Computational network theory

Table of Contents Cover Titles of the Series “Quantitative and Network Biology” Related Titles Title Page Copyright Dedication Preface List of Contributors Chapter 1: Model Selection for Neural Network Models: A Statistical Perspective 1.1 Introduction 1.2 Feedforward Neural Network Models 1.3 Model Selection 1.4 The Selection of the Hidden Layer Size 1.5 Concluding Remarks References Chapter 2: Measuring Structural Correlations in Graphs 2.1 Introduction 2.2 Related Work 2.3 Self Structural Correlation 2.4 Two-Event Structural Correlation 2.5 Conclusions References Chapter 3: Spectral Graph Theory and Structural Analysis of Complex Networks: An Introduction 3.1 Introduction 3.2 Graph Theory: Some Basic Concepts 3.3 Matrix Theory: Some Basic Concepts 3.4 Graph Matrices 3.5 Spectral Graph Theory: Some Basic Results 3.6 Computational Challenges for Spectral Graph Analysis 3.7 Conclusion References Chapter 4: Contagion in Interbank Networks 4.1 Introduction 4.2 Research Context 4.3 Models 4.4 Results 4.5 Stress Testing Applications 4.6 Conclusions References Chapter 5: Detection, Localization, and Tracking of a Single and Multiple Targets with Wireless Sensor Networks 5.1 Introduction and Overview 5.2 Data Collection and Fusion by WSN 5.3 Target Detection 5.4 Single Target Localization and Diagnostic 5.5 Multiple Target Localization and Diagnostic 5.6 Multiple Target Tracking 5.7 Applications and Case Studies 5.8 Final Remarks References Chapter 6: Computing in Dynamic Networks 6.1 Introduction 6.2 Preliminaries 6.3 Spread of Influence in Dynamic Graphs (Causal Influence) 6.4 Naming and Counting in Anonymous Unknown Dynamic Networks 6.5 Causality, Influence, and Computation in Possibly Disconnected Synchronous Dynamic Networks 6.6 Local Communication Windows 6.7 Conclusions References Chapter 7: Visualization and Interactive Analysis for Complex Networks by means of Lossless Network Compression 7.1 Introduction 7.2 Power Graph Algorithm 7.3 Validation—Edge Reduction Differs from Random 7.4 Graph Comparison with Power Graphs 7.5 Excursus: Layout of Power Graphs 7.6 Interactive Visual Analytics 7.7 Conclusion References Index End User License Agreement List of Illustrations Chapter 1: Model Selection for Neural Network Models: A Statistical Perspective Figure 1.1 Model M1 Results of the multiple testing procedure ( , , , , ) Figure in bold refer to the rejection of the corresponding hypotheses Figure 1.2 Model M2 Results of the multiple-testing procedure ( , , , , ) Figure in bold refer to the rejection of the corresponding hypotheses Figure 1.3 Model M3 Results of the multiple-testing procedure ( , , , , ) Figure in bold refer to the rejection of the corresponding hypotheses Figure 1.4 IVS for Ozone data via neural networks The relevance measure is the statistic The hidden layer size has been selected by -fold CV ( ) Subsample size selected by using minimum volatility method The nominal size is Figure 1.5 Model M1 Bootstrap confidence intervals for the maximum expected predictive performance of the neural networks with respect to the benchmark Figure 1.6 Model M2 Bootstrap confidence intervals for the maximum expected predictive performance of the neural networks with respect to the benchmark Figure 1.7 Model M3 Bootstrap confidence intervals for the maximum expected predictive performance of the neural networks with respect to the benchmark Figure 1.8 Joint confidence regions with nominal coverage probability Figure 1.9 Proportion of hidden layer size identification by using the testing procedure for superior predictive ability Figure 1.10 Bayesian information criterion values for different hidden layer sizes and different weight decay values (a) -fold CV values for and using a weight decay equal to zero (b) Figure 1.11 Joint confidence regions with nominal coverage probability (a) Absolute prediction error distributions computed on the test set for linear models and neural networks with hidden layer size ranging from 1 to 8 (b) Chapter 2: Measuring Structural Correlations in Graphs Figure 2.1 Structural correlation Figure 2.2 Two types of two-event structural correlation: (a) attraction and (b) repulsion Figure 2.3 Measuring SSC Figure 2.4 Bounds for one path Figure 2.5 Comparison of sampling and geometric distribution heuristic for estimating Figure 2.6 Exploring the convergence of (a) Iterative-alg, and (b) Sampling-alg Figure 2.7 Comparison of Iterative-alg and Sampling-alg with respect to the time used to estimate one DHT Figure 2.8 Applying gScore on synthetic events Figure 2.9 Comparison of DHT and pair-wise shortest distance as the proximity measure by adding noises into the cascade model Figure 2.10 Comparison of DHT and 1-neighborhood event fraction as the proximity measure by generating more general SSC in local neighborhoods Figure 2.11 Running times of Sampling-alg for