We introduce random directed acyclic graph and use it to model the information diffusion network. Subsequently, we analyze the cascade generation model (CGM) introduced by [19]. Until now only empirical studies of this model were done. In this paper, we present the first theoretical proof that the sizes of cascades generated by the CGM follow the power-law distribution, which is consistent with multiple empirical analysis of the large social networks. We compared the assumptions of our model with the Twitter social network and tested the goodness of approximation.
@cite_14: Cascades of information-sharing are a primary mechanism by which content reaches its audience on social media, and an active line of research has studied how such cascades, which form as content is reshared from person to person, develop and subside. In this paper, we perform a large-scale analysis of cascades on Facebook over significantly longer time scales, and find that a more complex picture emerges, in which many large cascades recur, exhibiting multiple bursts of popularity with periods of quiescence in between. We characterize recurrence by measuring the time elapsed between bursts, their overlap and proximity in the social network, and the diversity in the demographics of individuals participating in each peak. We discover that content virality, as revealed by its initial popularity, is a main driver of recurrence, with the availability of multiple copies of that content helping to spark new bursts. Still, beyond a certain popularity of content, the rate of recurrence drops as cascades start exhausting the population of interested individuals. We reproduce these observed patterns in a simple model of content recurrence simulated on a real social network. Using only characteristics of a cascade's initial burst, we demonstrate strong performance in predicting whether it will recur in the future.
@cite_1: Models for the processes by which ideas and influence propagate through a social network have been studied in a number of domains, including the diffusion of medical and technological innovations, the sudden and widespread adoption of various strategies in game-theoretic settings, and the effects of "word of mouth" in the promotion of new products. Recently, motivated by the design of viral marketing strategies, Domingos and Richardson posed a fundamental algorithmic problem for such social network processes: if we can try to convince a subset of individuals to adopt a new product or innovation, and the goal is to trigger a large cascade of further adoptions, which set of individuals should we target?We consider this problem in several of the most widely studied models in social network analysis. The optimization problem of selecting the most influential nodes is NP-hard here, and we provide the first provable approximation guarantees for efficient algorithms. Using an analysis framework based on submodular functions, we show that a natural greedy strategy obtains a solution that is provably within 63 of optimal for several classes of models; our framework suggests a general approach for reasoning about the performance guarantees of algorithms for these types of influence problems in social networks.We also provide computational experiments on large collaboration networks, showing that in addition to their provable guarantees, our approximation algorithms significantly out-perform node-selection heuristics based on the well-studied notions of degree centrality and distance centrality from the field of social networks.
@cite_0: 
@cite_15: 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.
@cite_16: How information spreads through a social network? Can we assume, that the information is spread only through a given social network graph? What is the correct way to compare the models of information flow? These are the basic questions we address in this work. We focus on meticulous comparison of various, well-known models of rumor propagation in the social network. We introduce the model incorporating mass media and effects of absent nodes. In this model the information appears spontaneously in the graph. Using the most conservative metric, we showed that the distribution of cascades sizes generated by this model fits the real data much better than the previously considered models.
In future, theoretical studies on cascades size distribution could explain phenomena in Gossip-based routing @cite_0 , rumor virality and influence maximization @cite_1 , recurrence of the cascades @cite_14 or assist in forecasting rumors @cite_15 . Currently the research in this area is purely empirical and we need more theoretical models to understand the process of information propagation @cite_16 .