Heat Kernels and Diffusion Processes Definition: University of Alicante (Spain) Matrix Computing (subject 3168 Degree in Maths) 30 hours (theory)) + 15 hours (practical assignment)
Contents 1. Solving the Discrete Heat Equation 1. Heat Equation and the Laplacian 2. The Heat Kernel and its Interpretation 3. Behavior depending on time 4. The Path-length Distribution 2. PageRank and the Diffusion Process 1. PageRank 2. Page-Rank and Diffusion Rank 3. Flow Complexity 1. From Polytopal Complexity to Flowing Complexity 2. The Flow Conjecture and Complexity 3. Examples from Bioinformatics and 3D object recognition
Solving the Discrete Heat Equation The discrete Heat Equation: Being L the Laplacian of a graph G with n vertices, and K a nxn matrix parameterized by β (time or inverse temperature). The solution: Is the matrix exponentiation of the Laplacian: Using Taylor expansion: Spectral decompositiontaking the eigenvectors of L and the eigenvalues of L:
The Heat Kernel and its Interpretation Example #2: 11 6 12 11 12 6 K(i,j) is the probability that a lazy random walk starting at i reaches j. Lazy random walks have a probability depending on β off staying at i. Simulation of a heat diffusion state starting by heat=1 at each vertex when β=0.
The Heat Kernel and its Interpretation More spectral definitions: K defines a doubly-stochastic matrix (sum of rows and cols = 1). As K is a kernel/gram matrix, K(i,j) represents a dot product (dissimilarity) in a given space. Behavior depending on time: Exercise #6 (proof) Path-length distribution: number of paths of length k:
PageRank Definition: [Page et al.,99][eiron et al,04] Idea: Quantify the average importance of a node (e.g. webpage) after a sequence of (probabilistic) transitions. Given a set V of vertices. let x a n= V indicator vector so that x i measures the importance of vertex i. Being A the adjacency matrix and α the probability of moving to another node (surf to another webpage) being 1 the vectors of all ones and g a randon vector, typically g=(1/n) 1 (no preferred starting node) and α=0.85.
PageRank Properties Over-democratic Internet: [Yang et al, 07] All nodes (pages) are born equally. This favors to manipulate the rank of a node by creating many links to it. Input-independent Given the transition matrix: independently of the (always non-zero) input, the iterative process will converge to the same stable distribution corresponding to the maximum eigenvalue 1 of P. This property makes impossible to set preferences (high initial values to trusted pages and low, even negative, for spam). Alternative? DiffussionRank!
DiffusionRank Definition: [Yang et al, 07] Assuming that the heat difference at a node between t and t + t, say f(t) and f(t + t) at a given node is the sum of the heat it receives from its neighboring nodes: we have that for t close to zero: For γ=0 no heat is diffused. Anti-manipulation ranking but network structure ignored For γ= DiffusionRank converges to PageRank. For γ=1 DiffusionRank works well in practice.
Polytopal Complexity: BvN Theorem Polytopal Complexity: [Escolano, Hancock and Lozano, 08] Following the Birkhoff-von Newmann theorem, any doublystochastic matrix (e.g. a diffusion kernel matrix) can be decomposed into a convex combination of permutation matrices:
Polytopal Complexity: Examples Star (40 nodes) Line (40 nodes)
Polytopal Complexity: Bvn Complexity Global Polytopal Complexity: It is given by the following multidimensional descriptor: When considered a function, it satisfies: Moreover and, are computed from:
Polytopal Complexity: Bvn Complexity Global Polytopal Complexity: The graph complexity trace is a signature of the interaction between the heat diffusion process and the structure/topology of the graph as the inverse temperature increases and thus the range of vertex interaction decreases. The signature can be also interpreted as a trajectory (or geodesic in the Polytope) between the vertex of the polytope encoding the identity matrix and the barycenter of the polytope. A typical singature is heavy tailed and monotonically increasing from 1 at until it reaches then a topological phase transition occurs and the signature descends towards zero at The interval encodes inter class variability whereas the other one encodes intra class variability.
Polytopal Complexity: PPIs
Polytopal Complexity: Diffusion process Line
The Flow Conjecture and Complexity Heat Flow (Definition): The flow of the heat kernel (DSM) at a given beta is defined by: The Flow Conjecture and Complexity The inverse temperature yields the maximum entropy of the pdf coming from the BvN decomposition and it is also a PTP, iff it also maximizes the heat flow and it is also a PTP. Flow Complexity
Polytopal Complexity: SHREC database
Polytopal to Flow: Definition Example of BvN decomposition: Polytopal Complexity: The Maximum Entropy BvN decomposition is unique but the problem is #P. The Constructive BvN decompostion is O(N 3 x γ).
Polytopal Complexity: 3D objects Extended Reeb Graphs (ERGs): [Biasotti, 04,05] Critical points -> Critical areas (maxima, minima, saddle) Track the evolution of level sets and form graphs. Different functions -> Different graphs (see conclusions).
Polytopal Complexity: 3D objects Integral geodesic distance: Computed for each v=vertex in the mesh The bi are an uniform sampling of all the vertices. The derived graph is invariant to translation at rotation (at least)
Polytopal Complexity: Similarity Matrix