DISTINGUISHING VERTICES OF INHOMOGENEOUS RANDOM GRAPHS. Paolo Codenotti. IMA Preprint Series #2419. (July 2013)

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1 DISTINGUISHING VERTICES OF INHOMOGENEOUS RANDOM GRAPHS By Paolo Codeotti IMA Preprit Series #2419 (July 2013) INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS UNIVERSITY OF MINNESOTA 400 Lid Hall 207 Church Street S.E. Mieapolis, Miesota Phoe: Fax: URL:

2 Distiguishig Vertices of Ihomogeeous Radom Graphs Paolo Codeotti Istitute for Mathematics ad its Applicatios Uiversity of Miesota 207 Church Street SE, 306 Lid Hall, Mieapolis, MN July 25, 2013 Abstract We explore uder what coditios simple combiatorial attributes ad algorithms such as the distace sequece ad degree-based partitioig ad refiemet ca be used to distiguish vertices of ihomogeeous radom graphs. I the classical settig of Erdős-Reyi graphs ad radom regular graphs it has bee prove that vertices ca be distiguished i a costat umber of rouds of degree-based refiemet or the distace sequece at a logarithmic distace. This yields a high-probability caoical labelig algorithm, ad hece a efficiet high-probability isomorphism test. I this paper we aalyze the same attributes i the cotext of radom graphs that come from distributios that more closely model real-world etworks. We first prove a techical result about the effects of oe refiemet step i the geeral settig were edges are chose idepedetly at radom. This allows us to prove that a algorithm based o distiguishig vertices yields a caoical labelig for graphs with scale-free degree distributio where edges are added idepedetly, ad for Stochastic Kroecker Product Graphs with certai settigs of the parameters. Alog the way we prove results o the degree distributio, coectedess ad diameter of Stochastic Kroecker Graphs with geeratig matrices of arbitrary size. 1 Itroductio Several radom graph models have bee proposed over the last two decades to model real-world etworks. For a more complete bibliograpy we refer the reader to the followig books [19, 33], ad review articles [11, 36]. These models have applicatios to a variety of fields. I computer sciece they are used for modelig the Iteret, the World Wide Web, ad social etworks [2, 3, 15, 25, 24, 27]. I biology they are used as models for protei-protei, gee-gee, ad gee-protei iteractio etworks [38]. Radom etwork models have also bee extesively used i statistics ad social scieces (see e. g. [17]). The ew radom graph models try to explai properties such as small-world, clusterig coefficiets, ad shrikig diameters, which are ot captured by the classical models of Erdős-Reyi This research was supported by the Istitute for Mathematics ad its Applicatios with fuds provided by the Natioal Sciece Foudatio. 1

3 graphs [20] ad radom regular graphs [10, 9]. Oe importat differece betwee the ew models ad the classical models is that the vertices are ot homogeeous. We explore uder what coditios simple combiatorial attributes which ca be computed efficietly such as the distace sequece ad a degree-based partitioig ad refiemet procedure ca be used to distiguish vertices of ihomogeeous radom graphs. I the classical settig of Erdős-Reyi graphs ad radom regular graphs it has bee prove that vertices ca be distiguished i a costat umber of rouds of degree-based refiemet ad by the distace sequece at a logarithmic distace [4, 5, 9, 10]. This yields a high-probability caoical labelig algorithm, ad hece a efficiet high-probability isomorphism test. I this paper we aalyze the same attributes i the cotext of radom graphs that come from distributios that more closely model real-world etworks. Our mai results are for settigs where edges are added idepedetly at radom. I geeral, let P be a matrix with etries 0 P [i, j] 1. We cosider a probability distributio G(, P ) o the set of all graphs o vertices. A graph G G(, P ) is selected by idepedetly addig each edge (i, j) with probability P [i, j]. I other words P(G) = P [i, j] (1 P [i, j]). (i,j) E(G) (i,j)/ E(G) The mai applicatio of our techique i this paper is the followig (see subsectio 2.2 for the precise defiitio of partitio-refiemet ad the distace sequece): Theorem 1.1 Let P [i, j] = Θ(1/ ij), ad pick the graph G G(, P ). Note that with high probability G will have scale-free degree distributio. With probability 1 o(1/), we have: deg(1) > deg(2) > > deg(k), for all k = o( 1/5 (l ) 2/5 ). I other words the degree of every vertex i = o( 1/5 (l ) 2/5 ) is uique i G. ( ) Every vertex i = O is uiquely determied by the degrees of its eighbors. l 8 We call two vertices equivalet if they have the same set of eighbors. Every vertex that is coected to the giat coected compoet will be distiguished from every other vertex that is ot equivalet to it by partitio-refiemet iitialized by the distace sequece. This theorem implies that our algorithm is a caoical labelig algorithm for almost every graph i the distributio G(, P ). I subsectio 3.1 we prove the more geeral techical results that we eed for the theorem above. These results ca be used i various other applicatios, for example they also uderly our results for Stochastic Kroecker Product Graphs [27]. We defer the defiitio of this model to sectio 4. Our mai result i this settig is the followig: i a stochastic Kroecker graph, every vertex that has expected degree at least ɛ, for ay costat ɛ will be distiguished after oe roud of aive partitio-refiemet. I fact, this gives us a caoical labelig algorithm for the settig where the Stochastic Kroecker Graphs are coected with high probability. The precise statemet is more ivolved, ad requires to first prove results about the degree distributio ad coectedess of Stochastic Kroecker Graphs with geeratig matrices of arbitrary size. These had previously oly bee aalyzed for 2 2 geeratig matrices [30]. We study the above radom graph models for two reasos: they are mathematically clea, ad several other models build upo them (see e.g. [25, 22]). Simulatios show that these attributes 2

