Finding Prominent Actors in Dynamic Affiliation Networks

Hossam ShararaComputer Science Department

University of Maryland, College Parkhossam@cs.umd.edu

Lisa SinghDepartment of Computer Science

Georgetown University, Washington DCsingh@cs.georgetown.edu

Lise GetoorComputer Science Department

University of Maryland, College Parkgetoor@cs.umd.edu

Janet MannDepartments of Biology and PsychologyGeorgetown University, Washington DC

mannj@georgetown.edu

ABSTRACT

Most networks contain embedded communities orgroups that impact the overall gathering and dis-semination of ideas and information. These groupsconsist of important or prominent individuals whoactively participate in network activities over time.In this paper, we introduce a new method for iden-tifying actors with prominent group membershipsin time-varying affiliation networks. We define aprominent actor to be one who participates in thesame group regularly (stable participation) and par-ticipates across different groups consistently (diverseparticipation), thereby having a position of struc-tural influence in the network. Our proposed meth-ods for quantifying stable and diverse participationtakes into consideration the underlying semantics forgroup participation as well as the level of impactof an actor’s history on his or her current behavior.We illustrate the semantics of our measures on bothsynthetic and real-world data sets with varying tem-poral connectivity structures. We also illustrate theirutility by demonstrating their complementary naturewhen compared to existing centrality measures.

I MOTIVATION

Much research has focused on identifying influentialindividuals and opinion leaders in a social network;that is, those who have high social capital, or whohelp maximize the spread of information and ideas[1]. While identifying these individuals is useful formacro-level diffusion analysis, it is less useful for un-derstanding the structural influence of individuals inembedded communities or groups in the network. Inthis context, we introduce the concept of prominentindividuals. Based on their position in the network,they can have the greatest influence on their under-lying groups. More specifically, a prominent individ-ual is one who participates regularly within a group(stable participation) and consistently across many

groups (diverse participation) compared to others inthe network. These individuals are important to thenetwork because they have access to more informa-tion than those that participate in only one or twogroups and they have the potential to disseminateinformation since they participate consistently acrossmany groups.

Our new method for identifying actors with promi-nent group membership incorporates measures thatquantify two specific types of actor behavior acrossdifferent groups: stable actors, those who partici-pate in the same groups regularly, and diverse actors,those who participate across different groups consis-tently. Together, these measures are used to identifyand rank actors with prominent group membershipsin any time-varying network, particularly affiliationnetworks. An affiliation network contains two differ-ent types of nodes, one for actors and one for events,and edges between actors and the events in whichactors participate [2]. In time-varying affiliation net-works, an actor’s participation in a particular eventis associated with a specific time, indicating whenthe participation occurred. Many interesting socialnetworks can be captured as affiliation networks, in-cluding organizational data describing peoples’ roleson teams or in companies; and social media data de-scribing users participation in blogs.

As an example, consider an epidemiological networkwhere groups are based on exposure to different dis-ease strains. Stable actors represent vulnerable indi-viduals who are consistently and recently exposed toa certain disease. Diverse actors represent those thathave repeatedly had exposure to a large number ofdisease groups. While each of these measures providemeaningful insight into the implications of differenttypes of disease exposures, studying the behavior ofindividuals who have elongated exposure to differenttypes of diseases is meaningful for understanding thevulnerability of the network as a whole, and the dy-namics of the spread of different disease strains.

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In this paper, we extend the work we proposed in [3]for quantifying user loyalty to a social group into amore general formulation for identifying prominentusers based on both their stability in their corre-sponding groups, as well their diversity across differ-ent groups in networks. The contributions of this pa-per are as follows. First, we propose a new method foridentifying individuals who have both stable and ex-tended reach in a network that is tunable for networkswith varying characteristics, including different se-mantics for network group participation and for pastparticipation. Second, we show how our method canbe used in time-varying affiliation networks; specifi-cally, how we can use the affiliation events to definethe groups that actors belong to. Third, we demon-strate the utility of the measures on both syntheticand real world data sets and compare them to exist-ing centrality measures of influence.

This paper is organized as follows. Section II presentsthe related work. Section III formally defines affili-ation networks and our grouping semantics. SectionIV describes our stability, diversity, and prominencemeasures and ranking. We evaluate our methods onboth real-world and synthetic data sets, and comparethem to existing centrality measures in Section V. Fi-nally, conclusions are presented in Section VI.

II RELATED WORK

Analyzing the dynamics of social group formationand user behavior within these groups is a growingarea of interdisciplinary research. The communitydetection literature has focused on using measures ofcohesion and clustering to identify subsets of users inthe network that are densely connected to each other,but less densely connected to users in other clusters.However, the majority of research conducted on com-munity detection focuses on static networks and con-sider only the case when an actor can be a memberof a single community [4–8].

