Adjacency Matrices Of Networks Research Articles

On inference for modularity statistics in structured networks

This paper revisits the classical concept of network modularity and its spectral relaxations used throughout graph data analysis. We formulate and study several modularity statistic variants for which we establish asymptotic distributional results in the large-network limit for networks exhibiting nodal community structure. Our work facilitates testing for network differences and can be used in conjunction with existing theoretical guarantees for stochastic blockmodel random graphs. Our results are enabled by recent advances in the study of low-rank truncations of large network adjacency matrices. We provide confirmatory simulation studies and real data analysis pertaining to the network neuroscience study of psychosis, specifically schizophrenia. Collectively, this paper contributes to the limited existing literature to date on statistical inference for modularity-based network analysis. Supplemental materials for this article are available online.

Risk contagion and diversification among sovereign CDS, stock, foreign exchange and commodity markets: Fresh evidence from G7 and BRICS countries

This paper investigates tail risk contagion and diversification effects among sovereign credit default swap (SCDS), foreign exchange, stock, and commodity markets by constructing a tail event driven network. Network adjacency matrices indicate more significant cross-border risk contagion effects compared to cross-market contagion. Besides, the systemic risk decomposition results demonstrate that SCDS markets play a dominant role in the aggregate risk level, while the foreign exchange markets effectively reduce systemic risk. Moreover, tail event driven network quantile regression analysis shows that the impact of network factors at different risk levels exhibits a coexistence of symmetry and asymmetry, accompanied by market heterogeneity.

Inferring networks from time series: A neural approach.

Network structures underlie the dynamics of many complex phenomena, from gene regulation and foodwebs to power grids and social media. Yet, as they often cannot be observed directly, their connectivities must be inferred from observations of the dynamics to which they give rise. In this work, we present a powerful computational method to infer large network adjacency matrices from time series data using a neural network, in order to provide uncertainty quantification on the prediction in a manner that reflects both the degree to which the inference problem is underdetermined as well as the noise on the data. This is a feature that other approaches have hitherto been lacking. We demonstrate our method's capabilities by inferring line failure locations in the British power grid from its response to a power cut, providing probability densities on each edge and allowing the use of hypothesis testing to make meaningful probabilistic statements about the location of the cut. Our method is significantly more accurate than both Markov-chain Monte Carlo sampling and least squares regression on noisy data and when the problem is underdetermined, while naturally extending to the case of nonlinear dynamics, which we demonstrate by learning an entire cost matrix for a nonlinear model of economic activity in Greater London. Not having been specifically engineered for network inference, this method in fact represents a general parameter estimation scheme that is applicable to any high-dimensional parameter space.

Detecting Plant-Wide Oscillation Propagation Effects of Disturbances and Faults in a Chemical Process Plant Using Network Topology of Variance Decompositions

This work demonstrates for the first time the application of network topology of variance decompositions in analyzing the connectedness of chemical plant process variable oscillations arising from disturbances and faults. Specifically, the time-based connectedness and frequency-based connectedness of variables can be used to compute the net pairwise dynamic connectedness (NPDC), which originated as a volatility spillover index for financial markets studies in the field of econometrics. This work used the anomaly-detection benchmark Tennessee-Eastman chemical process (TEP) dataset, which consists of 41 measured variables and 11 manipulated variables subjected to various faulty operating conditions. The data analytics was performed using key functions from the R-package ‘ConnectednessApproach’ that implements connectedness computations based on time and frequency. The NPDC coefficient matrices were then transformed into network adjacency matrices for the rendering of the network topology of connectedness for TEP. The resulting network topologies allow a comprehensive analysis of oscillation effects across all plant-measured and manipulated variables. Analyzing the directed connectedness of the system dynamics at short-range, mid-range, and long-range frequencies showed how the oscillation effects of disturbances and faults propagate and dissipate in the short-term, mid-term, and long-term periods.

Link Prediction with Continuous-Time Classical and Quantum Walks.

