Information cascades are ubiquitous in both physical society and online social media, taking on large variations in structures, dynamics and semantics. Although the dynamics and semantics of information cascades have been studied, the structural patterns and their correlations with dynamics and semantics are largely unknown. Here we explore a large-scale dataset including @math million information cascades with explicit records of spreading traces, spreading behaviors, information content as well as user profiles. We find that the structural complexity of information cascades is far beyond the previous conjectures. We first propose a ten-dimensional metric to quantify the structural characteristics of information cascades, reflecting cascade size, silhouette, direction and activity aspects. We find that bimodal law governs majority of the metrics, information flows in cascades have four directions, and the self-loop number and average activity of cascades follows power law. We then analyze the high-order structural patterns of information cascades. Finally, we evaluate to what extent the structural features of information cascades can explain its dynamic patterns and semantics, and finally uncover some notable implications of structural patterns in information cascades. Our discoveries also provide a foundation for the microscopic mechanisms for information spreading, potentially leading to implications for cascade prediction and outlier detection.
The dynamics of cascades have been empirically studied in marketing @cite_9 , Twitter @cite_3 , Facebook @cite_7 , QQ @cite_10 , and so on. Another line of works try to model the dynamics from differential equation and micro-process @cite_21 @cite_0 . The prediction of the dynamics are studied in Refs. @cite_16 @cite_11 @cite_30 . Time series clustering methods are applied to discover the dynamics patterns or shapes @cite_26 @cite_8 . Our work is based on the clustering method to find out the dynamics clusters in empirical cascade growth dynamics. However, in order to study the correlations between dynamics and structures, we propose new similarity intuitions and find the existing shift-variant and scale-variant algorithm can not do this job.
@cite_30 @cite_26 @cite_7 @cite_8 @cite_9 @cite_21 @cite_3 @cite_0 @cite_16 @cite_10 @cite_11
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Cascades are ubiquitous in various network environments. How to predict these cascades is highly nontrivial in several vital applications, such as viral marketing, epidemic prevention and traffic management. Most previous works mainly focus on predicting the final cascade sizes. As cascades are typical dynamic processes, it is always interesting and important to predict the cascade size at any time, or predict the time when a cascade will reach a certain size (e.g. an threshold for outbreak). In this paper, we unify all these tasks into a fundamental problem: cascading process prediction. That is, given the early stage of a cascade, how to predict its cumulative cascade size of any later time? For such a challenging problem, how to understand the micro mechanism that drives and generates the macro phenomena (i.e. cascading process) is essential. Here we introduce behavioral dynamics as the micro mechanism to describe the dynamic process of a node's neighbors getting infected by a cascade after this node getting infected (i.e. one-hop subcascades). Through data-driven analysis, we find out the common principles and patterns lying in behavioral dynamics and propose a novel Networked Weibull Regression model for behavioral dynamics modeling. After that we propose a novel method for predicting cascading processes by effectively aggregating behavioral dynamics, and present a scalable solution to approximate the cascading process with a theoretical guarantee. We extensively evaluate the proposed method on a large scale social network dataset. The results demonstrate that the proposed method can significantly outperform other state-of-the-art baselines in multiple tasks including cascade size prediction, outbreak time prediction and cascading process prediction. Cascades are a popular construct to observe and study information propagation (or diffusion) in social media such as Twitter. and are defined using notions of influence, activity, or discourse commonality (e.g., hashtags). While these notions of cascades lead to different perspectives, primarily cascades are modeled as trees. We argue in this paper an alternative viewpoint of cascades as forests (of trees) which yields a richer vocabulary of features to understand information propagation. We develop a framework to extract forests and analyze their growth by studying their evolution at the tree-level and at the node-level. Moreover, we demonstrate how the structural features of forests, properties of the underlying network, and temporal features of the cascades provide significant predictive value in forecasting the future trajectory of both size and shape of forests. We observe that the forecasting performance increases with observations, that the temporal features are highly indicative of cascade size, and that the features extracted from the underlying connected graph best forecast the shape of the cascade. On many social networking web sites such as Facebook and Twitter, resharing or reposting functionality allows users to share others' content with their own friends or followers. As content is reshared from user to user, large cascades of reshares can form. While a growing body of research has focused on analyzing and characterizing such cascades, a recent, parallel line of work has argued that the future trajectory of a cascade may be inherently unpredictable. In this work, we develop a framework for addressing cascade prediction problems. On a large sample of photo reshare cascades on Facebook, we find strong performance in predicting whether a cascade will continue to grow in the future. We find that the relative growth of a cascade becomes more predictable as we observe more of its reshares, that temporal and structural features are key