text: Many popular measures used in social network analysis, including centrality, are based on the random walk. The random walk is a model of a stochastic process where a node interacts with one other node at a time. However, the random walk may not be appropriate for modeling social phenomena, including epidemics and information diffusion, in which one node may interact with many others at the same time, for example, by broadcasting the virus or information to its neighbors. To produce meaningful results, social network analysis algorithms have to take into account the nature of interactions between the nodes. In this paper we classify dynamical processes as conservative and non-conservative and relate them to well-known measures of centrality used in network analysis: PageRank and Alpha-Centrality. We demonstrate, by ranking users in online social networks used for broadcasting information, that non-conservative Alpha-Centrality generally leads to a better agreement with an empirical ranking scheme than the conservative PageRank.
  @cite_24: Most methods proposed to uncover communities in complex networks rely on their structural properties. Here we introduce the stability of a network partition, a measure of its quality defined in terms of the statistical properties of a dynamical process taking place on the graph. The time-scale of the process acts as an intrinsic parameter that uncovers community structures at different resolutions. The stability extends and unifies standard notions for community detection: modularity and spectral partitioning can be seen as limiting cases of our dynamic measure. Similarly, recently proposed multi-resolution methods correspond to linearisations of the stability at short times. The connection between community detection and Laplacian dynamics enables us to establish dynamically motivated stability measures linked to distinct null models. We apply our method to find multi-scale partitions for different networks and show that the stability can be computed efficiently for large networks with extended versions of current algorithms.
  @cite_10: The behavior of complex systems is determined not only by the topological organization of their interconnections but also by the dynamical processes taking place among their constituents. A faithful modeling of the dynamics is essential because different dynamical processes may be affected very differently by network topology. A full characterization of such systems thus requires a formalization that encompasses both aspects simultaneously, rather than relying only on the topological adjacency matrix. To achieve this, we introduce the concept of flow graphs, namely weighted networks where dynamical flows are embedded into the link weights. Flow graphs provide an integrated representation of the structure and dynamics of the system, which can then be analyzed with standard tools from network theory. Conversely, a structural network feature of our choice can also be used as the basis for the construction of a flow graph that will then encompass a dynamics biased by such a feature. We illustrate the ideas by focusing on the mathematical properties of generic linear processes on complex networks that can be represented as biased random walks and their dual consensus dynamics, and show how our framework improves our understanding of these processes.
summary: The interplay between structural properties of networks and the diffusion processes occurring on them contribute to their complexity. This has been realized by several researchers in the past. For example, @cite_24 @cite_10 emphasized that dynamical processes play an important role in characterizing the structure of complex networks. In @cite_24 they measure the quality of a network partition in terms of the statistical property of the dynamic process taking place in the network. In @cite_10 they study the different equilibrium properties of these processes. However, their works focus on what we call conservative processes: unbiased and biased random walks, discrete and continuous time random walks. In contrast, we also study non-conservative dynamical processes. We also relate these processes to centrality. Although the relationship of PageRank to random walk-type processes is well known, we explain how Alpha-Centrality is related to a type of a non-conservative process. We also carry out an empirical study of different centrality measures, unlike previous works.