Link communities reveal multiscale complexity in networks

Link Communities Reveal Multiscale Complexity in Networks Yong-Yeol Ahn Center for Complex Network Research, Northeastern University

Sune Lehmann James P. Bagrow

Communities

“a group of densely interconnected nodes”

in understanding and visualizing the structure of networks. In this paper we show how this can be achieved. pr arXiv:cond-mat/0308 “a group of densely The study of community structure in networks has a long history. It is closely related to the ideas of graph nic era partitioning in graph theory and computer science, and th interconnected nodes” ing a op th rit ev sta if nit mu wh th mi be

arXiv:cond-m a op th rit ev sta if nit mu wh th mi be Hundreds of community FIG. 1: A small network with community structure of the type considered in this paper. In this case there are three us communities, denoted by the dashed circles, which have dense wi detection methods internal links but between which there are only a lower density of external links. ing div

Why bother?

Hierarchical organization Community overlap

Hierarchical organization

partitioning in graph theory and computer science, and FIG. 1: A small network with community structure of the

Hierarchical community structure Hierarchy Communities

Hierarchical Random Graph model Clauset et al., Nature (2008)

neously explain and to observed network data using the tools of statistical inf only observed topo- ence, combining a maximum likelihood approach [15] w as right-skewed de- a Monte Carlo sampling algorithm [16] on the space of fﬁcients, and short knowledge of hier- ict missing connec- high accuracy, and han competing tech- suggest that hierar- complex networks, network phenom- devoted to the study n networks [5, 6, 9, nd simple clustering, G. 1: A hierarchical network with structure on many scales and ation at hierarchical random graph. Each internal node r corresponding all scales in he dendrogram is associated with a probability p that a pair of r tices hierarchical struc- y, in the left and right subtrees of that node are connected. (The des of the internal nodes in the ﬁgure represent the probabilities.) am in which closely mmon ancestors that ore distantly related ability of a connec- Clauset et al., Nature (2008)

Community overlap

ciated large network we introduce the distributions of these four basic priori quantities. In particular we focus on their cumulative distribution ins5,6, o the es of e net- actual ps of main mmu- ucture eristic ﬁcient scale. ns we ies of raphs ns and nodes est of usters, ve no G. Palla, I. Derényi, I. Farkas & T. Vicsek, Nature, 2005

A B Multiple Contexts C overlap and hierarchy Family do not mix buildings in same neighborhood University home and work

A Multiple Contexts B Multiple Contexts Multiple Contexts C overlap and hierarchy Family do not mix Multiple Contexts buildings in same neighborhood University home and work

C overlap and hierarchy Family do not mix buildings in same neighborhood University home and work joint appointment D 1 2 F Single dendrogram cannot represent multiple hierarchical contexts 3 3! 4

Hierarchical community structure Hierarchy Communities

Hierarchy Communities Complex global structure

Colleagues Family Friends

Overlap is pervasive

Simple local structure

Complex global structure

Example:

What is this?

What the xxxx is this?