4 ca be used to distiguish vertices i very few rouds eve for preferetial attachmet graphs [3], where the edges are ot idepedet. Our algorithms ca be easily implemeted i a distributed settig. Wheever the algorithms require few rouds, the correspodig distributed implemetatios become very efficiet, with few commuicatio steps, ad low ruig time. Fially we ote that procedures to distiguish vertices ca be useful eve i cases where they do ot distiguish all the vertices i the graph (ad hece do ot obtai a caoical labelig). For example, they ca be used for leader selectio i distributed settigs, for ati-aliasig, or to reduce the search space for ode correspodece i maximum-likelihood parameter estimatio. Our mai result shows that i graphs that exhibit a core-periphery structure, vertices i the core will be easier to distiguish. This is good ews, sice these are the more iterestig vertices. The rest of the paper is orgaized as follows. I sectio 2 we discuss related work ad precisely describe the algorithms. I sectio 3 we preset our mai techical tools ad apply them to scalefree radom graphs with idepedet edges. The i sectio 4 we discuss our results for stochastic Kroecker graphs. I sectio 5 we preset our experimetal results for preferetial attachmet graphs. We coclude ad discuss further work i sectio 6. 2 Backgroud ad Prelimiaries 2.1 Graph Isomorphism The Graph Isomorphism problem asks whether two iput graphs are isomorphic or ot. Graph Isomorphism has a very iterestig computatioal complexity status: it is i co-am, ad hece is ot NP-complete uless the polyomial-time hierarchy collapses to the secod level [8]. However, we do ot kow ay polyomial-time algorithms for this problem. Graph Isomorphism is the oly importat atural problem with this complexity status. Graph Isomorphism has applicatios to several areas, icludig chemistry ad SAT-solvers, ad there is cosiderable iterest i practical algorithms to solve the problem. There are software packages [31, 21, 18] that perform well o most istaces that arise i practice, but take expoetial time o some istaces [32]. The fastest worst-case algorithm for Graph Isomorphism rus i time exp(o( log )) [7]. This algorithm was obtaied by combiig a combiatorial trick due to Zemlyacheko [39] ad Luks polyomial-time isomorphism test for bouded degree graphs [29]. Luks bouded degree result is the crowig achievemet of the group theoretic methods for graph isomorphism, first itroduced by Babai i 1979 [6]. 2.2 Algorithms We ow describe two simple algorithms to distiguish vertices of graphs: degree-based partitio ad refiemet, ad distace-sequece computatio. Degree sequece ad refiemet. Partitioig ad refiemet is a effective heuristic for graph isomorphism (cf. Read ad Coreil [35]). This heuristic is at the heart of most practical graph isomorphism algorithms (see e.g. [31, 21]). For almost all graphs this techique provably works i polyomial time, givig some theoretical explaatio for its success [4, 5]. 3

5 We cosider colored graphs. The refiemet procedure will work for ay colorig, but oe good example to keep i mid for this discussio is to iitially color vertices by their degree. The goal of the refiemet techique is to get closer to a colorig of the vertices such that each vertex gets a uique color. Give a graph X = (V, E) ad a colorig c : V C, where C is a set of colors, oe refiemet step computes a ew colorig c that is a refiemet of c. The ew color of a ode is defied by the multiset of colors of the eighbors, more precisely, c (v) := (c(v), {c(u) (u, v) E}) v V. We call the colorig c a ivariat if it is preserved by all automorphisms. Note that ay isomorphism that preserves c must preserve c. Hece if c was ivariat, the so is c. The set of potetial colors of c may be large, but at most colors actually appear, so we ca umber those colors that do appear, ad assume c : V {1,..., }. This trick allows us to recursively apply the refiemet step util we reach a stable colorig, that is, ay further refiemet does ot chage it. The umber of refiemet steps that ca occur is at most, sice if c c, at least oe color class must be partitioed. Therefore the overall ruig time is bouded by O(( + m)). If the umber of refiemet steps, r, is small, the the algorithm ca be implemeted as a r-roud distributed distributed algorithm, with ru time r, where is the maximum degree. This is a geeral procedure that works for ay iitial vertex-colorig of a graph. Ofte it is used as a subroutie i isomorphism algorithms. The Naive Refiemet algorithm starts by colorig each vertex accordig to its degree, ad recursively refies the colorig util a stable colorig is reached. Babai-Erdős-Selkow [4] who showed that apart from a 1/7 fractio of graphs, the Naive Refiemet procedure fids a caoical labelig i oe refiemet step. The heart of their proof is a estimate o the top degrees of vertices i a radom graph: they show that for every costat c, for almost all graphs, the highest c log degrees are distict. Usig a more complicated argumet, Naive Refiemet was show to caoically label all but a expoetially small fractio of graphs i just three refiemet steps [5]. Distace sequece. For radom graphs the aive refiemet procedure based o degrees described above does ot eve start. To distiguish vertices of radom regular graphs, it is useful to look at aother combiatorial attribute of vertices, the distace sequece [9, 10]. For a vertex v, let d i (v) be the umber of vertices at distace i from v. The distace sequece of vertex v is the sequece of d i (v). Bollobas proved that for (Erdős-Reyi) Radom Graphs, ad radom regular graphs, with probability 1, the distace sequece up to legth l 0 O(log ) uiquely determies every vertex i the graphs [9, 10]. I particular this gives a caoical labelig procedure for almost every regular graph, ad hece a isomorphism test for these graphs ad ay other graph. Our algorithm Our goal is to see how far these combiatorial techiques go for other distributios o the set of all graphs. The most geeral versio of our algorithm is to first color every vertex by its distace sequece, the ru partitio-refiemet. I fact, this will be overkill for may vertices. We show that high degree vertices will be distiguished by few rouds of aive partitio-refiemet (precise bouds for this statemet will deped o the model, see Theorem 1.1 for a example). Low degree vertices require computig the distace sequece ad possibly more rouds of refiemet. 4

