Math.Subj.Class.(2000): 05 A 18, 05 C 70, 05 C 85, 05 C 90, 68 R 10, 68 W 40, 92 H 30, 93 A 15.

Transcription

1 UNIVERSITY OF LJUBLJANA INSTITUTE OF MATHEMATICS, PHYSICS AND MECHANICS DEPARTMENT OF THEORETICAL COMPUTER SCIENCE JADRANSKA 19, LJUBLJANA, SLOVENIA Preprint series, Vol. 40 (2002), 799 GENERALIZED CORES Vladimir Batagelj, Matjaž Zaveršnik ISSN First version: November 24, 2001 Math.Subj.Class.(2000): 05 A 18, 05 C 70, 05 C 85, 05 C 90, 68 R 10, 68 W 40, 92 H 30, 93 A 15. Presented at Recent Trends in Graph Theory, Algebraic Combinatorics, and Graph Algorithms; September 24 27, 2001, Bled, Slovenia, Supported by the Ministry of Education, Science and Sport of Slovenia, Project J Ljubljana, December 29, 2001

2

3 Generalized Cores Vladimir Batagelj, Matjaž Zaveršnik University of Ljubljana, FMF, Department of Mathematics, and IMFM Ljubljana, Department of TCS, Jadranska ulica 19, Ljubljana, Slovenia vladimir.batagelj@uni-lj.si matjaz.zaversnik@fmf.unilj.si Abstract Cores are, besides connectivity components, one among few concepts that provides us with efficient decompositions of large graphs and networks. In the paper a generalization of the notion of core of a graph based on vertex property function is presented. It is shown that for the local monotone vertex property functions the corresponding cores can be determined in time. Key words: generalized cores, large networks, decomposition, algorithm. Math. Subj. Class. (2000): 05 A 18, 05 C 70, 05 C 85, 05 C 90, 68 R 10, 68 W 40, 92 H 30, 93 A Cores The notion of core was introduced by Seidman in 1983 [6]. is the set of vertices and $ is the set of lines Let! #"%$'& be a simple graph. (edges or arcs). We will denote ()+*,-* and./0* $ *. A subgraph 12+!34"%$ *,35& induced by the set 376+ is a 8 -core or a core of order 8 iff 9;:=<>3@?BACEDF:G&5HI8 and 1 is a maximum subgraph with this property. The core of maximum order is also called the main core. The core number of vertex : is the highest order of a core that contains this vertex. we also often call it a core. Since the set 3 determines the corresponding core J The degree ACED;:& can be the number of neighbors in an undirected graph or in-degree, out-degree, in-degree K out-degree,... determining different types of cores. The cores have the following important properties:

4 2 L -cores 2 Figure 1: 0, 1, 2 and 3 core M The cores are nested: NPORQSUT 1-V 6W1YX M Cores are not necessarily connected subgraphs. In this paper we present a generalization of the notion of core from degrees to other properties of vertices. 2 Z -cores Let [/+! #"%$\"^] & be a network, where _`!P"%$a& is a graph and ]b?c$ed IR is a function assigning values to lines. A vertex property function on [, or a L function for short, is a function Lf:"hgi&, :j<), gb6k with real values. Examples of vertex property functions: : in graph n, and lm:"hg &ople:&uqrg, gs6k. 1. LUt:"hgi&opACEDGuv:& 2. L;wx:"hgi&o indegu :G& 3. L;yx:"hgi&o outdegu :& 4. Lz{:"hgi&o indegu :G&K outdegu :G& 5. L; x:"hgi&o~}=x ƒ E ubˆ ] :;"^ŠB&, where ]b?{$ed IR Œ 6. L; x:"hgi&opž x x ƒ E ubˆ ] :"^ŠB&, where ]b?c$ed IR 7. L :"hgi&o number of cycles of length 8 through vertex : Let lm:g& denotes the set of neighbors of vertex

5 2 L -cores 3 The subgraph 1 `!34"%$ *,35& induced by the set 3`6k is a L -core at level ã< IR iff M 9;:j< 3b?{ãš=Lf:" 3& M 3 is maximal such set. The function L is monotone iff it has the property 3\t œ 3aw\Tž9 :j<)0? Lf:;" 3\th&vš=Lf:;" 3awŸ&^& All among functions L t LB are monotone. For monotone function the L -core at level can be determined by successively deleting vertices with value of L lower than. 3I? ~ ; while : <)3b?EL :" 35&'O> do 3`? ~3W \ : ; Theorem 1 For monotone function L the above procedure determines the L -core at level. Proof: The set 3 returned by the procedure evidently has the first property from the L -core definition. Let us also show that for monotone L the result of the procedure is independent of the order of deletions. Suppose the contrary there are two different L -cores at level, determined by sets 3 and. The core 3 was produced by deleting the sequence Š#t, Š;w, Š y,..., Š ; and by the sequence :Gt, :xw, :xy,..., :{ª. Assume that «43ž. We will show that this leads to contradiction. Take any =<W «3. To show that it also can be deleted we first apply the sequence : t, : w, : y,..., : ª to get. Since R< _ 3 it appears in the sequence Š t, Š w, Š y,..., ŠB \I. Let g Œ ` and gfx# gfx±²t ³ Š;X. Then, since 9;N <W µ µ L? LfŠ X " gfx±²t¹& O, we have, by monotonicity of L, also 9;N'<e µ µ L=?L ŠBX "! ' º&»gfX¼±²t½& Op. Therefore also all ŠX <¾ ` 3 are deleted ` 3 a contradiction. Since the result of the procedure is uniquely defined and vertices outside 3 have L value lower than, the final set 3 satisfies also the second condition from the definition of L -core it is the L -core at level. À Corolary 1 For monotone function L the cores are nested t OW w T1YÁû6W1¾ÁÅÄ Proof: Follows directly from the theorem 1. Since the result is independent of the order of deletions we first determine the 1)ÁÅÄ. In the following we eventually delete some additional vertices thus producing 1 ÁÃÂ. Therefore 1 Áà6W1 ÁÅÄ. À

