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Graphs serve as generic tools to encode the underlying relational structure of data. Often this graph is not given, and so the task of inferring it from nodal observations becomes important. Traditional approaches formulate a convex inverse…

The latent space approach to complex networks has revealed fundamental principles and symmetries, enabling geometric methods. However, the conditions under which network topology implies geometricity remain unclear. We provide a…

Inferring temporal interaction graphs and higher-order structure from neural signals is a key problem in building generative models for systems neuroscience. Foundation models for large-scale neural data represent shared latent structures…

Networks representing many complex systems in nature and society share some common structural properties like heterogeneous degree distributions and strong clustering. Recent research on network geometry has shown that those real networks…

Graph embedding is a transformation of nodes of a network into a set of vectors. A good embedding should capture the underlying graph topology and structure, node-to-node relationship, and other relevant information about the graph, its…

We study the richness of the ensemble of graphical structures (i.e., unlabeled graphs) of the one-dimensional random geometric graph model defined by $n$ nodes randomly scattered in $[0,1]$ that connect if they are within the connection…

We describe a very simple condition that is necessary for the universal rigidity of a complete bipartite framework $(K(n,m),p,q)$. This condition is also sufficient for universal rigidity under a variety of weak assumptions, such as general…

Understanding deep neural networks is a major research objective with notable experimental and theoretical attention in recent years. The practical success of excessively large networks underscores the need for better theoretical analyses…

Graph structure learning is a well-established problem that aims at optimizing graph structures adaptive to specific graph datasets to help message passing neural networks (i.e., GNNs) to yield effective and robust node embeddings. However,…

In this paper, we study the large-scale structure of dense regular graphs. This involves the notion of robust expansion, a recent concept which has already been used successfully to settle several longstanding problems. Roughly speaking, a…

Networks of elastic beams can deform either by stretching or bending of their members. The primary mode of deformation (bending or stretching) crucially depends on the specific details of the network architecture. In order to shed light on…

Real complex systems are not rigidly structured; no clear rules or blueprints exist for their construction. Yet, amidst their apparent randomness, complex structural properties universally emerge. We propose that an important class of…

In this paper we study the rank of planar rigidity matrix of 4-valent graphs, both in case of generic realizations and configurations in general position, under various connectivity assumptions on the graphs. For each case considered, we…

Represented as graphs, real networks are intricate combinations of order and disorder. Fixing some of the structural properties of network models to their values observed in real networks, many other properties appear as statistical…

The combinatorial characterization of generic rigidity for bar-joint frameworks in dimensions $d \ge 3$ has been a long-standing open problem in discrete geometry. While the two-dimensional case was resolved in 1927 by Pollaczek-Geiringer…

Lagrangian and Hamiltonian neural networks (LNNs and HNNs, respectively) encode strong inductive biases that allow them to outperform other models of physical systems significantly. However, these models have, thus far, mostly been limited…

This paper is first-line research expanding GANs into graph topology analysis. By leveraging the hierarchical connectivity structure of a graph, we have demonstrated that generative adversarial networks (GANs) can successfully capture…

A random algebraic graph is defined by a group $G$ with a uniform distribution over it and a connection $\sigma:G\longrightarrow[0,1]$ with expectation $p,$ satisfying $\sigma(g)=\sigma(g^{-1}).$ The random graph…

Graph neural networks (GNNs) have emerged as a fundamental tool for learning from graph-structured data, achieving strong performance across a wide range of applications. However, understanding their generalization capabilities remains…

A graph $G=(V,E)$ is called $d$-rigid if, for a generic embedding of its vertices in $\mathbb{R}^d$, every edge-length preserving continuous motion of the vertices preserves the distances between all pairs of non-adjacent vertices as well.…