Various important and useful quantities or measures that characterize the topological network structure are usually investigated for a network, then they are averaged over the samples. In this paper, we propose an explicit representation by the beforehand averaged adjacency matrix over samples of growing networks as a new general framework for investigating the characteristic quantities. It is applied to some network models, and shows a good approximation of degree distribution asymptotically. In particular, our approach will be applicable through the numerical calculations instead of intractable theoretical analyses, when the time-course of degree is a monotone increasing function like power law or logarithm.

We consider a network of interacting individuals, whose actions or transitions are determined by the states (behavior) of their neighbors as well as their own personal decisions. Specifically, we develop a model according to two simple decision-making rules that can describe the growth of the user population of a newly launched product or service. We analyze 22 sets of real-world historical growth data of a variety of products and services, and show that they all follow the growth equation. The numerical procedure for finding the model parameters allows the market size, and the relative effectiveness of customer service and promotional efforts to be estimated from the available historical growth data. We study the growth profiles of products and find that for a product or service to reach a mature stage within a reasonably short time in its user growth profile, the user growth rate corresponding to influenced transitions must exceed a certain threshold. Furthermore, results show that individuals in the group of celebrities having numerous friends become users of a new product or service at a much faster rate than those connected to ordinary individuals having fewer friends.

We introduce NetworKit, an open-source software package for analyzing the structure of large complex networks. Appropriate algorithmic solutions are required to handle increasingly common large graph data sets containing up to billions of connections. We describe the methodology applied to develop scalable solutions to network analysis problems, including techniques like parallelization, heuristics for computationally expensive problems, efficient data structures, and modular software architecture. Our goal for the software is to package results of our algorithm engineering efforts and put them into the hands of domain experts. NetworKit is implemented as a hybrid combining the kernels written in C++ with a Python frontend, enabling integration into the Python ecosystem of tested tools for data analysis and scientific computing. The package provides a wide range of functionality (including common and novel analytics algorithms and graph generators) and does so via a convenient interface. In an experimental comparison with related software, NetworKit shows the best performance on a range of typical analysis tasks.