Counting Arbitrary Subgraphs in Data Streams

Transcription

1 Counting Arbitrary Subgraphs in Data Streams Daniel M. Kane 1, Kurt Mehlhorn, Thomas Sauerwald, and He Sun 1 Department of Mathematis, Stanford University, USA Max Plank Institute for Informatis, Germany Abstrat. We study the subgraph ounting problem in data streams. We provide the first non-trivial estimator for approximately ounting the number of ourrenes of an arbitrary subgraph H of onstant size in a large) graph G. Our estimator works in the turnstile model, i.e., an handle both edge-insertions and edge-deletions, and is appliable in a distributed setting. Prior to this work, only for a few non-regular graphs estimators were known in ase of edge-insertions, leaving the problem of ounting general subgraphs in the turnstile model wide open f. 13, Problem 11). We further demonstrate the appliability of our estimator by analyzing its onentration for several graphs H and the ase where G is a power law graph. 1 Introdution Counting small) subgraphs in massive graphs is one of the fundamental tasks in algorithm design and has various appliations, inluding analyzing the onnetivity of networks, unovering the strutural information of large graphs, and indexing graph databases. The urrent best known algorithm for the simplest non-trivial version of the problem, ounting the number of triangles, is based on matrix multipliation, and is infeasible even for a graph of medium size. Therefore it is natural to onsider the problem in the data streaming setting, where the edges ome sequentially and the algorithm is required to approximate the number of subgraphs without storing the whole graph. Formally in this problem, we are given a set of items s 1, s,... in a data stream. These items arrive sequentially and represent edges of an underlying graph G = V, E). Two standard models 15 in this ontext are the Cash Register Model and the Turnstile Model. In the ash register model, eah item s i represents one edge and these arrived items form a graph G with edge set E := {s i }, where E = initially. The turnstile model generalizes the ash register model and is appliable to dynami situations. Speifially, eah item s i in the turnstile model is of the form e i, sign i ), where e i is an edge of G and sign i {+, } indiates that e i is inserted to or deleted from G. That is, after reading the ith item, E E {e i } if sign i = +, and E E \ {e i } otherwise. In a more general distributed setting, there are k distributed sites, eah reeiving a stream S i of elements over time, and every S i is proessed by a loal host. When the number of subgraphs is asked for, these k hosts ooperate to give an approximation for the underlying graph formed by k S i.

2 Daniel M. Kane, Kurt Mehlhorn, Thomas Sauerwald, and He Sun Our Results & Tehniques. We present the first sketh for ounting arbitrary subgraphs of onstant size in data streams. While most of the previous algorithms are based on sampling tehniques and annot be extended to ount more omplex subgraphs, our algorithm an approximately ount arbitrary possibly direted) subgraphs. Moreover, our algorithm runs in the turnstile model and is appliable in the distributed setting. More formally, for any fixed subgraph H with a onstant number of edges, we present an algorithm that 1 ± ε)-approximates #H, the number of ourrenes of H, in the underlying graph G. That is, for any onstant 0 < ε < 1, with probability at least /3 the output Z of our algorithm satisfies Z 1 ε) #H, 1 + ε) #H. For several families of graphs H, our algorithm ahieves a 1 ± ε)-approximation for the number of subgraphs H in G within sublinear spae. Our result generalizes the previous works whih an be only used for ounting yles in the turnstile model 11, 1, and answers the 11th open problem in the 006 IITK Workshop on Algorithms for Data Steams 13. Our sketh relies on a novel approah of designing random vetors that are based on different ombinations of omplex numbers. By using different roots of unity and random mappings from verties in G to omplex numbers, we obtain an unbiased estimator for #H. This partially answers Problem 4 of the survey by Muthukrishnan 15, whih asks for suitable appliations of omplex-valued hash funtions in data streaming algorithms. Apart from ounting subgraphs in streams, we believe that our new approah will find more appliations. We further onsider the problem of ounting stars in power law graphs, whih inlude many pratial networks from omputer siene, biology and soiology. We show that O 1 ε log n ) bits suffie to get a 1±ε)-approximation for ounting stars S k, while the exat ounting needs n log n bits of spae. Our main results are summarized in Table 1 on the next page. To demonstrate that for a large family of graphs G our algorithm ahieves a 1 ± ε)-approximation within sublinear spae, we onsider the random graph G = Gn, p), where eah edge is plaed independently with a fixed probability p 1 + ε) lnn)/n. Random graphs are of interest for the performane of our algorithm, as the independent appearane of the edges in G = Gn, p) redues the number of partiular patterns. In other words, if our algorithm has low spae omplexity for ounting a subgraph H in Gn, p), then the spae omplexity is even lower for ounting a more frequently ourring graph in a real-world graph G whih has the same density as Gn, p). Regarding the spae omplexity of our algorithm on random graphs, assume for instane that the subgraph H is a P 3 or S 3 i.e., a path or a star with 3 edges). The expeted number of ourrenes of suh a graph is of order n 4 p 3 1. It an be shown by standard tehniques f. 1, Setion 4.4) that the number of ourrenes is also of this order with probability 1 o1) as n. Assuming that this event ours, Theorem 3.3 along with the fats that m = Θn p) and G) = Θnp) imply a 1±ε)-approximation algorithm for P 3 or S 3 ) with spae omplexity O 1 ɛ n log n). For stars S k with any onstant k, the result from Theorem 4.1 yields a 1 ± ε)-approximation algorithm with spae omplexity

