Centrality in dynamic competition networks
Anthony Bonato, Nicole Eikmeier, David F. Gleich, Rehan Malik
CCentrality in dynamic competition networks
Anthony Bonato , Nicole Eikmeier , David F. Gleich , and Rehan Malik Ryerson University, Toronto, Ontario, Canada Grinnell College, Grinnell, IA, USA Purdue University, West Lafayette, IN, USA
Abstract.
Competition networks are formed via adversarial interactions between actors. TheDynamic Competition Hypothesis predicts that influential actors in competition networksshould have a large number of common out-neighbors with many other nodes. We empiricallystudy this idea as a centrality score and find the measure predictive of importance in severalreal-world networks including food webs, conflict networks, and voting data from Survivor.
While social networks are often studied from the perspective of positive interactions such as friend-ship or followers, the impact of negative social interaction on their structure and evolution cannotbe ignored. Structural balance theory posits positive and negative ties between actors in socialnetworks, and assumes such signed networks will stabilize so that triples of actors are either allmutually friends or possess common adversaries; see [12], and [9] for a modern treatment. The pre-diction of the signs of edges in a social network was previously studied [15,18,21]. Further, negativeinteractions as a model for edges was studied in the context of negatively correlated stocks in mar-ket graphs [4], and in the spatial location of cities as a model to predict the rise of conflicts andviolence [11]. Even in the highly cited Zachary Karate club network [22], the negative interactionbetween the administrator and instructor was the impetus for the split of the club participants intotwo communities. We propose that competition or negative interactions are critically important tothe study of social networks and more broadly, real-world complex networks, and are often hiddendrivers of link formation.In [6], we investigated properties inherent in social networks of competitors that evolve dynam-ically over time. Such networks are viewed as directed, where a directed edge from nodes u to v corresponds to some kind of negative social interaction. For example, a directed edge may representa vote by one player for another in a social game such as the television program Survivor. Directededges are added over discrete time-steps in what we call dynamic competition networks. Our maincontribution in [6] was the presentation of a hypothesis, referred to as the Dynamic CompetitionHypothesis, or (DCH), that served as a predictive tool to uncover alliances and leaders within dy-namic competition networks. We provided evidence for the hypothesis using U.S. voting record datafrom 35 seasons of Survivor.In the present paper, we focus on a particular implication of the DCH. Namely, the DCH pre-dicts that leaders and central actors in these networks should have large a large number of commonout-neighbors with other nodes in the network. Consequently, this score should constitute a moreaccurate and interesting centrality score in competition networks where edges have a negative con-notation. We study this score in terms of its ranking of leaders in various kinds of networks rangingfrom additional international seasons of Survivor, to conflict networks, and to food webs.We organize the discussion in this paper as follows. In Section 2, we formally define dynamiccompetition networks, and review the DCH as stated in [6], with a focus on the common out-neighborscores, called CON scores. In Section 3, we investigate using CON scores as centrality measures inthree distinct sources: i) voting data from all international (that is, non-U.S.) seasons of Survivor,ii) from conflict networks arising from the countries of Afghanistan, India, and Pakistan, and iii) a r X i v : . [ c s . S I] S e p Anthony Bonato, Nicole Eikmeier, David F. Gleich, and Rehan Malik in 14 food webs. We find that the CON scores predict influential actors in the networks with highprecision. The final section interprets our results for real-world complex networks, and suggestsfurther directions.We consider directed graphs with multiple directed edges throughout the paper. Additionalbackground on graph theory and complex networks may be found in the book [5] or [7].
