj. Refined Data Refined Data Refined Data Coarsened Data, Any Coarsened Data, Any Coarsened Data, Any

100 Coarsened Data, All 100 Coarsened Data, All 100 Coarsened Data, All 80 80 80 60 60 60 40 40 40 Concurrency Durations Concurrency Durations Concurrency Durations Concurrency 20 20 20

0 Refine Coarsen 0 Refine Coarsen 0 Refine Coarsen

6 3 2 2 3 6 12 6 3 2 2 3 6 12 6 3 2 2 3 6 12 Interval Adjustment (Months) Interval Adjustment (Months) Interval Adjustment (Months)

Figure 3.10: Eﬀects of Model Adjustment on Average Predicted Concurrency Durations

39 Similarly, Figures 3.8 and 3.7 show that only formation-plus-duration edges model adjustment preserves network churn seen in the ”all time points” data adjustment type. The formation adjustment also helps the formation and dissolution rates better match targets. When only adjusting the average duration of ties, formation rates increase when coarsening, due to the increased speed of dissolution and subsequent creation of isolates. Since isolates have higher propensity to form ties, more formation events are observed despite no adjustment of the model formation term. The unintended consequence of changing dissolution rates is the change in the composition of nodes in the formation network, which results in changes in the formation rates to satisfy speciﬁed isolate and concurrency penalties.

The second set of results is the effect of coarsening type on observed statistics. Surprisingly, duration, network churn, concurrency, and degree are only minimally affected by coarsening type. This surprisingly result is likely due to the relatively slow churn of sexual contact networks. In a given month with 1000 nodes and 750 edges, only 3.1 ties formation events are observed – this slow rate makes the aggregation of 6.2 events per two months, or 9.3 events per three months, not substantial enough to greatly affect most statistics. However, removing a few concurrent ties in the ”all time points” adjustment greatly influences the number of forward-reachable vertices in simulated data. The only substantially altered statistics are forward-reachable-vertices (FRVs) and concurrency durations. Though degree, concurrency rates, and durations stay nearly-identical between types, FRVs and concurrency durations are important for network-dependent public health processes, and further research should assess their effects upon epidemiological predictions.

3.6 Discussion

Overall, model adjustment of the duration only was suﬃcient to scale average durations, but many other statistics – such as degree, concurrency, and network churn – did not match

40 targets. The timepoints adjustment preserved statistics like degree and concurrency, but did not preserve duration-based statistics or network churn. The formation edges adjustment along with duration adjustment preserved all observed statistics in both reﬁnement and coarsening, with the exception of forward-reachable vertices at highly-coarsened timescales (6+ months).

The duration adjustment has been used in prior model adjustment for sexual contact networks. Depending upon the goals of the researcher, this adjustment is not sufficient to retain non-duration statistics, such as degree. It does match targets on duration, however. The duration-only adjustment increases degree at refined timescales, likely due to an increased rate of dissolution resulting in a greater number of isolates into the formation network at later time steps (isolates are penalized in the network, as nodes tend to seek ties when no sexual partnership is present). The duration-only adjusted model also increases the formation rate in networks, without any prior specification in the model to do so. With a large effect for partnership formation when nodes are isolates present, dissolving ties without adjusting the formation rates directly results in unintentionally higher degree.

Second, the effect of coarsening type on the observed statistics only substantially affected forward-reachable vertices, which is highly sensitive to ties being excluded from the network series. Degree, concurrency level, and durations were all only minimally affected by coarsening type, due to the relatively slow churn of the network relative to the scale of measurement. However, the statistics that were greatly affected – FRVs and concurrency durations – are especially important for HIV diffusion in sexual contact networks, and the impact on predictions of HIV spread could be severe. The effect of coarsening type on epidemiological predictions should be explored by future studies.

41 3.7 Limitations

The National Health and Social Life Survey is a nationally-representative sample of sexual partnerships. While it is often used for modeling of sexual partnerships, it does come with limitations: partnerships are aggregated to the monthly level, which aggregates all partnerships for the same ego that existed in the same month as having simultaneously existed. This could overestimate concurrency; as shown above, however, this is likely to only have minimal eﬀects on observed concurrency, since sexual partnerships churn slowly relative to the monthly scale. Second, the number of partnerships reported by men are higher than those reported by women, likely caused by a small number of over-reporting males (Mor- ris, 1993). Updating the collection to both collect instantaneous network data (to measure instantaneous concurrency rates) and observation of gender diﬀerences in sexual contact reporting (to determine if women are under-reporting, or men are over-reporting) would be fruitful.

3.8 Conclusion

We tested the ability of three possible timescale adjustments for sexual contact network models whose simulations reproduce statistics from adjusted data. The only network adjustment that preserved both cross-sectional, duration, and churn-rate statistics of adjusted data was a model adjustment of both the formation edge term and persistence term. Based on the above results, we suggest that researchers that seek to timescale-adjust SCN models apply both a duration and formation edges adjustment, to oﬀset the decreased network dissolution speed present in the duration-only adjustment.

We also introduced two diﬀerent targets of model timescale adjustments. One such target is to reproduce the statistics calculated from duration-adjusted data simulated from an

42 unadjusted model; the other target reproducing a cross-sectional slice of social activity at diﬀerent timescales, with requisite network churn between timepoints. Our method is primarily focused on achieving the ﬁrst target; future studies should address the second goal, which may be impossible without follow-up panel studies.

