Bias-Variance Tradeoff of Graph Laplacian Regularizer

This paper presents a bias-variance tradeoff of graph Laplacian regularizer, which is widely used in graph signal processing and semi-supervised learning tasks. The scaling law of the optimal regularization parameter is specified in terms of the spectral graph properties and a novel signal-to-noise ratio parameter, which suggests selecting a mediocre regularization parameter is often suboptimal. The analysis is applied to three applications, including random, band-limited, and multiple-sampled graph signals. Experiments on synthetic and real-world graphs demonstrate near-optimal performance of the established analysis.Comment: accepted by IEEE Signal Processing Letter

Revisiting Spectral Graph Clustering with Generative Community Models

The methodology of community detection can be divided into two principles: imposing a network model on a given graph, or optimizing a designed objective function. The former provides guarantees on theoretical detectability but falls short when the graph is inconsistent with the underlying model. The latter is model-free but fails to provide quality assurance for the detected communities. In this paper, we propose a novel unified framework to combine the advantages of these two principles. The presented method, SGC-GEN, not only considers the detection error caused by the corresponding model mismatch to a given graph, but also yields a theoretical guarantee on community detectability by analyzing Spectral Graph Clustering (SGC) under GENerative community models (GCMs). SGC-GEN incorporates the predictability on correct community detection with a measure of community fitness to GCMs. It resembles the formulation of supervised learning problems by enabling various community detection loss functions and model mismatch metrics. We further establish a theoretical condition for correct community detection using the normalized graph Laplacian matrix under a GCM, which provides a novel data-driven loss function for SGC-GEN. In addition, we present an effective algorithm to implement SGC-GEN, and show that the computational complexity of SGC-GEN is comparable to the baseline methods. Our experiments on 18 real-world datasets demonstrate that SGC-GEN possesses superior and robust performance compared to 6 baseline methods under 7 representative clustering metrics.Comment: Accepted by IEEE International Conference on Data Mining (ICDM) 2017 as a regular paper - full paper with supplementary materia

Attacking the Madry Defense Model with L1L_1-based Adversarial Examples

The Madry Lab recently hosted a competition designed to test the robustness of their adversarially trained MNIST model. Attacks were constrained to perturb each pixel of the input image by a scaled maximal $L_\infty$ distortion $\epsilon$ = 0.3. This discourages the use of attacks which are not optimized on the $L_\infty$ distortion metric. Our experimental results demonstrate that by relaxing the $L_\infty$ constraint of the competition, the elastic-net attack to deep neural networks (EAD) can generate transferable adversarial examples which, despite their high average $L_\infty$ distortion, have minimal visual distortion. These results call into question the use of $L_\infty$ as a sole measure for visual distortion, and further demonstrate the power of EAD at generating robust adversarial examples.Comment: Accepted to ICLR 2018 Workshop

Multilayer Spectral Graph Clustering via Convex Layer Aggregation: Theory and Algorithms

Multilayer graphs are commonly used for representing different relations between entities and handling heterogeneous data processing tasks. Non-standard multilayer graph clustering methods are needed for assigning clusters to a common multilayer node set and for combining information from each layer. This paper presents a multilayer spectral graph clustering (SGC) framework that performs convex layer aggregation. Under a multilayer signal plus noise model, we provide a phase transition analysis of clustering reliability. Moreover, we use the phase transition criterion to propose a multilayer iterative model order selection algorithm (MIMOSA) for multilayer SGC, which features automated cluster assignment and layer weight adaptation, and provides statistical clustering reliability guarantees. Numerical simulations on synthetic multilayer graphs verify the phase transition analysis, and experiments on real-world multilayer graphs show that MIMOSA is competitive or better than other clustering methods.Comment: Published at IEEE Transactions on Signal and Information Processing over Network

Deep Community Detection

A deep community in a graph is a connected component that can only be seen after removal of nodes or edges from the rest of the graph. This paper formulates the problem of detecting deep communities as multi-stage node removal that maximizes a new centrality measure, called the local Fiedler vector centrality (LFVC), at each stage. The LFVC is associated with the sensitivity of algebraic connectivity to node or edge removals. We prove that a greedy node/edge removal strategy, based on successive maximization of LFVC, has bounded performance loss relative to the optimal, but intractable, combinatorial batch removal strategy. Under a stochastic block model framework, we show that the greedy LFVC strategy can extract deep communities with probability one as the number of observations becomes large. We apply the greedy LFVC strategy to real-world social network datasets. Compared with conventional community detection methods we demonstrate improved ability to identify important communities and key members in the network.Comment: 15 pages, 13 figures, journal submission and supplementary file (Figures 11-13), to appear in IEEE Transactions on Signal Processin

