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Research ArticleExploring Structural Characteristics of Lattices in Real World

Yu Chen 1 Jinguo You 12 Benyuan Zou1 Guoyu Gan1 Ting Zhang3

and Lianyin Jia 12

1Kunming University of Science and Technology Yunnan 650504 China2Computer Technology Application Key Lab of Yunnan Province Yunnan 650504 China3Jiaozuo University Henan 454003 China

Correspondence should be addressed to Jinguo You jgyou126com

Received 19 September 2019 Accepted 26 December 2019 Published 21 January 2020

Academic Editor Rongqing Zhang

Copyright copy 2020YuChen et alis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

ere are two important models for data analysis and knowledge system data cube lattices and concept lattices ey bothessentially have lattice structures which are actually irregular in our real world However their structural characteristics andrelationship are not yet clear To the best of our knowledge no work has paid enough attention to this challenging issue from theperspective of graph data in spite of the importance of structures in lattice data In this paper we rst tackle the structural statisticsof lattice data from three aspects the degree distribution clustering coecient and average path length We demonstrated byvarious datasets that data cube lattices and concept lattices share similarities underlying their topology which are in generaldierent from random networks and complex networks Specically lattice data follow the Poisson distribution and have smallerclustering coecient and greater average path lengthWe further discuss and explain these characteristics intrinsically by buildingthe analytical model and the generating mechanism

1 Introduction

e data cube (lattice) [1] proposed by Gray et al is a core datamodel in data warehousing and online analytical processing(OLAP) [2] It plays a more critical role in data online analysisand processing especially in the era of big data It allows data tobe modeled and analyzed intuitively in the context of multipleanalysis dimensions and plays a vital role in business intelli-gence Based on data cubes online analysis operations such asrolling up drilling down slicing and rotation can be easilycarried out From a conceptual view users can analysis the dataalong a dimension hierarchy to various coarse-grained levels sothat a large amount of detailed data can be described in a moreconcise and summary fashion which facilitates users to obtainthe general view On the contrary they can also specialize dataalong a dimension hierarchy to the ne-grained dataAccording to the computational dependence relation of rollingup and drilling down the data cube lattice is generated

e concept lattice [3] proposed by German mathema-tician Wile in 1982 is an important model in formal concept

analysis (FCA) and is considered as an essential facility fordata analysis Formally the concept lattice takes the triple(UA I) as its formal background in which U is the objectset A is the attribute set and I is the binary relation betweenthe object set and the attribute set It derives the formalconcept (X B) where X is called the concept extent and B iscalled the concept intent each concept in the concept setforms the concept lattice according to the partial order re-lation e node of concept lattices denotes both the intentand the extent of the concept and the relationship betweennodes reiexclects the generalization and specialization of con-cepts According to the dependency of the knowledge body inthe intent and the extent a concept hierarchy model isestablished Concept lattices are widely used in machinelearning pattern recognition expert systems decision anal-ysis data mining information retrieval computer networksand software engineering and many other elds [4 5] Forexample most data mining tasks can generate a large numberof conceptse lattice structure as the organizational form ofthe concept has many advantages in knowledge discovery It

HindawiComplexityVolume 2020 Article ID 1250106 11 pageshttpsdoiorg10115520201250106

facilitates a deep understanding of the dependencies betweendifferent concepts selected from a data set

Essentially the instances of data cube lattices andconcept lattices all belong to lattice structure data As thecore model data cube lattices and concept lattices are widelyused in data analysis From a higher perspective their al-gebraic structures are intrinsically lattices which imply theyshare similarities or connections in external characteristics

Nedjar et al and Shi [6 7] demonstrated their relationshipin the generation mechanism Nedjar et al [6] pointed outthat in data warehousing and data mining the frequent closeditemset searching the concise representation of associationrules or data cube lattices can use the formal concept analysisfor feature depicting compress representation and effectivecalculation Furthermore they put forward Agree ConceptLattice and Quotient Agree Lattice to point out their closerelationship between them Shi [7] made a thorough study ofconcept lattices and finds that concept lattices are closelyrelated to data cube lattices )ey are all based on partial orderrelations When base tables are used as formal backgroundsthe covering equivalence class of the data cube lattice and theequivalent feature sets based on formal concept analysis theoryhave the same partition results and then the AggregateConcept Aggregate Concept Lattice (ACL) and ReductiveAggregate Concept (RAC) were proposed

)e structural analysis of these two crucial and repre-sentative models in data analysis under a unified frameworkwill facilitate the discovery of their essence and design moregeneralized algorithms Strogatz [8] discussed the reason whynetwork anatomy is so important to characterize is thatstructure always affects function Liu et al and Zhai et al[9 10] mentioned the application of degree distance andtopological statistics in some real networks Lattice topologycharacteristics like the degree distribution of nodes affect thecommunication overhead (virtually link numbers) acrossvarious nodes for splitting a lattice in the cloud environment)e clustering or partitioning of lattice structures from thegraph data perspective may result in space compression Untilrecently however very little attention has been devoted to thestructural characteristics and the relationship between them

To address the above challenges this paper explores andstudies the structural characteristics of data cube lattices andconcept lattices in terms of the degree distribution clusteringcoefficient average path length through the experimentaldemonstration and theoretical analysis By the analysis of thegraph structure we can find whether the data cube lattice andconcept lattice share similar structural characteristics whichare different from other networks like random networks orcomplex networks and why Note that the lattices discussed inour paper are generated from the real lifemdashthey are notcompletely regular lattices where all the nodes have the samedegrees and a plot of the degree distribution contains a singlesharp spike (delta distribution) [11] As we discovered inSection 3 the nodes of the real-world lattices have muchbroader range of degrees implying randomnesses are added

Our main contributions are as follows

(i) To the best of our knowledge we first study thegraph structural characteristics (such as the degree

distribution clustering coefficient and average pathlength) of data cube lattices and concept latticesanalyze and demonstrate them based on variousreal datasets

(ii) We present the structure relationship between datacube lattices and concept lattices Compared withrandom networks and complex networks real-world lattices follow the Poisson distribution andhave smaller clustering coefficient and greater av-erage path length

(iii) We discuss the analytical model and the generationmechanism of data cube lattices and conceptlattices

)is paper is structured as follows Section 2 introducesthe model of data cube lattices and concept lattices andrelevant definitions Section 3 presents the structuralcharacteristics of data cube lattices and concept latticesthrough the experimental study and validates the similaritybetween data cube lattices and concept lattices on structuralcharacteristics Section 4 discusses the analytical model andthe generation mechanism of data cube lattices and conceptlattices )e last section concludes this paper and describessome future works

2 Preliminary

21 Data Cube Lattices Data cube lattices generalize group-by operators and aggregate each combination of group-byattributes Among them the attributes grouped are calleddimensions D and the attributes aggregated are calledmeasurements M each grouped attribute combination iscalled cuboid (also called view) Correspondingly the cuboidcontaining i(0le ile n) dimensions called is i-dimensioncuboid )e data cell c (x1 x2 x3 xn M) is a tuple inthe cuboid where xi is the ith dimension attribute value

For a d-dimensional data cube lattice 2d group-by views(or cuboids) are generated since each combination of group-by attributes is computed

Definition 1 (partial order relation) If forallA isin Du[A] neALL⟶ u[A] v[A] then u v have partial orderrelation denoted by v≺ u or u≽ v If v≺ u and une v then it isdenoted as v≺ u or u≻ v We can say that u generalizes v or v

specializes u In other words u drills down to v or v rolls upto u If forallA isin D u[A] v[A] then u v

Definition 2 (data cube lattice) A data cube is aggregatedfrom a base table of data warehousing on various combi-nations of dimension attributes It contains the data cells Letc1 (x1 x2 x3 xn M) c2 (y1 y2 y3 yn M) bethe data cells)e partial order relation ≺ c2 ≺ c1 induces thedata cube lattice structure

Definition 3 (base tuple set) Given a data cell c isin C the basetuple set of c BTS(c) t | t isin r and t≺ c ie the set of allbase table tuples that roll up to c

2 Complexity

Definition 4 (covering equivalence) Suppose thatc1 c2 isin Cube and equiv cov lt c1 c2 gt | c1 c2 isin1113864 CubeBTS(c1) BTS(c2) is defined as the covering equivalence re-lation )e covered equivalence classes is the sets[c] equiv cov

x | x isin Cube c equiv covx1113864 1113865

Definition 5 (upper and lower bounds) Let ltA≺ gt be apartially ordered set Any element a in A is called an upperbound (lower bound) of the subsetM of A if for any elementm in M there is m≺ a(a≺m)

