Set-based Approach for Lossless Graph Summarization using Locality Sensitive Hashing

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1 Set-baed Approach for Lole Graph Summarization uing Locality Senitive Hahing Kifayat Ullah Khan Supervior: Young-Koo Lee Expected Graduation Date: Fall 0 Deptartment of Computer Engineering Kyung Hee Univerity Republic of Korea {kualizai, yklee}@khu.ac.kr Abtract Graph ummarization i a valuable approach for in-memory proceing of a big graph. A ummary graph i compact, yet it maintain the overall characteritic of the underlying graph, thu uitable for querying and viualization. To ummarize a big graph, the idea i to compre the imilar node in dene region of the graph. The exiting approache find thee imilar node either by node ordering or pair-wie imilarity computation. The former approache are calable but cannot imultaneouly conider the attribute and neighborhood imilarity among the node. In contrat, the pair-wie ummarization method can conider both the imilarity apect but are impractical for a big graph. In thi paper, we propoe a etbaed ummarization method that aggregate the et of imilar node in each iteration, thu provide calability. To find each et, we approximate the candidate imilar node without node ordering and explicit imilarity computation by uing Locality Senitive Hahing, LSH. In conjunction with an information theoretic approach, we preent the calable olution for lole ummarization of both attributed and non-attributed graph. I. INTRODUCTION With a continuou and overwhelming interet from people with divere background, the graph like ocial network, world wide web, academic, citation network, and chemical compound have become ubiquitou. Furthermore, with each blink of a human eye, the ize of thee graph i continuouly increaing. A a reult, thee graph have haped into knowledge warehoue ince their every change i triggered by the real life neceitie. However, retrieving the right information at right time i hard, ince the maive ize doe not allow their in-memory proceing. In thi ituation, graph ummarization i a valuable olution for in-memory proceing of a big graph. It compree a large ized graph into a compact ummary, and maintain the propertie of it underlying graph. Such compact verion i ueful to identify communitie and influential node, information propagation and viualization. The dene homogenou region in the graph play the key role for their ummarization. There exit a high overlap of common neighborhood among the node in uch region, which i directly exploited toward compreion. The exiting work i claified into two main tream to find uch dene region, i.e. compreion-baed and aggregation-baed. The 0 Correponding Author: Young-Koo Lee Total number of minute pent on Facebook each month: 0 Million. Lat acceed on 0//0 goal of compreion-baed method i to find an intrinic ordering of the node in a graph, o that the compreion algorithm can exploit the naturally occurring propertie like locality of reference and neighborhood imilarity. There exit a plethora of reearch work on thi apect [] [] [] [] [], however the ordering produced by combining the Layered Label Propagation and WebGraph framework in [] provide better compreion than all the exiting technique []. On the other hand, the aggregation-baed method perform pair-wie ummarization to collape a pair of node into a uper node and their correponding edge into a uper edge [] [] [0] []. Thee method create a highly compact ummary graph which i directly ueable for querying and viualization. We undertand that the trength and weaknee of both the compreion and aggregation-baed method, alo depend on the type of the graph. In cae of a non-attributed graph, the former approache are highly calable but doe not upport direct querying and viualization ince the compreed graph i tored in a compact data tructure. Furthermore, the low degree node are alo caue of in-compreibility, pecially in ocial network []. However, the pair-wie uper node creation technique i inenitive to the degree of the node and compree both the node and edge into a ummary graph. Unfortunately, the pair-wie aggregation method i impractical to ummarize a big graph. On the other hand, the node ordering cannot conider the attribute imilarity among the node when applied to a graph where each node i attached with multiple attribute. In thi ituation, the pair-wie method provide a uperior performance ince each pair-wie merge can invetigate both the neighborhood and attribute imilarity. Conidering thee fact, it i deirable to have a unified graph ummarization olution which i not only calable but alo applicable to variety of the graph, with ome minor modification. In thi paper, we preent a novel et-baed graph ummarization technique to iteratively aggregate the Set of Similar Node, SSN. A et-baed method i effective ince aggregating multiple node, facilitate efficient traveral of the entire graph for it fat ummarization. Since, the member node of a given et can have intra-connection apart from having common neighbor, o the propoed approach i uitable to identify any dene ubgraph tructure like clique, γ clique and bipartite ubgraph. However, locating uch et i the major challenge without node ordering and explicit imilarity computation. For thi purpoe, we find each SSN by approximating the