6 Ê 2 L -cores 4 Example of nonmonotone L function: where ]s?ó$ed Lf:;"hg &P ÇÇÆ È ÉÇÇ * le:;"hg &E* Ë x ƒ E ubˆ Consider the following L function ] :;"^ŠB& le:;"hg &P ÌxÍ%ÎCEÏ%Ð\Ñ Ò%C IR Œ on the network [/`!P"%$ "^] &,bs Ô;"¹ÕŸ"%ÖŸ"% "¹Øx"¹Ù, $ ÅÔ?cÕE&ÚÛÕ»?cÖ&ÚÅÖ»?Ó &ÚÛÕ»?ÓØŸ&ÜÛØi? ÙU& ] Ý Þ Þ We get different results depending on whether we first delete the vertex Õ or Ö (or Ø ) see Figure 2. Figure 2: Nonmonotone L function The original network is a L -core at level 2. Applying the algorithm to the network we have three choices for the first vertex to be deleted: Õ, Ö or Ø. Deleting Õ we get, after removing the isolated vertex Ô, the L -core 3 t ` ÖŸ"% "¹Øx"¹Ù at level 3. Note that the values of L in vertices Ö and Ø increased from 2 to 3.

7 3 Algorithms 5 Deleting Ö (or symmetrically Ø we analyze only the first case) we get the set 34w- Ô;"¹ÕŸ"¹Øx"¹Ù at level 2 the value at Õ increased to 2.5. In the next step we can delete either the vertex Õ, producing the set 3 y»` ŸØx"¹Ù at level 3, or the vertex Ø, producing the L -core 3 z s Ô;"¹Õ at level 4. As we see, the result of the algorithm depends on the order of deletions. The L -core at level 4 is not contained in the L -core at level 3. 3 Algorithms 3.1 Algorithm for ß -core at level à The L function is local iff Lf:"hgi&PRLf:;"%le:;"hg &^& The functions L t L from examples are local; LU is not local for 8áH>Ý. In the following we shall assume also that for the function L there exists a constant L Œ such that 9;:j<)0?hLf:;"¹ {&orl Œ For a local L function an â.~ž UÛã-"%ä ÌxDv( &^& algorithm for determining L -core at level exists (assuming that Lf:;"%le:;" 35&^& can be computed in â ÅACED;åa:&^& ). INPUT: graph ns`! #"%$'& represented by lists of neighbors and ṽ< IR OUTPUT: 3b6k, 3 is a L -core at level 1. 3b? ; 2. for : <) do Lfæ :cç? RLf:;"%le:;" 35&^& ; 3. Õ ŠNèÅ.ºN!( éøôlf:;"ålb& ; 4. while Lfæ ê LëçUO> do begin I? ~3 \ ìê LU ; 4.2. for :j<ylm ê¹l " 35& do begin Lfæ :{ç? RLf:"%lm:" 3&^& ; Š{L; ÓÔc ^Ø éøôl :"ÅLB& ; end; end; The step can often be speeded up by updating the L æ :{ç. This algorithm is straightforwardly extended to produce the hierarchy of L -cores. The hierarchy is determined by the core-number assigned to each vertex the highest level value of L -cores that contain the vertex.