Recently, researchers have begun to analyze the dy-namics of communities over time [9–16]. Much ofthis research focuses on two questions: what are thecommunities that exist in a particular data set, andhow do they change or evolve over time. The workby Tant et al. [16] is concerned with identifying acore set of communities of actors over time. Asuret al. [9] focused more on developing methods foridentifying significant changes to groups over time.Their proposed approach relies on partitioning tem-poral snapshots into groups and analyzing the cor-

responding group popularity and influence indices.In addition, Sun et al. [15] introduced a parameter-free approach for discovering communities and changepoints in community structure without relying ontemporal snapshots. Berger-Wolf and Saia [12] usedsocial groups to partition the data, and developed al-gorithms for generating meta-group statistics to finda “critical group set”, one whose removal leaves novisible meta-groups. Friedland and Jensen [13] intro-duced a method for detecting small groups of indi-viduals sharing unusual affiliations over time.

The above methods for modeling and understandingthe dynamics of the community formation and groupstructure can be categorized as a macro-level analysison the underlying social network. In contrast, the ap-proach we propose in this paper is a micro-level analy-sis technique that focuses on the dynamics of specificactors or individuals within different groups in thenetwork. Our analysis of actors can be conducted us-ing the groups identified by any of the aforementioneddynamic community detection algorithms on a singlemode social network. Once the social groups or com-munities are established, our goal is to understandthe dynamics of actors and their social relationshipsin the context of these predefined groups.

Backstrom et al. [11] discussed the notion of engagedusers in “thriving” social groups, and discovered thatthe core users of the group tend to receive preferentialtreatment from other members. This work empha-sizes the role and importance of these types of usersin online social groups. Another approach by Habibaet al. [17] proposed a set of methods for identifyingimportant actors in dynamic networks. The authorsidentified nodes in single mode networks that arelikely to be good “spread blockers”. To accomplishthis, they introduced dynamic measures for density,diameter, degree, betweenness, closeness and cluster-ing coefficient. Their measure of dynamic averagedegree is semantically meaningful in the context oftime-varying affiliation networks. We show that it is aspecial case of our diversity measure when there is nodiscount function. While all of these approaches areimportant for micro-level analysis of dynamic socialnetworks, none of them identify individuals that arestructurally well positioned in the network to gatherand disseminate information within different groupsor communities.

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III DEFINITIONS AND BACKGROUND

In this section, we define the notions of dynamicgroups and affiliation networks. We then explain anovel approach for defining groups from affiliationnetworks.

1 TEMPORAL SOCIAL GROUPS

The groups that we focus on are temporal, so an ac-tor’s participation varies with time. In addition, weallow actors to be members of multiple groups at eachtime step since this property is more typical in real-world networks. More formally, given a single modegraph G(A, E) containing a set of actor nodes A ={a1, a2, a3, . . . , an} and edges that connect actors,E = {(ai, aj)|ai and aj ∈ A}, we define a collectionof groups, G = {g1, g2, g3, . . . , gl}, and a boolean rela-tionship that describes temporal group membershipsof an actor, GroupMember(ai, gj , t). We also define atemporal social group as a subset of actors having thesame group value gj at time t: SocialGroup(gj , t) ={ai|ai ∈ A, GroupMember(ai, gj , t)}.

The above construct is general and can be applied ina variety of settings depending on the semantics ofthe underlying analysis task. One could group actorsbased on the output of the dynamic community de-tection algorithm discussed in the previous section.Another method would be to group actors based oncommon attribute values or common event participa-tion. We now describe this method in the context ofaffiliation networks.

2 EVENT-BASED GROUPING FROM AF-FILIATION NETWORKS

An affiliation network is represented as a graphG(A, E ,P) containing a set of actor nodes A ={a1, a2, a3, . . . , an}, a set of event nodes E ={e1, e2, e3, . . . , em}, and a set of participation edgesP that connect actors in A to events in E . Here,P = {(ai, ej)|ai ∈ A, ej ∈ E , and ai participates inej}. In addition, we assume that actors and eventshave attributes or features associated with them. Inorder to emphasize the event relation’s temporal com-ponent, it explicitly contains a time attribute, Etime,that can be used to associate specific time points toabstract groups of actors. An example of an affili-ation network is an author publication network. Insuch a network, the actors are the authors, the eventsare the publications, and the participation relation-ship indicates the authors of a publication. This affil-

iation network is temporal because each publicationevent has a date of publication. We will use this au-thor publication network as our running example.

Figure 1: Grouping abstractions based on publicationevent attributes at time t

The grouping construct we propose for affiliation net-works incorporates the semantics of events into thegrouping definition. Specifically, we propose defin-ing a group based on shared event attribute values.We consider any attribute (or combination of at-tributes) of the event relation to be a grouping ab-straction that can be used to define a set of groups.Assume each attribute Ek of Event has a domainof values, Domain(Ek) = {g1, g2, . . . .gp}. In or-der to construct an event-based grouping abstrac-tion, we define the temporal group membership asfollows: GroupMember(ai, gj , Etime) where gj is inDomain(Ek), and Etime is the time point when thegroup membership is valid. For simplicity, our defi-nition focuses on the case where groups are based ona single attribute Ek. It is straightforward to extendthis definition to consider multiple attributes.

