Effect of volume growth on the percolation threshold in random directed acyclic graphs with a given degree distribution.

In every network, a distance between a pair of nodes can be defined as the length of the shortest path connecting these nodes, and therefore one may speak of a ball, its volume, and how it grows as a function of the radius.
Spatial networks tend to feature peculiar volume scaling functions, as well as other topological features, including clustering, degree-degree correlation, clique complexes, and heterogeneity.
Here we investigate a nongeometric random graph with a given degree distribution and an additional constraint on the volume scaling function. We show that such structures fall into the category of m-colored random graphs and study the percolation transition by using this theory. We prove that for a given degree distribution the percolation threshold for weakly connected components is not affected by the volume growth function. Additionally, we show that the size of the giant component and the cyclomatic number are not affected by volume scaling.

Recombinant Obelin, 50UG

L001-50UG 50UG
EUR 258.00

L-Cysteine Monoclonal, 50UG

A002-50UG 50UG
EUR 263.00

ThioStar? Fluorescent Thiol Substrate, 50UG

L002-50UG 50UG
EUR 207.00

PGE2 Monoclonal Antibody (clone 5A2), 50UG

A011-50UG 50UG
EUR 587.00

DyLight® 488 Labeled Glutathione Monoclonal, 50UG

A001F-50UG 50UG
EUR 364.00

DyLight® 550 Labeled Glutathione Monoclonal, 50UG

A001T-50UG 50UG
EUR 364.00

Recombinant Mouse AK1 Protein, His, E.coli-50ug

QP10509-50ug 50ug
EUR 1261.00

Recombinant Mouse AK2 Protein, His, E.coli-50ug

QP10510-50ug 50ug
EUR 1261.00

Recombinant Human AKR1B1 Protein, Untagged, E.coli-50ug

QP10511-50ug 50ug
EUR 201.00

Recombinant Human ALDH2 Protein, Untagged, E.coli-50ug

QP10515-50ug 50ug
EUR 201.00

Recombinant Human ALDOC Protein, Untagged, E.coli-50ug

QP10517-50ug 50ug
EUR 1261.00

Recombinant Human BMP13 Protein, Untagged, E.coli-50ug

QP10537-50ug 50ug
EUR 201.00

Recombinant Mouse CES1D Protein, His, Insect-50ug

QP10550-50ug 50ug
EUR 1261.00

Recombinant Mouse CTSD Protein, His, Insect-50ug

QP10567-50ug 50ug
EUR 1261.00

Recombinant Human EGFR Protein, GST, Insect-50ug

QP10584-50ug 50ug
EUR 1161.00

Recombinant Human FKBP4 Protein, His, E.coli-50ug

QP10609-50ug 50ug
EUR 201.00

Recombinant Human Fumarase Protein, Untagged, E.coli-50ug

QP10621-50ug 50ug
EUR 201.00

Recombinant E.coli G6PD Protein, Untagged, E.coli-50ug

QP10622-50ug 50ug
EUR 201.00

Recombinant Mouse GDA Protein, His, E.coli-50ug

QP10627-50ug 50ug
EUR 1261.00

Recombinant Carp GH Protein, Untagged, E.coli-50ug

QP10631-50ug 50ug
EUR 201.00

Recombinant Zebrafish GH Protein, Untagged, E.coli-50ug

QP10635-50ug 50ug
EUR 201.00

Recombinant other GST Protein, Untagged, E.coli-50ug

QP10656-50ug 50ug
EUR 201.00

Recombinant Human GSTA1 Protein, Untagged, E.coli-50ug

QP10657-50ug 50ug
EUR 201.00

Recombinant Mouse GSTM1 Protein, Untagged, E.coli-50ug