2 0 (a ) 0 0 = {,,, } + (, ) - (, ) a (a ) v - (, ) Fig.. Set-baed approach to compre Graph G (V, E, A) Set of node with maximum ame attribute Compreion uing uper node Set of node having unique attribute for each node Compreion uing virtual node Candidate Set of Similar Node, CSSN, uing LSH. Since LSH i an approximate technique o each CSSN may contain node with varying imilaritie among each other. For thi purpoe, we propoe a pruning technique baed on a heuritic that focue the imilaritie of degree of node, to filter out a SSN from each CSSN. In cae of an attributed graph, LSH cannot imultaneouly conider both the neighborhood and attribute imilarity among the node. To olve thi problem, we modify it hahing cheme to generate SSN, containing node having high neighborhood a well a attribute imilaritie. For ummarization, we ue the information theoretic Minimum Decription Length, M DL, principle that produce a highly compact ummary graph with correction. In cae of nonattributed graph, MDL facilitate merging a SSN into a uper node with leat edge correction. Similarly for a multi-attributed graph, MDL help to compre a SSN into a uper node or aggregate it edge by adding a new virtual node, baed on highet compreion with leat attribute and edge correction. II. 0 MDL RULES AND PROBLEM STATEMENT A. MDL Rule for a Summary Graph MDL i an information theoretic principle ued to minimize the um of the ize of the theory and the aociated data to expre the knowledge. In cae of graph, the minimized theory i the ummary graph and the lit of correction i the aociated data to recontruct the original graph, if required. Our objective to ue MDL for graph ummarization i to create a compact ummary graph with leat correction. Below, we explain the MDL rule for an attributed graph which are applicable for a non-attributed graph, while ignoring the detail related to attribute and edge aggregation uing a virtual node. Following MDL explanation i adopted from [] but we modify it for attribute, virtual node and ummary graph with leat edge correction. We do not provide the comparative analyi due to lack of pace An undirected graph G(V, E, A) conit of node V and edge E where each v V i attached with the lit of attribute A. The MDL repreentation of G i formally repreented a G MDL = (S G, C r ) where S G i the ummary graph, S G (V, E, A ), and C r i the lit of correction. Each node v V i either a uper or virtual node. The uper node correpond to et A v where v A v i v V in G. A virtual node vn i a new node added to S G to compre the uper edge. Similarly each edge (A u, A v ) E i a uper edge and i the et of all edge between member of A u and A v in G, except the uper edge (A u, A vn ) E. In Figure and, we how the MDL repreentation of the graph in and after compreion uing uper and virtual node, along with the correction. The rule to create the uper edge and correction vary for uper node and virtual node. For edge correction uing uper node, we define Π a the et of all poible edge between (A u, A v ) V and A uv a the actual edge between member of A u and A v. A uper edge i created between A u and A v if A uv ( Π +)/ otherwie poitive edge correction are created between their member. The negative edge correction are created for the edge Π A uv. Since we are intereted in a compact ummary with leat correction, o we create either uper edge with poitive edge correction or only negative edge correction, which have minimum memory requirement. For each uper node, we et it attribute by picking the value which have maximum repetition in the given SSN. The remaining attribute value are marked a attribute correction. Uing virtual node, only edge correction are poible. In thi cae, we create negative edge correction for a pair of node, appearing a connected through virtual node and poitive edge correction in vice vera cae. The cot of the ummary graph depend upon it torage cot and correction. The torage cot of the mapping of uper node i mall compared to thoe of E and C r, o we ignore it. Thu the cot of ummary i computed a cot(g MDL ) = E + C r. B. Problem Statement Given an undirected graph G, our goal i to efficiently find the SSN to create a compact ummary graph with leat correction. In a non-attributed graph, there exit high neighborhood imilaritie among the member node of each SSN. The ummary graph of uch a graph i attached with edge correction only. On the other hand, there are attribute a well a edge correction for ummary of an attributed graph. Similarly, the member node of each SSN are imilar from both the neighborhood a well a attribute imilarity. III. SUMMARIZING A NON-ATTRIBUTED GRAPH In thi ection, we illutrate how we apply LSH on graph G(V, E) and briefly explain the propoed pruning technique. It hould be noted that the we identify the SSN uing a query node-baed approach, where we iteratively elect a query node q to identify it SSN. Thi approach provide each node at leat one chance of merger. We kip the explanation for bucket-baed SSN retrieval approach and related detail due to lack of pace.

