Algorithms and Dynamical Models for Communities and Reputation in Social Networks

237 Pages·2014·11.57 MB·English

by  Vincent Traag (auth.)

Springer Theses
Recognizing Outstanding Ph.D. Research
Vincent Traag
Algorithms and
Dynamical Models
for Communities
and Reputation in
Social Networks
Springer Theses
Recognizing Outstanding Ph.D. Research
For furthervolumes:
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includeanextendedintroduction,aswellasaforewordbythestudent’ssupervisor
explaining the special relevance of the work for the field. As a whole, the series
will provide a valuable resource both for newcomers to the research fields
described, and for other scientists seeking detailed background information on
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contributions made by today’s younger generation of scientists.
Theses are accepted into the series by invited nomination only
and must fulfill all of the following criteria
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accessible to scientists not expert in that particular field.
Vincent Traag
Algorithms and Dynamical
Models for Communities and
Reputation in Social
Networks
Doctoral Thesis accepted by
the Catholic University of Louvain, Belgium
123
Author Supervisors
Dr. Vincent Traag Prof.PaulVan Dooren
KITLV DepartmentofMathematicalEngineering—
Leiden ICTEAM
The Netherlands Université catholique de Louvain
Louvain-la-Neuve
Belgium
Prof.YuriiNesterov
Center forOperations Research and
Econometrics(CORE)
Université catholique de Louvain
Louvain-la-Neuve
Belgium
ISSN 2190-5053 ISSN 2190-5061 (electronic)
ISBN 978-3-319-06390-4 ISBN 978-3-319-06391-1 (eBook)
DOI 10.1007/978-3-319-06391-1
Springer ChamHeidelberg New YorkDordrecht London
LibraryofCongressControlNumber:2014939940
(cid:2)SpringerInternationalPublishingSwitzerland2014
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Supervisors’ Foreword
We are living in a world where the amount of data that is collected and stored is
just staggering. Moreover, the information and communication technology
required to have access to these data has become quite affordable so that every-
bodywhowishescanhaveaccesstoit,asfarasitisinthepublicdomain.Thishas
hadatremendousimpactnotonlyinscienceandtechnologybutalsoincommerce
and recreation, where having access to the right bit of information is crucial. An
obviousexampleofsuchasourceofinformationisthe‘‘internet,’’withwhichwe
mean the World Wide Web and search engines such as Google. But social net-
works have started to play a big role as well in getting access to data. Networks
suchasFacebook,LinkedIn,andTwitterhave attracted billions ofusersinavery
short time. These networks allow friends or colleagues to connect to each other
andretrieveordistributeinformationthatwouldbehardtofindotherwise.Butthe
networks themselves can also be viewed as data that can be analyzed to extract
valuableinformation about the ‘‘nodes’’ ofthe network, which can bepeople,but
also objects, pictures, texts, and so on.
The structure of such networks plays an important role in the type of infor-
mationonecanextractfromthem.Oneprominentfeatureofmanysocialnetworks
is the clustering of nodes (people in this case). Friends tend to have many friends
incommon,therebycreatingsocialgroupsinwhichmanypeopleknoweachother
(and often have the same taste, behavior or habits). Knowing these social groups
yields additional insight into the structure of these networks and can be used for
commercial purposes by companies or by providers of certain services. To find
these groups, the idea is to look for densely connected subgraphs in the network,
which are only loosely connectedamongeach other.These are commonlyknown
as‘‘communities’’andthefieldthatdealswithfindingsuchcommunitiesisknown
as ‘‘community detection.’’ Several more mathematical criteria have been pro-
posed to characterize these groups more precisely, such as the popular method
called ‘‘modularity,’’ introduced by Newman and Girvan. In this book, the author
analyzesindepththeproblemofcommunitydetectionandproposesanalternative
method, calledthe Constant PottsModel,andexplains thatitsmajor advantage is
that it has no resolution limit and hence can also detect relatively small commu-
nities in large networks. Although the proposed solution does not suffer from the
resolution limit, there are still some questions related to scale. The author then
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vi Supervisors’Foreword
introducestheconceptof‘‘significance’’whichhelpstodecidewhetherapartition
should be rather coarse of rather fine. Both these developments are important
contributions of his work.