predictors of cascade size, and that initially, breadth, rather than depth in a cascade is a better indicator of larger cascades. This prediction performance is robust in the sense that multiple distinct classes of features all achieve similar performance. We also discover that temporal features are predictive of a cascade's eventual shape. Observing independent cascades of the same content, we find that while these cascades differ greatly in size, we are still able to predict which ends up the largest. Cascades appear in many applications, including biological graphs and social media analysis. In a cascade, a dynamic attribute propagates through a graph, following its edges. We present the results of a formal user study that tests the effectiveness of different types of cascade visualisations on node-link diagrams for the task of judging cascade spread. Overall, we found that a small multiples presentation was significantly faster than animation with no significant difference in terms of error rate. Participants generally preferred animation over small multiples and a hierarchical layout to a force-directed layout. Considering each presentation method separately, when comparing force-directed layouts to hierarchical layouts, hierarchical layouts were found to be significantly faster for both presentation methods and significantly more accurate for animation. Representing the history of the cascade had no significant effect. Thus, for our task, this experiment supports the use of a small multiples interf... We introduce a class of (2+1)(2+1)-dimensional stochastic growth processes, that can be seen as irreversible random dynamics of discrete interfaces. “Irreversible” means that the interface has an average nonzero drift. Interface configurations correspond to height functions of dimer coverings of the infinite hexagonal or square lattice. The model can also be viewed as an interacting driven particle system and in the totally asymmetric case the dynamics corresponds to an infinite collection of mutually interacting Hammersley processes. When the dynamical asymmetry parameter (p−q)(p−q) equals zero, the infinite-volume Gibbs measures πρπρ (with given slope ρρ) are stationary and reversible. When p≠qp≠q, πρπρ are not reversible any more but, remarkably, they are still stationary. In such stationary states, we find that the average height function at any given point xx grows linearly with time tt with a nonzero speed: EQx(t):=E(hx(t)−hx(0))=V(ρ)tEQx(t):=E(hx(t)−hx(0))=V(ρ)t while the typical fluctuations of Qx(t)Qx(t) are smaller than any power of tt as t→∞t→∞. In the totally asymmetric case of p=0,q=1p=0,q=1 and on the hexagonal lattice, the dynamics coincides with the “anisotropic KPZ growth model” introduced by A. Borodin and P. L. Ferrari in [J. Stat. Mech. Theory Exp. 2009 (2009) P02009, Comm. Math. Phys. 325 603–684]. For a suitably chosen, “integrable”, initial condition (that is very far from the stationary state), they were able to determine the hydrodynamic limit and a CLT for interface fluctuations on scale logt−−−−√log⁡t, exploiting the fact that in that case certain space-time height correlations can be computed exactly. In the same setting, they proved that, asymptotically for t→∞t→∞, the local statistics of height fluctuations tends to that of a Gibbs state (which led to the prediction that Gibbs states should be stationary). We develop a canonical framework for the study of the problem of registration of multiple point processes subjected to warping, known as the problem of separation of amplitude and phase variation. The amplitude variation of a real random function Y(x) : x is an element of [0, 1] corresponds to its random oscillations in the y-axis, typically encapsulated by its (co) variation around a mean level. In contrast, its phase variation refers to fluctuations in the x-axis, often caused by random time changes. We formalise similar notions for a point process, and nonparametrically separate them based on realisations of i.i.d. copies Pi(i) of the phase-varying point process. A key element in our approach is to demonstrate that when the classical phase variation assumptions of Functional Data Analysis (FDA) are applied to the point process case, they become equivalent to conditions interpretable through the prism of the theory of optimal transportation of measure. We demonstrate that these induce a natural Wasserstein geometry tailored to the warping problem, including a formal notion of bias expressing over-registration. Within this framework, we construct nonparametric estimators that tend to avoid over-registration in finite samples. We show that they consistently estimate the warp maps, consistently estimate the structural mean, and consistently register the warped point processes, even in a sparse sampling regime. We also establish convergence rates, and derivev root n-consistency and a central limit theorem in the Cox process case under dense sampling, showing rate optimality of our structural mean estimator in that case. Information spreads across social and technological networks, but often the network structures are hidden from us and we only observe the traces left by the diffusion processes, called cascades. Can we recover the hidden network structures from these observed cascades? What kind of cascades and how many cascades do we need? Are there some network structures which are more difficult than others to recover? Can we design efficient inference algorithms with provable guarantees? Despite the increasing availability of cascade data and methods for inferring networks from these data, a thorough theoretical understanding of the above questions remains largely unexplored in the literature. In this paper, we investigate the network structure inference problem for a general family of continuous-time diffusion models using an @math -regularized likelihood maximization framework. We show that, as long as the cascade sampling process satisfies a natural incoherence condition, our framework can recover the correct network structure with high probability if we observe @math cascades, where @math is the maximum number of parents of a node and @math is the total number of nodes. Moreover, we develop a simple and efficient soft-thresholding inference algorithm, which we use to illustrate the consequences of our theoretical results, and show that our framework outperforms other alternatives in practice. We study an evolving spatial network in which sequentially arriving vertices are joined to existing vertices at random according to a rule that combines preference according to degree with preference according to spatial proximity. We investigate phase transitions in graph structure as the relative weighting of these two components of the attachment rule is varied. Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence of the resulting graph coincides with that of the standard Barab 'asi--Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence coincides with that of a purely geometric model, the on-line nearest-neighbour graph, which is of interest in its own right and for which we prove some extensions of known results. We also show the presence of an intermediate regime, in which the behaviour differs significantly from both the on-line nearest-neighbour graph and the Barab 'asi--Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence. Our results lend some mathematical support to simulation studies of Manna and Sen, while proving that the power law to stretched exponential phase transition occurs at a different point from the one conjectured by those authors. The problem of learning classifier cascades is considered. A new cascade boosting algorithm, fast cascade boosting (FCBoost), is proposed. FCBoost is shown to have a number of interesting properties, namely that it 1) minimizes a Lagrangian risk that jointly accounts for classification accuracy and speed, 2) generalizes adaboost, 3) can be made cost-sensitive to support the design of high detection rate cascades, and 4) is compatible with many predictor structures suitable for sequential decision making. It is shown that a rich family of such structures can be derived recursively from cascade predictors of two stages, denoted cascade generators. Generators are then proposed for two new cascade families, last-stage and multiplicative cascades, that generalize the two most popular cascade architectures in the literature. The concept of neutral predictors is finally introduced, enabling FCBoost to automatically determine the cascade configuration, i.e., number of stages and number of weak learners per stage, for the learned cascades. Experiments on face and pedestrian detection show that the resulting cascades outperform current state-of-the-art methods in both detection accuracy and speed. A review of dynamic nonlinear features of spatiotemporal congested patterns in freeway traffic is presented. The basis of the review is a comparison of theoretical features of the congested patterns that are shown by a microscopic traffic flow model in the context of the Kerner's three-phase traffic theory and empirical microscopic and macroscopic pattern characteristics measured on different freeways over various days and years. In this test of the microscopic three-phase traffic flow theory, a model of an "open" road is applied: Empirical time-dependence of traffic demand and drivers' destinations are used at the upstream model boundaries. At downstream model boundary conditions for vehicle freely leaving a modeling freeway section(s) are given. Spatiotemporal congested patterns emerge, develop, and dissolve in this open freeway model with the same types of bottlenecks as those in empirical observations. It is found that microscopic three-phase traffic models can explain all known macroscopic and microscopic empirical congested pattern features (e.g., probabilistic breakdown phenomenon as a first-order phase transition from free flow to synchronized flow, moving jam emergence in synchronized flow rather than in free flow, spatiotemporal features of synchronized flow and general congested patterns at freeway bottlenecks, intensification of downstream congestion due to upstream congestion at adjacent bottlenecks). It turns out that microscopic optimal velocity (OV) functions and time headway distributions are not necessarily qualitatively different, even if local congested traffic behavior is qualitatively different. Model performance with respect to spatiotemporal pattern emergence and evolution cannot be tested using these traffic characteristics. The reason for this is that important spatiotemporal features of congested traffic patterns are lost in these and many other macroscopic and microscopic traffic characteristics, which are widely used as the empirical basis for a test of traffic flow models. This is the second in a series of papers in which we derive a @math -expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In our previous paper, we computed the @math -limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order @math -limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg-Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing to the Euler-Lagrange equation, we establish for all configurations which have "almost minimal energy" the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of @math -equivalence, the obtained results also yield an expansion of the minimal energy for the original Ohta-Kawasaki energy. This leads to expecting to see triangular lattices of droplets as energy minimizers. Abstract Six learning algorithms are investigated and compared empirically. All of them are based on variants of the candidate training idea of the Cascade Correlation method. The comparison was performed using 42 different datasets from the PROBEN1 benchmark collection. The results indicate: (1) for these problems it is slightly better not to cascade the hidden units; (2) error minimization candidate training is better than covariance maximization for regression problems but may be a little worse for classification problems; (3) for most learning tasks, considering validation set errors during the selection of the best candidate will not lead to