Press Releases

6 pages
3 views

ISCoDe: A framework for interest similarity-based community detection in social networks

of 6
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Share
Description
This paper proposes a framework for node clustering in computerized social networks according to common interests. Communities in such networks are mainly formed by user selection, which may be based on various factors such as acquaintance, social
Transcript
   ISCoDe : a framework for interest similarity-basedcommunity detection in social networks Eva Jaho, Merkouris Karaliopoulos and Ioannis Stavrakakis Department of Informatics and TelecommunicationsNational & Kapodistrian University of AthensIlissia, 157 84 Athens, GreeceEmail: {ejaho, mkaralio, ioannis}@di.uoa.gr  Abstract —This paper proposes a framework for node clus-tering in computerized social networks according to commoninterests. Communities in such networks are mainly formed byuser selection, which may be based on various factors such asacquaintance, social status, educational background. However,such selection may result in groups that have a low degree of similarity. The proposed framework could improve the effective-ness of these social networks by constructing clusters of nodeswith higher interest similarity, and thus maximize the benefit thatusers extract from their participation. The framework is based onmethods for detecting communities over weighted graphs, wheregraph edge weights are defined based on measures of similaritybetween nodes’ interests in certain thematic areas. The capacityof these measures to enhance the sensitivity and resolutionof community detection is evaluated with concrete benchmarkscenarios over synthetic networks. We also use the frameworkto assess the level of common interests among sample users of a popular online social application. Our results confirm thatclusters formed by user selection have low degrees of similarity;our framework could, hence, be valuable in forming communitieswith higher coherence of interests. I. I NTRODUCTION The term  community  denotes a social group of people thathave one or more things in common. Whether this is residence,geographical neighborhood, traditions, or interests and ideals,communities have been long attracting the interest of sociolo-gists and psychologists thanks to their potential to motivate andshape human behavior. On the contrary,  virtual communities have emerged more recently and, almost always, transcenddistance barriers. Empowered by the Internet, these onlinecommunities socialize in virtual spaces provided by socialnetworking sites. A major research question is then how couldthe dynamics of these virtual worlds be exploited for more effi-cient design of networked communication protocols and whichfactors may shape the end-user (the network communicationsubject) behavior. It has been reported, for example, that highersimilarity in the interests/preferences of online social groupmembers favors collaborative, and even altruistic, behavior incontent replication [10] and content dissemination [1] scenar-ios. But is such similarity present in social networks, whereusers tend to select their friends/followers with very differentcriteria, including acquaintance, social status, educational andfamily background? To answer this question, we need to devisemechanisms and tools that can assess the similarity of interestsamong social group members and leverage the structure thissimilarity embeds in their social network.Our work in this paper addresses this requirement by poringover the interest-based community detection problem. Wepropose a framework, which we call “  ISCoDe ”, for assessingthe similarity in online social communities (Fig. 1). Input to  ISCoDe  are the interests of the communities’ member nodes incertain thematic areas, hereafter called “interest classes”, suchas music, sports, art. Each interest class could further be splitinto subcategories ( tags ). In section IV, we give an example of how the end-user interests can be inferred out of a real socialnetwork application.  