6 3 Idepedet edges I this sectio we prove our mai techical tools for distiguishig vertices. We cosider the followig radom graph model. Give a iteger, ad a matrix P of probabilities, we costruct a radom graph G(P, ) with vertices, ad edges added at radom with P((i, j) E) = P [i, j]. 3.1 Coi tosses with o-uiform probabilities Here we cosider the settig where we toss t idepedet cois, each with a (possibly differet) probability p i to come up heads. First we prove a specific cocetratio boud. Lemma 3.1 We toss t cois, each has probability p i to come up heads. Let λ = p 1 + +p t = E[# 2y l t of heads ]. Let s deote the umber of heads. ɛ > 0, y > 0, let z = max{λ, }, the ɛ 2 P[ s λ > ɛz] t y. Proof. The radom variable s is the sum of t idepedet Beroulli radom variables. By Beett s iequality, we have: P[ s λ > ɛz] exp [ ɛz ( 2 l 1 + ɛz )] λ exp [ z ] 2 ɛ l(1 + ɛ) [ exp y l t ] ɛ 2 ɛ l(1 + ɛ) exp [ y l t] = t y The first iequality is Beett s iequality. The secod follows sice z λ, hece z/λ 1. The 2y l t third iequality sice z, ad the fourth by the fact that l(1 + ɛ) ɛ. ɛ 2 Remark. I particular this applies for a vertex u, where λ = E[deg(u)], ad t =. Next we boud the probability of attaiig ay particular value. Lemma 3.2 We toss t cois, each has probability p i to come up heads. Let λ = p 1 + +p t = E[# of heads ], ad γ = max i p i. For every x, P(exactly x heads) eλγ 2π 1 λ Proof. Let f(x) = P(exactly x heads). The probability distributio f(x) will ot be too far from a Poisso distributio: f(x) λx e λ+xγ, (1) x! While the proof of Equatio (1) is elemetary, ad the settig is a commo settig i statistics, this was oly proved by Wag i 1993 [37, Equatio (31)]. 5

7 Next we use the followig form of Stirlig s approximatio(c.f. [1, ]): 2π +1/2 e! e 1/12 2π +1/2 e (2) Combiig Equatios (1) ad (2), we obtai: f(x) λx e λ+xγ 2πx x+1/2 e 1 λ λ e λ+λγ 1 x 2π λ λ e λ λ = eλγ 2π λ, where the secod iequality follows sice the fuctio is maximized for x = λ. Cosider the radom graph model G(P, ), ad fix a vertex v. Now suppose that there are k color classes i the graph. Let λ c = λ c (v) be the expected umber of eighbors of v that have color c, ad let γ c = γ c (v) be the maximum probability of ay edge from v to a vertex of color c: γ c = max u c(u)=c P [v, u]. Lemma 3.3 Fix a vertex v, ad a set of colors S C. Usig the otatio from above, let λ mi = mi c S λ c, ad K = max c S γ c λ c. The the probability that v is uiquely determied by the colors of its eighbors (i. e. that v has a uique color after oe step of refiemet) is 1 y, where y = S (l( 2πλ mi ) K) l. Proof. For a vertex v, let C(v) be the multi-set of colors of the eighbors of v. The for two vertices u, v, we have P(C(u) = C(v)) max P(C(v) = S) S max By Lemma 3.2, we have that: x 1,...,x k c k P(v has exactly x c eighbors of color c). P(C(u) = C(v)) e γcλc e K 2πλc 2πλmi c S c S = exp ( S (K l(2 π)) ) = exp( y l ) = y. Corollary 3.4 If there is a vertex v ad set of colors S with S 8 l, ad such that for all c S, λ c 2 ad γ c λ c < 1, the the probability of v beig uiquely determied by the colors of its eighbors is Applicatio to radom scale-free graphs I this subsectio we prove Theorem 1.1. If we choose P [i, j] = θ(1/ ij), the the expected umber of vertices of degree d will be θ(/d 3 ), ad with high probability, the umber of vertices of degree d will be cocetrated aroud its expected value (cf. [12]) Lemma 3.5 For ay costat y, with probability 1 o( y ), every vertex i = o( 1/5 (l ) 2/5 ) will have a uique degree. 6