QP10658-50ug 50ug
EUR 201.00

Recombinant Human GUSB Protein, His, Insect-50ug

QP10661-50ug 50ug
EUR 1261.00

Recombinant Rabbit IGF1 Protein, Untagged, E.coli-50ug

QP10679-50ug 50ug
EUR 201.00

Recombinant Mouse IGF1 Protein, Untagged, E.coli-50ug

QP10681-50ug 50ug
EUR 201.00

Recombinant Feline IL2 Protein, His, E.coli-50ug

QP10727-50ug 50ug
EUR 1261.00

Recombinant Human KLK13 Protein, His, Insect-50ug

QP10737-50ug 50ug
EUR 1261.00

Recombinant Mouse LDHB Protein, His, E.coli-50ug

QP10745-50ug 50ug
EUR 1261.00

Recombinant other LLO Protein, Untagged, E.coli-50ug

QP10773-50ug 50ug
EUR 663.00

Recombinant Human MEK1 Protein, His, Insect-50ug

QP10784-50ug 50ug
EUR 962.00

Recombinant Human NDRG1 Protein, His, E.coli-50ug

QP10791-50ug 50ug
EUR 201.00

Recombinant Human PPIF Protein, His, E.coli-50ug

QP10829-50ug 50ug
EUR 201.00

Recombinant Human PPIL1 Protein, His, E.coli-50ug

QP10831-50ug 50ug
EUR 201.00

Recombinant Human PPP1CA Protein, His, E.coli-50ug

QP10835-50ug 50ug
EUR 1261.00

Recombinant Human PPP1CC Protein, His, E.coli-50ug

QP10836-50ug 50ug
EUR 5251.00

Recombinant Rabbit Prolactin Protein, Untagged, E.coli-50ug

QP10850-50ug 50ug
EUR 201.00

Recombinant Human PTPN1 Protein, Untagged, E.coli-50ug

QP10856-50ug 50ug
EUR 201.00

Recombinant Human SAA Protein, His, E.coli-50ug

QP10860-50ug 50ug
EUR 201.00

Recombinant Human TNFRSF10D Protein, His, Insect-50ug

QP10888-50ug 50ug
EUR 1261.00

Recombinant Mouse TRAIL Protein, Untagged, E.coli-50ug

QP10892-50ug 50ug
EUR 201.00

Recombinant Human TSG Protein, Untagged, CHO-50ug

QP10900-50ug 50ug
EUR 201.00

Recombinant Human ACO1 Protein, His, E.coli-50ug

QP10940-50ug 50ug
EUR 1261.00

Recombinant Human AMD1 Protein, His, E.coli-50ug

QP11010-50ug 50ug
EUR 1261.00

Recombinant Human AMN Protein, His, E.coli-50ug

QP11013-50ug 50ug
EUR 1161.00

Recombinant Mouse ANTXR2 Protein, His, Insect-50ug

QP11030-50ug 50ug
EUR 1261.00

Recombinant Human ARC Protein, His, E.coli-50ug

QP11060-50ug 50ug
EUR 1261.00

Recombinant Human ARHGDIB Protein, GST, E.coli-50ug

QP11065-50ug 50ug
EUR 1161.00

Recombinant other AsnRS Protein, His, E.coli-50ug

QP11088-50ug 50ug
EUR 201.00

Recombinant Mouse HDAC8 Protein, His, E.coli-50ug

QP12203-50ug 50ug
EUR 1261.00

Recombinant Human HIF1AN Protein, Untagged, E.coli-50ug

QP12233-50ug 50ug
EUR 201.00

Recombinant Human HLF Protein, His, E.coli-50ug

QP12278-50ug 50ug
EUR 1161.00

Recombinant Human HNRNPC Protein, His, E.coli-50ug

QP12291-50ug 50ug
EUR 1261.00

Recombinant Mouse HPD Protein, His, E.coli-50ug

QP12299-50ug 50ug
EUR 1261.00

Recombinant Human HRASLS3 Protein, Untagged, E.coli-50ug

QP12312-50ug 50ug
EUR 201.00

Recombinant Human HSP27 Protein, Untagged, E.coli-50ug

QP12327-50ug 50ug
EUR 201.00

Recombinant Human HSPA9 Protein, His, E.coli-50ug

QP12330-50ug 50ug
EUR 201.00

Recombinant Human HTRA2 Protein, His, E.coli-50ug

QP12360-50ug 50ug
EUR 201.00

Recombinant Human CHMP4A Protein, His, E.coli-50ug

QP11421-50ug 50ug