measures, and second, these measures are, crucially, recalculated after each removal. We also propose a measure for the strength of the community structure found by our algorithms, which gives us an objective metric for choosing the number of communities into which a network should be divided. We demonstrate that our algorithms are highly effective at discovering community structure in both computer-generated and real-world network data, and show how they can be used to shed light on What the xxxx the sometimes dauntingly complex structure of networked systems. I. INTRODUCTION hierarchical clustering in sociology [18, 19]. Before pre- senting our own findings, it is worth reviewing some of this preceding work, to understand its achievements and is this? Empirical studies and theoretical modeling of networks have been the subject of a large body of recent research in where it falls short. statistical physics and applied mathematics [1, 2, 3, 4]. Graph partitioning is a problem that arises in, for ex- Network ideas have been applied with great success to ample, parallel computing. Suppose we have a num- topics as diverse as the Internet and the world wide ber n of intercommunicating computer processes, which web [5, 6, 7], epidemiology [8, 9, 10, 11], scientific ci- we wish to distribute over a number g of computer proces- tation and collaboration [12, 13], metabolism [14, 15], sors. Processes do not necessarily need to communicate and ecosystems [16, 17], to name but a few. A property with all others, and the pattern of required communica- that seems to be common to many networks is commu- tions can be represented by a graph or network in which nity structure, the division of network nodes into groups the vertices represent processes and edges join process within which the network connections are dense, but be- pairs that need to communicate. The problem is to allo- tween which they are sparser—see Fig. 1. The ability to cate the processes to processors in such a way as roughly find and analyze such groups can provide invaluable help to balance the load on each processor, while at the same in understanding and visualizing the structure of net- time minimizing the number of edges that run between works. In this paper we show how this can be achieved. processors, so that the amount of interprocessor commu- The study of community structure in networks has a nication (which is normally slow) is minimized. In gen- long history. It is closely related to the ideas of graph eral, finding an exact solution to a partitioning task of partitioning in graph theory and computer science, and this kind is believed to be an NP-complete problem, mak- ing it prohibitively difficult to solve for large graphs, but a wide variety of heuristic algorithms have been devel- oped that give acceptably good solutions in many cases, the best known being perhaps the Kernighan–Lin algorithm [20], which runs in time O(n3 ) on sparse graphs. A solution to the graph partitioning problem is however not particularly helpful for analyzing and understanding networks in general. If we merely want to find if and how a given network breaks down into communities, we probably don’t know how many such communities there are going to be, and there is no reason why they should be roughly the same size. Furthermore, the number of inter-community edges needn’t be strictly minimized either, since more such edges are admissible between large communities than between small ones. FIG. 1: A small network with community structure of the As far as our goals in this paper are concerned, a more useful approach is that taken by social network analysis

above, because none of the others in the literature satisfy all these of protein–protein interactions27 (Fig. 2c). These pictures ca requirements simultaneously21,24. tests or validations of the efﬁciency of our algorithm. In p Word association network: Network of “commonly associated English words” Figure 2 | The community structure around a particular node in three be associated with his ﬁelds of interest. b, The communities of t different networks. The communities are colourG. Palla, I. Derényi, I. Farkas & T. Vicsek, Nature, 2005 w* coded, the overlapping ‘bright’ in the South Florida Free Association norms list (for

a Link communities and Bob also work together b Spouses Alice Word Association examples Link communities COMBINE COMBINE JOIN Alice FRUIT BLENDER JOIN Alice FRUIT INTEGRATE BLENDER INTEGRATE Bob JUICE BLEND Bob JUICE BLEND MIX MIXTURE Family Work MIX MIXTURE Family Work Node communities Node communities Figure S16: Overlapping community structure around Acetyl-CoA in the E. coli metabolic network. Alice Alice different and important roles in metabolism. Shown are only communities with homogeneity score e DISAPPEAR inside each community share at least one pathway annotation); all other links, including those that Alice Alice LOOK structure, are omitted. Pathway annotations shared by all community members are displayed with c LOOK APPEAR DISAPPEAR two communities to the right of Acetyl-CoA are grouped since they share the same exact pathway an APPEAR VANISH Bob Bob SEE VANISH Bob Bob Work SEE REAPPEAR Family REAPPEAR Work SHOW ATTEND Family The Alice-Bob link was placed in family but both SHOW ATTEND The Alice-Bobwork was placed in are identiﬁed home and link relationships family but both home and work relationships are identiﬁed BROOM PAINT Figure S4: Overlapping links. In the link community framework, a link may beSWEEP assigned to only one community. By de gure S4: Overlapping links. In the link community framework, a link may be relationships betweencommunity. By derivi node communities, however, the problem of effectively discovering multiple assigned to only one nodes is effectively s PAINTER ode communities, however,many communities together regardless of the membership of the link betweenis effectively illust Two nodes can belong to the problem of effectively discovering multiple relationships between nodes them. Left: solv GROOM wo nodes can belong to manyexamples from word association network. In the upper example, Blend and blender belong to of the situation. Right: real communities together regardless of the membership of the link between them. Left: illustrati BRUSH PAINTING the situation.community and ‘mix’ from word association network. In thethe linkexample, Blend and blender belong tono ‘fruit juice’ Right: real examples community. In the bottom example, upper between appear and reappear does bo HAIR ruit juice’ communityother ‘mix’ community. they belong to several communities together. belong to any of the and communities, but In the bottom example, the COMB between appear and reappear does not ev link TOOTHBRUSH long to any of the other communities, but they belong to several communities together. HAIRSPRAY TOOTHPASTE link can simultaneously belong to multiple communities even though the link itself belongs to only

