text: To enjoy more social network services, users nowadays are usually involved in multiple online social networks simultaneously. The shared users between different networks are called anchor users, while the remaining unshared users are named as non-anchor users. Connections between accounts of anchor users in different networks are defined as anchor links and networks partially aligned by anchor links can be represented as partially aligned networks. In this paper, we want to predict anchor links between partially aligned social networks, which is formally defined as the partial network alignment problem. The partial network alignment problem is very difficult to solve because of the following two challenges: (1) the lack of general features for anchor links, and (2) the "one-to-one @math " (one to at most one) constraint on anchor links. To address these two challenges, a new method PNA (Partial Network Aligner) is proposed in this paper. PNA (1) extracts a set of explicit anchor adjacency features and latent topological features for anchor links based on the anchor meta path concept and tensor decomposition techniques, and (2) utilizes the generic stable matching to identify the non-anchor users to prune the redundant anchor links attached to them. Extensive experiments conducted on two real-world partially aligned social networks demonstrate that PNA can solve the partial network alignment problem very well and outperform all the other comparison methods with significant advantages.
  @cite_33: Abstract Two-sided matching markets of the kind known as the “college admissions problem” have been widely thought to be virtually equivalent to the simpler “marriage problem” for which some striking results concerning agents' preferences and incentives have been recently obtained. It is shown here that some of these results do not generalize to the college admissions problem, contrary to a number of assertions in the recent literature. No stable matching procedure exists that makes it a dominant strategy for colleges to reveal their true preferences, and some outcomes may be preferred by all colleges to the college-optimal stable outcome.
  @cite_52: Both rematching proof and strong equilibrium outcomes are stable with respect to the true preferences in the marriage problem. We show that not all rematching proof or strong equilibrium outcomes are stable in the college admissions problem. But we show that both rematching proof and strong equilibrium outcomes in truncations at the match point are all stable in the college admissions problem. Further, all true stable matchings can be achieved in both rematching proof and strong equilibrium in truncations at the match point. We show that any Nash equilibrium in truncations admits one and only one matching, stable or not. Therefore, the core at a Nash equilibrium in truncations must be small. But examples exist such that the set of stable matchings with respect to a Nash equilibrium may contain more than one matching. Nevertheless, each Nash equilibrium can only admit at most one true stable matching. If, indeed, there is a true stable matching at a Nash equilibrium, then the only possible equilibrium outcome will be the true stable matching, no matter how different are players' equilibrium strategies from the true preferences and how many other unstable matchings are there at that Nash equilibrium. Thus, we show that a necessary and sufficient condition for the stable matching rule to be implemented in a subset of Nash equilibria by the direct revelation game induced by a stable mechanism is that every Nash equilibrium profile in that subset admits one and only one true stable matching.
  @cite_43: A stable matching rule is used as the outcome function for the Admission game where colleges behave straightforwardly and the students’ strategies are given by their preferences over the colleges. We show that the college-optimal stable matching rule implements the set of stable matchings via the Nash equilibrium (NE) concept. For any other stable matching rule the strategic behavior of the students may lead to outcomes that are not stable under the true preferences. We then introduce uncertainty about the matching selected and prove that the natural solution concept is that of NE in the strong sense. A general result shows that the random stable matching rule, as well as any stable matching rule, implements the set of stable matchings via NE in the strong sense. Precise answers are given to the strategic questions raised.
  @cite_40: This book probes the stable marriage problem and its variants as a rich source of problems and ideas that illustrate both the design and analysis of efficient algorithms. It covers the most recent structural and algorithmic work on stable matching problems, simplifies and unifies many earlier proofs, strengthens several earlier results, and presents new results and more efficient algorithms.The authors develop the structure of the set of stable matchings in the stable marriage problem in a more general and algebraic context than has been done previously; they discuss the problem's structure in terms of rings of sets, which allows many of the most useful features to be seen as features of a more general set of problems. The relationship between the structure of the stable marriage problem and the more general stable roommates problem is demonstrated, revealing many commonalities.The results the authors obtain provide an algorithmic response to the practical, and political, problems created by the asymmetry inherent in the Gale Shapley solutions, leading to alternative methods and better compromises than are provided by the Gale Shapley method. And, in contrast to Donald Knuth's earlier work which primarily focused on the application of mathematics to the analysis of algorithms, this book illustrates the productive and almost inseparable relationship between mathematical insight and the design of efficient algorithms.Dan Gusfield is Associate Professor of Computer Science at the University of California, Davis. Robert W. Irving is Senior Lecturer in Computing Science at the University of Glasgow. The Stable Marriage Problem is included in the Foundations of Computing Series, edited by Michael Garey and Albert Meyer.
  @cite_23: We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributed-systems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. We apply our results to give a distributed (2+e)-approximation algorithm for maximum-weight matching in bicoloured graphs and a centralised randomised constant-time approximation scheme for estimating the size of a stable matching.
summary: College admission problem @cite_33 and stable marriage problem @cite_40 have been studied for many years and lots of works have been done in the last century. In recent years, some new papers have come out in these areas. @cite_43 propose to analyze the stability of the equilibrium outcomes in the admission games induced by stable matching rules. Ma @cite_52 analyzes the truncation in stable matching and the small core in nash equilibrium in college admission problems. Flor ' e @cite_23 propose to study the almost stable matching by truncating the Gale-Shapley algorithm.