ISCoDe  then proceeds in two steps. First,it quantifies the interest similarity between node pairs throughthe use of interest similarity metrics. Outcome of this step isa weighted graph representation of the social network, withedge weights corresponding to the similarity metric values.In a second step,  ISCoDe  can invoke standard communitydetection algorithms for weighted graphs (for example, [14][4]) to group nodes into disjoint clusters, connected internallyby high-weight edges and to other subsets’ nodes with small-or zero-weight edges. These algorithms also assess, in thesame time, the quality of this grouping through the modularitymetric [16]. SimilaritymetricsEdgeweightsWeightedcommunitydetectionClusters of  Input Output User interestdistributions users withsimilar interest Figure 1. The  ISCoDe  framework  We call  ISCoDe  a framework since there are more thanone options for its two main processing steps (namely, thederivation of the weighted graph edges and the communitydetection algorithm). Part of our work, hence, is devoted to theanalysis and assessment of these options. For the derivation of graph edge weights, we consider two metrics: the ProportionalSimilarity [20] and the inverse of the symmetrized Kullback-Leibler divergence [12]. Effectively, each metric could be seenas a different  transformation  from one data set (distributionof user interests over interest classes) to another (graph edgeweights). Comparing the outcomes of   ISCoDe  under syntheticuser interest distributions, we show that the choice of thesimilarity metric affects both the sensitivity and the resolutionproperties of our framework. Note that the similarity metricswe consider are different from the similarity indices that only This paper was presented as part of the Workshop on Network Science for Communication Networks (NetSciCom) 978-1-4244-9921-2/11/$26.00 ©2011 IEEE876  capture structural equivalence,  i.e. , same profile of relations toall other nodes in the network, such as the Pearson correlationand the Jaccard coefficient [5].  Related work  : In the literature, algorithms for detectingcommunity structure have largely been applied to a givennetwork structure, usually modeled as a graph. The mostprominent algorithm thereof is that of Girvan and Newman[16], which is highly efficient and overcomes many short-comings of previously proposed algorithms, such as graphpartitioning ( e.g. , spectral bisection [17], Kernighan-Lin algo-rithm [11]) and hierarchical methods ( e.g. , Euclidean distancesingle linkage clustering) [8]. These methods are not ideal foranalyzing general network data since usually it is not knownin advance in how many communities the network should besplit into and which is the best division. Newman furtherproposed a simple mapping from a weighted network to anunweighted multigraph and proposed an algorithm for detect-ing communities in weighted networks [14]. The graph edgeweights introduce another set of variables in the communitydetection process and it is shown in [7] that they can have biginfluence on the resulting community structure, especially ondense networks. Contribution of this paper  : As in Newman’s approach,current practice in community detection consists in apply-ing modularity-maximizing clustering algorithms over  given (weighted) graphs. Our work has a different starting point.In our paper the network structure,  e.g. , edge weight set, isnot given beforehand. It is rather generated by  ISCoDe  outof the distributions of user interests over different thematicareas, the only information we assume known and given tous. Since our framework uses the interest distributions as itsinput for community detection, in the same time it becomesa means of assessing the effectiveness of similarity metricsthat carry out the  mapping  of interest distribution differences.This paper, hence, explores how effectively different  mappings facilitate the detection of the underlying interest similaritystructure when the “commodity” community detection algo-rithms are applied on their  images ,  i.e. , the weighted edgesets they generate. Through the application of the framework and the presented results, the effectiveness of the proposedframework that advocates projecting distributional differencesto a weighted graph and using commodity approaches foridentifying communities thereof, is assessed and established.The paper proceeds as follows: Section II describes brieflythe scope and processing steps of the  ISCoDe  framework forinterest-based clustering. The evaluation methodology and ex-perimental results are presented in Section III. An applicationto a real network is described in Section IV. Finally, wesummarize the major conclusions of the paper in Section Vand point to interesting problems for future work.II. T HE  ISCoDe  FRAMEWORK FOR DETECTINGCOMMUNITIES OF NODES WITH SIMILAR INTERESTS In general, we want  ISCoDe  to satisfy three main require-ments: Correctness:  The framework should be able to distinguishcorrectly existing community structure. Whereas it may notalways be possible to conclude whether such structure reallyexists, the outcome of the framework should at least agreewith our intuition in certain benchmark scenarios, where thisstructure is evident. Sensitivity:  The framework should be able to adapt tochanges of the user interest distributions and reflect thestrength of the community structure. Resolution:  The framework should be able to identifyimportant community structure irrespective of its scale andthe overall network size.We evaluate  ISCoDe  along these lines in section III. In therest of this section, we detail the two processing steps of theframework and present baseline choices for populating them.  