Malliaros, Fragkiskos

Papadopoulos, Apostolos,

Seux, Damien

Vazirgiannis, Michalis

Papadopoulos, Apostolos, N

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HAL Id: hal-01672968https://hal.archives-ouvertes.fr/hal-01672968Submitted on 27 Dec 2017HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Core Decomposition of Uncertain Graphs UsingRepresentative InstancesDamien Seux, Fragkiskos Malliaros, Apostolos Papadopoulos, MichalisVazirgiannisTo cite this version:Damien Seux, Fragkiskos Malliaros, Apostolos Papadopoulos, Michalis Vazirgiannis. Core Decompo-sition of Uncertain Graphs Using Representative Instances. 6th International Conference on ComplexNetworks and Their Applications, Nov 2017, Lyon, France. ￿hal-01672968￿Core Decomposition of Uncertain Graphs UsingRepresentative InstancesDamien Seux1, Fragkiskos D. Malliaros2, Apostolos N. Papadopoulos3, andMichalis Vazirgiannis11 Computer Science Laboratory, École Polytechnique, France{damien.seux, michalis.vazirgiannis}@polytechnique.edu2 Center for Visual Computing, CentraleSupélec and Inria, France and UC San Diego, USAfmalliaros@eng.ucsd.edu3 Department of Informatics, Aristotle University of Thessaloniki, Greeceapostol@csd.auth.gr1 Introduction and Problem StatementIn many real-world applications, the corresponding graphs are inherently associatedwith uncertainty, which can be due to various reasons, such as uncertainty introducedby the data-collection process or for privacy-preserving reasons. For example, in thecase of protein-protein interaction networks (PPI) in the domain of biology, each nodecorresponds to a specific protein and the edges capture information about the interactionof two proteins. Since in many cases those interactions are indicated either by noisylaboratory experiments or by prediction algorithms based on features of the proteins(instead of being actually observed), a level of uncertainty is introduced in the edges ofthe graph. This uncertainty can be captured by the model of uncertain or probabilisticgraphs, where each edge is associated with a probability of existence.In this work, we are interested in a widely applied graph analytics tool, namely theone of k-core decomposition [4]. Let H be a subgraph of graph G. Subgraph H is de-fined to be a k-core of G, denoted by Gk, if it is a maximal connected subgraph of Gin which all vertices have degree at least k. Based on that, vertex i has core numbercoreG(i) = k, if it belongs to a k-core but not to any (k+ 1)-core. Due to its simplic-ity and computational efficiency, the k-core decomposition has been applied in manydomains, including community detection and identification of influential spreaders insocial networks. Then, the following questions arise, which also describe the goals ofthis work: how to define the concept of core decomposition in uncertain graphs and howto efficiently compute it? Bonchi et al. [2] proposed an extension of the k-core decom-position to uncertain graphs which requires that the probability that each vertex v withinthe core subgraph H has degree at least k, is greater than or equal to a parameter η . Nev-ertheless, this definition has two main weaknesses: (i) an extra probability threshold ηis required in order to define the core structure – making the resulting decompositiondependent on this user-defined parameter; (ii) the increased computational cost for per-forming the decomposition. Based on that, our goal is to define a simple-yet-effectivecore decomposition of uncertain graphs. To do so, we consider the expected degreeof each vertex in the uncertain graph, and in particular the concept of representativeinstance.2 Cores in Probabilistic GraphsLet G = (V,E, p) be an uncertain graph, where p : E→ (0,1] is a function that assignsprobabilities to the edges of the graph. A widely used approach to analyze uncertaingraphs is the one of possible worlds, where each possible world constitutes a determin-istic realization of G . According to this model, an uncertain graph G is interpreted asa set {G = (V,EG)}EG⊆E of 2|E| possible deterministic graphs [3]. Let Gv G indicatesthat G is a possible world of G . Then, the probability that G = (V,EG) is observed asa possible world of G is given by Pr(G|G ) = ∏e∈EG p(e) ∏e∈E\EG(1− p(e)). In ourapproach, we are using this general framework to derive an analogous of the k-core de-composition in uncertain graphs. In particular, we use the property of expected degree[d](v) of each node v ∈ G , leading to the concept of uncertain [k]-core decomposition.Definition 1 (Uncertain [k]-core). Given an uncertain graph G = (V,E, p), the uncer-tain [k]-core of G is the maximal subgraph H = (C,E|C, p) such that each vertex v∈Chas expected degree at least k in H , where k ∈ R+.According to this definition, in order to compute the decomposition, we can extract adeterministic representative instance Gv G that preserves the expected degree, i.e., thedegree of any vertex v ∈ G to be as close as possible to the