3 Counting Arbitrary Subgraphs in Data Streams 3 Conditions Spae Complexity Referene any graph G ) 1 O mk G) k log n Theorem 3.3 any graph H ε #H) ) 1 m O k log n Theorem 3.3 ε #H) any graph G H with δh) any graph G stars S k Power law graph G stars S k ) ) n O 1 1/k n k 3/ 1/k) G) + 1 log n ε #S k ) Theorem 4.1 O 1 ε log n ) Theorem 4. Table 1. Spae requirement for 1 ± ε)-approximately ounting an undireted and onneted graph H with k = O1) edges. Here δ and denote the minimum and maximum degree, respetively. Spae omplexity is measured in terms of bits. O 1 ɛ n log n ). Finally, for any yle with k = O1) edges, Theorem 3.3 gives an algorithm with spae omplexity O 1 ɛ p k log n), whih is sublinear for suffiiently large values of p, e.g., p = Θ1). Related Work. Bar-Yossef, Kumar and Sivakumar were the first to onsider the subgraph ounting problem in data streams and presented an algorithm for ounting triangles 3. After that, the problem of ounting triangles in data streams was studied extensively 4, 6, 11, 17. The problem of ounting other subgraphs was also addressed in the literature. Buriol et al. 7 onsidered the problem of estimating lustering indexes in data streams. Bordino et al. 5 extended the tehnique of ounting triangles 6 to all subgraphs on three and four verties. Manjunath et al. 1 onsidered ounting yles of onstant size in data streams. Among these results, only two algorithms 11, 1 work in the turnstile model and these only hold for yles. Apart from designing algorithms in the streaming model, the subgraph ounting problem has been studied extensively. Alon et al. presented an algorithm for ounting given-length yles. Gonen et al. 10 showed how to ount stars and other small subgraphs in sublinear time. In partiular, several small subgraphs in a network, named network motifs, have been identified as the simple building bloks of omplex biologial networks and the distribution of their ourrenes ould reveal answers to many important biologial questions 14, 19. Notation. Let G = V, E) be an undireted graph without self-loops and multiple edges. The set of verties and edges are represented by V G and EG, respetively. We will assume that V G = {1,..., n} and n is known in advane. For any vertex u V G, the degree of u is denoted by degu). The maximum and minimum degree of G are denoted by G) and δg), respetively. Given two direted graphs H 1 and H, we say that H 1 is homomorphi to H if there is a mapping i : V H 1 V H suh that u, v) EH 1 implies iu), iv)) EH. Graphs H 1 and H are said to be isomorphi if there is a bijetion i : V H 1 V H suh that u, v) EH 1 iff iu), iv)) EH. For any graph H, let autoh) be the number of automorphisms of H.

4 4 Daniel M. Kane, Kurt Mehlhorn, Thomas Sauerwald, and He Sun For any graph H, we all a not neessarily indued subgraph H 1 of G an ourrene of H, if H 1 is isomorphi to H. Let #H, G) be the number of ourrenes of H in G. When the referene to G is lear from the ontext, we simply write #H. A kth root of unity is any number of the form e πi j/k, 0 j < k. Organization. The paper is strutured as follows. Setion presents an unbiased estimator for ounting arbitrary subgraphs, followed by the orretness and onentration analysis in Setion 3. In Setion 4 we onsider two approahes to redue the spae omplexity and give an improved algorithm for ounting stars. Due to page limitations, some proofs and lemmas are omitted in this extended abstrat and an be found in the appendix. An Unbiased Estimator for Counting Subgraphs We present a framework for ounting general subgraphs. Suppose that H is a fixed graph with t verties and k edges, and we want to ount the number of ourrenes of H in G. For the notation, we denote verties of H by a, b and, and verties of G are denoted by u, v and w, respetively. Let the degree of vertex a in H be deg H a). We equip the edges of H with an arbitrary orientation, as this is neessary for the further analysis. Therefore, eah edge in H together with its orientation an be expressed as ab for some a, b V H. For simpliity and with slight abuse of notation we will use H to denote suh an oriented graph. At a high level, our estimator maintains k omplex variables Z ab G), ab EH, and these variables are set to be zero initially. For every arriving edge {u, v} EG we update eah Z ab G) aording to Z ab G) Z ab G) + M ab u, v) + M ab v, u), where M ab : V G V G C an be omputed in onstant time. Hene Z ab G) = {u,v} EG M ab u, v) + M ab v, u). Intuitively M ab u, v) expresses the event to give {u, v} the orientation uv and map uv to ab, and M ab u, v) + M ab v, u) is used to express two different orientations of edge {u, v}. For every query of the number of subgraphs, the estimator simply outputs the real part of α ab Z ab G), where α IR + is a saling fator. More formally, eah M ab u, v) is defined aording to the degree of verties a, b in graph H and onsists of the produt of three types of random variables Q, X w) and Y w), V H and w V G: Variable Q is a random τth root of unity, where τ := t 1. For vertex V H, w V G, X w) is random deg H )th root of unity, and for eah vertex V H, X : V G C is hosen independently and uniformly at random from a family of 4k-wise independent hash funtions. Variables Q and X, V H are hosen independently.

5 Counting Arbitrary Subgraphs in Data Streams 5 For every w V G, Y w) is a random element from S := { 1,, 4, 8,..., t 1} as part of a 4k-wise independent hash funtion. Variables Y w), w V G and Q are hosen independently. Given the notations above, we define eah funtion M ab as M ab u, v) := X a u) X b v) Q Y u) deg H a) Q Y v) deg H b). Estimator 1 gives the formal desription of the update and query proedure. Estimator 1 Counting #H, G) Step 1 Update): When an edge e = {u, v} EG arrives, update eah Z ab w.r.t. Z ab G) Z ab G) + M ab u, v) + M ab v, u). 1) Step Query): When #H, G) is required, output the real part of t t ZHG), ) t! autoh) where Z H is defined by Z HG) := ab EH Z ab G). 3) Estimator 1 is appliable in a quite general setting: First, the estimator runs in the turnstile model. For simpliity the update proedure above is only desribed for the edge-insertion ase. For every item of the stream that represents an edge-deletion, we replae + by in 1). Seond, our estimator also works in the distributed setting, where every loal host maintains variables Z ab, ab EH, and does the update for every arriving item in the loal stream. When the output is required, these variables loated at different hosts are summed up and we return the estimated value aording to 3). Third, the estimator above an be revised easily to ount the number of direted subgraphs in a direted graphs. Sine in this ase we need to hange the onstant of ) aordingly, in the rest of our paper we only fous on the ase of ounting undireted graphs. 3 Analysis of the Estimator In this setion we first prove that Z H G) defined by 3) is an unbiased estimator for #H, G). Then we analyze the variane of the estimator. The main result of our paper will be presented at the end of this setion. Let us first explain the intuition behind our estimator. By definition we have Z H G) = Z ab G) = ) M ab u, v) + M ab v, u). 4) ab EH ab EH {u,v} EG

