[IEEE 2012 International Conference on Advances in Social Networks Analysis and Mining (ASONAM 2012) - Istanbul (2012.08.26-2012.08.29)] 2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining - Global Similarity in Social Networks with Typed Edges

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<ul><li><p>Global Similarity in Social Networks with Typed Edges</p><p>D.B. Skillicorn and Q. ZhengSchool of ComputingQueens UniversityKingston. Canada</p><p>{skill,quan}@cs.queensu.ca</p><p>AbstractMost real-world social network analysis treats edges(relationships) as having different intensities (weights), but thesame qualitative properties. We address the problem of modellingedges of qualitatively different types that nevertheless interactwith one another. For example, influence flows along friend andcolleagues edges differently, but treating the two sets of differentkinds of edges as independent graphs surely misses interestingand useful structure.</p><p>We model the subgraph corresponding to each edge type as alayer, and show how to weight the edges connecting the layers toproduce a consistent spectral embedding, including for directedgraphs. This embedding can be used to compute social networkproperties of the combined graph, to predict edges, and to predictedge types. We illustrate with Padgetts dataset of Florentinefamilies in the 15th Century.</p><p>I. INTRODUCTION</p><p>Social networks capture the pairwise connections betweenindividuals (or, in general, nodes of any kind), and then induceknowledge from the global structure implied by the totality ofthese pairwise connections. Properties at three scales can bederived from such data:</p><p> Properties associated with individual nodes: betweenness,centrality;</p><p> Properties associated with the neighborhood of eachnode: closeness, clustering coefficient, density;</p><p> Properties of the graph as a whole: diameter, containedsubstructures.</p><p>These properties can be directly computed on the graph de-scribing the network, but direct algorithms are often expensiveand do not handle incremental settings well, since adding onlya single node to a graph can completely alter its path structure.It is common, therefore, to use spectral techniques to embeda graph in a geometric (usually Euclidean) space. Many ofthe desired properties can then be computed directly from thegeometry.</p><p>For example, the edges in a social network graph can beweighted with positive weights indicating the strength of thelocal similarity between the pair of nodes that each edgeconnects. The implicit similarity between unconnected nodesis a function of the strength, and often the number, of pathsbetween those nodes. Once embedded, all of these similaritiescan be computed easily based on the reciprocal of the distancesbetween the nodes: close nodes are considered to be highlysimilar, even if not originally connected. This is the basis ofedge prediction.</p><p>Spectral approaches to graphs, therefore, integrate local,pairwise similarity information over the entire graph, and usethe integrated information to embed the graph, from whichimproved local similarity information can then be determined.Even for a connected pairs of nodes, the distance betweenthem (and its implied similarity) may change from its originalvalue, indicating that the global structure of the graph hasimplications for what at first appeared to be purely local in-formation. The natural dimension of an embedding of a graphwith n nodes is n 1, so it is usual to follow an embeddingby a projection to a much lower dimensional space beforecalculating distances. Projection to a lower dimensionality alsomakes visualization possible.</p><p>In much social network analysis, edges are weighted, butof a single type. This rules out several interesting kinds ofanalysis. For example, the social network of an individualoften consists of two different kinds of relationships, thoseassociated with work (colleagues) and play (friends) with, ofcourse, potential overlaps. If we want to model the way inwhich, say, influence works in such a social network, treatingall of the edges the same seems inadequate. Presumably somekinds of influence flow better to (as it were) colleagues whileother kinds flow better to friends. On the other hand, it alsoseems inadequate to treat the colleague and friend networksas entirely separate. Presumably some influence can flow froman individual to colleagues, and then on to their friends.</p><p>The contribution of this paper is to extend the spectral graphapproach to networks where the edges have (one of a fixednumber of) types, as well as positive weights. This makes itpossible to combine the features of, for example, Facebookand LinkedIn into a single network framework that takesinto account the qualitative differences in the functionality ofedges.</p><p>As a side-effect of the construction, it is possible to answertwo kinds of edge prediction questions:</p><p>1) Should there be an edge between these two nodes; and,if so, what type should it be?</p><p>2) If there is a new edge between two nodes, what type ofedge is it?</p><p>This enables, for example, a refinement of the suggestionsmade by social network systems from this is someone youmight know to this is a potential colleague or this is apotential friend.