Beta-Skeleton Analysis of the Cosmic Web
Feng Fang, Jaime Forero-Romero, Graziano Rossi, Xiao-Dong Li, Long-Long Feng
MMNRAS , 1– ?? (2018) Preprint 7th September 2018 Compiled using MNRAS L A TEX style file v3.0 β -Skeleton Analysis of the Cosmic Web Feng Fang , Jaime Forero-Romero , Graziano Rossi , Xiao-Dong Li ,(cid:63) , Long-Long Feng School of Physics and Astronomy, Sun Yat-Sen University, Guangzhou 510297, P. R. China Departamento de F´ısica, Universidad de los Andes, Cra. 1 No. 18A-10 Edificio Ip, CP 111711, Bogot´a, Colombia Department of Physics and Astronomy, Sejong University, Seoul, 143-747, Korea (cid:63) corresponding author: [email protected]
ABSTRACT
The β -skeleton is a mathematical method to construct graphs from a set of pointsthat has been widely applied in the areas of image analysis, machine learning, visualperception, and pattern recognition. In this work, we apply the β -skeleton to studythe cosmic web. We use this tool on observed and simulated data to identify thefilamentary structures and characterize the statistical properties of the skeleton. Inparticular, we compare the β -skeletons built from SDSS-III galaxies to those obtainedfrom MD-PATCHY mocks, and also to mocks directly built from the Big MultiDark N -body simulation. We find that the β -skeleton is able to reveal the underlying structuresin observed and simulated samples without any parameter fine-tuning. A differentdegree of sparseness can be obtained by adjusting the value of β ; in addition, thestatistical properties of the length and direction of the skeleton connections showa clear dependence on redshift space distortions (RSDs) and galaxy bias. We alsofind that the N -body simulation accurately reproduces the RSD effect in the data,while the MD-PATCHY mocks appear to underestimate its magnitude. Our proof-of-concept study shows that the statistical properties of the β -skeleton can be used toprobe cosmological parameters and galaxy evolution. Key words:
Cosmology: cosmological parameters – observations – large-scale struc-ture of universe; Methods: statistical
The spatial distribution of the nearest galaxies on scales of afew hundred Megaparsecs follows a distinct filamentary mo-tif. This pattern is known as ‘cosmic web’ (Bardeen et al.1986), and it has been observed at different cosmic epochs(de Lapparent et al. 1986; Huchra et al. 2012; Tegmark et al.2004; Guzzo et al. 2014). The search of consistent and stablemethods to define this web-like structure has been the sub-ject of continuous research for the last ∼
40 years, since itsexistence was confirmed in early cosmic maps from galaxyredshift surveys. The cosmic web has also been detected inthe dark matter description provided by cosmological simu-lations – see Libeskind et al. (2018) for a recent review.The cosmic web is usually classified into four differentcomponents: halos, sheets, filaments, and voids. Many al-gorithms are focused in finding the two most prominent webfeatures present in redshift galaxy surveys: voids and fila-ments. Voids are regions with sizes in the range of 20 − −
100 Mpc.The emergence of the cosmic web can be understood asthe interplay of two conditions. First, the initial Gaussianrandom density field; second, its evolution under gravity. Infact, the initial anisotropies in the density field are ampli-fied by gravity to finally become filaments and voids. Thestructure of the cosmic web is thus expected to encode in-formation about the underlying cosmological model: namely,type of initial fluctuations, proportions of different kinds ofmatter, the expansion history of the Universe, and the rulesof gravity. Voids, for instance, can be used as cosmologicalprobes, as their structure is strongly influenced by dark en-ergy (Lee & Park 2009; Bos et al. 2012); and the statisticalisotropy of filaments can be used to perform the Alcock-Paczynski (AP) test (Li et al. 2014a).In this paper, we introduce the β -skeleton as an al-gorithm to characterize the cosmic web. The β -skeletonconcept stems from the fields of computational geometryand geometric graph theory and has been widely applied inthe areas of image analysis, machine learning, visual per-ception, and pattern recognition (Edelsbrunner et al. 1983;Amenta et al. 1998; Zhang & King 2002). In the context of c (cid:13) a r X i v : . [ a s t r o - ph . C O ] S e p Fang, Forero-Romero, Rossi, Li & Feng (2018) web finders, the β -skeleton belongs to a class of algorithmsthat, starting from a set of 3D spatial points, builds a graphdescribing the degree of connectedness. In this aspect, itis similar to the minimum spanning tree (MST) algorithm(Barrow et al. 1985), with the main difference that the res-ulting graph depends on the continous β parameter; it isalso related to web finders that are designed on the basis oftopological persistence, such as DisPerSE (Sousbie 2011).This paper is organized as follows. In Section 2, webriefly introduce the definition and the basic properties ofthe β -skeleton. In Section 3, we describe the Big MultiD-ark Planck (BigMDPL) simulation and the SDSS-III BOSSData Release 12 (DR12) galaxy sample, which are used lateron in the analysis. The application of the β -skeleton stat-istics is presented in Section 4, where we discuss the de-pendence of the skeleton on the values of β , on the red-shift of the various samples, on the redshift-space distortions(RSDs), and on the cosmological volume and AP effects; wealso graphically illustrate the β -skeleton constructed fromSDSS-III BOSS DR12 galaxies, and eventually compare theskeletons obtained from observational data and simulatedcatalogs. Finally, we summarize our findings and concludein Section 5. β -SKELETON: THEORY In what follows, we define the β -skeleton and briefly explainhow it is used to study the statistical properties of the large-scale structure (LSS) of the universe; for more details aboutthe β -skeleton in topology and in geometric graph theory,please refer to Kirkpatrick & Radke (1985); Correa & Lind-strom (2012).For a point set S in a n -dimensional Euclidean space,the β -skeleton defines an edge set so that for any two points p and q in S , those points are considered to be connectedif there is not a third point r in the various empty regions shown in Figure 1 with dotted lines. Specifically: • For 0 < β <
1, the empty region is the intersection ofall the spheres with diameter d pq /β , having p and q on theirboundary. • For β = 1, the empty region is the sphere with diameter d pq . • For β ≥
1, the empty region is defined in two differentways: namely, the
Circle-based definition and the
Lune-based definition (see again Figure 1 for details). In this paper, weadopt the latter one, according to which the empty region R pq is the intersections of two spheres with diameter βd pq and centered at p + β ( q − p ) / q + β ( p − q ) /
2, respectively.The β -skeleton defined above has several interestingmathematical properties. As β varies continuously from 0to ∞ , the constructed graphs change from a complete graphto an empty graph. The special case of β = 1 leads to the socalled ”Gabriel graph”, which is known to contain the Euc-lidean minimum spanning tree. The β -skeleton has severalimportant applications in computational science and graph-ical theory. For example, in image analysis, it was used to minimum spanning tree of a set of n points in the plane wherethe weight of the edge between each pair of points is the Euclideandistance between those two points. Figure 1.