A. From interest distributions to the weighted graph Let  N   =  { 1 , 2 ,...,N  }  be the set of the network nodes(online social network users) and  M = { 1 , 2 ,...,M  }  the setof interest areas (classes). We assume that for each node  n we can have an estimate of   F  n , the probability distribution of its preferences over the  M   interest areas, which takes discretevalues  F  n 1  ,F  n 2  ,...,F  nM   with ∑ m ∈M F  nm  = 1 . Practically,  F  nm could be measured through the normalized request rate of node n  for data objects (content) of type  m  or some other form of interest expression in a certain area ( e.g. , subscription to thiscategory’s tags). In section IV, we describe this process for aparticular online social application.From the node interest distributions, we can then computethe pairwise similarity in the interests of two nodes drawingon measures of distributional similarity. Hereafter, we describeand focus on two of the possible choices: a) the ProportionalSimilarity (PS) metric, which is shown in [20] to satisfy 11  criteria suggested as suitable for a measure of similaritybetween distributions; and b) the inverse of Kullback-Leiblersymmetrized divergence (InvKL) [12]. InvKL projects thedifference between two interest distributions to a significantlybroader range of values compared to the PS metric,  i.e. , (0 , + ∞ )  vs.  [0 , 1] , thus shaping the resolution properties of the framework, as we will see later in Section III. 1) Proportional Similarity (PS) metric:  With the PS metric,the interest similarity  PS  F  i ,F  j  between two nodes  i  and  j ,with interest distributions  F  i and  F  j , equals [20]: PS  F  i ,F  j  = 1 −  12 M  󲈑 m =1  F  im − F  jm  .  (1) 2) Inverse KL (InvKL) symmetrized divergence:  Our secondmetric is the inverse of the Kullback-Leibler (KL) sym-metrized divergence, a metric capturing the distance betweentwo distributions InvKL F  i ,F  j  = 􀀨 M  󲈑 m =1 F  im logF  im F  jm +  F  jm logF  jm F  im 􀀩 − 1 .  (2)The InvKL metric takes values in  (0 , + ∞ ) . The KL diver-gence goes to infinity in cases where there is no interest 877  in one interest category from one node, whereas there isnon-zero interest in it from another. In order to avoid suchproblems, smoothing methods ( e.g. , interpolation and backing-off schemes) can be used. These have been studied in statisticallanguage modeling in order to estimate the distribution of natural language as accurately as possible [13]. In our casenon-zero request rates for interest classes can be discountedwith different discounting methods (see [13]), whereas interestclasses for which there is no interest can be given a small  ϵ >  0 probability.  B. From weighted graphs to communities Out of the full population of clustering algorithms, relevantto our objectives are those carrying out density-based graphclustering [3]. Namely, they take as input a graph and partitionit in a way that some notion of density (in our case: the weightsof intra-cluster edges) is significantly higher within a partitionthan across different partitions (inter-cluster edges). Withinthe complex networking community the de-facto criterion forassessing the quality of the partitioning is modularity [14],[16]. Modularity sums across all partition clusters the fractionof within-cluster edges minus the expected fraction of edgesthat would fall within the same cluster were they selected atrandom. For a given partition of a weighted graph  G ( V,E  ) ,where  V   is the set of network nodes and  E   the edge setcapturing pairwise interest similarities, modularity  Q  equals[14] Q  = C  󲈑 c =1 󰁛 l c L  − 󰀨 d c 2 L 󰀩 2 󰁝 ,  (3)where the sum is over the  C   communities of the partition,  L  isthe sum of the weights of all edges in the graph,  l c  is the sumof weights over edges lying fully within community  c , and  d c the respective sum over the full set of edges incident to nodesin  c . Modularity takes values in the interval  [ − 1 / 2 , 1]  [2]. Itbecomes zero for community structures that do not differ thanwhat one would get by random chance, whereas values above 0 . 3  −  0 . 