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Register a new userStructure Amplification on Multi-layer Stochastic Block Models
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Much of the complexity of social, biological, and engineered systems arises from a network of complex interactions connecting many basic components. Network analysis tools have been successful at uncovering latent structure termed communities in such networks. However, some of the most interesting structure can be difficult to uncover because it is obscured by the more dominant structure. Our previous work proposes a general structure amplification technique called HICODE that uncovers many layers of functional hidden structure in complex networks. HICODE incrementally weakens dominant structure through randomization allowing the hidden functionality to emerge, and uncovers these hidden structure in real-world networks that previous methods rarely uncover. In this work, we conduct a comprehensive and systematic theoretical analysis on the hidden community structure. In what follows, we define multi-layer stochastic block model, and provide theoretical support using the model on why the existence of hidden structure will make the detection of dominant structure harder compared with equivalent random noise. We then provide theoretical proofs that the iterative reducing methods could help promote the uncovering of hidden structure as well as boosting the detection quality of dominant structure.
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We provide the first information theoretic tight analysis for inference of latent community structure given a sparse graph along with high dimensional node covariates, correlated with the same latent communities. Our work bridges recent theoretical breakthroughs in the detection of latent community structure without nodes covariates and a large body of empirical work using diverse heuristics for combining node covariates with graphs for inference. The tightness of our analysis implies in particular, the information theoretical necessity of combining the different sources of information. Our analysis holds for networks of large degrees as well as for a Gaussian version of the model.
We consider the problem of estimating common community structures in multi-layer stochastic block models, where each single layer may not have sufficient signal strength to recover the full community structure. In order to efficiently aggregate signal across different layers, we argue that the sum-of-squared adjacency matrices contains sufficient signal even when individual layers are very sparse. Our method features a bias-removal step that is necessary when the squared noise matrices may overwhelm the signal in the very sparse regime. The analysis of our method uses several novel tail probability bounds for matrix linear combinations with matrix-valued coefficients and matrix-valued quadratic forms, which may be of independent interest. The performance of our method and the necessity of bias removal is demonstrated in synthetic data and in microarray analysis about gene co-expression networks.
Hidden community is a new graph-theoretical concept recently proposed [4], in which the authors also propose a meta-approach called HICODE (Hidden Community Detection) for detecting hidden communities. HICODE is demonstrated through experiments that it is able to uncover previously overshadowed weak layers and uncover both weak and strong layers at a higher accuracy. However, the authors provide no theoretical guarantee for the performance. In this work, we focus on the theoretical analysis of HICODE on synthetic two-layer networks, where layers are independent of each other and each layer is generated by stochastic block model. We bridge their gap through two-layer stochastic block model networks in the following aspects: 1) we show that partitions that locally optimize modularity correspond to grounded layers, indicating modularity-optimizing algorithms can detect strong layers; 2) we prove that when reducing found layers, HICODE increases absolute modularities of all unreduced layers, showing its layer reduction step makes weak layers more detectable. Our work builds a solid theoretical base for HICODE, demonstrating that it is promising in uncovering both weak and strong layers of communities in two-layer networks.
With ever-increasing amounts of online information available, modeling and predicting individual preferences-for books or articles, for example-is becoming more and more important. Good predictions enable us to improve advice to users, and obtain a better understanding of the socio-psychological processes that determine those preferences. We have developed a collaborative filtering model, with an associated scalable algorithm, that makes accurate predictions of individuals preferences. Our approach is based on the explicit assumption that there are groups of individuals and of items, and that the preferences of an individual for an item are determined only by their group memberships. Importantly, we allow each individual and each item to belong simultaneously to mixtures of different groups and, unlike many popular approaches, such as matrix factorization, we do not assume implicitly or explicitly that individuals in each group prefer items in a single group of items. The resulting overlapping groups and the predicted preferences can be inferred with a expectation-maximization algorithm whose running time scales linearly (per iteration). Our approach enables us to predict individual preferences in large datasets, and is considerably more accurate than the current algorithms for such large datasets.
Methods for ranking the importance of nodes in a network have a rich history in machine learning and across domains that analyze structured data. Recent work has evaluated these methods though the seed set expansion problem: given a subset $S$ of nodes from a community of interest in an underlying graph, can we reliably identify the rest of the community? We start from the observation that the most widely used techniques for this problem, personalized PageRank and heat kernel methods, operate in the space of landing probabilities of a random walk rooted at the seed set, ranking nodes according to weighted sums of landing probabilities of different length walks. Both schemes, however, lack an a priori relationship to the seed set objective. In this work we develop a principled framework for evaluating ranking methods by studying seed set expansion applied to the stochastic block model. We derive the optimal gradient for separating the landing probabilities of two classes in a stochastic block model, and find, surprisingly, that under reasonable assumptions the gradient is asymptotically equivalent to personalized PageRank for a specific choice of the PageRank parameter $alpha$ that depends on the block model parameters. This connection provides a novel formal motivation for the success of personalized PageRank in seed set expansion and node ranking generally. We use this connection to propose more advanced techniques incorporating higher moments of landing probabilities; our advanced methods exhibit greatly improved performance despite being simple linear classification rules, and are even competitive with belief propagation.
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