Edge Weight Changes Research Articles

A network community structure similarity index for weighted networks.

Identification of communities in complex systems is an essential part of network analysis. Accordingly, measuring similarities between communities is a fundamental part of analysing community structure in different, yet related, networks. Commonly used methods for quantifying network community similarity fail to consider the effects of edge weights. Existing methods remain limited when the two networks being compared have different numbers of nodes. In this study, we address these issues by proposing a novel network community structure similarity index (NCSSI) based on the edit distance concept. NCSSI is proposed as a similarity index for comparing network communities. The NCSSI incorporates both community labels and edge weights. The NCSSI can also be employed to assess the similarity between two communities with varying numbers of nodes. We test the proposed method using simulated data and case-study analysis of New York Yellow Taxi flows and compare the results with that of other commonly used methods (i.e., mutual information and the Jaccard index). Our results highlight how NCSSI effectively captures the impact of both label and edge weight changes and their impacts on community structure, which are not captured in existing approaches. In conclusion, NCSSI offers a new approach that incorporates both label and weight variations for community similarity measurement in complex networks.

Parallel Hub Labeling Maintenance With High Efficiency in Dynamic Small-World Networks

Shortest path computation is a fundamental operation in many application domains and is especially challenging in frequently evolving small-world networks (i.e., graphs in which many nodes can be reached from every other node by a small number of hops). Index-based methods, especially ones based on 2-hop labeling are often used for high query efficiency. However, the evolvements of small-world networks in many realistic scenarios pose the challenge of efficient maintenance of the shortest path index. In this work, we adopt the state-of-the-art <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Parallel Shortest-distance Labeling (PSL)</i> as the underlying 2-hop labeling construction method, and design algorithms to support its efficient update given edge weight changes (increase and decrease). Specifically, we focus on weighted <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PSL (WPSL)</i> and propose a propagation-based update mechanism for both synchronous and asynchronous propagation. We also identify the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">curse of pruning power</i> in the edge weight increase case, and solve it with a balance between index size and effectiveness. Followed by, we extend the asynchronous propagation method to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Pruned Landmark Labeling (PLL)</i> for faster index maintenance and query processing with a smaller index size. Finally, we further optimize the index performance by reducing the index size through graph contraction and accelerating the index update through parallelized mix index update. Our experimental results on real-life and synthetic networks demonstrate the superiority of our algorithms over the relevant baselines on index maintenance.

Fear of Covid-19, anxiety, and depression during the pandemic: A network analysis

Introduction: Covid-19 has generated several psychological impacts on people’s lives and brought new challenges for dealing with the pandemic’s repercussions. Objective: The objective was to analyse the relationship between the symptoms of fear of Covid-19, anxiety, and depression using different samples extracted at three distinct time-points. This study performed a network analysis of the fear of Covid-19 and anxiety- and depression-related symptoms based on data collected during the Covid-19 pandemic. Method: Three yearly cross-sectional surveys were conducted based on independent samples (2020–2022) of Brazilian adults (Time-point 1, Time-point 2, and Time-point 3 with 1,520, 1,523, and 1,517 participants, respectively). The Fear of Covid-19 Scale, the Generalised Anxiety Disorder Scale (7 items), and the Patient Health Questionnaire (9 items) were applied. The network structure invariance, the global strength invariance, and the strength of centrality index between the samples were compared. Results: Items from the three instruments had high values (&gt;70%) in all samples (excluding the fear of Covid-19 at Time-point 3). Depression and anxiety symptoms increased, while fear-related behaviours decreased. Network analysis indicates that networks became progressively more parsimonious, and few edges connected different scales. Network comparisons revealed similar structures,but key differences were found: Time-point 1 showed significantly higher global connectivity,and changes in edge weights were observed only within the scales. Conclusions: Our findings suggest that the pandemic had a greater disorganising impact at its beginning, with a continuous increase in the mental health symptomatology and a reduction in the ratio between fear and anxiety and/or depression.