4  suggest strong community structure.Our framework lends to the use of different modularity-maximization algorithms. One example is the divisive cluster-ing algorithm Newman proposed in [14] for weighted graphs.The algorithm iteratively removes from the srcinal graph theedge with the highest “edge betweenness” (defined as thenumber of shortest paths between pairs of nodes traversingthe edge) and recalculates modularity and edge betweennessvalues till modularity does not increase any further. Thecomplexity of the algorithm is  O ( | E  | 2 | V   | ) , which for densegraphs yields  O ( | V   | 5 ) .More generally, the problem of finding a partition thatmaximizes modularity in general graphs has been formu-lated as an Integer Linear Program (ILP) and shown to beNP-hard [2]. Proposed heuristic algorithms for modularitymaximization draw on simulated annealing [19] or extremaloptimization [6]. More commonly used and computationallyfriendlier, however, is the greedy agglomerative clusteringalgorithm of Clauset  et al.  [4], [15]. We simply extend it toweighted graphs by directly relating it with the definition of modularity in weighted graphs in (3). Initially each vertexis viewed as a discrete cluster of size one. The algorithmthen iteratively merges the two clusters that yield the largestmodularity increase. The algorithm completes in at most | V   |− 1  steps and has an implementation cost of   O ( | V   | 2 log | V   | ) [2] permitting scalability for large graph sizes. We retain thegreedy algorithm as the baseline for the assessment of   ISCoDe in Section III-A.III.  ISCoDe  EVALUATION We work with synthetic networks of   N   member nodeswith  controllably  similar interests in order to evaluate thecorrectness, sensitivity, and resolution properties of the frame-work. With modularity as the fitness metric of the detectedcommunity structure, structures featuring tighter communitieswith cleaner separation from each other should see higher  Q values than equinumerous yet “looser" structures. Moreover,with respect to  ISCoDe ’s resolution, we recall the remarks byFortunato and Barthèlemy in [9] that algorithms seeking tomaximize modularity may fail to identify important structuressmaller than a scale. In concluding whether the identificationof further distinct communities within a single one is mean-ingful, we adopt the weak “community” condition by Radicchi[18],  i.e. , a community  c  is correctly identified as one if  l c L  − 󰀨 d c 2 L 󰀩 2 >  0 .  (4)Note that in  ISCoDe  the resulting modularity values aresignificantly affected by the choice of the similarity metric.Contrary to other studies in literature, where communitydetection algorithms maximizing modularity are studied ongiven complex weighted graphs,  ISCoDe  adds the additionaltransformation step of interests to graph edge weights. There-fore, another requirement from the evaluation process is toshow how the two interest similarity metrics affect the threeframework requirements.In the general setting, the network population is organizedinto  k  groups. Each group is interested in  M  , generallydifferent, interest classes, which are the same for all membernodes of a given group. We form  k  equal-size groups of  N/k  users: nodes  1 ..N/k  are assigned to group  1 , nodes N/k  + 1 .. 2 N/k  to group  2 , and so on (for the sake of theexample, we take  N/k  to be an integer). We then control thesimilarity within and across the  k  groups as follows: Interest similarity  across  groups.  This is controlled intwo ways. Firstly, through the number of common interestareas between groups, which may take any value  r  in  [0 ,M  ] .Secondly, and this relates to the way the similarity  within a single group is controlled, through the way the interestsoverlap. We consider two scenarios for the distribution of common interests between two groups: a) the  r  commoninterest areas are simultaneously the  r  least interesting forgroup  g  and the  r  most interesting for group  g +1 ,  0  < g < k (  L(ast)-F(irst) , Table I(a)); b) the  r  common interest areas arethe  r  most interesting for the users of all  k  groups ( F(irst)- 878  F(irst) , Table I(b)). These scenarios present two extreme casesregarding the interest similarity across groups. Given that thenumber of common interest areas and the distributions areheld fixed, the L-F(F-F) scenario yields the smallest(highest)similarity. Table IE XAMPLE OF THE TWO INTEREST - OVERLAP SCENARIOS FOR  k  = 3 , M   = 5 . T HE ORDER OF INTEREST CLASSES MARKS ALSO THE ORDER OFNODES ’  INTERESTS WITHIN A GROUP .(a) L-F with a single overlap in-terest class ( r  = 1 ) Group  1  Group  2  Group  31 5 92 6 103 7 114 8 125 9 13 (b) F-F with two overlap interestclasses ( r  = 2 ) Group  1  Group  2  Group  31 1 12 2 23 6 94 7 105 8 11 Interest similarity  within  groups.  