Most network data are collected from only partially observable networks with both missing nodes and edges, for example due to limited resources and privacy settings specified by users on social media. Thus, it stands to the reason that inferring the missing parts of the networks by performing completion should precede downstream mining or learning tasks on the networks. However, despite this need, the recovery of missing nodes and edges in such incomplete networks is an insufficiently explored problem. In this paper, we present DeepNC, a novel method for inferring the missing parts of a network that is based on a deep generative graph model. Specifically, our model first learns a likelihood over edges via a recurrent neural network (RNN)-based generative graph, and then identifies the graph that maximizes the learned likelihood conditioned on the observable graph topology. Moreover, we propose a computationally efficient DeepNC algorithm that consecutively finds a single node to maximize the probability in each node generation step, whose runtime complexity is almost linear in the number of nodes in the network. We empirically show the superiority of DeepNC over state-of-the-art network completion approaches on a variety of synthetic and real-world networks.
Network completion. Observing a partial sample of a network and inferring the remainder of the network is referred to as network completion . As the most influential study, KronEM, an approach based on Kronecker graphs to solving the network completion problem by applying the expectation-maximization (EM) algorithm, was suggested by Kim and Leskovec @cite_3 . For cases in which only a small number of edges are missing, vertex similarity @cite_5 was shown to be useful in recovering the underlying true network. Another method for inferring missing edges in social networks based on shared node neighborhoods was investigated by @cite_18 . MISC was developed to tackle the missing node identification problem when the information of connections between missing nodes and observable nodes is assumed to be available @cite_11 . A follow-up study of MISC @cite_7 attempted to incorporate side information such as the demographic information and the nodes’ historical behavior into the inference process.
@cite_18 @cite_7 @cite_3 @cite_5 @cite_11
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In recent years, social networks have surged in popularity. One key aspect of social network research is identifying important missing information that is not explicitly represented in the network, or is not visible to all. To date, this line of research typically focused on finding the connections that are missing between nodes, a challenge typically termed as the link prediction problem. This article introduces the missing node identification problem, where missing members in the social network structure must be identified. In this problem, indications of missing nodes are assumed to exist. Given these indications and a partial network, we must assess which indications originate from the same missing node and determine the full network structure. Toward solving this problem, we present the missing node identification by spectral clustering algorithm (MISC), an approach based on a spectral clustering algorithm, combined with nodes’ pairwise affinity measures that were adopted from link prediction research. We evaluate the performance of our approach in different problem settings and scenarios, using real-life data from Facebook. The results show that our approach has beneficial results and can be effective in solving the missing node identification problem. In addition, this article also presents R-MISC, which uses a sparse matrix representation, efficient algorithms for calculating the nodes’ pairwise affinity, and a proprietary dimension reduction technique to enable scaling the MISC algorithm to large networks of more than 100,000 nodes. Last, we consider problem settings where some of the indications are unknown. Two algorithms are suggested for this problem: speculative MISC, based on MISC, and missing link completion, based on classical link prediction literature. We show that speculative MISC outperforms missing link completion. In this paper, we consider the matrix completion problem when the observations are one-bit measurements of some underlying matrix M, and in particular the observed samples consist only of ones and no zeros. This problem is motivated by modern applications such as recommender systems and social networks where only "likes" or "friendships" are observed. The problem of learning from only positive and unlabeled examples, called PU (positive-unlabeled) learning, has been studied in the context of binary classification. We consider the PU matrix completion problem, where an underlying real-valued matrix M is first quantized to generate one-bit observations and then a subset of positive entries is revealed. Under the assumption that M has bounded nuclear norm, we provide recovery guarantees for two different observation models: 1) M parameterizes a distribution that generates a binary matrix, 2) M is thresholded to obtain a binary matrix. For the first case, we propose a "shifted matrix completion" method that recovers M using only a subset of indices corresponding to ones, while for the second case, we propose a "biased matrix completion" method that recovers the (thresholded) binary matrix. Both methods yield strong error bounds --- if M is n by n, the Frobenius error is bounded as O(1 ((1-rho)n), where 1-rho denotes the fraction of ones observed. This implies a sample complexity of O(n n) ones to achieve a small error, when M is dense and n is large. We extend our methods and guarantees to the inductive matrix completion problem, where rows and columns of M have associated features. We provide efficient and scalable optimization procedures for both the methods and demonstrate the effectiveness of the proposed methods for link prediction (on real-world networks consisting of over 2 million nodes and 90 million links) and semi-supervised clustering tasks. Network structures, such as social networks, web graphs and networks from systems biology, play important roles in many areas of science and our everyday lives. In order to study the networks one needs to first collect reliable large scale network data. While the social and information networks have become ubiquitous, the challenge of collecting complete network data still persists. Many times the collected network data is incomplete with nodes and edges missing. Commonly, only a part of the network can be observed and we would like to infer the unobserved part of the network. We address this issue by studying the Network Completion Problem: Given a network with missing nodes and edges, can we complete the missing part? We cast the problem in the Expectation Maximization (EM) framework where we use the observed part of the network to fit a model of network structure, and then we estimate the missing part of the network using the model, re-estimate the parameters and so on. We combine the EM with the Kronecker graphs model and design a scalable Metropolized Gibbs sampling approach that allows for the estimation of the model parameters as well as the inference about missing nodes and edges of the network. Experiments on synthetic and several real-world networks show that our approach can effectively recover the network even when about half of the nodes in the network are missing. Our algorithm outperforms not only classical link-prediction approaches but also the state of the art Stochastic block modeling approach. Furthermore, our algorithm easily scales to networks with tens of thousands of nodes. The discovery of networks is a fundamental problem arising in numerous fields of science and technology, including communication systems, biology, sociology, and neuroscience. Unfortunately, it is often difficult, or impossible, to obtain data that directly reveal network structure, and so one must infer a network from incomplete data. This paper considers inferring network structure from "co-occurrence" data: observations that identify which network components (e.g., switches, routers, genes) carry each transmission but do not indicate the order in which they handle the transmission. Without order information, the number of networks that are consistent with the data grows exponentially with the size of the network (i.e., the number of nodes). Yet, the basic engineering evolutionary principles underlying most networks strongly suggest that not all data-consistent networks are equally likely. In particular, nodes that co-occur in many observations are probably closely connected. With this in mind, we model the co-occurrence observations as independent realizations of a random walk on the network, subjected to a random permutation to account for the lack of order information. Treating permutations as missing data, we derive an expectation-maximization (EM) algorithm for estimating the random walk parameters. The model and EM algorithm significantly simplify the problem, but the computational complexity of the reconstruction process does grow exponentially in the length of each transmission path. For networks with long paths, the exact e-step may be computationally intractable. We propose a polynomial-time Monte Carlo EM algorithm based on importance sampling and derive conditions that ensure convergence of the algorithm with high probability. Simulations and experiments with Internet measurements demonstrate the promise of this approach.