Addressing quantum's "fine print" with efficient state preparation and information extraction for quantum algorithms and geologic fracture networks.

Quantum algorithms provide an exponential speedup for solving certain classes of linear systems, including those that model geologic fracture flow. However, this revolutionary gain in efficiency does not come without difficulty. Quantum algorithms require that problems satisfy not only algorithm-specific constraints, but also application-specific ones. Otherwise, the quantum advantage carefully attained through algorithmic ingenuity can be entirely negated. Previous work addressing quantum algorithms for geologic fracture flow has illustrated core algorithmic approaches while incrementally removing assumptions. This work addresses two further requirements for solving geologic fracture flow systems with quantum algorithms: efficient system state preparation and efficient information extraction. Our approach to addressing each is consistent with an overall exponential speed-up.

Clique densification in networks.

Real-world networks are rarely static. Recently, there has been increasing interest in both network growth and network densification, in which the number of edges scales superlinearly with the number of nodes. Less studied but equally important, however, are scaling laws of higher-order cliques, which can drive clustering and network redundancy. In this paper, we study how cliques grow with network size, by analyzing several empirical networks from emails to Wikipedia interactions. Our results show superlinear scaling laws whose exponents increase with clique size, in contrast to predictions from a previous model. We then show that these results are in qualitative agreement with a model that we propose, the local preferential attachment model, where an incoming node links not only to a target node, but also to its higher-degree neighbors. Our results provide insights into how networks grow and where network redundancy occurs.

The transsortative structure of networks.

Network topologies can be highly non-trivial, due to the complex underlying behaviours that form them. While past research has shown that some processes on networks may be characterized by local statistics describing nodes and their neighbours, such as degree assortativity, these quantities fail to capture important sources of variation in network structure. We define a property called transsortativity that describes correlations among a node's neighbours. Transsortativity can be systematically varied, independently of the network's degree distribution and assortativity. Moreover, it can significantly impact the spread of contagions as well as the perceptions of neighbours, known as the majority illusion. Our work improves our ability to create and analyse more realistic models of complex networks.

Author Correction: Neighbor-Neighbor Correlations Explain Measurement Bias in Networks.

A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has not been fixed in the paper.

Degree correlations amplify the growth of cascades in networks.

Networks facilitate the spread of cascades, allowing a local perturbation to percolate via interactions between nodes and their neighbors. We investigate how network structure affects the dynamics of a spreading cascade. By accounting for the joint degree distribution of a network within a generating function framework, we can quantify how degree correlations affect both the onset of global cascades and the propensity of nodes of specific degree class to trigger large cascades. However, not all degree correlations are equally important in a spreading process. We introduce a new measure of degree assortativity that accounts for correlations among nodes relevant to a spreading cascade. We show that the critical point defining the onset of global cascades has a monotone relationship to this new assortativity measure. In addition, we show that the choice of nodes to seed the largest cascades is strongly affected by degree correlations. Contrary to traditional wisdom, when degree assortativity is positive, low degree nodes are more likely to generate largest cascades. Our work suggests that it may be possible to tailor spreading processes by manipulating the higher-order structure of networks.

Neighbor-Neighbor Correlations Explain Measurement Bias in Networks.

In numerous physical models on networks, dynamics are based on interactions that exclusively involve properties of a node's nearest neighbors. However, a node's local view of its neighbors may systematically bias perceptions of network connectivity or the prevalence of certain traits. We investigate the strong friendship paradox, which occurs when the majority of a node's neighbors have more neighbors than does the node itself. We develop a model to predict the magnitude of the paradox, showing that it is enhanced by negative correlations between degrees of neighboring nodes. We then show that by including neighbor-neighbor correlations, which are degree correlations one step beyond those of neighboring nodes, we accurately predict the impact of the strong friendship paradox in real-world networks. Understanding how the paradox biases local observations can inform better measurements of network structure and our understanding of collective phenomena.

Multiclass Data Segmentation Using Diffuse Interface Methods on Graphs.

We present two graph-based algorithms for multiclass segmentation of high-dimensional data on graphs. The algorithms use a diffuse interface model based on the Ginzburg-Landau functional, related to total variation and graph cuts. A multiclass extension is introduced using the Gibbs simplex, with the functional's double-well potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm uses a graph adaptation of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates between diffusion and thresholding. We demonstrate the performance of both algorithms experimentally on synthetic data, image labeling, and several benchmark data sets such as MNIST, COIL and WebKB. We also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments indicate that the results are competitive with or better than the current state-of-the-art in multiclass graph-based segmentation algorithms for high-dimensional data.

Spectral clustering with epidemic diffusion.

Spectral clustering is widely used to partition graphs into distinct modules or communities. Existing methods for spectral clustering use the eigenvalues and eigenvectors of the graph Laplacian, an operator that is closely associated with random walks on graphs. We propose a spectral partitioning method that exploits the properties of epidemic diffusion. An epidemic is a dynamic process that, unlike the random walk, simultaneously transitions to all the neighbors of a given node. We show that the replicator, an operator describing epidemic diffusion, is equivalent to the symmetric normalized Laplacian of a reweighted graph with edges reweighted by the eigenvector centralities of their incident nodes. Thus, more weight is given to edges connecting more central nodes. We describe a method that partitions the nodes based on the componentwise ratio of the replicator's second eigenvector to the first and compare its performance to traditional spectral clustering techniques on synthetic graphs with known community structure. We demonstrate that the replicator gives preference to dense, clique-like structures, enabling it to more effectively discover communities that may be obscured by dense intercommunity linking.

Extremal optimization at the phase transition of the three-coloring problem.

We investigate the phase transition in vertex coloring on random graphs, using the extremal optimization heuristic. Three-coloring is among the hardest combinatorial optimization problems and is equivalent to a 3-state anti-ferromagnetic Potts model. Like many other such optimization problems, it has been shown to exhibit a phase transition in its ground state behavior under variation of a system parameter: the graph's mean vertex degree. This phase transition is often associated with the instances of highest complexity. We use extremal optimization to measure the ground state cost and the "backbone," an order parameter related to ground state overlap, averaged over a large number of instances near the transition for random graphs of size n up to 512. For these graphs, benchmarks show that extremal optimization reaches ground states and explores a sufficient number of them to give the correct backbone value after about O (n(3.5)) update steps. Finite size scaling yields a critical mean degree value alpha(c) =4.703 (28). Furthermore, the exploration of the degenerate ground states indicates that the backbone order parameter, measuring the constrainedness of the problem, exhibits a first-order phase transition.

Scaling and universality in continuous length combinatorial optimization.

We consider combinatorial optimization problems defined over random ensembles and study how solution cost increases when the optimal solution undergoes a small perturbation delta. For the minimum spanning tree, the increase in cost scales as delta2. For the minimum matching and traveling salesman problems in dimension d >/= 2, the increase scales as delta3; this is observed in Monte Carlo simulations in d = 2, 3, 4 and in theoretical analysis of a mean-field model. We speculate that the scaling exponent could serve to classify combinatorial optimization problems of this general kind into a small number of distinct categories, similar to universality classes in statistical physics.