Sequential Defense Against Random and Intentional Attacks in Complex Networks

Network robustness against attacks is one of the most fundamental researches in network science as it is closely associated with the reliability and functionality of various networking paradigms. However, despite the study on intrinsic topological vulnerabilities to node removals, little is known on the network robustness when network defense mechanisms are implemented, especially for networked engineering systems equipped with detection capabilities. In this paper, a sequential defense mechanism is firstly proposed in complex networks for attack inference and vulnerability assessment, where the data fusion center sequentially infers the presence of an attack based on the binary attack status reported from the nodes in the network. The network robustness is evaluated in terms of the ability to identify the attack prior to network disruption under two major attack schemes, i.e., random and intentional attacks. We provide a parametric plug-in model for performance evaluation on the proposed mechanism and validate its effectiveness and reliability via canonical complex network models and real-world large-scale network topology. The results show that the sequential defense mechanism greatly improves the network robustness and mitigates the possibility of network disruption by acquiring limited attack status information from a small subset of nodes in the network.Comment: 13 pages, 14 figure

Phase Transitions and a Model Order Selection Criterion for Spectral Graph Clustering

One of the longstanding open problems in spectral graph clustering (SGC) is the so-called model order selection problem: automated selection of the correct number of clusters. This is equivalent to the problem of finding the number of connected components or communities in an undirected graph. We propose automated model order selection (AMOS), a solution to the SGC model selection problem under a random interconnection model (RIM) using a novel selection criterion that is based on an asymptotic phase transition analysis. AMOS can more generally be applied to discovering hidden block diagonal structure in symmetric non-negative matrices. Numerical experiments on simulated graphs validate the phase transition analysis, and real-world network data is used to validate the performance of the proposed model selection procedure.Comment: Accepted to IEEE Transactions on Signal Processin

Analysis of Information Delivery Dynamics in Cognitive Sensor Networks Using Epidemic Models

To fully empower sensor networks with cognitive Internet of Things (IoT) technology, efficient medium access control protocols that enable the coexistence of cognitive sensor networks with current wireless infrastructure are as essential as the cognitive power in data fusion and processing due to shared wireless spectrum. Cognitive radio (CR) is introduced to increase spectrum efficiency and support such an endeavor, which thereby becomes a promising building block toward facilitating cognitive IoT. In this paper, primary users (PUs) refer to devices in existing wireless infrastructure, and secondary users (SUs) refer to cognitive sensors. For interference control between PUs and SUs, SUs adopt dynamic spectrum access and power adjustment to ensure sufficient operation of PUs, which inevitably leads to increasing latency and poses new challenges on the reliability of IoT communications. To guarantee operations of primary systems while simultaneously optimizing system performance in cognitive radio ad hoc networks (CRAHNs), this paper proposes interference-aware flooding schemes exploiting global timeout and vaccine recovery schemes to control the heavy buffer occupancy induced by packet replications. The information delivery dynamics of SUs under the proposed interference-aware recovery-assisted flooding schemes is analyzed via epidemic models and stochastic geometry from a macroscopic view of the entire system. The simulation results show that our model can efficiently capture the complicated data delivery dynamics in CRAHNs in terms of end-to-end transmission reliability and buffer occupancy. This paper sheds new light on analysis of recovery-assisted flooding schemes in CRAHNs and provides performance evaluation of cognitive IoT services built upon CRAHNs.Comment: 10 page

Universal Phase Transition in Community Detectability under a Stochastic Block Model

We prove the existence of an asymptotic phase transition threshold on community detectability for the spectral modularity method [M. E. J. Newman, Phys. Rev. E 74, 036104 (2006) and Proc. National Academy of Sciences. 103, 8577 (2006)] under a stochastic block model. The phase transition on community detectability occurs as the inter-community edge connection probability $p$ grows. This phase transition separates a sub-critical regime of small $p$, where modularity-based community detection successfully identifies the communities, from a super-critical regime of large $p$ where successful community detection is impossible. We show that, as the community sizes become large, the asymptotic phase transition threshold $p^*$ is equal to $\sqrt{p_1\cdot p_2}$, where $p_i~(i=1,2)$ is the within-community edge connection probability. Thus the phase transition threshold is universal in the sense that it does not depend on the ratio of community sizes. The universal phase transition phenomenon is validated by simulations for moderately sized communities. Using the derived expression for the phase transition threshold we propose an empirical method for estimating this threshold from real-world data.Comment: 9 pages, 7 figures, to appear in Physical Review

Is Ordered Weighted ℓ1\ell_1 Regularized Regression Robust to Adversarial Perturbation? A Case Study on OSCAR

Many state-of-the-art machine learning models such as deep neural networks have recently shown to be vulnerable to adversarial perturbations, especially in classification tasks. Motivated by adversarial machine learning, in this paper we investigate the robustness of sparse regression models with strongly correlated covariates to adversarially designed measurement noises. Specifically, we consider the family of ordered weighted $\ell_1$ (OWL) regularized regression methods and study the case of OSCAR (octagonal shrinkage clustering algorithm for regression) in the adversarial setting. Under a norm-bounded threat model, we formulate the process of finding a maximally disruptive noise for OWL-regularized regression as an optimization problem and illustrate the steps towards finding such a noise in the case of OSCAR. Experimental results demonstrate that the regression performance of grouping strongly correlated features can be severely degraded under our adversarial setting, even when the noise budget is significantly smaller than the ground-truth signals.Comment: Accepted to IEEE GlobalSIP 2018. Pin-Yu Chen and Bhanukiran Vinzamuri contribute equally to this work; v2 fixes missing citatio