The interests of nodeswithin a group are spread over the ordered  M   interest classesinline with the Zipf distribution 1 . The skewness parameter  s of the distribution differs for each group node. The interest of the first node of each group are uniformly distributed ( s 1  = 0 )and  s  increases with constant step  p  so that for node  n , s n  =  p ( n − 1) ,p ∈R . Higher  p  values increase the skewnessin the interest distribution and concentrate the node interests’mass in the higher-order interest classes. Interestingly, changesof   p  also affect the similarity of interests between nodes be-longing to different groups depending on the overlap scenario(Table I): higher  p  values result in weaker (stronger) inter-group similarity in the  L − F   ( F   − F  ) overlap scenario.In summary, by calibrating  p , the overlap scenario and thenumber of common interest classes, we can produce networkswith community structures of variable discernibility.  A. Experimental results and discussion We show and discuss representative results from our exper-imentation with  ISCoDe  on synthetic networks that outline themain behavior of the framework. All experiments are carriedout with the greedy agglomeration algorithm in [4] since ityields significantly faster run times than its competitors 2 . 1) Correctness and sensitivity experiments:  In this set of experiments,  N   = 80  and  k  = 4 . The impact of   M   wasfound to be minimal, thus we present herein results only for M   = 20 . We vary the interest overlap scenarios (as in TableI), the number of common interest classes, one (Tables II(a),II(c)) or half of them (Tables II(b), II(d)), and the skewness of the interest distributions, smaller  p  values representing moreuniform distributions within a single group nodes.The first remark is that both metrics produce the sameintuitive community partitions in the presence of strong com-munity structure, as in Tables II(a) and II(c) for low  p  values. 1 Zipf distributions have been used in the recent past to capture preferencesfor web objects. Furthermore, they exhibit high modelling simplicity andflexibility, in that proper manipulation of their single parameter  s , gives riseto a wide set of distributions ranging from uniform ( s  = 0 ) to highly skewedones with power-law characteristics ( s >>  0 ). 2 We run experiments also with the divisive clustering algorithm in [14] butwe had to restrict to small group sizes. In these cases, we obtained similarresults with respect to community structure and modularity values. On the contrary, when such structure is not evident, the twometrics result in considerably different partitions.The second remark has to do with the higher sensitivity of the framework when the PS metric is used in its first pro-cessing step. The modularity of the resulting partitions underthe PS metric decreases when the interests of nodes are morerandomly diffused over the different interest classes and goesdown to zero when the similarity structure tends to disappear,as in Table II(d). On the contrary, the resulting modularityvalues under the InvKL metric are almost insensitive to thechanges in the input interest distributions. With InvKL themodularity values are dominated by the edge weights betweenindividual node pairs; these tend to be very high ( ≫  1 ) forhighly similar nodes and very low for highly dissimilar nodes.Finally, as  p  increases, the interest distributions of most nodestend to be more concentrated on the first group objects, andthe interest distributions become less uniform. For cases shownin Tables II(a) and II(b), this results in increasing modularityunder the PS metric, thanks to the decreasing weights of inter-group edges,  i.e. , nodes initially assigned to different groups.It has the opposite effect for cases shown in Tables II(c) andII(d), where increasing  p  leads to stronger ties between nodesin different groups. On the contrary, InvKL does not adapt toany of these changes. 2) Resolution experiments:  We run two additional exper-iments focusing on the impact of the two similarity metricsupon the overall framework resolution. The first experimentinvolves nodes with  highly similar interests . All nodes areinterested in the same  M   objects, in the same order. Theydifferentiate only slightly in how their interests are spread overthe  M   interest classes, modelled by Zipf(s) distributions with s  varying from 0 to its maximum value in steps of   p  = 0 . 