Finally, we assessed the diﬀerence in coarsening methods on statistics. We found that forward-reachable vertices and concurrency durations greatly varied according to how one coarsened data, suggesting that observed network activity could vary substantially according to how data is collected. Aggregating all network ties that existed within an interval as having existed for the entire interval greatly increased the number of forward-reachable vertices due to increased connectivity of the graph. The over-estimation of concurrency durations could have implications for epidemiological modeling, which should be explored in future work.

43 Chapter 4

Practical Methods for Imputing Follower Count Dynamics

4.1 Introduction

Microblogging sites like Twitter have become an important setting for online social interaction, both among members of the public at large and between organizations (e.g., government agencies, ﬁrms) and the public. Members of the general class of publish/subscribe (pub/sub) systems, microblogging sites typically allow users to post (publish) information to a stream, to which other users may subscribe (or follow). Since the followers of a given user are exposed to his or her messages, follower counts (i.e., indegree in the following or subscrip- tion network) are an important indicator of initial audience size for users’ posts. Moreover, retransmission of users’ messages by followers can further extend their reach, and/or increase the number of times recipients are exposed to them (multiple exposure being known to enhance e.g. behavior change (Rodgers et al., 2005; Vilella et al., 2004; Franklin et al., 2003; Fjeldsoe et al., 2009); and recall (Haber and Haber, 2000). Follower counts have been

44 shown to be an important factor in such retransmission. These effects can be large, with one study finding that doubling of follower counts resulted in 2-6 times the number of predicted retweets (Sutton et al., 2015). Other studies also report positive effects for follower count on retransmission (Son et al., 2017; Suh et al., 2010; Sutton et al., 2014a,b). Not surprisingly, growing the number of followers is a core interest of firms, organizations, or individuals on Twitter (Bakshy et al., 2011; Scanfeld et al., 2010); as follower counts are a key component of influence (Cha et al., 2010; Weng et al., 2010). Follower counts are also interesting in and of themselves as a measure of prominence, the variation therein being an example of attentional dynamics (Almquist et al., 2016).

While follower counts are of clear importance, recording them through time has challenges. For many systems of interest (including Twitter), follower counts can only be measured at query time, and cannot be obtained retroactively; hence, gathering data on follower count dynamics requires regular querying of targeted accounts via a prospective design throughout the intended observation period. Such designs offer many opportunities for missingness to arise. For instance, API limits (e.g., exhaustion of available queries) and system failures may generate intervals of missingness internal to the data collection period. Limits on the number of queries also leaves gaps of time between a query and tweet activity. Alternately, where key users are identified only after an event of interest has begun, follower counts early in the period may be unknown. Similarly, data collection that ends prior to the completion of a period of interest (as may occur, e.g., when reusing data collected for a different purpose) leaves counts at later intervals unobserved.

Such gaps in follower count observation are problematic for applications such as prediction of information transmission, estimation of message exposure, or even the modeling of attentional dynamics itself. While latent missing data models (see, e.g. Clark and Bjørnstad, 2004; Dufouil et al., 2004; Van Dongen, 2001), can be employed for this purpose, such models are typically complex and reliant on computationally intensive approaches such as Markov

45 chain Monte Carlo (MCMC) that are challenging to scale to large data sets. Particularly for exploratory and large-scale analyses, it is useful to have relatively simple, scalable procedures for imputation that can eﬀectively predict missing values from readily available time series data. Such methods may serve as basic adjuncts to more complex procedures, or may in some cases be suﬃciently accurate to be used as stand-alone techniques.

This paper explores several approaches to the imputation of missing follower count data and evaluates these approaches in the context of a case study involving public health organizations communicating during the 2014 Ebola outbreak. Our focus is on relatively simple, scalable methods that can be easily employed on large data sets and that are not closely tied to speciﬁc applications. We oﬀer an assessment of method performance and a consideration of implications for practical use. As we show, simple procedures can work extremely well for both imputation and short-term extrapolation of follower counts in realistic settings.

4.2 Background

The online social media platform Twitter is a widely used pub/sub system (with over 328 million monthly active users as of 2017 (Twitter, 2017)) and is the canonical example of the “microblogging” genre (Dogruel, 2014; Dewey et al., 2012). Microblogging systems combine the publicly oriented, topical aspects of blogs with an emphasis on terse messaging (Sutton et al., 2014a, 2015; Son et al., 2017), encouraging users to share news, observations, commen- tary, or information in extremely brief messages intended for immediate consumption (Lin and Chitty, 2012). The brevity of the microblogging format lends itself well to use during rapidly unfolding events such as political protests (Gerbaudo, 2012; Eltantawy and Wiest, 2011; Wolfsfeld et al., 2013; Steinert-Threlkeld, 2017) and natural disasters (Gao et al., 2011; Fraustino et al., 2012; Sutton et al., 2015), where it has seen extensive use by state and corporate actors as well as the general public (Metallo and Gesuele, 2016; Cook, 2017; Chauhan

46 and Hughes, 2015). Microblogging services have also attracted substantial interest from researchers, both because of growing interest in understanding online behavior per se (see, e.g. Pan and You, 2017; Barber´a,2014; Varol et al., 2014) and because of the potential of data from such services to serve as time-resolved measurements of social dynamics (e.g., Mønsted et al., 2017; Borge-Holthoefer et al., 2015).

