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1. The Fréchet Mean of Inhomogeneous Random Graphs

   https://doi.org/10.1007/978-3-030-93409-5_18  

   Meyer, Francois G. (January 2022, Complex Networks & Their Applications X)

   Benito, Rosa Maria; Cherifi, Chantal; Cherifi, Hocine; Moro, Esteban; Rocha, Luis M. (Ed.)

   To characterize the “average” of a set of graphs, one can compute the sample Fr ́echet mean. We prove the following result: if we use the Hamming distance to compute distances between graphs, then the Fr ́echet mean of an ensemble of inhomogeneous random graphs is obtained by thresholding the expected adjacency matrix: an edge exists between the vertices i and j in the Fr ́echet mean graph if and only if the corresponding entry of the expected adjacency matrix is greater than 1/2. We prove that the result also holds for the sample Fr ́echet mean when the expected adjacency matrix is replaced with the sample mean adjacency matrix. This novel theoretical result has some significant practical consequences; for instance, the Fr ́echet mean of an ensemble of sparse inhomogeneous random graphs is the empty graph. 

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2. Sampling random graph homomorphisms and applications to network data analysis

   Hanbaek Lyu, Facundo Mémoli (April 2023, Journal of machine learning research)

   Francis Bach (Ed.)

   A graph homomorphism is a map between two graphs that preserves adjacency relations. We consider the problem of sampling a random graph homomorphism from a graph into a large network. We propose two complementary MCMC algorithms for sampling random graph homomorphisms and establish bounds on their mixing times and the concentration of their time averages. Based on our sampling algorithms, we propose a novel framework for network data analysis that circumvents some of the drawbacks in methods based on independent and neighborhood sampling. Various time averages of the MCMC trajectory give us various computable observables, including well-known ones such as homomorphism density and average clustering coefficient and their generalizations. Furthermore, we show that these network observables are stable with respect to a suitably renormalized cut dis- tance between networks. We provide various examples and simulations demonstrating our framework through synthetic networks. We also demonstrate the performance of our frame- work on the tasks of network clustering and subgraph classification on the Facebook100 dataset and on Word Adjacency Networks of a set of classic novels. 

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3. Sampling random graph homomorphisms and applications to network data analysis

   Lyu, Hanbaek; Memoli, Facundo; Sivakoff, David (January 2023, Journal of machine learning research)

   Bach, Francis (Ed.)

   A graph homomorphism is a map between two graphs that preserves adjacency relations. We consider the problem of sampling a random graph homomorphism from a graph into a large network. We propose two complementary MCMC algorithms for sampling random graph homomorphisms and establish bounds on their mixing times and the concentration of their time averages. Based on our sampling algorithms, we propose a novel framework for network data analysis that circumvents some of the drawbacks in methods based on independent and neighborhood sampling. Various time averages of the MCMC trajectory give us various computable observables, including well-known ones such as homomorphism density and average clustering coefficient and their generalizations. Furthermore, we show that these network observables are stable with respect to a suitably renormalized cut dis- tance between networks. We provide various examples and simulations demonstrating our framework through synthetic networks. We also demonstrate the performance of our frame- work on the tasks of network clustering and subgraph classification on the Facebook100 dataset and on Word Adjacency Networks of a set of classic novels. 

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4. Joint inference of multiple graphs from matrix polynomials

   Novarro, M.; Wang, Y.; Marques, A.G.; Uhler, C.; Segarra, S. (January 2022, Journal of machine learning research)

   Inferring graph structure from observations on the nodes is an important and popular network science task. Departing from the more common inference of a single graph, we study the problem of jointly inferring multiple graphs from the observation of signals at their nodes (graph signals), which are assumed to be stationary in the sought graphs. Graph stationarity implies that the mapping between the covariance of the signals and the sparse matrix representing the underlying graph is given by a matrix polynomial. A prominent example is that of Markov random fields, where the inverse of the covariance yields the sparse matrix of interest. From a modeling perspective, stationary graph signals can be used to model linear network processes evolving on a set of (not necessarily known) networks. Leveraging that matrix polynomials commute, a convex optimization method along with sufficient conditions that guarantee the recovery of the true graphs are provided when perfect covariance information is available. Particularly important from an empirical viewpoint, we provide high-probability bounds on the recovery error as a function of the number of signals observed and other key problem parameters. Numerical experiments demonstrate the effectiveness of the proposed method with perfect covariance information as well as its robustness in the noisy regime. 

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5. WMGCN: Weighted Meta-Graph Based Graph Convolutional Networks for Representation Learning in Heterogeneous Networks

   JINLI ZHANG, ZONGLI JIANG (April 2020, IEEE access)

   Network embedding has been an effective tool to analyze heterogeneous networks (HNs) by representing nodes in a low-dimensional space. Although many recent methods have been proposed for representation learning of HNs, there is still much room for improvement. Random walks based methods are currently popular methods to learn network embedding; however, they are random and limited by the length of sampled walks, and have difculty capturing network structural information. Some recent researches proposed using meta paths to express the sample relationship in HNs. Another popular graph learning model, the graph convolutional network (GCN) is known to be capable of better exploitation of network topology, but the current design of GCN is intended for homogenous networks. This paper proposes a novel combination of meta-graph and graph convolution, the meta-graph based graph convolutional networks (MGCN). To fully capture the complex long semantic information, MGCN utilizes different meta-graphs in HNs. As different meta-graphs express different semantic relationships, MGCN learns the weights of different meta-graphs to make up for the loss of semantics when applying GCN. In addition, we improve the current convolution design by adding node self-signicance. To validate our model in learning feature representation, we present comprehensive experiments on four real-world datasets and two representation tasks: classication and link prediction. WMGCN's representations can improve accuracy scores by up to around 10% in comparison to other popular representation learning models. What's more, WMGCN'feature learning outperforms other popular baselines. The experimental results clearly show our model is superior over other state-of-the-art representation learning algorithms. 

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