Trilogy on Computing Maximal Eigenpair

The eigenpair here means the twins consist of eigenvalue and its eigenvector. This paper introduces the three steps of our study on computing the maximal eigenpair. In the first two steps, we construct efficient initials for a known but dangerous algorithm, first for tridiagonal matrices and then for irreducible matrices, having nonnegative off-diagonal elements. In the third step, we present two global algorithms which are still efficient and work well for a quite large class of matrices, even complex for instance.Comment: Updated versio

Reducing the Effects of Unequal Number of Games on Rankings

Ranking is an important mathematical process in a variety of contexts such as information retrieval, sports and business. Sports ranking methods can be applied both in and beyond the context of athletics. In both settings, once the concept of a game has been defined, teams (or individuals) accumulate wins, losses, and ties, which are then factored into the ranking computation. Many settings involve an unequal number of games between competitors. This paper demonstrates how to adapt two sports rankings methods, the Colley and Massey ranking methods, to settings where an unequal number of games are played between the teams. In such settings, the standard derivations of the methods can produce nonsensical rankings. This paper introduces the idea of including a super-user into the rankings and considers the effect of this fictitious player on the ratings. We apply such techniques to rank batters and pitchers in Major League baseball, professional tennis players, and participants in a free online social game. The ideas introduced in this paper can further the scope that such methods are applied and the depth of insight they offer

Thermodynamic formalism for dissipative quantum walks

We consider the dynamical properties of dissipative continuous-time quantum walks on directed graphs. Using a large-deviation approach we construct a thermodynamic formalism allowing us to define a dynamical order parameter, and to identify transitions between dynamical regimes. For a particular class of dissipative quantum walks we propose a quantum generalization of the the classical PageRank vector, used to rank the importance of nodes in a directed graph. We also provide an example where one can characterize the dynamical transition from an effective classical random walk to a dissipative quantum walk as a thermodynamic crossover between distinct dynamical regimes.Comment: 8 page

Influence, originality and similarity in directed acyclic graphs

We introduce a framework for network analysis based on random walks on directed acyclic graphs where the probability of passing through a given node is the key ingredient. We illustrate its use in evaluating the mutual influence of nodes and discovering seminal papers in a citation network. We further introduce a new similarity metric and test it in a simple personalized recommendation process. This metric's performance is comparable to that of classical similarity metrics, thus further supporting the validity of our framework.Comment: 6 pages, 4 figure

PageRank Optimization by Edge Selection

The importance of a node in a directed graph can be measured by its PageRank. The PageRank of a node is used in a number of application contexts - including ranking websites - and can be interpreted as the average portion of time spent at the node by an infinite random walk. We consider the problem of maximizing the PageRank of a node by selecting some of the edges from a set of edges that are under our control. By applying results from Markov decision theory, we show that an optimal solution to this problem can be found in polynomial time. Our core solution results in a linear programming formulation, but we also provide an alternative greedy algorithm, a variant of policy iteration, which runs in polynomial time, as well. Finally, we show that, under the slight modification for which we are given mutually exclusive pairs of edges, the problem of PageRank optimization becomes NP-hard.Comment: 30 pages, 3 figure

Adiabatic quantum algorithm for search engine ranking

We propose an adiabatic quantum algorithm for generating a quantum pure state encoding of the PageRank vector, the most widely used tool in ranking the relative importance of internet pages. We present extensive numerical simulations which provide evidence that this algorithm can prepare the quantum PageRank state in a time which, on average, scales polylogarithmically in the number of webpages. We argue that the main topological feature of the underlying web graph allowing for such a scaling is the out-degree distribution. The top ranked $\log(n)$ entries of the quantum PageRank state can then be estimated with a polynomial quantum speedup. Moreover, the quantum PageRank state can be used in "q-sampling" protocols for testing properties of distributions, which require exponentially fewer measurements than all classical schemes designed for the same task. This can be used to decide whether to run a classical update of the PageRank.Comment: 7 pages, 5 figures; closer to published versio

Schroedinger-like PageRank equation and localization in the WWW

The WorldWide Web is one of the most important communication systems we use in our everyday life. Despite its central role, the growth and the development of the WWW is not controlled by any central authority. This situation has created a huge ensemble of connections whose complexity can be fruitfully described and quantified by network theory. One important application that allows to sort out the information present in these connections is given by the PageRank alghorithm. Computation of this quantity is usually made iteratively with a large use of computational time. In this paper we show that the PageRank can be expressed in terms of a wave function obeying a Schroedinger-like equation. In particular the topological disorder given by the unbalance of outgoing and ingoing links between pages, induces wave function and potential structuring. This allows to directly localize the pages with the largest score. Through this new representation we can now compute the PageRank without iterative techniques. For most of the cases of interest our method is faster than the original one. Our results also clarify the role of topology in the diffusion of information within complex networks. The whole approach opens the possibility to novel techniques inspired by quantum physics for the analysis of the WWW properties.Comment: 5 page

Nonlinear Diffusion Through Large Complex Networks Containing Regular Subgraphs

Transport through generalized trees is considered. Trees contain the simple nodes and supernodes, either well-structured regular subgraphs or those with many triangles. We observe a superdiffusion for the highly connected nodes while it is Brownian for the rest of the nodes. Transport within a supernode is affected by the finite size effects vanishing as $N\to\infty.$ For the even dimensions of space, $d=2,4,6,...$, the finite size effects break down the perturbation theory at small scales and can be regularized by using the heat-kernel expansion.Comment: 21 pages, 2 figures include

Googling the brain: discovering hierarchical and asymmetric network structures, with applications in neuroscience

Hierarchical organisation is a common feature of many directed networks arising in nature and technology. For example, a well-defined message-passing framework based on managerial status typically exists in a business organisation. However, in many real-world networks such patterns of hierarchy are unlikely to be quite so transparent. Due to the nature in which empirical data is collated the nodes will often be ordered so as to obscure any underlying structure. In addition, the possibility of even a small number of links violating any overall “chain of command” makes the determination of such structures extremely challenging. Here we address the issue of how to reorder a directed network in order to reveal this type of hierarchy. In doing so we also look at the task of quantifying the level of hierarchy, given a particular node ordering. We look at a variety of approaches. Using ideas from the graph Laplacian literature, we show that a relevant discrete optimization problem leads to a natural hierarchical node ranking. We also show that this ranking arises via a maximum likelihood problem associated with a new range-dependent hierarchical random graph model. This random graph insight allows us to compute a likelihood ratio that quantifies the overall tendency for a given network to be hierarchical. We also develop a generalization of this node ordering algorithm based on the combinatorics of directed walks. In passing, we note that Google’s PageRank algorithm tackles a closely related problem, and may also be motivated from a combinatoric, walk-counting viewpoint. We illustrate the performance of the resulting algorithms on synthetic network data, and on a real-world network from neuroscience where results may be validated biologically