The Dynamic Competition Hypothesis (DCH) provides a quantitative framework for the structureof dynamic competition networks. We recall the statement of the DCH as first stated in [6]. Beforewe state the DCH, we present some terminology.A competition network G is one where nodes represent actors, and there is directed edge betweennodes u and v in G if actor u is in competition with actor v . A dynamic competition network is acompetition network where directed edges are added over discrete time-steps. For example, nodesmay consist of individuals and edges correspond to conflicts between them; as another example,we may consider species in an ecological community, and directed edges correspond to predation.Observe that dynamic competition networks may have multiple edges if there were multiple conflicts;further, not all edges need be present.The central piece of the DCH we study here are the common out-neighbor scores. Without loss ofgenerality, we assume that the node correspond to integers such that we can use the nodes to addressan adjacency matrix as well. Consequently, let A be the adjacency matrix of given competitionnetwork. Entries in the matrix are 0 or positive integers for the number of competition interactions.For nodes u, v, and w, we say that w is a common out-neighbor of u and v if ( u, w ) and ( v, w ) aredirected edges. Alternatively, A uv A vw ≥
1. For a pair of distinct nodes u, v , we define CON( u, v )to be the number of common out-neighbors of u and v . Note that this common out-neighbor scorecounts multiplicities based on the minimum number of interactions: CON( u, v ) = (cid:80) k min( A uk , A vk ),which corresponds to a multiset intersection. For a fixed node u , defineCON( u ) = (cid:88) v ∈ V ( G ) CON( u, v ) . We call CON( u ) the CON score of u. For a set of nodes S with at least two nodes, we defineCON( S ) = (cid:88) u,v ∈ S CON( u, v ) . Observe that CON( S ) is a non-negative integer.In the DCH, leaders are defined as members of a competition network with high standing inthe network, and edges emanating from leaders may influence edge creation in other actors. In thecontext of conflict networks within a country, leaders may be actors who exert the strongest politicalinfluence within the country; note that these may not be the largest or most powerful actors. Asanother example, leaders in a food web would naturally have higher trophic levels (that is, higherposition in a food chain). The DCH characterizes leaders as those nodes with high CON scores, lowin-degree, high out-degree and high closeness. Recall that for a strongly connected digraph G and anode v , we define the closeness of u by C ( u ) = (cid:88) v ∈ V ( G ) \{ u } d ( u, v ) − where d ( u, v ) corresponds to the distance measured by one-way, directed paths from u to v . entrality in dynamic competition networks 3 In this paper, we focus on the implication that leaders in competition networks should have highCON scores, which suggests this is a natural centrality measure for these networks. The DCH alsoinvolves the notion of alliances, that does not factor into our present study.
Alliances are defined asgroups of agents who pool capital towards mutual goals. In the context of social game shows such asSurvivor, alliances are groups of players who work together to vote off players outside the alliance.Members of an alliance are typically less likely to vote for each other, and this is the case in strongalliances. This is characterized in terms of near independent sets ; see [6] for the formalism.In summary, the
Dynamic Competition Hypothesis (or
DCH ) asserts that dynamic competitionnetworks satisfy the following four properties.1. Alliances are near independent sets.2. Strong alliances have low edge density.3. Members of an alliance with high CON scores are more likely leaders.4. Leaders exhibit high closeness, high CON scores, low in-degree, and high out-degree.Our focus in this work will be on the validation of the DCH with regards to detecting leaders;in particular, we will focus on items (3) and (4) of the DCH. Note that while we expect leaders tobe in alliances (that is, have prominent local influence), leaders are determined via global metricsof the network.
In [6], we studied the voting history of U.S. seasons of Survivor, which is a social game show whereplayers compete by voting each other out. In Survivor, strangers called survivors are placed in alocation and forced to provide shelter and food for themselves, with limited support from the outsideworld. Survivors are split into two or more tribes which cohabit and work together. Tribes competefor immunity and the losing tribe goes to tribal council where one of their members is voted off.At some point during the season, tribes merge and the remaining survivors compete for individualimmunity. Survivors voted off may be part of the jury. When there are a small number of remainingsurvivors who are finalists (typically two or three), the jury votes in favor of one of them to becomethe Sole Survivor who receives a cash prize of one million dollars. Figure 1 represents a graphicaldepiction of the voting history of a season of U.S. Survivor.We extend the analysis of the 35 U.S. seasons in [6] to 82 international seasons of Survivor.Data used in our analysis was obtained from the Survivor wiki pages https://survivor.fandom.com/wiki/Main Page. Several seasons (beyond the 82) were excluded for varying reasons. In some cases,a wiki page exists, but there was no voting data. In other cases, much of the voting informationwas missing, or the rules are significantly different than the traditional version of the game shows.Nevertheless, the number of seasons collected exceeds the number in [6].In Table 1 we display some of the CON scores for a few example seasons. We distinguish whichplayers are finalists, since the rules change in determining who is the last player eliminated. Forexample, instead of eliminating the last player via votes against players, in survivor many playersmay return for a final vote for who they would like to win.In Table 2, we detail relevant statistics on these networks. For each network, we consider whetherthe winner of the season had one of the top three or top five CON scores and list the percentageof such networks. For example from Table 2 we see that 81.7 percent of Survivor winners had oneof the largest 5 CON scores. For comparison, we also compute PageRank (on the reversed-edge network, where we change the orientation of directed edge) and Jaccard similarity scores, whichare both standard ranking scores. Jaccard similarity is a type of normalized CON score; see [10].We find that the CON scores are a more accurate predictor for determining finalists of Survivorthan both PageRank and Jaccard similarity. We observe that these results are consistent with
Anthony Bonato, Nicole Eikmeier, David F. Gleich, and Rehan Malik
Fig. 1.
