Related papers: Characteristic Power Series of Graph Limits
The symmetric tensor power of graphs is introduced and its fundamental properties are explored. A wide range of intriguing phenomena occur when one considers symmetric tensor powers of familiar graphs. A host of open questions are…
We give criteria for graded ideals to have the property that all their powers are componentwise linear. Typical examples to which our criteria can be applied include the vertex cover ideals of certain finite graphs.
We compute correlation functions of inverse powers and ratios of characteristic polynomials for random matrix models with complex eigenvalues. Compact expressions are given in terms of orthogonal polynomials in the complex plane as well as…
Random graphs offer a useful mathematical representation of a variety of real world complex networks. Exponential random graphs, for example, are particularly suited towards generating random graphs constrained to have specified statistical…
We investigate families of graphs and graphons (graph limits) that are defined by a finite number of prescribed subgraph densities. Our main focus is the case when the family contains only one element, i.e., a unique structure is forced by…
We characterize the graphs with loops whose degree sequences have no repeated values and find their adjacency spectrum. In the case of simple graphs, such graphs are called anti-regular graphs and are examples of threshold graphs. The…
Using the theory of $L^p$-graphons (Borgs et al, 2014), we derive and rigorously justify the continuum limit for systems of differential equations on sparse random graphs. Specifically, we show that the solutions of the initial value…
Graph neural networks (GNNs) are effective machine learning models for many graph-related applications. Despite their empirical success, many research efforts focus on the theoretical limitations of GNNs, i.e., the GNNs expressive power.…
To achieve control objectives for extremely large-scale complex networks using standard methods is essentially intractable. In this work a theory of the approximate control of complex network systems is proposed and developed by the use of…
In this work, some combinatorial lower bound for regularity of powers of the edge ideal of a uniform hypergarph is gained. A family of hypergraphs whose regularity of edge ideal attains this bound and has a significant difference from the…
Pruning at Initialisation methods discover sparse, trainable subnetworks before training, but their theoretical mechanisms remain elusive. Existing analyses are often limited to finite-width statistics, lacking a rigorous characterisation…
Let K be an algebraically closed field of characteristic p. We exhibit a counterexample against a theorem asserted in one of our earlier papers, which claims to characterize the integral closure of K((t)) within the field of…
Current methods of graph signal processing rely heavily on the specific structure of the underlying network: the shift operator and the graph Fourier transform are both derived directly from a specific graph. In many cases, the network is…
A graphon is a limiting object used to describe the behaviour of large networks through a function that captures the probability of edge formation between nodes. Although the merits of graphons to describe large and unlabelled networks are…
Graph polynomials are deemed useful if they give rise to algebraic characterizations of various graph properties, and their evaluations encode many other graph invariants. Algebraic: The complete graphs $K_n$ and the complete bipartite…
Graphons, as limits of graph sequences, provide an operator-theoretic framework for analyzing the asymptotic behavior of graph neural operators. Spectral convergence of sampled graphs to graphons induces convergence of the corresponding…
Building on the limit theory for set functions, we prove that the limit of convergent sequence of bounded-degree graphs' cycle matroids can be represented as the cycle matroid of a graphing, analogous to the completeness result for…
Power grids are undergoing major changes from a few large producers to smart grids build upon renewable energies. Mathematical models for power grid dynamics have to be adapted to capture, when dynamic nodes can achieve synchronization to a…
This survey concerns regular graphs that are extremal with respect to the number of independent sets, and more generally, graph homomorphisms. More precisely, in the family of of $d$-regular graphs, which graph $G$ maximizes/minimizes the…
The paper considers the spectral determinant of quantum graph families with chaotic classical limit and no symmetries. The secular coefficients of the spectral determinant are found to follow distributions with zero mean and variance…