Although most methods for community detection focus on networks that have
positivelinks,negativelinksalsoappearnaturallyandmayrepresentanimosityor
distrust. Incorporating these negative links can be done in a relatively natural
mannerbyinsistingonaslittlenegativelinksaspossiblewithinacommunity.This
isillustratedhereusinganetworkofinternationalrelationsandacitationnetwork.
Thestructureofnegativelinkshasbeenstudiedbythesocialsciencesbeforeinthe
contextof‘‘socialbalance’’andisbasedontheadagethat‘‘theenemyofanenemy
is a friend.’’ The main observation in that literature was that socially balanced
networks can be split into at most two factions where each faction has only
positive links within and negative links between the factions. Besides the impor-
tantquestionofdetectingsuchfactions innetworks, theauthoralsoanalyzes how
social balance may emerge and why it is observed so often. This is done using a
new dynamical model that explains the emergence of social balance. In addition,
there is a natural connection between negative links and the problem of the evo-
lution of cooperation that one finds in the area of dynamical games. The author
uses ideas borrowed from this literature to explain that social balance can lead to
cooperation.Finally,theauthoralsolooksathowtodeterminewhowillcooperate
with whom. This is especially pertinent in online markets such as eBay or Ama-
zon,whereonewantstomakesureonecantrustones‘‘friends.’’Theauthorshows
howtousethenetworkconsistingoflocallinks(whicharepositivefor‘‘trust’’and
negativefor‘‘distrust’’)tocalculateaglobaltrustvalue,whichisthe‘‘reputation’’
of the corresponding node.
This book makes the bridge between two distinct areas: (i) community detec-
tion in large sparse graphs and (ii) social balance and evolution of cooperation.
The author covers quite a wide range of topics in it since the two distinct areas
requiredifferentbackgrounds.Thesynthesisofthestateoftheartintheseareasis
well equilibrated and all the important concepts are well described. The book
makes important novel contributions in a very competitive area of research.
Louvain-la-Neuve, April 2014 Prof. Paul Van Dooren
Prof. Yurii Nesterov
Preface
The first presentation ever of my research was on February 2009, Friday the
13th—how scary is that—and was in front of mathematicians in Louvain-la-
Neuve—how scary is that. Having only a Master’s in Sociology in my pocket I
arrived there to apply for a position as a Ph.D. candidate (although, if memory
serves me well, that was not entirely clear for everyone). Of course, I was no
complete stranger to mathematics, yet not having studied it and still wanting to
pursue a Ph.D. in that direction did not quite seem to add up. Fortunately, my
advisors Paul Van Dooren and Yurii Nesterov were happy to take me on board. I
amgratefultothisdatethattheydidso.Theleewaytheyallowedmetopursuemy
own interest is much appreciated. I have learned a lot from them, and both are
impressively (if not intimidatingly) fast when doing mathematics. I was fortunate
enough to be funded by the Actions de recherche concertées, Large Graphs and
Networks of the Communauté Française de Belgique and the Belgian Network
Dynamical Systems, Control, and Optimization (DYSCO), funded by the Inter-
university Attraction Poles Programme, initiated by the Belgian State, Science
Policy Office.
My fellow Ph.D. students have also taught me a lot. Not having had the exact
same training as most other Ph.D. candidates, I could borrow their expertise in
trying to understand something. For some courses I was the designated teaching
assistant, without actually ever having taken the course myself, making it some-
what of a challenge. For example, I had to learn integer programming. Before
beingabletolearnintegerprogramming,Ihadtolearnlinearprogramming,which
also involved doing the simplex algorithm. If I say I will never forget that, it is
probably true, but I would like to never make another simplex tableau again.