6 6 Daniel M. Kane, Kurt Mehlhorn, Thomas Sauerwald, and He Sun Sine H has k edges, Z H G) is a produt of k terms and eah term is a sum over all edges of G eah with two possible orientations. Hene, in the expansion of Z H G) any k-tuple e 1,..., e k ) E k G ontributes k different terms to Z H G) and eah term orresponds to a ertain orientation of e 1,..., e k ). Let T = e1,..., e k ) be an arbitrary orientation of e 1,..., e k ) and G T be the graph indued from T. At a high level, we use three types of variables to test if G T is isomorphi to H. However, the roles of these variables are different. i) Through funtion Y every vertex u V T maps to one element in S randomly. If V T = S = t, then with onstant probability, verties in V T map to different t numbers in S. Otherwise, V T < t and verties in V T annot map to different t elements. Sine Q has the property that EQ i 0 1 i τ) iff i = τ, where τ = l S l, the ombination of Q and Y guarantees that G T ontributes to EZ H G) only if graph H and G T have the same number of verties. ii) For V H, w V G, random variable X w) guarantees that G T ontributes to EZ H G) only if there is a homomorphism from H to G T. Now we analyze our sketh and prove that Z H G) is an unbiased estimator for #H, G). Theorem 3.1. Let H be a graph with t verties and k edges. Assume that variables X w), Y w), V H, w V G and Q are as defined above. Then EZ H G) = t! autoh) t t #H, G). Proof. For a k-tuple T = e 1,..., e k ) E k G, let G T = V T, E T ) be the indued multi-graph from set E T, i.e., G T has edge multi-set E T = {e 1,..., e k }. Let T = e 1,..., e k ) be an arbitrary orientation of e 1,..., e k ), where e i = u i v i. Consider the expansion of Z H G) below: Z H G) = Z ab G) = ) M ab u, v) + M ab v, u). ab EH ab EH {u,v} EG The term orresponding to e 1,..., e k ) in the expansion of Z H G) is k M aib i u i, v i ) = k X ai u i ) X bi v i ) Q Y u i ) deg H a i) Q Y v i ) deg H b i ), 5) where a i b i is the ith edge of H where we assume any order) and u i v i is the ith edge in T. We show that the expetation of 5) is non-zero if and only if the graph indued by T is an ourrene of H in G. Moreover, if the expetation of 5) is non-zero, then its value is a onstant. For a vertex w of G and a vertex of H, let θ T, w) = { i u i = w and a i = ) or v i = w and b i = ) }

7 Counting Arbitrary Subgraphs in Data Streams 7 be the number of edges in T with head or tail) w mapping to the edges in H with head or tail). Sine every vertex of H is inident to deg H ) edges, for any V H it holds that w V θ T T, w) = deg H ). By the definition of θ T, we rewrite 5) as Therefore Z H G) = w V T e 1,...,e k e T = e1,..., e i EG k ) X θ T,w) w V T w) X θ T,w) w V T w) θ,w)y w) T deg Q H ) w V T. θ,w)y w) T deg Q H ) where the first summation is over all k-tuples of edges in EG and the seond summation is over all their possible orientations. By linearity of expetations of these random variables and the assumption that X w) V H, w V G), Y and Q have suffiient independene, we have EZ H G) = E = Let e 1,...,e k e T = e1,..., e i EG k ) e 1,...,e k e T = e1,..., e i EG k ) α T := E w V T w V T X E θ T,w) w V T X θ T,w) } {{ } A w) X θ T,w) w) E w V T Q w) E w V T θ T,w)Y w) deg H ) w V T Q θ,w)y w) T deg Q H ) θ T,w)Y w) deg H ) } {{ } B,. 6) We will next show that α T is either zero or a nonzero onstant independent of T. The latter is the ase if and only if GT, the undireted graph indued from edge set T, is an ourrene of H in G. We onsider the produt A at first. Assume A 0. Using the same tehnique as 1, we onstrut a homomorphism from H to G T under the ondition A 0. Remember that: i) For any V H and w V T, θ T, w) deg H ), and ii) E Xw) i 0 if and only if i = deg H ) or i = 0. Therefore for any.

8 8 Daniel M. Kane, Kurt Mehlhorn, Thomas Sauerwald, and He Sun fixed T and V H, E w V T X θ T,w) w) {0, deg H )} for all w. Now assume that E w V T 0 if and only if θ T, w) X θ T,w) w) 0 for every V H. Then θ T, w) {0, deg H )} for all V H and w V G. Sine w θ, w) = deg T H) for any V H, there must be a unique vertex w V T suh that θ T, w) = deg H ). Define ϕ : V H V T as ϕ) = w for the vertex w satisfying θ T, w) = deg H ). Then ϕ is a homomorphism, i.e. ab EH implies ϕa)ϕb) EG T. Hene A 0 implies H is homomorphi to G T, and E w V T X θ T,w) w) = E X deg H ) ϕ)) = 1. 7) Seond we onsider the produt B. Our task is to show that, under the ondition A 0, G T is an ourrene of H if and only if B 0. Observe that E w V T θ,w)y w) T deg Q H ) = E Q θ,w)y w) T deg w V T H ). Case 1: Assume that G T is an ourrene of H in G. Then V H = V T and funtion ϕ onstruted above is a bijetion, whih implies that w V T = θ, w)y w) T deg H ) θ T, ϕ))y ϕ)) = deg H ) Y ϕ)) = w V T Y w). Without loss of generality, let V T = {w 1,..., w t }. By onsidering all possible hoies for Y w 1 ),..., Y w t ), denoted by yw 1 ),..., yw t ) S, and independene between Q and Y w), w V G, τ 1 B = j=0 yw 1),...,yw t) S τ 1 = 1 τ j=0 yw 1),...,yw t) S ϑ:=yw 1)+ +yw t),τ ϑ τ 1 j=0 yw 1),...,yw t) S ϑ:=yw 1)+ +yw t),τ ϑ t ) πij Pr Y w i ) = yw i ) exp τ 1 τ 1 τ ) t ) 1 πi exp t τ ϑ j + ) t ) 1 πi exp t τ ϑ j. ) t yw l ) l=1

