Decomposition of the Google pagerank and optimal linking strategy

We provide the analysis of the Google PageRank from the perspective of the Markov Chain Theory. First we study the Google PageRank for a Web that can be decomposed into several connected components which do not have any links to each other. We show that in order to determine the Google PageRank for a completely decomposable Web, it is sufficient to compute a subPageRank for each of the connected components separately. Then, we study incentives for the Web users to form connected components. In particular, we show that there exists an optimal linking strategy that benefits a user with links inside its Web community and on the other hand inappropriate links penalize the Web users and their Web communities. \u

A scaling analysis of a cat and mouse Markov chain

Motivated by an original on-line page-ranking algorithm, starting from an arbitrary Markov chain $(C_n)$ on a discrete state space ${\cal S}$, a Markov chain $(C_n,M_n)$ on the product space ${\cal S}^2$, the cat and mouse Markov chain, is constructed. The first coordinate of this Markov chain behaves like the original Markov chain and the second component changes only when both coordinates are equal. The asymptotic properties of this Markov chain are investigated. A representation of its invariant measure is in particular obtained. When the state space is infinite it is shown that this Markov chain is in fact null recurrent if the initial Markov chain $(C_n)$ is positive recurrent and reversible. In this context, the scaling properties of the location of the second component, the mouse, are investigated in various situations: simple random walks in $\mathbb{Z}$ and $\mathbb{Z}^2$, reflected simple random walk in $\mathbb{N}$ and also in a continuous time setting. For several of these processes, a time scaling with rapid growth gives an interesting asymptotic behavior related to limit results for occupation times and rare events of Markov processes.\u

Markov chains and optimality of the Hamiltonian cycle

We consider the Hamiltonian cycle problem (HCP) embedded in a controlled Markov decision process. In this setting, HCP reduces to an optimization problem on a set of Markov chains corresponding to a given graph. We prove that Hamiltonian cycles are minimizers for the trace of the fundamental matrix on a set of all stochastic transition matrices. In case of doubly stochastic matrices with symmetric linear perturbation, we show that Hamiltonian cycles minimize a diagonal element of a fundamental matrix for all admissible values of the perturbation parameter. In contrast to the previous work on this topic, our arguments are primarily based on probabilistic rather than algebraic methods

Cost-efficient vaccination protocols for network epidemiology

We investigate methods to vaccinate contact networks -- i.e. removing nodes in such a way that disease spreading is hindered as much as possible -- with respect to their cost-efficiency. Any real implementation of such protocols would come with costs related both to the vaccination itself, and gathering of information about the network. Disregarding this, we argue, would lead to erroneous evaluation of vaccination protocols. We use the susceptible-infected-recovered model -- the generic model for diseases making patients immune upon recovery -- as our disease-spreading scenario, and analyze outbreaks on both empirical and model networks. For different relative costs, different protocols dominate. For high vaccination costs and low costs of gathering information, the so-called acquaintance vaccination is the most cost efficient. For other parameter values, protocols designed for query-efficient identification of the network's largest degrees are most efficient

Predicting the long-term citation impact of recent publications

A fundamental problem in citation analysis is the prediction of the long-term citation impact of recent publications. We propose a model to predict a probability distribution for the future number of citations of a publication. Two predictors are used: The impact factor of the journal in which a publication has appeared and the number of citations a publication has received one year after its appearance. The proposed model is based on quantile regression. We employ the model to predict the future number of citations of a large set of publications in the field of physics. Our analysis shows that both predictors (i.e., impact factor and early citations) contribute to the accurate prediction of long-term citation impact. We also analytically study the behavior of the quantile regression coefficients for high quantiles of the distribution of citations. This is done by linking the quantile regression approach to a quantile estimation technique from extreme value theory. Our work provides insight into the influence of the impact factor and early citations on the long-term citation impact of a publication, and it takes a step toward a methodology that can be used to assess research institutions based on their most recently published work.Comment: 17 pages, 17 figure

Convergence of rank based degree-degree correlations in random directed networks

We introduce, and analyze, three measures for degree-degree dependencies, also called degree assortativity, in directed random graphs, based on Spearman's rho and Kendall's tau. We proof statistical consistency of these measures in general random graphs and show that the directed configuration model can serve as a null model for our degree-degree dependency measures. Based on these results we argue that the measures we introduce should be preferred over Pearson's correlation coefficients, when studying degree-degree dependencies, since the latter has several issues in the case of large networks with scale-free degree distributions

Degree-degree correlations in random graphs with heavy-tailed degrees

We investigate degree-degree correlations for scale-free graph sequences. The main conclusion of this paper is that the assortativity coefficient is not the appropriate way to describe degree-dependences in scale-free random graphs. Indeed, we study the infinite volume limit of the assortativity coefficient, and show that this limit is always non-negative when the degrees have finite first but infinite third moment, i.e., when the degree exponent $\gamma + 1$ of the density satisfies $\gamma \in (1,3)$. More generally, our results show that the correlation coefficient is inappropriate to describe dependencies between random variables having infinite variance. We start with a simple model of the sample correlation of random variables $X$ and $Y$, which are linear combinations with non-negative coefficients of the same infinite variance random variables. In this case, the correlation coefficient of $X$ and $Y$ is not defined, and the sample covariance converges to a proper random variable with support that is a subinterval of $(-1,1)$. Further, for any joint distribution $(X,Y)$ with equal marginals being non-negative power-law distributions with infinite variance (as in the case of degree-degree correlations), we show that the limit is non-negative. We next adapt these results to the assortativity in networks as described by the degree-degree correlation coefficient, and show that it is non-negative in the large graph limit when the degree distribution has an infinite third moment. We illustrate these results with several examples where the assortativity behaves in a non-sensible way. We further discuss alternatives for describing assortativity in networks based on rank correlations that are appropriate for infinite variance variables. We support these mathematical results by simulations

Phase transitions for scaling of structural correlations in directed networks

Analysis of degree-degree dependencies in complex networks, and their impact on processes on networks requires null models, i.e. models that generate uncorrelated scale-free networks. Most models to date however show structural negative dependencies, caused by finite size effects. We analyze the behavior of these structural negative degree-degree dependencies, using rank based correlation measures, in the directed Erased Configuration Model. We obtain expressions for the scaling as a function of the exponents of the distributions. Moreover, we show that this scaling undergoes a phase transition, where one region exhibits scaling related to the natural cut-off of the network while another region has scaling similar to the structural cut-off for uncorrelated networks. By establishing the speed of convergence of these structural dependencies we are able to asses statistical significance of degree-degree dependencies on finite complex networks when compared to networks generated by the directed Erased Configuration Model