The Survivor Heroes vs. Healers vs. Hustlers co-voting network. Nodes are scaled by closeness, andcolor-coded according to their original tribe. Thicker edges represent multiple votes.Australian Survivor (2002)Name ID OD C CONRobert 5 10 0.714 44Sciona 1 9 0.652 37Joel 7 8 0.625 35Katie 3 9 0.652 38Sophie 3 8 0.652 38Jane 9 6 0.625 36Lance 8 5 0.577 27Craig 8 8 0.577 18Naomi 8 7 0.5 25Caren 10 6 0.5 25Sylvan 3 5 0.417 30Deborah 4 4 0.395 23Jeff 5 1 0.395 4David 6 3 0.441 23Tim 4 2 0.294 10Lucinda 8 1 0.0 7 Robinson 2009Name ID OD C CONEllenor 0 6 0.563 36Jarmo 7 8 0.557 34Anna 2 10 0.645 54Nina 4 7 0.557 38Erik Bl. 7 9 0.612 47Lukas 4 7 0.557 31Angela 6 8 0.612 46Ranjit 5 5 0.51 30Christian 3 4 0.51 24Rafael 5 4 0.49 28Erik Bi. 9 5 0.438 26Erik R. 5 4 0.422 18Mika 5 3 0.306 13Josefine 0 2 0.265 17Erika 7 2 0.306 15Beatrice 6 2 0.35 12Micha 12 1 0.299 7 Survivor South Africa MalaysiaName ID OD C CONLorette 4 9 0.653 35Grant 5 8 0.653 33Amanda 4 8 0.622 26Mandla 0 8 0.652 32Angie 6 9 0.688 31Angela 4 6 0.594 28Dyke 4 6 0.568 17Hein 5 4 0.484 22Irshaad 3 5 0.544 25Lisa 11 4 0.484 16Rijesh 4 3 0.408 13Nichal 6 2 0.363 12Elsie 8 2 0.436 9Viwe 5 2 0.344 11Nicola 5 1 0.304 6Nomfundo 4 1 0.335 8
Table 1.
Three international Survivor seasons. Players are listed by first name, in order from top to bottomwith the winner at the top, and the first eliminated player at the bottom. For each player we list the in-degree, out-degree, closeness, and CON score. The horizontal line separates finalists from the rest of thegroup. the analysis in [6] for the U.S. seasons. As an added comparison, we list the probability of thewinner appearing in a random set of three or five players; note that there is a range of percentagesdepending on how many players are in a given season. In the interest of space, we refer the reader tohttps://eikmeier.sites.grinnell.edu/uncategorized/competition-show-data/ where we house all dataon these seasons. entrality in dynamic competition networks 5
CON PageRank JaccardSimilarity random setSurvivor Top 3 57.3
Top 5 81.7
Table 2.
Statistics on international Survivor seasons.
For our second competition network, we extracted data from
The Armed Conflict Location and EventData Project (or
ACLED conflict networks ,nodes correspond to actors in a given region, and edges correspond to conflicts between the actors.Many types of metadata are recorded corresponding to each event. Our particular interest is in theactors involved in each event, and where the event took place. More information about this projectcan be found on the project website.An important note about the ACLED data is that we do not know which actor initiated a givenevent. Therefore, we do not consider the majority of edges (events) to be directed. The only eventswhich we assume knowledge about directed-ness is when civilians are involved. We restricted ourstudy to a set of events to a particular country; keeping the scale at the country level allows usto keep a larger set of actors. We selected three countries that have a large number of actors andevents to analyze: Afghanistan, India, and Pakistan.We first consider the rankings for Pakistan with commentary.
