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Neuronal networks constitute a special class of dynamical systems, as they are formed by individual geometrical components, namely the neurons. In the existing literature, relatively little attention has been given to the influence of…

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…

We show that if a graph $G$ satisfies certain conditions then the connectivity of neighbourhood complex $\mathcal{N}(G)$ is strictly less than the vertex connectivity of $G$. As an application, we give a relation between the connectivity of…

Graph neural networks (GNN) have been ubiquitous in graph node classification tasks. Most of GNN methods update the node embedding iteratively by aggregating its neighbors' information. However, they often suffer from negative disturbance,…

This article presents an analysis of the structural resilience of a fragment of Internet topology against both targeted and random attacks, based on empirical data obtained from the iThena project. Using a processed visualization of the…

Graph clustering aims to partition nodes into distinct clusters based on their similarity, thereby revealing relationships among nodes. Nevertheless, most existing methods do not fully utilize these edge weights. Leveraging edge weights in…

In recent years, graph neural networks (GNNs) have shown great potential in addressing various graph structure-related downstream tasks. However, recent studies have found that current GNNs are susceptible to malicious adversarial attacks.…

Whilst grading neurovascular abnormalities is critical for prompt surgical repair, no statistical markers are currently available for predicting the risk of adverse events, such as stroke, and the overall resilience of a network to vascular…

The emerging field of optogenetics allows for optical activation or inhibition of neurons and other tissue in the nervous system. In 2005 optogenetic proteins were expressed in the nematode C. elegans for the first time. Since then, C.…

We investigate how the graph topology influences the robustness to noise in undirected linear consensus networks. We measure the structural robustness by using the smallest possible value of steady state population variance of states under…

Perturbations targeting the graph structure have proven to be extremely effective in reducing the performance of Graph Neural Networks (GNNs), and traditional defenses such as adversarial training do not seem to be able to improve…

We present a novel theoretical framework connecting k-component edge connectivity with spectral graph theory and homology theory to pro vide new insights into the resilience of real-world networks. By extending classical edge connectivity…

Graph neural networks are becoming increasingly popular in the field of machine learning due to their unique ability to process data structured in graphs. They have also been applied in safety-critical environments where perturbations…

Connectivity and controllability of a complex network are two important issues that guarantee a networked system to function. Robustness of connectivity and controllability guarantees the system to function properly and stably under various…

The connectome, or the entire connectivity of a neural system represented by network, ranges various scales from synaptic connections between individual neurons to fibre tract connections between brain regions. Although the modularity they…

Let A be a minor-closed class of labelled graphs, and let G_n be a random graph sampled uniformly from the set of n-vertex graphs of A. When n is large, what is the probability that G_n is connected? How many components does it have? How…

The essential graph is a distinguished member of a Markov equivalence class of AMP chain graphs. However, the directed edges in the essential graph are not necessarily strong or invariant, i.e. they may not be shared by every member of the…

A random graph model with prescribed degree distribution and degree dependent edge weights is introduced. Each vertex is independently equipped with a random number of half-edges and each half-edge is assigned an integer valued weight…

An important structural feature of a graph is its set of edges, as it captures the relationships among the nodes (the graph's topology). Existing node label noise models like Symmetric Label Noise (SLN) and Class Conditional Noise (CCN)…

In this paper, we study the connectivity of a one-dimensional soft random geometric graph (RGG). The graph is generated by placing points at random on a bounded line segment and connecting pairs of points with a probability that depends on…