Primarily focused on public interaction, Twitter allows users to post short, public messages (“tweets”) to a stream which other users can subscribe (“follow”). Users often follow political or government organizations, celebrities, journalists, offline social contacts, or other accounts that suit their interests (Wang et al., 2017; Dubois and Gaffney, 2014). These follower relations comprise an attentional network (Spiro et al., 2016; Almquist et al., 2016), in which the sender of a tie attends to the communications of the recipient. Measurement of follower relations can be performed via a Twitter-provided application program interface (API), that allows users to collect information on the activity of public accounts. Queries supported by the API include current follower counts (as of query time), and potentially list of follower account ID numbers; such queries are subject to “budget” limitations, with only a specified number of queries being allowed within a fixed time period. Exceeding the API limits may lead to temporary or permanent restrictions on a user’s ability to place new queries. By prospectively querying an account at regular intervals, it is possible to obtain longitudinal data on follower counts (indegree). This approach has been successfully used to conduct a number of studies (e.g. Weng et al., 2010; Suh et al., 2010; Sutton et al., 2014b). However, such data collection is not without challenges, and numerous problems can arise that result in missing data.

There are several mechanisms through which missingness can arise when collecting data on follower counts, despite the obvious advantages of automated querying versus manual data collection techniques. First, collecting data on all possible Twitter accounts (or any other social media accounts) poses prohibitive storage and bandwidth costs, even absent limi-

47 tations imposed by the service provider; thus, selection of accounts (including the special case in which a census of a focal account set is attempted) is inevitable. Depending on the method employed (see: Aghababaei and Makrehchi, 2017), and the manner in which it is implemented, all accounts of interest may not be identiﬁed at the same time. This may lead to missingness during the initial observation period. A particularly common example of this problem arises when accounts are selected in response to an emergent event (e.g., a political protest or natural disaster). To the extent that the event is unanticipated, it may be impossible to begin observing some or all accounts of interest until after the event has begun. More subtly, changes in the composition of relevant accounts may occur as an event unfolds, necessitating the addition of previously irrelevant (or even non-existent) accounts. For example, previously obscure actors joining the response to an ongoing political or en- vironmental event may become important research targets, despite the lack of prospective data on their accounts. In subsequent analyses, imputation of such accounts’ follower counts early in the event is of obvious interest.

Even where the challenges of account selection itself are not an issue, the underlying technology on which social media data collection depends is not without ﬂaws. For instance, API limits require queries that are spaced apart in time; during periods of rapid growth of follower counts, hours between query times will miss large growths of follower counts. In addition, collection of data from social media services is often reliant upon service provider support for queries made through their APIs (as in the case of Twitter), and providers may have problems with accurate and consistent query handling in real time. For example, Twit- ter’s API has occasional periods of outage while Twitter activity is still ongoing, causing data from the aﬀected periods to be missing, and in some cases individual queries may fail for idiosyncratic reasons (e.g., a momentary server load increase or other technical issues).

Similarly, collection of ongoing behavior such as follower dynamics typically requires special- ized software and hardware infrastructure on the researcher’s side to make queries, process

48 and store data, and respond to faults (e.g., API failures, network outages, etc.). Such data collection infrastructure can be complex and subject to numerous possible points of failure, resulting in periods during which account behavior is not observed.

Finally, another source of “missingness” arises when forecasts of future account behavior are necessary or desired. At the time of this writing, it is generally impossible to observe data from future time points. Forward projection of follower account growth can nevertheless be important for many applications, such as predicting diﬀusion of information at the onset of a disaster event, predicting the attention that will be given to an announcement, or adaptively sampling accounts gaining prominence due to involvement in an ongoing political crisis. Forecasting follower counts from current and past count data is closely related to the imputation problem, and we consider it here as well.

Despite the importance of the issue, we are unaware of any research to date that has exam- ined the problem of imputing follower counts on Twitter or related systems. However, there do exist approaches to related problems in the time series literature. For example, Kalman ﬁltering (Kalman et al., 1960) has been used in numerous settings to adjust noisy or missing observations based upon a weighted average of previous observations, with higher weight given to observations deemed more accurately measured. Regression-based multiple imputation is another common imputation technique when a highly predictive set of covariates is available, including a time variable (Barnard and Meng, 1999; Royston, 2005; Rubin, 2004; Seaman et al., 2012; Van Buuren et al., 1999; Van Buuren and Groothuis-Oudshoorn, 2011; Von Hippel, 2009); this broad approach imputes missing data by predicting an outcome variable using relevant covariates in the set, then applies the prediction to unobserved (missing) cases. Other forms of missing data imputation include those designed for forecasting applications. For instance, the Lee-Carter model for forecasting mortality data uses a singular value decomposition of age distribution and yearly mortality data to predict mortality (and life expectancy). Simple linear, polynomial, or spline extrapolations of time series data also

49 can be used (Vink and van Buuren, 2013; Demirtas and Hedeker, 2008). Here, we employ a variety of techniques from the time series literature to determine a simple and accurate method for imputing missing follower counts on Twitter.

4.3 Imputation for Follower Count Data

As stated, follower counts can be missing for a number of reasons. Here, our goal is to formulate a relatively simple, practical method for imputing follower counts, that does not depend on detailed knowledge of the source of missingness and that can be deployed in a wide range of circumstances. We assess several candidate techniques in the context of a simulation study in which we impose missingness on an observed follower count series (i.e. we hold out observed data and treat as if it were missing). We evaluate each method by its ability to recover the held out data. Here, we discuss each imputation technique in turn.

Orthogonal Polynomials

Polynomials are a common and natural choice for functional approximation, particularly in an interpolative context. They are also easily and eﬃciently estimated. Here, we consider use of polynomial basis functions to impute follower counts, with ordinary least squares regression used for estimation.