Unidentified Armed Group (Pakistan)TTP: Tehreek-i-Taliban PakistanRioters (Pakistan)Police Forces of Pakistan (2013-2018)Military Forces of Pakistan (2013-2018)Police Forces of Pakistan (2008-2013)LeI: Lashkar-e-IslamMilitary Forces of Pakistan (2008-2013)Aman LashkarUnidentified Armed Group (Afghanistan)Pakistan RangersFrontier CorpsPolice Forces of Pakistan (2018-)Islamic State (Pakistan)BLF: Baloch Liberation FrontCON Unidentified Armed Group (Pakistan)TTP: Tehreek-i-Taliban PakistanPolice Forces of Pakistan (2013-2018)Military Forces of Pakistan (2013-2018)Police Forces of Pakistan (2008-2013)Military Forces of Pakistan (2008-2013)Rioters (Pakistan)Police Forces of Pakistan (2018-)ProtestersBLF: Baloch Liberation FrontJatoi Communal Militia (Pakistan)Chandio Communal Militia (Pakistan)Private Security Forces (Pakistan)Pakistan RangersMilitary Forces of United StatesPageRank
Fig. 2.
A Slope Graph to compare the rankings via CON and PageRank. On the left, the top actors in theconflict network Pakistan via CON metrics, while on the right, the top actors in Pakistan via PageRank onthe reverse network. Actors are labeled in black if the difference in rankings is less than or equal to three.Actors are labeled in red if the CON ranking is at least four places higher than the PageRank, and in greenif the PageRank is at least 4 places higher than the CON ranking. Note that no line appears to connect theleft and right side if the actor does not show up in the top 15 of the other ranking.
Pakistan has faced terrorism activities since 2000, with many militant groups attacking civiliansand Pakistan armed forces. TTP (Pakistan) is one of the largest radical extremist groups, which is
Anthony Bonato, Nicole Eikmeier, David F. Gleich, and Rehan Malik an umbrella organization of many militant groups such as Lashkar-e-Islam, Islamic State (Pakistan),and Jamaat-ul-Ahrar. In Figure 2, we find that TTP has one of the highest CON scores. TTP hasalliances with another terrorist organizations in Pakistan and neighboring countries, which lendsto its prominence. In addition, due to the Afghan war, TTP has a strong influence and hold overmany Islamic institutions in Pakistan. The Police Forces of Pakistan and Military Forces of Pakistanensure national security, and they share information for achieving their goals. The Police Forces ofPakistan are an influential actor in the conflict network with another one of the highest CON scores.They perform their duties in all provinces of Pakistan with the help of their paramilitary forces suchas Pakistan Rangers and Frontier Corps, and they maintain law and order, as well as border control.Military Forces of Pakistan (2013-2018) has one of the largest CON scores owing to their increasedactivities against terrorist groups in recent years.We also offer commentary on some of the lower ranked actors. The Baloch Liberation Front(or BLF) is an ethnic-separatist political front and militant organization that is currently fightingagainst the Pakistani government for an independent Balochi state. The BLF is the strongest andmost influential militant group of Baluchistan, but there has been no confirmed coordination betweenthe BLF and other Balochi and non-Balochi groups, and they operate independently of one another.This is a large reason that BLF have low CON and closeness scores. The Islamic State is a partof the militant Islamist group: Islamic State of Iraq and Levant (ISIS). The Islamic State wasformed by some of the TTP leaders and is more successful in Afghanistan. This organization hashad less success in Pakistan largely carrying out isolated, small scale attacks. The Police Forces ofPakistan actively participated with the support of paramilitary forces of Pakistan in 2008-2018 forwar against terror. The Police Forces of Pakistan mostly work to maintain the daily law and orderin their respective provinces. Likely for these reasons, they have lower CON scores than the yearsbetween 2008-2018.
