Large Complex Networks Research Articles

Network models of financial systemic risk: a review

The global financial system can be represented as a large complex network in which banks, hedge funds and other financial institutions are interconnected to each other through visible and invisible financial linkages. Recently, a lot of attention has been paid to the understanding of the mechanisms that can lead to a breakdown of this network. This can happen when the existing financial links turn from being a means of risk diversification to channels for the propagation of risk across financial institutions. In this review article, we summarize recent developments in the modeling of financial systemic risk. We focus, in particular, on network approaches, such as models of default cascades due to bilateral exposures or to overlapping portfolios, and we also report on recent findings on the empirical structure of interbank networks. The current review provides a landscape of the newly arising interdisciplinary field lying at the intersection of several disciplines, such as network science, physics, engineering, economics, and ecology.

Positive role of multiplication noise in attaining complete synchronization on large complex networks of dynamical systems

• The dynamical networks with complex structure are investigated. • The condition of synchronization is acquired by theoretical analyses. • Theoretical result shows noise plays a positive role in attaining synchronization. • The key role of eigenvalues of coupling matrix in attaining synchronization is revealed. In this paper, the role of multiplicative noise in attaining complete synchronization on large complex networks of dynamical systems is investigated by theoretical analysis and numerical simulations. Based on the stability theory of stochastic differential equation, we prove that the multiplicative noise plays a positive role in attaining synchronization if the complex networks of dynamical systems are bounded. Moreover, the theoretical result shows that smaller second eigenvalue of coupling matrix is of benefit in attaining complete synchronization. To demonstrate the correctness of theoretical results, the coupled Lorenz systems, Hindmarsh–Rose neuronal systems and Rössler-like systems are performed as numerical examples.

Approximation and hardness results for the Max k-Uncut problem

In the study of the homophily law of large scale complex networks, we get a combinatorial optimization problem which we call the Maxk-Uncut problem. Given an n-vertex undirected graph G=(V,E) with nonnegative weights {we|e∈E} defined on edges, and a positive integer k, the Maxk-Uncut problem asks to find a partition {V1,V2,⋯,Vk} of V such that the total weight of edges that are not cut is maximized. Intuitively, an edge that is not cut connects two vertices with the same or similar attributes since they are in the same part of the partition. Interestingly, the Maxk-Uncut problem is just the complement of the classic Mink-Cut problem. For Maxk-Uncut, we present a randomized (1−kn)2-approximation algorithm, a greedy (1−2(k−1)n)-approximation algorithm, and an Ω(12α)-approximation algorithm by reducing it to Densestk-Subgraph, where α is the approximation ratio of the Densestk-Subgraph problem. More importantly, we show that Maxk-Uncut and Densestk-Subgraph are in fact equivalent in approximability up to a factor of 2. We also prove an approximation hardness result for Maxk-Uncut under the assumption P≠NP.

Parallel Graph Partitioning for Complex Networks

Processing large complex networks like social networks or web graphs has attracted considerable interest. To do this in parallel, we need to partition them into pieces of roughly equal size. Unfortunately, previous parallel graph partitioners originally developed for more regular mesh-like networks do not work well for complex networks. Here we address this problem by parallelizing and adapting the label propagation technique originally developed for graph clustering. By introducing size constraints, label propagation becomes applicable for both the coarsening and the refinement phase of multilevel graph partitioning. This way we exploit the hierarchical cluster structure present in many complex networks. We obtain very high quality by applying a highly parallel evolutionary algorithm to the coarsest graph. The resulting system is both more scalable and achieves higher quality than state-of-the-art systems like ParMetis or PT-Scotch. For large complex networks the performance differences are very big. As an example, our algorithm partitions a web graph with 3.3 G edges in 16 seconds using 512 cores of a high-performance cluster while producing a high quality partition—none of the competing systems can handle this graph on our system.

A New Method for Extracting the Hierarchical Organization of Networks

Extracting the hierarchical organization of networks is currently a pressing task for understanding complex networked systems. The hierarchy of a network is essentially defined by the heterogeneity of link densities of communities at different scales. Here, we define a top-level partition (TLP) as a bipartition of the network (or a sub-network) such that no top-level community (TLC) runs across the two parts. It has been found that a TLP generally has a higher modularity than a non-top-level (TLP) partition when their TLCs have similar sizes and when the link densities of neighboring levels are well separated from each other. A spectral TLP procedure is proposed here to search for TLPs of a network (or sub-network). To extract the hierarchical organization of large complex networks, an algorithm called TLPA has been developed based on the TLP. Experiments have shown that the method developed in this research extract hierarchy accurately from network data.