9 Counting Arbitrary Subgraphs in Data Streams 9 Applying Lemma A. with R = exp ) πi τ, the seond summation is zero. Hene by Lemma A.3 we have B = yw 1),...,yw t) S τ yw 1)+ +yw t) ) t 1 = t yw 1),...,yw t) S yw 1)+ +yw t)=τ ) t 1 = t ) t 1 t! = t! t t t. Case : Assume that G T is not an ourrene of H in G and let V T = {w 1,..., w t }, t < t. Then there is a vertex w V T and different b, V H, suh that ϕb) = ϕ) = w. As before we have w V T θ, w)y w) T = Y ϕ)). deg H ) By Lemma A.3, τ Y ϕ)) regardless of the hoies of Y w 1),..., Y w t ). Hene τ 1 B = j=0 yw 1),...,yw t ) S ϑ:= yϕ)) τ 1 = j=0 yw 1),...,yw t ) S ϑ:= yϕ)) 1 τ 1 τ ) t ) 1 πij exp ϑ t τ ) t ) 1 πi exp t τ ϑ j = 0, where the last equality follows from Lemma A. with R = exp ) πi τ. Let 1 G T H be the indiator expression that is one if G T and H are isomorphi and zero otherwise. By the definition of graph automorphism and 7) EZ H G) = e 1,...,e k e T = e1,..., e i EG k ) t! ) t t t! autoh) 1 G T H = t t #H, G). We an use a similar tehnique to analyze the variane of Z H G) and then use Chebyshev s inequality to upper bound the number of trials required for the desired approximation. Sine Z H G) is omplex-valued, we need to upper bound Z H G) Z H G). Lemma 3.. Let G be any graph with m edges, H be any graph with k edges for a onstant k. Random variables X w), V H, w V G and Q are defined as above. Then the following statements hold: 1. If δh), then E Z H G) Z H G) = O m k).. Let H be a onneted graph with k edges and H be the set of all subgraphs H in G with the following properties: i) H has k edges, and

10 10 Daniel M. Kane, Kurt Mehlhorn, Thomas Sauerwald, and He Sun ii) every onneted omponent of H ontains at least two edges. Then E Z H G) Z H G) = O H ). In partiular, E Z H G) Z H G) = O m k G) k). By using Chebyshev s inequality, we an get a 1 ± ε)-approximation by running our estimator in parallel and returning the average of the output of these individual estimators. This leads to our main result for ounting #H, G). Theorem 3.3. Let G be any graph with m edges and H be any graph with k = O1) edges. For any onstant 0 < ε < 1, there is an algorithm to 1 ± ε)- approximate #H, G) using i) O 1 m ε k #H) log n ) bits if δh), and ii) using O 1 ε mk G)) k #H) log n ) bits for any H. Disussion. Statement i) of Theorem 3.3 extends the main result of 1, Theorem 1 whih requires H to be a yle. Note that a naïve sampling-based approah would hoose a random k-tuple of edges and require m k /#H) spae. Theorem 3.3 improves upon this approah, in partiular if the graph G is sparse and the number of ourrenes of H is a growing funtion in n. 4 Extensions We have developed a general framework for ounting arbitrary subgraphs of onstant size. Our algorithm is general in the sense that, for any fixed subgraph H, one only needs to i) give verties of H an arbitrary labeling, and edges of H an arbitrary orientation; ii) hoose hash funtions aording to H, and iii) run Estimator 1 multiple times in parallel. A reasonable number of exeutions guarantees a 1 ± ε)-approximation. For several typial appliations we an further improve the spae omplexity by grouping the skethes or using ertain properties of the underlying graph G. For the ease of the disussion we only fous on ounting stars throughout the setion. Grouping Skethes. The spae omplexity in Theorem 3.3 relies on the number of edges that the sketh reads. To redue the variane, a natural way is to use multiple opies of the skethes, and every sketh is only responsible for the updates of the edges from a ertain subgraph. To formulate this intuition, we partition V = {1,..., n} into g := n 1 1/k) subsets V 1,..., V g, and V i := { j : i 1) n 1/k) + 1 j i n 1/k)}. Without loss of generality we assume that n 1/k) N. Assoiated with every V i, we maintain a sketh C i, whose desription is essentially the same as Estimator 1 and an be found in the appendix. For every arriving edge e = {u, v} in the stream, we update sketh C i if u V i or v V i. Sine i) the entral vertex of every ourrene of S k is in exatly one subset V i, and ii) every edge adjaent to

