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RICHARDS SHE-HULK M/TIO@ MEPHV. SPIDER-MAN STACY SENSM SM:FA FRANKLIN B MICRO SPIDER-MAN CLONE GWEN SM:PL SILVER SURFER M/TIO THOMPSON S-M PP ROBERTSON EUGENE FLA SPECSM MS. MARVEL II RANDY M/SHSW URICH PWJ ST BEN UTSM SUB-MARINER ROBERTSON M/FAN JOE PPTSS@ MR. FANTASTIC WATSON SVTU ANNA UTSM:SE PARKER THING IW MAY PPTSS WATSON-PARKER TB MARY WOSM MEDUSA OSBORN IM2 LIZ ALLAN PPSM2

BLACK BOLT A BLACK CAT ASM OSBORN MASTERS HR:R HARRY MSM ALICIA REIS CUSHING A@ KATE ASM2 Active set CRYSTAL H2 LEEDS M/TU@ NED ASM ASM:FO LYJA LAZERFIST DAREDEVIL GAL LEEDS M/TU2 BETTY BRANT DR. DOOM DAZZ DR. OCTOPUS ASM@ KARNAK FFV.X KINGPIN M/TU DF GRANT ASMU GLORIA GLORY GORGON FF@ JAMESON GSSH INVISIBLE WOMAN FF MARLA MADISJAMESONGREEN GOBLIN ASMV.W HUMAN TORCH FF3 FF2 J. JONAH GSS-M DB

BLACK BOLT A BLACK CAT ASM OSBORN MASTERS HR:R HARRY MSM ALICIA REIS CUSHING A@ KATE ASM2 CRYSTAL H2 LEEDS M/TU@ NED ASM ASM:FO LYJA LAZERFIST DAREDEVIL GAL LEEDS M/TU2 BETTY BRANT random compression DR. DOOM DAZZ DR. OCTOPUS ASM@ Active set - Structured KARNAK FFV.X KINGPIN M/TU DF GRANT ASMU GLORIA GLORY GORGON FF@ JAMESON GSSH INVISIBLE WOMAN FF MARLA MADISJAMESONGREEN GOBLIN ASMV.W HUMAN TORCH FF3 FF2 J. JONAH GSS-M DB

OSBORNSPIDER-MAN NW PPTSS SCHEMERSCREAMING MIMI SLEEP SM:MM LIZ ALLAN STARFOX Q ROM SPIDER-MAN SQS MR. FANTASTIC NM SPACEKNIGHT S-M THING T STARSHINE II TB ROBERTSON MEDUSA MARVELS JOE Q THOR TB THING UX MASTERS M/TIO POWER MAN PPTSS ALICIA REIS WASP WCA2 WASP W2

JAMESON NIGHTCRAWLER PPSM2 J. JONAH M/SHSW WATSON-PARKER WATSON-PARKER MARY X:HY MARY WCA2 IRON MAN IM3 MEDUSA PM&IF

ANT-MAN A CH'OD A

IRON MAN IV HR:R LILANDRA NERAMANI M/TU

BLACK BOLT AA2 CLOAK A3 LEEDS INVISIBLE WOMAN compression FW NED INH3

BLACK KNIGHT V ASM COLOSSUS II ASM JAMESON HUMAN TORCH FS J. JONAH IG

Active set - Gaussian BUCKY CA3 CRYSTAL ASM2 HAWK FF@ IRON MAN H2 CAPTAIN AMERICA DAGGER DD HAMMER ASM@ GORGON FF3 JUSTIN DUGAN FW FURY CAPTAIN MARVEL II DEF FURY TIMOTHY ALOYI C2 COL. NICHOLASDAREDEVIL FF2 FF COL. NICHOLASFALCON DHKF CA