Empty region of the β -skeleton under the Lune-baseddefinition. Left: β <
1, Middle: β =1, Right: β > reconstruct the shape of a two-dimensional object given aset of sample points on the boundary of the object: this isbecause the β = 1 . Circle-based graphs have been provento correctly reconstruct the entire boundary of any smoothsurface, without generating any edges that do not belong tothe boundary – as long as the samples are sufficiently densewith respect to the local curvature of the surface. The β -skeleton has also been applied in machine learning systems,in order to solve geometric classification problems (Zhang &King 2002; Toussaint 2005). In wireless ad hoc networks, forcontrolling the communication complexity, the β -skeletonwas used as a mechanism to choose a subset of the pairsof wireless stations that can communicate with each other(Bhardwaj et al. 2005). In visual perception and pattern re-cognition, it was used to find families of proximity graphs(Ersoy et al. 2011). For more details about the applicationof the β -skeleton, see e.g. Bose et al. (2002); Wang (2008);Lafarge & Alliez (2013). First, we test our method using the BigMDPL simulation.The BigMDPL belongs to the series of MultiDark N -bodysimulations with Planck 2015 cosmology, thoroughly de-scribed in Klypin et al. (2016). It is characterized by abox with 2 . h − Gpc on a side, with 3840 dark matterparticles, providing a mass resolution of 2 . × h − M (cid:12) .The initial conditions, based on primordial Gaussian fluc-tuations, are generated via the Zel’dovich approximationat z init = 100. The cosmology assumed is a flat ΛCDMmodel with Ω m = 0 . b = 0 . σ = 0 . n s = 0 . H = 67 .
77 km s − Mpc − .We then apply the β -skeleton statistics to the BaryonOscillation Spectroscopic Survey (BOSS) DR12 CMASSgalaxy sample. BOSS (Dawson et al. 2012; Smee et al. 2013),is the cosmological counterpart of the Sloan Digital Sky Sur-vey III (SDSS-III; Eisenstein et al. 2011), and it is still one ofthe largest spectroscopic galaxy surveys to date. It has ob-tained spectra and redshifts of about 1 .
37 million galaxiesselected from the SDSS imaging up to z = 0 .
7. The North-ern and Southern Sky footprints cover an area of ∼ , z ≤ .
43 and the CMASScatalog covering the redshift interval 0 . ≤ z ≤ . In experimental testing, β = 1 . , 1– ?? (2018) -Skeleton Analysis et al. 2015). In this work, we only use the CMASS sampleat 0 . ≤ z ≤ .
7, which contains ∼ .
