Related papers: Local Homophily on Bicolored Graphs is $\mathbf{P}…

A graph homomorphism is a vertex map which carries edges from a source graph to edges in a target graph. The instances of the Weighted Maximum H-Colourable Subgraph problem (MAX H-COL) are edge-weighted graphs G and the objective is to find…

In 1973, Erd\H{o}s and Simonovits asked whether every $n$-vertex triangle-free graph with minimum degree greater than $1/3 \cdot n$ is 3-colourable. This question initiated the study of the chromatic profile of triangle-free graphs: for…

In this work, we give a unifying view of locality in four settings: distributed algorithms, sequential greedy algorithms, dynamic algorithms, and online algorithms. We introduce a new model of computing, called the online-LOCAL model: the…

Existing multiplex graph models often assume homophily, where connected nodes tend to belong to the same class or share similar attributes. Consequently, these models may struggle with graphs exhibiting heterophily, where connected nodes…

Local complement is a graph operation formalized by Bouchet which replaces the neighborhood of a chosen vertex with its edge-complement. This operation induces an equivalence relation on graphs; determining the size of the resulting…

Graph Neural Networks (GNNs) have proven to be useful for many different practical applications. However, many existing GNN models have implicitly assumed homophily among the nodes connected in the graph, and therefore have largely…

Real-world graphs generally have only one kind of tendency in their connections. These connections are either homophily-prone or heterophily-prone. While graphs with homophily-prone edges tend to connect nodes with the same class (i.e.,…

Heterophily has been considered as an issue that hurts the performance of Graph Neural Networks (GNNs). To address this issue, some existing work uses a graph-level weighted fusion of the information of multi-hop neighbors to include more…

Suppose in a graph $G$ vertices can be either red or blue. Let $k$ be odd. At each time step, each vertex $v$ in $G$ polls $k$ random neighbours and takes the majority colour. If it doesn't have $k$ neighbours, it simply polls all of them,…

We study a variation of the graph colouring problem on random graphs of finite average connectivity. Given the number of colours, we aim to maximise the number of different colours at neighbouring vertices (i.e. one edge distance) of any…

Real-world complex systems are often better modeled as hypergraphs, where edges represent group interactions involving multiple entities. Understanding and quantifying homophily (similarity-driven association) in such networks is essential…

We study the task of node classification for graph neural networks (GNNs) and establish a connection between group fairness, as measured by statistical parity and equal opportunity, and local assortativity, i.e., the tendency of linked…

Graphs constructed to translate some graph problem into another graph problem are usually called auxiliary graphs. Specifically total graphs of simple graphs are used to translate the total colouring problem of the original graph into a…

Since heterogeneity presents a fundamental challenge in graph federated learning, many existing methods are proposed to deal with node feature heterogeneity and structure heterogeneity. However, they overlook the critical homophily…

Lattice-based cryptography is not only for thwarting future quantum computers, and is also the basis of Fully Homomorphic Encryption. Motivated from the advantage of graph homomorphisms we combine graph homomorphisms with graph total…

In this paper, the problem of matching pairs of correlated random graphs with multi-valued edge attributes is considered. Graph matching problems of this nature arise in several settings of practical interest including social network…

The problems studied in this paper originate from Graph Motif, a problem introduced in 2006 in the context of biological networks. Informally speaking, it consists in deciding if a multiset of colors occurs in a connected subgraph of a…

We consider constrained variants of graph homomorphisms such as embeddings, monomorphisms, full homomorphisms, surjective homomorpshims, and locally constrained homomorphisms. We also introduce a new variation on this theme which derives…

Homophily principle, \ie{} nodes with the same labels or similar attributes are more likely to be connected, has been commonly believed to be the main reason for the superiority of Graph Neural Networks (GNNs) over traditional Neural…

A linear connection on a Lie algebroid is called a Cartan connection if it is suitably compatible with the Lie algebroid structure. Here we show that a smooth connected manifold $M$ is locally homogeneous - i.e., admits an atlas of charts…