Returning to our publication network with authorsas actors and publications as events, we focus onthe three attributes associated with the publicationevent - publication topic, publication venue, and pub-lication authors. Each attribute is an abstractionthrough which actors can be grouped. Using this for-mulation, we have multiple ways an actor relates toother actors through a particular event. An exam-ple social group belonging to the topic grouping ab-straction is data mining, e.g., actors who have pub-lished on the topic data mining. Figure 1 shows apartial lattice structure for the publication networkat a particular time point t. We see that authors areconnected to publications through similar values ofthree different grouping abstractions: topic, author-ship and venue. This lattice structure emphasizes the

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connectivity between the temporal social groups andthe actors and events in the network.

There are several advantages to our group formula-tion. First, our approach, while very simple, is sur-prisingly flexible. Second, actors can belong to mul-tiple affiliation-based groups at a particular time. Inother words, membership in different groups can beoverlapping. Third, actors are not required to be partof an event (or group) at every time t.

IV QUANTIFYING ACTOR PARTICIPA-TION

Formally, we are interested in the following problem:Given a dynamic affiliation network G, identify thetop-k prominent actors P in the network. Intuitively,an actor that is prominent has two characteristics.First, the actor participates within a dynamic socialgroup consistently over time and hence, is consid-ered a stable actor. Second, the actor participatesacross many different groups consistently over timeand therefore, is considered a diverse actor. We nowdefine actor stability, diversity, and prominence in theremainder of this section.

1 ACTOR STABILITY

Different static and temporal definitions can exist forstability. We consider two in this paper: frequency-based stability and consistency-based stability.

1.1 FREQUENCY-BASED STABILITY

One possible definition for stability considers thenumber of times an actor participates in a group. Letns(ai, gj) be the number of time points that actor aiparticipates in group gj . Let Tmax be the maximumnumber of time points that any of the actors in thenetwork participated in any group in G. Then thestability of actor ai in a group gj is defined as the ra-tio between the number of time points ai participatesin a particular group gj and the maximum number oftime points Tmax:

S(ai, gj) =ns(ai, gj)

Tmax

While this definition takes into consideration the dy-namics between actors and groups, it does not cap-ture the temporal component of the actor participa-tion.

1.2 CONSISTENCY-BASED STABILITY

In some domains, it is important to favor consistentand recent actor participation over irregular and out-dated participations. This is especially relevant indata sets containing a large number of time periods,in which it is valuable to highlight the duration(s)for which a stable member possesses this property.For this reason, we introduce a discount function thatserves as the mechanism for taking the temporal com-ponent of the actor’s participation into account. Asa result, instead of using the total number of a givenactor’s participations in a corresponding group whencalculating stability, we can alter the effect over timedepending on the discount function.

The main inputs to the discount function are the pre-vious value of the actor’s participation in group gjand the difference in time between the current timepoint and the previous time point where the actor’slast activity was monitored, t − tprev. The outputof the function is the discounted value of the actor’sparticipation at the current time step t. Examples ofcommon discount functions are linear (F(x, y) = x

α.y )

and exponential decay (F(x, y) = xeα.y ). We place no

constraint on the type of discount function used inthe model. Any model of decay that suits the do-main being studied is reasonable.

We calculate the discounted sum of the actor partic-ipation at each time point as follows:

Ns(ai, gj , t) = δ(ai, gj , t) t = t0= δ(ai, gj , t)

+(F(Ns(ai, gj , tprev), t− tprev))t > t0

whereNs(ai, gj , t) is the discounted value of actor ai’sparticipation in group gj up to time point t, F is theuser-defined discount function, t0 is the initial timepoint where actor ai participated in group gj , tprev isthe last time point the actor participated in before t,and δ(ai, gj , t) is a participation function that eval-uates to one when actor ai participates in group gjat time t or zero when actor ai does not. We thenevaluate stability at the time point of interest, tf , us-ing the discounted value of the actor participation upuntil that point by augmenting the original stabilitymeasure as follows:

S(ai, gj , tf ) =Ns(ai, gj , tf )

Tmax

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where Tmax is the maximum number of time pointsfrom t0 until time point tf that any actor in A par-ticipated in any group in G. Here, we use Tmax tonormalize the scores. We could simply use the to-tal number of time points; however, if most actorsparticipate in a small fraction of time steps, usingthe maximum participation of any actor results in awider distribution of stability values.

2 ACTOR DIVERSITY

Similar to stability, different static and temporal def-initions can exist for diversity. Two that we con-sider in this paper are frequency-based diversity andconsistency-based diversity.