</p><p>2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining</p><p>978-0-7695-4799-2/12 $26.00 2012 IEEEDOI 10.1109/ASONAM.2012.23</p><p>79</p><p>2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining</p><p>978-0-7695-4799-2/12 $26.00 2012 IEEEDOI 10.1109/ASONAM.2012.23</p><p>79</p></li><li><p>Validation in this setting is, of course, problematic since itis not clear how influence should flow across heterogeneousedges. We illustrate the approach by applying it to a well-studied dataset. Our results do not support the most-popularunderstanding of the rise of the Medici family.</p><p>II. RELATED WORK</p><p>We frame the problem of interest as combining the infor-mation implicit in subgraphs whose edges represent differentkinds of local similarity. The problem of finding severaldifferent clusterings in a dataset that are somehow consistentwith each other has also been addressed and solutions to oneproblem serve also as solutions to the other. However, thedifferent ways of framing the problem have led to differentalgorithmic strategies.</p><p>Most attempts to represent graphs with different edge typesbegin with the separate subgraphs of each type. embed eachone independently into a geometric space and then attemptto paste these spaces together into a consistent whole. Forexample, Zhou et al. [13] combine normalized Laplacianmatrices of different views with user-determined weights foreach. Xia et al. [11] and Kumar and Daum [5] use iterativetechniques, using each representation of the graph in turn toconverge to a common representation. Cheng and Zhao [1]combine the distances in each separate embedding to create asingle similarity matrix and then repeat the spectral clusteringon this matrix to produce the final embedding. Tang et al.[9] use an approach they call Linked Matrix Factorizationto decompose each graph and then combine pieces of thedecompositions with weights to create a global fusion of thedata. The problem with strategies of these kinds is decidinghow the individual representations should be combined, aboutwhich there is usually not enough information to decide in aprincipled way.</p><p>Another line of attack is to connect the subgraphs represent-ing each view of the data with edges whose weights are thendetermined using a regularization framework [4, 7]. De Sa [3]considers the two subgraph case, models it as a bipartite graphand uses a spectral partition of the product matrices from thegraph. Regularization is problematic because it is not clearwhether to build a global regularizer that applies equally toedges of each type and to the edges that connect differentsubgraphs; or whether each subgraph should have its ownregularizer and there should also be another regularizer forthe connecting edges. There does not seem to be a compellingargument for either possibility, let along a way to relate themto one another.</p><p>III. APPROACH</p><p>Embeddings of undirected graphs attempt to place nodesthat are well connected close together so that the (heavy)edges that join them are short. The implicit circularity inthis description is resolved, as it often is, by the use of aneigendecomposition whose result is a geometric fixed point.An alternate, but equivalent view, is that the length of each</p><p>edge in the embedding should reflect the frequency with whichit is traversed in a random walk on the graph, so that longedges connect parts of the graph that are difficult to reachfrom one another, and so represent good places to cut thegraph into clusters.</p><p>Direct use of an eigendecomposition of an adjacency matrixfails because well-connected nodes correspond to rows withlarger entries. An eigendecomposition embeds such nodes farfrom the origin when, from a graph perspective, they should becentral. Adjacency matrices are therefore transformed into oneof a number of Laplacian matrices [6], the choices embodyingdecisions about how edges are integrated into paths, whetherdifferences in local degrees should affect the distance scale inthe embedding, and turning the graph structure inside-out sothat well-connected points are placed centrally (and sparselyconnected points are placed peripherally).</p><p>For directed graphs, the intuition behind spectral techniquesis less obvious. Approaches take the asymmetric adjacencymatrix of a directed graph and turn it, in one of a number ofways, into a symmetric matrix which can then be embeddedin the standard way [2].</p><p>Consider a set of n nodes connected by edges of c differenttypes which, for convenience we will consider to be colors.The set of edges of each particular type (color) forms asubgraph on the set of n nodes. There could, of course, beedges of more than one type between a particular pair of nodes,and there could be nodes connected by edges of only one type;in general, nodes will have incident edges of several types.Edges are positively weighted with a value that represents thetyped pairwise similarity between the nodes they connect.