estimating one DHT when varying the graph size 2.4 Two-Event Structural Correlation Figure 2.12 Four illustrative examples showing that density changes of the two events between two reference nodes show an evidence of correlation Figure 2.13 and when we incorporate nodes whose -vicinities do not contain any occurrence of or Figure 2.14 -vicinities of event nodes Figure 2.15 (a–c) Performance of three reference node sampling algorithms on simulated positively correlated event pairs Results for various noise levels are reported under different vicinity levels Figure 2.16 (a–c) Performance of three reference node sampling algorithms on simulated negatively correlated event pairs Results for various noise levels are reported under different vicinity levels Figure 2.17 Performance of sampling different number of reference nodes from each for Importance sampling Figure 2.18 Impact of randomly removing or adding edges on the correlation results Figure 2.19 Running time of reference node sampling algorithms with increasing number of event nodes Figure 2.20 Running time of one -hop BFS search and computation Chapter 3: Spectral Graph Theory and Structural Analysis of Complex Networks: An Introduction Figure 3.1 The graph , drawn using its adjacency matrix and randomly chosen positions for the vertices Figure 3.2 The graph , drawn using its adjacency matrix and its two Laplace eigenvectors and Figure 3.4 The graph , drawn using its adjacency matrix and its three Laplace eigenvectors , , and Chapter 4: Contagion in Interbank Networks Figure 4.1 A generated interbank network Note: an arrow between bank A and B indicates an interbank deposit of bank B placed in bank A; the width of an arrow reflects the size of the exposure; the lighter the green color of an arrow, the lower the probability that the arrow joins a given pair of banks Figure 4.2 The sequential four-round procedure of the interbank formation Figure 4.3 Betweenness-centrality measures: distribution on the simulated networks versus the average network Note: Blue line: distribution on the simulated networks; red (vertical) line: measure for the average simulated network: green (vertical) line: measure for he entropy maximising network Only internationally active banks are presented Figure 4.4 Distribution of the average CAR reduction (in p.p.) Figure 4.5 Decomposition of the distribution of individual banks' CAR reduction into first-and second-round contagion (in p.p.) Note: blue area–aggregate effect of firstround contagion; red area–second-round contagion Only internationally active banks are presented Figure 4.6 Endogenous networks versus random graphs generated with parameters inherited from the endogenous ones Note: -axis: banks in the sample -axis: statistical measure of topological properties Blue-wide lines: referring to endogenous networks (average in a random sample of 100 networks) Red-thin lines: referring to random graphs (top row: random degree graphs; bottom row: randomly clustered graph NetworkX library in Python was used to generate and analyze the random graphs.) Figure 4.7 Incompleteness of the interbank network structure with the estimated correlation of risks The darker the line, the higher the probability that the link exists The circles around the nodes indicate bank sizes (proportional to log of total assets) Figure 4.8 Worst-case bp reduction in CT1 ratio due to interbank contagion–crosscountry dispersion Note: -axis: basis point CT1 ratio reduction; interquartile range represents 25th and 75th percentiles of cross-country contagion effects under the most severe of the simulated networks Figure 4.9 Counterparty credit quality and the impact of LE limits on the losses incurred due to contagion Note: -axis: CDS spread (in bps) -axis: difference of CAR after adverse stress testing shock estimated for CVA ( ), estim CVA ( ), estim CVA ( ) versus no CVA regime (in pp, negative number means that introduction of CVA charge decreases contagion losses) No CVA adjustment (i.e., ) The size of a circle is proportional to a bank's total assets Figure 4.10 Counterparty credit quality and the impact of CVA capital charge on the losses incurred due to contagion -axis: CDS spread (in bps) -axis: difference of CAR after adverse stress testing shock between no CVA regime calculations and CVA regime with: estimated CVA ( ), estimated CVA ( ) and estimated CVA ( ), in pp, positive number means that introduction of CVA charge increases contagion losses The size of a circle is proportional to a bank's total assets Chapter 5: Detection, Localization, and Tracking of a Single and Multiple Targets with Wireless Sensor Networks Figure 5.1 (a) Target signal generated by the model for a target at location with , (b) Target energy contaminated by Gaussian noise of variance (signal-to-noise ratio ) (c) Sensor decisions based on individual false alarm probability (the same for