Protein-protein interaction (PPI) networks consist of the physical and/or functional interactions between the proteins of an organism, and they form the basis for the field of network medicine. Since the biophysical and high-throughput methods used to form PPI networks are expensive, time-consuming, and often contain inaccuracies, the resulting networks are usually incomplete. In order to infer missing interactions in these networks, we propose a novel class of link prediction methods based on continuous-time classical and quantum walks. In the case of quantum walks, we examine the usage of both the network adjacency and Laplacian matrices for specifying the walk dynamics. We define a score function based on the corresponding transition probabilities and perform tests on six real-world PPI datasets. Our results show that continuous-time classical random walks and quantum walks using the network adjacency matrix can successfully predict missing protein-protein interactions, with performance rivalling the state-of-the-art.

Why Does Tardive Dyskinesia Have Oro-facial Predominance? A Network Analysis

Tardive dyskinesia is a involuntary hyperkinetic disorder which usually occurs in older patients after long-term treatment with antipsychotic drugs. These dyskinesias are mostly irreversible and are frequently expressed in the tongue, cheeks, mandible, perioral area and other regions of the face. In this theoretical study we asked the question, why does tardive dyskinesia often have orofacial predominance? What might be the underlying neural network structure which contributes to this propensity? Graph analysis of high-level cortico-striato-thalamo-cortical network structure suggests a connectivity bottleneck. The number of walks of different lengths from the substantia nigra pars reticulata (SNr) to other vertices, as well as the returning cycles are the lowest in the network, which may indicate a higher damage susceptibility of this node. Analysis was also performed on published data from a recent high resolution histological study on cortico-striato-thalamo-cortical networks in rodents. Finer network partitioning and adjacency matrices demonstrated that the SNr has a heterogeneous connectivity structure and the number of local walks from nodes neighboring orofacial neural representation is higher, indicating possible early compensatory escape routes. However, with more extensive SNr damage the larger circuit compensation might be limited. This area of inquiry is important for future research, because identifying key vulnerable structures may provide more targeted therapeutical interventions.

Doubly stochastic scaling unifies community detection

Graph partitioning, or community detection, has been widely investigated in network science. Yet, the correct community structure on a given network is essentially data-driven. Thus, instead of a formal definition, diverse measures have been conceived to capture intuitive desirable properties shared by most of the community structures. In this work, we propose a preprocessing based on a doubly stochastic scaling of network adjacency matrices, to highlight these desirable properties. By investigating a range of community detection measures, and carefully generalising them to doubly stochastic graphs, we show that such a scaling unifies a whole category of these measures—namely, the so-called linear criteria—onto two unique measures to set up. Finally, to help practitioners setting up these measures, we provide an extensive numerical comparison of the capacity of these measures to uncover community structures within stochastic block models, using the Louvain algorithm.

Estimating scale-free dynamic effective connectivity networks from fMRI using group-wise spatial–temporal regularizations

Estimating dynamic effective connectivity (dEC) networks is crucial to understand the time-varying directional interconnections among brain regions. It is now widely understood that brain networks have the property of being scale-free. However, this property has seldom been considered and is often inadequately preserved using conventional dEC estimation methods. As a result, important hubs and network graphical characteristics cannot be accurately obtained. In this work, we develop a new method to use a group-wise penalty together with spatial sparsity and temporal smoothness regularizations (namely Group-wise Spatial–Temporal Regularizations, GSTR) for the inference of scale-free dEC networks from functional magnetic resonance imaging (fMRI). The method employs a time-varying vector autoregressive (VAR) model, where the model coefficients can be formed as adjacency matrices of the dEC networks. Meanwhile, the proposed group-wise regularization is able to preserve the connectivities of potential hubs in scale-free networks by grouping them as an entire set. To deal with the complexity of optimization with multiple regularizations, we propose an effective algorithm based on the augmented Lagrangian multiplier. The accuracy of the GSTR method is validated using a variety of synthetic datasets with the scale-free property. Furthermore, we apply the GSTR method to an open fMRI dataset recorded from a block design visual task-related experiment containing 255 healthy participants to estimate visual-induced dEC networks and find GSTR can achieve reasonable and interpretable dEC estimates. Results from both synthetic and real-world datasets suggest that the proposed GSTR method could serve as a powerful analytical tool to accurately infer scale-free dEC patterns.