</p><p>Each typed subgraph can be represented by a weighted nnadjacency matrix. But how should these matrices be combinedto represent the whole graph? For simplicity of description, weassume that c = 2 and call the two types of edges red andgreen. Our strategy for representing the multicolored graph isto replicate the nodes of the graph c times, and to representeach subgraph as a layer, connecting the versions of the samenodes. Thus each node in the original graph is represented by cvirtual copies of itself, one in each layer. The adjacency matrixof the entire graph is cn cn, with the adjacency matrices foreach color as submatrices down the main diagonal.</p><p>But how then are the layers connected to one another? Or,to put it another way, what should appear in the off-diagonalpart of the larger adjacency matrix? It is conceivable thatdownweighted versions of single-color edges could be addedto represent diagonal paths from, say, a red version of onenode to a green version of another. However, it turns out to besimpler to add only vertical edges connecting the layers,that is edges from the version of a node of one color tothe matching version of the same node of another color. Inother words, the off-diagonal n n submatrices of the largeadjacency matrix are all diagonal matrices.</p><p>Now the question becomes: what weights should be asso-ciated with these vertical edges? The basis for choosing these</p><p>8080</p></li><li><p>weights comes from lazy random walks. In such models, arandom walker stays at the current node with probability 1/2or moves with probability 1/2, choosing among the outgoingedges with probability proportional to their weights. Lazyrandom walks are better behaved than random walks sincethey always have a stationary distribution.</p><p>In the layered model, then, a random walker at a node inone of the layers makes a move that stays within the samelayer with probability 1/2 (the actual move depending on theweights of the outgoing edges of the node as usual), or a moveto another layer with total probability 1/2, divided uniformlyacross the c 1 other layers. Thus, from a monochromeperspective, the random walker behaves like an ordinaryrandom walker in the monochrome graph that is the unionof the colored subgraphs.</p><p>Given a graph with c colored subgraphs, therefore, we builda cn cn adjacency matrix. The submatrices on the maindiagonal are the c different colored subgraphs. The remainingsubmatrices in each block row are diagonal matrices whoseentries are the row sums of the matching colored matrixdivided by c 1. Thus the row sum of the adjacency matrixis twice the row sum of each row of a color submatrix.Henceforth, we ignore the constant factor which does notaffect the eigenvectors.</p><p>Let R be the adjacency matrix of the graph with red edges,and G the adjacency matrix of the graph with green edges.Then we construct</p><p>A =</p><p>(R Dr</p><p>Dg G</p><p>)</p><p>where Dr (respectively Dg) is the matrix with the degrees(or row sums) of R (respectively G) on its diagonal. In otherwords, the two versions of each node of the original graphare connected by vertical edges whose weight depends on theedge weights of that node in the red and green subgraphs.</p><p>Embedding such a matrix constructed by using two copiesof R, for example, is straightforward because the matrix issymmetric; it embeds the two copies of each node exactly atthe same locations, demonstrating why these values for theedge weights between layers are appropriate.</p><p>In general, A will not be symmetric because the total rededge weight and the total green edge weight are not the sameat each node. Hence the graph described by the adjacencymatrix is actually a directed graph because the downwardsand upwards edges between layers have different weights.</p><p>It is possible to explicitly symmetrize the entries so that thedownward and upward edges between versions of the samenode have the same weight. However, we use directed-graphspectral techniques directly, constructing a symmetric graph toembed, but in a different way. One of the advantages of usingdirected-graph spectral techniques is that it makes it possiblefor the red and green subgraphs also to be directed, increasingthe space of possible models.</p><p>The social-network intuition behind this model is as follows:</p><p>Each individual plays a number of different roles, and ineach role is connected to a different set of others with role-specific intensities or closeness. Properties such as influenceflow along edges of a single color exactly as we assumethey do in existing social-network modelling. However, pathsthat involve multiple colors encounter a kind of resistancethat is proportional to how easily each individual can moveamong their roles. In other words, the cost associated with flowchanging from one colored edge to another is modelled as thecost of individuals changing gear to convey some propertysuch as influence from one of their networks to another.In practice, this cost might be associated with contexts: anindividual has to remember something they learned at work intheir home context in order to pass it on to a friend.</p><p>For a directed graph, the edge weight on an individual edgeis not a good estimate of its global importance in connectingthe grap...</p></li></ul>

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