pping community structure around Acetyl-CoA in the E. coli metabolic network. Acetyl-CoA plays several tant roles in metabolism. Shown are only communities with homogeneity score equal to 1 (all compounds nity share at least one pathway annotation); all other links, including those that contribute to community Simple Complex ed. Pathway annotations shared by all community members are displayed with corresponding colors. The the right of Acetyl-CoA are grouped since they share the same exact pathway annotations. BROOM PAINT SWEEP PAINTER GROOM PAINTING BRUSH HAIR TOOTHBRUSH COMB HAIRSPRAY TOOTHPASTE Global • SUNSET, SUNRISE, ORANGE Local • SUNSET, SUNRISE, RED • SUNSET, SUNRISE, PRETTY, BEAUTIFUL • SUNSET, SUNRISE, MOON • SUNSET, SUNRISE, BEACH • SUNSET, SUNRISE, SUN, DAWN, DUSK, SUNSHINE • SUNSET, SUNRISE, DAWN, DUSK, AFTERNOON, EVENING

Then, how can we ﬁnd hierarchical community structure in COMPLEX networks with pervasive overlap?

Our solution: Use Links

Our solution: Use Links “a group of densely interconnected nodes”

Our solution: Use Links “a group ofTopologically densely Similar interconnected nodes” LINKS

Colleagues Family Friends

Colleagues ‘Family’ links Family Friends

Colleagues ‘Family’ links Friends Family ‘Friends’ links

‘Nerds & geeks’ links Colleagues ‘Family’ links Friends Family ‘Friends’ links

Nodes: multiple membership Links: unique membership

Overlap

Hierarchy Communities

Reconciliation

So, How?

Similarity between links Hierarchical Clustering

Hierarchical Link Clustering (HLC)

A B ei k ejk c i k j a S(eac , ebc ) Figure S1: (A) The similarity measure S(eik , ejk ) between edges For this example, |n+ (i) ∪ n+ (j)| = 12 and |n+ (i) ∩ n+ (j)| = 4, cases: (B) an isolated (ka = kb = 1), connected triple (a,c,b) has S triangle has S = 1. structure can become radically different.) Thus, we neglect the ne ﬁrst deﬁne the inclusive neighbors of a node i as:

A B ei k ejk c i k j a S(eac , ebc ) Figure S1: (A) The similarity measure S(eik , ejk ) between edges For this example, |n+ (i) ∪ n+ (j)| = 12 and |n+ (i) ∩ n+ (j)| = 4, cases: (B) an isolated (ka = kb = 1), connected triple (a,c,b) has S triangle has S = 1. 4 structure can become radically different.) Thus, we neglect12 ne the ﬁrst deﬁne the inclusive neighbors of a node i as:

(a) 1 2 (c) 3 3!4 9 2!4 4 7 1!4 6 8 2!3 1!2 5 1!3 (b) 1 2 4!7 5!6 4!6 3 4!5 9 7!9 4 7 7!8 6 8!9 8 5 3!4 2!4 1!4 2!3 1!2 1!3 4!7 5!6 4!6 4!5 7!9 7!8 8!9

Partition Density Community c has mc edges and nc induced nodes c mc nc

Partition Density Community c has mc edges and nc induced nodes c mc nc mc = 8 nc = 5

Partition Density Community c has mc edges and nc induced nodes c mc nc = mc

Partition Density Community c has mc edges and nc induced nodes c mc nc − = mc − (nc − 1)

Partition Density Community c has mc edges and nc induced nodes c mc nc − mc − (nc − 1) = nc (nc −1) − 2 − (nc − 1)

Partition Density − mc − (nc − 1) = nc (nc −1) − 2 − (nc − 1) mc − (nc − 1) =2 (nc − 2)(nc − 1)

Partition Density − mc − (nc − 1) = nc (nc −1) − 2 − (nc − 1) mc − (nc − 1) =2 (nc − 2)(nc − 1) 2 mc − (nc − 1) D≡ mc M c (nc − 2)(nc − 1)