EUR 1161.00

Recombinant Human CPLX1 Protein, His, E.coli-50ug

QP11495-50ug 50ug
EUR 201.00

Recombinant Human CRK Protein, His, E.coli-50ug

QP11508-50ug 50ug
EUR 1161.00

Recombinant Mouse CTSL Protein, His, Insect-50ug

QP11562-50ug 50ug
EUR 1261.00

Recombinant Human DDR1 Protein, His, Insect-50ug

QP11610-50ug 50ug
EUR 1261.00

Recombinant Human DHRS4 Protein, His, E.coli-50ug

QP11665-50ug 50ug
EUR 1161.00

Recombinant other DMGO Protein, His, E.coli-50ug

QP11673-50ug 50ug
EUR 201.00

Recombinant other DnaJ Protein, Untagged, E.coli-50ug

QP11674-50ug 50ug
EUR 201.00

Recombinant Human DNAJB11 Protein, His, E.coli-50ug

QP11675-50ug 50ug
EUR 1261.00

Recombinant Human DNAJB6 Protein, His, E.coli-50ug

QP11677-50ug 50ug
EUR 1261.00

Recombinant other DnaK Protein, Untagged, E.coli-50ug

QP11682-50ug 50ug
EUR 201.00

Recombinant Human DnaK Protein, His, E.coli-50ug

QP11683-50ug 50ug
EUR 201.00

Recombinant other DsbA Protein, Untagged, E.coli-50ug

QP11703-50ug 50ug
EUR 201.00

Recombinant other DsbC Protein, Untagged, E.coli-50ug

QP11704-50ug 50ug
EUR 201.00

Recombinant Human SUMO1 Protein, Untagged, E.coli-50ug

QP13648-50ug 50ug
EUR 201.00

Recombinant Human SUMO2 Protein, Untagged, E.coli-50ug

QP13649-50ug 50ug
EUR 201.00

Recombinant Human SYF2 Protein, His, E.coli-50ug

QP13655-50ug 50ug
EUR 1161.00

Recombinant Human TBC1D22B Protein, His, E.coli-50ug

QP13694-50ug 50ug
EUR 1261.00

Recombinant Human TCL1A Protein, Untagged, E.coli-50ug

QP13711-50ug 50ug
EUR 201.00

Recombinant Human TEK Protein, His, Insect-50ug

QP13720-50ug 50ug
EUR 1261.00

Recombinant Human TFAM Protein, His, E.coli-50ug

QP13726-50ug 50ug
EUR 1261.00

Recombinant Human TFPI2 Protein, His, Insect-50ug

QP13730-50ug 50ug
EUR 1261.00

Recombinant Mouse TLR2 Protein, His, Insect-50ug

QP13759-50ug 50ug
EUR 1261.00

Recombinant Human TLR3 Protein, His, Insect-50ug

QP13760-50ug 50ug
EUR 1261.00

Recombinant Human TNIP1 Protein, His, E.coli-50ug

QP13774-50ug 50ug
EUR 1161.00

Recombinant Human TNNI3 Protein, Untagged, E.coli-50ug

QP13777-50ug 50ug
EUR 201.00

Recombinant Human TnT Protein, Untagged, Human-50ug

QP13779-50ug 50ug
EUR 201.00

Recombinant Rat Tpc1808 Protein, His, E.coli-50ug

QP13791-50ug 50ug
EUR 201.00

Recombinant Human TYRP1 Protein, His, Insect-50ug

QP13846-50ug 50ug
EUR 1261.00

Recombinant Human UBA2 Protein, His, Insect-50ug

QP13849-50ug 50ug
EUR 1261.00

Recombinant Human UBE2L3 Protein, Untagged, E.coli-50ug

QP13863-50ug 50ug
EUR 201.00

Recombinant other UPP1ella Protein, Untagged, E.coli-50ug

QP13901-50ug 50ug
EUR 201.00

Recombinant Mouse VAMP2 Protein, His, E.coli-50ug

QP13916-50ug 50ug
EUR 1261.00

Recombinant Human VHL Protein, His, E.coli-50ug

QP13934-50ug 50ug
EUR 201.00

Recombinant Human VPS28 Protein, Untagged, E.coli-50ug

QP13943-50ug 50ug
EUR 201.00

Recombinant Human VPS29 Protein, His, E.coli-50ug

QP13945-50ug 50ug
EUR 1161.00

Recombinant Mouse EFNB2 Protein, His, Insect-50ug

QP11755-50ug 50ug