Around the time I started, there were a few other students coming in from the
privatesector:Pierre,François-Xavier,andArnaud,whichreassuredmethatIwas
not the only one that had tried the private sector and returned to academia.
Throughout the years, Arnaud and I collaborated on various projects, I have
enjoyed our cooperation very much. Similarly for Pierre Deville and Adeline
Decuyper,itwasapleasureworkingwithyou,andgoodluckorganizingNetMob
next time around, for which Vincent Blondel was kind enough to invite us last
year.Finally,IwouldliketothankeverybodyelseintheEulerbuilding(toomany
peopletolist)forthegreatatmosphereduringcoffeebreaksandlunchtime.Ihave
vii
viii Preface
enjoyed theconversationsinthecafeteriaverymuch,although forthemostpartI
have only listened instead of actually engaging in the discussions.
Iwouldliketothanktheothermembersofthejury,FrançoisGlineur,Vincent
Blondel, Marco Saerens, and Patrick De Leenheer. Their comments and remarks
have greatly improved this thesis. I have had the pleasure to collaborate with
PatrickwhilehewasBelgiumin2012.Hishelpwasquintessentialtotheprogress
on the social balance project, for which I am much obliged.
Many friends and family have come to visit in Brussels, and it was always a
pleasure having you. Bas, Hans-Hein, and Mathijs, you have always had that
fingerspitzengefühl for coming to Brussels. Merijn, despite your busy job, two
kids, moving two times, and an entire renovation, you still managed to come to
Brussels: so good you could make it. Roel, our discussions on the balcony of the
Rue Lebeau were marvellous—as always—I hope to continue many of them in
Amsterdam.ManyaSundaymorningwasspentattheVossenplein/PlaceduJeude
Balle when my family-in-law came over. Fortunately, due to long breakfasts we
never arrived that early, you’re always welcome for such long breakfasts. From
Brussels,IhaveverymuchenjoyedclimbingwithyouTom,Ihopetoseeyoustill
after moving. Frank, our lunches were a pleasant distraction from the daily Ph.D.
grind. Many friends go unnamed, but not forgotten: I hope to see you all more
often when I am back in Amsterdam. Likewise for my parents, my brother and
sister, Ernst and Susan, I hope to see you more often, Marco, Carlijn and Niels
included of course. I hold you all very dear. Mom and dad, you have always
supported me—both before and during my Ph.D.—I will always be grateful for
your care and love.
Finally,somebodythatmeritsaparagraphinitsown.Thefirsttwoyearsofmy
Ph.D. our time together largely loomed in the shadow of the loss of your mother.
Although such a loss will always leave a void, together I believe we have over-
come.Afterhavingbeenpartedbyover200kmofrailforover3years,wefinally
spentthelastyeartogetherinBrussels.Itwasablisstofinallylivetogether,andI
hopetocontinuetoenjoyyourcompanyformanyyearstocome!Lio,youaremy
true love.
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Part I Communities in Networks
2 Community Detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Modularity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Canonical Community Detection . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Reichardt and Bornholdt . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Arenas, Fernández and Gómez. . . . . . . . . . . . . . . . . 18
2.2.3 Ronhovde and Nussinov . . . . . . . . . . . . . . . . . . . . . 19
2.2.4 Constant Potts Model . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.5 Label Propagation. . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.6 Random Walker. . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.7 Infomap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.8 Alternative Clustering Methods . . . . . . . . . . . . . . . . 27
2.3 Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.1 Simulated Annealing. . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.2 Greedy Improvement . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.3 Louvain Method. . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.4 Eigenvector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 Benchmarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.1 Test Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Comparing Partitions . . . . . . . . . . . . . . . . . . . . . . . 39
2.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Scale Invariant Community Detection. . . . . . . . . . . . . . . . . . . . . 49
3.1 Issues with Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.1 Resolution Limit. . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.2 Non-locality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.3 Spuriously High Modularity. . . . . . . . . . . . . . . . . . . 55
ix