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In recent years, game theory has provided insights into the behavior of distributed systems by modeling the players as utility-maximizing agents. In particular, it has been shown that selfishness causes many systems to perform in a globally suboptimal fashion. Such systems are said to have a large price of anarchy. In this article, we extend this field of research by allowing some players to be malicious rather than selfish. What, we ask, is the impact of malicious players on the system consisting of otherwise selfish players? In particular, we introduce the price of malice as a measure that captures how much the system's efficiency degrades in the presence of malicious players, compared to a purely selfish environment. As a specific example, we analyze the price of malice of a game that models the containment of the spread of viruses. In this game, each player or node can choose whether to install antivirus software. Then, a virus starts from a random node and recursively infects all neighboring nodes that are not inoculated. We establish various results about this game. For instance, we quantify how much the presence of malicious players can deteriorate or---in case of highly risk-averse selfish players---improve the social welfare of the distributed system.
In this paper we consider the problem of computing the PageRank vector in an efficient way. By combining some of the existing techniques and different approaches, including the power method, linear systems, iterative aggregation/disaggregation, and matrix reorderings, we propose algorithms that decrease the number of iterations to reach the desired solution, thus accelerating convergence to the vector that contains the importance of web pages.
We suggest a revision to the PageRank random surfer model that considers the influence of a population of random surfers on the PageRank vector. In the revised model, each member of the population has its own teleportation parameter chosen from a probability distribution, and consequently, the ranking vector is random. We propose three algorithms for computing the statistics of the random ranking vector based respectively on (i) random sampling, (ii) paths along the links of the underlying graph, and (iii) quadrature formulas. We find that the expectation of the random ranking vector produces similar rankings to its deterministic analogue, but the standard deviation gives uncorrelated information (under a Kendall-tau metric) with myriad potential uses. We examine applications of this model to web spam.
We give an analysis of a variant of the contact process on finite graphs, allowing for nonuniform cure rates, modeling antidote distribution. We examine an inoculation scheme using PageRank vectors that quantify the correlations among vertices in the contact graph. We show that for a contact graph on $n$ nodes we can select a set $H$ of nodes to inoculate such that with probability at least $1-2\ep$, any infection from any starting infected set of $s$ nodes will die out in $c \log s + c'$ time, where $c$ and $c'$ depend only on the probabilistic error bound $\ep$ and the infection rate, and the size of $H$ depends only on $s$, $\ep$, and the topology around the initially infected nodes, independent of the size of the whole graph.