2.1 FREQUENCY-BASED DIVERSITY

One possible definition for actor diversity considersthe number of groups in which an actor participates.Let nd(ai) represent the number of groups that actorai participates in over all time points and let Gmaxbe the total number of groups with at least one ac-tor participation in the network, where Gmax ≤ |G|.Then the diversity of actor ai is defined as the numberof groups ai actually participates in over the numberof groups ai can participate in:

D(ai) =nd(ai)

Gmax

2.2 CONSISTENCY-BASED DIVERSITY

Similar to stability, we are interested in favoring re-cent and consistent diversity. Therefore, we will alsouse a discount function for the actor’s diversity. Wefirst calculate the discounted sum of actor participa-tions at each time point:

Nd(ai, t) = nd(ai, t) t = t0= nd(ai, t)

+(F(Nd(ai, tprev), t− tprev))t > t0

where Nd(ai, t) is the discounted value of actor ai’snumber of group participations up to time point t,F is the user-defined discount function, and tprev isthe last time point the actor participated in prior tot. Then, the diversity at the time point of interest,tf , is calculated using the discounted value of the ac-tor’s number of group participations up to tf divided

by the number of groups in the network times thenumber of time points the actor participated in.

D(ai, tf ) =Nd(ai, tf )

Gmax × Tmaxwhere Tmax is the maximum number of time pointsthat any of the actors in Actor participated in anygroup in G until time point tf .

3 ACTOR PROMINENCE

Once we have the stability for all actors in their cor-responding groups, as well as their overall diversity,we can use these measures to determine the set ofprominent actors in the affiliation network.

DEFINITION 1. A prominent actor P has both ahigh stability S within groups in G and a high diver-sity D across groups in G over time.

We define SAk(gj , t) to be the top-k stable ac-tors for group gj at time point t and SAk(G, t) =⋃gj∈G SAk(gj , t). Similarly, we define DAk(t) to be

the top-k diverse actors at time point t. Then promi-nence P (t) is calculated as follows:

1. Calculate D(ai, t) and S(ai, gj , t) for all ai andgj .

2. Determine SAk(G, t) and DAk(t).

3. Intersect top-k stability and diversity sets tofindprominent actors Pk(t) = SAk(G, t))∩DAk(t)).

This final set will contain the actors who possess bothhigh stability and high diversity measures. Noticethat it is possible for a particular data set to containno prominent actors. For example, there may be af-filiation networks that contain stable members thatare not diverse. In such cases, the intersection willyield an empty set of prominent actors. To avoid el-evating non-prominent actors in data sets containinglow diversity and stability values for all the actors,we can include a minimum threshold so that only ac-tors above the minimum thresholds for stability anddiversity are candidates for prominence.

4 EXAMPLE CALCULATIONS

In order to further illustrate these measures, considerthe simple example in Figure 2 containing 5 actorsparticipating in 3 groups over 6 time points, with alinear discount function. First, we consider stability.

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Figure 2: Example for calculating actor stability and diversity.

We assign a color to each group in the figure. Groupg1 is (orange), g2 is (yellow), and g3 is (blue). Themost stable actor in each group at every time pointis assigned the same color of the group. For example,at time point 1, actors a1 and a5 are the most stableactors in group g1. Actors a2 and a3 are most stablein group g2. Actor a5 is also most stable in group g3.Notice that actor a5 is stable in two groups at timepoint 1 and therefore, is two colors. The first threerows under each time period show each actor’s dis-counted sum for stability and the stability for groupsg1, g2, and g3, respectively. The last row shows eachactor’s discounted sum for diversity and the diversityfor each actor.

Let’s first consider the stability of the actors. By ex-amining the evolution of actor stability, we see thatuntil the third time point, the stable members ofeach group do not change because of the persistentparticipation of those actors in their correspondinggroups. At the third time point, the stability of actora2 drops. By the fourth time point, the stability scoreof actor a2 in group g2 is lower than that of actor a4since actor a2 has not participated in group g2 fortwo time points and actor a4, a previous participantof group g2, begins participating in group g2 again.

At the fifth time point, the stable members of bothgroups g1 and g2 change. Actor a3, who consistently

participated in group g1 during the last 3 time points,has a higher stability score than both actor a1, whohas stopped participating in the group since the thirdtime point, and actor a5, who did not participate inthe group in the previous time point. Thus, actor a3becomes the most stable actor in group g1. As forgroup g2, since actor a2 returns, a2’s stability scoreincreases over that of actor a4 who missed the partic-ipation at this step. Finally, at the final time point,actor a4 participates in group g2 and becomes morestable than actor a2.

At the final time point, we can see how the temporalaspect of our proposed measures impacts the results.For group g1, the most stable member is actor a3 whoparticipated in the group consistently and recentlyover the last 4 time points. The second most stablemember is actor a5 who also participated in the groupat 4 time points, but in earlier time points than thatof a3. The least stable member of the group is ac-tor a1 who participated in only 3 earlier time points.As for group g2, we find that the scores of both ac-tors a2 and a4 are very close. Actor a4 is consideredmore stable because he or she participates more re-cently and more consistently than actor a2. Finally,for group g3, actor a5 is the most stable actor. Sincehe or she participated in group g3 at every time point,his or her stability score is 1.