Latent structure blockmodels for Bayesian spectral graph clustering

Spectral embedding of network adjacency matrices often produces node representations living approximately around low-dimensional submanifold structures. In particular, hidden substructure is expected to arise when the graph is generated from a latent position model. Furthermore, the presence of communities within the network might generate community-specific submanifold structures in the embedding, but this is not explicitly accounted for in most statistical models for networks. In this article, a class of models called latent structure block models (LSBM) is proposed to address such scenarios, allowing for graph clustering when community-specific one-dimensional manifold structure is present. LSBMs focus on a specific class of latent space model, the random dot product graph (RDPG), and assign a latent submanifold to the latent positions of each community. A Bayesian model for the embeddings arising from LSBMs is discussed, and shown to have a good performance on simulated and real-world network data. The model is able to correctly recover the underlying communities living in a one-dimensional manifold, even when the parametric form of the underlying curves is unknown, achieving remarkable results on a variety of real data.

Two-dimensional phase lag index image representation of electroencephalography for automated recognition of driver fatigue using convolutional neural network

Driving in fatigue state will increase the occurrence probability of related traffic accidents and cause severe economic and societal problems. To tackle the issue, a deep learning approach is proposed for the automated recognition of driver fatigue using electroencephalography (EEG) signals obtained from real driving. The methodology here proposed consists of converting the multi-channel EEG recording into functional brain network (FBN) adjacency matrices based on phase lag index (PLI) and feeding them into various convolutional neural networks (CNN) as input. These CNN models with convolutional layer, rectifier linear activation unit (ReLU), pooling layer and fully connected layer are designed to extract hidden features from images representing FBN adjacency matrices and then to achieve the two-ways classification task. The experimental results indicate that the highest classification accuracy of 95.4 ± 2.0%, highest sensitivity of 93.9 ± 3.1%, highest precision of 95.5 ± 2.4%, highest F1 score of 94.7 ± 2.0% and highest value of area under the receiver operating curve (AUC-ROC = 0.9953) are achieved using Model 4 based on PLI adjacency matrices as input with the 10-fold cross validation strategy. Indeed, all the CNN models considered in this research achieved accuracy higher than 94.40%. It is hence concluded that the proposed CNN models have the ability to self-learn and pick up more distinguishable features from the input data without a separate feature extraction or feature selection procedure. The experimental results also confirmed the effectiveness of the combination of FBN and CNN for the recognition of driver fatigue.

Statistical properties of mutualistic-competitive random networks

Mutualistic networks are used to study the structure and processes inherent to mutualistic relationships. In this paper, we introduce a random matrix ensemble (RME) representing the adjacency matrices of mutualistic networks composed by two vertex sets of sizes n and m−n. Our RME depends on three parameters: the network size n, the size of the smaller set m, and the connectivity between the two sets α, where α is the ratio of current adjacent pairs over the total number of possible adjacent pairs between the sets. We focus on the spectral, eigenvector and topological properties of the RME by computing, respectively, the ratio of consecutive eigenvalue spacings r, the Shannon entropy of the eigenvectors S, and the Randić index R. First, within a random matrix theory approach (i.e. a statistical approach), we identify a parameter ξ≡ξ(n,m,α) that scales the average normalized measures <X¯> (with X representing r, S and R). Specifically, we show that (i) ξ∝αn with a weak dependence on m, and (ii) for ξ<1/10 most vertices in the mutualistic network are isolated, while for ξ>10 the network acquires the properties of a complete network, i.e., the transition from isolated vertices to a complete-like behavior occurs in the interval 1/10<ξ<10. Then, we demonstrate that our statistical approach predicts reasonably well the properties of real-world mutualistic networks; that is, the universal curves <X¯> vs. ξ show good correspondence with the properties of real-world networks.