A feasible graph partition framework for parallel computing of big graph

With the emerging of large scale complex networks, graph computation, such as community detection, meets new technology challenges of extremely large computational cost. In order to deal with these challenges, the parallelism is becoming necessary. Graph partition is a fundamental problem of parallel computing for big graph data. The challenges of graph partition include large numbers of communications between partitions, extreme replication of vertices, and unbalanced partition. In this paper, we propose a feasible graph partition framework for parallel computing of big graph. The framework is based on an objective optimization problem to reduce the bandwidth, memory and storage cost on condition that the load balance could be guaranteed. In this framework, three greedy graph partition algorithms are proposed. By running the algorithms on the different kinds of graphs, including real-world graphs and synthetic graphs, the experimental results show that our algorithms surpass the state of the art heuristic algorithms. For example, running time is reduced more than 21.56% and the communication cost is decreased by more than 17.90% for weighted graph. The adequate experiments verify that our algorithms are capable of solving the problem of graph partition with different needs.

Finding the Most Influential Nodes in Pinning Controllability of Complex Networks

Identifying the best drivers (i.e., the nodes to apply the control signals in a large complex network), which gives the fastest synchronization to the reference state, is a challenge in pinning control of a network. There is not yet a method that exactly predicts a set of best drivers. In this brief, we introduce a novel method that gives first-order approximation for the importance of nodes in pinning control. A spectral measure (the largest eigenvalue of the augmented Laplacian matrix divided by the smallest eigenvalue) is considered as a pinning controllability metric. We develop this method based on the sensitivity analysis of the Laplacian eigenratio, resulting in the scoring of nodes based on their importance in pinning control. The method is rather simple to compute and needs a single eigendecomposition of the Laplacian matrix of the connection graph. Applying this technique on a number of model networks reveals its effectiveness over heuristic approaches.

Bounded-Degree Connected Approximations of Stochastic Networks

We propose a method to find optimal sparse connected approximations for large complex networks of nodes interacting over time. Optimality is measured by Kullback–Leibler divergence. The sparsity is controlled by the user through specifying the in-degrees. The approximations have spanning tree subgraphs, enabling them to depict flow through a network. They can also depict feedback. The approximations can capture both significant statistical dependencies in, and preserve salient topological features of, the full (unknown) network. The proposed method can be applied to networks without restrictions on the dynamics beyond stationarity. We analyze computational and sample complexity. We demonstrate their efficacy using a large-scale simulation of the turtle visual cortex and experimental data from the primate motor cortex.

Optimal control of complex networks: Balancing accuracy and energy of the control action.

Recently, it has been shown that the control energy required to control a large dynamical complex network is prohibitively large when there are only a few control inputs. Most methods to reduce the control energy have focused on where, in the network, to place additional control inputs. We also have seen that by controlling the states of a subset of the nodes of a network, rather than the state of every node, the required energy to control a portion of the network can be reduced substantially. The energy requirements exponentially decay with the number of target nodes, suggesting that large networks can be controlled by a relatively small number of inputs as long as the target set is appropriately sized. Here, we see that the control energy can be reduced even more if the prescribed final states are not satisfied strictly. We introduce a new control strategy called balanced control for which we set our objective function as a convex combination of two competitive terms: (i) the distance between the output final states at a given final time and given prescribed states and (ii) the total control energy expenditure over the given time period. We also see that the required energy for the optimal balanced control problem approximates the required energy for the optimal target control problem when the coefficient of the second term is very small. We validate our conclusions in model and real networks regardless of system size, energy restrictions, state restrictions, input node choices, and target node choices.

QRE: Quick Robustness Estimation for large complex networks

Robustness estimation is critical for the design and maintenance of resilient networks. Existing studies on network robustness usually exploit a single network metric to generate attack strategies, which simulate intentional attacks on a network, and compute a metric-induced robustness estimation, called R. While some metrics are easy to compute, e.g. degree, others require considerable computation efforts, e.g. betweenness centrality. We propose Quick Robustness Estimation (QRE), a new framework and implementation for estimating the robustness of a network in sub-quadratic time, i.e., significantly faster than betweenness centrality, based on the combination of cheap-to-compute network metrics. Experiments on twelve real-world networks show that QRE estimates the robustness better than betweenness centrality-based computation, while being at least one order of magnitude faster for larger networks. Our work contributes towards scalable, yet accurate robustness estimation for large complex networks.