8 Proof. First we show that vertices i for i = o( 1/3 / l ) will all have distict degrees with high probability. Let ( ) λ i = E[deg(i)] = θ, i ad ( ) δ i = (λ i λ i+1 )/2 = Θ = Θ( i 3/2 ). i i + 1 We have P[deg(i) > deg(i + 1)] (1 P[deg(i) < λ i δ i ])(1 P[deg(i + 1) > λ i+1 + δ i ]) Sice δ i > δ i+1, it is eough to show that. P[ deg(i) λ i > δ i+1 ] < y, y = y + 1/3 We apply Lemma 3.1, with t =,, ad set ɛ = C y l i3/2 for some (small) costat C depedig o the Θs above. Note that sice i = o( 1/5 (l ) 2/5 ), we have that ɛ < 1/i. By Lemma 3.1, with probability 1 y, we have ( deg(i) λ i < ɛz < max ɛλ i, ) 2y l < C ɛ i 3/2 < δ i+1. (C is aother costat we ca make as small as we wat by settig C). Lemma 3.6 With probability 1 o( 1 ) every vertex i / log 8 is uiquely determied by the degrees of its eighbors. Proof. Now cosider the eighbors of vertex i. We wat to apply Corollary 3.4 i this case, so we are iterested i which values of d ad i are such that E[ N c (i) ] 2, where N c (i) stads for the eighbors of i of degree (color) c. Suppose i log 8. So if i log 8, the E[ N c (i) ] = P [i, j] = 1 ij j : d(j)=c j : d(j)=c ( ) ( ) = Ω d 3 = Ω i d 3. i E[ N c (i) ] Ω c 3 log 8 ( log 4 ) = Ω c 3 which will be at least 2 for c T log, for ay costat T. Notice further that all vertices j have expected degree Ω ( 1/4), ad hece, by Lemma 3.1 have degree greater tha T log. Therefore for the colors c we are cosiderig, we have γ c 1/, ad λ c γ c 1. So by Corollary 3.4, with probability o( 1 ), all vertices i will be uiquely determied by the degrees of their log 8 eighbors. 7

9 We ow tur our attetio to other vertices. Notice that there may be some isolated vertices, or vertices that are part of very small coected compoets, ad we caot hope to distiguish these. Notice further that a costat fractio of vertices will have a costat umber of eighbors, so we caot hope to distiguish them just by the degrees of the eighbors. Moreover some of the vertices of low degree will share all their eighbors. We call such sets of vertices equivalet. However we ca still say somethig about our geeral algorithm. Recall this algorithm first colors vertices by their degree sequece, ad the rus partitio-refiemet startig from this colorig. We ca show that with high probability all vertices that are coected to the giat coected compoet of the graph will be distiguished by this algorithm from all other vertices they are ot equivalet to. Lemma 3.7 If v is a vertex that is coected to the giat coected compoet, the with probability 1 o( 1 ), the mai algorithm will distiguish v from all other vertices exceptio those equivalet to v. Sketch of the proof. This proof is more techical tha the previous oes. First of all, we oly eed to cosider vertices i >. The idea is to look at two such vertices u, ad v, ad grow two log 8 BFS startig at u ad v. If ay BFS layer for v has a differet umber of odes tha u the we distiguished u ad v. If at ay poit the BFS layer is large eough, the with high probability it will cotai some vertices i, ad with high probability these will be differet for u ad log 8 v, ad sice these are already uiquely labelled we are doe. The importat detail that we swept uder the rug is that we eed to maitai the property that the BFS levels for u ad v remai disjoit eough uless they are equal after oe step (i which case u ad v are equivalet). 4 Stochastic Kroecker Graphs 4.1 Motivatio Kroecker prodcut graphs ad stochastic Kroecker product graphs were itroduced to model real world etworks [26, 27]. From the theoretical side, Mahdia ad Xu [30] studied the case of 2 2 iitiator matrices, fidig thresholds for coectivity, the emergece of a giat coected compoet, ad provig that the effective diameter is shrikig for certai rages of the parameters. While the case of larger iitiator matrices has bee used i may cotexts(see [27]), we are ot aware of theoretical results i this case. Further motivatio to study iitiator matrices of arbitrary size arises from the relatioship to a differet radom graph model. It has bee show that, i the limit to a ifiite umber of odes, the adjacecy matrix of a dese graph is well approximated by a cotiuous fuctio o the uit square [28, 16] (see also [11, 14] for similar results i the sparse graph case). This fact has bee used to geerate multifractal radom etworks by a cotiuous aalog to the Kroecker product [34]. A iitiator matrix of arbitrary size ca be used as a approximatio to a cotiuous fuctio o the uit square, ad hece stochastic Kroecker product graphs with large iitiator matrices ca model multifractal radom etworks. Oe importat differece betwee the two models is that multifractal radom etworks require row sums of 1, ad hece i the ifiite limit, the graphs are ot iterestig: o() vertices will be isolated [34, Supportig Iformatio]. The stochastic Kroecker graph model is more geeral i the sese that row sums ca be arbitrary. As we will see, the graphs will be coected exactly whe the row sums are greater tha 1. 8