improved networks, but for a few tasks it will. © 1997 Elsevier Science Ltd. The percolation phase transition and the mechanism of the emergence of the giant component through the critical scaling window for random graph models, has been a topic of great interest in many dierent communities ranging from statistical physics, combinatorics, computer science, social networks and probability theory. The last few years have witnessed an explosion of models which couple random aggregation rules, that specify how one adds edges to existing congurations, and choice, wherein one selects from a ‘limited’ set of edges at random to use in the conguration. These models exhibit fascinating new phenomenon, ranging from delaying or speeding up the emergence of the giant component, to percolation", where the diameter of the scaling window is several orders of magnitude smaller than that for standard random graph models. While an intense study of such models has ensued, understanding the actual emergence of the giant component and merging dynamics in the critical scaling window has remained impenetrable to a rigorous analysis. In this work we take an important step in the analysis of such models by studying one of the standard examples of such processes, namely the Bohman-Frieze model, and provide rst results on the asymptotic dynamics, through the critical scaling window, that lead to the emergence of the giant component for such models. We identify the scaling window and show that through this window, the component sizes properly rescaled converge to the standard multiplicative coalescent. Proofs hinge on a careful analysis of certain innite-type branching processes with types taking values in the space of RCLL paths, and stochastic analytic techniques to estimate susceptibility functions of the components all the way through the scaling window where these functions explode. Previous approaches for analyzing random graphs at criticality have relied largely on classical breadth-rst search techniques that exploit asymptotic connections with Brownian excursions. For dynamic random graph models evolving via general Markovian rules, such approaches fail and we develop a quite dierent set of tools that can potentially be used for the study of critical dynamics for all bounded size rules. We study a regularized version of Hastings-Levitov planar random growth that models clusters formed by the aggregation of diffusing particles. In this model, the growing clusters are defined in terms of iterated slit maps whose capacities are given by c_n=c| n-1 '(e^ +i )|^ - , 0, where c>0 is the capacity of the first particle, _n are the composed conformal maps defining the clusters of the evolution, _n are independent uniform angles determining the positions at which particles are attached, and >0 is a regularization parameter which we take to depend on c. We prove that under an appropriate rescaling of time, in the limit as c converges to 0, the clusters converge to growing disks with deterministic capacities, provided that does not converge to 0 too fast. We then establish scaling limits for the harmonic measure flow, showing that by letting tend to 0 at different rates it converges to either the Brownian web on the circle, a stopped version of the Brownian web on the circle, or the identity map. As the harmonic measure flow is closely related to the internal branching structure within the cluster, the above three cases intuitively correspond to the number of infinite branches in the model being either 1, a random number whose distribution we obtain, or unbounded, in the limit as c converges to 0. We also present several findings based on simulations of the model with parameter choices not covered by our rigorous analysis. We study the entropy landscape of solutions for the bicoloring problem in random graphs, a representative difficult constraint satisfaction problem. Our goal is to classify which types of clusters of solutions are addressed by different algorithms. In the first part of the study we use the cavity method to obtain the number of clusters with a given internal entropy and determine the phase diagram of the problem--e.g., dynamical, rigidity, and satisfiability-unsatisfiability (SAT-UNSAT) transitions. In the second part of the paper we analyze different algorithms and locate their behavior in the entropy landscape of the problem. For instance, we show that a smoothed version of a decimation strategy based on belief propagation is able to find solutions belonging to subdominant clusters even beyond the so-called rigidity transition where the thermodynamically relevant clusters become frozen. These nonequilibrium solutions belong to the most probable unfrozen clusters. Based on the empirical analysis of the dependency network in 18 Java projects, we develop a novel model of network growth which considersboth: an attachment mechanism and the addition of new nodes with a heterogeneous distribution of their initial degree, k0. Empirically we find that the cumulative degree distributions of initial degrees and of the final network, follow power-law behaviors: P(k0) ∝ k01−α, and P(k) ∝ k1−γ, respectively. For the total number of links as a function of the network size, we find empirically K(N) ∝ Nβ, where β is (at the beginning of thenetwork evolution) between 1.25 and 2, while converging to ∼ 1 for large N. This indicates a transition from a growth regime with increasing network density towards a sustainable regime, which revents a collapse because of ever increasing dependencies. Our theoretical framework is able to predict relations between the exponents α, β, γ, which also link issues of software engineering and developer activity. These relations are verified by means of computer simulations and empirical investigations. They indicate that the growth of real Open Source Software networks occurs on the edge between two regimes, which are either dominated by the initial degree distribution of added nodes, or by the preferential attachment mechanism. Hence, the heterogeneous degree distribution of newly added nodes, found empirically, is essential