It’s just density No resolution limit

Boulatruelle Jondrette Brujon Anzelma Blacheville Dahlia Gueulemer Favourite MmeBurgon Fameuil Babet Child1 Zephine Eponine Listolier Child2 Montparnasse Tholomyes MotherPlutarch Claquesous Perpetue Fantine Mabeuf Marguerite Brevet Thenardier MmeThenardier Javert Combeferre Gavroche Simplice Champmathieu Judge Bahorel Courfeyrac Toussaint Chenildieu Joly Bamatabois Marius Enjolras Woman2 Cochepaille Grantaire Feuilly Cosette Valjean Bossuet Woman1 Gribier Prouvaire Magnon Fauchelevent MmeHucheloup LtGillenormand Scaufflaire MotherInnocent Gillenormand MlleBaptistine Gervais Pontmercy Isabeau BaronessT MlleGillenormand MmeDeR MmeMagloire CountessDeLo Labarre Myriel MmePontmercy Napoleon Geborand MlleVaubois OldMan Count Cravatte Champtercier 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Does it really work?

Quantitative Evaluation Framework

Quantitative Evaluation Framework How homogeneous each Community quality community is? How accurate the # of Overlap quality overlap is? How many nodes are Community coverage covered? How many memberships Overlap coverage are assigned?

Metadata Figure R11: Example of the network and available metadata for the Amazon.com product co-purchases network. Here we show a particular book (upper left), some of the books it is often bought with (lower left), the set of subjects it is classiﬁed into by Amazon.com (upper right), and the set of popular “tags” Amazon.com users have chosen to describe or annotate the book’s content (lower right). We can use shared tags to quantify how similar pairs of books are, and the more subjects a book has, the more communities it is expected to belong to. Other combinations of metadata are certainly possible. Other networks used here have analogous metadata.

Quantitative Evaluation Framework Community quality Amazon.com Community coverage no membership Subjects Subjects HIV / AIDS Medical Africa - General Africa Africa History Subjects HIV / AIDS Medical Nonfiction / General Infectious Diseases high coverage low coverage Overlap quality Metabolic network Overlap coverage community memberships Acetyl-CoA 1. Glycolysis / Gluconeogenesis 2. TCA cycle 3. Fatty acid biosynthesis 4. ... Many pathway Memberships high overlap IDP (Inosine diphosphate) 1. Purine metaboilsm Few pathway Memberships low overlap high overlap coverage low overlap coverage

and topologies (for example, the network range from sparse (average degree 6.34) to dense (average degree 38.95)). metadata network description N k community overlap PPI (Y2H) PPI network of S. cerevisiae 1647 3.06 Set of each protein’s The number of GO obtained by yeast two-hybrid known functions (GO terms (Y2H) experiment [3] terms)a PPI (AP/MS) Affinity purification mass 1004 16.57 GO terms GO terms spectrometry (AP/MS) experiment PPI (LC) Literature curated (LC) 1213 4.21 GO terms GO terms PPI (all) Union of Y2H, AP/MS, and LC 2729 8.92 GO terms GO-terms PPI networksb Metabolic Metabolic network (metabolites 1042 16.81 Set of each The number of connected by reactions) of E. metabolite’s pathway KEGG pathway coli annotations (KEGG)c annotations Phone Social contacts between mobile 885989 6.34 Each user’s most likely Call activity phone users [15, 16, 17] geographic location (number of phone callsd ) Actor Film actors that appear in the 67411 8.90 Set of plot keywords Length of career same movies during for all of the actor’s (year of first role) 2000–2009 [18] films US Congress Congressmen who co-sponsor 390 38.95 Political ideology, Seniority (number bills during the 108th US from the common of congresses Congress [19, 20] space score [21, 22] served) Philosopher Philosophers and their 1219 9.80 Set of (wikipedia) Number of philosophical influences, from hyperlinks exiting in wikipedia subject the English Wikipediae the philosopher’s page categories Word Assoc. English words that are often 5018 22.02 Set of each word’s Number of senses mentally associated [23] senses, as documented by WordNet f Amazon.com Products that users frequently 18142 5.09g Set of each product’s Number of product buy together user tags (annotations) categories

