The success of Graph Neural Networks (GNNs) has led to a need for understanding their decision-making process and providing explanations for their predictions, which has given rise to explainable AI (XAI) that offers transparent explanations for black-box models. Recently, the use of prototypes has successfully improved the explainability of models by learning prototypes to imply training graphs that affect the prediction. However, these approaches tend to provide prototypes with excessive information from the entire graph, leading to the exclusion of key substructures or the inclusion of irrelevant substructures, which can limit both the interpretability and the performance of the model in downstream tasks. In this work, we propose a novel framework of explainable GNNs, called interpretable Prototype-based Graph Information Bottleneck (PGIB), that incorporates prototype learning within the information bottleneck framework to provide prototypes with the key subgraph from the input graph that is important for the model prediction. This is the first work that incorporates prototype learning into the process of identifying the key subgraphs that have a critical impact on the prediction performance. Extensive experiments, including qualitative analysis, demonstrate that PGIB outperforms state-of-the-art methods in terms of both prediction performance and explainability.

We use the notation G= (V,E) to represent unweighted graphs, and G= (V,E,w) for weighted graphs. We use lowercase letters u,v to refer to vertices in V, and given a vertex v, we use dG(v) to refer to its degree in graph G. We use capital letters S,T to represent subsets of vertices, and given a vertex set S V, we use |S|to refer to its cardinality, S:= V \S to refer to its complement, and G[S] to refer to the subgraph of Ginduced by vertex set S. Furthermore, given two disjoint vertex sets S,T, we use wG(S,T):= P Given a graph G = (V,E), we use T to refer to a hierarchical clustering (tree) of the vertex set V, and costG(T) to refer to the cost of this clustering in graph G. Without loss of generality, we restrict our attention to just full binary hierarchical clustering trees, since the optimal tree is binary [20].

Through the introduction of a diffusion-convolution operation, we show how diffusion-based representations can be learned from graphstructured data and used as an effective basis for node classification. DCNNs have several attractive qualities, including a latent representation for graphical data that is invariant under isomorphism, as well as polynomial-time prediction and learning that can be represented as tensor operations and efficiently implemented on a GPU. Through several experiments with real structured datasets, we demonstrate that DCNNs are able to outperform probabilistic relational models and kernel-on-graph methods at relational node classification tasks.

Message Passing Neural Networks (MPNNs) are a staple of graph machine learning. MPNNs iteratively update each node's representation in an input graph by aggregating messages from the node's neighbors, which necessitates a memory complexity of the order of the number of graph edges

Graph clustering is a fundamental task in many data-mining and machine-learning pipelines. In particular, identifying a good hierarchical structure is at the same time a fundamental and challenging problem for several applications. The amount of data to analyze is increasing at an astonishing rate each day. Hence there is a need for new solutions to efficiently compute effective hierarchical clusterings on such huge data. The main focus of this paper is on minimum spanning tree (MST) based clusterings. In particular, we propose affinity, a novel hierarchical clustering based on Boruvka's MST algorithm. We prove certain theoretical guarantees for affinity (as well as some other classic algorithms) and show that in practice it is superior to several other state-of-the-art clustering algorithms.

Remark Since we apply the row-wise softmax in Eq. (7), P jpij = 1 i and pij 0 (i,j) is alwaysfulfilled.If C(P)=0,allbutoneentryinacolumn pi, are0andtheotherentryis1. Hence,P ipij = 1 j isfulfilled. Synthetic random graph generation To generate train and test graph datasets we utilized the pythonpackage NetworkX[1]. Ego graphs extracted from Binominal graphs (p (0.2,0.6))selecting all neighbours of onerandomnode. Training Details We did not perform an extensive hyperparameter evaluation for the different experiments and mostly followed [2]for hyperparameter selection. We set the graph embedding dimension to 64.