Example 1 Table 1 is a base table of product sales which hasthree dimension attributes of Product Time and Store andone measurement attribute Sales )e data cube latticegenerates eight group-by views by aggregation which formthe cube lattice structure based on the relationship (as shownby the connection between nodes) between rolling up(generalization) and drilling down (specialization)

Figure 1 is a data cube lattice generated by aggregation ofthe Sum operation on sales in Table 1 Note that lowast denotesthe dimension attribute value ALL )e data cells(P1lowast S1 10) (P1 T1 S1 4) and (P1 T2 S1 6) havethe semantic relations of rolling up and drilling down eachother )e partial order relationship between them forms adata cube lattice

22 Concept Lattices

Definition 6 (concept lattice) Let I be a binary relationshipbetween U and A where U is the object set and A is theattribute set Given that x isin U y isin A when (x y) isin I wesay the object x owns the attribute y and the triple tuple(U A I) was called formal background On a triple tuple(U A I) we take a subset of U and a subset of A and get aconcept set under relation I L (A B) | A isin P(U)

B isin P(A) (A B) is the concept among them A is the ex-tent of L and B is the intent of L We define a partial orderingrelation on L (A1 B1)≺ (A2 B2)⟺A1 subeA2 (or B2 subeB1)And then (L≺ ) is a complete Galois lattice called a conceptlattice about formal background (U A I)

Definition 7 (equivalent feature set) Let K (U A I) be aformal background NsubeU MsubeA If MsubeNprime and Mprime N

are satisfied thenM is an equivalent feature set ofN denotedby equ(N) 1113864M | MsubeNprime Mprime subeN1113865

Definition 8 (Hasse diagram) A Hasse diagram is a kind ofmathematical diagram used to represent a finite partiallyordered set in the form of a drawing of its transitive re-duction Concretely for a partially ordered set (A≺ ) onerepresents each element of S as a vertex in the plane anddraws a line segment or curve that goes upward from vi to vj

whenever vj covers vi (that is whenever vi≺vj and there is novk such that vi ≺ vk ≺ vj) )ese curves may cross each otherbut must not touch any vertices other than their endpointsSuch a diagram with labeled vertices uniquely determinesits partial order

According to the definition of Concept Lattice theconcept lattice generated by a formal background given inTable 2 is shown in Figure 2

Figure 2 shows a Hasse diagram representation ofconcept lattices corresponding to the formal background ofTable 2 Each node in the graph represents a concept eachconcept is identified by its extent and intent and the orderrelationship between concepts is represented by the edgesbetween nodes Among them the concept with the largestextent (corresponding to the smallest intent) in the conceptlattice is the largest concept in the concept lattice which islocated at the top of the concept lattice the concept with thelargest intent (corresponding to the smallest extent) in theconcept lattice is the smallest concept in the concept latticewhich is located at the bottom of the concept lattice

3 Lattice Structure Characteristics

Since the three spectacular conceptsmdashthe degree distribu-tion clustering coefficient and average path lengthmdashplay amore important role in networks than other quantities andmeasures [11] we verify them on various real datasets forlattices particularly the main representationsmdashdata cubelattices and concept lattices

31 Data Cube Latticesrsquo Structural Characteristics We usedthe two classic datasets Foodmart andWeather (httpcdiacesdomlgovcdiacndpsndp026bhtml) for the data cubelattices )en we calculated its topology statistics to analyzewhether the data cube lattices generated by different data setshave similar structural characteristics

We randomly extracted 10000 tuples from Foodmartand generated a data cube lattice Foodmart-1w by adaptingthe data cube construction algorithm [12] )en we ran-domly extracted 10000 tuples from Weather data andgenerated the data cube lattice Weather-1w by the samelattice structure algorithm )ey are shown in Table 3

For the two datasets in Table 3 the structural charac-teristics of the degree distribution clustering coefficient andaverage path length are calculated

Figure 3(a) shows the degree distribution of Foodmart-1w and Weather-1w )e horizontal axis represents thedegree value of the node while the vertical axis representsthe total number of nodes when the degree is that value Bycomparison it can be found that the two curves in the graphfirst jump sharply and then decrease exponentially )eaverage degree of each node of Foodmart-1w is 88 theaverage degree of each node of Weather-1w is 79 It can beseen that the average of data cube lattices of different datasets is not very different

Figure 3(b) shows the clustering coefficient of two datasets in the data cube lattice )e horizontal axis representsthe degree value of the node while the vertical axis repre-sents the average clustering coefficient of nodes when thedegree is that value )e average clustering coefficient ofFoodmart-1w and Weather-1w is 00231 and 00042 re-spectively All of them have relatively small average clus-tering coefficients

Complexity 3

Figure 3(c) shows the average path length distribution oftwo data sets in the data cube lattice )e horizontal axisrepresents the length of the path (hops) and the vertical axisrepresents the number of pairs of nodes when the pathlength is that value)e average path length of Foodmart-1wand Weather-1w is 525 and 721 respectively Figure 3(c)shows the distribution of the average path length is similarbetween two structures

32 Concept Latticesrsquo Structural Characteristics For theconcept lattices we used the Mushroom data in the UCImachine learning library (httparchiveicsuciedumldatasets) which is the common benchmark dataset in cal-ibrating various concept lattice algorithmsWe also analyzedwhether the concept lattices generated by different data setshave similar structural characteristics

We randomly selected 600 tuples from the Mushroomdata set of UCI and divided them into two parts each with300 tuples )en we used In-Close algorithm [13] to gen-erate concept lattices Mushroom-1 and Mushroom-2 andused FcaStone (httpfcastonesourceforgenet) to transformthe concept lattices into the graph structures furtherforming the second group dataset as shown in Table 4

)e degree distribution clustering coefficient averagepath length and other structural characteristics are calcu-lated by using the data in Table 4 )e results are shown inFigures 4(a)ndash4(c)

Figure 4(a) shows the degree distribution of Mushroom-1 and Mushroom-2 By comparison it is an interestingdiscovery that the two curves in the graph jump sharply firstand then decrease exponentially )e average degree of eachnode ofMushroom-1 is 81 while that of Mushroom-2 is 79It can be seen that the average degree of concept lattices ofdifferent data sets is not very different

Figure 4(b) shows the clustering coefficient of the twodata sets in the concept lattice in which the horizontal axisrepresents the degree value of the node and the vertical axisrepresents the average clustering coefficient of nodes whenthe degree is the value )e average clustering coefficient ofMushroom-1 and Mushroom-2 is 01064 and 00842 re-spectively )ey all have relatively small average clusteringcoefficient

Figure 4(c) shows the average path length distribution ofthe two data sets in the concept lattice in which the hori-zontal axis represents the path length (hops) and the verticalaxis represents the number of node pairs when path length isthat value )e average path length of Mushroom-1 andMushroom-2 is 614 and 515 respectively Both of themhave smaller average path length

33 Relationship between Data Cube Lattices and ConceptLattices We used the data cube lattice generated byFoodmart data in the first group and the concept latticegenerated by Mushroom data in the second group as shownin Table 5 to analyze whether they share the similarstructural characteristics

For the two kinds of lattice structure data in Table 5 thedegree distribution clustering coefficient and average pathlength are calculated )e data cube lattice and conceptlattice are abbreviated as CubeLattice and ConceptLatticerespectively

Table 1 Base table of product sales

Product Time Store Sales1 P1 T1 S1 42 P1 T2 S1 63 P2 T1 S2 2

(P1 T1 S1 4) (P1 T2 S1 6) (P2 T1 S2 2)

(P1 lowast S1 10)

(P1 T1 lowast 4)

(lowast T1 S1 4)

(P1 T2 lowast 6)

(lowast T2 S1 6)

(P2 lowast S2 2)

(P2 T1 lowast 2)

(lowast T1 S2 2)

(P1 lowast lowast 10)(lowast lowast S1 10)

(lowast T1 lowast 6)(lowast T2 lowast 6)

(lowast lowast S2 2)

(P2 lowast lowast 2)

(lowast lowast lowast 12)

Figure 1 Data cube lattice

Table 2 A formal background

Object a b c d1 1 0 1 12 1 1 0 13 0 1 1 04 0 1 1 05 1 0 1 1

(12345 Φ)

(125 ad) (2 abd) (1345 c)