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As for diversity, we can see that actor a5 is also themost consistently diverse actor over time, having thehighest diversity score at the final time point. Ac-tor a5 regularly appears in two of the three groups.In contrast, actors a1 and a2 have the lowest diver-sity scores since they appear in only a single groupthroughout the example. In order to realize the im-portance of including the temporal discount factor,we notice that if the discount factor is not included,the diversity of both actors a3 and a5 would be thesame even though actor a5 is consistently diverse for4 time points and actor a3 participated in two groupsonly once.

Suppose we set k = 2 to determine the k prominentactors. The two most stable actors are a5 and a3.These two actors also have the highest diversity scoresand are therefore, both prominent actors.

V EXPERIMENTAL RESULTS

We begin by analyzing our proposed measures, sta-bility, diversity, and prominence, on three affiliationnetworks: a scientific publication network, a senatebill sponsorship network, and a dolphin social net-work. We analyze the distribution of values for eachdata set and illustrate meaningful characteristics ofthe actors in the networks. We then compare stabil-ity and diversity to well known centrality measuresand show that these measures capture a different dy-namic than existing measures. To show how our mea-sures perform on a broader range of actor behaviors,we generate a set of synthetic data sets that containvarying probabilities for generating different types ofactors and use our measures to see if they adequatelyidentify stable and diverse actors. Finally, we showhow the different temporal discount functions can beused, illustrating the value of using dynamic measuresrather than static ones.

1 DATA SETS

Scientific publication network: This network isbased on publications in the ACM Computer-HumanInteraction (ACMCHI) conference from 1982 until2004. Similar to our running example, this dataset describes an author/publication affiliation net-work. It was extracted from the ACM Digital Libraryand contains 4,073 publications and 6,358 authors.There are 12,727 participation relationships (edges)between authors and publications. Since we are inter-ested in the temporal dynamics of the actors, singleactor participations are removed as a preprocessing

step for all the data sets. We grouped publicationsusing the topic attribute. There are 15 values for thisattribute.

Senate bill sponsorship network: This networkis based on data collected about senators and the billsthey sponsor [18]. The data contains each senator’sdemographic information and the bills each senatorsponsored or co-sponsored from 1993 through Febru-ary 2008. Each bill has a date and topics associ-ated with it. We group the bills using their generaltopic. After removing the senators that do not spon-sor a bill, the bills that do not have a topic, and pre-processing the data, our analysis uses 181 senators,28,372 bills, and 188,040 participation relationshipsspanning 100 general topics. While we used all thegroups for our analysis, due to space limitations weillustrate the results using only a subset of the 100topics.

Dolphin behavioral network: This network isbased on a data set accumulated over the last 25years on a population of wild bottle-nose dolphinsin Shark Bay, Australia. The dolphin populationhas been monitored annually since 1984 by membersof the Shark Bay Dolphin Research Project. Theyhave collected 13,400 observation surveys of dolphingroups. Each observation of a group of dolphins rep-resents a ’snapshot’ of associations and behaviors. Inthis affiliation network, dolphins are defined as actorsand surveys as events. Dolphins observed in a sur-vey constitutes the participation relationship. Ouranalysis includes 560 dolphins, 10,731 surveys, and36,404 relationships between dolphins and surveys.We group survey observations together by the loca-tion (latitude-longitude) of the survey. Seven differ-ent predetermined areas of approximately equal size(75 sq km.) were used as groups for this data set.

2 MEASURING STABILITY

The results of measuring the stability of actors todifferent groups in each network are summarized inFigures 3(a) - 3(c). Here the x-axis represents the sta-bility value and the y-axis contains the group names.Except where otherwise noted, we use a linear dis-count function with α = 1 for all the results reported.As can be seen from the figures, because the seman-tics and evolution of each network are different, theoverall actor stability varies across the data sets withthe average stability being lowest for the publicationdata set and highest for the senate data set. The lowaverage stability of the authors to publication topics

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(a) Publication network group stability

(b) Senate network group stability

(c) Dolphin network group stability

Figure 3: Stability of actors across various groups

in the scientific publication data set results becausemost of the authors do not publish in this venue ev-ery year on the same topic. This may lead one tobelieve that these authors are diverse. As we will seelater, while there are some diverse authors, the ma-jority are not. In fact, the majority of authors do notconsistently stay a member of a single group and do

Figure 4: Group stability changes in senate network

not consistently publish across groups. In this dataset, the low stability scores is an indication that veryfew people in the network are well positioned to havea strong, continual impact on authors in these differ-ent groups.