Intuitively, a major motivation for the use of a polynomial basis is the fact that all continuous functions can be well-approximated over a ﬁxed interval by a polynomial of suﬃciently high order; at the other extreme, very low-order polynomials (linear and quadratic functions) are typical workhorse approximators familiar to the working sociologist. Given a k-order

Pk i polynomial Pk(x) = i=0 βix in variable x, we may think of the function Pk as the inner product XT β, where X = (0, x, x2, . . . , xk) is a vector of basis functions; for some arbitrary function g(x), the projection of g into the space spanned by X is analogous to a linear re-

50 ˆ gression of g(x) on X, and yields the k-order polynomial Pk that minimizes the squared error ˆ between g(x) and Pk(x). While follower counts are obviously discontinuous, the simplicity and ﬂexbility of polynomial approximation makes it an approach worth investigating.

While we do not here know g – the follower count as a function of time – we can approximate it at each query time by the observed count at that time point. The imputation process then reduces to performing an OLS regression of the follower count on the first k powers of the observation time. One important question when implementing this procedure is the order of k needed for optimum imputation (i.e., does one use just a squared term, or a squared term and a cubed term, and so on). While standard regression diagnostics can be helpful here, they suffer in practice from the well-known problem that the powers of an input variable are strongly correlated: X is a basis for the kth order polynomial, but it is not an orthogonal basis. This introduces numerical instability into coefficient estimates (betaˆ ), which degrades extrapolation quality, and makes model selection more difficult. One way to reduce this problem is to seek an alternative basis set for Pk(x) that is orthogonal with respect to some inner product; in this application, we can think of this as choosing a new set of polynomials that span Pk(x) (i.e., any member of Pk(x) can be constructed by taking a linear combination of them), but that are uncorrelated with respect to some sampling distribution over x. Such polynomials are known as orthogonal polynomials, and are widely employed in functional approximation (Gautschi, 2016; Scott et al., 2017; Szeg, 1939).

Our ﬁrst imputation method is thus to ﬁt a series of orthogonal polynomials to the data by least squares (Abramowitz et al., 1972), using the resulting function to impute unobserved counts. In general, the orthogonal polynomials fn(x) with respect to the weight function w(x), where degree [fn(x)] = n, and n = 0, 1, 2, ... are characterized by the condition

Z b w(x)fn(x)fm(x)dx = 0; (4.1) a

51 For convenience to the working sociologist, we use the poly() function in R, which is a base function for computing orthogonal polynomials. The function uses the following recursion algorithm to calculate monic orthogonal polynomials of degree k, represented by P0(x),

P1(x), P2(x), ... Pk(x):

• Center x byx ˆ

• Set P−1(x) = 0, P0(x) = 1, P1(x) = x.

• For i = 0, 1, 2, ...k, deﬁne li = hPi(x),Pi(x)i and αi = hPi(x)x, x)i/li. For i = 1, 2, ..., k,

deﬁne βi = li/li−1;

• For i = 2, 3, ..., k, use recursion to ﬁnd Pi(x):

Pi(x) = (x − αi−1)Pi−1(x) − βi−1Pi−2(x) √ • For i = 0, 1, 2, ...k, scale Pi(x) by its L2-norm, Pi(x) := Pi(x)/ li

The resulting linear predictor matrix is then placed into a least squares regression for parameter estimation. The polynomial order to be used, k, is chosen by experiment – imputation performance via imputed values compared to target values (see section 4.1).

Smoothing Spline Interpolation

While polynomials are a relatively flexible and efficient approximation tool, even greater flexibility can be obtained using smoothing splines (i.e., piecewise polynomial functions). Here, we use a generalized additive model (GAM) with thin plate regression splines to both interpolate and extrapolate follower counts as a function of time. GAMs are generalized linear models whose linear predictors depend partially upon some smooth function of the covariates (Hastie and Tibshirani, 1990; Wood, 2017), making them semi-parametric. For a smoothing function, we use thin plate regression splines, which we found to perform better than cubic splines. Thin plate regression splines use a lower-rank parameterization of thin

52 plate splines, which in turn are a form of splines with a smoothness function that tempers the ‘wiggliness’ often found in polynomial functions, also known as Runge’s phenomenon (Runge, 1901). Thin plate regression splines lower the rank (i.e. the number of coeﬃcients relative to the amount of data) of thin plate spines using a truncated eigen-decomposition. For a more formal treatment of thin plate regression splines, see Wood (2003). Generalized additive models with thin plate regression splines are computationally eﬃcient and readily available in R in the package mgcv (Wood, 2017).

Regression Techniques

Finally, we use familiar regression techniques – specifically, negative binomial and Poisson regression models, which are both natural choices for count data in general. Poisson regression (also called log-linear regression) assumes the data is Poisson distributed, which is often the case with count data. Negative binomial regression is similar, except it relaxes the assumption made by the Poisson model that the variance of the modeled variable is equal to its expectation. In particular, the negative binomial can be viewed as a gamma mixture of Poissons, and can thus be viewed as arising from a (fully latent) hierarchical structure in which there is uncertainty in both the expected count to be observed at a given time point and in the count that arises given the expectation. In our analysis, we employ a simple and universally applicable model using only time as a continuous covariate, together with account-level fixed effects.

4.3.1 Evaluation Study: U.S. Public Health Accounts in the 2014

Ebola Outbreak

To provide a realistic evaluation of our candidate imputation methods, we employ a simulation study based on an observed data set. Speciﬁcally, we draw our observed data from a large set of U.S. public health agency accounts that were involved with public communica-

53 tion during the 2014 Ebola outbreak. By artiﬁcially inducing missingness and assessing our ability to correctly recover it, we are able to assess how well each method performs under realistic conditions.