3 Node Attribute Neighbor Id a, b,, a, b,,, a, b,,,, a, b,, a, b, a, b,, π π π π π π π π Degree (n i ) = Degree (n ) =... = Degree (n c ) () Degree (n i ) = Degree (q) () Band Band π π π π (e) Hah Table Hah Table,,,,, Fig.. LSH applied on graph G (V, E, A) Binary repreentation of neighbor lit and hah function Minhah matrix Diviion of matrix into band (e) Combined hah code for each portion of column (f) Hah table containing bucket of candidate imilar node (f) Let evaluate a cae where we obtain ome edge correction, when the imilaritie between the node in a SSN are lightly varying. Conider the bipartite graph in Figure which on aggregation, produce a uper node with two edge correction. Thi illutrate that there are no edge correction when the degree are ame, Figure, but they occur when a change happen in the degree, Figure. So it i evident that the edge correction from a uper node, depend upon the deviation in the degree of each n i from that of q (). Edge Correction Fig.. Relationhip between the neighborhood imilarity and degree of node. Node having ame degree and ame imilaritie Variation in degree of node, producing edge correction A. Identifying SSN We create the uper node uing SSN. To find the SSN, we apply LSH on G uing tep in Figure, and retrieve CSSN againt each q. Since each SSN i a ubet of it correponding CSSN, o we propoe a heuritic baed on imilaritie among degree of node and ue it to propoe a pruning technique. ) Similar Degree Heuritic: Given a CSSN, the imilar degree heuritic aim to identify it ubet of node by chooing the node having imilar degree to that of q. By having the imilar degree, the imilarity among the member node of each SSN increae, reulting in uper node with leat edge correction. Note that the node having ame degree to that of q, are conidered by default. Normally, the node having varying degree reduce the neighborhood overlap. Thu we find that there i a relationhip between the imilarity of node and their degree. Conider a CSSN = {,,, } in Figure. From thi CSSN, the SSN contain all the node which on aggregation produce a uper node with no edge correction. We oberve that in thi SSN, the neighborhood of any given node n i i ame to every other node () and alo to that of q (). Nbr (n i ) = Nbr (n )... Nbr (n c ) () where n i SSN and n i q - (, ) - (, ) Nbr (n i ) = Nbr (q) () Invetigating the degree of each node in the given SSN, we find that degree of any n i SSN i alo equal to every other node in it () and alo to that of q (). Degree (q) Degree (n i ) δ Degree (q) () We add any n i CSSN to SSN if it degree difference with q atifie δ Degree (q) (). Otherwie, there are more edge correction from a uper node compared to the deired compreion ratio. We bound δ by 0 δ 0., where the lower limit target each n i having complete neighborhood overlap with q. Similarly, the upper limit of δ enforce to have more common to that of non-common neighbor between n i and q. Thu we oberve that the node having imilar degree in CSSN, have higher probability to be more imilar. By defining the range of δ, pruning the leat imilar node in a CSSN i facilitated. Uing (), we alo obtain the lower and upper bound to extract a SSN from a given CSSN (). δ min Degree (q) Degree ni δ max Degree (q) () When each n i have imilar degree to that of q then there i a high probability to have large neighborhood overlap between n i and q. However, imply relying on imilar degree, produce inconitent reult in ome of the cae. Fortunately, we reolve thi inconitency and identify the required node uing propoed pruning method. ) Auto Pruning: The objective of auto pruning i to filter the node from a CSSN, which minimize the overlap of common neighbor. The imilar degree heuritic directly help prune the node having large deviation from the degree of q. To prune the leat imilar node, we firt ort the CSSN uing degree of node. Thi ort bring the node having imilar degree to q cloer to it in the orted order. We then tart the earch of SSN by iteratively comparing the overlap ratio from the nearby node in the orted order. We add a node into SSN if it neighbor atify the bound in (). On narrowing the bound, we tend to extract only the highly imilar node which reult in lower compreion ratio. Having uch bound, the likely imilar node at leat get the chance for overlap computation. An end to the traveral happen either when we find the node violating the bound or entire lit i exhauted if each member of CSSN ha degree within the bound.