Figure 3(b) shows the results on the senate bill spon-sorship network. Here we notice that the average sta-bility of senators in different groups is much higherthan that of the publication network with some hav-ing a score above 0.8. This means that senators areregularly sponsoring bills of a certain type. This isparticularly true for bills sponsored within certaintopics, e.g., commemorations, foreign policy, taxa-tion. However, other topics, including environmentalpolicy and women, have significantly less stable mem-bership (average stability less than 0.1). A more de-tailed temporal comparison is illustrated in Figure 4.It shows the changing dynamics of the average ac-tor stability in different groups over time. Initially,they are relatively similar and relatively low. How-ever, since the late 1990’s, the average stability val-ues have increased rapidly for six groups includingdefense, commemorations, business, and criminal jus-tice, while the remaining average stability values havebeen relatively consistent, with only a slight increaseor decrease, each year. The increase in stability ofsenators in these areas is consistent with historicalevents, e.g. wars in Afghanistan and Iraq.

Figure 3(c) shows the results on the dolphin network.Here we see that dolphin stability is more variablethan the previous data sets. There are three loca-tions with more stable membership than the others.The average stability is highest for the ’East’ loca-tion, but the most stable dolphins in the data set are

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Figure 5: Actor diversity

in ’Red Cliff Bay’ and ’Whale Bight’. While someof this may be explained by heavier sampling in cer-tain regions, biologists believe this is likely to be aresult of habitat structure in the region [19, 20]. Forexample, ’East’, which has the highest average sta-bility, is mostly deep channels bisected by shallowsea grass banks. Dolphins with high stability in the’East’ have certain foraging specializations (channelforagers or sea grass bed foragers). Since many dol-phins spend a large amount of time foraging, a highstability in regions where specialized foraging is nec-essary is consistent with biologists’ interpretation ofdolphin behavior. Dynamic measures like stabilityprovide observational scientists with a tool for mea-suring and comparing social variability throughoutan animal’s life history.

3 MEASURING DIVERSITY

In Figure 5, we compute the diversity distributionamong the actors of each network. To make the fig-ure easier to read, we sorted the diversity values foreach data set from highest to lowest along the x-axis.The figure shows that the average diversity is high-est for senators, while the range of diversity valuesis widest for the dolphins. The diversity of actors inthe scientific publication network is very low (aver-age < 0.05). This is an indication that authors arenot publishing consistently across topics. The diver-sity values for the senator sponsorship network mayseem low since they are all below 0.5 and intuitively,we expect senators to sponsor bills across a range oftopics. However, this is not surprising because thereare 100 different bill topics, and the number of topicsis part of the denominator of the diversity equation.Finally, the range in dolphin diversity is much higherthan the other two data sets. Again, this is consis-tent with biologists’ interpretation of dolphin behav-ior. While many dolphins settle in some areas (bightsor bays), others spend more time in adjacent bays atspecific stages in their life history (e.g., juvenile pe-

riod or adulthood), thereby increasing their diversityscore with respect to location.

4 PROMINENT ACTORS

Recall that prominent actors are structurally well po-sitioned in the network to both gather new informa-tion and ideas from different groups (diversity), aswell as disseminate them to members of groups theyactively participate in (stability). In order to find theprominent actors, we apply the method discussed inSection IV using (k=10). We highlight some inter-esting findings. First, none of the data sets have 10prominent actors. In other words, few actors in thedata sets are both stable and diverse. The dolphindata set, which has the highest stability and diversityscores, returns the fewest prominent actors (4). Thesenator network has the largest number of prominentactors (8), and the publication data set is in the mid-dle (6).

Focusing on the senator data set, the following ac-tors are considered prominent: Sen. Jeff Bingaman,Sen. Barbara Boxer, Sen. Diane Fienstein, Sen. Ed-ward Kennedy, Sen. John Kerry, Sen. Patrick Leahy,Sen. Joseph Lieberman, and Sen. Patricia Murray -all well-known Democratic senators. To gain furtherinsight we determined whether or not the prominentactors are stable in the same groups. Figure 6 shows

Figure 6: Stable group membership of prominent sen-ators

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(a) Authors

(b) Senators

(c) Dolphins

Figure 7: Stability vs Diversity

the groups in which these senators are considered sta-ble. The size of each pie slice represents the numberof prominent actors in the topic group. While thereis definite overlap (5 out of 8 are stable in defense),there are clear differences as well (Sen. Patty Murrayfrom Washington is stable in Agriculture).

We have similar findings for the publication data set.We also further analyze the prominent actors in thepublication network by checking the DBLP listing ofpublications for each prominent author. We find thatfor all the authors, 31% to 53% of their total publica-tions are in ACM CHI. Thus, this conference repre-sents a very important venue in their research portfo-lio. Finally, prominent actors in the dolphin network

all have high stability and diversity in the same lo-cation groups. Scientists who monitor the dolphinsknow these dolphins to be highly sociable, i.e. sincethe 1980s, their rate of contact with other dolphins ishigh. These dolphins are sighted regularly in manydifferent locations and are rarely sighted alone. Thisis consistent with the definition of prominence.

In order to better understand the relationships be-tween stability and diversity of all the actors in thedata sets, we plotted stability vs. diversity in Fig-ure 7. The variations across data sets is evident.For the publication network, high stability gener-ally correlates with low diversity. The senator net-work is much more varied, while in the dolphin net-work highly stable actors also tend to be more di-verse. These figures reconfirm that prominence is nota common characteristic for individuals in differentnetworks and prominent individuals hold a uniqueposition in the network that provides them an op-portunity to be very influential in groups within thenetwork.

