We define the spreading time in the SIS process as the average time between the start of the outbreak and the time that the number of infected nodes first reaches the average number of infected nodes in the metastable state. We show that the spreading time can be computed using a uniformised embedded Markov chain and give numerical results for the complete graph and the star graph. For the complete graph we derive, using the same method, an analytical expression for the spreading time starting from a single infected node. We show that the spreading time is only significantly larger for a single initially infected than when a few nodes are infected, and scales logarithmically as a function of the network size for a fixed fraction of infected nodes in the metastable state. We also show that mean-field methods predict that the spreading time in regular graphs is independent of the degree. For graphs with a high epidemic threshold, the spreading time is lower than for graphs with a low epidemic threshold. The spreading time seems to be related to the average hop count in the graph. For graphs that have a relatively low average hop count, the spreading time scales logarithmically, but for graphs with a high average hop count, such as the rectangular grid and the ring graph, this is not the case.

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Multiplayer Online Games (MOGs) like Defense of the Ancients and StarCraft II have attracted hundreds of millions of users who communicate, interact, and socialize with each other through gaming. In MOGs, rich social relationships emerge and can be used to improve gaming services such as match recommendation and game population retention, which are important for the user experience and the commercial value of the companies who run these MOGs. In this work, we focus on understanding social relationships in MOGs. We propose a graph model that is able to capture social relationships of a variety of types and strengths. We apply our model to real-world data collected from three MOGs that contain in total over ten years of behavioral history for millions of players and matches. We compare social relationships in MOGs across different game genres and with regular online social networks like Facebook. Taking match recommendation as an example application of our model, we propose SAMRA, a Socially Aware Match Recommendation Algorithm that takes social relationships into account. We show that our model not only improves the precision of traditional link prediction approaches, but also potentially helps players enjoy games to a higher extent.@en

Mean-field approximations (MFAs) are frequently used in physics. When a process (such as an epidemic or a synchronization) on a network is approximated by MFA, a major hurdle is the determination of those graphs for which MFA is reasonably accurate. Here, we present an accuracy criterion for Markovian susceptible-infected-susceptible (SIS) epidemics on any network, based on the spectrum of the adjacency and SIS covariance matrix. We evaluate the MFA criterion for the complete and star graphs analytically, and numerically for connected Erdős-Rényi random graphs for small size N≤14. The accuracy of MFA increases with average degree and with N. Precise simulations (up to network sizes N=100) of the MFA accuracy criterion versus N for the complete graph, star, square lattice, and path graphs lead us to conjecture that the worst MFA accuracy decreases, for large N, proportionally to the inverse of the spectral radius of the adjacency matrix of the graph.
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We introduce an extension of the SIS epidemic model that describes infection, mutation and curing for a whole hierarchy of viruses, resembling a nested spreading process. In our model, high level viruses are only allowed to spread to nodes that have acquired a lower level of infection before. The simplest case of two viruses, in which one "superinfects" the other, shows already rich dynamics that are difficult to predict by common mean-field approximation techniques in certain cases. We derive an exact Markovian description for superinfection in the complete network and the star network showing that the steady state of the epidemic process is highly sensitive to the spreading rate of both viruses. Taking the spreading rates into account, we outline conditions for epidemic outbreaks, coexistence of both viruses and extinction cycles.@en