expected degree of v ∈ G– therefore, casting the problem to a weighted version of the k-core decompositionon deterministic graphs. That way, the proposed algorithm comprises of two phases:(i) extraction of a representative instance of the uncertain graph; (ii) apply a modifiedversion of the k-core decomposition, suitable for fractional degree values.For the first step of the algorithm that converts the uncertain graph to a determin-istic one by preserving the expected degree, we rely on conversion algorithms that aimat minimizing the discrepancy of each vertex of the graph [3]. In particular, the dis-crepancy disG(v) of a vertex v in the representative instance G v G , is defined as thedifference between the degree in the representative instance and expected degree inthe uncertain graph, i.e., disG(v) = d(v)− [d](v). As the existence of the edges of thegraph are independent of each other, the expected degree [d](v) of a vertex v ∈V is thesum of the probabilities of the incident edges, i.e., [d](v) = ∑e=(v,u)∈E p(e). The overalldiscrepancy of the representative instance Gv G is defined as ∆(G) = ∑v∈V |disG(v)|.Then, the problem of finding a “good” representative instance G∗ can be expressed as aminimization optimization problem: G∗ = argminGvG ∆(G).After extracting an average degree-preserving representative instance of the uncer-tain graph, the core number of a vertex is not an integer anymore but a real number.Thus, the second phase of the proposed technique consists of a modified k-core decom-position algorithm that operates on a deterministic graph with fractional node degrees.3 Experimental Results and DiscussionWe have performed preliminary experiments on a co-authorship network (DBLP) de-rived from DBLP (http://dblp.uni-trier.de), that consists of 404,892 nodes and 1,422,263edges. Since the DBLP dataset is not inherently uncertain, we are using a method toconvert the graph to uncertain by examining the similarity between the neighborhoodTable 1. Properties of the [k]-core decomposition on the DBLP graph.Correlation Partial FullτB 0.66 0.12r 0.79 −0.02of two nodes – towards computing the probability of being linked. In particular, wehave applied the Jaccard similarity coefficient to quantify the similarity of two nodes,following two different approaches. The first one, denoted by Partial, is adding a prob-ability of existence only to current edges of the graph. However, it does not allow toconsider more pairs of nodes, than the already existing edges in the graph. The secondapproach, denoted by Full, is computing the Jaccard similarity coefficient for all pairsof nodes. To overcome the complexity of examining all possible pairs, we restrict ourinterest to pairs that have at least one neighbor in common. Then, we compute the coredecomposition on both the original graph and the uncertain one, and examine the cor-relation among them using the Kendall rank correlation coefficient τB (which measureshow much the ranking output is the same in both cases), and Pearson’s correlation r(which measures how much the core numbers are linearly related).Table 1 depicts the results. As we observe, computing the similarity only for existingedges retains the structural properties of the decomposition in both cases. Nevertheless,notice that in this case the original graph can easily be recovered from the uncertainone (for example, when obfuscating the graph for privacy preserving reasons makingit uncertain, this approach is not possible). However, computing the probabilities forall pair of nodes (Full), erases all the structural information of cores (both correlationmeasures are close to zero).Currently, we are working towards examining practical applications of the proposed[k]-core decomposition in uncertain graphs. For example, in the DBLP graph, it wouldbe interesting to conduct an exploratory study comparing the authors (i.e., nodes) be-longing to the maximal core subgraph extracted by the algorithms on the deterministicand uncertain graphs respectively. Moreover, we plan to examine the performance ofthe high core number nodes detected by the proposed decomposition, in the task ofinfluence maximization.References1. Batagelj V., Zaversnik M.: An O(m) algorithm for cores decomposition of networks. arXiv(2003)2. Bonchi, F., Gullo, F., Kaltenbrunner, A., Volkovich, Y.: Core decomposition of uncertaingraphs. The 20th ACM SIGKDD International Conference on Knowledge Discovery andData Mining, (KDD), pp. 1316–1325 (2014)3. Parchas, P., Gullo, F., Papadias, D., Bonchi, F.: Uncertain graph processing through repre-sentative instances. ACM Trans. Database Syst., 40(3):20:1–20:39 (2015)4. Seidman, S. B.: Network Structure and Minimum Degree. Social Networks, 5:269–287(1983)

Core Decomposition of Uncertain Graphs Using Representative Instances

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Core Decomposition of Uncertain Graphs Using Representative Instances

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