In-Degree and PageRank of Web pages: Why do they follow similar power laws?

The PageRank is a popularity measure designed by Google to rank Web pages. Experiments confirm that the PageRank obeys a `power law' with the same exponent as the In-Degree. This paper presents a novel mathematical model that explains this phenomenon. The relation between the PageRank and In-Degree is modelled through a stochastic equation, which is inspired by the original definition of the PageRank, and is analogous to the well-known distributional identity for the busy period in the M/G/1 queue. Further, we employ the theory of regular variation and Tauberian theorems to analytically prove that the tail behavior of the PageRank and the In-Degree differ only by a multiplicative factor, for which we derive a closed-form expression. Our analytical results are in good agreement with experimental data.Comment: 20 pages, 3 figures; typos added; reference adde

PageRank in scale-free random graphs

We analyze the distribution of PageRank on a directed configuration model and show that as the size of the graph grows to infinity it can be closely approximated by the PageRank of the root node of an appropriately constructed tree. This tree approximation is in turn related to the solution of a linear stochastic fixed point equation that has been thoroughly studied in the recent literature

Thermodynamics of reaction of praseodymium with gallium-indium eutectic alloy

Thermodynamic properties of Ga-In eutectic alloys saturated with praseodymium were determined for the first time employing the electromotive force method. The equilibrium potentials of the Pr-In alloys saturated with praseodymium (8.7-12.1 mol.% Pr) and Pr-Ga-In alloys (containing 0.0012-6.71 mol.% Pr) were measured between 573-1073 K. Pr-In alloy containing solid PrIn3 with known thermodynamic properties was used as the reference electrode when measuring the potentials of ternary Pr-In-Ga alloys. Activity, partial and excessive thermodynamic functions of praseodymium in alloys with indium and Ga-In eutectic were calculated. Activity (a), activity coefficients (γ) and solubility (X) of praseodymium in the studied temperature range can be expressed by the following equations: lgaα-Pr(In) = 4.425 - 11965/T ± 0.026. lgα-Pr(Ga-In) = 5.866 - 14766/T ± 0.190. lgγα-Pr(Ga-In) = 2.351 - 9996/T ± 0.39. lgPr(Ga-In) = 3.515 - 4770/T ± 0.20. © 2013 Elsevier B.V. All rights reserved

The Impact of Geographic Distance on Online Social Interactions

Online social networking services entice millions of users to spend hours every day interacting with each other. The focus of this work is to explain the effect that geographic distance has on online social interactions and, simultaneously, to understand the interplay between the social characteristics of friendship ties and their spatial properties. We analyze data from a large-scale online social network, Tuenti, with about 10 million active users: our sample includes user profiles, user home locations and online social interactions among Tuenti members. Our findings support the idea that spatial distance constraints whom users interact with, but not the intensity of their social interactions. Furthermore, friendship ties belonging to denser connected groups tend to arise at shorter spatial distances than social ties established between members belonging to different groups. Finally, we show that our findings mostly do not depend on the age of the users, although younger users seem to be slightly more constrained to shorter geographic distances. Augmenting social structure with geographic information adds a new dimension to social network analysis and a large number of theoretical investigations and practical applications can be pursued for online social systems, with many promising outcomes. As the amount of available location-based data is increasing, our findings and results open the door to future possibilities: researchers would benefit from these insights when studying online social services, while developers should be aware of these additional possibilities when building systems and applications related to online social platforms

Excessive thermodynamic properties of praseodymium in a gallium-indium alloy

The equilibrium potentials of praseodymium-diluted homogeneous Pr-Ga-In alloys in a (Li-K-Cs)Cleut-based salt electrolyte were measured between 573-1073 K by the emf method. These potentials are used to calculate the activity coefficients of α-praseodymium in liquid Ga-In eutectic alloys. PrIn3 alloy with well-known thermodynamic characteristics and without phase transitions in the temperature range 428-1483 K was employed as the reference electrode. © 2013 Pleiades Publishing, Ltd

Electrochemical Properties of Molybdenum in Alkali Chloride Melts

Behavior of molybdenum was studied in individual fused alkali chlorides and their mixtures employing electrochemical and spectroscopic techniques. Formal standard electrode potentials of molybdenum and thermodynamics of molybdenum chloride were obtained in molten LiCl, NaCl, KCl, RbCl, CsCl at 1123 K, NaCl- CsCl eutectic at 793-1023 K, NaCl-KCl equimolar mixture at 973- 1123 K, LiCl-KCl-CsCl eutectic at 633-1173 K. Diffusion coefficients of molybdenum ions were determined in LiCl-KCl- CsCl eutectic melt. Electronic absorption spectra of Mo(III) ions were measured in individual salts and their mixtures and spectroscopic parameters of MoCl6 3- complex ions determined. Copyright © 2014 by The Electrochemical Society. All rights reserved

ELECTRONIC ABSORPTION SPECTRA OF URANIUM(V) IONS IN ALKALI CHLORIDE MELTS

Electronic absorption spectra of uranium(V) ions were recorded in the melts based on individual alkali chlorides and their mixtures. The spectra were measured between 350 and 850 oC. The effect of temperature and alkali cations forming the second coordination sphere on the spectra is discussed

Random walk centrality for temporal networks

Nodes can be ranked according to their relative importance within a network. Ranking algorithms based on random walks are particularly useful because they connect topological and diffusive properties of the network. Previous methods based on random walks, for example the PageRank, have focused on static structures. However, several realistic networks are indeed dynamic, meaning that their structure changes in time. In this paper, we propose a centrality measure for temporal networks based on random walks under periodic boundary conditions that we call TempoRank. It is known that, in static networks, the stationary density of the random walk is proportional to the degree or the strength of a node. In contrast, we find that, in temporal networks, the stationary density is proportional to the in-strength of the so-called effective network, a weighted and directed network explicitly constructed from the original sequence of transition matrices. The stationary density also depends on the sojourn probability q, which regulates the tendency of the walker to stay in the node, and on the temporal resolution of the data. We apply our method to human interaction networks and show that although it is important for a node to be connected to another node with many random walkers (one of the principles of the PageRank) at the right moment, this effect is negligible in practice when the time order of link activation is included