A Deep Learning Approach Using Graph Neural Networks for Anomaly Detection in Air Quality Data Considering Spatiotemporal Correlations

The ambient air pollution problem has become more severe as the social economy develops. Abnormal event detection in air quality data can prevent property loss and protect human health. The majority of existing anomaly detection models in air quality data are based on a single variable or a single monitoring station, ignoring spatial correlation and multivariate features of regional air pollutant concentrations. To address the aforementioned issues, this paper proposes a new deep learning approach that combines spatial correlation and temporal correlation of air quality data to detect air quality anomalies. The essential idea of this approach is to use the correlation degree between node information to fuse the spatial correlational feature and temporal correlational feature of air quality data to construct spatiotemporal graph structure data, which are used for anomaly detection in air quality data. In this proposed approach, (1) the weighted adjacency matrix is established to characterize the spatial correlation, and the feature matrix is constructed to characterize the temporal correlation; (2) changes in node information are transformed into changes in edge weights using the correlation degree between node information, and spatiotemporal graph structure data are constructed to characterize the spatiotemporal correlation; and (3) an advanced deep learning model, Context augmented Graph Autoencoder (Con-GAE), is utilized to handle the above spatiotemporal graph structure data and detect abnormal air quality events. The efficiency of the approach is demonstrated by anomaly detection on synthetic test sets produced from real-world datasets.

TrendNets: mapping emerging research trends from dynamic co-word networks via sparse representation

Mapping the knowledge structure from word co-occurrences in a collection of academic papers has been widely used to provide insight into the topic evolution in an arbitrary research field. In a traditional approach, the paper collection is first divided into temporal subsets, and then a co-word network is independently depicted in a 2D map to characterize each period’s trend. To effectively map emerging research trends from such a time-series of co-word networks, this paper presents TrendNets, a novel visualization methodology that highlights the rapid changes in edge weights over time. Specifically, we formulated a new convex optimization framework that decomposes the matrix constructed from dynamic co-word networks into a smooth part and a sparse part: the former represents stationary research topics, while the latter corresponds to bursty research topics. Simulation results on synthetic data demonstrated that our matrix decomposition approach achieved the best burst detection performance over four baseline methods. In experiments conducted using papers published in the past 16 years at three conferences in different fields, we showed the effectiveness of TrendNets compared to the traditional co-word representation. We have made our codes available on the Web to encourage scientific mapping in all research fields.

A Combinatorial Methodology for Optimizing Non-Binary Graph-Based Codes: Theoretical Analysis and Applications in Data Storage

Non-binary (NB) low-density parity-check (LDPC) codes are graph-based codes that are increasingly being considered as a powerful error correction tool for modern dense storage devices. Optimizing NB-LDPC codes to overcome their error floor is one of the main code design challenges facing storage engineers upon deploying such codes in practice. Furthermore, the increasing levels of asymmetry incorporated by the channels underlying modern dense storage systems, e.g., multi-level Flash systems, exacerbate the error floor problem by widening the spectrum of problematic objects that contribute to the error floor of an NB-LDPC code. In a recent research, the weight consistency matrix (WCM) framework was introduced as an effective combinatorial NB-LDPC code optimization methodology that is suitable for modern Flash memory and magnetic recording (MR) systems. The WCM framework was used to optimize codes for asymmetric Flash channels, MR channels that have intrinsic memory, in addition to canonical symmetric additive white Gaussian noise channels. In this paper, we provide an in-depth theoretical analysis needed to understand and properly apply the WCM framework. We focus on general absorbing sets of type two (GASTs) as the detrimental objects of interest. In particular, we introduce a novel tree representation of a GAST called the unlabeled GAST tree, using which we prove that the WCM framework is optimal in the sense that it operates on the minimum number of matrices, which are the WCMs, to remove a GAST. Then, we enumerate WCMs and demonstrate the significance of the savings achieved by the WCM framework in the number of matrices processed to remove a GAST. Moreover, we provide a linear-algebraic analysis of the null spaces of WCMs associated with a GAST. We derive the minimum number of edge weight changes needed to remove a GAST via its WCMs, along with how to choose these changes. In addition, we propose a new set of problematic objects, namely oscillating sets of type two (OSTs), which contribute to the error floor of NB-LDPC codes with even column weights on asymmetric channels, and we show how to customize the WCM framework to remove OSTs. We also extend the domain of the WCM framework applications by demonstrating its benefits in optimizing column weight 5 codes, codes used over Flash channels with additional soft information, and spatially coupled codes. The performance gains achieved via the WCM framework range between 1 and nearly 2.5 orders of magnitude in the error floor region over interesting channels.