Data and Experimental Design

Data were collected using the Twitter Search Application Programming Interface (API). We collected all available tweet and account metadata over the month of October, 2014, for 236 local and state health departments, and U.S. federal agencies who engage in public communication during public health threats. We selected these accounts by drawing on publicly available lists of public health Twitter accounts maintained by the CDC, the Nations Health (a publication of the American Public Health Association), and a public health researcher (Harris et al., 2013). These accounts were associated with state, local, and federal health agencies in the United States. Those in the dataset produced at least one message containing the word Ebola between October 1st and 29th. During this month, Ebola was becoming a well-known public health threat in the United States, and increased attention was given to public health organizations dispersing information about the disease, the potential hazards it posed, and appropriate actions to mitigate risk (Piot et al., 2017). Disseminating information was a key intervention for agencies attempting to prevent the spread of Ebola, and such information was aggressively sought by members of the public concerned with how to prevent its spread to their own households. The case is also notable for its widespread impact (with Ebola cases in Dallas, Ohio, and New York), and for having several distinct points during the event in which public attention surged due to new Ebola cases being reported by media. The 2014 Ebola outbreak is hence a case with the potential for both slow and sudden growth in follower counts within a large and diverse population of accounts (from small, local agencies to prominent national organizations), making it an excellent setting to test the ability of our imputation methods to capture both regular and irregular behavior in system heterogeneous in spatial and temporal scales.

54 There are two sources of follower counts from Twitter’s API: the follower count included in the query of user tweets, and a second source of follower counts included in a query of metadata for each account. In some cases, tweets are retroactively queried, but metadata cannot be retroactively queried. Follower counts are drawn from both sources, matched as closely as possible to timestamps of tweets from our accounts. If follower counts were observed more than an hour away from the tweet, the follower count is considered missing. This reflects a dataset most likely to be useful to researchers matching follower count covariates with characteristics of tweets (e.g., diffusion of the tweet). A quarter of accounts’ follower counts were first observed before October 7th, and three-quarters of accounts were first observed before the 15th. A broad-based growth in follower counts occurred during this time for these accounts, with few instances of follower count decay. This case is characterized by sustained attention over a period of weeks, concerning a high-uncertainty, latent threat with severe consequences and an unknown but potentially high risk of escalating into a mass casualty event.

Figure 4.1a shows the diﬀerence between posting time for a tweet (its creation time) versus the closest observation time for follower count in our data set. The posting time for a tweet is often what researchers require when measuring retweet rates, assuming that the primary application of interest is to study the relationship between follower count and the spread of information online (e.g., through its impact on retweets and tweet exposure). The further our observation time is from tweet creation time, the more likely it is for observed follower count at posting time to be inaccurate, and thus the initial measure of tweet exposure to be inaccurate. In Figure 4.1a, the cyan histogram represents the distribution of this time gap for “non-missing” observation, meaning the API query was successful just before or after the tweet posting time. About 17% of cases were missing: red bars represent frequencies of missing observations, meaning either the account was not in our system at the time, or a query failed to observe their follower count. As our data collection continued, some accounts were added late in the period, resulting in some observations being days or weeks

55 removed from tweet posting times. In general, queries retrieved follower counts within 24 hours before or after the tweet was posted. However, as Figure 4.1b shows, 24 hours can in some cases be long enough for rapid growth of follower counts to occur, in which case much of this growth will be missed between queries. This shows that even with successful queries, the time distance between tweet posting and observation time may be large relative to the timescale of follower growth. Such rapid growth of follower counts means that even when successfully querying Twitter, large gaps might exist between the measured follower count at query time and the unknown follower count at the time at which a tweet was ﬁrst posted. The inevitable presence of such gaps strongly motivates the use of imputation methods, even for high-quality data sets.

To place the above growth rates in perspective, Figure 4.1c shows the follower growth of accounts over the observation period. There is an overall growth in follower count over time for all accounts, with not one account experiencing a net loss over the entire time series.1 Interest in Ebola was increasing throughout the period, with more users paying attention to public health accounts due to the new risk of this frightening disease. (This is an important observation in public health contexts, as some followers gained during a crisis are likely to persist during the subsequent period.) Many accounts tweeted only once during the period (see 4.1d), making our observation lengths for those accounts only a day long. Even so, those accounts did not show an overall loss of followers over the period.

To determine the eﬀectiveness of all imputation methods tried, we imposed missingness in a manner one might encounter while collecting Twitter data, then used each method to try to recover the missing values. Each method is assessed according to how well it predicted held-out (“missing”) data. We ﬁrst selected a complete set of follower count observations for all accounts. We then imposed a random count (between 1–30) of “system outages,”

1Loss of individual followers does occur, but where present is outweighed by gain of new followers in our data. Because API limits prevent us from examining this phenomenon in detail, we examine only the net count at each point in time.

56 70 400 Missing 60 Not Missing 50 300 40 Freq. Freq. 200 30 20 100 10 0 0

0 100 200 300 400 1 5 50 100 1000 10000 Query Time Minus Creation Time (in hours) Follower Growth per Account, Daily

(a) (b) 60 20 50 15 40 30 Freq. Freq. 10 20 5 10 0 0

1 5 50 100 1000 10000 0 5 10 15 20 25 30 Follower Growth per Account, Entire Series Observation Length per Account, Days

Figure 4.1: Follower Count Descriptives for Ebola-responding Public Health Accounts

57 each of random length (between 1–48 hours), on the full set of accounts. This procedure thus randomly varied whether the imputation would require interpolation (both endpoints of the observation period were observed) and/or extrapolation (one or both of the endpoints needed estimation). In some cases, this procedure left very little or no data, which made imputation impossible. Generally, we were able to form estimates even with as much as 85% of the data missing from the analysis. Finally, we determine how well the values predicted from the imputation model recovers what is observed (using correlations between observed versus imputed values, and the absolute relative error). For example, if one account posted 100 total tweets in the monthly period, and we withhold the follower counts for tweets 20– 35, we impute the follower counts for those query points and compare the imputed values with the held out observations. We repeat this procedure for all imputation types over one thousand trials.