TalibanPolice Forces of Afghanistan (2004-)Unidentified Armed Group (Afghanistan)Military Forces of Afghanistan (2004-)Islamic State (Afghanistan)NATOMilitary Forces of Afghanistan (2004-) Special ForcesHQN: Haqqani NetworkAfghan Local PolicePrivate Security Forces (Afghanistan)Militia (Pro-Government)Military Forces of Pakistan (2008-)National Directorate of SecurityTaliban and/or Islamic State AfghanistanHezbi IslamiCON TalibanMilitary Forces of Afghanistan (2004-)Islamic State (Afghanistan)Police Forces of Afghanistan (2004-)NATOUnidentified Armed Group (Afghanistan)Military Forces of Afghanistan (2004-) Special ForcesNational Directorate of SecurityHQN: Haqqani NetworkTaliban and/or Islamic State AfghanistanAfghan and/or NATO ForcesAfghan Local PolicePrivate Security Forces (Afghanistan)Militia (Pro-Government)TTP: Tehreek-i-Taliban PakistanPageRank
Fig. 3.
Top actors in Afghanistan via CON score and PageRank.
We note that the ranking of the top actors using the CON score (on the left in Figure 2) is notdissimilar to the one using PageRank on the reversed-edge network (on the right in Figure 2). Toquantify the difference in the rankings we used Spearman’s rank correlation coefficient. Note thatwe cannot use Pearson correlation because our data is not at all Gaussian. Recall that Spearman’scorrelation coefficient is defined as 1 − (cid:80) Ni =1 d i N ( N − , entrality in dynamic competition networks 7 Rioters (India)Police Forces of India (2014-)Unidentified Armed Group (India)Military Forces of India (2014-)Protestors (India)CPI (Maoist): Communist Party of India (Maoist)Police Forces (2014-) Rashtriya RiflesBJP: Bharatiya Janata PartyPolices Forces (2014-) Assam RiflesTMC: Trinamool Congress PartyINC: Indian National CongressJJMP: Jharkhand Jan Mukti ParishadNSCN-K: National Socialist Council of Nagaland-KhaplangPolice Forces (2014-) Central Reserve ForceUnidentified Armed Group (Pakistan)CON Police Forces of India (2014-)Rioters (India)Military Forces of India (2014-)Unidentified Armed Group (India)CPI (Maoist): Communist Party of India (Maoist)Police Forces (2014-) Border Security ForceProtestors (India)Police Forces (2014-) Central Reserve ForcePolices Forces (2014-) Assam RiflesPolice Forces (2014-) Rashtriya RiflesNSCN-IM: National Socialist Council of Nagaland-Isak MuivahINC: Indian National CongressBJP: Bharatiya Janata PartyPolice Forces (2014-) Forest Range OfficerSAD: Shiromani Akali DalPageRank
Fig. 4.
Top actors in India via CON score and PageRank. where N is the total number of actors, and d i is the difference in rankings between actor i . A valueclose to 1 means that the two rankings are very well positively correlated. The Spearman correlationfor Pakistan is -0.341, which suggests that the rankings are not that similar. In fact, the negativevalue implies that as the CON ranking decreases, the PageRank score increases. There are 741 totalactors we consider in the Pakistan data set, and the later rankings clearly vary greatly.We finish this section with the rankings for Afghanistan and India in Figures 3 and 4. TheSpearman coefficients are 0.604 and -.267 respectively, indicating that the rankings provided byCON matches more similarly to PageRank in the Afghanistan dataset. While we do not providein-depth commentary on these rankings, we find influential actors in both countries with the largestrankings against DCH metrics. As a third and final type of data that we analyzed against the DCH, we studied food web datasetsfrom the Pajek website: http://vlado.fmf.uni-lj.si/pub/networks/data/bio/foodweb/foodweb.htm [3].These are 14 food webs in total. In food webs, the nodes are species, and a weighted edge ( u, v )exists with weight w if u inherits carbon from v (that is, u preys on v ) [2]. We interpret this as adirected negative interaction from node u to node v . A noteworthy difference in these networks (vs.Survivor, say) is that the movement of energy is balanced , meaning the in-degree and out-degree foreach species is equal.Rankings of selected food web datasets are in Figures 5 and 6; the rankings for all the datasetsmay be found at https://eikmeier.sites.grinnell.edu/uncategorized/competition-show-data/ alongwith the computed CON scores, closeness, and PageRank on the reversed-edge network.In studying the rankings of these 14 food webs, we see a difference between the CON rank-ings and PageRank. PageRank has been used to study the importance of species in regards toco-extinction [1,16,14], which