A scalable algorithm for structure identification of complex gene regulatory network from temporal expression data

BackgroundGene regulatory interactions are of fundamental importance to various biological functions and processes. However, only a few previous computational studies have claimed success in revealing genome-wide regulatory landscapes from temporal gene expression data, especially for complex eukaryotes like human. Moreover, recent work suggests that these methods still suffer from the curse of dimensionality if a network size increases to 100 or higher.ResultsHere we present a novel scalable algorithm for identifying genome-wide gene regulatory network (GRN) structures, and we have verified the algorithm performances by extensive simulation studies based on the DREAM challenge benchmark data. The highlight of our method is that its superior performance does not degenerate even for a network size on the order of 104, and is thus readily applicable to large-scale complex networks. Such a breakthrough is achieved by considering both prior biological knowledge and multiple topological properties (i.e., sparsity and hub gene structure) of complex networks in the regularized formulation. We also validate and illustrate the application of our algorithm in practice using the time-course gene expression data from a study on human respiratory epithelial cells in response to influenza A virus (IAV) infection, as well as the CHIP-seq data from ENCODE on transcription factor (TF) and target gene interactions. An interesting finding, owing to the proposed algorithm, is that the biggest hub structures (e.g., top ten) in the GRN all center at some transcription factors in the context of epithelial cell infection by IAV.ConclusionsThe proposed algorithm is the first scalable method for large complex network structure identification. The GRN structure identified by our algorithm could reveal possible biological links and help researchers to choose which gene functions to investigate in a biological event. The algorithm described in this article is implemented in MATLAB Ⓡ, and the source code is freely available from https://github.com/Hongyu-Miao/DMI.git.

Brittleness analysis and important nodes discovery in large time-evolving complex networks

The brittleness analysis and important nodes detection have been a hot spot in the complex networks. How to get the overall feature of the whole network and how to find out some important nodes are requisites to solve these problems. In this paper, we adopt the trace of the adjacency matrix and the centrality of the complex networks to give a quantitative and qualitative analysis of networks being studied. Results show that the k-shell plays a more important role than the degree centrality and the betweenness in finding important nodes, and it can also be used to give direction on the immunization and maintenance of complex networks.

On the Emergence of Islands in Complex Networks

Most growth models for complex networks consider networks comprising a single connected block or island, which contains all the nodes in the network. However, it has been demonstrated that some large complex networks have more than one island, with an island size distribution (Is) obeying a power-law function Is~s-α. This paper introduces a growth model that considers the emergence of islands as the network grows. The proposed model addresses the following two features: (i) the probability that a new island is generated decreases as the network grows and (ii) new islands are created with a constant probability at any stage of the growth. In the first case, the model produces an island size distribution that decays as a power-law Is~s-α with a fixed exponent α=1 and in-degree distribution that decays as a power-law Qi~i-γ with γ=2. When the second case is considered, the model describes island size and in-degree distributions that decay as a power-law with 2<α<∞ and 2<γ<∞, respectively.

Network Models of Financial Systemic Risk: A Review

The global financial system can be represented as a large complex network in which banks, hedge funds and other financial institutions are interconnected to each other through visible and invisible financial linkages. Recently, a lot of attention has been paid to the understanding of the mechanisms that can lead to a breakdown of this network. This can happen when the existing financial links turn from being a means of risk diversification to channels for the propagation of risk across financial institutions. In this review article, we summarize recent developments in the modeling of financial systemic risk. We focus in particular on network approaches, such as models of default cascades due to bilateral exposures or to overlapping portfolios, and we also report on recent findings on the empirical structure of interbank networks. The current review provides a landscape of the newly arising interdisciplinary field lying at the intersection of several disciplines, such as network science, physics, engineering, economics, and ecology.

Community detection by label propagation with compression of flow

The label propagation algorithm (LPA) has been proved to be a fast and effective method for detecting communities in large complex networks. However, its performance is subject to the non-stable and trivial solutions of the problem. In this paper, we propose a modified label propagation algorithm LPAf to efficiently detect community structures in networks. Instead of the majority voting rule of the basic LPA, LPAf updates the label of a node by considering the compression of a description of random walks on a network. A multi-step greedy agglomerative strategy is employed to enable LPAf to escape the local optimum. Furthermore, an incomplete update condition is also adopted to speed up the convergence. Experimental results on both synthetic and real-world networks confirm the effectiveness of our algorithm.