11 Counting Arbitrary Subgraphs in Data Streams 11 one vertex in V i is taken into aount by sketh C i, we know that every ourrene of S k in G is only ounted by one sketh C i. More formally, let #S k, G Vi ) be the number of S k whose entral vertex is in V i. Then it holds that #S k, G) = g #S k, G Vi ). This indiates that if every C i is unbiased for #S k, G Vi ), then we an use the sum of returned values of different C i to approximate #S k, G). See Setion B.1 for a detailed analysis. Theorem 4.1. Let G be a graph with n verties. For any onstants 0 < ε < 1 and k, there is an algorithm to 1 ± ε)-approximate #S k, G) with spae omplexity n 1 1/k) n 3/ 1/k) G) k ) ) O ε #S k ) + 1 log n. To illustrate Theorem 4.1, let us onsider graphs G with G)/δG) = on 1/4k) ) and δg) k. Sine #S k, G) = Ω n δg) k), Theorem 4.1 implies that o 1 ε n log n ) bits suffie to give a 1 ± ε)-approximation. Counting on Power Law Graphs. Besides organizing the skethes into groups, we will see now that the spae omplexity an be also redued if we assume the underlying graph G to have ertain properties. One of the most important properties shared by many biologial, soial or tehnologial networks is the soalled Power Law degree distribution, i.e., the number of verties with degree d, denoted by fd) := {v V : degv) = d}, satisfies fd) d β, where β > 0 is the power law exponent. For many tehnologial networks, experimental studies indiate that β is between and 3 see 16). Formally, we use the following model from 18 based on the umulative degree distribution. For given onstants σ 1 and d min IN, we say that G has an approximate power law degree distribution with exponent β, 3), if n 1 fd) σ 1 n k β+1, σ n k β+1 k d min. d=k Hene, the degree distribution below the threshold d min is arbitrary. Note that for this model, the maximum degree G) is of order n 1/β 1) see also 8, 16 for some justifiation why this should be the right order of the maximum degree). We now state our result for ounting stars on power law graphs. Theorem 4.. Assume that G has an approximate power law degree distribution with exponent β, 3). Then, for any onstants 0 < ε < 1 and k, we an 1 ± ε)-approximate #S k, G) using O 1 ε log n ) bits. Bibliography 1 N. Alon and J. Spener. The Probabilisti Method. Wiley-Intersiene Series in Disrete Mathematis and Optimization. John Wiley & Sons, 3rd edition, 008.

12 1 Daniel M. Kane, Kurt Mehlhorn, Thomas Sauerwald, and He Sun N. Alon, R. Yuster, and U. Zwik. Finding and ounting given length yles. Algorithmia, 173):09 3, Z. Bar-Yossef, R. Kumar, and D. Sivakumar. Redutions in streaming algorithms, with an appliation to ounting triangles in graphs. In Pro. 13th Symp. on Disrete Algorithms SODA), pages 63 63, L. Behetti, P. Boldi, C. Castillo, and A. Gionis. Effiient semi-streaming algorithms for loal triangle ounting in massive graphs. In Pro. 14th Intl. Conf. Knowledge Disovery and Data Mining KDD), pages 16 4, I. Bordino, D. Donato, A. Gionis, and S. Leonardi. Mining large networks with subgraph ounting. In Pro. 8th Intl. Conf. on Data Mining ICDM), pages , L. S. Buriol, G. Frahling, S. Leonardi, A. Marhetti-Spaamela, and C. Sohler. Counting triangles in data streams. In Pro. 5th Symp. Priniples of Database Systems PODS), pages 53 6, L. S. Buriol, G. Frahling, S. Leonardi, and C. Sohler. Estimating lustering indexes in data streams. In Pro. 15th European Symposium on Algorithms ESA), pages , R. Cohen, K. Erez, D. ben-avraham, and S. Havlin. Resiliene of the internet to random breakdowns. Phys. Rev. Lett., 85: , S. Ganguly. Estimating frequeny moments of data streams using random linear ombinations. In Pro. 8th Intl. Workshop on Randomization and Comput. RANDOM), pages , M. Gonen, D. Ron, and Y. Shavitt. Counting stars and other small subgraphs in sublinear-time. SIAM J. Dis. Math., 53): , H. Jowhari and M. Ghodsi. New streaming algorithms for ounting triangles in graphs. In Pro. 11th Intl. Conf. Computing and Combinatoris COCOON), pages , M. Manjunath, K. Mehlhorn, K. Panagiotou, and H. Sun. Approximate ounting of yles in streams. In Pro. 19th European Symposium on Algorithms ESA), pages , A. MGregor. Open Problems in Data Streams and Related Topis, IITK Workshop on Algorithms For Data Sreams a.in/users/sganguly/data-stream-probs.pdf. 14 R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, and U. Alon. Network motifs: Simple building bloks of omplex networks. Siene, ):84 87, S. Muthukrishnan. Data Streams: Algorithms and Appliations. Foundations and Trends in Theoretial Computer Siene, 1), M. E. J. Newman. The struture and funtion of omplex networks. SIAM Review, 45:167 56, R. Pagh and C. E. Tsourakakis. Colorful triangle ounting and a mapredue implementation. Inf. Proess. Lett., 117):77 81, R. van der Hofstad. Random Graphs and Complex Networks. win.tue.nl/~rhofstad/notesrgcn.pdf, 01. Book in preparation. 19 E. Wong, B. Baur, S. Quader, and C. Huang. Biologial network motif detetion: priniples and pratie. Briefings in Bioinformatis, pages 1 14, June 011.

13 A Omitted Details from Setion 3 Counting Arbitrary Subgraphs in Data Streams 13 In this setion, we provide the omitted details from Setion 3. In Setion A.1, we list three auxiliary lemmas that are used to show that Z H G) is an unbiased estimator. Setion A. gives several upper bounds on the number of subgraphs, whih are used to bound the variane in Setion A.3. A.1 Auxiliary Algebrai Tools The following three lemmas are used to prove Theorem 3.1. Lemma A.1 9). Let X, V H be a randomly hosen deg H )th root of unity. Then for any 1 < i deg H ), it holds that EX i = In partiular, EX = 1 if deg H ) = 1. { 1, i = degh ), 0, 1 i < deg H ). Lemma A.. Let R be a primitive τth root of unity and k N. Then τ 1 { τ, τ k, R k ) l = 0, τ k. l=0 Proof. 1) If τ k, then R k = 1 and τ 1 l=0 Rk ) l = τ 1 l=0 1 = τ. ) If τ k, then R k 1 and τ 1 R k ) l = 1 Rk ) τ 1 R k = 1 Rk τ = 0. 1 Rk l=0 Lemma A.3. Let x i ZZ 0 and t 1 i=0 x i = t. Then t 1 t 1 i=0 i x i if and only if x 0 = = x t 1 = 1. Proof. Let x 0,..., x t 1 be sequene so that i) Not all x i = 0, ii) t 1 t 1 i=0 i x i, and iii) t 1 i=0 x i is minimal among all sequenes {x i } satisfying the onditions i) and ii). It is easy to see that x i < for eah i. This is beause otherwise dereasing x i by and inreasing x i+1 by 1 or dereasing x t 1 by and inreasing x 0 by 1) preserves t 1 i=0 i x i mod t 1 and dereases t 1 i=0 x i. On the other hand, if x i 1 then t 1 i=0 i x i t 1 with equality iff x i = 1 for all i. Thus x 0 =... = x t 1 = 1 is the unique solution to t 1 t 1 i=0 i x i for x i 0 and 0 < t 1 i=0 x i t.