10 4.2 Defiitios We will deote by A[i, j] or A ij the etry i row i, colum j of a matrix A. The Kroecker product of a m matrix A with etries a i,j = A[i, j], ad a p q matrix B with etries B[k, l] is the mp q matrix C = A B. We will idex the rows of C by pairs (ik) [m] [p], ad the colums by pairs (jl) [] [q]. The etries of C are defied by: C[(ik), (jl)] = A[i, j]b[k, l]. I other words, A B = a 11 B a 1 B..... a m1 B a m B Give a r r iitiator matrix θ, with etries i [0, 1], ad a iteger k, we costruct a radom graph G(r k, θ) with = r k vertices, each vertex labelled by a uique strig of legth k over the alphabet [r]. We add each edge idepedetly at radom. Give two vertices u ad v with labels u 1 u 2... u k ad v 1 v 2... v k resp., the probability with which we chose edge (u, v) is P [u, v] = k i=1 θ[u i, v i ]. I other words, the matrix P = θ k = θ θ θ is the Kroecker product of θ with itself k times. For a vertex u with label u 1 u 2... u k, ad j [r], let w j be the umber of coordiates i such that u i = j. We say that u has weight vector (w 1,..., w r ). 4.3 Expected Degrees ad Cocetratio Results First we prove that the expected i- ad out-degrees of the vertices are distributed like multiomials. Lemma 4.1 The expected i- ad out-degree of a vertex u with weight vector (w 1,..., w r ) are: E[d + u ] = (θ θ 1r ) w 1 (θ θ 2r ) w 2... (θ r1 + + θ rr ) wr ; ad E[d u ] = (θ θ r1 ) w 1 (θ θ r2 ) w 2... (θ 1r + + θ rr ) wr. Proof. We prove the result for the out-degree, d + u, the result for i-degree is prove aalogously. For otatioal coveiece, let θ i = θ r1 + + θ rr be the row-sums of the iitiator matrix θ. For every vertex v, let I (u,v) be the idicator fuctio of the evet that (u, v) is a edge. The E[d + u ] = E[ v I (u,v)]. By the defiitio of the Kroecker product, E[I (u,v) ] = P[(u, v)] = k i=1 θ[u i, v i ]. Therefore we have E[d + u ] = k v i=1 θ[u i, v i ]. For l, m [r], let l,m be the umber of coordiates i such that u i = l, ad v i = m. Note that P[(u, v)] = l,m [r] θ[l, m] l,m; therefore we ca re-write the above sum as: 9

11 ( ) E[d + w 1 u ] = 1, ,r =w 1,1,... 1,r 1 ( ) r... w r r θ[l, m] l,m... r,1 + + r,1,... r,r r,r=w r l=1 m=1 ( ) r w 1 = θ[1, m] 1,m 1, ,r =w 1,1,... 1,r 1 m=1 ( ) r... w r θ[r, m] r,m... r,1 + + r,1,... r,r r,r=w r m=1 ( ) r w 1 = θ[1, m] 1,m 1, ,r =w 1,1,... 1,r 1 m=1... (θ[r, 1] + + θ[r, r]) wr 2, ,r =w 2 = (θ 1 ) w 1 (θ 2 ) w 2... (θ r ) wr. Lemma 3.1 gives us cocetratio results for the degrees of the vertices. 4.4 Coectedess We ow tur our attetio to determiig for what iitiator matrices the stochastic Kroecker graphs will be coected. As above, we have P = θ k. For subsets S, T V, let P [S, T ] = s S,t T P [s, t]. We use the followig result: Theorem 4.2 ([30, Theorem 1]) There exists a (absolute) costat c such that if S V, S, P (S, V \ S) c l, the with high probability G(, P ) is coected. For u, v i V, let e(u, v) be the edit distace betwee u ad v, that is, the umber of coordiates i such that u i v i. First we cosider the settig where the iitiator matrix has o zero etries. I this case, we defie θ mi = mi i,j θ[i, j], θ max = max i,j θ[i, j], ad θ R = θ max /θ mi. The followig lemma plays the role of [30, Lemma 2]. Lemma 4.3 Suppose all etries of the iitiator matrix are o-zero. The for ever u, v V, ad every S V, P [v, S] P [u, S] θ R P [v, S]. θ R 10

12 Proof. Without loss of geerality, u ad v differ i the last coordiate, r. P [u, S] = P [u, w] = r r 1 θ ui w i θ mi w S w S i=1 w S i=1 r 1 r = θ mi θ vi,w i θ mi θ vi w i /θ max w S i=1 w S i=1 = w S P [v, w]/θ R. θ ui w i We are ow able to describe sufficiet ad ecessary coditios for coectivity. Theorem 4.4 Let θ be symmetric, with o zero etries. The G(, P ) is coected with high probability if ad oly if the row sums satisfy θ ij > 1 for every i. j Proof. We first prove the oly if. If i such that θ i < 1, the with costat probability, the vertex i = (i, i,..., i) has expected degree θi k = o(1), hece it will be isolated with probability 1. If θ i = 1, the either i is discoected with probability at least e 2, or row i is determiistic (exactly oe 1 ad every other positio 0). But the secod case caot happe sice we assumed θ does t have ay zero etries. The proof follows alog the lies as the proof that 0 is isolated i [30, Theorem 4]: P[ i isolated] = (1 P [ i, v]) v = w 1 + +w r=k w 1 + +w r=k = exp 2 (1 θ[i, 1] w 1 θ[i, 2] w 2... θ[i, r] wr ) ( k w 1,...,wr ) ( ( ) k exp 2 )θ[i, 1] w 1... θ[i, r] wr w 1,..., w r ( ) k θ[i, 1] w 1... θ[i, r] wr w 1,... w k w 1 + +w r=k ( = exp 2(θ[i, 1] + + θ[i, r]) k) ( = exp 2(θ i ) k). So, as claimed, if θ i < 1, the probability that vertex i is isolated is 1 o(1), if θ i = 1, the vertex i is isolated with probability at least e 2. Now for the other directio. Suppose θ i > 1 for every i. Our goal is to show that P satisfies the coditios of Theorem 4.2. Let S V, S. There exist u S ad v V \ S that differ i oly 1 coordiate. We have [u, V \ {u}] = E[deg(u)] = r (θ i ) wi(u) (θ mi ) k. i=1 11