Dynamic Physarum Solver: a bio-inspired shortest path method of dynamically changing graphs

In dynamic graphs, edge weights of the graph change with time and solving the shortest path problem in such graphs is an important real-world problem. The studies in the literature require excessive computational time for computing the dynamic shortest path since determining changing edge weights is difficult especially when the graph size becomes large. In this paper, we propose a dynamic bio-inspired algorithm for finding the dynamic shortest path for large graphs based on Physarum Solver, which is a shortest path algorithm for static graphs. The proposed method is evaluated using three different large graph models representing diverse real-life applications. The effect of changing edge weights on the solution time is evaluated for each graph model separately and compared against ∆-stepping, which is the most representative implementation of Dijkstra’s algorithm. Experimental results show that the proposed method easily adapts edge weight changes and computes the dynamic shortest path efficiently.

Transduction on Directed Graphs via Absorbing Random Walks.

In this paper we consider the problem of graph-based transductive classification, and we are particularly interested in the directed graph scenario which is a natural form for many real world applications. Different from existing research efforts that either only deal with undirected graphs or circumvent directionality by means of symmetrization, we propose a novel random walk approach on directed graphs using absorbing Markov chains, which can be regarded as maximizing the accumulated expected number of visits from the unlabeled transient states. Our algorithm is simple, easy to implement, and works with large-scale graphs on binary, multiclass, and multi-label prediction problems. Moreover, it is capable of preserving the graph structure even when the input graph is sparse and changes over time, as well as retaining weak signals presented in the directed edges. We present its intimate connections to a number of existing methods, including graph kernels, graph Laplacian based methods, and spanning forest of graphs. Its computational complexity and the generalization error are also studied. Empirically, our algorithm is evaluated on a wide range of applications, where it has shown to perform competitively comparing to a suite of state-of-the-art methods. In particular, our algorithm is shown to work exceptionally well with large sparse directed graphs with e.g., millions of nodes and tens of millions of edges, where it significantly outperforms other state-of-the-art methods. In the dynamic graph setting involving insertion or deletion of nodes and edge-weight changes over time, it also allows efficient online updates that produce the same results as of the batch update counterparts.

Modeling dynamic social networks using spectral embedding

Dynamic social network analysis aims to understand the structures in networks as they evolve, as nodes appear and disappear, and as edge weights change. Working directly with a social network graph is difficult, and it has become standard to use spectral techniques that embed a graph in a geometry. Analysis can then be done in the geometry where distance approximates dissimilarity. Recently, spectral techniques have been extended to model directed graphs; we build on these techniques to model directed graphs that change over time. The snapshots of the social network at each time period are bound together into a single graph in a way that keeps structures aligned over time, and this global graph is then spectrally embedded. The similarities among a set of nodes can be tracked over time, so that changing relationships and clusters can be seen; and the concept of the trajectory of a node across time also becomes meaningful. We illustrate how these approaches can be used to understand the changing social network of the Caviar drug-trafficking network under both internal dynamics and response to law-enforcement actions.

Spectral Theory for Dynamics on Graphs Containing Attractive and Repulsive Interactions

Many applied problems can be posed as a dynamical system defined on a network with attractive and repulsive interactions. Examples include synchronization of nonlinear oscillator networks; the behavior of groups, or cliques, in social networks; and the study of optimal convergence for consensus algorithm. It is important to determine the index of a matrix, i.e., the number of positive and negative eigenvalues, and the dimension of the kernel. In this paper we consider the common examples where the matrix takes the form of a signed graph Laplacian. We show that the there are topological constraints on the index of the Laplacian matrix related to the dimension of a certain homology group. When the homology group is trivial, the index of the operator is determined only by the topology of the network and is independent of the strengths of the interactions. In general, these constraints give bounds on the number of positive and negative eigenvalues, with the dimension of the homology group counting the number of eigenvalue crossings. The homology group also gives a natural decomposition of the dynamics into “fixed” degrees of freedom, whose index does not depend on the edge weights, and an orthogonal set of “free” degrees of freedom, whose index changes as the edge weights change. We also explore the spectrum of the Laplacians of signed random matrices.

Box-covering algorithm for fractal dimension of weighted networks

Box-covering algorithm is a widely used method to measure the fractal dimension of complex networks. Existing researches mainly deal with the fractal dimension of unweighted networks. Here, the classical box covering algorithm is modified to deal with the fractal dimension of weighted networks. Box size length is obtained by accumulating the distance between two nodes connected directly and graph-coloring algorithm is based on the node strength. The proposed method is applied to calculate the fractal dimensions of the “Sierpinski” weighted fractal networks, the E.coli network, the Scientific collaboration network, the C.elegans network and the USAir97 network. Our results show that the proposed method is efficient when dealing with the fractal dimension problem of complex networks. We find that the fractal property is influenced by the edge-weight in weighted networks. The possible variation of fractal dimension due to changes in edge-weights of weighted networks is also discussed.