(2 abd)(34 bc) (15 acd)

(Φ abcd)

Figure 2 Concept lattice

Table 3 )e first group dataset

Data cube lattices Node number Edge numberFoodmart-lw 17579 31676Weather-lw 13230 53424

4 Complexity

Figure 5(a) shows the degree distribution of two differentlattice structure data CubeLattice and ConceptLattice )edegree distributions of both of them rise sharply reach apeak value and then decay exponentially In the degreedistribution of CubeLattice when the degree of a node is 5the number of nodes reaches a peak value which is 5450 inthe degree distribution of ConceptLattice when the degreeof a node is 7 the number of nodes reaches a peak valuewhich is 2440 By comparison we have found that thedegree distribution of CubeLattice is similar to that ofConceptLattice

Figure 5(b) shows the distribution of the clusteringcoefficient for CubeLattice and ConceptLattice FromFigure 5(b) we can see that both of them rise sharply at firstthen decline slowly after a peak Although there are fluc-tuations in amplitude during the decline process the overalltrend is downward )e average clustering coefficient ofCubeLattice and ConceptLattice is 00231 and 01064 re-spectively So the average clustering coefficient of latticestructure data is relatively small

Figure 5(c) shows the distribution of the average pathlength for the two different lattice structures CubeLatticeand ConceptLattice By comparison it has been found thatthe average path length distribution of them increases slowlyat first reaches a peak value and then decreases slowly Aftercalculation we found that the average path length ofCubeLattice is 525 and that of ConceptLattice is 614Obviously the distribution of the average path length aboutCubeLattice is similar to that of ConceptLattice

34 Comparison with Other Main Networks In order tocheck whether the latticesrsquo structural statistics are different

with other main networks such as random networks andcomplex networks [8 11] we first extracted the 10000 tuplesfrom Foodmart and generated a data cube lattice )en wegenerated a random graph based on the ERmodel [14] usingSNAP [15] For the complex network the data set is a realsocial network Facebook collected in SNAP data sets

)e comparison results are as follows

(1) )e degree distribution of lattices is obviously dif-ferent from that of Facebook but shares certainsimilarity with the ER random graph )e degreedistribution of Facebook generally follows the heavytail distribution which indicates the occurrence ofnodes with a much higher degree than most othernodes On the contrary Lattice and ER follow thePoisson-like degree distribution (Figure 6)

(2) )e average clustering coefficient of lattices is dif-ferent from that of complex networks According tothe calculation the average clustering coefficient ofLattice is 00231 and the average clustering coeffi-cient of ER is 00011 )ey both have a small clus-tering coefficient )e clustering coefficient in ERrandom graphs is evenly and randomly distributedDifferent from ER the nodes with a smaller degree inLattice have the larger clustering coefficient in lat-tices )e average clustering coefficient of Facebookis 05225 which accords with the large clusteringcoefficient of the small-world network )at is alsodifferent from the average clustering coefficient ofLattice (Figure 7)

(3) )e average path length of Lattice and ER is 721 and426 respectively Both of them have smaller averagepath length )e longest path between two points inthe lattice structure is related to the number of di-mensions while the longest path between two nodesin the ER random graph is related to the number ofnodes Facebookrsquos average path length is 37 and thelongest path between two points in the network is 6Obviously the average path length distribution of

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Node degree

Weather-1wFoodmart-1w

(a)

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1 3 5 7 9 11 13 15 17 19 21 23 25 27

Clus

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(c)

Figure 3)e structural characteristics of cube lattices on Foodmart-1w andWeather-1w (a) Degree distribution (b) Clustering coefficient(c) Average path length

Table 4 )e second group dataset

Concept lattices Node number Edge numberMushroom-1 10486 46289Mushroom-2 10547 60146

Complexity 5

the lattice structure is different from that of theFacebook social network (Figure 8)

In summary on the degree distribution lattices andrandom networks all follow the Poisson distribution Be-sides lattices and random networks have the smaller clus-tering coefficient But lattices have the rather larger averagepath length while random networks are moderately large inthe average path length For the typical complex networksthey have the larger average clustering coefficient and thesmaller average path length and their degree distributionsatisfies the heavy tail distribution

4 Some Discussions

41 Analytical Model Data cube lattices have differentstructural characteristics from random networks andcomplex networks First in the process of generating lattice-structured data nodes do not connect according to a ran-dom probability but there are partial ordering relationshipsamong some different nodes which generate edgesaccording to the partial ordering relationship secondlythere is no partial order relationship between data cells in thesame layer that is the lattice structure data have a clearhierarchical structure finally the lattice structure has itsunique regular structure and every two nodes have upperand lower bounds

We elaborate the analytical model of data cube latticesand concept lattices from three aspects

411 Degree Distribution From the above experiments wecan see that the degree distribution curves of the latticestructure data first jump sharply then drop exponentiallywhich tends to be similar to the Poisson distribution curve

In order to prove this point we use polynomial distributionand Poisson distribution curves to fit the degree distributionof the lattice structure data

Firstly the degree distribution curve of the data cubelattice (it has 10000 base tuples 15 dimensions 13230nodes and 53424 edges) is fitted by the polynomial curveIn the process of fitting when the polynomial is 6th powerand more the R-square reaches the maximum value Asshown in Figure 9 the vertical axis represents the probabilityof the occurrence of nodes with different degree values thediamond points represent the discrete points of the degreedistribution of the data cube lattice and the triangle curvesrepresent the fitting polynomial curves of 6th power At thistime the value of R-square is about 089 which fits theoriginal degree distribution curve better

In order to evaluate the accuracy of curve fitting thesquare of the error between the discrete point of the degreedistribution curve and the discrete point of the polynomialcurve is calculated )e result is about 00084

)en the Poisson distribution curve is fitted to thedegree distribution curve and the probability expression ofPoisson distribution is as follows

P(X) ϑx

xe

minus ϑ (1)

where ϑ denotes the expected value )e degree values of thenodes in the data cube lattice are multiplied by their cor-responding probabilities and then the sum can be calculatedas follows ϑ 1113936 dlowastP(d) )en the Poisson distributioncurve can be drawn as shown in Figure 10 )e vertical axisrepresents the probability of the occurrence of nodes withdifferent degree values the curve with diamonds representsthe curve simulating Poisson distribution the curve withsquares represents the degree distribution curve of the wholelattice structure data and the curve with triangles is thedegree distribution curve of the nodes in the eleventh layerof the lattice structure It can be seen that both the degreedistribution of the whole structure and the degree part of oneof the layers tightly fit the Poisson distribution curve Alsoanother data cube lattice (it has 10000 base tuples 12088

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Mushroom-2Mushroom-1

(a)

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(b)


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1 2 3 4 5 6 7 8 9 10 11 12 13

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t pat

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Number of hops

(c)

Figure 4 )e structural characteristics of concept lattices on the mushroom date set (a) Degree distribution (b) Clustering coefficient(c) Average path length

Table 5 )e third group dataset

Lattices Node number Edge numberData cube lattice 17579 31676Concept lattice 10486 46289

6 Complexity

nodes and 54658 edges) is also fitted with the Poissondistribution curve As shown in Figure 11 the degree dis-tribution of different lattice structure data is tightly fittedwith the Poisson distribution curve

)e square of the errors of the discrete points of thedegree distribution curve and the Poisson distribution iscalculated )e result is about 00038 and the error is muchsmaller than the fitting curve of the polynomial Combinedwith the additivity of Poisson distribution it is furtherconfirmed that the degree distribution of the lattice structureis similar to Poisson distribution

412 Clustering Coefficient If a node i is chosen arbitrarilyin the lattice and the degree of i is di then the number ofpossible connected edges of the di adjacent nodes of node i isdi(di minus 1)2 If the actual number of edges between the di

adjacent nodes is r the clustering coefficient of node i is asfollows

Ci r

di di minus 1( 11138572 (2)

)e average clustering coefficient of all nodes in thewhole lattice structure is as follows

C 1N

1113944

N

i1Ci (3)

where N denotes the total number of nodes in the latticestructure

It is assumed that the node i is connected with the node jand the node k in the same layer but there is no edgebetween j and k so the value of r is rather smaller

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CubeLatticeConceptLattice

(a)

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(c)

Figure 5 )e structural characteristics between cube lattices and concept lattices (a) Degree distribution (b) Clustering coefficient(c) Average path length

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(a)

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1 21 41 61 81 101 122 142 163 183 203 291

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Facebook

(b)