14 14 Daniel M. Kane, Kurt Mehlhorn, Thomas Sauerwald, and He Sun A. Bounding the Number of Subgraphs In the following, we list four lemmas that all give upper bounds on the number of subgraphs with ertain properties in G. These bounds are used later to analyze the variane of our unbiased estimator. Lemma A.4 1). Let G be a graph with m edges and H be a set of subgraphs of G suh that every H H has the following properties: 1) H has k edges, where k is a onstant. ) Eah onneted omponent of H is an Eulerian iruit. Then H = Om k/ ). Lemma A.5. Let G be a graph with m edges and H be any fixed graph with k edges and δh), where k is a onstant. Then #H, G) = O m k/). Proof. Case 1: Every vertex in H has even degree. Assume that H has l onneted omponents H 1,..., H l where eah H i has k i edges. Note that #H, G) l #H i, G). Sine every H i is Eulerian, by Lemma A.4 we have #H i, G) = O m ki/), and #H, G) = l O m ki/) = O m k/). Case : There is a vertex in H with odd degree. We prove the statement by indution on EH. By the handshake lemma, at least two verties must be of odd degree. Let P = v 0,..., v l ) be a path of minimal length between verties of odd degree in H. Thus degv 0 ), degv l ) are odd, and degv j ) is even for all 1 j l 1. Let H be the graph obtained from H by removing the edges in P and the verties v j of degree exatly whih would upon removal of edges in P have degree 0). Then δh ). Thus by our indutive hypothesis, #H, G) = Om k l)/ ). We split into two ases based upon whether l is even or odd. If l is even, then a homomorphism H G is determined by a homomorphism H G along with the images of the edges {v 1, v }, {v 3, v 4 },..., {v l 1, v l }. Thus the number of suh homomorphisms is at most Om k l)/ ) m l/ = Om k/ ). If l is odd, then a homomorphism H G is determined by a homomorphism H G along with the images of the edges {v 1, v }, {v 3, v 4 },..., {v l, v l 1 }. Hene #H, G) #H, G) m l 1)/ = Om k l)/ ) m l 1)/ = Om k/ ). We now prove a general upper bound on the number of onneted subgraphs with k edges in terms of the k-th moment of the degree distribution. Lemma A.6. Let G be any graph and H be an arbitrary onneted subgraph of G with k edges possibly with multiple edges), where k is a onstant. Then #H, G) = O u V degu))k). Proof. For any u V G, define H u to be the set onsisting of all subgraphs H with k edges in G so that u = argmax w V H {degw)} If there is more than one suh vertex u, pik an arbitrary one). Sine we an speify any H H u by

15 Counting Arbitrary Subgraphs in Data Streams 15 i) hoosing all edges inident to u, ii) hoosing all edges inident to u, or the other endpoint of the edges hosen in i), and so on, we have H u k i degu)) = O degu)) k). Let H 1 be an ourrene of H in G and let u = argmax w V H1 {degw)}. Note that H 1 H u. Therefore #H, G) ) H u = O degu)) k u V G By partitioning the subgraph H into onneted omponents, we easily obtain the following result. Lemma A.7. Let G be any graph with m edges and H be an arbitrary subgraph of G with k edges possibly with multiple edges), where k is a onstant and every onneted omponent of H ontains at least two edges. Then #H, G) = Om k/ k/ ). Proof. Assume w.l.o.g. that k is even. Partitioning all subgraphs with k edges into at most k/ onneted omponents with l 1,..., l k/ edges eah and applying Lemma A.6 to every omponent separately, we onlude that ) #H, G) = O degu) lj = = = l 1,...,l k/ {0,,3,...,k} k/ j=1 lj=k l 1,...,l k/ {0,,3,...,k} k j=1 lj=k l 1,...,l k/ {0,,3,...,k} k/ j=1 lj=k l 1,...,l k/ {0,,3,...,k} k/ j=1 lj=k = O k/ m k/). j : l j j : l j u V u V O lj 1 ) degu) u V O k {j : lj } {j : m lj } ) O k {j : lj } ) m/ ) A.3 Bounding the Variane After the preparations in Setion A.1 and Setion A., we are now ready to prove Lemma 3. whih provides two upper bounds on the variane of our estimator. Lemma 3. from page 9). Let G be any graph with m edges, H be any graph with k edges for a onstant k. Random variables X w), V H, w V G and Q are defined as above. Then the following statements hold:

16 16 Daniel M. Kane, Kurt Mehlhorn, Thomas Sauerwald, and He Sun 1. If δh), then E Z H G) Z H G) = O m k).. Let H be a onneted graph with k edges and H be the set of all subgraphs H in G with the following properties: i) H has k edges, and ii) every onneted omponent of H ontains at least two edges. Then E Z H G) Z H G) = O H ). In partiular, E Z H G) Z H G) = O m k G) k). Proof. Note that there is a trivial bound E Z H G) Z H G) = Om k ), sine in the expansion of Z H G) Z H G), eah term orresponds to a tuple of k edges with an arbitrary orientation and eah suh term is bounded by a onstant. To obtain the two statements of the lemma, we have to bound the number of ourring subgraphs more arefully. We start to prove the first statement now. As in our analysis for EZ H G), we analyze the subgraphs that ontribute to E Z H G) Z H G). However, instead of onsidering subgraphs of G with k edges, we need to analyze the subgraphs in G with k edges. By the definition of Z H G), we express Z H G) Z H G) as follows: Z H G) Z H G) = e 1,...,e k T 1= e 1,..., e k ) = w V T1 e 1,...,e k T = e 1,..., e k ) X θ T1,w) w V T e 1,...,e k T 1= e 1,..., e k ) e 1,...,e k T = e 1,..., e k ) Q X w) θ T,w) w V T1 Q w) X w V V T1 T V θ H w V T1,w)Y w) θ w V T,w)Y w) T1 T θ T1,w) Y w) deg H ) w V T θ T1,w) θ T,w) )/ θ,w)y w) T deg Q H ) deg H ) w). By the linearity of expetations and the ondition that random variables X w) V H, w V G) are 4k-wise independent, and X, V H, Q are hosen inde-

