Center clusters in the Yang-Mills vacuum

Properties of local Polyakov loops for SU(2) and SU(3) lattice gauge theory at finite temperature are analyzed. We show that spatial clusters can be identified where the local Polyakov loops have values close to the same center element. For a suitable definition of these clusters the deconfinement transition can be characterized by the onset of percolation in one of the center sectors. The analysis is repeated for different resolution scales of the lattice and we argue that the center clusters have a continuum limit.Comment: Table added. Final version to appear in JHE

The egalitarian effect of search engines

Search engines have become key media for our scientific, economic, and social activities by enabling people to access information on the Web in spite of its size and complexity. On the down side, search engines bias the traffic of users according to their page-ranking strategies, and some have argued that they create a vicious cycle that amplifies the dominance of established and already popular sites. We show that, contrary to these prior claims and our own intuition, the use of search engines actually has an egalitarian effect. We reconcile theoretical arguments with empirical evidence showing that the combination of retrieval by search engines and search behavior by users mitigates the attraction of popular pages, directing more traffic toward less popular sites, even in comparison to what would be expected from users randomly surfing the Web.Comment: 9 pages, 8 figures, 2 appendices. The final version of this e-print has been published on the Proc. Natl. Acad. Sci. USA 103(34), 12684-12689 (2006), http://www.pnas.org/cgi/content/abstract/103/34/1268

Distributed Graph Clustering using Modularity and Map Equation

We study large-scale, distributed graph clustering. Given an undirected graph, our objective is to partition the nodes into disjoint sets called clusters. A cluster should contain many internal edges while being sparsely connected to other clusters. In the context of a social network, a cluster could be a group of friends. Modularity and map equation are established formalizations of this internally-dense-externally-sparse principle. We present two versions of a simple distributed algorithm to optimize both measures. They are based on Thrill, a distributed big data processing framework that implements an extended MapReduce model. The algorithms for the two measures, DSLM-Mod and DSLM-Map, differ only slightly. Adapting them for similar quality measures is straight-forward. We conduct an extensive experimental study on real-world graphs and on synthetic benchmark graphs with up to 68 billion edges. Our algorithms are fast while detecting clusterings similar to those detected by other sequential, parallel and distributed clustering algorithms. Compared to the distributed GossipMap algorithm, DSLM-Map needs less memory, is up to an order of magnitude faster and achieves better quality.Comment: 14 pages, 3 figures; v3: Camera ready for Euro-Par 2018, more details, more results; v2: extended experiments to include comparison with competing algorithms, shortened for submission to Euro-Par 201

Stable and Efficient Structures for the Content Production and Consumption in Information Communities

Real-world information communities exhibit inherent structures that characterize a system that is stable and efficient for content production and consumption. In this paper, we study such structures through mathematical modelling and analysis. We formulate a generic model of a community in which each member decides how they allocate their time between content production and consumption with the objective of maximizing their individual reward. We define the community system as "stable and efficient" when a Nash equilibrium is reached while the social welfare of the community is maximized. We investigate the conditions for forming a stable and efficient community under two variations of the model representing different internal relational structures of the community. Our analysis results show that the structure with "a small core of celebrity producers" is the optimally stable and efficient for a community. These analysis results provide possible explanations to the sociological observations such as "the Law of the Few" and also provide insights into how to effectively build and maintain the structure of information communities.Comment: 21 page

Testing Cluster Structure of Graphs

We study the problem of recognizing the cluster structure of a graph in the framework of property testing in the bounded degree model. Given a parameter $\varepsilon$, a $d$-bounded degree graph is defined to be $(k, \phi)$-clusterable, if it can be partitioned into no more than $k$ parts, such that the (inner) conductance of the induced subgraph on each part is at least $\phi$ and the (outer) conductance of each part is at most $c_{d,k}\varepsilon^4\phi^2$, where $c_{d,k}$ depends only on $d,k$. Our main result is a sublinear algorithm with the running time $\widetilde{O}(\sqrt{n}\cdot\mathrm{poly}(\phi,k,1/\varepsilon))$ that takes as input a graph with maximum degree bounded by $d$, parameters $k$, $\phi$, $\varepsilon$, and with probability at least $\frac23$, accepts the graph if it is $(k,\phi)$-clusterable and rejects the graph if it is $\varepsilon$-far from $(k, \phi^*)$-clusterable for $\phi^* = c'_{d,k}\frac{\phi^2 \varepsilon^4}{\log n}$, where $c'_{d,k}$ depends only on $d,k$. By the lower bound of $\Omega(\sqrt{n})$ on the number of queries needed for testing graph expansion, which corresponds to $k=1$ in our problem, our algorithm is asymptotically optimal up to polylogarithmic factors.Comment: Full version of STOC 201

Critical Droplets and Phase Transitions in Two Dimensions

In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point T_c of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability p_B<1, the corresponding site-bond clusters keep on percolating at T_c and the exponents do not change, until p_B=p_CK=1-exp(-2J/kT): for this special expression of the bond weight the critical percolation exponents switch to the 2D Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability p_c such that, for any p_B>=p_c, the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for p_c<p_B<=1 but, for p_B=p_c, they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.Comment: Final version for publication, minor changes, figures adde

A Geometrical Interpretation of Hyperscaling Breaking in the Ising Model

In random percolation one finds that the mean field regime above the upper critical dimension can simply be explained through the coexistence of infinite percolating clusters at the critical point. Because of the mapping between percolation and critical behaviour in the Ising model, one might check whether the breakdown of hyperscaling in the Ising model can also be intepreted as due to an infinite multiplicity of percolating Fortuin-Kasteleyn clusters at the critical temperature T_c. Preliminary results suggest that the scenario is much more involved than expected due to the fact that the percolation variables behave differently on the two sides of T_c.Comment: Lattice2002(spin

Implementation of the Quantum Fourier Transform

The quantum Fourier transform (QFT) has been implemented on a three bit nuclear magnetic resonance (NMR) quantum computer, providing a first step towards the realization of Shor's factoring and other quantum algorithms. Implementation of the QFT is presented with fidelity measures, and state tomography. Experimentally realizing the QFT is a clear demonstration of NMR's ability to control quantum systems.Comment: 6 pages, 2 figure

Motif-based communities in complex networks

Community definitions usually focus on edges, inside and between the communities. However, the high density of edges within a community determines correlations between nodes going beyond nearest-neighbours, and which are indicated by the presence of motifs. We show how motifs can be used to define general classes of nodes, including communities, by extending the mathematical expression of Newman-Girvan modularity. We construct then a general framework and apply it to some synthetic and real networks

Solution of the Bohr hamiltonian for soft triaxial nuclei

The Bohr-Mottelson model is solved for a generic soft triaxial nucleus, separating the Bohr hamiltonian exactly and using a number of different model-potentials: a displaced harmonic oscillator in $\gamma$, which is solved with an approximated algebraic technique, and Coulomb/Kratzer, harmonic/Davidson and infinite square well potentials in $\beta$, which are solved exactly. In each case we derive analytic expressions for the eigenenergies which are then used to calculate energy spectra. Here we study the chain of osmium isotopes and we compare our results with experimental information and previous calculations.Comment: 13 pages, 9 figure