77 million galaxies.In order to compare observational data with N -bodysimulation predictions, we use the MD-PATCHY mocksavailable for the BOSS survey. The MD-PATCHY mocks(Kitaura et al. 2016; Rodr´ıguez-Torres et al. 2016) adoptan halo abundance matching technique to reproduce thetwo- and three-point clustering measurements of BOSS. Theredshift evolution of the biased tracers is matched to thecorresponding observations by applying the aforementionedtechnique in a number of redshift bins, with the resultingmock catalogs being combined together to form a contigu-ous lightcone. The MD-PATCHY mocks are constructed toreproduce the number density, selection function, and sur-vey geometry of the BOSS DR12 catalog; moreover, the two-point correlation function (2PCF) of the observational datais correctly recovered down to a few Mpc scales, in generalwithin 1 σ error (Kitaura et al. 2016). The MD-PATCHYmocks have been carefully tested and subsequently adoptedfor the statistical analysis of BOSS data in a series of works– see for example Alam et al. (2017), and references therein. As an illustrative example, we first apply the β -skeletonstatistics to a set of LSS mock samples using β = 1 , , z = 0 halo catalog of the BigMDPL simulationand apply a mass cut M > × M (cid:12) and a radial cut r cut < h − Mpc. This procedure allows us to create ashell-shaped sample containing 30 ,
000 dark matter halos.In order to make comparisons with an unclustered distri-bution, we also built a random sample with the same size,shape, and number of points as the previous mock realiza-tions.Results of this test are displayed in Figures 2 and 3,where we show the skeletons of the mock samples using β =1 , ,
10, respectively (from top to bottom) – as well as theskeleton of the random sample when β = 3. In all cases,the left panels display a 200 × × h − Mpc slice of thesamples with connections (red lines), while the right panelsshow histograms of the length of the connections L (upperpart) and the cosine of the angle between the line-of-sight(LOS) and the connection line, µ ≡ | cosθ | (lower part).Clearly, the amount of connections is smaller when β islarger. This is evident from the definition for the β -skeletonpresented in Figure 1, which shows an increment of theempty region with β ; namely, the threshold for having twoparticles connected becomes more strict. In particular, when β = 1 we find ∼ ,
000 connections, far more than the num-ber of points of the sample, while we detect only 15 , β = 10.The β -skeleton automatically generates filament-likestructures from the point sample; this is most clearly detec-ted when β = 3, as can be seen in Figure 2. For example, inthe upper-left panel one can notice that ∼
20 galaxies natur-ally arise from a long straight filament-shape structure: thisstructure is then identified, and those galaxies are linked to-gether. The straight line ends at ( x, y ) ≈ (25 , h − Mpc, while the structure continues and extends up to y = 60 h − Mpc. It then bifurcates at y ≈ h − Mpc, and furtherextends to the left, lower-left, and right side of the graph,forming a larger connected structure which captures ≈ x, y ) ≈ (25 , h − Mpc links together the up-downfilament at its left to the galaxies at the right. Moreover,there are also isolated structures having a relatively smallnumber of group members – see for instance ∼
25 galaxiesdistributed around ( x, y ) ≈ (100 , h − Mpc that form an“A”-shaped structure.Altering the values of β has a strong influence on theoverall shape of the skeleton graphs. For example, the caseof β = 1 roughly corresponds to computing the 2PCF, inthe sense that many connections are generated, regardlessof whether or not those connections lie within a filamentWhen β = 3, the set of structures generated is much closerto the observed cosmic web, meaning that the number ofconnections is comparable to the number of actual galax-ies. For β = 10, one gets a very sparse graph as expected,since only the small and relatively isolated compact groupsof galaxies are identified and connected.The statistical properties of the connection length L also vary with β . For larger values of β , L gets smaller andappears to be more concentrated – this is because, due to atight threshold, it is difficult to connect two points separatedby a large distance. From the figure, we infer that the meanlength is ¯ L = 6 . , . , . h − Mpc when β = 1 , , β = 3, we then compare the results obtained fromthe mock samples with those derived from the unclustered(random) distribution. As expected, we find that the randomsample exhibits “structures” chaotic in shape; moreover, dueto a lack of compact structures, the distribution of L inferredfrom the random sample has a mean ¯ L = 4 .
55, a value muchlarger than those obtained from the mock samples.Finally, as shown in all the bottom right panels of Fig-ures 2 and 3, we find that µ ∼ . Next, we study in detail the statistical properties of the β -skeletons constructed from N -body simulations. We analyze4 BigMDPL snapshots at redshifts 0 , . , .
6, and 0 .
9, re-spectively, and consider both cases with and without RSDeffects. Moreover, we impose a mass cut
M < × M (cid:12) and a radial cut r < h − Mpc, yielding a number ofgalaxies N gal = 3.85, 3.37, 2.71, 2.07 (in units of millions)at those 4 redshifts, respectively.Results are displayed in Figure 4. Specifically, theupper-right panel shows the histogram of the connectinglength L , assuming that the real space positions of galaxiesare used to construct the skeleton (i.e., no RSD involved).With this assumption, our main findings are summarized asfollows: • The distribution of the connecting length peaks at1 . − . h − Mpc. This represents the typical separation
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0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169̄μ = 0̄49769 ± 0̄00169 x (h −1 Mpc) y ( h − M p c ) β = 10 L (h −1 Mpc) ̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078̄L = 1̄97396 ± 0̄01078 μ ̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235̄μ = 0̄49603 ± 0̄00235 Figure 2.
An illustrative example . Application of the β -skeleton statistics to a set of LSS mock samples when β = 1 , ,
10, respectively.In the figure, the left panels show the skeletons of the mock samples for different values of β , while the right panels present the statisticsof the length of the connections (upper parts) and the orientations of those connections (lower parts). See the main text for more details.MNRAS , 1– ?? (2018) -Skeleton Analysis x (h −1 Mpc) y ( h − M p c ) β = 3, ran L (h −1 Mpc) ̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975̄L = 4̄54963 ± 0̄00975 μ ̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169̄μ = 0̄49876 ± 0̄00169 Figure 3.
Same as the previous figure when β = 3, but for an unclustered (random) distribution. length between galaxies in the skeleton. Above (below) thepeak scale, N decreases with increasing (decreasing) L ; asecondary peak appears at 0 . − . h − Mpc, due to thefact that there is a large number of compact clusters at thisscale. • As the redshift increases, the number of connectionsdecreases with decreasing N gal . Again, the total number ofconnections, which is found to be 3 . , . , . , .
00 at z =0 . , . , . , .
