Kernels of Directed Graph Laplacians

Let G denote a directed graph with adjacency matrix Q and in- degree matrix D. We consider the Kirchhoff matrix L = D − Q, sometimes referred to as the directed Laplacian. A classical result of Kirchhoff asserts that when G is undirected, the multiplicity of the eigenvalue 0 equals the number of connected components of G. This fact has a meaningful generalization to directed graphs, as was observed by Chebotarev and Agaev in 2005. Since this result has many important applications in the sciences, we offer an independent and self-contained proof of their theorem, showing in this paper that the algebraic and geometric multiplicities of 0 are equal, and that a graph-theoretic property determines the dimension of this eigenspace--namely, the number of reaches of the directed graph. We also extend their results by deriving a natural basis for the corresponding eigenspace. The results are proved in the general context of stochastic matrices, and apply equally well to directed graphs with non-negative edge weights

Kernels of directed graph Laplacians

Abstract. Let G denote a directed graph with adjacency matrix Q and in-degree matrix D. We consider the Kirchhoff matrix L = D−Q, sometimes referred to as the directed Laplacian. A classical result of Kirchhoff asserts that when G is undirected, the multiplicity of the eigenvalue 0 equals the number of connected components of G. This fact has a meaningful generalization to directed graphs, as was recently observed by Chebotarev and Agaev in 2005. Since this result has many important applications in the sciences, we offer an independent and self-contained proof of their theorem, showing in this paper that the algebraic and geometric multiplicities of 0 are equal, and that a graph-theoretic property determines the dimension of this eigenspace -namely, the number of reaches of the directed graph. We also extend their results by deriving a natural basis for the corresponding eigenspace. The results are proved in the general context of stochastic matrices, and apply equally well to directed graphs with non-negative edge weights. AMS 2002 Subject Classification: Primary 05C50. Keywords: Kirchhoff matrix. Eigenvalues of Laplacians. Graphs. Stochastic matrix. Definitions Let G denote a directed graph with vertex set V = {1, 2, ..., N } and edge set E ⊆ V ×V . To each edge uv ∈ E, we allow a positive weight ω uv to be assigned. The adjacency matrix Q is the N × N matrix whose rows and columns are indexed by the vertices, and where the ij-entry is ω ji if ji ∈ E and zero otherwise

Making Laplacians commute

In this paper, we construct multimodal spectral geometry by finding a pair of closest commuting operators (CCO) to a given pair of Laplacians. The CCOs are jointly diagonalizable and hence have the same eigenbasis. Our construction naturally extends classical data analysis tools based on spectral geometry, such as diffusion maps and spectral clustering. We provide several synthetic and real examples of applications in dimensionality reduction, shape analysis, and clustering, demonstrating that our method better captures the inherent structure of multi-modal data

Graph Laplacians and their convergence on random neighborhood graphs

Given a sample from a probability measure with support on a submanifold in Euclidean space one can construct a neighborhood graph which can be seen as an approximation of the submanifold. The graph Laplacian of such a graph is used in several machine learning methods like semi-supervised learning, dimensionality reduction and clustering. In this paper we determine the pointwise limit of three different graph Laplacians used in the literature as the sample size increases and the neighborhood size approaches zero. We show that for a uniform measure on the submanifold all graph Laplacians have the same limit up to constants. However in the case of a non-uniform measure on the submanifold only the so called random walk graph Laplacian converges to the weighted Laplace-Beltrami operator.Comment: Improved presentation, typos corrected, to appear in JML

A Spectral Framework for a Class of Undirected Hypergraphs

We extend the graph spectral framework to a new class of undirected hypergraphs with bipartite hyperedges. A bipartite hyperedge is a pair of disjoint sets of nodes in which every node is associated with a weight. A bipartite hyperedge can be viewed as a relation between two teams of nodes in which every node has a weighted contribution to its team. Undirected hypergraphs generalize over undirected graphs. Consistently with the case of graphs, we define the notions of hypergraph gradient, hypergraph Laplacian, and hypergraph kernel as the Moore-Penrose pseudoinverse of a hypergraph Laplacian. Therefore, smooth labeling of (teams of) nodes and hypergraph regularization methods can be performed. Contrary to the graph case, we show that the class of hypergraph Laplacians is closed by the pseudoinverse operation (thus it is also the class of hypergraphs kernels), and is closed by convex linear combination. Closure properties allow us to define (hyper)graph combinations and operations while keeping a hypergraph interpretation of the result. We exhibit a subclass of signed graphs that can be associated with hypergraphs in a constructive way. A hypergraph and its associated signed graph have the same Laplacian. This property allows us to define a distance between nodes in undirected hypergraphs as well as in the subclass of signed graphs. The distance coincides with the usual definition of commute-time distance when the equivalent signed graph turns out to be a graph. We claim that undirected hypergraphs open the way to solve new learning tasks and model new problems based on set similarity or dominance. We are currently exploring applications for modeling games between teams and for graph summarization

Hypernode Graphs for Spectral Learning on Binary Relations over Sets

Paper accepted for publication at ECML/PKDD 2014International audienceWe introduce hypernode graphs as weighted binary relations between sets of nodes: a hypernode is a set of nodes, a hyperedge is a pair of hypernodes, and each node in a hypernode of a hyperedge is given a non negative weight that represents the node contribution to the relation. Hypernode graphs model binary relations between sets of individuals while allowing to reason at the level of individuals. We present a spectral theory for hypernode graphs that allows us to introduce an unnormalized Laplacian and a smoothness semi-norm. In this framework, we are able to extend spectral graph learning algorithms to the case of hypernode graphs. We show that hypernode graphs are a proper extension of graphs from the expressive power point of view and from the spectral analysis point of view. Therefore hypernode graphs allow to model higher order relations whereas it is not true for hypergraphs as shown in~\cite{Agarwal2006}. In order to prove the potential of the model, we represent multiple players games with hypernode graphs and introduce a novel method to infer skill ratings from game outcomes. We show that spectral learning algorithms over hypernode graphs obtain competitive results with skill ratings specialized algorithms such as Elo duelling and TrueSkill