13 Sice k = log (i some base), for k large eough, we have (θ mi ) k (θ R + 1)c l. Now either P [u, V \ S] c l or P [u, S] c θ R l = P [v, S] c l. The last implicatio follows by Lemma 4.3. Ad therefore P [S, V \ S] c l. I Appedix A we discuss possible extesios of the coectedess result to the settig where the iitiator matrix has zero etries. 4.5 Distiguishig vertices Theorem 4.5 I a stochastic Kroecker graph, with high probability every vertex that has expected degree at least ɛ, for ay costat ɛ will be distiguished after oe roud of aive partitiorefiemet. Sketch of the proof. By Lemma 3.1, ad Lemma 4.1, with high probability all vertices of degree at least ɛ will have degree at least ɛ/2. The result follows alog the lies of the proofs of Lemmas 3.5 ad 3.6. Remark. I the settig where the iitiator matrix has o zero etries ad the graph is coected with high probability, the by Lemma 4.1 the expected degree of every vertex is at least ɛ for some epsilo, so the theorem above allows us to distiguish all vertices. I particular we obtai a caoical labelig algorithm. We expect that it is possible to exted the above theorem to other vertices, usig a more careful aalysis. 5 Experimetal results for preferetial-attachmet graphs We ow cosider preferetial attachmet graphs. This model was first suggested by Albert- Barabasi [3], ad aalyzed more carefully by Bollobas et al. [15, 12, 13]. We cosider the model of Bollobas et al. [15]. There is oe parameter, m, which is half the average degree. A preferetial attachmet (multi)graph with vertices is costructed as follows. Start with two vertices ad m edges betwee them. Now add the remaiig vertices oe at a time. Whe addig a vertex v, add m edges to it, where each edge (v, u) has probability proportioal to deg(u). The idea of this model is that popular vertices are more likely to be liked i the future. For m = 1, the graph will be a tree, ad it is kow that trees (o-radom) are caoically labelled by the partitio-refiemet techique (see [35]). For m = 2, with costat probability there will be at least oe pair of vertices of degree 2 with exactly the same eighbors; hece these vertices caot be distiguished. This is easy to compute directly (see [23]). For m 3 with high probability there are o such pairs (agai see [23]). For both m = 2 ad m = 3 we geerated 10 5 graphs with 10 5 vertices by preferetial attachmet. We the ra the aive partitio-refiemet algorithm ad computed distace sequeces (recall the defiitios from subsectio 2.2). Both algorithms distiguished all vertices of the graphs, except (i the case m = 2) those pairs of degree 2 vertices that share the same eighbors. Ideed aive partitio refiemet coverged i at most three rouds o all graphs. I Appedix B we discuss some prelimiary thoughts o provig that partitio refiemet ad the distace sequece distiguish all vertices for almost all preferetial attachmet graphs, as the experimets suggest. 12

14 6 Coclusios ad future work The gap betwee mathematically ice radom models ad real-world etworks is a iterestig ad promisig area of research. We show that we ca distiguish vertices ad the develop efficiet algorithms for several importat families of real-world etworks. O the other had, there are examples of real-world etworks for which caoical labelig is difficult for these simple algorithms. It is iterestig to study the questio of distiguishig vertices i other radom graph models. We believe that the results for stochastic Kroecker graphs apply i other settigs of the parameters; extedig our results requires uderstadig the behavior of the low-degree vertices i this settig. Aother directio is to prove theoretical results for models where edges are ot added idepedetly, such as preferetial attachmet. We suspect that for most models vertices will be distiguished i a few rouds of the combiatorial algorithms described i this paper. Fidig hard examples for graph isomorphism is a otoriously difficult task. If there were models where distiguishig vertices is difficult, this would provide a class of such graphs. So both positive ad egative results i this directio are iterestig. Refereces [1] M. Abramowitz ad I.A. Stegu, editors. Hadbook of Mathematical Fuctios. Number 55 i Applied Mathematics Series. Natioal Bureau of Stadards, teth editio, [2] William Aiello, Fa Chug, ad Liyua Lu. A radom graph model for massive graphs. I Proceedigs of the thirty-secod aual ACM symposium o Theory of computig, STOC 00, pages , New York, NY, USA, ACM. [3] R. Albert ad A.L. Barabási. Statistical mechaics of complex etworks. Reviews of moder physics, 74(1):47, [4] L. Babai, P. Erdős, ad S.M. Selkow. Radom graph isomorphism. SIAM Joural o Computig, 9(3): , [5] L. Babai ad L. Kucera. Caoical labellig of graphs i liear average time. I Foudatios of Computer Sciece, 1979., 20th Aual Symposium o, pages IEEE, [6] László Babai. Mote carlo algorithms i graph isomorphism testig. I Uiversité de Motréal Techical Report, volume 70 of DMS, page 42, laci/lasvegas79.pdf. [7] László Babai ad Eugee M. Luks. Caoical labelig of graphs. I Proc. 15th STOC, pages ACM Press, [8] László Babai ad Shlomo Mora. Arthur-Merli games: A radomized proof system, ad a hierarchy of complexity classes. Joural of Computer ad System Scieces, 36(2): , [9] B. Bollobás. Distiguishig vertices of radom graphs. North-Hollad Mathematics Studies, 62:33 49, [10] B. Bollobás. Radom graphs, volume 73. Cambridge Uiversity Press,