17 Counting Arbitrary Subgraphs in Data Streams 17 pendently, we have E Z H G) Z H G) = e 1,...,e k T 1= e 1,..., e k ) e 1,...,e k T = e 1,..., e k ) E Q E w V V T1 T V θ H w V T1,w)Y w) θ w V T,w)Y w) T1 T X θ,w) θ T 1 T,w) w) )/ deg H ). Let us define Sine E Q α T1, T := E w V V T1 T X θ,w) θ T 1 T,w) w). V θ H w V T1,w)Y w) θ w V T,w)Y w) T1 T )/ deg H ) = O1), we only need to onsider the number of graphs indued by T 1, T with the property α T1, 0. Remember that: i) For any V T H and w V V1, EX i V w) 0 if and only if i is divisible by deg H ), and ii) for any V H and w V V1, V it holds that 0 θ T1, w) deg H ) and 0 θ T, w) deg H ). Therefore α T1, 0 if and only if for every V H and every w V T V1 one of the V following three onditions holds: i) θ T1, w) = deg H ) and θ T, w) = 0; ii) θ T1, w) = 0 and θ T, w) = deg H ); iii) θ T1, w) = θ T, w). We partition into three disjoint subsets A, B and C defined by V T1 T A := V T1 \ V T, B := V T \ V T1, C := V T1 V T. Sets A, B, C are defined orresponding to the above onditions i), ii) and iii). By the assumption δh), the degree of every vertex in sets A and B is at least two. By ondition iii), every vertex in C has nonzero and even degree, and hene the degree is also at least two. Hene the edges in T 1 T form a homomorphi opy of some graph H with k edges and δh ). Clearly there are at most O1) isomorphism lasses of suh graph H, and by Lemma A.5 eah isomorphism lass embeds in at most Om k ) ways. Therefore the number of suh subgraphs in G is bounded by Om k ) and E Z H G) Z H G) = O m k). This finishes the proof of the first statement. We now show the seond statement of the lemma. Fix T 1 and T with α T1, T 0. We first show that there is no onneted omponent onsisting of a single edge in G T1. T

18 18 Daniel M. Kane, Kurt Mehlhorn, Thomas Sauerwald, and He Sun Assume for ontradition that there is one onneted omponent in G T1 T that onsists of a single edge e := {u, v}. Sine every vertex in set C has even degree, it holds {u, v} A B. By the proof of the first statement, there is no edge in G T1 T onneting A and B diretly. Therefore verties u, v are in the same set. For simpliity assume {u, v} A. Then there is a mapping between edges in EH and edges in G T1. Beause H is onneted, one of u, v has degree at least. Therefore there is another edge inident to edge e, whih ontradits the assumption. Sine α T1, = O1), E T Z H G) Z H G) = O H ) by the definition of set H. Using this and Lemma A.7, we obtain the seond statement. Theorem 3.3 from page 10). Let G be any graph with m edges and H be any graph with k = O1) edges. For any onstant 0 < ε < 1, there is an algorithm to 1 ± ε)-approximate #H, G) using i) O 1 m ε k #H) log n ) bits if δh), and ii) using O 1 ε mk G)) k #H) log n ) bits for any H. Proof. We run s parallel and independent opies of our estimator, and take the average value Z = 1 s s Z i, where eah Z i is the output of the i-th instane of the estimator. Therefore EZ = EZ H G) and a straightforward alulation shows EZ Z E Z = 1 s EZ H G) Z H G) EZ H G) ). By Chebyshev s inequality for omplex-valued random variables see, e.g., 1, Lemma 3), we have Pr Z EZ ε EZ EZ HG) Z H G) EZ H G) EZ H G) s ε EZ H G). For Statement i), by Lemma 3., Statement 1, we have EZ H G) Z H G) EZ H G) EZ H G) EZ H G) Z H G) = Om k ). 1 m By hoosing s = O ε k #H) ), we get Pr Z E Z ε EZ 1/3. ) Hene the overall spae omplexity is O log n. By using the same 1 m ε k #H) k analysis and Lemma 3., Statement, we obtain ii). B Omitted Details from Setion 4 B.1 Grouping Skethes This part gives the omitted details of ounting stars by grouping skethes.

19 Counting Arbitrary Subgraphs in Data Streams 19 Lemma B.1 Deomposition Lemma). Let V 1,..., V g be a partition of V. For any fixed V i V, define G Vi = V i, E i ) as the subgraph of G suh that: i) V i V i V i, where S is the set of neighbors of verties in S; ii) For every {u, v} E i iff u V i or v V i. Then for any star S k it holds that #S k, G) = g #S k, G Vi ), where #S k, G Vi ) is the number of stars in G Vi whose entral point is in V i. Proof. Assume that H is an ourrene of S k in G and the entral vertex of star H is u. Sine V 1,..., V g is a partition of V, vertex u is in exatly one set V i. By definition, H appears in G Vi. Hene graph H is an ourrene of S k in G iff H is an ourrene of S k in exatly one G Vi. By the definition of #S k, G Vi ), the statement follows. With the help of Lemma B.1, we revise our estimator as follows. In the initialization step, we have g k random variables, where g := n 1 1/k). Every k variables form a group S i) whih is used for ounting #H, G Vi ). That is, we have g idential opies of the estimator and the i-th opy only uses random variables in group S i). Then for every oming edge u, v) EG, we update the random variables of group S i) if u V i or v V i. In the end, instead of omputing Z Sk G), we alulate Z Sk G V1 ),..., Z Sk G Vg ) and output Z Sk G) := g Z Sk G Vi ) 8) as the approximation of #H, G). Estimator presents a revised estimator for ounting #S k, G Vi ), where vertex a is the entral vertex of the star. Estimator Counting #S k, G Vi ), update proedure Step 1 Update): When an edge e = {u, v} EG arrives, update eah variable Z ab : a) If u V i and v V i, then b) If u V i and v V i, then ) If u V i and v V i, then Z ab G) Z ab G) + M ab u, v) + M ab v, u). Z ab G) Z ab G) + M ab u, v). Z ab G) Z ab G) + M ab v, u).

