Associate Professor (Reader) in Machine Learning
School of Mathematics, The University of Edinburgh
The Maxwell Institute for Mathematical Sciences
School of Mathematics, Gran Sasso Science Institute
JCMB, King’s Buildings, Edinburgh EH93FD UK
email: f dot tudisco at ed.ac.uk
This project is funded by the
You can find below a list of news and publications that are related to this project.
Publication
Abstract: We present a unifying Perron–Frobenius theory for nonlinear spectral problems defined in terms of nonnegative tensors. By using the concept of tensor shape partition, our results include, as a special case, a wide variety of particular tensor spectral problems considered in the literature and can be applied to a broad set of problems involving tensors (and matrices), including the computation of operator norms, graph and hypergraph matching in computer vision, hypergraph spectral theory, higher-order network analysis, and multimarginal optimal transport. ... Read more
Publication
Abstract: Core-periphery detection is a key task in exploratory network analysis where one aims to find a core, a set of nodes well-connected internally and with the periphery, and a periphery, a set of nodes connected only (or mostly) with the core. In this work we propose a model of core-periphery for higher-order networks modeled as hypergraphs and we propose a method for computing a core-score vector that quantifies how close each node is to the core. ... Read more
Publication
Abstract: We analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate $\ell^p$ matrix norms. In particular, exploiting the Birkoff–Hopf contraction ratio of nonnegative matrices, we obtain novel and explicit global convergence guarantees for a range of matrix norms whose computation has been recently proven to be NP-hard in the general case, including the case of mixed-subordinate norms induced by the vector norms made by the sum of different $\ell^p$-norms of subsets of entries. ... Read more
Publication
Abstract: Many graph mining tasks can be viewed as classification problems on high dimensional data. Within this class we consider the issue of discovering core-periphery structure, which has wide applications in the economic and social sciences. In contrast to many current approaches, we allow for weighted and directed edges and we do not assume that the overall network is connected. Our approach extends recent work on a relevant relaxed nonlinear optimization problem. ... Read more
Publication
Abstract: Nonnegative tensors arise very naturally in many applications that involve large and complex data flows. Due to the relatively small requirement in terms of memory storage and number of operations per step, the (shifted) higher-order power method is one of the most commonly used technique for the computation of positive Z-eigenvectors of this type of tensors. However, unlike the matrix case, the method may fail to converge even for irreducible tensors. ... Read more
Publication
Abstract: Signed graphs encode positive (attractive) and negative (repulsive) relations between nodes. We extend spectral clustering to signed graphs via the one-parameter family of Signed Power Mean Laplacians, defined as the matrix power mean of normalized standard and signless Laplacians of positive and negative edges. We provide a thorough analysis of the proposed approach in the setting of a general Stochastic Block Model that includes models such as the Labeled Stochastic Block Model and the Censored Block Model. ... Read more
Publication
Abstract: We introduce a ranking model for temporal multi-dimensional weighted and directed networks based on the Perron eigenvector of a multi-homogeneous order-preserving map. The model extends to the temporal multilayer setting the HITS algorithm and defines five centrality vectors: two for the nodes, two for the layers, and one for the temporal stamps. Nonlinearity is introduced in the standard HITS model in order to guarantee existence and uniqueness of these centrality vectors for any network, without any requirement on its connectivity structure. ... Read more
Publication
Abstract: We derive and analyse a new iterative algorithm for detecting network core–periphery structure. Using techniques in nonlinear Perron-Frobenius theory, we prove global convergence to the unique solution of a relaxed version of a natural discrete optimization problem. On sparse networks, the cost of each iteration scales linearly with the number of nodes, making the algorithm feasible for large-scale problems. We give an alternative interpretation of the algorithm from the perspective of maximum likelihood reordering of a new logistic core–periphery ... Read more
Publication
Abstract: With the notion of mode-$j$ Birkhoff contraction ratio, we prove a multilinear version of the Birkhoff-Hopf and the Perron-Fronenius theorems, which provide conditions on the existence and uniqueness of a solution to a large family of systems of nonlinear equations of the type $$f_i(x_1,…,x_ν)= λ_i x_i, ,$$ being $x_i$ and element of a cone $C_i$ in a Banach space $V_i$. We then consider a family of nonlinear integral operators $f_i$ with positive kernel, acting on product of spaces of continuous real valued functions. ... Read more
Publication
Abstract: Multilayer graphs encode different kind of interactions between the same set of entities. When one wants to cluster such a multilayer graph, the natural question arises how one should merge the information form different layers. We introduce in this paper a one-parameter family of matrix power means for merging the Laplacians from different layers and analyze it in the stochastic block model. We show that this family allows to recover ground truth clusters under different settings and verify this in real world data. ... Read more
Publication
Abstract: We provide explicit expressions for the eigenvalues and eigenvectors of matrices that can be written as the Hadamard product of a block partitioned matrix with constant blocks and a rank one matrix. Such matrices arise as the expected adjacency or modularity matrices in certain random graph models that are widely used as benchmarks for community detection algorithms. Please cite this work as: @article{fasino2018expected, title={The expected adjacency and modularity matrices in the degree corrected stochastic block model}, author={Fasino, Dario and Tudisco, Francesco}, journal={Special Matrices}, volume={6}, pages={110--121}, year={2018} }
Publication
Abstract: We introduce the concept of shape partition of a tensor and formulate a general tensor eigenvalue problem that includes all previously studied eigenvalue problems as special cases. We formulate irreducibility and symmetry properties of a nonnegative tensor $T$ in terms of the associated shape partition. We recast the eigenvalue problem for $T$ as a fixed point problem on a suitable product of projective spaces. This allows us to use the theory of multihomogeneous order-preserving maps to derive a new and unifying Perron–Frobenius theorem for nonnegative tensors which either implies earlier results of this kind or improves them, as weaker assumptions are required. ... Read more
Publication
Abstract: Eigenvector-based centrality measures are among the most popular centrality measures in network science. The underlying idea is intuitive and the mathematical description is extremely simple in the framework of standard, mono-layer networks. Moreover, several efficient computational tools are available for their computation. Moving up in dimensionality, several efforts have been made in the past to describe an eigenvector-based centrality measure that generalizes Bonacich index to the case of multiplex networks. In this work, we propose a new definition of eigenvector centrality that relies on the Perron eigenvector of a multi-homogeneous map defined in terms of the tensor describing the network. ... Read more
Publication
Abstract: Identifying important components in a network is one of the major goals of network analysis. Popular and effective measures of importance of a node or a set of nodes are defined in terms of suitable entries of functions of matrices $f(A)$. These kinds of measures are particularly relevant as they are able to capture the global structure of connections involving a node. However, computing the entries of $f(A)$ requires a significant computational effort. ... Read more