all sensors) Figure 5.2 Ordinary versus local vote decision fusion under a square grid design (a,b) and random deployment (c,d) The network is comprised of 100 sensors, with individual sensor false alarm probability , system-wide false alarm probability and a target located at the center of the monitored region The signal is generated by the model , with , and the measured energy is corrupted by Gaussian noise with Figure 5.3 Example of sensor neighborhoods with , and Figure 5.4 Square (a), hexagonal (b), and diamond-shaped (c) neighborhoods on a regular grid Figure 5.5 True trajectories (solid lines) and positions estimated by ML(Z) at each time point for three targets with SNR = 5 (a) The signal from the second target is briefly lost; (b) Two targets come close together and the third target briefly loses signal; (c) Another noise realization/solution for (b) 5.7 Applications and Case Studies Figure 5.6 (a) The activation pattern of NEST sensors by a person traversing the monitored area (b) The trajectory of a single zebra in the monitored area Figure 5.7 Estimated and true trajectories for (a) one, (b) two, and (c) three NEST targets Figure 5.8 (a) Random sensor deployment (b) The recorded locations of the four zebras scaled and plotted on the unit square 6.5 Causality, Influence, and Computation in Possibly Disconnected Synchronous Dynamic Networks Figure 6.1 The alternating matchings dynamic graph for The solid lines appear every odd round ( ) while the dashed lines every even round ( ) Figure 6.2 Soifer's dynamic graph for and In particular, in round 1, the graph consists of the black solid edges, then in round 2 the center becomes connected via a dotted edge to the next peripheral node clockwise and all edges perpendicular to it (the remaining dotted ones) become available, and so on, always moving clockwise Figure 6.3 A partitioning of into two sets The left set is , that is, the set of nodes whose -state has influenced by time All nodes in also belong to Looking back in time at the interval , there should be an edge from some in the left set to some in the right set This implies that has heard from by time and as has heard from the -state of it has also heard from the initial state of This implies that is a strict superset of as long as the right set is not empty Figure 6.4 If there are still nodes that have not heard from , then if is an upper bound on the , in at most rounds another node will hear from (by definition of the ) Chapter 7: Visualization and Interactive Analysis for Complex Networks by means of Lossless Network Compression Figure 7.1 Difficulties of current graph drawing approaches (a) Network with 279 nodes and 4849 edges appears as black lump (b) Co-occurrence network of the New Testament Details in the appearing clusters (encircled) are difficult to see (c) Small graph with a biclique that would not be detected with modular decomposition, together with an equivalent power graph representation Figure 7.2 Co-occurrence network of the New Testament (characters that appear together in verses are connected) The network data comes from the Stanford GraphBase (Knuth, 1993) (a) Power graph, Jesus appears as central hub, nodes that are not connected to Jesus can be identified easily (b) The 12 apostles can be found in an onion-like structured clique of 14 (c) Underlying network (d) Legend Cliques are colored grey for readability (instead of drawing the reflexive power edges as loops) Figure 7.3 (a) Power graph semantics: biclique, star, and clique motifs (b) Power graph conditions and their equivalent decompositions Figure 7.4 A graph can be transformed in different power graph representations, power graphs (b-k) are a selection for graph (a) (which is a power graph representation by itself) (h–k) are minimal, no equivalent power graph with fewer power edges exists Figure 7.5 Power graph similarity (PG-similarity) (a) Two distinct graphs and on the same set of nodes (b) Power graphs and for and , respectively, after applying the power graph algorithm (c) Power node matching as basis of the similarity measure Each power node in is matched with the power node in with the highest F-measure, and vice versa Precision and recall of those matchings are summarized to precision, recall and F-measure between and Figure 7.6 The influence of nesting level (depth) in power graphs on the layout, the edge–edge and edge–power node crossing count, and the edge reduction Figure 7.7 (a/b) Deterministic patterns for power nodes without outgoing edges (a— circular patterns, b—phyllotactic patterns (Shipman and Newell, 2004)) (c) Additional to attractive and repulsive forces, twisting forces are applied Figure 7.8 Power edge filtering (a) Unfiltered power graph (b) Filtered by size, only power edges abstracting at least 13 edges are kept (c) All power edges are removed, only power nodes remain which provide information