5 COMPARISON WITH CENTRALITYMEASURES

A natural direction is to understand how stabilityand diversity compare to existing centrality measures.Do they capture the same information, or do theyprovide additional insight? We begin by compar-ing stability to the most common centrality mea-sures. In order to do this, we generated the under-lying single-mode, co-membership network for actorsparticipating in a certain affiliation group, and com-puted various centrality measures on the generatednetworks. We show the results for the publicationdata in Table 1 using the ‘Information Visualization’topic as a sample affiliation group. The table showsthe measure values, followed by the ranking of theactor for each measure. For example, Benjamin Bed-erson ranked as the author with highest betweennessand eigenvector centrality. However, by examiningthe publications pattern we note that they are nei-ther consistent across time nor numerous, and thesame is true for Robert Spence who was ranked firstaccording to the closeness centrality. On the otherhand, the time-consistent, recent and numerous pub-lications of the most stable author, namely StuartCard, illustrates exactly what our proposed stabil-ity measure captures that the other centrality mea-sures missed. A similar experiment was performedusing the original affiliation network to compare ourproposed diversity measure with the same centrality

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Actor Stability Closeness Betweenness Eigenvector

Raw Value(Rank)

Participation [Publication Year ]

Stuart K. Card 0.255(1) 0.305(85) 0.445(6) 0.003(9)1987, 1991, 1993, 1994, 1995, 1996,

1998, 2000, 2001, 2003, 2004Robert Spence 0.038(49) 1(1) 0.006(50) 0(88)

1993, 1996, 1999, 2001Benjamin B. Bederson 0.117(18) 0.321(82) 1(1) 0.276(1)

1995, 1999, 2000, 2003

Table 1: Stability vs. Centrality

Actor Diversity Closeness Betweenness Eigenvector

Raw Value(Rank)

Participation [Year(Group Count)]

Allison Druin 0.209(1) 0.395(10) 0.013() 0.002()1994(2), 1995(1), 1996(2), 1997(1), 1998(2), 1999(4),

2000(2), 2001(2), 2002(4), 2003(1), 2004(1)Ben Shneiderman 0.127(6) 0.446(1) 0.3(2) 0.0027(1)

1982(1), 1987(1), 1991(2), 1992(3), 1993(1), 1994(3),1995(2), 1996(1), 1998(5), 1999(2), 2001(1), 2002(4)

Brad A. Myers 0.106(9) 0.428(3) 0.306(1) 0.0025(6)1985(1), 1987(1), 1990(1), 1991(3), 1992(1), 1993(4),1994(3), 1995(3), 1996(3), 1998(1), 2000(2), 2002(3)

Table 2: Diversity vs. Centrality

measures. Although the results reported in Table 2have more similarity, the effects of recency and con-sistency over time capture a dynamic that the othercentrality measures miss.

6 EVALUATION ON SYNTHETIC DATA

To illustrate that our measure effectively captures thesemantics we expect, we developed a synthetic datagenerator that allows us control the behavior of actorswithin and across affiliation groups. Our data gen-erator uses an evolutionary approach to emulate thedynamics of a time-varying affiliation network. Weallow the actors in the network to exhibit 4 differentbehaviors: normal, stable, diverse and persistent.

The generator starts by creating an initial networkcontaining ng affiliation groups and na actors usinga predefined percentage of each type of actor. Whenan actor is created, it samples its lifetime from thedistribution DL, and picks its type according to theinput probabilities. Then, the network is allowed toevolve through the simulation time T , with specific

evolution rules according to each actor’s type. A nor-mal actor samples the number of affiliation links tobe created at this time from the distribution DN ,then picks a random set of groups corresponding tothis number. The other types of actors sample theirnumber of participations from the (higher) distribu-tion DF , where a stable actor creates its participationin the one group it was affiliated with in a prior timepoint, a diverse actor chooses a random set of groupsto participate in, and a prominent actor establishesa set of groups that it continues participating in overtime. Actors are generated throughout the simula-tion according to the same probabilities to keep theaverage number of active nodes and the distributionof the actor types fixed over time.

The data generator is very flexible, allowing users tospecify the following parameters: the total period ofsimulation T , the average number of active actors atany time point na, the number of affiliation groupsng, the probability of generating a stable actor ps, adiverse actor pd, or a prominent actor pp, a distribu-tion of the actors’ lifetime DL, and two distributions

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ps pd pp F1(diversity) F1(stability) F1(prominence)0.2 0.2 0.04 0.992 0.976 0.9310.1 0.1 0.01 0.99 0.974 0.9850.01 0.01 0.0001 0.945 0.989 1

Table 3: F1 measures on synthetic data

of actors’ participations per time step DN and DF ,where the first is used for normal actors (with rela-tively lower participation) and the second is used forthe other types of actors (with higher participation).