EUR 1261.00

Recombinant Mouse EIF4E Protein, His, E.coli-50ug

QP11772-50ug 50ug
EUR 1261.00

Recombinant Human EIF4EBP1 Protein, His, E.coli-50ug

QP11773-50ug 50ug
EUR 201.00

Recombinant Mouse En Protein, His, Insect-50ug

QP11785-50ug 50ug
EUR 1261.00

Recombinant Mouse EPHA4 Protein, His, Insect-50ug

QP11797-50ug 50ug
EUR 1261.00

Recombinant Mouse EPHB4 Protein, His, Insect-50ug

QP11799-50ug 50ug
EUR 1261.00

Recombinant Human ETNK2 Protein, His, E.coli-50ug

QP11813-50ug 50ug
EUR 1261.00

Recombinant Human EXOSC3 Protein, His, E.coli-50ug

QP11819-50ug 50ug
EUR 1261.00

Recombinant Human FCAR Protein, His, Insect-50ug

QP11857-50ug 50ug
EUR 1261.00

Recombinant Human FGFR4 Protein, His, Insect-50ug

QP11877-50ug 50ug
EUR 1261.00

Recombinant Human FUCA1 Protein, His, Insect-50ug

QP11909-50ug 50ug
EUR 1261.00

Recombinant Human GAD2 Protein, His, Insect-50ug

QP11919-50ug 50ug
EUR 663.00

Recombinant Human GADD45B Protein, Untagged, E.coli-50ug

QP11920-50ug 50ug
EUR 201.00

Recombinant Mouse GALNT1 Protein, His, Insect-50ug

QP11928-50ug 50ug
EUR 1261.00

Recombinant other Ganirelix Protein, Untagged, E.coli-50ug

QP11930-50ug 50ug
EUR 218.00

Recombinant Human GAS7 Protein, His, E.coli-50ug

QP11934-50ug 50ug
EUR 201.00

Recombinant Human GDA Protein, Untagged, E.coli-50ug

QP11946-50ug 50ug
EUR 1161.00

Recombinant Rat GFRA1 Protein, His, Insect-50ug

QP11958-50ug 50ug
EUR 1261.00

Recombinant Denis GH Protein, Untagged, E.coli-50ug

QP11961-50ug 50ug
EUR 201.00

Recombinant Rabbit GH Protein, Untagged, E.coli-50ug

QP11963-50ug 50ug
EUR 201.00

Recombinant other GLB1 Protein, Untagged, E.coli-50ug

QP11980-50ug 50ug
EUR 201.00

Recombinant Human GNMT Protein, His, E.coli-50ug

QP12022-50ug 50ug
EUR 201.00

Recombinant Human GPX2 Protein, His, E.coli-50ug

QP12043-50ug 50ug
EUR 1161.00

Recombinant other GroEL Protein, Untagged, E.coli-50ug

QP12051-50ug 50ug
EUR 201.00

Recombinant other GroES Protein, Untagged, E.coli-50ug

QP12052-50ug 50ug
EUR 201.00

Recombinant Human B3GNT2 Protein, His, Insect-50ug

QP11119-50ug 50ug
EUR 1261.00

Recombinant other Batroxobin Protein, Untagged, E.coli-50ug

QP11130-50ug 50ug
EUR 201.00

Recombinant Human BCKDHA Protein, His, E.coli-50ug

QP11137-50ug 50ug
EUR 1261.00

Recombinant Human BHMT Protein, His, E.coli-50ug

QP11159-50ug 50ug
EUR 201.00

Recombinant Human BIN2 Protein, His, Insect-50ug

QP11163-50ug 50ug
EUR 1261.00

Recombinant Human BLVRB Protein, Untagged, E.coli-50ug

QP11170-50ug 50ug
EUR 201.00

Recombinant Human BRAK Protein, His, E.coli-50ug

QP11201-50ug 50ug
EUR 1261.00

Recombinant Human BTD Protein, His, Insect-50ug

QP11209-50ug 50ug
EUR 1161.00

Recombinant Human CABP7 Protein, His, E.coli-50ug

QP11251-50ug 50ug
EUR 1261.00

Recombinant Human CALB1 Protein, Untagged, E.coli-50ug

QP11258-50ug 50ug
EUR 201.00

Recombinant Human CALN1 Protein, His, E.coli-50ug

QP11263-50ug 50ug
EUR 1161.00

Recombinant Human CANX Protein, Untagged, E.coli-50ug

QP11271-50ug 50ug