15 [11] B. Bollobás, S. Jaso, ad O. Riorda. The phase trasitio i ihomogeeous radom graphs. Radom Structures & Algorithms, 31(1):3 122, [12] B. Bollobás ad O. Riorda. The diameter of a scale-free radom graph. Combiatorica, 24(1):5 34, [13] B. Bollobás ad O. Riorda. Robustess ad vulerability of scale-free radom graphs. Iteret Mathematics, 1(1):1 35, [14] B. Bollobás ad O. Riorda. Sparse graphs: Metrics ad radom models. Radom Structures & Algorithms, 39(1):1 38, [15] Béla Bollobás, Oliver Riorda, Joel Specer, ad Gábor Tusády. The degree sequece of a scale-free radom graph process. Radom Struct. Algorithms, 18(3): , May [16] C. Borgs, J.T. Chayes, L. Lovász, V.T. Sós, ad K. Vesztergombi. Coverget sequeces of dese graphs i: Subgraph frequecies, metric properties ad testig. Advaces i Mathematics, 219(6): , [17] Nicholas A. Christakis ad James H. Fowler. The spread of obesity i a large social etwork over 32 years. New Eglad Joural of Medicie, 357(4): , [18] P. Cordella, L. P. Foggia, C. Sasoe, ad M. Veto. Evaluatig performace of the vf graph matchig algorithm. I Proc. of the 10th Iteratioal Coferece o Image Aalysis ad Processig, pages IEEE Computer Society Press, [19] David Easley ad Jo Kleiberg. Networks, Crowds, ad Markets: Reasoig About a Highly Coected World. Cambridge Uiversity Press, [20] P. Erdős ad A. Réyi. O the evolutio of radom graphs. Magyar Tud. Akad. Mat. Kutató It. Közl, 5:17 61, [21] H. Katebi, K. Sakallah, ad I. Markov. Coflict aticipatio i the search for graph automorphisms. I Logic for Programmig, Artificial Itelligece, ad Reasoig, pages Spriger, [22] Myughwa Kim ad Jure Leskovec. Multiplicative attribute graph model of real-world etworks. Iteret Mathematics, 8(1-2): , [23] R.D. Kleiberg ad J.M. Kleiberg. Isomorphism ad embeddig problems for ifiite limits of scale-free graphs. I Proceedigs of the sixteeth aual ACM-SIAM symposium o Discrete algorithms, pages Society for Idustrial ad Applied Mathematics, [24] R. Kumar, P. Raghava, S. Rajagopala, D. Sivakumar, A. Tomkis, ad E. Upfal. Stochastic models for the web graph. I Foudatios of Computer Sciece, Proceedigs. 41st Aual Symposium o, pages IEEE, [25] S. Lattazi ad D. Sivakumar. Affiliatio etworks. I Proceedigs of the 41st aual ACM symposium o Theory of computig, pages ACM,

16 [26] J. Leskovec, D. Chakrabarti, J. Kleiberg, ad C. Faloutsos. Realistic, mathematically tractable graph geeratio ad evolutio, usig kroecker multiplicatio. Kowledge Discovery i Databases: PKDD 2005, pages , [27] J. Leskovec, D. Chakrabarti, J. Kleiberg, C. Faloutsos, ad Z. Ghahramai. Kroecker graphs: A approach to modelig etworks. The Joural of Machie Learig Research, 11: , [28] L. Lovász ad B. Szegedy. Limits of dese graph sequeces. Joural of Combiatorial Theory, Series B, 96(6): , [29] Eugee M. Luks. Isomorphism of graphs of bouded valece ca be tested i polyomial time. J. Comp. Sys. Sci., 25:42 65, [30] Mohammad Mahdia ad Yig Xu. Stochastic kroecker graphs. I Proc. 5th iteratioal coferece o algorithms ad models for the web-graph, WAW 07, pages , [31] Breda D. McKay. Practical graph isomorphism. I Cogressus Numeratium, 30, pages 45 87, [32] T. Miyazaki. The complexity of McKay s caoical labelig algorithm. I L. Fikelstei ad W.M. Kator, editors, Groups ad Computatio, II, pages Amer. Math. Soc., [33] M. Newma. Networks: A Itroductio. Oxford Uiversity Press, Ic., [34] G. Palla, L. Lovász, ad T. Vicsek. Multifractal etwork geerator. Proceedigs of the Natioal Academy of Scieces, 107(17): , [35] R. C. Read ad D. G. Coreil. The graph isomorphism disease. Joural of Graph Theory, pages , [36] M. Salter-Towshed, A. White, I. Gollii, ad T. B. Murphy. Review of statistical etwork aalysis: models, algorithms, ad software. Statistical Aalysis ad Data Miig, 5(4): , [37] H Wag, Y. O the umber of successes i idepedet trials. Statistica Siica, 3: , [38] Gezhi Weg, Upider S Bhalla, ad Ravi Iyegar. Complexity i biological sigalig systems. Sciece, 284:92 96, [39] V. Zemlyacheko, N. Korieko, ad R. Tyshkevich. Graph isomorphism problem. The Theory of Computatio I, 118, A About extesios of the coectedess result for stochastic Kroecker graphs If θ has zero etries, the situatio is more complicated. 15

