Pan Zhang

Spectral redemption in clustering sparse networks

Significance Spectral algorithms are widely applied to data clustering problems, including finding communities or partitions in graphs and networks. We propose a way of encoding sparse data using a “nonbacktracking” matrix, and show that the corresponding spectral algorithm performs optimally for some popular generative models, including the stochastic block model. This is in contrast with classical spectral algorithms, based on the adjacency matrix, random walk matrix, and graph Laplacian, which perform poorly in the sparse case, failing significantly above a recently discovered phase transition for the detectability of communities. Further support for the method is provided by experiments on real networks as well as by theoretical arguments and analogies from probability theory, statistical physics, and the theory of random matrices. Spectral algorithms are classic approaches to clustering and community detection in networks. However, for sparse networks the standard versions of these algorithms are suboptimal, in some cases completely failing to detect communities even when other algorithms such as belief propagation can do so. Here, we present a class of spectral algorithms based on a nonbacktracking walk on the directed edges of the graph. The spectrum of this operator is much better-behaved than that of the adjacency matrix or other commonly used matrices, maintaining a strong separation between the bulk eigenvalues and the eigenvalues relevant to community structure even in the sparse case. We show that our algorithm is optimal for graphs generated by the stochastic block model, detecting communities all of the way down to the theoretical limit. We also show the spectrum of the nonbacktracking operator for some real-world networks, illustrating its advantages over traditional spectral clustering.

Ultrahigh compression of water using intense heavy ion beams: laboratory planetary physics

Intense heavy ion beams offer a unique tool for generating samples of high energy density matter with extreme conditions of density and pressure that are believed to exist in the interiors of giant planets. An international accelerator facility named FAIR (Facility for Antiprotons and Ion Research) is being constructed at Darmstadt, which will be completed around the year 2015. It is expected that this accelerator facility will deliver a bunched uranium beam with an intensity of 5x10(11) ions per spill with a bunch length of 50-100 ns. An experiment named LAPLAS (Laboratory Planetary Sciences) has been proposed to achieve a low-entropy compression of a sample material like hydrogen or water (which are believed to be abundant in giant planets) that is imploded in a multi-layered target by the ion beam. Detailed numerical simulations have shown that using parameters of the heavy ion beam that will be available at FAIR, one can generate physical conditions that have been predicted to exist in the interior of giant planets. In the present paper, we report simulations of compression of water that show that one can generate a plasma phase as well as a superionic phase of water in the LAPLAS experiments.

Model selection for degree-corrected block models

The proliferation of models for networks raises challenging problems of model selection: the data are sparse and globally dependent, and models are typically high-dimensional and have large numbers of latent variables. Together, these issues mean that the usual model-selection criteria do not work properly for networks. We illustrate these challenges, and show one way to resolve them, by considering the key network-analysis problem of dividing a graph into communities or blocks of nodes with homogeneous patterns of links to the rest of the network. The standard tool for undertaking this is the stochastic block model, under which the probability of a link between two nodes is a function solely of the blocks to which they belong. This imposes a homogeneous degree distribution within each block; this can be unrealistic, so degree-corrected block models add a parameter for each node, modulating its overall degree. The choice between ordinary and degree-corrected block models matters because they make very different inferences about communities. We present the first principled and tractable approach to model selection between standard and degree-corrected block models, based on new large-graph asymptotics for the distribution of log-likelihood ratios under the stochastic block model, finding substantial departures from classical results for sparse graphs. We also develop linear-time approximations for log-likelihoods under both the stochastic block model and the degree-corrected model, using belief propagation. Applications to simulated and real networks show excellent agreement with our approximations. Our results thus both solve the practical problem of deciding on degree correction and point to a general approach to model selection in network analysis.

Scalable detection of statistically significant communities and hierarchies, using message passing for modularity

Significance Most work on community detection does not address the issue of statistical significance, and many algorithms are prone to overfitting. We address this using tools from statistical physics. Rather than trying to find the partition of a network that maximizes the modularity, our approach seeks the consensus of many high-modularity partitions. We do this with a scalable message-passing algorithm, derived by treating the modularity as a Hamiltonian and applying the cavity method. We show analytically that our algorithm succeeds all the way down to the detectability transition in the stochastic block model; it also performs well on real-world networks. It also provides a principled method for determining the number of groups or hierarchies of communities and subcommunities. Modularity is a popular measure of community structure. However, maximizing the modularity can lead to many competing partitions, with almost the same modularity, that are poorly correlated with each other. It can also produce illusory ‘‘communities’’ in random graphs where none exist. We address this problem by using the modularity as a Hamiltonian at finite temperature and using an efficient belief propagation algorithm to obtain the consensus of many partitions with high modularity, rather than looking for a single partition that maximizes it. We show analytically and numerically that the proposed algorithm works all of the way down to the detectability transition in networks generated by the stochastic block model. It also performs well on real-world networks, revealing large communities in some networks where previous work has claimed no communities exist. Finally we show that by applying our algorithm recursively, subdividing communities until no statistically significant subcommunities can be found, we can detect hierarchical structure in real-world networks more efficiently than previous methods.

