Modeling/Predicting the Evolution of User Activity Graphs on OSN-based Applications Han Liu * , Atif Nazir * , Jinoo Joung † , Chen-Nee Chuah * , * Department of Electrical & Computer Engineering, University of California, Davis, CA, USA † Department of Computer Science, Sangmyung University, Seoul, Korea {bhgliu,anazir}@ucdavis.edu, jinoo.joung@gmail.com, chuah@ucdavis.edu Abstract—While various models have been proposed for gen- erating social/friendship network graphs, the dynamics of user interactions through online social network (OSN) based ap- plications remain largely unexplored. We bridge this gap by using unique user activity data collected from three home-grown, popular gifting applications on Facebook to study the long-term evolution of user activity graphs (UAGs). This paper presents a new dynamic graph evolution model aimed to capture micro- scopic user-level behaviors that govern the growth of the UAG and collectively define the overall graph structure. We demonstrate the utility of our model by applying it to forecast the number of active users over time as the application transition from initial growth to peak/mature and decline/fatique phase. Using empirical evaluations, we show that our model can accurately reproduce the evolution trend of active user population for OSN applications in the same genre. We also demonstrate that the predictions from our model can guide the generation of synthetic graphs, which accurately forecast the structure of real UAG snapshots at different evolution stage. I. I NTRODUCTION The growing popularity of online social networks (OSNs) such as Facebook has led to extensive research on OSN friendship graphs [1], [2], [3]. Arguably, however, the worth of an OSN resides in how much activity its users generate, rather than simply how connected its users are. Unlike online friend- ships that are mostly static, the amount of activity between user pairs varies over time [4], [5]. Citing this difference, many researchers have highlighted the importance of studying user activity as opposed to simple OSN friendship graphs [6], [7]. User activity data from OSN-based applications hence provides a gateway to study the nature of user dynamics on OSNs. In particular, a user activity graph (UAG) can be constructed for each application where a node represents a user and a directed edge represents an action initiated by one user, targeting another (e.g., user A sends a gift to user B). Unlike friendship graphs, UAGs consist of directed, transient edges and may not be reciprocal. In this paper, we consider the problem of measuring and modeling the evolution of UAGs from OSN-based applica- tions. Such an evolution model can provide important insights into patterns of user interactions and how they morph across different stages (e.g., initial growth, peak/mature, and decline) of OSN application life span. It forms a basis for investigating factors contributing to the viral growth of application and cascading effects. Although useful in many aspects, modeling UAGs is a very challenging task given their dynamic nature and the lack of user activity data due to privacy concerns. Related Work:Most existing graph models for social networks are designed specifically for friendship graphs [8], [3], [2], [9]. Those studies generate edges between isolate nodes according to two structural properties of social networks, preferential attachment and triadic closure, and thereby create a graph topologically similar to real networks. In addition, latest studies consider more information, such as social attributes [10] and user location [11], when applying the preferential attachment and triadic closure, so as to generate more rep- resentative synthetic graphs. Some recent work also allows new nodes to arrive at a given constant rate, and then adds edges between new nodes and exiting nodes. This approach can partially mimic the expansion of friendship graphs [2], [9], [10], [11]. For the case of UAGs, however, since both nodes and edges are transient, these models can only be used to generate static snapshots, which represent user activities accumulated in a given time window, instead of the continual evolution process. Moreover, for every snapshot, empirical parameters including the number of active nodes (or the nodes joining rate) are required as the inputs of these models. The study of dynamic graphs [12] is an emergent topic [13], and characterizing temporal networks through parametrized, generative models remains an open problem. So far, two categories of modeling strategies have been proposed. The first category involves discretizing a temporal network by generating static snapshots of the network in consecutive time windows. The snapshots are then used to model how graph characteristics change with time [14], [15]. The second approach adopts a continuous process that predicts each node’s activity based on its current and previous states [16], [17]. In order to model the temporal sequence of user interactions, some studies use small time granularities so as to capture only one user activity in each time slot [18]. The first approach is easier to apply and can leverage existing studies on static graphs, while the second is more accurate in capturing de- tails of dynamic graph evolution. Unfortunately, current work adopting either approach is incapable of modeling temporal networks with both dynamic edges and nodes. In most recent studies, researchers assume nodes neither join nor leave the graph to reduce the modeling complexity [19], [20]. Although such studies can find practical use in many scenarios, e.g., phone call communications [16] and face-to-face interactions