on the architecture of the network In average, each power node is derived from the information of 11 edges Figure 7.9 Interactive visual analysis of the Florida Food Chain Network (a) The largest power nodes correspond to relevant groups of animals in the food chain (b) Selecting power edges around a region of interest –for example here a group of predators—helps to locally explore the network (c) These predators share many fish species in their diet and are thus in competition in the food chain Yet it can be seen that crocodiles and raptors prefer larger predatory fish and pelicans, cormorants and dolphins prefer smaller fish Note: The food chain sink corresponds to the output of carbon from the ecological system studied, it is thus not a species but represent exchanges with the outside List of Tables Chapter 1: Model Selection for Neural Network Models: A Statistical Perspective Table 1.1 Comparison of variable selection procedures on the Ozone data Table 1.2 Values of the test statistics for different input neuron sets and different hidden layer size Chapter 2: Measuring Structural Correlations in Graphs Table 2.1 SSC for top five correlated products in category “Laptops and tablets” in TaoBao Table 2.2 SSC for top-five correlated products in category “Other” in TaoBao Table 2.3 SSC for the five most uncorrelated products in category “Other” in TaoBao 2.4 Two-Event Structural Correlation Table 2.4 Five keyword pairs exhibiting high 1-hop positive correlation (DBLP) All scores are -scores Table 2.5 Five keyword pairs exhibiting high 3-hop negative correlation (DBLP) All scores are -scores Table 2.6 Five alert pairs exhibiting high 1-hop positive correlation (Intrusion) All scores are -scores Table 2.7 Five alert pairs exhibiting high 2-hop negative correlation (Intrusion) All scores are -scores Table 2.8 Two rare alert pairs with positive 1-hop TESC which are not discovered by proximity pattern mining 5.7 Applications and Case Studies Table 5.1 Average distances from the true trajectories and estimated SNR Table 5.2 Average distance from the true zebra trajectories (one unit of distance is approximately 5 km), for the case of isotropic signal attenuation ( ) Table 5.3 The distribution of the estimated number of targets for zebra tracking (%), for the case of isotropic signal attenuation Chapter 7: Visualization and Interactive Analysis for Complex Networks by means of Lossless Network Compression Table 7.1 Pseudocode for the power graph algorithm Table 7.2 Edge reduction and relative edge reduction of diverse networks of complex systems ( denotes average degree) Edge reduction is generally in the range of 45–82%, relative edge reduction in the range 15–50% TESC EIS incidence matrix independence number individual sensor's false alarm probability influence time incoming outgoing instantaneous graphs interactive visual analysis interactive visual analytics interbank network data endogenous networks literature models simulated networks stress testing applications Systemic Probability Index 1-interval connectivity model intrusion dataset iterative-alg iterative approximation j Jaccard's coefficient k k-d tree indices Kendall's τ rank correlation measure Krylov subspace methods l labeling Laplace matrix Laptops and tablets LDAP Auth Failed leader node linear model line graph local communication windows local connectivity loops m Mabinogi matrix adjacency cut-set degree and diffusion matrix eigenvalues and eigenvectors incidence Laplace path trace and determinant maximum cut problem maximum outgoing influence (moi) definition oit unit Menger's theorem minimal (consecutive) naming mining cohesive graph pattern problem mobility pattern modular decomposition moi, see maximum outgoing influence (moi) Monte Carlo experiment Monte Carlo sampling method mother communities, in social network Motif mining method multigraph multiple target localization and diagnostic from binary decisions from corrected decisions from energies hybrid estimation starting values Multiple testing algorithm n naming, dynamic networks naming problem negative correlation 1-neighborhood event fraction measure Netscape Enterprise Server software network coding network construction Network Embedded Systems Technology (NEST) project network null model neural network models Akaike information criterion feedforward neural network model hidden layer size selection logistic activation function model selection in radially symmetric function real data application relevance measure Schwarz information criterion (SIC) square loss function superior predictive ability approach test non linear analysis nonlinear least squares non parametric analysis null hypothesis o optimization problem ordinary decision fusion (ODF) out-of-sight nodes Ozone data, neural network p pairwise measures parametric analysis path matrix personalized PageRank PG-similarity population protocol (PP) model positive correlation power edge filtering power graph algorithm analysis conditions definition edge reduction and relative edge reduction extraction layout range of semantics similarity power grid network powerset probability law pruning strategy p-value, test q quadratic loss function queue r radially symmetric function randomization technique random walk and hitting time reality check approach bootstrap procedure covariance matrix linear model reference node sampling algorithms BFS search complexity analysis global sampling importance regular graph RejectSamp relative edge reduction R-tree indices s sampling-alg scalability Schwarz information criterion (SIC) second limit theorem self structural correlation (SSC) description estimation novel measure problem formulation random walk and hitting time real event semilinearity sensitivity measures sigmoidal activation function simulated interbank network characteristics contagion mechanism contagion results fire sales interbank network probability map simultaneous consensus single target localization and diagnostics accuracy computational cost hybrid maximum likelihood estimates from local vote decision fusion maximum likelihood estimates from ordinary decision fusion robustness to model misspecification starting values for localization from value fusion Soifer's dynamic graph South Florida Ecosystem network spanning line spanning ring spanning star spanning subgraph spectral graph theory bipartite graph complete graph computational challenges cospectral graph and graph colouring and graph drawing line graph regular graph tree and walks square loss function static network static networks with broadcast stepM procedure stress testing applications structural correlations pattern mining problem self, see self structural correlation (SSC) TESC, see two-event structural correlation (TESC) subsampling procedure superior predictive ability approach synchronous dynamic graph alternating matchings connectivity time continuous disconnectivity synchronous dynamic networks broadcast influence time moi, see maximum outgoing influence (moi) termination and computation, see Termination and computation synchronous message passing, see communication Systemic Probability Index t TaoBao dataset temporal connectivity conditions temporal graph termination and computation communication network hear from known known upper bound on the oit optimal termination criterion talk to known termination criteria ThinkPad Tibshirani model time-varying graph tokens transaction correlation transition probability matrix truncated hitting time (THT) turing machine (TM) Twitter dataset SSC TESC two-event structural correlation (TESC) aims description efficiency and scalability event simulation methodology graph datasets graph density importance sampling novel measure performance comparison preliminaries and problem formulation real events reference node(s) reference node sampling testing test u undirected graph unknown, dynamic networks unweighted graph upper bound of theorem v validation value fusion vertex-connectivity of graph edge visual analytics w weighted graph Wilcoxon rank sum test wireless sensor networks (WSN) applications of data collection and fusion ordinary decision fusion value fusion multiple target localization and diagnostic from binary decisions from corrected decisions from energies hybrid estimation starting values multiple target tracking single target localization and diagnostics accuracy computational cost hybrid maximum likelihood estimates from local vote decision fusion maximum likelihood estimates from ordinary decision fusion robustness to model misspecification starting values for localization from value fusion structure and the design target detection accuracy of detection performance from local vote decision fusion from ordinary decision fusion quality of approximation radar-based applications from value fusion worst-case adversary worst-case dynamicity x XE dynamic networks communication z zooming and network expansion WILEY END USER LICENSE AGREEMENT Go to www.wiley.com/go/eula to access Wiley's ebook EULA ... This calls for a definition and treatment of computational network theory as a branch of network theory to classify the methods developed in this volume correctly The definition we would like to form views computational network theory as a tool to derive or verify... Exploratory and computational analysis of networks on a large scale Model selection strategies for computational network analysis Computational approaches to graph spectra Computational methods for network visualization... To date, no book dedicated exclusively to computational network theory has been produced Existing books dealing with related topics such as complex networks and computational analysis of social networks have limited scope, considering only specialized graph classes and