Optimized annealing of traveling salesman problem from the nth-nearest-neighbor distribution

We report a new statistical general property in traveling salesman problem, that the nth-nearest-neighbor distribution of optimal tours verifies with very high accuracy an exponential decay as a function of the order of neighbor n. Defining the energy function as deviation λ from this exponential decay, which is different to the tour length d in normal annealing processes, we propose a distinct highly optimized annealing scheme which is performed in λ-space and d-space by turns. The simulation results of some standard traveling salesman problems in TSPLIB95 are presented. It is shown that our annealing recipe is superior to the canonical simulated annealing.

Detectability Thresholds and Optimal Algorithms for Community Structure in Dynamic Networks

We study the fundamental limits on learning latent community structure in dynamic networks. Specifically, we study dynamic stochastic block models where nodes change their community membership over time, but where edges are generated independently at each time step. In this setting (which is a special case of several existing models), we are able to derive the detectability threshold exactly, as a function of the rate of change and the strength of the communities. Below this threshold, we claim that no algorithm can identify the communities better than chance. We then give two algorithms that are optimal in the sense that they succeed all the way down to this limit. The first uses belief propagation (BP), which gives asymptotically optimal accuracy, and the second is a fast spectral clustering algorithm, based on linearizing the BP equations. We verify our analytic and algorithmic results via numerical simulation, and close with a brief discussion of extensions and open questions.

Community detection in networks with unequal groups

Recently, a phase transition has been discovered in the network community detection problem below which no algorithm can tell which nodes belong to which communities with success any better than a random guess. This result has, however, so far been limited to the case where the communities have the same size or the same average degree. Here we consider the case where the sizes or average degrees differ. This asymmetry allows us to assign nodes to communities with better-than-random success by examining their local neighborhoods. Using the cavity method, we show that this removes the detectability transition completely for networks with four groups or fewer, while for more than four groups the transition persists up to a critical amount of asymmetry but not beyond. The critical point in the latter case coincides with the point at which local information percolates, causing a global transition from a less-accurate solution to a more-accurate one.

Comparative study for inference of hidden classes in stochastic block models

Inference of hidden classes in stochastic block models is a classical problem with important applications. Most commonly used methods for this problem involve naive mean field approaches or heuristic spectral methods. Recently, belief propagation was proposed for this problem. In this contribution we perform a comparative study between the three methods on synthetically created networks. We show that belief propagation shows much better performance when compared to naive mean field and spectral approaches. This applies to accuracy, computational efficiency and the tendency to overfit the data.

Frequency and phase synchronization of two coupled neurons with channel noise

Abstract.We study the frequency and phase synchronization in two coupled identical and nonidentical neurons with channel noise. The occupation number method is used to model the neurons in the context of stochastic Hodgkin-Huxley model in which the strength of channel noise is represented by ion channel cluster size of neurons. It is shown that channel noise allows the two neurons to achieve both frequency and phase synchronization in the regime where the deterministic Hodgkin-Huxley neuron is unable to be excited. In particular, the identical channel noises lead to frequency synchronization in weak-coupling regime. However, if the coupling is strong, the two neurons could be frequency locked even though the channel noises are not identical. We also show that the relative phase of neurons displays profuse dynamical regimes under the combined action of coupling and channel noise. Those regimes are characterized by the distribution of the cyclic relative phase corresponding to antiphase locking, random switching between two or more states. Both qualitative and quantitative descriptions are applied to describe the transitions to perfect phase locking from no synchronization states.

Fast and simple decycling and dismantling of networks

Decycling and dismantling of complex networks are underlying many important applications in network science. Recently these two closely related problems were tackled by several heuristic algorithms, simple and considerably sub-optimal, on the one hand, and involved and accurate message-passing ones that evaluate single-node marginal probabilities, on the other hand. In this paper we propose a simple and extremely fast algorithm, CoreHD, which recursively removes nodes of the highest degree from the 2-core of the network. CoreHD performs much better than all existing simple algorithms. When applied on real-world networks, it achieves equally good solutions as those obtained by the state-of-art iterative message-passing algorithms at greatly reduced computational cost, suggesting that CoreHD should be the algorithm of choice for many practical purposes.