Steffen Staab 1WeST Web Science & Technologies University of Koblenz ▪ Landau, Germany Network Theory and Dynamic Systems Link Prediction Jérôme Kunegis

Jérôme Kunegis 2WeST Link Prediction Examples Friend recommender (person–person network) Search engine (word–document network) Product recommender (person–product network) Rating prediction (person–item network) Iteraction prediction (person–thing network) Communication prediction (person–person network)

Jérôme Kunegis 3WeST Bipartite Graphs Example: person–product recommender Bipartite graphs: paths have odd length Example: person–product graph Compute sum of odd powers of A The resulting polynomial is odd αA ³ + βA⁵ + … Does not work: number of common neighbors

Jérôme Kunegis 4WeST Rating Graphs d B c D C b a E A User \ ItemABCDE alike b dislikelike c dislikelike ddislike Predict ratings using the multiplication rule Examples: b~B~a~A = +1 × +1 × +1 = +1 = like b~C~c~E = −1 × +1 × +1 = −1 = dislike The matrix A contains the ratings (±1) Powers of A implement the multiplication rule

Jérôme Kunegis 5WeST Looking at Real Facebook DataLooking at Real Facebook Data Dataset:Facebook New Orleanshttp://konect.uni-koblenz.de/networks/facebook-wosn-links63,731 persons1,545,686 friendship links with creation datesAdjacency matrix At at time t (t = )Compute all eigenvalue decompositions At = Ut Λt UtTDataset:Facebook New Orleanshttp://konect.uni-koblenz.de/networks/facebook-wosn-links63,731 persons1,545,686 friendship links with creation datesAdjacency matrix At at time t (t = )Compute all eigenvalue decompositions At = Ut Λt UtT

Jérôme Kunegis 6WeST Evolution of EigenvaluesEvolution of Eigenvalues ( Λ t ) ii

Jérôme Kunegis 7WeST Eigenvector EvolutionEigenvector Evolution C o s i n e s i m i l a r i t y b e t w e e n ( U t ) i a n d ( U t + x ) i

Jérôme Kunegis 8WeST Eigenvector PermutationEigenvector Permutation Time split:old edges A = U Λ UT new edges B = V D VTTime split:old edges A = U Λ UT new edges B = V D VT Eigenvectors permuteEigenvectors permute | U i · V j |

Jérôme Kunegis 9WeST a) Learning by Extrapolationa) Learning by Extrapolation Extrapolate the growth of the spectrum Potential problem:overfittingPotential problem:overfitting Good when growthis irregularGood when growthis irregular

Jérôme Kunegis 10WeST b) Learning by Curve Fittingb) Learning by Curve Fitting f f AB UΛUTUΛUT B ΛUTBUUTBU DiagonalDiagonal

Jérôme Kunegis 11WeST Curve FittingCurve Fitting ΛiiΛii (UTBU)ii(UTBU)ii

Jérôme Kunegis 12WeST Polynomial Curve FittingPolynomial Curve Fitting Fit a polynomial a + bx + cx2+ dx3 + ex4Fit a polynomial a + bx + cx2+ dx3 + ex4

Jérôme Kunegis 13WeST Evaluation MethodologyEvaluation Methodology 3-way split of edge set by edge creation time Training set E a ∪ E b Test set E c Source set E a Target set E b All edges E Learn Apply Edge creation time ˙