Figure 6 Degree distribution of three kinds of network structures (a) Lattice and ER (b) Facebook

Complexity 7

Fewer nodes are connected across layers so there arefewer edges between the nodes connected with the samenode

In concept lattices there are few nodes connected acrosslayers and few edges between the nodes connected with thesame node)us the clustering coefficient of concept latticesin the graph structure is also very small

)erefore combining equations 2 and 3 we can see thatdata cube lattice and concept lattice have smaller clusteringcoefficients

413 Average Path Length If the dimension number ofnodes in the lattice structure is h the number of layers is alsoh since it rolls up from the bottom cuboid to the top cuboid

Taking any two nodes i and j the minimum number of edgesto pass from i to j is called the shortest distance from i to jand is denoted as lij )e average distance of all pairs ofnodes in the data cube lattice is as follows

l 2

M(M minus 1)1113944iltj

lij (4)

where M is the sum-up node of the data cube lattice andthere must be lij le h )erefore the data cube lattice has arelatively small average path length

For concept lattices the following definitions are givento facilitate the analysis of their graph structure model

)e Hasse diagram of an n-layer concept lattice consistsof a triple G (N E m) whereN is the set of nodes E is the

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Facebook

(b)

Figure 7 Clustering coefficient of three kinds of network structures (a) Lattice and ER (b) Facebook

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Facebook

(b)

Figure 8 Average path length of three kinds of network structures (a) Lattice and ER (b) Facebook

8 Complexity

set of edges and m is the number of layers of the conceptlattice graph )erefore in this hierarchical structure thereare the following properties

(1) N consists of m nonempty subsetsN N1 cupN2 cup middot middot middot cupNm and Ni cupNj empty whereine j Ni is the ith layer in the graph

(2) If the edge (x y) isin E then x isin Ni y isin Nj and ile jthen j minus i is the span of edge (x y) that is thedistance between the node x and the node j

Let lxy be the distance from the node x to the node y wehave lxy lem Combining equation 4 it can be deduced thatthe graph structure of the concept lattice also has a relativelysmall average path length

42 Generation Mechanism According to the definition ofdata cube lattices in Definition 2 and the definition ofconcept lattices in Definition 6 data cube lattices are derivedfrom base tables in the data warehouse and concept latticesare established based on formal background When the basetable is stored in the relational model it describes the re-lationship between attributes and tuples and formal back-ground describes the relationship between attributes andobjects Attributes Product Time and Store in Figure 1 canbe mapped to attributes a b and c in Figure 2 and the tupleldquo1rdquo in Figure 1 can be mapped to the object ldquo1rdquo in Figure 2and so on By this attributes in the base table can correspondto attributes and objects in the formal background one byone )erefore the structure of the base table in the datacube lattice and the formal background in the concept latticeare the same

According to [16] when the base table in the datawarehouse is taken as the formal background there is one-to-one correspondence between the covering equivalenceclass in the data cube lattice and the equivalent feature set inthe concept lattice )ey both have the same covering tupleset and the upper bound of each covering equivalence classcorresponds to the concept contained in the equivalentfeature set Combined with the definition of equivalentfeature sets in Definition 7 the following theorem holds in[16]

Theorem 1 When the base table is taken as the formalbackground ie K (U A I) where U corresponds to thetuple set of the base table A corresponds to the dimensionattribute set of the base table (without measurement attri-butes) Let NsubeU c isin Cube (Nprime NPrime) be the concept of thecorresponding N where Cube is a data cube lattice derivedfrom the base table and c is a data cell If BTS(c) N is

y = 00000 x6 ndash 00000 x5 + 00003 x4 ndash 00047x3 + 00314 x2 ndash 00596 x + 00336

R2 = 08857

ndash005

0

005

01

015

02

0 10 20 30 40

Prob

abili

ty o

f occ

urre

nce

Degree of node

Figure 9 Polynomial curve fitting

0

002

004

006

008

01

012

014

016

018

02

1 4 7 10 13 16 19 22 25 29

Prob

abili

ty o

f occ

urre

nce

Degree of node

PoissonLattice-allLattice-layer11

Figure 10 Poisson distribution curve fitting I

0

002

004

006

008

01

012

014

016

018

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 34Degree of node

Prob

abili

ty o

f occ

urre

nce

PoissonLattice-all

Figure 11 Poisson distribution curve fitting II

Complexity 9

satisfied then [c] equiv cov equ(N) and the upper bound u of

[c] equiv covis the corresponding conceptual intent Nprime of N

Based on Ceorem 1 we have the following corollary

Corollary 1 Let L u | u is the upper bound of [c] equiv cov1113966 1113967 and

Lprime (A B) | A isin P(U) B isin P(A) (A B) is the concept1113864 1113865 thenthe corresponding data cube lattice (L≺ ) is equivalent to theconcept lattice (Lprime≺ ) denoted by (L≺ ) equiv (Lprime≺ )

For example let L 1(P1 T1 S1) 2(P1 T2 S1) thenthe data cell c may take any value of (P1lowastlowast) (lowastlowast S1)and (P1lowast S1) in Figure 1 )e corresponding concept of Lis ((1 2) (P1 S1)) and (P1lowast S1) is not only the upperbound of [c] equiv cov

but also the intent of the concept((1 2) (P1 S1)) Corollary 1 proves that if the base table ofthe data warehouse is taken as the formal background theconcept lattice derived from the base table is the same as thestructure of the data cube lattice which only preserves theupper bounds in equivalence classes (as shown in Figure 12)

5 Related Work

In order to improve the performance of online analysis anddecision-making of data warehousing Gray et al proposedthe data cube operator CUBE [1] which generalizes group-by cross-tab and subtotal operators and preaggregates andmaterializes the attributes (ie dimensions) of group-bys

Laks et al [12] proposed quotient cubes to compress datacubes efficiently It uses cover partition to partition data cellswith the same upper bound into classes to preserve thesemantics of the data cube )e upper and lower bounds ofeach class are preserved to achieve data cube compression)e closed data cube is proposed in [17] which also im-plements compression of data cubes through equivalentclasses )e difference is that the closed data cube only keepsits upper bound for each class so it is more efficient tocompress the data cube

In 1982 German mathematician Wile first proposed thetheory of formal concept analysis based on concepts andconceptual levels )e core data model is the concept latticealso known as Galois lattice which is used to discover sortand display concepts [3] At present the constructionmethods of concept lattices are divided into the batchprocessing algorithm [18] and the progressive algorithm[13] Zhang et al [19] studied how to quickly and effectivelyadjust the original concept lattice to get the concept lattice ofthe new formal background after some attribute reduction ofthe formal concept rather than the reconstruction algorithmin the traditional way Sarmah et al [20] proposed an in-cremental algorithm to reduce multiple attributes Com-pared with the incremental algorithm to reduce singleattributes the algorithm only needs to be executed onceWith the further improvement and development of theconcept lattice theory and methods the fields of fuzzytheory spatial clustering granular computing and otherfields have been intersected and integrated with formalconcept analysis and concept lattices resulting in new ap-plications [16 21]

Until now none of the above work studies the structuralcharacteristics of data cube lattices and concept lattices inview of the importance of structures

6 Conclusion and Future Work

)is paper analyzes and demonstrates the structural char-acteristics of data cube lattices and concept lattices based onthe synthetic and real data sets We found the similaritiesbetween them in the degree distribution clustering coeffi-cient and average path length We further discuss theirsimilarity of the analytical model and the generationmechanism in the intrinsic perspective

Our results demonstrate initial promise for exploring thestructural characteristics of data cube lattices and conceptlattices in data analysis however there are many directionsof future research Next we will investigate if data cubelattices and concept lattices can be unified under the same

(lowast lowast lowast)12

(P1 lowast S1)10 (lowast T1 lowast)6

(P2 T1 S2)2 (P1 T1 S1)4 (P1 T2 S1)6

False

(a)

(123 Φ)

(12 P1 S1) (13 T1)

(2 P1 T2 S1) (1 P1 T1 S1) (3 P2 T1 S2)

(Φ P1 P2 T1 T2 S1 S2)

(b)

Figure 12 Data cube lattice and the corresponding concept lattice(a) Data cube lattice (b) Concept lattice

10 Complexity

framework since they are externally and intrinsically similar)us some efficient algorithms such as construction re-duction or query in data cube lattices or concept latticescould be applied to each other or be generalized Besides weintend on utilizing the lattice structural characteristics suchas the degree distribution to facilitate the lattice structuredata partitioning in the distributed cloud environment as thestructure is the key factor in the load balancing and thecommunication cost minimizing

