The mixing time of random walks on a graph has found broad applications across both theoretical and practical aspects of computer science, with the application effects depending on the behavior of mixing time. It is extensively believed that real-world networks, especially social networks, are fast mixing with their mixing time at most $O(\log N)$ where $N$ is the number of vertices. However, the behavior of mixing time in the real-life networks has not been examined carefully, and exactly analytical research for mixing time in models mimicking real networks is still lacking. In this paper, we first experimentally evaluate the mixing time of various real-world networks with scale-free small-world properties and show that their mixing time is much higher than anticipated. To better understand the behavior of the mixing time for real-world networks, we then analytically study the mixing time of the Apollonian network, which is simultaneously scale-free and small-world. To this end, we derive the recursive relations for all eigenvalues, especially the second largest eigenvalue modulus of the transition matrix, based on which we deduce a lower bound for the mixing time of the Apollonian network, which approximately scales sublinearly with $N$. Our results indicate that real-world networks are not always fast mixing, which has potential implications in the design of algorithms related to mixing time.
Concomitant with the tremendous prevalence of online social media platforms, the interactions among individuals are unprecedentedly enhanced. People are free to interact with acquaintances, express and exchange their own opinions through commenting, liking, retweeting on online social media, leading to resistance, controversy and other important phenomena over controversial social issues, which have been the subject of many recent works. In this paper, we study the problem of minimizing risk of conflict in social networks by modifying the initial opinions of a small number of nodes. We show that the objective function of the combinatorial optimization problem is monotone and supermodular. We then propose a naïve greedy algorithm with a (1 − 1/𝑒) approximation ratio that solves the problem in cubic time. To overcome the computation challenge for large networks, we further integrate several effective approximation strategies to provide a nearly linear time algorithm with a (1 − 1/𝑒 − 𝜖) approximation ratio for any error parameter 𝜖 > 0. Extensive experiments on various real-world datasets demonstrate both the efficiency and effectiveness of our algorithms. In particular, the fast one scales to large networks with more than two million nodes, and achieves up to 20× speed-up over the state-of-the-art algorithm.
Simplicial complexes are a popular tool used to model higher-order interactions between elements of complex social and biological systems. In this paper, we study some combinatorial aspects of a class of simplicial complexes created by a graph product, which is an extension of the pseudo-fractal scale-free web. We determine explicitly the independence number, the domination number, and the chromatic number. Moreover, we derive closed-form expressions for the number of acyclic orientations, the number of root-connected acyclic orientations, the number of spanning trees, as well as the number of perfect matchings for some particular cases.
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