0, respectively, scales with N gal . • The 4 distributions (indicated in the panel with differ-ent colors) merge at L ≈ h − Mpc. Above this scale, the z = 0 . N gal is significantly smaller com-pared to the other three samples), which is a clear signal thatthe constructed structures in this sparse sample have largersizes – namely, at lower redshifts, objects become more com-pact and the distribution shifts to smaller L as structuresgrow.The upper-left panel in Figure 4 displays the L -distribution, but now using the redshift space positions ofthe same galaxies considered before. In this case, the pecu-liar velocity of galaxies perturbs their observed redshifts via∆ z = (1 + z ) v LOS c , (1)where v LOS is the line-of-sight (LOS) component of the velo-city. The distortion of z leads to a corresponding distortionin the inferred galaxy distances, known as the RSD effect.At small scales ( (cid:46) h − Mpc), this leads to the finger of god(FOG) feature (Jackson 1972) (i.e., a stretch of structuresalong the LOS) due to chaotic small-scale motions of galax-ies in the non-linear regime. At large scales ( (cid:38) h − Mpc),the RSD effect is known as the ‘Kaiser effect’ (Kaiser 1987)(i.e., a compression of structures along the LOS), due to thecoherent motions of galaxies driven by gravity.Considering the previously reported measurements of L ,we can infer that the skeletons constructed from the BigM-DPL simulation are mainly affected by the small-scale FOGeffect. As a consequence, the number of short connections characterized by L (cid:46) h − Mpc is heavily suppressed, be-cause of the stretch of distances among galaxies due to theFOG feature. Also, the secondary peak – found in the casewhere no RSD are considered – now disappears. The dis-tribution still peaks around 1 . h − Mpc, but the height is ∼
20% higher than the one found in the no RSD case; thisis because there is an extra contribution from the ‘spikes’created by the FOG effect.Finally, the lower panel in Figure 4 shows the histogramof µ at those four different redshifts previously specified,when RSD effects are present. As expected, we find a non-flat distribution due to anisotropies induced by RSDs; theFOG leads to a sharp increment of N at µ →
1, and theeffect is stronger at lower redshifts.
We then consider the effect of cosmological parameters onthe β -skeleton statistics. To this end, suppose we are probingboth the shape and volume of a celestial object by measuringits redshift span ∆ z and angular size ∆ θ . We can computeits LOS dimensions in the radial (∆ r (cid:107) ) and transverse (∆ r ⊥ )directions using the relations:∆ r (cid:107) = cH ( z ) ∆ z, ∆ r ⊥ = (1 + z ) D A ( z )∆ θ, (2)where H is the Hubble parameter and D A is the angulardiameter distance. For a flat ΛCDM model with constantdark energy equation of state (DE EoS) parameter w , H and D A are simply expressed by: H ( z ) = H (cid:113) Ω m a − + (1 − Ω m )(1 + z ) w ) ,D A ( z ) = 11 + z r ( z ) = 11 + z (cid:90) z dz (cid:48) H ( z (cid:48) ) , (3)with H the present value of the Hubble constant, and r ( z )the comoving distance.If an incorrect set of cosmological parameters is chosenin the conversion defined by Equations (2) and (3), then theinferred ∆ r (cid:107) and ∆ r ⊥ would be both incorrect, resulting ina distorted shape (this is known as the ‘AP effect’) and in a MNRAS , 1– ????
We then consider the effect of cosmological parameters onthe β -skeleton statistics. To this end, suppose we are probingboth the shape and volume of a celestial object by measuringits redshift span ∆ z and angular size ∆ θ . We can computeits LOS dimensions in the radial (∆ r (cid:107) ) and transverse (∆ r ⊥ )directions using the relations:∆ r (cid:107) = cH ( z ) ∆ z, ∆ r ⊥ = (1 + z ) D A ( z )∆ θ, (2)where H is the Hubble parameter and D A is the angulardiameter distance. For a flat ΛCDM model with constantdark energy equation of state (DE EoS) parameter w , H and D A are simply expressed by: H ( z ) = H (cid:113) Ω m a − + (1 − Ω m )(1 + z ) w ) ,D A ( z ) = 11 + z r ( z ) = 11 + z (cid:90) z dz (cid:48) H ( z (cid:48) ) , (3)with H the present value of the Hubble constant, and r ( z )the comoving distance.If an incorrect set of cosmological parameters is chosenin the conversion defined by Equations (2) and (3), then theinferred ∆ r (cid:107) and ∆ r ⊥ would be both incorrect, resulting ina distorted shape (this is known as the ‘AP effect’) and in a MNRAS , 1– ???? (2018) Fang, Forero-Romero, Rossi, Li & Feng (2018)
L (h −1 Mpc) N With RSD z=0.0z=0.3z=0.6z=0.9
L (h −1 Mpc) No RSD z=0.0z=0.3z=0.6z=0.9 μ N With RSD z=0.0z=0.3z=0.6z=0.9
Figure 4.
Histograms of the connecting lengths L with (top-left panel) and without (top-right panel) RSD effects. Those lengths areused to construct the corresponding skeleton structures, as explained in the main text. The lower panel shows the histogram of thedirections of the connections µ at 4 different redshifts – as indicated in the plot with different colors – when RSD effects are included. wrongly estimated volume (this is termed as ‘volume effect’)of the cosmological object. We can describe the magnitudeof this combined effect via the relations:[∆ r (cid:107) / ∆ r ⊥ ] wrong [∆ r (cid:107) / ∆ r ⊥ ] true = [ D A ( z ) H ( z )] true [ D A ( z ) H ( z )] wrong , (4)[∆ r (cid:107) (∆ r ⊥ ) ] wrong [∆ r (cid:107) (∆ r ⊥ ) ] true = Vol wrong Vol true = [ D A ( z ) /H ( z )] wrong [ D A ( z ) /H ( z )] true , (5)where ‘true’ and ‘wrong’ denote the values of those measuredquantities in the actual (‘true’) cosmology and in the incor-rectly assumed cosmology, respectively. From the AP andthe volume effects, we can therefore constrain the two quant-ities D A ( z ) H ( z ) and D A ( z ) /H ( z ). Clearly, these two effectswill impact the statistical properties of the β -skeleton, whichis sensitive to both the number density and the anisotropyof the cosmological sample in question. In order to quantify the sensitivity of the AP andvolume effects on the β -skeleton, we next apply the β -skeleton statistics to the z = 0 . different cosmologies . Namely,we adopt five cosmological models characterized by Ω m =0 . w = − . , − . , − .