01 .The second experiment involves nodes with  highly dissimilar interests ; there is a single common interest class betweensuccessively ordered nodes. The experiment resembles the L-Foverlap scenario shown in Table I(a), if each group containedonly one node. The results from the two experiments arereported in Table III and clearly demonstrate the capacityof the two metrics to illuminate better different parts of theinterest similarity range.Mapping highly similar interest distributions to a muchbroader edge weight value range (Figure 2(a)), InvKL canresolve more communities than PS in the first experiment,all of which satisfy Radicchi’s weak community condition of (4). On the contrary, PS tends to group small communitiestogether. Notably, the communities produced by both metricsdo not satisfy the inequality l c  < √  2 L,  (5)which according to [9] suggests that community  c  may bethe combination of two or more smaller communities thatcannot simply be detected when pursuing the optimization of modularity due to their small size.The situation is reversed in the second experiment: it isnow PS that can recognize smaller communities, as shownin Table III(b). Moreover, (5) is satisfied, implying that there 879  Table IIC ORRECTNESS AND SENSITIVITY EXPERIMENTS : M ODULARITY AND COMMUNITIES FORMED FOR DIFFERENT VALUES OF  p , N   = 80 , M   = 20 (a) L-F,  1  common objectPS InvKL Q C   partition  Q C   partition  p = 0 . 02 0 . 6849 4  {1..20}...{61..80}  0 . 7498 4  {1..20}...{61..80}  p = 0 . 04 0 . 6925 4  {1..20}...{61..80}  0 . 7493 4  {1..20}...{61..80}  p = 0 . 06 0 . 6992 4  {1..20}...{61..80}  0 . 7483 4  {1..20}...{61..80}  p = 0 . 08 0 . 7048 4  {1..20}...{61..80}  0 . 7698 8  {1..10}...{71..80}  p = 0 . 10 0 . 7095 4  {1..20}...{61..80}  0 . 7745 8  {1..10}...{71..80}(b) L-F,  M/ 2  common objectsPS InvKL Q C   partition  Q C   partition  p = 0 . 02 0 . 3594 2  {1..40} {41..80}  0 . 7667 4  {1..20}...{61..80}  p = 0 . 04 0 . 3669 2  {1..40} {41..80}  0 . 7490 4  {1..20}...{61..80}  p = 0 . 06 0 . 3756 2  {1..40} {41..80}  0 . 7475 4  {1..20}...{61..80}  p = 0 . 08 0 . 3938 3  {1..20} {21..40} {41..80}  0 . 7687 8  {1..10}...{71..80}  p = 0 . 10 0 . 4142 3  {1..20} {21..40} {41..80}  0 . 7730 8  {1..10}...{71..80}(c) F-F,  1  common objectPS InvKL Q C   partition  Q C   partition  p = 0 . 02 0 . 5711 4  {1..20}...{61..80}  0 . 7498 4  {1..20}...{61..80}  p = 0 . 04 0 . 5146 4  {1..20}...{61..80}  0 . 7492 4  {1..20}...{61..80}  p = 0 . 06 0 . 4465 4  {1..20}...{61..80}  0 . 7480 4  {1..20}...{61..80}  p = 0 . 08 0 . 3734 4  {1..20}...{61..80}  0 . 7692 8  {1..10}...{71..80}  p = 0 . 10 0 . 3038 4  {1..20}...{61..80}  0 . 7730 8  {1..10}...{71..80}(d) F-F,  M/ 2  common objectsPS InvKL Q C   partition  Q C   partition  p = 0 . 02 0 . 1103 4  {1..20}...{61..80}  0 . 7496 4  {1..20}...{61..80}  p = 0 . 04 0 . 0841 4  {1..20}...{61..80}  0 . 7481 4  {1..20}...{61..80}  p = 0 . 06 0 . 0610 4  {1..20}...{61..80}  0 . 7444 4  {1..20}...{61..80}  p = 0 . 08 0 . 0422 4  {1..20}...{61..80}  0 . 7611 8  {1..10}...{71..80}  p = 0 . 10 0 . 0485 5  {1..15}...{61..75} {16..20,36..40,56..60,76..80}  0 . 7549 8  {1..10}...{71..80}Table IIIR ESOLUTION EXPERIMENTS : M ODULARITY AND COMMUNITIES FORMED , N   = 80 , M   = 20 (a) Similar nodesPS InvKL Q C   partition  Q C   partition 0 . 0215 2  {1..38} {39..80}  0 . 6740 5  {1..14} {15..28} {29..44} {45..61} {62..80}(b) Dissimilar nodesPS InvKL Q C   partition  Q C   partition 0 . 7860 10  {1..8}..{73..80}  0 1  {1..80} 0.97 0.975 0.98 0.985 0.99 0.995 1051015 weight   w  e   i  g   h   t   d   i  s   t  r   i   b  u   t   i  o  n PS   1020304050607080 10 4 10 5 10 6 10 7 020406080 InvKLweight   w  e   i  g   h   t   d   i  s   t  r   i   b  u   t   i  o  n   1020304050607080 (a) Similar nodes 0 0.01 0.02 0.03 0.04 0.05020406080 weight   w  e   i  g   h   t   d   i  s   t  r   i   b  u   t   i  o  n PS   1020304050607080 0.0245 0.025 0.0255 0.026 0.0265 0.027020406080 weight   w  e   i  g   h   t   d   i  s   t  r   i   b  u   t   i  o  n InvKL   1020304050607080 (b) Dissimilar nodesFigure 2. Resolution experiments: Edge weight distributions are more non-detected communities. InvKL, on the contrary, cannot   since it squeezes all edge weight values that resultfrom the first processing step within an interval of   0 . 012  width(Figure 2(b)).However, an important question is regarding the level of resolution,  i.e. , in which cases communities should be moreresolved. Intuitively, it seems more important to identify finercommunity structure in a network with more similar nodes,than in case of dissimilar ones. Hence, the resolution advan-tage of InvKL at high similarity scenarios may overweigh itsdisadvantage at low similarity ones.IV. A PPLICATION TO A REAL NETWORK We apply  ISCoDe  to data traces extracted from the Deliciouswebsite (www.delicious.com). Delicious is a social bookmark-ing application where users can save all their web bookmarks(annotated with tags) online, share them with other users,and track what other users are bookmarking themselves. Each 880
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks
SAVE OUR EARTH

We need your sign to support Project to invent "SMART AND CONTROLLABLE REFLECTIVE BALLOONS" to cover the Sun and Save Our Earth.

More details...

Sign Now!

We are very appreciated for your Prompt Action!

x