Minority Becomes Majority in Social Networks

It is often observed that agents tend to imitate the behavior of their neighbors in a social network. This imitating behavior might lead to the strategic decision of adopting a public behavior that differs from what the agent believes is the right one and this can subvert the behavior of the population as a whole. In this paper, we consider the case in which agents express preferences over two alternatives and model social pressure with the majority dynamics: at each step an agent is selected and its preference is replaced by the majority of the preferences of her neighbors. In case of a tie, the agent does not change her current preference. A profile of the agents' preferences is stable if the preference of each agent coincides with the preference of at least half of the neighbors (thus, the system is in equilibrium). We ask whether there are network topologies that are robust to social pressure. That is, we ask if there are graphs in which the majority of preferences in an initial profile always coincides with the majority of the preference in all stable profiles reachable from that profile. We completely characterize the graphs with this robustness property by showing that this is possible only if the graph has no edge or is a clique or very close to a clique. In other words, except for this handful of graphs, every graph admits at least one initial profile of preferences in which the majority dynamics can subvert the initial majority. We also show that deciding whether a graph admits a minority that becomes majority is NP-hard when the minority size is at most 1/4-th of the social network size.Comment: To appear in WINE 201

Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?

In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of αGW + ∈, for all ∈ \u3e 0; here αGW ≈ .878567 denotes the approximation ratio achieved by the Goemans-Williamson algorithm [26]. This implies that if the Unique Games Conjecture of Khot [37] holds then the Goemans-Williamson approximation algorithm is optimal. Our result indicates that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-CUT problem. Our reduction relies on a theorem we call Majority Is Stablest. This was introduced as a conjecture in the original version of this paper, and was subsequently confirmed in [45]. A stronger version of this conjecture called Plurality Is Stablest is still open, although [45] contains a proof of an asymptotic version of it. Our techniques extend to several other two-variable constraint satisfaction problems. In particular, subject to the Unique Games Conjecture, we show tight or nearly tight hardness results for MAX-2SAT, MAX-q-CUT, and MAX-2LIN(q). For MAX-2SAT we show approximation hardness up to a factor of roughly .943. This nearly matches the .940 approximation algorithm of Lewin, Livnat, and Zwick [41]. Furthermore, we show that our .943... factor is actually tight for a slightly restricted version of MAX-2SAT. For MAX-q-CUT we show a hardness factor which asymptotically (for large q) matches the approximation factor achieved by Frieze and Jerrum [25], namely 1 − 1/q + 2(ln q)/q2 . For MAX-2LIN(q) we show hardness of distinguishing between instances which are (1−∈)-satisfiable and those which are not even, roughly, (q−∈/2)-satisfiable. These parameters almost match those achieved by the recent algorithm of Charikar, Makarychev, and Makarychev [10]. The hardness result holds even for instances in which all equations are of the form xi − xj = c. At a more qualitative level, this result also implies that 1 − ∈ vs. ∈ hardness for MAX-2LIN(q) is equivalent to the Unique Games Conjecture

A PCP Characterization of AM

We introduce a 2-round stochastic constraint-satisfaction problem, and show that its approximation version is complete for (the promise version of) the complexity class AM. This gives a `PCP characterization' of AM analogous to the PCP Theorem for NP. Similar characterizations have been given for higher levels of the Polynomial Hierarchy, and for PSPACE; however, we suggest that the result for AM might be of particular significance for attempts to derandomize this class. To test this notion, we pose some `Randomized Optimization Hypotheses' related to our stochastic CSPs that (in light of our result) would imply collapse results for AM. Unfortunately, the hypotheses appear over-strong, and we present evidence against them. In the process we show that, if some language in NP is hard-on-average against circuits of size 2^{Omega(n)}, then there exist hard-on-average optimization problems of a particularly elegant form. All our proofs use a powerful form of PCPs known as Probabilistically Checkable Proofs of Proximity, and demonstrate their versatility. We also use known results on randomness-efficient soundness- and hardness-amplification. In particular, we make essential use of the Impagliazzo-Wigderson generator; our analysis relies on a recent Chernoff-type theorem for expander walks.Comment: 18 page

Quantum Algorithms for Learning and Testing Juntas

In this article we develop quantum algorithms for learning and testing juntas, i.e. Boolean functions which depend only on an unknown set of k out of n input variables. Our aim is to develop efficient algorithms: - whose sample complexity has no dependence on n, the dimension of the domain the Boolean functions are defined over; - with no access to any classical or quantum membership ("black-box") queries. Instead, our algorithms use only classical examples generated uniformly at random and fixed quantum superpositions of such classical examples; - which require only a few quantum examples but possibly many classical random examples (which are considered quite "cheap" relative to quantum examples). Our quantum algorithms are based on a subroutine FS which enables sampling according to the Fourier spectrum of f; the FS subroutine was used in earlier work of Bshouty and Jackson on quantum learning. Our results are as follows: - We give an algorithm for testing k-juntas to accuracy $\epsilon$ that uses $O(k/\epsilon)$ quantum examples. This improves on the number of examples used by the best known classical algorithm. - We establish the following lower bound: any FS-based k-junta testing algorithm requires $\Omega(\sqrt{k})$ queries. - We give an algorithm for learning $k$-juntas to accuracy $\epsilon$ that uses $O(\epsilon^{-1} k\log k)$ quantum examples and $O(2^k \log(1/\epsilon))$ random examples. We show that this learning algorithms is close to optimal by giving a related lower bound.Comment: 15 pages, 1 figure. Uses synttree package. To appear in Quantum Information Processin

Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms

Constrained submodular maximization problems have long been studied, with near-optimal results known under a variety of constraints when the submodular function is monotone. The case of non-monotone submodular maximization is less understood: the first approximation algorithms even for the unconstrainted setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC '09, APPROX '09) show how to approximately maximize non-monotone submodular functions when the constraints are given by the intersection of p matroid constraints; their algorithm is based on local-search procedures that consider p-swaps, and hence the running time may be n^Omega(p), implying their algorithm is polynomial-time only for constantly many matroids. In this paper, we give algorithms that work for p-independence systems (which generalize constraints given by the intersection of p matroids), where the running time is poly(n,p). Our algorithm essentially reduces the non-monotone maximization problem to multiple runs of the greedy algorithm previously used in the monotone case. Our idea of using existing algorithms for monotone functions to solve the non-monotone case also works for maximizing a submodular function with respect to a knapsack constraint: we get a simple greedy-based constant-factor approximation for this problem. With these simpler algorithms, we are able to adapt our approach to constrained non-monotone submodular maximization to the (online) secretary setting, where elements arrive one at a time in random order, and the algorithm must make irrevocable decisions about whether or not to select each element as it arrives. We give constant approximations in this secretary setting when the algorithm is constrained subject to a uniform matroid or a partition matroid, and give an O(log k) approximation when it is constrained by a general matroid of rank k.Comment: In the Proceedings of WINE 201

10-year follow-up of patients with rheumatoid arthritis and secondary Sjogren's syndrome or sicca symptoms in daily clinical practice

Objective. To evaluate the presence of sicca symptoms and secondary Sjogren's syndrome (SS) and the association with clinical characteristics, functional tests and patient-reported outcomes in patients with rheumatoid arthritis (RA) at baseline and after 10 years of follow-up. Methods. A cohort of RA patients was evaluated in 2008 and re-evaluated in 2018 with respect to sicca symptoms, presence of secondary SS according to AECG classification criteria, disease activity of RA and patient-reported outcomes. Patient characteristics were compared between the RA-non-sicca, RA-sicca and RA-SS groups. Results. Of the original 2008 cohort of 96 RA patients, 32 (33%) had sicca symptoms and 6 (6.3%) secondary SS. Of the 36 patients who agreed to be reevaluated in 2018, 6 (17%) had sicca symptoms and 2 (6%) developed secondary SS. In the majority of patients, sicca symptoms were reversible while the functional tests of salivary and lacrimal glands significantly decreased. 67% of RA-sicca patients had no sicca complaints at the second screening, while only two RA-sicca patients developed secondary SS. RA-SS patients and, to a slightly lesser extent, RA-sicca patients had significantly higher RA disease activity (DAS-28), lower lacrimal (Schirmer's test) and salivary gland function, more limitations in daily activities (HAQ), worse health-related quality of life (RAND-36), more fatigue (MFI) and more patient symptoms (ESSPRI) compared to RA-non-sicca patients. Conclusion. Secondary SS was found in a minor subset of the RA patients. Sicca symptoms of the eyes or mouth were more frequent, but their presence varied over time. Higher RA disease activity was associated with SS and sicca symptoms. These patients had lower gland function and worse patient-reported outcomes

10-year follow-up of patients with rheumatoid arthritis and secondary Sjogren's syndrome or sicca symptoms in daily clinical practice

Objective. To evaluate the presence of sicca symptoms and secondary Sjogren's syndrome (SS) and the association with clinical characteristics, functional tests and patient-reported outcomes in patients with rheumatoid arthritis (RA) at baseline and after 10 years of follow-up. Methods. A cohort of RA patients was evaluated in 2008 and re-evaluated in 2018 with respect to sicca symptoms, presence of secondary SS according to AECG classification criteria, disease activity of RA and patient-reported outcomes. Patient characteristics were compared between the RA-non-sicca, RA-sicca and RA-SS groups. Results. Of the original 2008 cohort of 96 RA patients, 32 (33%) had sicca symptoms and 6 (6.3%) secondary SS. Of the 36 patients who agreed to be reevaluated in 2018, 6 (17%) had sicca symptoms and 2 (6%) developed secondary SS. In the majority of patients, sicca symptoms were reversible while the functional tests of salivary and lacrimal glands significantly decreased. 67% of RA-sicca patients had no sicca complaints at the second screening, while only two RA-sicca patients developed secondary SS. RA-SS patients and, to a slightly lesser extent, RA-sicca patients had significantly higher RA disease activity (DAS-28), lower lacrimal (Schirmer's test) and salivary gland function, more limitations in daily activities (HAQ), worse health-related quality of life (RAND-36), more fatigue (MFI) and more patient symptoms (ESSPRI) compared to RA-non-sicca patients. Conclusion. Secondary SS was found in a minor subset of the RA patients. Sicca symptoms of the eyes or mouth were more frequent, but their presence varied over time. Higher RA disease activity was associated with SS and sicca symptoms. These patients had lower gland function and worse patient-reported outcomes

A Contour Method on Cayley tree

We consider a finite range lattice models on Cayley tree with two basic properties: the existence of only a finite number of ground states and with Peierls type condition. We define notion of a contour for the model on the Cayley tree. By a contour argument we show the existence of $s$ different (where $s$ is the number of ground states) Gibbs measures.Comment: 12 page