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Given a friendship network, how certain are we that Smith is a progressive (vs. conservative)? How can we propagate these certainties through the network? While Belief propagation marked the beginning of principled label-propagation to classify nodes in a graph, its numerous variants proposed in the literature fail to take into account uncertainty during the propagation process. As we show, this limitation leads to counter-intuitive results for even simple graphs. Motivated by these observations, we formalize axioms that any node classification algorithm should obey and propose NetConf which satisfies these axioms and handles arbitrary network e↵ects (homophily/heterophily) at scale. Our contributions are: (1) Axioms: We state axioms that any node classification algorithm should satisfy; (2) Theory: NetConf is grounded in a Bayesian-theoretic framework to model uncertainties, has a closed-form solution and comes with precise convergence guarantees; (3) Practice: Our method is easy to implement and scales linearly with the number of edges in the graph. On experiments using real world data, we always match or outperform BP while taking less processing time.

How do we spot interesting events from e-mail or transportation logs? How can we detect port scan or denial of service attacks from IP-IP communication data? In general, given a sequence of weighted, directed or bipartite graphs, each summarizing a snapshot of activity in a time window, how can we spot anomalous graphs containing the sudden appearance or disappearance of large dense subgraphs (e.g., near bicliques) in near real-time using sublinear memory? To this end, we propose a randomized sketching-based approach called SpotLight, which guarantees that an anomalous graph is mapped 'far' away from 'normal' instances in the sketch space with high probability for appropriate choice of parameters. Extensive experiments on real-world datasets show that SpotLight (a) improves accuracy by at least 8.4% compared to prior approaches, (b) is fast and can process millions of edges within a few minutes, (c) scales linearly with the number of edges and sketching dimensions and (d) leads to interesting discoveries in practice. CCS CONCEPTS • Information systems → Data stream mining; • Theory of computation → Graph algorithms analysis;