5, and Ω m = 0 . , . w = − .
0, and infer the actual positions of the galaxysample using those five cosmologies in turn. For all thosecases, we then analyze the statistical properties of the con-nection length L and of the cosine of the orientation angle µ . Our main results are shown in Figures 5 and 6. Specific-ally, Figure 5 displays the histograms of L : the left panelpresents the statistical distribution in redshift space (withRSDs), while the right panel shows the analogous distribu-tion but in real space (no RSDs). Also in these cases, wefind similar properties as those highlighted in Figure 4 (e.g., MNRAS , 1– ?? (2018) -Skeleton Analysis L (h −1 Mpc) N With RSD Ω m = 0.3071, w = − 1.0Ω m = 0.3071, w = − 1.5Ω m = 0.3071, w = − 0.5Ω m = 0.2, w = − 1.0Ω m = 0.4, w = − 1.0 L (h −1 Mpc) No RSD Ω m = 0.3071, w = − 1.0Ω m = 0.3071, w = − 1.5Ω m = 0.3071, w = − 0.5Ω m = 0.2, w = − 1.0Ω m = 0.4, w = − 1.0 Figure 5.
Sensitivity of the AP and volume effects on the β -skeleton: connecting length . Distribution of L in different cosmologicalmodels defined by Ω m and w , as indicated in the various panels. The β -skeleton statistics is applied to the BigMDPL simulation snapshotat z = 0 .
6, when RSDs are considered (redshift space – left panel) or excluded (real space – right panel). Cosmological effects due tosimultaneous variations in Ω m and w are clearly detected. μ N No RSD Ω m = 0.3071, w = − 1.0Ω m = 0.3071, w = − 1.5Ω m = 0.3071, w = − 0.5Ω m = 0.2, w = − 1.0Ω m = 0.4, w = − 1.0 Figure 6.
Sensitivity of the AP and volume effects on the β -skeleton: connecting direction . Distribution of µ in different cos-mological models as in the right panel of the previous figure (noRSD effects), but now for the connecting direction. See the maintext for more details. two peaks when RSDs are not present, and one peak at1 . h − Mpc if RSDs are added).Moreover, cosmological effects of varying Ω m and w areclearly detected: the two cosmologies with w = − . m = 0 . L shifted to larger length; on the other hand,the overall LSS is stretched along the LOS because of theAP effect. Hence, the distribution of µ is enhanced (sup-pressed) in the region where µ > . µ < . In the other two considered cosmologies, the effect is theopposite: a shrinking of the volume size shifts L to smallerscales, and the compression of structures along the LOS tiltsthe distribution of µ , as expected. In the previous analysis we imposed a fixed mass-cut toall the BigMDPL snapshots considered, namely
M > × M (cid:12) . We now explore the effect of a different mass-cuton the β -skeleton statistics. To this end, Figure 7 shows res-ults of varying the mass cut (indicated with different colorlines) at z = 0, when RSDs are also accounted for. In par-ticular, we highlight the following findings: • When selecting galaxies in the mass intervals [4 × − × ] M (cid:12) , [6 × − × ] M (cid:12) , [2 × − × ] M (cid:12) ,and [5 × − × ] M (cid:12) , we obtain N gal = [6101837,3142136, 2318205, 706430] and L = [3 . , . , . , .
23] –respectively. The relation ¯ L ∝ N − / holds well. • Samples characterized by a relatively smaller galaxymass are dominated by satellite galaxies, and therefore theyare more affected by the small-scale FOG effect – resultingin a significant peak around µ (cid:39) • On the contrary, samples with relatively larger galaxymass are more dominated by central galaxies. Hence, theyare more affected by the Kaiser effect, and thus present amore significant tilt when µ < .
8. The peak near µ ∼ See Figure 1 of Li et al. (2014b, 2015) for a clearer explanationson the volume and AP effects in cosmologies with incorrect Ω m or w values.MNRAS , 1– ????
8. The peak near µ ∼ See Figure 1 of Li et al. (2014b, 2015) for a clearer explanationson the volume and AP effects in cosmologies with incorrect Ω m or w values.MNRAS , 1– ???? (2018) Fang, Forero-Romero, Rossi, Li & Feng (2018)
L (h −1 Mpc) N With RSD − 6 × 10 M ⊙ − 8 × 10 M ⊙ − 5 × 10 M ⊙ − 1 × 10 M ⊙ μ ̄ N With RSD − 6 × 10 M ⊙ − 8 × 10 M ⊙ − 5 × 10 M ⊙ − 1 × 10 M ⊙ Figure 7.
Mass-cut effects on the β -skeleton statistics . [Left] Distribution of the connecting length L as a function of different masscuts, as indicated in the panel with different line-colors, when RSDs are included and z = 0. [Right] Same as in the left panel, but nowfor the connecting direction µ . Finally, we apply the β -skeleton statistics to observationalgalaxy data, obtained from the SDSS-III BOSS Data Re-lease 12 (DR12); in particular, we consider only the CMASSgalaxy sample within 0 . ≤ z ≤ .
