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Recent work suggests that changing Convolutional Neural Network (CNN) architecture by introducing a bottleneck in the second layer can yield changes in learned function. To understand this relationship fully requires a way of quantitatively…

In this paper, we introduce a magneto-spectral invariant for finite graphs. This invariant vanishes on trees and is maximized by complete graphs. We compute this invariant for cycles, complete graphs, wheel graphs, hypercubes, complete…

The comparison of graphs is a vitally important, yet difficult task which arises across a number of diverse research areas including biological and social networks. There have been a number of approaches to define graph distance however…

We introduce a new methodology for model selection in the context of modeling network data. The statistical network analysis literature has developed many different classes of network data models, with notable model classes including…

Resonance is a well-known phenomenon that happens in systems with second order dynamics. In this paper we address the fundamental question of making a network robust to signal being periodically pumped into it at or near a resonant…

Symmetry -- invariance to certain operators -- is a fundamental concept in many branches of physics. We propose ways to measure symmetric properties of vertices, and their surroundings, in networks. To be stable to the randomness inherent…

In recent years, methods from network science are gaining rapidly interest in economics and finance. A reason for this is that in a globalized world the interconnectedness among economic and financial entities are crucial to understand and…

An invariant for cospectral graphs is a property shared by all cospectral graphs. In this paper, we establish three novel arithmetic invariants for cospectral graphs, revealing deep connections between spectral properties and combinatorial…

This paper presents methods to compare high order networks, defined as weighted complete hypergraphs collecting relationship functions between elements of tuples. They can be considered as generalizations of conventional networks where only…

Spectral embedding finds vector representations of the nodes of a network, based on the eigenvectors of a properly constructed matrix, and has found applications throughout science and technology. Many networks are multipartite, meaning…

The spectrum of network is an important tool to study the function and dynamic properties of network, and graph operation and product is an effective mechanism to construct a specific local and global topological structure. In this study, a…

We propose to test, and when possible establish, an equivalence between two different artificial neural networks by attempting to construct a data-driven transformation between them, using manifold-learning techniques. In particular, we…

Measuring similarity of neural networks to understand and improve their behavior has become an issue of great importance and research interest. In this survey, we provide a comprehensive overview of two complementary perspectives of…

Many real-world complex networks contain a significant amount of structural redundancy, in which multiple vertices play identical topological roles. Such redundancy arises naturally from the simple growth processes which form and shape many…

This article deals with the spectra of Laplacians of weighted graphs. In this context, two objects are of fundamental importance for the dynamics of complex networks: the second eigenvalue of such a spectrum (called algebraic connectivity)…

We investigate the equivalence between spectral characteristics of the Laplace operator on a metric graph, and the associated unitary scattering operator. We prove that the statistics of level spacings, and moments of observations in the…

To describe the flow of a miscible quantity on a network, we introduce the graph wave equation where the standard continuous Laplacian is replaced by the graph Laplacian. This is a natural description of an array of inductances and…

We define a number of natural (from geometric and combinatorial points of view) deformation spaces of valuations on finite graphs, and study functions over these deformation spaces. These functions include both direct metric invariants…

Typically, graph structures are represented by one of three different matrices: the adjacency matrix, the unnormalised and the normalised graph Laplacian matrices. The spectral (eigenvalue) properties of these different matrices are…

Network theory and inverse modeling are two standard tools of applied physics, whose combination is needed when studying the dynamical organization of spatially distributed systems from indirect measurements. However, the associated…