Data Availability

)e data used to support the findings of this study areavailable

Conflicts of Interest

)ere are no conflicts of interest regarding the publication ofthis paper

Acknowledgments

)e authors are grateful to Yang Wang for his early work inpart related to this paper when he is a graduate in our lab)is work was partially supported by the Natural ScienceFoundation of China (61462050 and 61562054) and theNatural Science Foundation of Yunnan Province(KKSY201603016)

References

[1] J Gray S Chaudhuri A Bosworth et al ldquoData cube a re-lational aggregation operator generalizing group-by cross-tab and sub-totalsrdquo Data Mining and Knowledge Discoveryvol 1 no 1 pp 29ndash53 1997

[2] J Xin G Wang G Li Y Gao and Z Zhang ldquoState of the artdata model and its research progressrdquo Journal of Softwarevol 30 no 1 pp 142ndash163 2019

[3] R Wile ldquoRestructuring lattice theory an approach based onhierarchies of conceptsrdquo in Ordered Sets I Rival EdSpringer Dordrecht Netherlands pp 445ndash470 1982

[4] J Han M Kamber and J Pei Data Mining Concepts andTechniques China Machine Press Beijing China 2012

[5] X Zhang Y Yao and Y Liang Rough Set and Concept LatticeXirsquoan JiaoTong University Press Xirsquoan China 2006

[6] S Nedjar F Pesci L Lakhal and R Cicchetti ldquo)e agreeconcept lattice for multidimensional database analysisrdquo inProceedings of the International Conference on Formal ConceptAnalysis Nicosia Cyprus May 2011

[7] Z Shi Research on High Performance Data Cube and ItsSemantics PhD thesis Beijing Jiaotong University BeijingChina 2010

[8] S H Strogatz ldquoExploring complex networksrdquo Naturevol 410 pp 268ndash276 2001

[9] J-B Liu S Wang C Wang and S Hayat ldquoFurther results oncomputation of topological indices of certain networksrdquo IETControl Ceory amp Applications vol 11 no 13 pp 2065ndash20712017

[10] Y Zhai J-B Liu and S Wang ldquoStructure properties of kochnetworks based on networks dynamical systemsrdquo Complexityvol 2017 Article ID 6210878 7 pages 2017

[11] X F Wang and G Chen ldquoComplex networks small-worldscale-free and beyondrdquo IEEE Circuits and Systems Magazinevol 3 no 1 pp 6ndash20 2003

[12] V S Laks J Pei and J Han ldquoQuotient cube how to sum-marize the semantics of a data cuberdquo in Proceedings of the 28thInternational Conference on Very Large Data Basespp 778ndash789 VLDB Endowment Hong Kong August 2002

[13] S Anderws ldquoIn-close a fast algorithm for computing formalconceptsrdquo in Proceedings of the International Conference onConceptual Structures (ICCS) Moscow Russia July 2009

[14] Bela Bollobas Random Graphs Cambridge University PressCambridge UK 2001

[15] J Leskovec and R Sosic ldquoSnaprdquo ACM Transactions on In-telligent Systems and Technology vol 8 no 1 pp 1ndash20 2016

[16] X Kang and D Miao ldquoA knowledge acquisition methodbased on concept lattice in set-valued information systemsrdquoCAAI Transactions on Intelligent Systems vol 11 no 3pp 287ndash293 2016

[17] S Li and S Wang ldquoResearch on closed data cube technologyrdquoJournal of Software vol 15 no 8 pp 1165ndash1171 2004

[18] J P Bordat ldquoCalcul pratique du treillis de galois drsquounecorrespondancerdquo Mathematiques et Sciences Humainesvol 24 no 96 pp 31ndash47 1986

[19] L Zhang H Zhang and L Yin ldquo)eory and algorithms ofattribute decrement for concept latticerdquo Journal of ComputerResearch and Development vol 50 no 2 pp 248ndash259 2013

[20] Y Ma and W Ma ldquoConstruction of multi-attributes decre-ment for concept latticerdquo Journal of Software vol 26 no 12pp 3162ndash3173 2015

[21] A K Sarmah S M Hazarika and S K Sinha ldquoFormalconcept analysis current trends and directionsrdquo ArtificialIntelligence Review vol 44 no 1 pp 47ndash86 2015

Complexity 11

Hindawiwwwhindawicom Volume 2018

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facilitates a deep understanding of the dependencies betweendifferent concepts selected from a data set

Essentially the instances of data cube lattices andconcept lattices all belong to lattice structure data As thecore model data cube lattices and concept lattices are widelyused in data analysis From a higher perspective their al-gebraic structures are intrinsically lattices which imply theyshare similarities or connections in external characteristics

Nedjar et al and Shi [6 7] demonstrated their relationshipin the generation mechanism Nedjar et al [6] pointed outthat in data warehousing and data mining the frequent closeditemset searching the concise representation of associationrules or data cube lattices can use the formal concept analysisfor feature depicting compress representation and effectivecalculation Furthermore they put forward Agree ConceptLattice and Quotient Agree Lattice to point out their closerelationship between them Shi [7] made a thorough study ofconcept lattices and finds that concept lattices are closelyrelated to data cube lattices )ey are all based on partial orderrelations When base tables are used as formal backgroundsthe covering equivalence class of the data cube lattice and theequivalent feature sets based on formal concept analysis theoryhave the same partition results and then the AggregateConcept Aggregate Concept Lattice (ACL) and ReductiveAggregate Concept (RAC) were proposed

)e structural analysis of these two crucial and repre-sentative models in data analysis under a unified frameworkwill facilitate the discovery of their essence and design moregeneralized algorithms Strogatz [8] discussed the reason whynetwork anatomy is so important to characterize is thatstructure always affects function Liu et al and Zhai et al[9 10] mentioned the application of degree distance andtopological statistics in some real networks Lattice topologycharacteristics like the degree distribution of nodes affect thecommunication overhead (virtually link numbers) acrossvarious nodes for splitting a lattice in the cloud environment)e clustering or partitioning of lattice structures from thegraph data perspective may result in space compression Untilrecently however very little attention has been devoted to thestructural characteristics and the relationship between them

To address the above challenges this paper explores andstudies the structural characteristics of data cube lattices andconcept lattices in terms of the degree distribution clusteringcoefficient average path length through the experimentaldemonstration and theoretical analysis By the analysis of thegraph structure we can find whether the data cube lattice andconcept lattice share similar structural characteristics whichare different from other networks like random networks orcomplex networks and why Note that the lattices discussed inour paper are generated from the real lifemdashthey are notcompletely regular lattices where all the nodes have the samedegrees and a plot of the degree distribution contains a singlesharp spike (delta distribution) [11] As we discovered inSection 3 the nodes of the real-world lattices have muchbroader range of degrees implying randomnesses are added

Our main contributions are as follows

(i) To the best of our knowledge we first study thegraph structural characteristics (such as the degree

distribution clustering coefficient and average pathlength) of data cube lattices and concept latticesanalyze and demonstrate them based on variousreal datasets

(ii) We present the structure relationship between datacube lattices and concept lattices Compared withrandom networks and complex networks real-world lattices follow the Poisson distribution andhave smaller clustering coefficient and greater av-erage path length

(iii) We discuss the analytical model and the generationmechanism of data cube lattices and conceptlattices

)is paper is structured as follows Section 2 introducesthe model of data cube lattices and concept lattices andrelevant definitions Section 3 presents the structuralcharacteristics of data cube lattices and concept latticesthrough the experimental study and validates the similaritybetween data cube lattices and concept lattices on structuralcharacteristics Section 4 discusses the analytical model andthe generation mechanism of data cube lattices and conceptlattices )e last section concludes this paper and describessome future works

2 Preliminary

21 Data Cube Lattices Data cube lattices generalize group-by operators and aggregate each combination of group-byattributes Among them the attributes grouped are calleddimensions D and the attributes aggregated are calledmeasurements M each grouped attribute combination iscalled cuboid (also called view) Correspondingly the cuboidcontaining i(0le ile n) dimensions called is i-dimensioncuboid )e data cell c (x1 x2 x3 xn M) is a tuple inthe cuboid where xi is the ith dimension attribute value

For a d-dimensional data cube lattice 2d group-by views(or cuboids) are generated since each combination of group-by attributes is computed