7, which contains ∼ . . < z < .
48 – the specific redshift rangehas been chosen just for visualization purposes, and in theplot different colors indicate galaxies with different angu-lar directions and distances. The right panel is a sub-patchenlargement of the left panel, where the coordinate cut isdefined by 170 < RA <
210 and 30 < DEC < β -skeleton as inferred fromthe SDSS-III BOSS CMASS galaxy sample, and of 4 mockMD-PATCHY realizations that are constructed to mimicthe BOSS CMASS sample, plus a BigMDPL snapshot at z = 0 .
6. The main findings are as follows: • The observed and simulated distributions of the con-nection lengths L are in good agreement. They both peakat L ∼ . h − Mpc, and decrease outside of this interval. • The MD-PATCHY mocks generally underestimate theFOG effect, a fact evident if one looks towards the µ → • The µ distribution of BOSS galaxies is much closer tothe one derived from the BigMDPL mock at z = 0 .
6, indicat-ing that N -body simulations are capable of well-reproducingthe RSD effect present in the data. In this work, we performed a first investigation of the applic-ation of β -skeleton statistics to cosmic web data. We use theBigMDPL simulation as a testing sample, and study how the constructed skeleton depends on the values of β , redshifts,RSD, AP and volume effects, and different and mass cuts.We find a significant variation of the length and direction ofthe cosmic web connections under different parameters andassumptions.We then apply the β -skeleton method to SDSS-IIIBOSS DR12 CMASS galaxies, and compare our measure-ments with MD-PATCHY mocks. We find that the N -bodysample provides a rather similar µ -distribution to the oneof the data, implying that RSD effects of the sample are ac-curately reconstructed. On the contrary, the MD-PATCHYmocks appear to underestimate the magnitude of the FOGeffect, although they are designed to correctly reproduce the2- and 3-point correlation functions of the data.The β -skeleton clearly reveals the underlying structuresencoded in the sample of points. From its definition, we seethat it does not require us to pre-select a specific scale (suchas the linking length in the FoF algorithm). One can in factadjust the value of β , and obtain a skeleton-like structurewith different magnitudes of sparseness. Furthermore, thestatistical properties of the β -skeleton depend on the RSDeffect, on the AP and volume effects, and on galaxy bias.Hence, in turn they could be used as a statistical tool tocharacterize the magnitude of these effects. A standard cosmological analysis generally involves the com-putation of the 2PCF, and of 2-point-related statistics. Incomputing the 2PCF, one considers all the possible pairs ofgalaxies (restricted to some specific scale), and study theirmain clustering properties. Instead, the β -skeleton statisticsfocuses only on the small fraction of pairs which traces thestructure; hence, the physical information is actually con-centrated on a subset of galaxies. Also, the computation ofthe β -skeleton is much faster than the 2PCF, so it can beused as a complementary fast statistical tool to study thebasic properties of a given sample.Although the pairs that define the β -skeleton constitute MNRAS , 1– ?? (2018) -Skeleton Analysis −500 0 500 1000 −1200−1000−800−600−400−2000200020040060080010001200
200 300 400 500 600 700 800 −1100−1000−900−800−700−600−500650700750800850900950
Figure 8.
Application of the β -skeleton to observational data . [Left] Visualization of the skeleton mapped by SDSS-III CMASS BOSSgalaxies within the redshift range 0 . < z < .
48 in the Northern sky (in units of h − Mpc). Different colors indicate galaxies withdifferent angular directions and distances. [Right]. Zoom into a sub-patch of the left panel, which clearly shows the structure of theobserved β -skeleton. See the main text for more details. a subset of those involved in the 2PCF calculations, onecannot conclude that the information derived from the β -skeleton analysis is just a subset of the one inferred from2PCF measurements. For example, Figure 9 already revealsthat the MD-PATCHY mocks, constructed to reproduce the2PCF of the data, have instead a rather different β -skeletonstatistics from the actual data. This is also one main reason to pursue a β -skeleton ana-lysis: the 2-point statistics, although powerful, essentiallycompresses all the LSS information into histograms, whilethe cosmic web presents a much richer and complex struc-ture that can only be revealed with higher-order, more de-tailed analysis. This work is a first attempt to apply the β -skeleton statist-ics to describe the cosmic web. Of course, our study can befurther expanded in several directions. For instance, in thispaper we only focused on the distribution of L and µ , in or-der to characterize the size and anisotropy of the LSS, butadditional quantities can be used in future investigations.An example is represented by the number of connectionslinked at every galaxy, which allows us to study and weightthe ‘knots’ (which connect together different filaments). An-other possibility is to study how the connection lengths ofgalaxies differ depending on their environment. Namely, if The β -skeleton distribution can be thought as a ‘weighted’2PCF statistics, in which galaxy pairs are weighted by 0 or 1,respectively, based on a graphical criterion. One may be ableto extract additional information from this particular weightingscheme. they are within a homogeneous structure such as cluster,their connection length values should be statistically closeto unity, while for galaxies lying at the boundary of clustersand filaments we expect those values to deviate from unity;the magnitude of the deviation describes how sharp the LSSare transformed from cluster-like to filament-like structures.Another possibility is to compare the β -skeleton statist-ics with other cosmic web structure finders – e.g., friends-of-friends (FoF) (Davis et al. 1985), density-based techniques(Klypin & Holtzman 1997; Springel et al. 2001; Knollmann& Knebe 2009), T-web (Hahn et al. 2007; Forero-Romeroet al. 2009), V-web (Hoffman et al. 2012; Forero-Romeroet al. 2014), etc. Interesting points to address include thefollowing: finding a value of β that yields a cosmic web real-ization similar to the one obtained with a different method;finding β for which the connections best trace the filament-like structures identified by a different realization of the cos-mic web; using an alternative method to classify the cosmicweb into clusters, filaments, walls, and voids, and study thestatistical properties of the β -skeleton in those regions; usingthe β -skeleton statistics to study how the RSD effect variesin cluster, filament, wall, and void regions; etc.Moreover, the β -skeleton can have several other applic-ations in galaxy clustering analysis – being fast to computeand particularly sensitive to clustering properties. For ex-ample, it can be used to assess how well mocks can reproducethe properties of the observational sample, since it is sensit-ive to the strength and anisotropy of clustering. It can alsobe directly used to derive quantitative constraints on cosmo-logical parameters, as the β -skeleton statistics are sensitiveto the AP, volume, and RSD effects. This could be quanti-fied by a β -correlation that compares the length of skeletonwedges built from data, randoms, and joint data/randoms: MNRAS , 1– ????