to describe the laws of sustainable growth in networks. Information cascades on social networks, such as retweet cascades on Twitter, have been often viewed as an epidemiological process, with the associated notion of virality to capture popular cascades that spread across the network. The notion of structural virality or average path length has been posited as a measure of global spread. In this paper, we argue that this simple epidemiological view, though analytically compelling, is not the entire story. We first show empirically that the classical SIR diffusion process on the Twitter graph, even with the best possible distribution of infectiousness parameter, cannot explain the nature of observed retweet cascades on Twitter. More specifically, rather than spreading further from the source as the SIR model would predict, many cascades that have several retweets from direct followers, die out quickly beyond that. We show that our empirical observations can be reconciled if we take interests of users and tweets into account. In particular, we consider a model where users have multi-dimensional interests, and connect to other users based on similarity in interests. Tweets are correspondingly labeled with interests, and propagate only in the subgraph of interested users via the SIR process. In this model, interests can be either narrow or broad, with the narrowest interest corresponding to a star graph on the interested users, with the root being the source of the tweet, and the broadest interest spanning the whole graph. We show that if tweets are generated using such a mix of interests, coupled with a varying infectiousness parameter, then we can qualitatively explain our observation that cascades die out much more quickly than is predicted by the SIR model. In the same breath, this model also explains how cascades can have large size, but low "structural virality" or average path length. New phase transition phenomena have recently been discovered for the stochastic block model, for the special case of two non-overlapping symmetric communities. This gives raise in particular to new algorithmic challenges driven by the thresholds. This paper investigates whether a general phenomenon takes place for multiple communities, without imposing symmetry. In the general stochastic block model @math , @math vertices are split into @math communities of relative size @math , and vertices in community @math and @math connect independently with probability @math . This paper investigates the partial and exact recovery of communities in the general SBM (in the constant and logarithmic degree regimes), and uses the generality of the results to tackle overlapping communities. The contributions of the paper are: (i) an explicit characterization of the recovery threshold in the general SBM in terms of a new divergence function @math , which generalizes the Hellinger and Chernoff divergences, and which provides an operational meaning to a divergence function analog to the KL-divergence in the channel coding theorem, (ii) the development of an algorithm that recovers the communities all the way down to the optimal threshold and runs in quasi-linear time, showing that exact recovery has no information-theoretic to computational gap for multiple communities, in contrast to the conjectures made for detection with more than 4 communities; note that the algorithm is optimal both in terms of achieving the threshold and in having quasi-linear complexity, (iii) the development of an efficient algorithm that detects communities in the constant degree regime with an explicit accuracy bound that can be made arbitrarily close to 1 when a prescribed signal-to-noise ratio (defined in term of the spectrum of @math ) tends to infinity. We study the condensation phenomenon in a zero-range process on scale-free networks. We show that the stationary state property depends only on the degree distribution of underlying networks. The model displays a stationary state phase transition between a condensed phase and an uncondensed phase, and the phase diagram is obtained analytically. As for the dynamical property, we find that the relaxation dynamics depends on the global structure of underlying networks. The relaxation time follows the power law @math with the network size @math in the condensed phase. The dynamic exponent @math is found to take a different value depending on whether underlying networks have a tree structure or not. We study the statistical properties of the endpoint of stochastic gradient descent (SGD). We approximate SGD as a stochastic differential equation (SDE) and consider its Boltzmann Gibbs equilibrium distribution under the assumption of isotropic variance in loss gradients.. Through this analysis, we find that three factors – learning rate, batch size and the variance of the loss gradients – control the trade-off between the depth and width of the minima found by SGD, with wider minima favoured by a higher ratio of learning rate to batch size. In the equilibrium distribution only the ratio of learning rate to batch size appears, implying that it’s invariant under a simultaneous rescaling of each by the same amount. We experimentally show how learning rate and batch size affect SGD from two perspectives: the endpoint of SGD and the dynamics that lead up to it. For the endpoint, the experiments suggest the endpoint of SGD is similar under simultaneous rescaling of batch size and learning rate, and also that a higher ratio leads to flatter minima, both findings are consistent with our theoretical analysis. We note experimentally that the dynamics also seem to be similar under the same rescaling of learning rate and batch size, which we explore showing that one can exchange batch size and learning rate in a cyclical learning rate schedule. Next, we illustrate how noise affects memorization, showing that high noise levels lead to better generalization. Finally, we find experimentally that the similarity under simultaneous rescaling of learning rate and batch size breaks down if the learning rate gets too large or the batch size gets too small.