Definition 1 (partial order relation) If forallA isin Du[A] neALL⟶ u[A] v[A] then u v have partial orderrelation denoted by v≺ u or u≽ v If v≺ u and une v then it isdenoted as v≺ u or u≻ v We can say that u generalizes v or v

specializes u In other words u drills down to v or v rolls upto u If forallA isin D u[A] v[A] then u v

Definition 2 (data cube lattice) A data cube is aggregatedfrom a base table of data warehousing on various combi-nations of dimension attributes It contains the data cells Letc1 (x1 x2 x3 xn M) c2 (y1 y2 y3 yn M) bethe data cells)e partial order relation ≺ c2 ≺ c1 induces thedata cube lattice structure

Definition 3 (base tuple set) Given a data cell c isin C the basetuple set of c BTS(c) t | t isin r and t≺ c ie the set of allbase table tuples that roll up to c

2 Complexity






22 Concept Lattices
















Complexity 3












(P1 T1 S1 4) (P1 T2 S1 6) (P2 T1 S2 2)

(P1 lowast S1 10)

(P1 T1 lowast 4)

(lowast T1 S1 4)

(P1 T2 lowast 6)

(lowast T2 S1 6)

(P2 lowast S2 2)

(P2 T1 lowast 2)

(lowast T1 S2 2)








Object a b c d1 1 0 1 12 1 1 0 13 0 1 1 04 0 1 1 05 1 0 1 1

(12345 Φ)

(125 ad) (2 abd) (1345 c)

(2 abd)(34 bc) (15 acd)

(Φ abcd)




4 Complexity










0

1

2

3

4

5

6

1 4 7 10 13 16 19 22 25 28

Coun

t103

Node degree


(a)

0

001

002

003

004

005

006

007

008

1 3 5 7 9 11 13 15 17 19 21 23 25 27

Clus

terin

g co

effic

ient

Node degree


(b)

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Num

ber o

f sho

rtes

t pat

hs1

06

Number of hops


(c)




Complexity 5



4 Some Discussions








P(X) ϑx

xe

minus ϑ (1)


0

05

1

15

2

25

3

1 4 7 10 13 16 19 22

Cou

nt1

03

Node degree


(a)

0

002

004

006

008

01

012

014

1 3 5 7 9 11 13 15 17 19 21 23 25 29

Aver

age c

luste

ring

coef

ficie

nt

Node degree


(b)


0

05

1

15

2

25

3

35

1 2 3 4 5 6 7 8 9 10 11 12 13

Num

ber o

f sho

rtes

t pat

hs1

06

Number of hops

(c)




6 Complexity





Ci r

di di minus 1( 11138572 (2)


C 1N

1113944

N

i1Ci (3)



0

1

2

3

4

5

6

1 3 5 7 9 11 13 15 17 19 21 23 25 29

Coun

t103

Node degree


(a)

0

002

004

006

008

01

012

014

1 6 11 16 21 26 35 42 52

Ave

rage

clus

terin

g co

effic

ient

Node degree


(b)


0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11

Num

ber o

f sho

rtes

t pat

hs1

06

Number of hops

(c)


0

500

1000

1500

2000

2500

1 4 7 10 13 16 19 22 25 29

Cou

nt

Node degree

LatticeER

(a)

0

20

40

60

80

100

120

1 21 41 61 81 101 122 142 163 183 203 291

Cou

nt

Node degree

Facebook

(b)


Complexity 7






l 2


lij (4)




0

0001

0002

0003

0004

0005

0006

0007

0008

1 4 7 10 13 16 19 22 25 29

Aver

age c

luste

ring

coef

ficie

nt

Node degree

LatticeER

(a)

0

02

04

06

08

1

12

1 22 43 64 85 106 128 150 171 192 222

Aver

age c

luste

ring

coef

ficie

nt

Node degree

Facebook

(b)


0

1000000

2000000

3000000

4000000

5000000

6000000

0 2 4 6 8 10 12 14

Num

ber o

f sho

rtes

t pat

hs

Number of hops

LatticeER

(a)

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

0 1 2 3 4 5 6 7 8

Num

ber o

f sho

rtes

t pat

hs

Number of hops

Facebook

(b)


8 Complexity









R2 = 08857

ndash005

0

005

01

015

02

0 10 20 30 40

Prob

abili

ty o

f occ

urre

nce

Degree of node


0

002

004

006

008

01

012

014

016

018

02

1 4 7 10 13 16 19 22 25 29

Prob

abili

ty o

f occ

urre

nce

Degree of node



0

002

004

006

008

01

012

014

016

018

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 34Degree of node

Prob

abili

ty o

f occ

urre

nce

PoissonLattice-all


Complexity 9








5 Related Work










(P2 T1 S2)2 (P1 T1 S1)4 (P1 T2 S1)6

False

(a)

(123 Φ)

(12 P1 S1) (13 T1)

(2 P1 T2 S1) (1 P1 T1 S1) (3 P2 T1 S2)

(Φ P1 P2 T1 T2 S1 S2)

(b)


10 Complexity


Data Availability




Acknowledgments


References






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018













22 Concept Lattices
















Complexity 3












(P1 T1 S1 4) (P1 T2 S1 6) (P2 T1 S2 2)

(P1 lowast S1 10)

(P1 T1 lowast 4)

(lowast T1 S1 4)

(P1 T2 lowast 6)

(lowast T2 S1 6)

(P2 lowast S2 2)

(P2 T1 lowast 2)

(lowast T1 S2 2)








Object a b c d1 1 0 1 12 1 1 0 13 0 1 1 04 0 1 1 05 1 0 1 1

(12345 Φ)

(125 ad) (2 abd) (1345 c)

(2 abd)(34 bc) (15 acd)

(Φ abcd)




4 Complexity










0

1

2

3

4

5

6

1 4 7 10 13 16 19 22 25 28

Coun

t103

Node degree


(a)

0

001

002

003

004

005

006

007

008

1 3 5 7 9 11 13 15 17 19 21 23 25 27

Clus

terin

g co

effic

ient

Node degree


(b)

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Num

ber o

f sho

rtes

t pat

hs1

06

Number of hops


(c)




Complexity 5



4 Some Discussions








P(X) ϑx

xe

minus ϑ (1)


0

05

1

15

2

25

3

1 4 7 10 13 16 19 22

Cou

nt1

03

Node degree


(a)

0

002

004

006

008

01

012

014

1 3 5 7 9 11 13 15 17 19 21 23 25 29

Aver

age c

luste

ring

coef

ficie

nt

Node degree


(b)


0

05

1

15

2

25

3

35

1 2 3 4 5 6 7 8 9 10 11 12 13

Num

ber o

f sho

rtes

t pat

hs1

06

Number of hops

(c)




6 Complexity





Ci r

di di minus 1( 11138572 (2)


C 1N

1113944

N

i1Ci (3)



0

1

2

3

4

5

6

1 3 5 7 9 11 13 15 17 19 21 23 25 29

Coun

t103

Node degree


(a)

0

002

004

006

008

01

012

014

1 6 11 16 21 26 35 42 52

Ave

rage

clus

terin

g co

effic

ient

Node degree


(b)


0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11

Num

ber o

f sho

rtes

t pat

hs1

06

Number of hops

(c)


0

500

1000

1500

2000

2500

1 4 7 10 13 16 19 22 25 29

Cou

nt

Node degree

LatticeER

(a)

0

20

40

60

80

100

120

1 21 41 61 81 101 122 142 163 183 203 291

Cou

nt

Node degree

Facebook

(b)


Complexity 7






l 2


lij (4)




0

0001

0002

0003

0004

0005

0006

0007

0008

1 4 7 10 13 16 19 22 25 29

Aver

age c

luste

ring

coef

ficie

nt

Node degree

LatticeER

(a)

0

02

04

06

08

1

12

1 22 43 64 85 106 128 150 171 192 222

Aver

age c

luste

ring

coef

ficie

nt

Node degree

Facebook

(b)


0

1000000

2000000

3000000

4000000

5000000

6000000

0 2 4 6 8 10 12 14

Num

ber o

f sho

rtes

t pat

hs

Number of hops

LatticeER

(a)

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

0 1 2 3 4 5 6 7 8

Num

ber o

f sho

rtes

t pat

hs

Number of hops

Facebook

(b)