48 in the Northern sky (in units of h − Mpc). Different colors indicate galaxies withdifferent angular directions and distances. [Right]. Zoom into a sub-patch of the left panel, which clearly shows the structure of theobserved β -skeleton. See the main text for more details. a subset of those involved in the 2PCF calculations, onecannot conclude that the information derived from the β -skeleton analysis is just a subset of the one inferred from2PCF measurements. For example, Figure 9 already revealsthat the MD-PATCHY mocks, constructed to reproduce the2PCF of the data, have instead a rather different β -skeletonstatistics from the actual data. This is also one main reason to pursue a β -skeleton ana-lysis: the 2-point statistics, although powerful, essentiallycompresses all the LSS information into histograms, whilethe cosmic web presents a much richer and complex struc-ture that can only be revealed with higher-order, more de-tailed analysis. This work is a first attempt to apply the β -skeleton statist-ics to describe the cosmic web. Of course, our study can befurther expanded in several directions. For instance, in thispaper we only focused on the distribution of L and µ , in or-der to characterize the size and anisotropy of the LSS, butadditional quantities can be used in future investigations.An example is represented by the number of connectionslinked at every galaxy, which allows us to study and weightthe ‘knots’ (which connect together different filaments). An-other possibility is to study how the connection lengths ofgalaxies differ depending on their environment. Namely, if The β -skeleton distribution can be thought as a ‘weighted’2PCF statistics, in which galaxy pairs are weighted by 0 or 1,respectively, based on a graphical criterion. One may be ableto extract additional information from this particular weightingscheme. they are within a homogeneous structure such as cluster,their connection length values should be statistically closeto unity, while for galaxies lying at the boundary of clustersand filaments we expect those values to deviate from unity;the magnitude of the deviation describes how sharp the LSSare transformed from cluster-like to filament-like structures.Another possibility is to compare the β -skeleton statist-ics with other cosmic web structure finders – e.g., friends-of-friends (FoF) (Davis et al. 1985), density-based techniques(Klypin & Holtzman 1997; Springel et al. 2001; Knollmann& Knebe 2009), T-web (Hahn et al. 2007; Forero-Romeroet al. 2009), V-web (Hoffman et al. 2012; Forero-Romeroet al. 2014), etc. Interesting points to address include thefollowing: finding a value of β that yields a cosmic web real-ization similar to the one obtained with a different method;finding β for which the connections best trace the filament-like structures identified by a different realization of the cos-mic web; using an alternative method to classify the cosmicweb into clusters, filaments, walls, and voids, and study thestatistical properties of the β -skeleton in those regions; usingthe β -skeleton statistics to study how the RSD effect variesin cluster, filament, wall, and void regions; etc.Moreover, the β -skeleton can have several other applic-ations in galaxy clustering analysis – being fast to computeand particularly sensitive to clustering properties. For ex-ample, it can be used to assess how well mocks can reproducethe properties of the observational sample, since it is sensit-ive to the strength and anisotropy of clustering. It can alsobe directly used to derive quantitative constraints on cosmo-logical parameters, as the β -skeleton statistics are sensitiveto the AP, volume, and RSD effects. This could be quanti-fied by a β -correlation that compares the length of skeletonwedges built from data, randoms, and joint data/randoms: MNRAS , 1– ???? (2018) Fang, Forero-Romero, Rossi, Li & Feng (2018)
L (h −1 Mpc) N MDPATCHYBOSS DR12 galaxies μ N MDPATCHYBOSS DR12 galaxies μ N BigMDPL simulationBOSS DR12 galaxies
Figure 9.
Comparisons between observed and simulated β -skeleton statistics . [Top left] Connection length distributions as measuredfrom SDSS-III BOSS galaxy data (green solid line), and as derived from the Patchy mocks (dashed yellow line). [Top right] Same asin the left panel (also with identical line styles), but now for the distribution of the orientation directions; note that the Patchy mocksgenerally underestimate the FOG effect. [Bottom] Distribution of µ for SDSS-III BOSS galaxies (solid green line), and for the BigMDPLmock at z = 0 . N -body simulations arecapable of well-reproducing the RSD effect. that function can be defined in such a way that in the limit β → β -skeleton stat-istics to study the LSS, but this method can be refined anddeveloped further along with other techniques in order tobetter characterize the properties of the cosmic web, andextract useful cosmological information. ACKNOWLEDGEMENTS
J.E. F-R acknowledges support from COLCIENCIAS Con-tract No. 287-2016, Project 1204-712-50459. G.R. acknow-ledges support from the National Research Foundation ofKorea (NRF) through Grant No. 2017R1E1A1A01077508funded by the Korean Ministry of Education, Science andTechnology (MoEST), and from the faculty research fundof Sejong University in 2018. F.L.L. acknowledges supportfrom Key Program of National Natural Science Foundationof China (NFSC) through grant 11733010 and 11333008, and the State Key Development Program for Basic Research ofChina (2015CB857000).We greatly acknowledge Changbom Park for many help-ful discussions.