8 Ê 3.2 Determining the hierarchy of L -cores Determining the hierarchy of ß -cores INPUT: graph ns`! #"%$'& represented by lists of neighbors OUTPUT: table Öhê í{ø with core number for each vertex 1. 3b? ; 2. for : <) do Lfæ :cç? RLf:;"%le:;" 35&^& ; 3. Õ ŠNèÅ.ºN!( éøôlf:;"ålb& ; 4. while în cøê Ù#Ûé;ØÔLB&aï do begin I? ~3 \ ìê LU ; 4.2. ÖhêŸí{ØÓæ ìê Lëç? elfæ ìê Lëç ; 4.3. for :j<ylm ê¹l " 35& do begin Lfæ :{ç? pž ^Lfæ ê¹lç!"ålf:"%lm:" 3&^&¹ ; Š{L; ÓÔc ^Ø éøôl :"ÅLB& ; end; end; Let us assume that ð is a maximum time needed for computing the value of Lf:"hgi&, : <>, g_6+. Then the complexity of statements is ñpt^±yò«â ( &fkkâjåð ( &fk âj(iä ÌxDP( &aiâ (¾ó Ž ²Åð "%ä ÌxDo( &^&. Let us now look at the body of the while loop. Since at each repetition of the body the size of the set 3 is decreased by 1 there are at most ( repetitions. Statements 4.1 and 4.2 can be implemented to run in constant time, thus contributing ñ zhô,t zhô w «â ( & to the loop. In all combined repetitions of the while and for loops each line is considered at most once. Therefore the body of the for loop (statements 4.3,1 and 4.3.2) is executed at most. times contributing at most ñfzhô y\k.õóåð K5âjÅä ÌxDa( &^& to the while loop. Often the value Lf:;"%le:;" 35&^& can be updated in constant time ð âj &. Summing up all the contributions we get the total time complexity of the algorithm ñ Wñ t^±y KRñ zhô,t zhô w KRñ zhô y ~â.@óžj ²Åð "%ä ÌxDo( &^&. For a local L function, for which the value of Lf:;"%le:;" 35&^& can be computed in âjåacedóå :&^& ), we have ðs â Ûãj&. The described algorithm is partially implemented in program for large networks analysis Pajek (Slovene word for Spider) for Windows (32 bit) [1]. It is freely available, for noncommercial use, at its homepage: A standalone implementation of the algorithm in C is available at For the property functions Lft L;z a quicker âj.r& core determining algorithm can be developed [4].

9 Ê 4 Example Internet Connections 7 Table 1: L -cores of the Routing Data Network at Different Levels. 8 ( 8 ( Example Internet Connections As an example of application of the proposed algorithm we applied it to the routing data on the Internet network. This network was produced from web scanning data (May 1999) available from It can be obtained also as a Pajek s NET file from It has ö Ý xøg vertices, ùxø öxù arcs (loops were removed), ã@ øg, and average degree is Þ µ Þ. The arcs have as values the number of traceroute paths which contain the arc. Using Pajek implementation of the proposed algorithm on 300 MHz PC we obtained in 3 seconds the LB -cores segmentation presented in Table 1 there are ( ú vertices with L -core number in the interval %ú ±²t "^ ìúç. The program also determined the L² -core number for every vertex. Figure 3 shows a L -core at level of the Internet network every vertex inside this core is visited by at least traceroute paths. In the figure the sizes of circles representing vertices are proportional to (the square roots of) their LU -core numbers. Since the arcs values span from 1 to they can not be displayed directly. We recoded them according to the thresholds ÊxÊxÊ arcs. ócö ú ±²t, 8 "¹ö "¹ÞaµEµEµ. These class numbers are represented by the thickness of the

10 4 Example Internet Connections 8 Figure 3: L -core of the Routing Data Network.

11 5 Conclusions 9 5 Conclusions The cores, because they can be efficiently determined, are one among few concepts that provide us with meaningful decompositions of large networks [5]. We expect that different approaches to the analysis of large networks can be built on this basis. For example, the sequence of vertices in sequential coloring can be determined by descending order of their core numbers (combined with their degrees). We obtain on this basis the following upper bound on the chromatic number of a given graph û º&ašsóK core º& Cores can also be used to localize the search for interesting subnetworks in large networks [3, 2]: M If it exists, a 8 -component is contained in a 8 -core. M If it exists, a 8 -clique is contained in a 8 -core. ü»º&aš core ¾&. Acknowledgment This work was supported by the Ministry of Education, Science and Sport of Slovenia, Project J It is a detailed version of the part of the talk presented at Recent Trends in Graph Theory, Algebraic Combinatorics, and Graph Algorithms, September 24 27, 2001, Bled, Slovenia,

12 REFERENCES 10 References [1] BATAGELJ, V. & MRVAR, A. (1998). Pajek A Program for Large Network Analysis. Connections 21 (2), [2] BATAGELJ, V. & MRVAR, A. (2000). Some Analyses of Erdős Collaboration Graph. Social Networks 22, [3] BATAGELJ, V., MRVAR, A. & ZAVERŠNIK, M. (1999). Partitioning approach to visualization of large graphs. In KRATOCHVÍL, Jan (ed.). Proceedings of 7th International Symposium on Graph Drawing, September 15-19, 1999, Štiřín Castle, Czech Republic. (Lecture notes in computer science, 1731). Berlin [etc.]: Springer, [4] BATAGELJ, V. & ZAVERŠNIK, M. (2001). An â.r& Algorithm for Cores Decomposition of Networks. Manuscript, Submitted. [5] GAREY, M. R. & JOHNSON, D. S. (1979). Computer and intractability. San Francisco: Freeman. [6] SEIDMAN, S. B. (1983). Network structure and minimum degree. Social Networks 5, [7] WASSERMAN, S. & FAUST, K. (1994). Social Network Analysis: Methods and Applications. Cambridge: Cambridge University Press.