8 Complexity









R2 = 08857

ndash005

0

005

01

015

02

0 10 20 30 40

Prob

abili

ty o

f occ

urre

nce

Degree of node


0

002

004

006

008

01

012

014

016

018

02

1 4 7 10 13 16 19 22 25 29

Prob

abili

ty o

f occ

urre

nce

Degree of node



0

002

004

006

008

01

012

014

016

018

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 34Degree of node

Prob

abili

ty o

f occ

urre

nce

PoissonLattice-all


Complexity 9








5 Related Work










(P2 T1 S2)2 (P1 T1 S1)4 (P1 T2 S1)6

False

(a)

(123 Φ)

(12 P1 S1) (13 T1)

(2 P1 T2 S1) (1 P1 T1 S1) (3 P2 T1 S2)

(Φ P1 P2 T1 T2 S1 S2)

(b)


10 Complexity


Data Availability




Acknowledgments


References






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018



















(P1 T1 S1 4) (P1 T2 S1 6) (P2 T1 S2 2)

(P1 lowast S1 10)

(P1 T1 lowast 4)

(lowast T1 S1 4)

(P1 T2 lowast 6)

(lowast T2 S1 6)

(P2 lowast S2 2)

(P2 T1 lowast 2)

(lowast T1 S2 2)








Object a b c d1 1 0 1 12 1 1 0 13 0 1 1 04 0 1 1 05 1 0 1 1

(12345 Φ)

(125 ad) (2 abd) (1345 c)

(2 abd)(34 bc) (15 acd)

(Φ abcd)




4 Complexity










0

1

2

3

4

5

6

1 4 7 10 13 16 19 22 25 28

Coun

t103

Node degree


(a)

0

001

002

003

004

005

006

007

008

1 3 5 7 9 11 13 15 17 19 21 23 25 27

Clus

terin

g co

effic

ient

Node degree


(b)

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Num

ber o

f sho

rtes

t pat

hs1

06

Number of hops


(c)




Complexity 5



4 Some Discussions








P(X) ϑx

xe

minus ϑ (1)


0

05

1

15

2

25

3

1 4 7 10 13 16 19 22

Cou

nt1

03

Node degree


(a)

0

002

004

006

008

01

012

014

1 3 5 7 9 11 13 15 17 19 21 23 25 29

Aver

age c

luste

ring

coef

ficie

nt

Node degree


(b)


0

05

1

15

2

25

3

35

1 2 3 4 5 6 7 8 9 10 11 12 13

Num

ber o

f sho

rtes

t pat

hs1

06

Number of hops

(c)




6 Complexity





Ci r

di di minus 1( 11138572 (2)


C 1N

1113944

N

i1Ci (3)



0

1

2

3

4

5

6

1 3 5 7 9 11 13 15 17 19 21 23 25 29

Coun

t103

Node degree


(a)

0

002

004

006

008

01

012

014

1 6 11 16 21 26 35 42 52

Ave

rage

clus

terin

g co

effic

ient

Node degree


(b)


0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11

Num

ber o

f sho

rtes

t pat

hs1

06

Number of hops

(c)


0

500

1000

1500

2000

2500

1 4 7 10 13 16 19 22 25 29

Cou

nt

Node degree

LatticeER

(a)

0

20

40

60

80

100

120

1 21 41 61 81 101 122 142 163 183 203 291

Cou

nt

Node degree

Facebook

(b)


Complexity 7






l 2


lij (4)




0

0001

0002

0003

0004

0005

0006

0007

0008

1 4 7 10 13 16 19 22 25 29

Aver

age c

luste

ring

coef

ficie

nt

Node degree

LatticeER

(a)

0

02

04

06

08

1

12

1 22 43 64 85 106 128 150 171 192 222

Aver

age c

luste

ring

coef

ficie

nt

Node degree

Facebook

(b)


0

1000000

2000000

3000000

4000000

5000000

6000000

0 2 4 6 8 10 12 14

Num

ber o

f sho

rtes

t pat

hs

Number of hops

LatticeER

(a)

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

0 1 2 3 4 5 6 7 8

Num

ber o

f sho

rtes

t pat

hs

Number of hops

Facebook

(b)


8 Complexity









R2 = 08857

ndash005

0

005

01

015

02

0 10 20 30 40

Prob

abili

ty o

f occ

urre

nce

Degree of node


0

002

004

006

008

01

012

014

016

018

02

1 4 7 10 13 16 19 22 25 29

Prob

abili

ty o

f occ

urre

nce

Degree of node



0

002

004

006

008

01

012

014

016

018

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 34Degree of node

Prob

abili

ty o

f occ

urre

nce

PoissonLattice-all


Complexity 9








5 Related Work










(P2 T1 S2)2 (P1 T1 S1)4 (P1 T2 S1)6

False

(a)

(123 Φ)

(12 P1 S1) (13 T1)

(2 P1 T2 S1) (1 P1 T1 S1) (3 P2 T1 S2)

(Φ P1 P2 T1 T2 S1 S2)

(b)


10 Complexity


Data Availability




Acknowledgments


References






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018

















0

1

2

3

4

5

6

1 4 7 10 13 16 19 22 25 28

Coun

t103

Node degree


(a)

0

001

002

003

004

005

006

007

008

1 3 5 7 9 11 13 15 17 19 21 23 25 27

Clus

terin

g co

effic

ient

Node degree


(b)

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Num

ber o

f sho

rtes

t pat

hs1

06

Number of hops


(c)




Complexity 5



4 Some Discussions








P(X) ϑx

xe

minus ϑ (1)


0

05

1

15

2

25

3

1 4 7 10 13 16 19 22

Cou

nt1

03

Node degree


(a)

0

002

004

006

008

01

012

014

1 3 5 7 9 11 13 15 17 19 21 23 25 29

Aver

age c

luste

ring

coef

ficie

nt

Node degree


(b)


0

05

1

15

2

25

3

35

1 2 3 4 5 6 7 8 9 10 11 12 13

Num

ber o

f sho

rtes

t pat

hs1

06

Number of hops

(c)




6 Complexity





Ci r

di di minus 1( 11138572 (2)


C 1N

1113944

N

i1Ci (3)



0

1

2

3

4

5

6

1 3 5 7 9 11 13 15 17 19 21 23 25 29

Coun

t103

Node degree


(a)

0

002

004

006

008

01

012

014

1 6 11 16 21 26 35 42 52

Ave

rage

clus

terin

g co

effic

ient

Node degree


(b)


0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11

Num

ber o

f sho

rtes

t pat

hs1

06

Number of hops

(c)


0

500

1000

1500

2000

2500

1 4 7 10 13 16 19 22 25 29

Cou

nt

Node degree

LatticeER

(a)

0

20

40

60

80

100

120

1 21 41 61 81 101 122 142 163 183 203 291

Cou

nt

Node degree

Facebook

(b)


Complexity 7






l 2


lij (4)




0

0001

0002

0003

0004

0005

0006

0007

0008

1 4 7 10 13 16 19 22 25 29

Aver

age c

luste

ring

coef

ficie

nt

Node degree

LatticeER

(a)

0

02

04

06

08

1

12

1 22 43 64 85 106 128 150 171 192 222

Aver

age c

luste

ring

coef

ficie

nt

Node degree

Facebook

(b)


0

1000000

2000000

3000000

4000000

5000000

6000000

0 2 4 6 8 10 12 14

Num

ber o

f sho

rtes

t pat

hs

Number of hops

LatticeER

(a)

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

0 1 2 3 4 5 6 7 8

Num

ber o

f sho

rtes

t pat

hs

Number of hops

Facebook

(b)


8 Complexity









R2 = 08857

ndash005

0

005

01

015

02

0 10 20 30 40

Prob

abili

ty o

f occ

urre

nce

Degree of node


0

002

004

006

008

01

012

014

016

018

02

1 4 7 10 13 16 19 22 25 29

Prob

abili

ty o

f occ

urre

nce

Degree of node



0

002

004

006

008

01

012

014

016

018

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 34Degree of node

Prob

abili

ty o

f occ

urre

nce

PoissonLattice-all


Complexity 9








5 Related Work










(P2 T1 S2)2 (P1 T1 S1)4 (P1 T2 S1)6

False

(a)

(123 Φ)

(12 P1 S1) (13 T1)

(2 P1 T2 S1) (1 P1 T1 S1) (3 P2 T1 S2)

(Φ P1 P2 T1 T2 S1 S2)

(b)


10 Complexity


Data Availability




Acknowledgments


References






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018










4 Some Discussions








P(X) ϑx

xe

minus ϑ (1)