17 It still holds that if θ i < 1 for some i, the i is discoected with probability 1 ad hece G is ot coected. It is also still true that if θ i = 1, the the ith row must be determiistic. Moreover, it is ot hard to see that if θ is periodic, the P will be discoected. Therefore the followig are ecessary (but ot sufficiet) coditios for G to be coected with high probability: 1. θ i 1 i ad if θ i = 1, the the ith row is determiistic. 2. θ is aperiodic. I view of Theorem 4.2 [30, Theorem 1], a way to prove coectedess is to boud the size of ay cut. Oe way to do this is to look at the secod eigevalue of the Laplacia. Ufortuately, while eigevalues behave icely uder Kroecker product, eigevalues of the Laplacia do ot. A more promisig approach is to prove the cojecture below: Cojecture A.1 There exist costats c 1 ad c 2 so that if θ is aperiodic ad for every vertex v [r] k, the cut that separates v from the rest of the graph has weight at least c 1 log, the every cut of P has weight at least c 2 log. If the cojecture holds, it would be sufficiet to prove that with high probability there are o isolated vertices. B Towards caoical labelig of preferetial attachmet graphs The geeral idea for the algorithm is as follows. The first step of the proof is to show that all vertices that have degree at least ɛ, for some costat ɛ, will be distiguished after oe roud of refiemet. The top vertices will have degree at least ɛ. Ideed Bollobas ad Riorda [13] ote that with probability 1 o( δ ) (for a fixed costat δ), every vertex i with expected degree E[d(i)] 2 ɛ has d(i) ɛ. Bollobas ad Riorda do ot make ay attempt to optimize δ. However, based o the proof of [12, Lemma 7] (specifically property E 1 ) the statemet should be true for δ = 5 or 6. I particular, we will have may vertices that will be distiguished after oe roud of refiemet. The secod step is to take care of vertices of small degree. There are two lies of attack, usig ideas from [12, 13], ad it is likely that usig a combiatio will be useful. The first idea is to use the fact that with probability 1 o( δ ), all vertices are coected by a path of legth at most 8 log log to early vertices [12, Lemma 8] (i this case δ is small: Bollobas-Riorda prove their results for δ = 1/1000). The we use the fact that early vertices are already distiguished, ad ideas about eighborhood growth from [13] to show that vertices will be distiguished either by their distace sequece up to some small value (O(log log )), or by the closest early vertices. B.1 Degree distributio of vertices from t o Bollobas, Riorda, Specer ad Tusady [15] studied the degree distributio of graphs geerated by preferetial attachmet. We are iterested i the degree distributio of the eighbors of a vertex v. The out-degree distributio (vertices that were added before v) is give by the costructio. The i-degree distributio requires looki at vertices added after v. As a first step i that directio we characterize the degree distributio of vertices added after time t. 16

18 Extedig the otatio of [15], let #,t m (d) be the umber of vertices i G m that have degree m + d ad are added at or after time t. Let r = t/. Lemma B.1 E[#,t m (d)] 2m(m + 1) (d + m)(d + m + 1)(d + m + 2) I 1 k (d + 1, m + 2) = E[# m(d)] I 1 k (d + 1, m + 2) where I s (a, b) = B s (a, b)/b(a, b) is the regularized icomplete beta fuctio. We should be able to get high cofidece bouds like the oes i [15], provig that the error is at most a additive factor of log with probability 1. Sice we have: I s (a, b) = a+b 1 j=a E[#,t m (d)] d+m+2 j=d+1 j ( a + b 1 j ) s j (1 s) a+b 1 j, 2m(m + 1) (d + m)(d + m + 1)(d + m + 2) ( ) d + m + 2 (1 r) j k d+m+2 j. Remark. Our mai use of this lemma will be for relatively small values of r (r 1 ɛ ). It is likely that the estimates we eed for these values of r ca be obtaied more immediately, but we keep the result as geeral as possible. Proof. The proof follows alog the lies of Bollobas et al. Here is a sketch. They obtai P[d k+1 = d + m] ( ) d + m 1 = o( 1 ) + (1 + o(1)) k m/2 (1 k) d, m 1 ad compute this by computig the itegral 1 0 k m/2 (1 k) d dk. We wat to compute the same itegral, but stop early, amely start at r istead of 0. As i Bollobas et al., we substitute k = (1 u) 2. 1 r k m/2 (1 k) d dk = 2 = 2B 1 r (d + 1, m + 2) 1 r 0 (1 u) m+1 u d du Where B s (a, b) is the icomplete Beta fuctio. Therefore the estimate is the same as the estimate for E[# m(d)] i [15], except the beta fuctio is replaced by the icomplete beta fuctio. The result follows. 17