Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion Multiscale Processing on Networks and Community Mining Part 2 - Spectral Graph Wavelets and Multiscale Community Detection Pierre Borgnat CR1 CNRS – Laboratoire de Physique, ENS de Lyon, Université de Lyon Équipe S I S Y P HE : Signaux, Systèmes et Physique 05/2014 p. 1
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion Overview of the lecture • General objective: revisit the classical question of finding communities in networks using multiscale processing methods on graphs. • The things we will discuss: • Recall the notion of community in networks and brief survey of some aspects of community detection • Introduce you to the emerging field of graph signal processing • Show a connexion between the two: detection of communities with graph signal processing • Organization: 1. A (short) lecture about communities in networks 2. Signal processing on networks; Spectral graph wavelets 3. Multiscale community mining with wavelets p. 2
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion Graph Wavelets • Fourier is a global analysis. Fourier modes (eigenvectors of the laplacian) are used in classical spectral clustering, but do not enable a jointly local and scale dependent analysis. • For that classical signal processing (or harmonic analysis) teach us that we need wavelets . • Wavelets : local functions that act as well as a filter around a chosen scale. A wavelet: – Translated: p. 3 – Scaled by analogy
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion The classical wavelets Each wavelet ψ s , a is derived by translating and scaling a mother wavelet ψ : ψ s , a ( x ) = 1 � x − a � s ψ s Equivalently, in the Fourier domain: � ∞ 1 � x − a � exp − i ω x dx ˆ ψ s , a ( ω ) = s ψ s −∞ � ∞ 1 � X � exp − i ω X dX = exp − i ω a s ψ s −∞ � ∞ exp − i ω X ′ dX ′ = exp − i ω a X ′ � � ψ −∞ = ˆ δ a ( ω ) ˆ where ψ ( s ω ) δ a = δ ( x − a ) � ∞ ψ ( s ω ) exp i ω x d ω One possible definition: ψ s , a ( x ) = −∞ ˆ δ a ( ω ) ˆ p. 4
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion The classical wavelets � ∞ ψ ( s ω ) exp i ω x d ω −∞ ˆ δ a ( ω ) ˆ ψ s , a ( x ) = • In this definition, ˆ ψ ( s ω ) acts as a filter bank defined by scaling by a factor s a filter kernel function defined in the Fourier space: ˆ ψ ( ω ) • The filter kernel function ˆ ψ ( ω ) is necessarily a bandpass filter with: • ˆ ψ ( 0 ) = 0 : the mean of ψ is by definition null lim ψ ( ω ) = 0 ˆ : the norm of ψ is by definition finite • ω → + ∞ (Note: the actual condition is the admissibility property) p. 5
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion by analogy Classical wavelets → Graph wavelets − − − − − − [Hammond et al. ACHA ’11] Classical (continuous) world Graph world Real domain x node a Fourier domain eigenvalues λ i ω Filter kernel ˆ g ( λ i ) ⇔ ˆ ψ ( ω ) G ˆ Filter bank g ( s λ i ) ⇔ ˆ ψ ( s ω ) G s Fourier modes exp − i ω x eigenvectors χ i � ∞ −∞ f ( x ) exp − i ω x dx f = χ ⊤ f ˆ ˆ Fourier transf. of f f ( ω ) = The wavelet at scale s centered around node a is given by: � ∞ ψ ( s ω ) exp i ω x d ω − G s χ ⊤ δ a δ a ( ω ) ˆ ˆ → ψ s , a = χ ˆ G s ˆ δ a = χ ˆ ψ s , a ( x ) = − −∞ p. 6
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion Examples of graph wavelets A WAVELET : T RANSLATING : S CALING : p. 7
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion Examples of wavelets: they encode the local topology ψ s = 1 , a ψ s = 25 , a ψ s = 50 , a ψ s = 35 , a p. 8
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion Complement: the graph scaling functions • Consider the following lowpass filter kernel h : � 1 / 2 �� ∞ | g ( ω ′ ) | 2 d ω ′ h ( ω ) = ω ′ ω Classically, if g corresponds to a wavelet filter kernel, this equation defines the associated scaling function filter kernel. • With the same arguments as previously, we define the graph scaling function at scale s centered around a as: H s χ ⊤ δ a H s ˆ φ s , a = χ ˆ δ a = χ ˆ p. 9