0

05

1

15

2

25

3

1 4 7 10 13 16 19 22

Cou

nt1

03

Node degree


(a)

0

002

004

006

008

01

012

014

1 3 5 7 9 11 13 15 17 19 21 23 25 29

Aver

age c

luste

ring

coef

ficie

nt

Node degree


(b)


0

05

1

15

2

25

3

35

1 2 3 4 5 6 7 8 9 10 11 12 13

Num

ber o

f sho

rtes

t pat

hs1

06

Number of hops

(c)




6 Complexity





Ci r

di di minus 1( 11138572 (2)


C 1N

1113944

N

i1Ci (3)



0

1

2

3

4

5

6

1 3 5 7 9 11 13 15 17 19 21 23 25 29

Coun

t103

Node degree


(a)

0

002

004

006

008

01

012

014

1 6 11 16 21 26 35 42 52

Ave

rage

clus

terin

g co

effic

ient

Node degree


(b)


0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11

Num

ber o

f sho

rtes

t pat

hs1

06

Number of hops

(c)


0

500

1000

1500

2000

2500

1 4 7 10 13 16 19 22 25 29

Cou

nt

Node degree

LatticeER

(a)

0

20

40

60

80

100

120

1 21 41 61 81 101 122 142 163 183 203 291

Cou

nt

Node degree

Facebook

(b)


Complexity 7






l 2


lij (4)




0

0001

0002

0003

0004

0005

0006

0007

0008

1 4 7 10 13 16 19 22 25 29

Aver

age c

luste

ring

coef

ficie

nt

Node degree

LatticeER

(a)

0

02

04

06

08

1

12

1 22 43 64 85 106 128 150 171 192 222

Aver

age c

luste

ring

coef

ficie

nt

Node degree

Facebook

(b)


0

1000000

2000000

3000000

4000000

5000000

6000000

0 2 4 6 8 10 12 14

Num

ber o

f sho

rtes

t pat

hs

Number of hops

LatticeER

(a)

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

0 1 2 3 4 5 6 7 8

Num

ber o

f sho

rtes

t pat

hs

Number of hops

Facebook

(b)


8 Complexity









R2 = 08857

ndash005

0

005

01

015

02

0 10 20 30 40

Prob

abili

ty o

f occ

urre

nce

Degree of node


0

002

004

006

008

01

012

014

016

018

02

1 4 7 10 13 16 19 22 25 29

Prob

abili

ty o

f occ

urre

nce

Degree of node



0

002

004

006

008

01

012

014

016

018

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 34Degree of node

Prob

abili

ty o

f occ

urre

nce

PoissonLattice-all


Complexity 9








5 Related Work










(P2 T1 S2)2 (P1 T1 S1)4 (P1 T2 S1)6

False

(a)

(123 Φ)

(12 P1 S1) (13 T1)

(2 P1 T2 S1) (1 P1 T1 S1) (3 P2 T1 S2)

(Φ P1 P2 T1 T2 S1 S2)

(b)


10 Complexity


Data Availability




Acknowledgments


References






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018












Ci r

di di minus 1( 11138572 (2)


C 1N

1113944

N

i1Ci (3)



0

1

2

3

4

5

6

1 3 5 7 9 11 13 15 17 19 21 23 25 29

Coun

t103

Node degree


(a)

0

002

004

006

008

01

012

014

1 6 11 16 21 26 35 42 52

Ave

rage

clus

terin

g co

effic

ient

Node degree


(b)


0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11

Num

ber o

f sho

rtes

t pat

hs1

06

Number of hops

(c)


0

500

1000

1500

2000

2500

1 4 7 10 13 16 19 22 25 29

Cou

nt

Node degree

LatticeER

(a)

0

20

40

60

80

100

120

1 21 41 61 81 101 122 142 163 183 203 291

Cou

nt

Node degree

Facebook

(b)


Complexity 7






l 2


lij (4)




0

0001

0002

0003

0004

0005

0006

0007

0008

1 4 7 10 13 16 19 22 25 29

Aver

age c

luste

ring

coef

ficie

nt

Node degree

LatticeER

(a)

0

02

04

06

08

1

12

1 22 43 64 85 106 128 150 171 192 222

Aver

age c

luste

ring

coef

ficie

nt

Node degree

Facebook

(b)


0

1000000

2000000

3000000

4000000

5000000

6000000

0 2 4 6 8 10 12 14

Num

ber o

f sho

rtes

t pat

hs

Number of hops

LatticeER

(a)

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

0 1 2 3 4 5 6 7 8

Num

ber o

f sho

rtes

t pat

hs

Number of hops

Facebook

(b)


8 Complexity









R2 = 08857

ndash005

0

005

01

015

02

0 10 20 30 40

Prob

abili

ty o

f occ

urre

nce

Degree of node


0

002

004

006

008

01

012

014

016

018

02

1 4 7 10 13 16 19 22 25 29

Prob

abili

ty o

f occ

urre

nce

Degree of node



0

002

004

006

008

01

012

014

016

018

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 34Degree of node

Prob

abili

ty o

f occ

urre

nce

PoissonLattice-all


Complexity 9








5 Related Work










(P2 T1 S2)2 (P1 T1 S1)4 (P1 T2 S1)6

False

(a)

(123 Φ)

(12 P1 S1) (13 T1)

(2 P1 T2 S1) (1 P1 T1 S1) (3 P2 T1 S2)

(Φ P1 P2 T1 T2 S1 S2)

(b)


10 Complexity


Data Availability




Acknowledgments


References






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018













l 2


lij (4)




0

0001

0002

0003

0004

0005

0006

0007

0008

1 4 7 10 13 16 19 22 25 29

Aver

age c

luste

ring

coef

ficie

nt

Node degree

LatticeER

(a)

0

02

04

06

08

1

12

1 22 43 64 85 106 128 150 171 192 222

Aver

age c

luste

ring

coef

ficie

nt

Node degree

Facebook

(b)


0

1000000

2000000

3000000

4000000

5000000

6000000

0 2 4 6 8 10 12 14

Num

ber o

f sho

rtes

t pat

hs

Number of hops

LatticeER

(a)

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

0 1 2 3 4 5 6 7 8

Num

ber o

f sho

rtes

t pat

hs

Number of hops

Facebook

(b)


8 Complexity









R2 = 08857

ndash005

0

005

01

015

02

0 10 20 30 40

Prob

abili

ty o

f occ

urre

nce

Degree of node


0

002

004

006

008

01

012

014

016

018

02

1 4 7 10 13 16 19 22 25 29

Prob

abili

ty o

f occ

urre

nce

Degree of node



0

002

004

006

008

01

012

014

016

018

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 34Degree of node

Prob

abili

ty o

f occ

urre

nce

PoissonLattice-all


Complexity 9








5 Related Work










(P2 T1 S2)2 (P1 T1 S1)4 (P1 T2 S1)6

False

(a)

(123 Φ)

(12 P1 S1) (13 T1)

(2 P1 T2 S1) (1 P1 T1 S1) (3 P2 T1 S2)

(Φ P1 P2 T1 T2 S1 S2)

(b)


10 Complexity


Data Availability




Acknowledgments


References






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018
















R2 = 08857

ndash005

0

005

01

015

02

0 10 20 30 40

Prob

abili

ty o

f occ

urre

nce

Degree of node


0

002

004

006

008

01

012

014

016

018

02

1 4 7 10 13 16 19 22 25 29

Prob

abili

ty o

f occ

urre

nce

Degree of node



0

002

004

006

008

01

012

014

016

018

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 34Degree of node

Prob

abili

ty o

f occ

urre

nce

PoissonLattice-all


Complexity 9








5 Related Work










(P2 T1 S2)2 (P1 T1 S1)4 (P1 T2 S1)6

False

(a)

(123 Φ)

(12 P1 S1) (13 T1)

(2 P1 T2 S1) (1 P1 T1 S1) (3 P2 T1 S2)

(Φ P1 P2 T1 T2 S1 S2)

(b)


10 Complexity


Data Availability




Acknowledgments


References






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018















5 Related Work










(P2 T1 S2)2 (P1 T1 S1)4 (P1 T2 S1)6

False

(a)

(123 Φ)

(12 P1 S1) (13 T1)

(2 P1 T2 S1) (1 P1 T1 S1) (3 P2 T1 S2)

(Φ P1 P2 T1 T2 S1 S2)

(b)


10 Complexity


Data Availability




Acknowledgments


References






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018









Data Availability




Acknowledgments


References






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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