4.4 Results

Generally, orthogonal polynomial functions were sufficient for interpolation but not for extrapolation of follower counts. The other methods – smoothing splines, modeling the follower account changes using a negative binomial model using fixed effects for time and account, and modeling each account separately in a negative binomial model, were all sufficient for prediction in both the interpolative and extrapolative cases. Though more complicated, the spline and regression methods used here were able to impute missing data in the substantive cases where 1) data was missing from a late start in data collection and 2) future prediction of follower count (albeit with some obvious limitations). After discussion of results for interpolative imputation versus extrapolative imputation, we discuss a method for obtaining prediction intervals for each imputed value.

58 Imputation versus Extrapolation of Follower Accounts

As noted above, all methods were excellent at interpolation, but not all methods excel at extrapolating follower counts prior to or after the observation period. When imposing missing follower counts, orthogonal polynomial functions recovered missing values with great success when interpolating. Figure 4.2 shows the absolute relative error between the imputed values and the true answer when using orthogonal polynomials. When interpolating, relative error clusters around 0.009 (i.e., approximately 1% of the true value); when extrapolating, error is more than 16 times the size of the true value, on average. This diﬀerence stems from the compromise between the relatively high polynomial order needed to obtain good interpolative performance and the unstable behavior of such polynomials as one leaves the interval of the observed data, and is a common limitation of this approach.

59 Kernel Density of Imputation Error

1.1 Interpolation

1.0 Extrapolation 0.9 0.8 0.7 0.6 Density 0.5 0.4 0.3 0.2 0.1 0.0

1e−04 0.01 1 100 10000 1e+07 Relative Error

Figure 4.2: Kernel Density of Relative Error when Extrapolating v. Interpolating

The difference in performance between extrapolation and interpolation is attenuated using natural spline functions to impute missing follower counts. Our spline models had much better performance across extrapolation and interpolation results (approximately .026 and .006 absolute relative errors, respectively). Although smoothing splines are less efficient than global polynomial fits, they are able to obtain flexibility without requiring high-order terms (with the associated fast-growing derivatives). Though, in principle, global polynomial information can aid in extrapolation (relative to the local information guiding extrapolation of the spline function), that turned out not to be a factor in this case. Indeed, stability was

60 far more important, as we also see with our regression-based results.

Table 4.1: Mean Absolute Relative Errors Across Methods, Interpolation v. Extrapolation

Extrapolation Interpolation Polynomial 16.598 0.009 Spline 0.026 0.006 Regression, Accounts Together 0.013 0.012 Regression, Accounts Separate 0.009 0.007

In both Poisson and negative binomial models, we see similar results: extremely high accuracy (absolute relative error between missing and imputed values below .01 for both models). Accurate point estimates were achieved in all models, regardless of whether fixed effects for all accounts were included in the model (with time and account as predictors), or if each account’s follower counts were modeled individually (with only time as a predictor). All predictions were practically identical across models. Negative binomial models had better fit than Poisson, however. In the negative binomial model predicting follower counts of public health accounts during the onset of Ebola, deviance was reduced from 42140000 to 2151 (AIC is 33210) – a reduction most likely due to account effects. The model had no substantial differences in error for imputation versus extrapolation (0.013 for extrapolation, 0.012 for interpolation), unlike the spline and polynomial models.

Predictions are uncertain, and it is hence important to consider not only predictive accuracy but also the coverage of prediction intervals that indicate the level of uncertainty in the imputed values. While the regression and spline models provide excellent predictive performance, all models - polynomials, splines, and regression models – were insufficient in providing classical 95% prediction intervals that covered 95% of the missing values. In every case, the model was not conservative enough in providing prediction intervals with sufficient coverage. Though, for all models except polynomial extrapolation, point estimates were extremely accurate, the intervals surrounding each point estimate were so small that the actual missing value did not fall within the interval 95% of the time. Next, we show that a bootstrapped prediction interval is sufficient to provide a conservative prediction interval

61 broad enough to obtain good coverage for imputed data.

Predicted v. Observed, Follower Counts 15

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5 ● ●●● ●● ●●●● ●● ●●●●● ●●●●● ●●●●●● ●●●●●●●● ●●● ●●●●●● ●●● ● ●●●●●●●●●●●●●●● ●●● ●●● ●●● ● ●● ●● ●● ●● ●●● ●● Follower Count (logscale) Follower ● ●● ●●● ● ● ●●●●●● Observed Counts ●●● ●●●●

●●● ●●● ●●●●●● Predicted Counts

●● ●●●

●●● 0

Figure 4.3: Observed Follower Counts Versus Counts Predicted by Negative Binomial Re- gression (All Accounts Included)

Prediction Intervals

In Poisson and negative binomial models, predicted values and true values are nearly identical, and in most cases, the point estimate is enough for imputation of values. However, beyond mere point estimates of the true value, coverage of the “true value” by a speciﬁed prediction interval is also desirable. A prediction interval for a predicted value (as opposed to a conﬁdence interval for a predicted value) takes into account the variation of each estimated parameter, i.e., prediction intervals do not assume the model is precisely correct regarding point estimates of parameters.