EUR 201.00

Recombinant Human CD160 Protein, His, Insect-50ug

QP11302-50ug 50ug
EUR 1261.00

Recombinant Porcine CD163 Protein, His, E.coli-50ug

QP11303-50ug 50ug
EUR 201.00

Recombinant Human CD163 Protein, His, Insect-50ug

QP11304-50ug 50ug
EUR 1261.00

Recombinant Human CD1A Protein, His, Insect-50ug

QP11308-50ug 50ug
EUR 1261.00

Recombinant Human CD1D Protein, His, Insect-50ug

QP11310-50ug 50ug
EUR 1261.00

Recombinant Human CD80 Protein, His, Insect-50ug

QP11350-50ug 50ug
EUR 1261.00

Recombinant Human CD90 Protein, His, Insect-50ug

QP11355-50ug 50ug
EUR 1261.00

Recombinant Human CD93 Protein, His, Insect-50ug

QP11357-50ug 50ug
EUR 1161.00

Recombinant Human CDC25A Protein, His, E.coli-50ug

QP11361-50ug 50ug
EUR 1261.00

Recombinant Human CDC42 Protein, Untagged, E.coli-50ug

QP11365-50ug 50ug
EUR 201.00

Recombinant Mouse CDH5 Protein, His, Insect-50ug

QP11371-50ug 50ug
EUR 1261.00

Recombinant Human CGB3 Protein, His, Insect-50ug

QP11399-50ug 50ug
EUR 1261.00

Recombinant Human IGSF8 Protein, His, Insect-50ug

QP12389-50ug 50ug
EUR 1261.00

Recombinant Rat IL11RA Protein, His, Insect-50ug

QP12403-50ug 50ug
EUR 1261.00

Recombinant Human IL17RA Protein, His, Insect-50ug

QP12408-50ug 50ug
EUR 1261.00

Recombinant Human IL18RAP Protein, His, Insect-50ug

QP12411-50ug 50ug
EUR 1261.00

Recombinant Canine IL2 Protein, GST, E.coli-50ug

QP12414-50ug 50ug
EUR 1261.00

Recombinant Human IL3RA Protein, His, Insect-50ug

QP12424-50ug 50ug
EUR 1261.00

Recombinant Human IL7R Protein, His, Insect-50ug

QP12429-50ug 50ug
EUR 1261.00

Recombinant Human ISG15 Protein, Untagged, E.coli-50ug

QP12460-50ug 50ug
EUR 201.00

Recombinant Human JDP2 Protein, His, E.coli-50ug

QP12470-50ug 50ug
EUR 1161.00

Recombinant Human KAAG1 Protein, His, E.coli-50ug

QP12475-50ug 50ug
EUR 1261.00

Recombinant Human KIR2DL4 Protein, His, Insect-50ug

QP12489-50ug 50ug
EUR 1261.00

Recombinant Human KIR3DL2 Protein, His, Insect-50ug

QP12493-50ug 50ug
EUR 1261.00

Recombinant Human KLK13 Protein, His, E.coli-50ug

QP12503-50ug 50ug
EUR 1261.00

Recombinant Human KLRC1 Protein, His, Insect-50ug

QP12509-50ug 50ug
EUR 1261.00

Recombinant Human KLRD1 Protein, His, Insect-50ug

QP12511-50ug 50ug
EUR 1261.00

Recombinant Human LGALS14 Protein, His, E.coli-50ug

QP12560-50ug 50ug
EUR 1161.00

Recombinant Human LGALS16 Protein, Untagged, E.coli-50ug

QP12561-50ug 50ug
EUR 201.00

Recombinant Human LIN28B Protein, His, E.coli-50ug

QP12572-50ug 50ug
EUR 1261.00

Recombinant Human LY9 Protein, His, Insect-50ug

QP12603-50ug 50ug
EUR 1161.00

Recombinant Human MAGED1 Protein, His, E.coli-50ug

QP12628-50ug 50ug
EUR 1261.00

Recombinant Human MAP2K2 Protein, His, E.coli-50ug

QP12638-50ug 50ug
EUR 1261.00

Recombinant Human MAPK11 Protein, His, E.coli-50ug

QP12640-50ug 50ug
EUR 1161.00

Recombinant Human MAVS Protein, His, E.coli-50ug

QP12652-50ug 50ug
EUR 1261.00

Recombinant Human MEP1A Protein, His, Insect-50ug

QP12678-50ug 50ug
EUR 1261.00