References
Alam S., et al., 2017, Monthly Notices of the Royal AstronomicalSociety, 470, 2617Amenta N., Bern M., Eppstein D., 1998, Graphical models andimage processing, 60, 125Bardeen J. M., Bond J. R., Kaiser N., Szalay A. S., 1986, ApJ,304, 15Barrow J. D., Bhavsar S. P., Sonoda D. H., 1985, MNRAS, 216,17Bhardwaj M., Misra S., Xue G., 2005, in High PerformanceSwitching and Routing, 2005. HPSR. 2005 Workshop on. pp371–375Bos E. G. P., van de Weygaert R., Dolag K., Pettorino V., 2012,MNRAS, 426, 440 MNRAS , 1– ?? (2018) -Skeleton Analysis Bose P., Devroye L., Evans W., Kirkpatrick D., 2002, in LatinAmerican Symposium on Theoretical Informatics. pp 479–493Correa C. D., Lindstrom P., 2012, in Proceedings of the 18thACM SIGKDD international conference on Knowledge dis-covery and data mining. pp 1330–1338Davis M., Efstathiou G., Frenk C. S., White S. D., 1985, TheAstrophysical Journal, 292, 371Dawson K. S., et al., 2012, The Astronomical Journal, 145, 10Edelsbrunner H., Kirkpatrick D., Seidel R., 1983, IEEE Transac-tions on information theory, 29, 551Ersoy O., Hurter C., Paulovich F., Cantareiro G., Telea A., 2011,IEEE Transactions on Visualization and Computer Graphics,17, 2364Forero-Romero J., Hoffman Y., Gottl¨ober S., Klypin A., YepesG., 2009, Monthly Notices of the Royal Astronomical Society,396, 1815Forero-Romero J. E., Contreras S., Padilla N., 2014, MonthlyNotices of the Royal Astronomical Society, 443, 1090Guzzo L., et al., 2014, A&A, 566, A108Hahn O., Porciani C., Carollo C. M., Dekel A., 2007, MonthlyNotices of the Royal Astronomical Society, 375, 489Hoffman Y., Metuki O., Yepes G., Gottl¨ober S., Forero-RomeroJ. E., Libeskind N. I., Knebe A., 2012, Monthly Notices ofthe Royal Astronomical Society, 425, 2049Huchra J. P., et al., 2012, ApJS, 199, 26Jackson J., 1972, Monthly Notices of the Royal Astronomical So-ciety, 156, 1PKaiser N., 1987, Monthly Notices of the Royal Astronomical So-ciety, 227, 1Kirkpatrick D. G., Radke J. D., 1985, in , Vol. 2, Machine Intel-ligence and Pattern Recognition. Elsevier, pp 217–248Kitaura F.-S., et al., 2016, Monthly Notices of the Royal Astro-nomical Society, 456, 4156Klypin A., Holtzman J., 1997, arXiv preprint astro-ph/9712217Klypin A., Yepes G., Gottl¨ober S., Prada F., Hess S., 2016,Monthly Notices of the Royal Astronomical Society, 457, 4340Knollmann S. R., Knebe A., 2009, The Astrophysical JournalSupplement Series, 182, 608Lafarge F., Alliez P., 2013, in Computer Graphics Forum. pp 225–234Lee J., Park D., 2009, ApJ, 696, L10Li X.-D., Park C., Forero-Romero J. E., Kim J., 2014a, ApJ, 796,137Li X.-D., Park C., Forero-Romero J. E., Kim J., 2014b, The As-trophysical Journal, 796, 137Li X.-D., Park C., Sabiu C. G., Kim J., 2015, Monthly Notices ofthe Royal Astronomical Society, 450, 807Libeskind N. I., et al., 2018, MNRAS, 473, 1195Reid B., et al., 2015, Monthly Notices of the Royal AstronomicalSociety, 455, 1553Rodr´ıguez-Torres S. A., et al., 2016, Monthly Notices of the RoyalAstronomical Society, 460, 1173Smee S. A., et al., 2013, The Astronomical Journal, 146, 32Sousbie T., 2011, MNRAS, 414, 350Springel V., White S. D., Tormen G., Kauffmann G., 2001,Monthly Notices of the Royal Astronomical Society, 328, 726Tegmark M., et al., 2004, ApJ, 606, 702Toussaint G., 2005, International Journal of Computational Geo-metry & Applications, 15, 101Wang Y., 2008, in , Wireless sensor networks and applications.Springer, pp 113–147Zhang W., King I., 2002, in Neural Information Processing, 2002.ICONIP’02. Proceedings of the 9th International Conferenceon. pp 1423–1427de Lapparent V., Geller M. J., Huchra J. P., 1986, ApJ, 302, L1van de Weygaert R., 2016, in van de Weygaert R.,Shandarin S., Saar E., Einasto J., eds, IAU SymposiumVol. 308, The Zeldovich Universe: Genesis and Growth of the Cosmic Web. pp 493–523 ( arXiv:1611.01222 ),doi:10.1017/S1743921316010504MNRAS , 1– ????