4 IV. SUMMARIZING AN ATTRIBUTED GRAPH In thi ection, we preent the propoed approach to unify the topological and attribute information in LSH, ince the LSH tep in Figure doe not conider the attribute imilarity. For each node in an attributed graph G(V, E, A), the adjacency lit N br and attribute et Attrib are emantically two different entitie. Wherea, the adjacency lit of each node hold it tructural information, the et of attribute decribe it identity. Apart from their emantic difference, uually the neighborhood of each node ha much larger ize compared to it attribute et, N br >> Attrib. Since in real world graph like ocial network, uer often do not provide all the attribute information. On the other hand, the graph having power law propertie have large number of low degree node, leading to N br < Attrib. Both of thee information can be conidered a the two dimenion of each node and the problem i how to unify or reduce them uch that their output preerve the intrinic propertie from both the apect. In our cenario, we refer thi problem a the cure of dimenionality due to the emantic difference between the two dimenion, rather than upon their count. Since the exiting dimenion reduction method, both linear and non-linear, operate on n-dimenional data where n can be large but have emantic homogeneity like patial information. So we oberve that thee technique are not applicable for G(V, E, A). However, influenced by the projection technique we propoe to project the two dimenion of each node into a ingle unified granularity level. To be precie, we union the attribute et of each node with it adjacency lit and perform LSH on the unified lit. By uch unification, the ubequent minhah ignature of each node contain the repreentation from both the apect. Thi i o ince any value in the combined lit can become the minimum value againt a given hah function π k. Thu, the reulting CSSN have high probability to be imilar from both the apect. We now formally define the propoed unified information in Definition. Definition. [N eighbor Attribute Lit] Given a node v i V along with it lit of neighbor N i and attribute A i, the Neighbor Attribute Lit (NAL) i a unified lit that concatenate N i and A i. Figure how a graph G (V, E, A) where each v i V i attached with two attribute. In Figure, we map the attribute into numeric value to unite them with the adjacency lit. The Figure illutrate the NAL for node and. With the unification of attribute and neighborhood information, we can compute the minhah value uing hah function. Since now we conider both the information for minhah computation, it i deirable that their repreentation mut be preerved in hah function. We now formally define the hah function in Definition. Definition. [U nif ied Hah F unction] The random permutation {π, π,..., π m } i a et of hah function where π i i a permutation of {,,..., n, n +,..., k}. The et {,,..., n} are the total node in G and {n +,..., k} i the union of all cardinal value from each attribute. We illutrate the ample hah function in Figure. Here each function i the random permutation from total node in Node Id Attribute Neighbor a, b,, a, b,,, a, b,,,, a, b,, a, b, a, b,, 0 A B a b a b Mapping A B 0 π 0 π 0 Fig.. Illutrating the Unified LSH concept Graph G (V,E,A) Attribute Mapping Sample Node from G with their NAL Hah function the G and poible attribute value from all the attribute, after mapping. A. Balance between Neighborhood and Attribute Information Having NAL for each node and unified hah function, we can find the CSSN. However, thi procedure may till not reveal the node imilar from both the apect. For intance, conider the NAL i of a node v i. Uing (), we find that each element in NAL i ha the equal probability to be minimum againt a given hah function, π k. On the other hand, ince the ize of adjacency lit and attribute et are not ame. So v i V, whoe Nbr >> Attrib, their attribute have very low probability to be minimum in any of the hah function, compared to that of their neighbor, (). Thi reult in creation of minhah ignature column preerving the neighborhood information only, thu minimizing the influence of attribute information. In wort cae, a pair of node (u, v) V having a complete overlap of neighbor but no attribute imilarity or vice vera, will produce exactly ame minhah ignature. For example, conider the node in Figure. Uing (), the probability of each element in it NAL to be conidered a minimum i 0.. However, the probability of any neighbor in NAL to become minimum i larger than that of any of it attribute value, 0. and 0. repectively. Now in Figure conider the node, a wort cae cenario, where both hare all of their neighbor but have no attribute in common. Concluively, their Jaccard imilarity doe not depict the real ituation and may produce more attribute correction on aggregation into a uper node. P r (min (π (NAL i )) = x NAL i ) = NAL i P r (min (π (NAL i )) = x Nbr i ) P r (min (π (NAL i )) = y Attrib i ) () Ideally the minhah ignature of each node mut have the repreentation from it neighbor a well attribute. For thi purpoe, we propoe to enforce repreentation from both the ()

