Many online social networks feature restrictive web interfaces which only allow the query of a user's local neighborhood through the interface. To enable analytics over such an online social network through its restrictive web interface, many recent efforts reuse the existing Markov Chain Monte Carlo methods such as random walks to sample the social network and support analytics based on the samples. The problem with such an approach, however, is the large amount of queries often required (i.e., a long "mixing time") for a random walk to reach a desired (stationary) sampling distribution. In this paper, we consider a novel problem of enabling a faster random walk over online social networks by "rewiring" the social network on-the-fly. Specifically, we develop Modified TOpology (MTO)-Sampler which, by using only information exposed by the restrictive web interface, constructs a "virtual" overlay topology of the social network while performing a random walk, and ensures that the random walk follows the modified overlay topology rather than the original one. We show that MTO-Sampler not only provably enhances the efficiency of sampling, but also achieves significant savings on query cost over real-world online social networks such as Google Plus, Epinion etc.
@cite_21 found that the mixing time of typical online social networks is much larger than anticipated, which validates our motivation to shorten the mixing time of random walk. @cite_13 derived the fastest mixing random walk on a graph by convex optimization on second largest eigenvalue of the transition matrix, but it need the whole topology of the graph, and its high time complexity make it inapplicable in large graphs.
@cite_21 @cite_13
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Social networks provide interesting algorithmic properties that can be used to bootstrap the security of distributed systems. For example, it is widely believed that social networks are fast mixing, and many recently proposed designs of such systems make crucial use of this property. However, whether real-world social networks are really fast mixing is not verified before, and this could potentially affect the performance of such systems based on the fast mixing property. To address this problem, we measure the mixing time of several social graphs, the time that it takes a random walk on the graph to approach the stationary distribution of that graph, using two techniques. First, we use the second largest eigenvalue modulus which bounds the mixing time. Second, we sample initial distributions and compute the random walk length required to achieve probability distributions close to the stationary distribution. Our findings show that the mixing time of social graphs is much larger than anticipated, and being used in literature, and this implies that either the current security systems based on fast mixing have weaker utility guarantees or have to be less efficient, with less security guarantees, in order to compensate for the slower mixing. We consider a symmetric random walk on a connected graph, where each edge is labeled with the probability of transition between the two adjacent vertices. The associated Markov chain has a uniform equilibrium distribution; the rate of convergence to this distribution, i.e., the mixing rate of the Markov chain, is determined by the second largest eigenvalue modulus (SLEM) of the transition probability matrix. In this paper we address the problem of assigning probabilities to the edges of the graph in such a way as to minimize the SLEM, i.e., the problem of finding the fastest mixing Markov chain on the graph. We show that this problem can be formulated as a convex optimization problem, which can in turn be expressed as a semidefinite program (SDP). This allows us to easily compute the (globally) fastest mixing Markov chain for any graph with a modest number of edges (say, @math ) using standard numerical methods for SDPs. Larger problems can be solved by exploiting various types of symmetry and structure in the problem, and far larger problems (say, 100,000 edges) can be solved using a subgradient method we describe. We compare the fastest mixing Markov chain to those obtained using two commonly used heuristics: the maximum-degree method, and the Metropolis--Hastings algorithm. For many of the examples considered, the fastest mixing Markov chain is substantially faster than those obtained using these heuristic methods. We derive the Lagrange dual of the fastest mixing Markov chain problem, which gives a sophisticated method for obtaining (arbitrarily good) bounds on the optimal mixing rate, as well as the optimality conditions. Finally, we describe various extensions of the method, including a solution of the problem of finding the fastest mixing reversible Markov chain, on a fixed graph, with a given equilibrium distribution.