we expect is likely reflected in the rankings we see here using vanillaPageRank. However, we find a substantially different ranking when using the CON scores; for exam-ple, see the placement of Heteroflagellates in Figure 5. The average Spearman correlation coefficientacross these 14 datasets is 0.271, and the range is between 0.004 and 0.554. (Recall that a value closeto 1 means very well correlated.) Therefore, we suggest that the CON scores are giving a different ranking, which is much closer to trophic levels of species. In particular, the CON scores reflect anatural hierarchical structure in ecosystems, and this is consistent with the DCH. Anthony Bonato, Nicole Eikmeier, David F. Gleich, and Rehan Malik
Bay AnchovyWeakfishHogchokerChrysaoraCtenophoresHerrings and ShadsBlue CrabWhite PerchAmerican EelCatfishMeroplanktonOystersSuspension Feeding BenthosBluefishCroakerStriped BassMeiofaunaMenhadenSpotMicrophytobenthosCiliatesMesozooplanktonDeposit Feeding BenthosRotifersSAVHeteroflagellatesCON HeteroflagellatesDeposit Feeding BenthosRotifersOystersMeroplanktonChrysaoraBlue CrabSuspension Feeding BenthosCiliatesWhite PerchCatfishHerrings and ShadsHogchokerCtenophoresMicrophytobenthosAmerican EelBay AnchovyWeakfishBluefishCroakerMeiofaunaMenhadenMesozooplanktonSAVSpotStriped BassPageRank
Fig. 5.
The Chesapeake Bay Lower food web dataset. On the left, organisms are listed in decreasing order,with the largest CON score at the top. On the right, organisms are listed by decreasing PageRank score.
We studied an implication of the Dynamic Competition Hypothesis (DCH) for competition networksacross several different types of real-world networks. We found that the DCH prediction that highCON scores should correspond to leaders is supported in predicting winners in international seasonsof Survivor, in predicting species with high trophic level species in food web, and for determininginfluential actors in conflict networks in Afghanistan, India, and Pakistan. Metrics such as CONscores outperformed PageRank as an indicator of influential actors in the competition networks westudied.While our results provide support for the DCH, more work needs to be done. We did not addressitems (1) and (2) of the DCH regarding alliances in our data sets, and that would be an importantnext step. Another direction is to consider an aggregate score, based on the CON score, closeness,and in- and out-degree, as a measure of detecting leaders in competition networks. An interestingdirection would be to study more closely the dynamic aspects of competition networks, analyzingthem over time to predict leaders. For example, we could analyze the co-voting network of Survivorof each episode of a season, and determine if temporal trends in network statistics predict finalists.An open question is whether CON score centrality is applicable to large-scale networks exhibitingadversarial interactions, such as in Epinions and Slashdot (which give rise to signed data sets with entrality in dynamic competition networks 9
SilversideGoldspotted KillifishGulf FlounderLongnosed KillifishSheepshead KillifishNeedlefishPinfishSheepsheadSilver JennyGulf KillifishBay AnchovyMulletMoharraBlacktip SharkStingrayStriped AnchovyBenthic InvertebratesZooplanktonCON StingrayBlacktip SharkSilversideBenthic InvertebratesSilver JennyGoldspotted KillifishGulf FlounderBay AnchovyMoharraSheepsheadStriped AnchovyLongnosed KillifishZooplanktonMulletSheepshead KillifishNeedlefishGulf KillifishPinfishPageRank
Fig. 6.
The CrystalC food web dataset. On the left, organisms are listed in decreasing order, with the largestCON score at the top. On the right, organisms are listed by decreasing PageRank score. tens of thousands of nodes, and available from [13]). Epinions was an on-line consumer review site,where users could trust or distrust each other. Slashdot is a social network that contains friend andfoe links. A challenge with these data sets from the view of validating the DCH is that there is noinherently defined ranking, as there is in Survivor (via the order contestants were voted off), foodwebs (trophic level), and in conflict graphs (via political and strategic influence).
The research for this paper was supported by grants from NSERC and Ryerson University. Gleichand Eikmeier acknowledge the support of NSF Awards IIS-1546488, CCF-1909528, the NSF Centerfor Science of Information STC, CCF-0939370, and the Sloan Foundation.
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