In our tests, the classical 95% prediction interval for missing values in the negative binomial model only covered the true value around 65% of the time, due to its being extremely narrow

62 (rather than the estimates being generally inaccurate). Classical prediction intervals can be overconﬁdent for a number of reasons (including deviations from distributional assumptions). We employ a parametric bootstrap to obtain prediction intervals that are more robust to heterogeneity, with good results.

In the regression case, a bootstrapped interval for follower count Yr can be obtained as follows(Davison and Hinkley, 1997):

For n = 1, ..., N,

1 Compute a regression model as described in Section 3.3.

2 Compute predicted valuesy ˆ and residuals r1...n = y1 − yˆ1, ..., yn − yˆn

3 Add randomly-sampled residuals to each predicted value of yn, creating yn∗

4 Reestimating the model using the modiﬁed yn∗ values; then

5 For m = 1, ...M,

(a) Sample +,m from r1 − r,¯ ..., rn − r¯, and

T ˆ∗ T ˆ ∗ (b) Compute prediction error δ∗nm = x+βn − (x+β + +,m)

6 Compute 95% prediction interval P95(l < yˆn < u) by setting l and u equal to the

2.5th and 97.5th percentiles ofy ˆn + δ∗nm , respectively.

Intuitively, this method employs a parametric bootstrap to estimate the joint sampling distribution of the model parameters, and combines this distribution with the estimated residual distribution to estimate the prediction error distribution. This, in turn, is used to construct the prediction interval.

63 Table 4.2: 95% Prediction Interval Coverage across Methods

Method Coverage Polynomial 0.439 Spline 0.378 Regression, Together (Hessian) 0.648 Regression, Together (Bootstrap) 1.000 Regression, Separately (Hessian) 0.766 Regression, Separately (Bootstrap) 0.891

0.15 Neg. Bin. Model, Accounts Together Neg. Bin. Model, Accounts Separate 0.10 0.05 Proportion of Prediction Intervals 0.00

0 2 4 6 8 Width of Prediction Interval, logged

Figure 4.4: Bootstrapped Prediction Interval Width v. Regression Type

For the negative binomial model, observed follower counts are within the 95% prediction interval of predicted accounts 100% of the time, meaning conservative estimation of prediction intervals. Poisson models had 100% coverage as well, but intervals were much wider than negative binomial models; indeed, the Poisson-predicted boostrapped prediction intervals often reached 20+ logged follower counts. Negative binomial models had much small intervals with the same coverage; intervals spanning less than three units on log scale.

64 ● 1.0

●●●● ●●●●●● ● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●● ● ●●●●●●●●●●●●●●●●● ● ●● ●●●●●●●●●●●●● ●●●●●● 0.4 ●● ●● ●● ● ● 0.8 ● ● ●● 0.3

● 0.6 0.2 Coverage 0.4

●

● 0.1 Width of Prediction Interval ● ● 0.2 ● ● ● ● ● ●● ● ●● ● ●● ●●● ● ● ● ●● ●● 0.0 0.0

0 1 2 3 4 0 20 40 60 80 100 Total Growth over Period, Logged Number of Observations (a) Follower Growth per Account (b) Coverage of Prediction Interval versus Prediction Interval Width by Number of Observations

Figure 4.5: Performance of Negative Binomial Regression Imputation Prediction Intervals, with Accounts Modeled Separately

On the other hand, modeling each account separately using regression models resulted in bootstrapped prediction intervals that are extremely narrow. Though the interval varied, the average interval width of only .2 logged units. The trade-oﬀ is a slightly decreased coverage (89%). For separately-modeled regression models, prediction intervals varied widely according to the growth rates of individual accounts; thus, for these models, more tolerance for the variation in individual accounts’ growth rates is observed. One caveat is that modeling each account separately takes more than a few observations; with more than twenty observations, we get about 90% coverage across all accounts.

Finally, we find that the method works well even when approaching 80% missingness. Though the variation of error between imputed and observed increases, it is still on a small scale (not exceeding .02). We thus are confident in the method up to 85% missingness, at which point the model has trouble fitting with so few cases per account.

65 Relative Error as a function of Proportion Missing

● ● 0.017

●

● ●

0.016 ●

● ● ● ● 0.015

● ● ● ● ● ● ●

ARE ● ● 0.014 ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.013 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●

0.012 ● ●● ● ● ● ● ● ●● ● ●● ● ● ●● ● ●● ●● ● ● ● ● ● 0.011

0.0 0.2 0.4 0.6 0.8

Proportion Missing

Figure 4.6: Relative Error as a Function of Proportion of Cases Missing

4.5 Discussion

Overall, imputation of follower counts was accurate for polynomial functions and regressions when interpolating, but negative binomial regression did the best across both interpolation and extrapolation. Splines showed greater stability than orthogonal polynomials, but without high coverage for prediction intervals. Although naive prediction intervals from the regression methods were overconfident (true coverage less than nominal coverage), bootstrapped prediction intervals for models with all accounts included (with account-level fixed effects) were conservative enough to cover the true value 100% of the time in our sample while being narrow enough to be useful; bootstrapped prediction intervals for models with accounts modeled separately had substantially narrower prediction intervals, with 90% coverage, using 20 or more observations. While coverage is higher for models with all accounts

66 included, prediction intervals will be more precise when accounts are modeled separately.