5 apect in the minhah ignature o that the bia toward either apect i minimized. In our approach, we ue a parameter p to preerve the neighborhood and attribute information in the minhah ignature for each node. Specifying p guarantee that certain proportion of minhah value are computed from each component, retricting the large percentage of minhahe from either component. Uing p, the equation () and (0) tate the fraction of minhah value from each of the two component in NAL i for each v i. We undertand that the right election of p play a key role to aign certain weight to each of the two component in NAL i. For intance, if p = and total hah function are 0 then uing () and (0), the minhah ignature column for the node in Figure, contain 0% minhah value from each of the component, (). Thu, we find with the incluion of p, the probability to have minhah value from attribute increae from 0. to 0.. Minhah Nbr v i Minhah Attrib v i = m p m = p m () (0) P r (min (π (NAL i )) = x Nbr i ) = P r (min (π (NAL i )) = y Attrib i ) () We oberve that the enforced incluion of neighborhood and attribute information in the minhah ignature, may produce erroneou reult when we generate CSSN in hah table. Figure diplay the tep when we plit the minhah matrix into b band of r row each, to create CSSN on the bai of hah code. If a given band get a portion of minhah column where all the minhah value are either from neighbor or attribute then the correponding bucket will contain the node imilar from that very apect only. There i a higher probability for thi ituation in the cae when we ue higher count of hah function, ince then the length of reulting minhah column for each node i alo high. Thi ituation i poible in both the random and equential election of minhah value for each band. To overcome thi limitation, we propoe a concept called Bi M inhahing which produce minhah ignature preerving the repreentation from both the apect from NAL i of each v i. B. Bi-Minhahing In bi-minhahing, we compute the minhah ignature of each v i where both of it dimenion (neighbor or attribute) ha the equal probability to be choen a minimum againt any hah function, π k. Uing () and (0), we firt compute the minhah ignature M NAL from neighbor and attribute. Next, we again apply minhaing on M NAL to compute it minhah ignature uing the ame hah function to create M / NAL. Thi econd level hahing completely randomize the minhah repreentation of both the apect from NAL i of each v i. Thu it rightly approximate the true Jaccard imilarity between any pair of node (v i, v j ). We find the ignificance of bi-minhahing i two-folded. Firt, it create a balance between neighborhood and attribute information, i.e., when N br >> Attrib or vice vera. Since etting p to 0%, the initial hahing create the minhah ignature where each of the two component ha balanced repreentation. Secondly, the equal minhah repreentation i itelf the olution of electing the appropriate value of p. V. CONCLUSION AND FUTURE WORK In thi paper, we propoe a et-baed approach for graph ummarization. A et-baed approach i calable ince it aggregate multiple node in each iteration. We undertand that the propoed approach add a new dimenion in aggregation-baed ummarization ince we directly target the inherent occurrence of phenomenon like friend of friend have many common friend in ocial network without node ordering or explicit imilarity computation. Currently, our approach i applicable to both attributed and non-attributed graph, but we plan to build a unified olution to ummarize further type of graph like heterogeneou and weighted graph. VI. ACKNOWLEDGMENT Thi work wa upported by the National Reearch Foundation of Korea (NRF) grant funded by the Korea government(mest) (No. 0RAAA00). REFERENCES [] F. Chierichetti, R. Kumar, S. Lattanzi, M. Mitzenmacher, A. Panconei, and P. Raghavan, On compreing ocial network, in Proceeding of the th ACM SIGKDD international conference on Knowledge dicovery and data mining. ACM, 00, pp.. [] P. Boldi and S. Vigna, The webgraph framework i: compreion technique, in Proceeding of the th international conference on World Wide Web. ACM, 00, pp. 0. [] U. Kang and C. Falouto, Beyond caveman communitie : Hub and poke for graph compreion and mining, in Data Mining (ICDM), 0 IEEE th International Conference on. IEEE, 0, pp [] G. Buehrer and K. Chellapilla, A calable pattern mining approach to web graph compreion with communitie, in Proceeding of the 00 International Conference on Web Search and Data Mining. ACM, 00, pp. 0. [] I. Safro and B. Temkin, Multicale approach for the network compreion-friendly ordering, Journal of Dicrete Algorithm, vol., no., pp. 0 0, 0. [] P. Boldi, M. Roa, M. Santini, and S. Vigna, Layered label propagation: A multireolution coordinate-free ordering for compreing ocial network, in Proceeding of the 0th international conference on World Wide Web. ACM, 0, pp.. [] P. Liako, K. Papakontantinopoulou, and M. Siouti, Puhing the envelope in graph compreion, in Proceeding of the rd ACM International Conference on Information and Knowledge Management. ACM, 0, pp.. [] S. Navlakha, R. Ratogi, and N. Shrivatava, Graph ummarization with bounded error, in Proceeding of the 00 ACM SIGMOD international conference on Management of data. ACM, 00, pp.. [] H. Toivonen, F. Zhou, A. Hartikainen, and A. Hinkka, Compreion of weighted graph, in Proceeding of the th ACM SIGKDD international conference on Knowledge dicovery and data mining. ACM, 0, pp.. [0] K. LeFevre and E. Terzi, Gra: Graph tructure ummarization. in SDM. SIAM, 00, pp.. [] Y. Tian, R. A. Hankin, and J. M. Patel, Efficient aggregation for graph ummarization, in Proceeding of the 00 ACM SIGMOD international conference on Management of data. ACM, 00, pp. 0.