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

,$-+.%$+ 2S89C;4(12S89;F8 1233Q<67R(12S89;F8 2S89C;4(PQ;C67R 1233Q<67R(PQ;C67R ,$"#)/+ - . / 0 - . / 0 N -6<O5 - . / 0 - . / 0 - - . / 0 ?29@(=5521% =3;>2<%123 .C6PQ8(A8912C;762< =1729 DE (.2<F9855 AB6C2524B89 . N !"#$%&'$"()%*+ / N /988@R(J2@QC;967R( 5)41-2&'$"()%*+ 0( N 0<:23;4 "*)! )!)'+ &#* )+)# $,')) ++%*+ "%*# &! %#" #%!* !%#*

d as soc. Wor ,$-+.%$+ 2S89C;4(12S89;F8 1233Q<67R(12S89;F8 2S89C;4(PQ;C67R 1233Q<67R(PQ;C67R ,$"#)/+ - . / 0 - . / 0 N -6<O5 - . / 0 - . / 0 - - . / 0 ?29@(=5521% =3;>2<%123 .C6PQ8(A8912C;762< =1729 DE (.2<F9855 AB6C2524B89 . N !"#$%&'$"()%*+ / N /988@R(J2@QC;967R( 5)41-2&'$"()%*+ 0( N 0<:23;4 "*)! )!)'+ &#* )+)# $,')) ++%*+ "%*# &! %#" #%!* !%#*

Examples

BROOM PAINT SWEEP PAINTER GROOM PAINTING BRUSH HAIR TOOTHBRUSH COMB HAIRSPRAY TOOTHPASTE

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

/ 34(5&,-.."(46417&'8-"()&&'"$9!"#$&:8-.&;;<&='00> %&'()*+, !"#$% !"#$ &'"$(%5(-A(& 7"?"(46&-:&?-6@& !"#$%&'()&*#+# -#.# ,-./012

B proteasome core complex (GO:005839, C) threonine-type endopeptidase activity (GO:0004298, F) Signalosome (GO:0008180, C) ubiquitin-dependent protein catabolic process (GO:0006511, P) Protein deneddylation (GO:0000338, P) proteasome regulatory particle (GO:0005838, C) ubiquitin-dependent protein catabolic process (GO:0006511, P) endopeptidase activity (GO:0004175, F) Core Regulatory particle Proteasome ure S16: Another example of overlapping community structure. (A) The subnetwork

B 103 number of communities 103 metabolites ATP number of 2 10 ADP 2 H2O, H+ 10 1 Pi 10 0 10 101 0 50 100 150 200 number of communities per metabolite 0 E. coli 10 101 102 103 number of metabolites per community 106 mmunities 6 10 ber of users 105 105 4 10 104 10 3 102

u rren cies B 3 C number of communities 10 103 metabolites ATP number of 2 10 ADP 2 H2O, H+ 10 1 Pi 10 0 10 101 0 50 100 150 200 number of communities per metabolite 0 E. coli 10 101 102 103 number of metabolites per community 106 mmunities 6 10 ber of users 105 105 4 10 104 10 3 102

Hierarchical organization

~600k nodes ~3M edges

threshold = 0.20

A B threshold = 0.23 50 km thr = 0.24 C D threshold thr = 0.27 = 0.20 thr = 0.27 F 0.4 0.5 0.6 0.7 0.8 E 0.9 1

Remaining hierarchy e 1 Phone Metabolic Word association 0.8 Q/Qmax 0.6 0.4 0.2 Actual Control 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Link dendrogram threshold, t Figure 4 | Meaningful communities at multiple levels of the link dendrogram. a–c, The social network of mobile phone users displays co- located, overlapping communities on multiple scales. a, Heat map of the most likely locations of all users in the region, showing several cities. b, Cutting the dendrogram above the optimum threshold yields small, intra-