Generally, most of our regression models worked well, but bootstrapped prediction intervals worked best with the negative binomial model, suggesting a reasonable distribution of residuals; still, if growth rates substantially vary across accounts, it may be more precise to model them separately. We suggest modeling all accounts jointly if researchers need to impute counts for accounts with fewer than 20 observations and/or if growth rates are not substantially diﬀerent across accounts. If an account has more than 20 observations and growth rates vary, modeling it separately from other accounts will result in similar point estimates with more precise bootstrapped prediction intervals than the joint model. Point estimates for all negative binomial models, whether accounts are modeled jointly or not, may be good enough for researchers looking for a quick way to impute missing follower counts. Further, if computational ability or some other reason prevents researchers from calculating a bootstrapped prediction interval, point estimates for the negative binomial model seems to be accurate enough for practical use.

4.6 Conclusion

In many cases, researchers encounter instances where a large amount of missing values need to be recovered by imputation. This can be especially true when new data sources such as Twitter have limits on API bandwidth or limit retrospective querying. In some cases, researchers would like to predict values going into the future. Here, we accurately impute follower counts for Twitter accounts during the beginnings of the Ebola crisis, using a negative binomial model with accounts modeled separately, and time as an independent variable. For very small samples, a model with all accounts modeled together will produce ﬁne point estimates, but with less precise prediction intervals. Future work could look into other cases, or a set of accounts not related to one another. For now, we hope researchers will ﬁnd this

67 procedure helpful in the unfortunate case of having missing data.

68 Chapter 5

Conclusion

Essential to understanding social systems is to understand the social networks that comprise these systems (Mayhew, 1984; Moreno, 1934). Due to advances in data collection and techniques for modeling, analyzing longitudinal social networks has become easier than ever, but not without its continuing challenges. This thesis aimed to improve our ability to model relational dynamics through a substantive exploration of a dynamic social systems, a purely predictive imputation of edge dynamics among social media accounts, and an exploration of how a dynamic model handles temporal adjustments.

In the first chapter, I found that dissolution dynamics differ substantially from formation dynamics in sexual contact networks. Race matching, age matching, and gender matching had no measurable effect on dissolution hazard above and beyond controls. However, concurrency dynamics varied – dissolution hazard was highly dependent upon concurrency rates, and whether partnerships were considered primary or secondary by the interviewee.

In the second chapter, I found that relatively simple methods can reproduce observed follower count dynamics on social media. Typically, account followers are only gained (not lost), and build slowly over time. Even so, rapid changes are also successfully predicted with great

69 accuracy by regression on each per-account follower count time series. Researchers can now impute missing follower counts or, with limits, predict future counts.

Finally, the third chapter reveals substantial differences in timescale correction methods for social network models, as well as differences in coarsening strategies for observed data. A correction used in some network literature is not fully sufficient to ensure simulated networks that preserve statistics that matter for disease diffusion modeling. We suggest that, in addition to adjusting the duration of ties, adjusting also the formation rates of ties to account for altered timescales.

This thesis provides many ﬁndings related to social phenomena in dynamic social systems. Among them include three broad ﬁndings that improve our understanding of social dynamics, particularly that of sexual contact networks:

• Many processes of sexual contact network formation do not apply to their dissolution, and many processes that result in the existence of SCN structure can only be speciﬁed in parameters of the dissolution network. This could greatly improve our ability to predict HIV diﬀusion, especially through modeling more precise concurrency dynamics.

• Subtle, non-obvious eﬀects of adjusting only durations of ties – including the model being forced into (unintentionally by the researcher) adjusting formation rates – has substantial consequences for the simulation of networks that could aﬀect our ability to predict disease.

• Tiny changes in observed degree and concurrency, caused by coarsening method, resulted in substantial changes in forward-reachable vertices, which is vitally important for many network-related processes.

The above ﬁndings could help improve our understanding of social systems. However, this work was limited by a series of practical limitations:

70 • The computational ineﬃciency of STERGMs hamstrung my ability to both model dissolution dynamics, and to more precisely calibrate timescale corrections. Methods like dynamic network logistic regression or other more eﬃcient techniques should be fully explored as a permanent replacement to STERGM, due to Moore’s Law prohibitive slowness relative to resource-hungry routines in STERGM, combined with the size of networks we encounter with social media.

• No follow-up panel study of US-representative sexual partnerships exists to my knowledge. Measurements of instantaneous concurrency are possibly wrong. Due to the importance of this for diﬀusion of disease, this should be rectiﬁed.

• A single question on a sexual contact network – whether the respondent considers a partnership to be ongoing, or whether sex with the potential partner is expected in the future – could help identify some ambiguous partnership dynamics (such as separated but not divorced partnerships).

The above ﬁndings, combined with the limitations of the studies, suggest a series of future steps for this research in the future:

• As mentioned above, separable DNR applied to sexual contact networks is an important – perhaps urgent – advancement, for its potential in eﬃciently modeling sexual contact networks at arbitrary population sizes.

• Sociologists – and not state or corporate inﬂuencers – should be at the forefront of research on the mechanics of public inﬂuence and political action, especially in regards to domains easily studied (like Twitter).

• Researchers collecting data on sexual partnerships should consider a hybrid follow-up panel plus retrospective-life-history collection method, to get both precise instanta-

71 neous network cross-sections and time-aggregated networks. Both contain important durational information, and a comparison of methods would be fruitful.

Through a series of substantive, methodological, and predictive developments, I have aimed to improve our understanding of social dynamics. Through the above ﬁndings, I hope to have contributed to the technical and theoretical development of social science, speciﬁcally through the analysis of the networks that comprise social activity.

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