RICHARDS RUNNER SS3 ROBERTSONSANDMAN SM:RGG SHALLA BAL II RANDY FRANKLIN B ROBERTSON SPIDER-MAN SM:MM Q TQ MARTHA THOMPSON SSWP REPTYL SILVER SURFER ROBERTSON EUGENE FLA JOE TOMBSTONE SM:L NOVA II STARFOX ROBERTSON AMANDA INV@ WCA@ WATSON ANNA WOSM RANDI MR. FANTASTIC TRADER WATSON-PARKER MARY SM/M OSBORN LIZ ALLAN

MANTIS ASTRONOMER GSDEF A ARRANGER ASM OSBORN HARRY BAMBI PPTSS@

Active set KAMO THARNN CHAMPION MUGGINS MAMIE BANNON ASM2 COC A@ LANCE MERCADO Random sampling INVISIBLE WOMAN COLLECTOR JOY CANDY PPTSS MAULER GRANDMASTER CONTEMPLATOR FOREIGNER ASM@ CM5 ASM LEEDS BETTY BRANT GRANT GARDENER FIRELORD JAMESON GLORIA GLORY PPSM2 GALACTUS CM4 J. JONAHHOBGOBLIN V MK

SIF STARFOX T T2 SIRYN SUNSPOT XCAL XF STORM T@ RICHARDS THUNDERBIRD II XF@ SHADOWCAT FRANKLIN B X&M TALISMAN II ST WARLOCK III XFOR SASQUATCH TS PUCK UX@ THOR SS3 WHITE QUEEN XFOR@ PAGE PUCK TTA KAREN UX VOLSTAGG WOLFSBANE XTERM SS PROFESSOR X NAMORITA SWII WM2 WONG WONDER MAN XU

ODIN Q MIRAGE II SEX

BALDER A BANSHEE AF2

MAGMA PP MR. FANTASTIC PP BOX IV BOOMER C2 AF LOKI MAGIK NMSE M/SH3 CAPTAIN AMERICA CANNONBALL II FA

KARNILLA BIZADV KARMA NM@

approximation (Zhou, CAPTAIN MARVEL II M/GN CHAMBER FS INVISIBLE WOMAN DAZZ IRON MAN NM:TD

Cichocki, Xie. TSP, 2012) COLOSSUS II GENX DR. STRANGE M/CP HUDSON HUSK NM Fast NMF based on low-rank HEATHER FANDRAL H2 HUDSON CRYSTAL M/AGE@ HOGUN HEIMDALL JIM IW HEATHER CYPHER MGK M/GN

Figure 1. Biclustering the Marvel Universe collaboration network (http://www.chronologyproject.com/). The network links Marvel characters and the Marvel comic books in which they appear, and exhibits most characteristics of real-life collaboration networks. It can be represented as an m n matrix, where m =6445andn =12850arethenumberofcharactersandcomics,respectively.WebiclusterthismatrixusingNMFwithr =10,aimingatobtaining⇥ 10 very representative groups of characters appearing jointly in di↵erent comic books. The ith bicluster (i =1...10) is formed by the ith column of X and the ith row of Y (small entries were set to zero). The radar plots represent the coecients of these vectors. We show two biclusters that we identify with characters from the Fantastic Four (first two columns of the figure) and the Spider-Man (last two columns of the figure) comics. Active set NMF correctly identifies that Mr Fantastic, The Thing, the Invisible Woman, and the Human Torch are the four most recurring characters in the “Fantastic Four” (FF) series. Similarly, active set NMF correctly identifies that Spider-Man/Peter Parker, Mary Jane Watson-Parker (Peter Parker’s wife), and Jonah Jameson (Peter Parker’s boss) are the most recurring characters in the “Amazing Spider-Man” (ASM) and “Peter Parker, The Spectacular Spider-Man” (PPTSS) series. It is clear that the biclusters recovered using structured random compression (SC) are very close to the biclusters found with no compression; contrarily, the other methods (GC and RS respectively stand for Gaussian compression and random sampling) significantly a↵ect the biclustering result. All 10 biclusters can be found at http://www.marianotepper.com.ar/research/cnmf.