20 0 Daniel M. Kane, Kurt Mehlhorn, Thomas Sauerwald, and He Sun Lemma B.. Let H be an S k for any onstant k. Then EZ H G) = #H, G) and EZ H G) Z H G) = O n 1/k) k) + #H, G)). Proof. Sine for any star S k in G with entral point w, S k is only ounted by exatly one G Vi with the property that w V i, it holds that #H, G) = g #H, G Vi ). 9) On the other hand, by linearity of expetations we know that g E Z H G) = E Z H G Vi ). 10) Combing 9), 10) and the fat that E Z H G Vi ) = #H, G Vi ) yields E Z H G) = #H, G). We bound EZ H G) Z H G) as follows. EZ H G) Z H G) g ) g ) = E Z H G Vi ) Z H G Vi ) g = E Z H G Vi ) Z H G Vi ) + = g 1 i j g E Z H G Vi ) Z H G Vi ) + 1 i j g ) Z H G Vi ) Z H G Vj ) E Z H G Vi ) Z H G Vj. Note that every G Vi has at most n 1/k) edges. Sine H = S k, we know by Lemma 3., Statement, that every graph ontributing to E Z H G Vi ) Z H G Vi ) only onsist of onneted omponents having at least two edges. Hene, ) k E Z H G Vi ) Z H G Vi ) = O n 1/k) k) = O n 1/ k). Therefore EZ H G) Z H G) = O n 1 1/k) n 1/ k) + = O n 1 1/k) n 1/ k) + 1 i j g ) E Z H G Vi ) E Z H G Vj g # H, G Vi ) = O n 3/ 1/k) k) + #H, G)), ) whih ompletes the proof.

21 Counting Arbitrary Subgraphs in Data Streams 1 Theorem 4.1 from page 11). Let G be a graph with n verties. For any onstants 0 < ε < 1 and k, there is an algorithm to 1 ± ε)-approximate #S k, G) with spae omplexity O n 1 1/k) ε n 3/ 1/k) G) k ) ) #S k ) + 1 log n. Proof. Sine the spae requirement of a single exeution of all g estimators is Og log n) = On 1 1/k) log n) bits, following the proof of Theorem 3.3 we know that )) k 1 n 3/ 1/k) in order to ahieve 1 ± ε)-approximation, we run O ε #S k ) + 1 independent exeutions of eah estimator and the total spae requirement is n 1 1/k) n 3/ 1/k) k ) ) O ε #S k ) + 1 log n. B. Counting Stars on Power Law Graphs We apply Lemma A.6 to graphs with approximate power law degree distribution See Setion 4 for the preise definition). As it turns out, the number of onneted subgraphs with k edges is Θ k ), whih is mathed for stars with k edges. Lemma B.3. Let G be a graph that has an approximate power law degree distribution with exponent β, 3). Let H be any onneted graph with k edges some of whih may be multi-edges). Then #H, G) = On k/β 1) ). Moreover, if H = S k, then #H, G) = Θ n k/β 1)). Proof. We use Lemma A.6 and the approximate Power law degree distribution to obtain that #H, G) = O degu) k )) = O u V dmin n d min ) k + d=1 = O n + = O n + = O n + log /d min) r=1 log /d min) r=1 log /d min) r=1 d=d min+1 d min r d=d min r 1 +1 d min r ) k fd) d k ) fd) d k d=d min r 1 +1 fd) d min r ) k n d min r 1 ) β+1 = O n + n k β 1)) = O n k/β 1)),

22 Daniel M. Kane, Kurt Mehlhorn, Thomas Sauerwald, and He Sun where in the last line we used our assumptions k, β, 3) and the fat that = Θn 1/β 1) ). Similarly, we an prove if H = S k, then #H, G) = Ω k) = Ω n k/β 1)). We point out that Lemma B.3 implies that we an get a onstant fator approximation for #S k, G) by simply estimating the maximum degree of G. However, even if we knew the maximum degree of G exatly, then this would only yield an approximation of #S k, G) up to some large onstant fator. Based on Lemma B.3 and the results from Setion 3, we an now easily prove that the unbiased estimator from Setion requires only logarithmi spae for approximately ounting the number of ourrenes of S k in power law graphs. Theorem 4. from page 11). Assume that G has an approximate power law degree distribution with exponent β, 3). Then, for any onstants 0 < ε < 1 and k, we an 1 ± ε)-approximate #S k, G) using O 1 ε log n ) bits. Proof. Let us onsider E Z H G) Z H G). If H = S k, then by Lemma 3., this term is upper bounded by the number of subgraphs of G with k edges so that every onneted omponent onsists of at least two edges possibly, multi-edges). By partitioning all graphs with that property into the at most k onneted omponents and applying Lemma B.3 to eah onneted omponent with l i edges separately, we onlude that E Z H G) Z H G) = O n li/β 1)) = O n k/β 1)). l 1,l,...,l k {0,,3,...,k} k li=k i : l i Sine #S k, G) = Ω n k/β 1)), we obtain the result as in Theorem 3.3 by taking the average of O ) 1 ε independent exeutions and applying Chebyshev s inequality.