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  1. Free, publicly-accessible full text available August 7, 2024
  2. Massoulie, Laurent (Ed.)
    Spreading processes on graphs arise in a host of application domains, from the study of online social networks to viral marketing to epidemiology. Various discrete-time probabilistic models for spreading processes have been proposed. These are used for downstream statistical estimation and prediction problems, often involving messages or other information that is transmitted along with infections caused by the process. These models generally model cascade behavior at a small time scale but are insufficiently flexible to model cascades that exhibit intermittent behavior governed by multiple scales. We argue that the presence of such time scales that are unaccounted for by a cascade model can result in degradation of performance of models on downstream statistical and time-sensitive optimization tasks. To address these issues, we formulate a model that incorporates multiple temporal scales of cascade behavior. This model is parameterized by a \emph{clock}, which encodes the times at which sessions of cascade activity start. These sessions are themselves governed by a small-scale cascade model, such as the discretized independent cascade (IC) model. Estimation of the multiscale cascade model parameters leads to the problem of \emph{clock estimation} in terms of a natural distortion measure that we formulate. Our framework is inspired by the optimization problem posed by DiTursi et al, 2017, which can be seen as providing one possible estimator (a maximum-proxy-likelihood estimator) for the parameters of our generative model. We give a clock estimation algorithm, which we call FastClock, that runs in linear time in the size of its input and is provably statistically accurate for a broad range of model parameters when cascades are generated from any spreading process model with well-concentrated session infection set sizes and when the underlying graph is at least in the semi-sparse regime. We exemplify our algorithm for the case where the small-scale model is the discretized independent cascade process and extend substantially to processes whose infection set sizes satisfy a general martingale difference property. We further evaluate the performance of FastClock empirically in comparison to the state of the art estimator from DiTursi et al, 2017. We find that in a broad parameter range on synthetic networks and on a real network, our algorithm substantially outperforms that algorithm in terms of both running time and accuracy. In all cases, our algorithm's running time is asymptotically lower than that of the baseline. 
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  3. Abstract

    The problem of reverse-engineering the evolution of a dynamic network, known broadly as network archaeology, is one of profound importance in diverse application domains. In analysis of infection spread, it reveals the spatial and temporal processes underlying infection. In analysis of biomolecular interaction networks (e.g., protein interaction networks), it reveals early molecules that are known to be differentially implicated in diseases. In economic networks, it reveals flow of capital and associated actors. Beyond these recognized applications, it provides analytical substrates for novel studies – for instance, on the structural and functional evolution of the human brain connectome. In this paper, we model, formulate, and rigorously analyze the problem of inferring the arrival order of nodes in a dynamic network from a single snapshot. We derive limits on solutions to the problem, present methods that approach this limit, and demonstrate the methods on a range of applications, from inferring the evolution of the human brain connectome to conventional citation and social networks, where ground truth is known.

     
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  4. Graph symmetries intervene in diverse applications, from enumeration, to graph structure compression, to the discovery of graph dynamics (e.g., node arrival order inference). Whereas Erdős‐Rényi graphs are typically asymmetric, real networks are highly symmetric. So a natural question is whether preferential attachment graphs, where in each step a new node withmedges is added, exhibit any symmetry. In recent work it was proved that preferential attachment graphs are symmetric form = 1, and there is some nonnegligible probability of symmetry form = 2. It was conjectured that these graphs are asymmetric whenm ≥ 3. We settle this conjecture in the affirmative, then use it to estimate the structural entropy of the model. To do this, we also give bounds on the number of ways that the given graph structure could have arisen by preferential attachment. These results have further implications for information theoretic problems of interest on preferential attachment graphs.

     
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