Recommendation Subgraphs for Web Discovery

Recommendations are central to the utility of many websites including YouTube, Quora as well as popular e-commerce stores. Such sites typically contain a set of recommendations on every product page that enables visitors to easily navigate the website. Choosing an appropriate set of recommendations at each page is one of the key features of backend engines that have been deployed at several e-commerce sites. Specifically at BloomReach, an engine consisting of several independent components analyzes and optimizes its clients' websites. This paper focuses on the structure optimizer component which improves the website navigation experience that enables the discovery of novel content. We begin by formalizing the concept of recommendations used for discovery. We formulate this as a natural graph optimization problem which in its simplest case, reduces to a bipartite matching problem. In practice, solving these matching problems requires superlinear time and is not scalable. Also, implementing simple algorithms is critical in practice because they are significantly easier to maintain in production. This motivated us to analyze three methods for solving the problem in increasing order of sophistication: a sampling algorithm, a greedy algorithm and a more involved partitioning based algorithm. We first theoretically analyze the performance of these three methods on random graph models characterizing when each method will yield a solution of sufficient quality and the parameter ranges when more sophistication is needed. We complement this by providing an empirical analysis of these algorithms on simulated and real-world production data. Our results confirm that it is not always necessary to implement complicated algorithms in the real-world and that very good practical results can be obtained by using heuristics that are backed by the confidence of concrete theoretical guarantees

A Quantum Lovasz Local Lemma

The Lovasz Local Lemma (LLL) is a powerful tool in probability theory to show the existence of combinatorial objects meeting a prescribed collection of "weakly dependent" criteria. We show that the LLL extends to a much more general geometric setting, where events are replaced with subspaces and probability is replaced with relative dimension, which allows to lower bound the dimension of the intersection of vector spaces under certain independence conditions. Our result immediately applies to the k-QSAT problem: For instance we show that any collection of rank 1 projectors with the property that each qubit appears in at most $2^k/(e \cdot k)$ of them, has a joint satisfiable state. We then apply our results to the recently studied model of random k-QSAT. Recent works have shown that the satisfiable region extends up to a density of 1 in the large k limit, where the density is the ratio of projectors to qubits. Using a hybrid approach building on work by Laumann et al. we greatly extend the known satisfiable region for random k-QSAT to a density of $\Omega(2^k/k^2)$. Since our tool allows us to show the existence of joint satisfying states without the need to construct them, we are able to penetrate into regions where the satisfying states are conjectured to be entangled, avoiding the need to construct them, which has limited previous approaches to product states.Comment: 19 page

Zooming in on local level statistics by supersymmetric extension of free probability

We consider unitary ensembles of Hermitian NxN matrices H with a confining potential NV where V is analytic and uniformly convex. From work by Zinn-Justin, Collins, and Guionnet and Maida it is known that the large-N limit of the characteristic function for a finite-rank Fourier variable K is determined by the Voiculescu R-transform, a key object in free probability theory. Going beyond these results, we argue that the same holds true when the finite-rank operator K has the form that is required by the Wegner-Efetov supersymmetry method of integration over commuting and anti-commuting variables. This insight leads to a potent new technique for the study of local statistics, e.g., level correlations. We illustrate the new technique by demonstrating universality in a random matrix model of stochastic scattering.Comment: 38 pages, 3 figures, published version, minor changes in Section

Using graph concepts to understand the organization of complex systems

Complex networks are universal, arising in fields as disparate as sociology, physics, and biology. In the past decade, extensive research into the properties and behaviors of complex systems has uncovered surprising commonalities among the topologies of different systems. Attempts to explain these similarities have led to the ongoing development and refinement of network models and graph-theoretical analysis techniques with which to characterize and understand complexity. In this tutorial, we demonstrate through illustrative examples, how network measures and models have contributed to the elucidation of the organization of complex systems.Comment: v(1) 38 pages, 7 figures, to appear in the International Journal of Bifurcation and Chaos v(2) Line spacing changed; now 23 pages, 7 figures, to appear in the Special Issue "Complex Networks' Structure and Dynamics'' of the International Journal of Bifurcation and Chaos (Volume 17, Issue 7, July 2007) edited by S. Boccaletti and V. Lator

A Hamiltonian approach for explosive percolation

We introduce a cluster growth process that provides a clear connection between equilibrium statistical mechanics and an explosive percolation model similar to the one recently proposed by Achlioptas et al. [Science 323, 1453 (2009)]. We show that the following two ingredients are essential for obtaining an abrupt (first-order) transition in the fraction of the system occupied by the largest cluster: (i) the size of all growing clusters should be kept approximately the same, and (ii) the inclusion of merging bonds (i.e., bonds connecting vertices in different clusters) should dominate with respect to the redundant bonds (i.e., bonds connecting vertices in the same cluster). Moreover, in the extreme limit where only merging bonds are present, a complete enumeration scheme based on tree-like graphs can be used to obtain an exact solution of our model that displays a first-order transition. Finally, the proposed mechanism can be viewed as a generalization of standard percolation that discloses an entirely new family of models with potential application in growth and fragmentation processes of real network systems.Comment: 4 pages, 4 figure

Synchronization in Complex Systems Following the Decision Based Queuing Process: The Rhythmic Applause as a Test Case

Living communities can be considered as complex systems, thus a fertile ground for studies related to their statistics and dynamics. In this study we revisit the case of the rhythmic applause by utilizing the model proposed by V\'azquez et al. [A. V\'azquez et al., Phys. Rev. E 73, 036127 (2006)] augmented with two contradicted {\it driving forces}, namely: {\it Individuality} and {\it Companionship}. To that extend, after performing computer simulations with a large number of oscillators we propose an explanation on the following open questions (a) why synchronization occurs suddenly, and b) why synchronization is observed when the clapping period ($T_c$) is $1.5 \cdot T_s < T_c < 2.0 \cdot T_s$ ($T_s$ is the mean self period of the spectators) and is lost after a time. Moreover, based on the model, a weak preferential attachment principle is proposed which can produce complex networks obeying power law in the distribution of number edges per node with exponent greater than 3.Comment: 16 pages, 5 figure

Topology and Computational Performance of Attractor Neural Networks

To explore the relation between network structure and function, we studied the computational performance of Hopfield-type attractor neural nets with regular lattice, random, small-world and scale-free topologies. The random net is the most efficient for storage and retrieval of patterns by the entire network. However, in the scale-free case retrieval errors are not distributed uniformly: the portion of a pattern encoded by the subset of highly connected nodes is more robust and efficiently recognized than the rest of the pattern. The scale-free network thus achieves a very strong partial recognition. Implications for brain function and social dynamics are suggestive.Comment: 2 figures included. Submitted to Phys. Rev. Letter