NetworKit: A tool suite for large-scale complex network analysis

AbstractWe introduce NetworKit, an open-source software package for analyzing the structure of large complex networks. Appropriate algorithmic solutions are required to handle increasingly common large graph data sets containing up to billions of connections. We describe the methodology applied to develop scalable solutions to network analysis problems, including techniques like parallelization, heuristics for computationally expensive problems, efficient data structures, and modular software architecture. Our goal for the software is to package results of our algorithm engineering efforts and put them into the hands of domain experts. NetworKit is implemented as a hybrid combining the kernels written in C++ with a Python frontend, enabling integration into the Python ecosystem of tested tools for data analysis and scientific computing. The package provides a wide range of functionality (including common and novel analytics algorithms and graph generators) and does so via a convenient interface. In an experimental comparison with related software, NetworKit shows the best performance on a range of typical analysis tasks.

The Local Edge Machine: inference of dynamic models of gene regulation.

We present a novel approach, the Local Edge Machine, for the inference of regulatory interactions directly from time-series gene expression data. We demonstrate its performance, robustness, and scalability on in silico datasets with varying behaviors, sizes, and degrees of complexity. Moreover, we demonstrate its ability to incorporate biological prior information and make informative predictions on a well-characterized in vivo system using data from budding yeast that have been synchronized in the cell cycle. Finally, we use an atlas of transcription data in a mammalian circadian system to illustrate how the method can be used for discovery in the context of large complex networks.Electronic supplementary materialThe online version of this article (doi:10.1186/s13059-016-1076-z) contains supplementary material, which is available to authorized users.

Generating Random Networks Without Short Cycles

Random graph generation is an important tool for studying large complex networks. Despite abundance of random graph models, constructing models with application-driven constraints is poorly understood. To advance state-of-the-art in this area, we focus on random graphs without short cycles as a stylized family of graphs, and we propose the RandGraph algorithm for randomly generating them. For any constant k, when m = O(n1+1/[2k(k+3)]), RandGraph generates an asymptotically uniform random graph with n vertices, m edges, and no cycle of length at most k using O(n2m) operations. We also characterize the approximation error for finite values of n. To the best of our knowledge, this is the first polynomial-time algorithm for the problem. RandGraph works by sequentially adding m edges to an empty graph with n vertices. Recently, such sequential algorithms have been successful for random sampling problems. Our main contributions to this line of research include introducing a new approach for sequentially approximating edge-specific probabilities at each step of the algorithm and providing a new method for analyzing such algorithms.

Data Analytics for Fault Localization in Complex Networks

We consider the problem of identifying the source of failure in a network after receiving alarms or having observed symptoms. To locate the root cause accurately and timely in a large communication system is challenging because a single fault can often result in a large number of alarms, and multiple faults can occur concurrently. In this paper, we present a new fault localization method using a machine-learning approach. We propose to use logistic regression to study the correlation among network events based on end-to-end measurements. Then based on the regression model, we develop fault hypothesis that best explains the observed symptoms. Unlike previous work, the machine-learning algorithm requires neither the knowledge of dependencies among network events, nor the probabilities of faults, nor the conditional probabilities of fault propagation as input. The “low requirement” feature makes it suitable for large complex networks where accurate dependencies and prior probabilities are difficult to obtain. We then evaluate the performance of the learning algorithm with respect to the accuracy of fault hypothesis and the concentration property. Experimental results and theoretical analysis both show satisfactory performance.

Ivga: A fast force-directed method for interactive visualization of complex networks

Complex networks play a very important role in various fields of science as data structures, which aggregate information about mutual relationships between numerous objects. The structural properties of these large graphs can be scrutinized throughout their interactive visualization. However, visual analysis of complex networks consisting of |V|∼106+ vertices represents a great challenge for nowadays computer systems both from computational and storage perspective. Therefore, the existing graph drawing methods involving greater than O(|V|) time and space complexity cannot be regarded as promising tools in the advent of the Big Data era. We present here a new and very fast graph drawing method with O(|V|) time and space complexity – ivga (interactive visualization of graphs). We evaluate its usefulness and performance by testing ivga on the large complex networks from the Stanford Large Network Dataset Collection. We demonstrate that ivga allows for very fast interactive visualization of large graphs consisting of up to a few million vertices on a regular laptop what makes it very competitive to other state-of-art graph drawing methods. Particularly, we recommend ivga method for interactive visualization of large non-planar complex networks such as small-world and scale-free social networks. The main concept of ivga can be seriously considered in developing tools for visualization and analysis of really huge networks, with billions of vertices and edges, on Big Data systems.