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The popular Random Dot Product Graph (RDPG) generative model postulates that each node has an associated (latent) vector, and the probability of existence of an edge between two nodes is their inner-product (with variants to consider directed and weighted graphs). In any case, the latent vectors may be estimated through a spectral decomposition of the adjacency matrix, the so-called Adjacency Spectral Embedding (ASE). Assume we are monitoring a stream of graphs and the objective is to track the latent vectors. Examples include recommender systems or monitoring of a wireless network. It is clear that performing the ASE of each graph separately may result in a prohibitive computation load. Furthermore, the invariance to rotations of the inner product complicates comparing the latent vectors at different time-steps. By considering the minimization problem underlying ASE, we develop an iterative algorithm that updates the latent vectors' estimation as new graphs from the stream arrive. Differently to other proposals, our method does not accumulate errors and thus does not requires periodically re-computing the spectral decomposition. Furthermore, the pragmatic setting where nodes leave or join the graph (e.g. a new product in the recommender system) can be accommodated as well. Our code is available at https://github.com/marfiori/efficient-ASE 

Advances in graph signal processing for network neuroscience offer a unique pathway to integrate brain structure and function, with the goal of revealing some of the brain's organizing principles at the system level. In this direction, we develop a supervised graph representation learning framework to model the relationship between brain structural connectivity (SC) and functional connectivity (FC) via a graph encoder-decoder system. Specifically, we propose a Siamese network architecture equipped with graph convolutional encoders to learn graph (i.e., subject)-level embeddings that preserve application-dependent similarity measures between brain networks. This way, we effectively increase the number of training samples and bring in the flexibility to incorporate additional prior information via the prescribed target graph-level distance. While information on the brain structure-function coupling is implicitly distilled via reconstruction of brain FC from SC, our model also manages to learn representations that preserve the similarity between input graphs. The superior discriminative power of the learnt representations is demonstrated in downstream tasks including subject classification and visualization. All in all, this work advocates the prospect of leveraging learnt graph-level, similarity-preserving embeddings for brain network analysis, by bringing to bear standard tools of metric data analysis. 

The Random Dot Product Graph (RDPG) is a popular generative graph model for relational data. RDPGs postulate there exist latent positions for each node, and specifies the edge formation probabilities via the inner product of the corresponding latent vectors. The embedding task of estimating these latent positions from observed graphs is usually posed as a non-convex matrix factorization problem. The workhorse Adjacency Spectral Embedding offers an approximate solution obtained via the eigendecomposition of the adjacency matrix, which enjoys solid statistical guarantees but can be computationally intensive and is formally solving a surrogate problem. In this paper, we bring to bear recent non-convex optimization advances and demonstrate their impact to RDPG inference. We develop first-order gradient descent methods to better solve the original optimization problem, and to accommodate broader network embedding applications in an organic way. The effectiveness of the resulting graph representation learning framework is demonstrated on both synthetic and real data. We show the algorithms are scalable, robust to missing network data, and can track the latent positions over time when the graphs are acquired in a streaming fashion. 

We propose a deep learning solution to the inverse problem of localizing sources of network diffusion. Invoking graph signal processing (GSP) fundamentals, the problem boils down to blind estimation of a diffusion filter and its sparse input signal encoding the source locations. While the observations are bilinear functions of the unknowns, a mild requirement on invertibility of the graph filter enables a convex reformulation that we solve via the alternating-direction method of multipliers (ADMM). We unroll and truncate the novel ADMM iterations, to arrive at a parameterized neural network architecture for Source Localization on Graphs (SLoG-Net), that we train in an end-to-end fashion using labeled data. This way we leverage inductive biases of a GSP model-based solution in a data-driven trainable parametric architecture, which is interpretable, parameter efficient, and offers controllable complexity during inference. Experiments with simulated data corroborate that SLoG-Net exhibits performance in par with the iterative ADMM baseline, while attaining significant (post-training) speedups. 

Given a sequence of random graphs, we address the problem of online monitoring and detection of changes in the underlying data distribution. To this end, we adopt the Random Dot Product Graph (RDPG) model which postulates each node has an associated latent vector, and inner products between these vectors dictate the edge formation probabilities. Existing approaches for graph change-point detection (CPD) rely either on extensive computation, or they store and process the entire observed time series. In this paper we consider the cumulative sum of a judicious monitoring function, which quantifies the discrepancy between the streaming graph observations and the nominal model. This reference distribution is inferred via spectral embeddings of the first few graphs in the sequence, and the monitoring function can be updated in an efficient, online fashion. We characterize the distribution of this running statistic, allowing us to select appropriate thresholding parameters that guarantee error-rate control. The end result is a lightweight online CPD algorithm, with a proven capability to flag distribution shifts in the arriving graphs. The novel method is tested on both synthetic and real network data, corroborating its effectiveness in quickly detecting changes in the input graph sequence. 

We consider network topology identification subject to a signal smoothness prior on the nodal observations. A fast dual-based proximal gradient algorithm is developed to efficiently tackle a strongly convex, smoothness-regularized network inverse problem known to yield high-quality graph solutions. Unlike existing solvers, the novel iterations come with global convergence rate guarantees and do not require additional step-size tuning. Reproducible simulated tests demonstrate the effectiveness of the proposed method in accurately recovering random and real-world graphs, markedly faster than state-of-the-art alternatives and without incurring an extra computational burden. 

The growing success of graph signal processing (GSP) approaches relies heavily on prior identification of a graph over which network data admit certain regularity. However, adaptation to increasingly dynamic environments as well as demands for real-time processing of streaming data pose major challenges to this end. In this context, we develop novel algorithms for online network topology inference given streaming observations assumed to be smooth on the sought graph. Unlike existing batch algorithms, our goal is to track the (possibly) time-varying network topology while maintaining the memory and computational costs in check by processing graph signals sequentially-in-time. To recover the graph in an online fashion, we leverage proximal gradient (PG) methods to solve a judicious smoothness-regularized, time-varying optimization problem. Under mild technical conditions, we establish that the online graph learning algorithm converges to within a neighborhood of (i.e., it tracks) the optimal time-varying batch solution. Computer simulations using both synthetic and real financial market data illustrate the effectiveness of the proposed algorithm in adapting to streaming signals to track slowly-varying network connectivity.