Random matrix analysis of multiplex networks

We investigate the spectra of adjacency matrices of multiplex networks under random matrix theory (RMT) framework. Through extensive numerical experiments, we demonstrate that upon multiplexing two random networks, the spectra of the combined multiplex network exhibit superposition of two Gaussian orthogonal ensemble (GOE)s for very small multiplexing strength followed by a smooth transition to the GOE statistics with an increase in the multiplexing strength. Interestingly, randomness in the connection architecture, introduced by random rewiring to 1D lattice, of at least one layer may govern nearest neighbor spacing distribution (NNSD) of the entire multiplex network, and in fact, can drive to a transition from the Poisson to the GOE statistics or vice versa. Notably, this transition transpires for a very small number of the random rewiring corresponding to the small-world transition. Ergo, only one layer being represented by the small-world network is enough to yield GOE statistics for the entire multiplex network. Spectra of adjacency matrices of underlying interaction networks have been contemplated to be related with dynamical behavior of the corresponding complex systems, the investigations presented here have implications in achieving better structural and dynamical control to the systems represented by multiplex networks against structural perturbation in only one of the layers.

Quantitative analysis of trade networks: data and robustness

A common issue in trade network analysis is missing data, as some countries do not report trade flows. This paper explores what constitutes suitable data, how to deal with missing data, and demonstrates the results using key network measures. All-to-all potential connectivity of trade between countries is considered as a starting point, in contrast to the common approach of analyzing trade networks using only the countries that actually report trade flows. In order to fill the gap between the two approaches, a more complete dataset than just the dataset of trade between reporting countries is reconstructed and the robustness of studying this bigger dataset is examined. The difference between imputed and actual network adjacency matrices is evaluated based on several centrality measures. The results are illustrated using ten commodity groups from the United Nations Database, which demonstrate that under the proposed reconstruction procedure the ranks of the countries do not change significantly as the size of the imputed network becomes bigger or smaller. Further, the degree distributions of networks based on reporting countries and trading partners are the same to within their uncertainties. So, it is robust to study the imputed bigger network that provides richer insights into trade relations, particularly for nonreporting countries.

Multifractal analysis of eigenvectors of small-world networks

Many real-world complex systems have small-world topology characterized by the high clustering of nodes and short path lengths.It is well-known that higher clustering drives localization while shorter path length supports delocalization of the eigenvectors of networks. Using multifractals technique, we investigate localization properties of the eigenvectors of the adjacency matrices of small-world networks constructed using Watts-Strogatz algorithm. We find that the central part of the eigenvalue spectrum is characterized by strong multifractality whereas the tail part of the spectrum have Dq->1. Before the onset of the small-world transition, an increase in the random connections leads to an enhancement in the eigenvectors localization, whereas just after the onset, the eigenvectors show a gradual decrease in the localization. We have verified an existence of sharp change in the correlation dimension at the localization-delocalization transition

Spectral density of dense random networks and the breakdown of the Wigner semicircle law

Although the spectra of random networks have been studied for a long time, the influence of network topology on the dense limit of network spectra remains poorly understood. By considering the configuration model of networks with four distinct degree distributions, we show that the spectral density of the adjacency matrices of dense random networks is determined by the strength of the degree fluctuations. In particular, the eigenvalue distribution of dense networks with an exponential degree distribution is governed by a simple equation, from which we uncover a logarithmic singularity in the spectral density. We also derive a relation between the fourth moment of the eigenvalue distribution and the variance of the degree distribution, which leads to a sufficient condition for the breakdown of the Wigner semicircle law for dense random networks. Based on the same relation, we propose a classification scheme of the distinct universal behaviours of the spectral density in the dense limit. Our theoretical findings should lead to important insights on the mean-field behaviour of models defined on graphs.

Predicting Network Controllability Robustness: A Convolutional Neural Network Approach.

Network controllability measures how well a networked system can be controlled to a target state, and its robustness reflects how well the system can maintain the controllability against malicious attacks by means of node removals or edge removals. The measure of network controllability is quantified by the number of external control inputs needed to recover or to retain the controllability after the occurrence of an unexpected attack. The measure of the network controllability robustness, on the other hand, is quantified by a sequence of values that record the remaining controllability of the network after a sequence of attacks. Traditionally, the controllability robustness is determined by attack simulations, which is computationally time consuming. In this article, a method to predict the controllability robustness based on machine learning using a convolutional neural network (CNN) is proposed, motivated by the observations that: 1) there is no clear correlation between the topological features and the controllability robustness of a general network; 2) the adjacency matrix of a network can be regarded as a grayscale image; and 3) the CNN technique has proved successful in image processing without human intervention. Under the new framework, a fairly large number of training data generated by simulations are used to train a CNN for predicting the controllability robustness according to the input network-adjacency matrices, without performing conventional attack simulations. Extensive experimental studies were carried out, which demonstrate that the proposed framework for predicting controllability robustness of different network configurations is accurate and reliable with very low overheads.