a b Planets Diving, Swim, Marine life SPLASH DUCK Water and aquatic animals MARSH Astronomy SAILING Astronomy MARS Scuba diving, DROWN SINKER DRIFT BOG SWAMP (more general terms) PLUTO URANUS Scuba diving Coral reef LAGOON SWAN CROCODILE SATURN GALAXY REEF SWIMMER EARTH UNDERWATER SAIL JUPITER OVERFLOW DIVER CORAL FLOAT POND REPTILE PLANET PLANETS NEPTUNE DIVING STARS MOAT UNIVERSE SNORKEL SWIM DUCKS VENUS DIVE RAFT ALLIGATOR ASTRONOMY SCUBA LAKE METEORITE ASTROLOGY CANOE PIER MOON MERMAID BROOK DOCK COMET FIN PADDLE CREEK METEOR STAR FLIPPER BAY FISHING OBSERVE UPSTREAM ASTEROID DOLPHIN RIVER CANAL ROCKET PORPOISE SKY ASTRONAUT Diving with animals OTTER FLOOD WHALE DOWNSTREAM SHUTTLE TELESCOPE SEAL TANK STREAM DAM SALMON MARINE FLOW WALRUS MAMMAL TROUT INLET c d SATURN MERMAID URANUS NEPTUNE DIVING JUPITER MARS SWIMMER PLUTO CORAL TELESCOPE VENUS SWIM STARS UNDERWATER FIN REEF MOON PLANETS SNORKEL MARINE GALAXY DIVE PLANET SCUBA METEOR UNIVERSE DOLPHIN DIVER ASTRONOMY FLIPPER ASTEROID METEORITE WHALE PORPOISE A community at threshold = 0.20, A community at threshold = 0.20, and sub-communities at threshold = 0.28 COMET and sub-communities at threshold = 0.28 WALRUS Figure 23: Examples of hierarchical structure in the word association network. The word association network is a nice example for this purpose, since it is easy to appreciate the meanings and contexts of the individual words and communities. (a) Here we pick a link and follow how the link merges with others as we climb the hierarchical tree. (b) We start from the link MARS–

Conclusion • Link viewpoint eﬀectively removes the problem of overlap. • Global hierarchical structure can be found by clustering links. • doi:10.1038/nature09182 • http://barabasilab.neu.edu/projects/ linkcommunities/

Acknowledgements

Acknowledgements A.-L. Barabási, H. Yu, S. Ahnert, J. Park, D.- S. Lee, P.-J. Kim, M. A. Yildirim,

Acknowledgements A.-L. Barabási, H. Yu, S. Ahnert, J. Park, D.- S. Lee, P.-J. Kim, M. A. Yildirim, T. S. Evans, R. Lambiotte, Line Graphs, Link Partitions and Overlapping Communities, http://sites.google.com/site/linegraphs/

xkcd.com

a Spouses Alice and Bob also work together b Word Association examples Link communities COMBINE JOIN Alice FRUIT BLENDER INTEGRATE Bob JUICE BLEND MIX MIXTURE Family Work Node communities Alice Alice LOOK DISAPPEAR APPEAR VANISH Bob Bob SEE REAPPEAR Work Family SHOW ATTEND The Alice-Bob link was placed in family but both home and work relationships are identiﬁed ultiple relationships between nodes be found by link communities that assume one membe hemselves “inherit” multiple memberships from their links. Two nodes can belong to many c

link communities a Internal groups without distinguishing features are undetectable to ALL methods i e communities language class basketball team f d project g b prob. p a j c h students a b c d e f g h i j all students are identical one community, D = 0.750 b subtle structural differences are found by link communities g c coach e communities language class basketball team a f project prob. p i d j b h students a b c d e f g h i j coach coach separates them two communities, D = 0.756 c juniors basketball team seniors 1 2 3 4 5 6 7 8 9 10 21 22 23 24 25 26 27 28 29 30 6 10 1 3 11 12 13 14 15 16 17 18 19 20 14 2 20 7 project 24 12 prob. p 8 13 18 28 5 23 15 three communities, D = 0.745 4 17 25 9 26 11 16 Multiple relationships are found: 22 juniors and 19 29 The link between students 18 and 20 basketball players is senior but both 18 and 20 belong to 30 27 21 both seniors and basketball players! seniors and basketball players Figure 5: Some small, illustrative examples of the subtle structural changes that link communities detect, using the bipartite social model of [21] with p = 0.8, followed by our link communities algorithm. In (a) there are no distinguishing structural features to separate the “subsumed” basketball team from the language class. Detecting the team is impossible for all methods.

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Link communities reveal multiscale complexity in networks

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