Recombinant Mouse MEP1B Protein, His, Insect-50ug

QP12679-50ug 50ug
EUR 1261.00

Recombinant Mouse METRN Protein, His, Insect-50ug

QP12686-50ug 50ug
EUR 1261.00

Recombinant Mouse MFGE8 Protein, His, Insect-50ug

QP12692-50ug 50ug
EUR 1261.00

Recombinant Human MOG Protein, His, E.coli-50ug

QP12718-50ug 50ug
EUR 201.00

Recombinant other MSRA Protein, His, E.coli-50ug

QP12742-50ug 50ug
EUR 1161.00

Recombinant other MutY Protein, His, E.coli-50ug

QP12757-50ug 50ug
EUR 1161.00

Recombinant Mouse MYDGF Protein, His, E.coli-50ug

QP12766-50ug 50ug
EUR 1261.00

Recombinant Human NAGA Protein, His, Insect-50ug

QP12785-50ug 50ug
EUR 1261.00

Recombinant Human NDUFAF4 Protein, His, E.coli-50ug

QP12815-50ug 50ug
EUR 1261.00

Recombinant Human NDUFS3 Protein, His, E.coli-50ug

QP12819-50ug 50ug
EUR 1261.00

Recombinant Human NECTIN3 Protein, His, Insect-50ug

QP12824-50ug 50ug
EUR 1261.00

Recombinant Human NHLH1 Protein, His, E.coli-50ug

QP12843-50ug 50ug
EUR 1161.00

Recombinant Human NKIRAS1 Protein, His, E.coli-50ug

QP12848-50ug 50ug
EUR 1261.00

Recombinant Mouse NKp46 Protein, His, Insect-50ug

QP12851-50ug 50ug
EUR 1161.00

Recombinant Mouse NMNAT1 Protein, His, E.coli-50ug

QP12858-50ug 50ug
EUR 1261.00

Recombinant Mouse NRP1 Protein, His, Insect-50ug

QP12883-50ug 50ug
EUR 1261.00

Recombinant Rat NRP1 Protein, His, Insect-50ug

QP12884-50ug 50ug
EUR 1261.00

Recombinant Human NSDHL Protein, His, E.coli-50ug

QP12885-50ug 50ug
EUR 1261.00

Recombinant Human NT5M Protein, His, E.coli-50ug

QP12890-50ug 50ug
EUR 1261.00

Recombinant other NusA Protein, Untagged, E.coli-50ug

QP12911-50ug 50ug
EUR 201.00

Recombinant Mouse OGG1 Protein, His, E.coli-50ug

QP12922-50ug 50ug
EUR 1261.00

Recombinant Human OSTF1 Protein, His, E.coli-50ug

QP12936-50ug 50ug
EUR 201.00

Recombinant Human OVCA2 Protein, His, E.coli-50ug

QP12940-50ug 50ug
EUR 1261.00

Recombinant Porcine OXM Protein, Untagged, E.coli-50ug

QP12942-50ug 50ug
EUR 201.00

Recombinant Human PCYT2 Protein, His, E.coli-50ug

QP12983-50ug 50ug
EUR 1261.00

Recombinant Human PDGFRA Protein, His, E.coli-50ug

QP12993-50ug 50ug
EUR 1261.00

Recombinant Human PDK1 Protein, His, E.coli-50ug

QP12998-50ug 50ug
EUR 201.00

Recombinant Human PEBP1 Protein, Untagged, E.coli-50ug

QP13001-50ug 50ug
EUR 201.00

Recombinant Human RPS27A Protein, Untagged, E.coli-50ug

QP13356-50ug 50ug
EUR 201.00

Recombinant Human RSG1 Protein, His, E.coli-50ug

QP13364-50ug 50ug
EUR 1261.00

Recombinant Human RXRA Protein, Untagged, E.coli-50ug

QP13378-50ug 50ug
EUR 201.00

Recombinant Human SDC2 Protein, His, Insect-50ug

QP13438-50ug 50ug
EUR 1261.00
These findings may explain the surprisingly good performance of network models that neglect volume scaling. Even though this paper focuses on the implications of the volume growth, the model is generic and might lead to insights in the field of random directed acyclic graphs and their applications.
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