To better understand how well our measures capturethe different actor behaviors, we generated a numberof synthetic networks with different distributions ofactor types. The synthetic networks used for theseexperiments were obtained by setting the parametersof the generator as follows: (T = 25 time points,na = 1000 actors, ng = 25 groups, DL N(12, 3),DN N(0, 1), and DF N(3, 1)). It is important tonote that the distributions for the user participationsshould follow the same distribution, but with a lowermean for the ‘normal’ actors. Doing so is necessary togenerate the required behavior for diverse and promi-nent actors. We carried out the experiments by vary-ing the probabilities for generating different types ofactors, and then using our measures on the differ-ent networks to see if we could correctly identify thestable, diverse and prominent actors at the differenttime points.

For calculating stability, diversity and prominence,we used a linear discount function with a value of(α = 1), and a threshold equal to the correspondingpercentage of active actors that should be present atthe network at the end of the simulation. Each exper-iment for a certain set of parameters was carried out100 times and the average F-1 measure for stability(as a group average), diversity, and prominence arereported in Table 3. The table shows that for net-works with different distributions of actor behaviors,our measures accurately identify stable, diverse andprominent actors, with an F-1 measure of over 0.9 inall cases.

7 COMPARISON OF DISCOUNT FUNC-TIONS

In order to demonstrate the flexibility introduced inour model through the discount function, we illus-trate the results of using different temporal discountfunctions to calculate the diversity of authors in thepublication data set. We consider three different

Figure 8: Exponential discount (F(x, y) = xey )

Figure 9: Identity discount (F(x, y) = x)

Figure 10: Linear discount (F(x, y) = xy )

models for decay - an exponential, the identity (i.e.,none) and a linear model. The results are illustratedin Figures 8, 9 and 10 respectively. The left part ofthe figures shows the authors participation over time.The right side shows the average degree of those ac-tors. The actors in both graphs are sorted by increas-ing diversity.

The exponential decay discount function favors re-cency, as illustrated Figure 8. Actors that partici-pate through the entire time period, including mostrecently, have the highest diversity. The diversityof those actors that appear in the same number oftime periods differs, depending upon recency. The

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top diverse actors are determined based on the re-cency of their last group participation since the effectof their older participations is diminished exponen-tially. This model is useful when we need to capturethe recent behavior of actors with only up-to-dateparticipations, with limited influence from their pastbehavior.

At the other extreme, the model could use the iden-tify function and effectively do no temporal discount-ing. In this case, the order of occurrence of actorparticipations in different groups is ignored, and thetop actors are determined only by the number of par-ticipations in their corresponding groups. Using thismodel is equivalent to using the dynamic average de-gree proposed by Habiba et al. [17]. Analyzing Fig-ure 9, we see that actors having the same number ofparticipations at different time periods have the samediversity value. This model is appropriate when thoseanalyzing the network are more concerned with themagnitude of the actors’ group participations overtime.

Finally, the linear model attempts to account for bothrecency and frequency in determining top actors. Aswe can see in Figure 10, the time graph shows thatthe model does still favor recency, but it also capturesthe magnitude of group participations (as shown inthe right-hand graph). Such a model can be usedwhen we need to account for recent actor behavior,older behavior, and consistent behavior.

VI CONCLUSIONS AND FUTUREWORK

In this paper, we introduce the concepts of stable, di-verse and prominent actors in a network and exhibitmethods for identifying them in the case of dynamicaffiliation networks. Because these networks are morenuanced than traditional static social networks, themeasures are more complex, and capture both thetemporal aspects of the networks and the variety ofways of defining groups within an affiliation network.We illustrate the utility of our measures of stabilityand diversity on several real-world networks, comparethem to other well known measures of centrality, andshow how they highlight important subtleties thatare not captured traditionally. We also show how ourproposed measures can be used to accurately captureprominent actors that are persistent within groupsand diverse across groups on different synthetic datasets. Finally, we analyze prominent actors in differentdomains and highlight the importance of capturing

both stability and diversity.

One direction for future work is to adapt the pro-posed method for characterizing the dynamic evolu-tion of different ties between actors. Another inter-esting direction is to investigate the exact role thatthese prominent actors play in the dissemination ofinformation across a network, as well as the generaldynamic underlying network formation in the pres-ence of such actors. It would also be interesting tointegrate these measures into a learning algorithmthat predicts future network dynamic. Last but notleast, analyzing the variations in the proposed mea-sures could lead to valuable insights about the specificnature of different network types.

VII ACKNOWLEDGEMENT

The authors would like to thank the anonymous re-viewers for their valuable feedback and the mem-bers of the Shark Bay Dolphin Research Project (SB-DRP). This work was supported in part by ONRGrant #10230702 and NSF Grants #0941487 and#0746930.

A summary version based on this paper appears inASE Social Informatics 2012. This paper was origi-nally submitted to that conference and accepted bythe ASE Social Informatics 2012 PC as a journal pa-per.

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