Null-eigenvalue localization of quantum walks on complex networks

First we report that the adjacency matrices of real-world complex networks systematically have null eigenspaces with much higher dimensions than that of random networks. These null eigenvalues are caused by duplication mechanisms leading to structures with local symmetries which should be more present in complex organizations. The associated eigenvectors of these states are strongly localized. We then evaluate these microstructures in the context of quantum mechanics, demonstrating the previously mentioned localization by studying the spread of continuous-time quantum walks. This null-eigenvalue localization is essentially different from the Anderson localization in the following points: first, the eigenvalues do not lie on the edges of the density of states but at its center; second, the eigenstates do not decay exponentially and do not leak out of the symmetric structures. In this sense, it is closer to the bound state in continuum.

Identifying critical nodes in complex networks via graph convolutional networks

Critical nodes of complex networks play a crucial role in effective information spreading. There are many methods have been proposed to identify critical nodes in complex networks, ranging from centralities of nodes to diffusion-based processes. Most of them try to find what kind of structure will make the node more influential. In this paper, inspired by the concept of graph convolutional networks(GCNs), we convert the critical node identification problem in complex networks into a regression problem. Considering adjacency matrices of networks and convolutional neural networks(CNNs), a simply yet effectively method named RCNN is presented to identify critical nodes with the best spreading ability. In this approach, we can generate feature matrix for each node and use a convolutional neural network to train and predict the influence of nodes. Experimental results on nine synthetic and fifteen real networks show that under Susceptible–Infected–Recovered (SIR) model, RCNN outperforms the traditional benchmark methods on identifying critical nodes under spreading dynamic.

A subset selection based approach to structural reducibility of complex networks

Most of the real world networks such as the internet network, collaboration networks, brain networks, citation networks, powerline and airline networks are very large and to study their structure and dynamics one often requires working with large connectivity (adjacency) matrices. However, it is almost always true that a few or sometimes most of the nodes and their connections are not very crucial for network functioning or that the network is robust to a failure of certain nodes and their connections to the rest of the network. In the present work, we aim to extract the size reduced representation of complex networks such that new representation has the most relevant network nodes and connections and retains its spectral properties. To achieve this, we use the Subset Selection (SS) procedure. The SS method, in general, is used to retrieve maximum information (based on Frobenius norm) from a matrix in terms of its most informative columns. The retrieved matrix, typically known as subset has columns of an original matrix that have the least linear dependency. We present the application of SS procedure to many adjacency matrices of real-world networks and model network types to extract their subset. The subset owing to its small size can play a crucial role in analyzing spectral properties of large complex networks where space and time complexity of analyzing full adjacency matrices are too expensive. The adjacency matrix constructed from the obtained subset has a smaller size and represents the most important network structure. We observed that the subset network which is almost half the size of the original network has better information flow efficiency than the original network. Also, we found that the contribution to the Inverse Participation ratio of the network comes almost entirely from nodes that are there in the subset. This implies that the SS procedure can extract the top most influential nodes without the need for analyzing the full adjacency matrix.

Core community structure recovery and phase transition detection in temporally evolving networks

Community detection in time series networks represents a timely and significant research topic due to its applications in a broad range of scientific fields, including biology, social sciences and engineering. In this work, we introduce methodology to address this problem, based on a decomposition of the network adjacency matrices into low-rank components that capture the community structure and sparse & dense noise perturbation components. It is further assumed that the low-rank structure exhibits sharp changes (phase transitions) at certain epochs that our methodology successfully detects and identifies. The latter is achieved by averaging the low-rank component over time windows, which in turn enables us to precisely select the correct rank and monitor its evolution over time and thus identify the phase transition epochs. The methodology is illustrated on both synthetic networks generated by various network formation models, as well as the Kuramoto model of coupled oscillators and on real data reflecting the US Senate’s voting record from 1979–2014. In the latter application, we identify that party polarization exhibited a sharp change and increased after 1993, a finding broadly concordant with the political science literature on the subject.