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion Examples of scaling functions φ s = 25 , a φ s = 1 , a φ s = 50 , a φ s = 35 , a p. 10
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion Example of filters For each graph under study, we automatically find the good filter parameters for g by imposing: • The coarsest scale needs to be focused on the eigenvector χ 1 (Fiedler vector). • All scales need at least to keep some information from χ 1 . • The finest scale needs to use the information from all eigenvectors (i.e., Fourier modes). 8 8 6 6 g(s λ ) h(s λ ) 4 4 2 2 0 0 0 10 20 0 10 20 λ λ p. 11
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion Example of wavelet filters • More precisely, we will use the following kernel: x − α x α for x < x 1 1 for p ( x ) x 1 ≤ x ≤ x 2 g ( x ; α, β, x 1 , x 2 ) = x β 2 x − β for x > x 2 . • The parameters will be: s min = 1 x 2 = 1 s max = 1 � λ 3 � x 1 = 1 , β = 1 / log 10 , , , λ 2 λ 2 λ 2 λ 2 2 • This leads to: ( α = 2) 10 α =1 8 8 α =2 s=7 6 α =10 6 s=13 g(x) α =50 g(s λ ) s=25 4 4 s=47 2 2 p. 12 0 0 0 1 2 x 1 x 2 λ x
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion Nota Bene In the following, we will not actually focus on the Wavelet Transform of a signal. We will rather focus on the wavelets ψ i themselves We take advantage of the local topological information at their scale encoded in them. p. 13
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion Purpose of the last part of the lecture Develop a scale dependent community mining tool using concepts from graph signal processing. Why ? For joint processing of graph signals and networks. General Ideas • Take advantage of local topological information encoded in Graph Wavelets. Wavelet = ego-centered vision from a node • Group together nodes whose local environments are similar at the description scale • This will naturally offer a multiscale vision of communities p. 14
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion Let us recall: objective of community detection Three examples of community detections: • (A) A complex sensor network (non-uniform swiss roll topology) • (B) A contact network in a primary school [Stehle ’11] • (C) A hierarchical graph benchmark [Sales-Pardo ’07] C B A p. 15
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion or multiscale community detection ? p. 16
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion Multiscale community structure in a graph Classical community detection algorithm based (for instance) on modularity optimisation only finds one solution: C B A Where the modularity function reads: � � 1 A ij − d i d j � Q = δ ( c i , c j ) 2 N 2 N ij p. 17
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion A new method for multiscale community detection [N. Tremblay, P . Borgnat, 2013] The problem of community mining is considered as a problem of clustering. We then need to decide upon: 1. feature vectors for each node 2. a distance to measure two given vectors’ closeness 3. a clustering algorithm to separate nodes in clusters The method uses: 1. wavelets (resp. scaling functions) as feature vectors 2. the correlation distance 3. the complete linkage clustering algorithm p. 18
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion 1) Wavelets as features Each node a has feature vector ψ s , a . Globally, one will need Ψ s , all wavelets at a given scale s , i.e. = χ G s χ ⊤ . � � Ψ s = ψ s , 1 | ψ s , 2 | . . . | ψ s , N A T SMALL SCALE : A T LARGE SCALE : N ODE A: N ODE B: p. 19
Introduction Spectral Graph Wavelets Multiscale community mining Developments; Stability of communities Conclusion 2) Correlation distances ψ ⊤ s , a ψ s , b D s ( a , b ) = 1 − . || ψ s , a || 2 || ψ s , b || 2 N ODE A: N ODE B: C ORR . -0.50 0.97 C OEF .: Far appart in the Close to each other R ESULT : dendrogram in the dendrogram p. 20
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