Shortest Path Tree Computation in Dynamic Graphs

Let G = (V, E, omega) be a simple digraph, in which all edge weights are nonnegative real numbers. Let G' be obtained from G by an application of a set of edge weight updates to G. Let sisinV and let Ts and Ts' be Shortest Path Trees (SPTs) rooted at s in G and G', respectively. The Dynamic Shortest Path (DSP) problem is to compute Ts' from Ts. Existing work on this problem focuses on either a single edge weight change or multiple edge weight changes in which some of them are incorrect or are not optimized. We correct and extend a few state-of-the-art dynamic SPT algorithms to handle multiple edge weight updates. We prove that these algorithms are correct. Dynamic algorithms may not outperform static algorithms all the time. To evaluate the proposed dynamic algorithms, we compare them with the well-known static Dijkstra algorithm. Extensive experiments are conducted with both real-life and artificial data sets. The experimental results suggest the most appropriate algorithms to be used under different circumstances.

An efficient dynamic algorithm for maintaining all-pairs shortest paths in stochastic networks

This paper presents a new solution to the dynamic all-pairs shortest path routing problem, using a linear reinforcement learning scheme. The particular instance of the problem that we have investigated concerns finding the all-pairs shortest paths in a stochastic graph, where there are continuous probabilistically-based updates in edge-weights. We present the details of the algorithm with an illustrative example. The algorithm can be used to find the all-pairs shortest paths for the "statistical" average graph, and the solution converges irrespective of whether there are new changes in edge-weights or not. On the other hand, the existing algorithms will fail to exhibit such a behavior and would recalculate the affected shortest paths after each edge-weight update. There are two important contributions of the proposed algorithm. The first contribution is that not all the edges in a stochastic graph are probed and, even if they are, they are not all probed equally often. Indeed, the algorithm attempts to almost always probe only those edges that will be included in the final list involving all pairs of nodes in the graph, while probing the other edges minimally. This increases the performance of the proposed algorithm. The second contribution is designing a data-structure, the elements of which represent the probability that a particular edge in the graph lies in the shortest path between a pair of nodes in the graph. All the algorithms were tested in environments where edge-weights change stochastically and where the graph topologies undergo multiple simultaneous edge-weight updates. Its superiority in terms of the average number of processed nodes, scanned edges, and the time per update operation, when compared with the existing algorithms, was experimentally established.

Planar graphs, negative weight edges, shortest paths, and near linear time

In this paper, we present an O ( n log 3 n ) time algorithm for finding shortest paths in an n-node planar graph with real weights. This can be compared to the best previous strongly polynomial time algorithm developed by Lipton, Rose, and Tarjan in 1978 which runs in O ( n 3 / 2 ) time, and the best polynomial time algorithm developed by Henzinger, Klein, Subramanian, and Rao in 1994 which runs in O ˜ ( n 4 / 3 ) time. We also present significantly improved data structures for reporting distances between pairs of nodes and algorithms for updating the data structures when edge weights change.

Dynamic Algorithms for the Shortest Path Routing Problem: Learning Automata-Based Solutions

This paper presents the first Learning Automaton-based solution to the dynamic single source shortest path problem. It involves finding the shortest path in a single-source stochastic graph topology where there are continuous probabilistic updates in the edge-weights. The algorithm is significantly more efficient than the existing solutions, and can be used to find the "statistical" shortest path tree in the "average" graph topology. It converges to this solution irrespective of whether there are new changes in edge-weights taking place or not. In such random settings, the proposed learning automata solution converges to the set of shortest paths. On the other hand, the existing algorithms will fail to exhibit such a behavior, and would recalculate the affected shortest paths after each weight-change. The important contribution of the proposed algorithm is that all the edges in a stochastic graph are not probed, and even if they are, they are not all probed equally often. Indeed, the algorithm attempts to almost always probe only those edges that will be included in the shortest path graph, while probing the other edges minimally. This increases the performance of the proposed algorithm. All the algorithms were tested in environments where edge-weights change stochastically, and where the graph topologies undergo multiple simultaneous edge-weight updates. Its superiority in terms of the average number of processed nodes, scanned edges and the time per update operation, when compared with the existing algorithms, was experimentally established. The algorithm can be applicable in domains ranging from ground transportation to aerospace, from civilian applications to military, from spatial database applications to telecommunications networking.