Measure Of Network Robustness Research Articles

Gene interaction network analysis in multiple myeloma detects complex immune dysregulation associated with shorter survival

The plasma cell cancer multiple myeloma (MM) varies significantly in genomic characteristics, response to therapy, and long-term prognosis. To investigate global interactions in MM, we combined a known protein interaction network with a large clinically annotated MM dataset. We hypothesized that an unbiased network analysis method based on large-scale similarities in gene expression, copy number aberration, and protein interactions may provide novel biological insights. Applying a novel measure of network robustness, Ollivier-Ricci Curvature, we examined patterns in the RNA-Seq gene expression and CNA data and how they relate to clinical outcomes. Hierarchical clustering using ORC differentiated high-risk subtypes with low progression free survival. Differential gene expression analysis defined 118 genes with significantly aberrant expression. These genes, while not previously associated with MM, were associated with DNA repair, apoptosis, and the immune system. Univariate analysis identified 8/118 to be prognostic genes; all associated with the immune system. A network topology analysis identified both hub and bridge genes which connect known genes of biological significance of MM. Taken together, gene interaction network analysis in MM uses a novel method of global assessment to demonstrate complex immune dysregulation associated with shorter survival.

Network Modeling of Apoptosis-Related Genes Reveal a High-Risk Cluster in Multiple Myeloma

The plasma cell cancer multiple myeloma (MM) varies significantly in genomic characteristics, response to therapy, and long-term prognosis. Recent studies have shown the complexity of biological systems associated with poor outcomes in MM. One hallmark of cancer pathogenesis, applicable to MM, is the dysregulation of apoptotic cell death. Understanding the relationship between known drivers of MM and critical pathways in the apoptosis gene network is key towards identifying novel therapeutic targets. Recently, there has been a surge of interest in developing network-based approaches to study gene interactions and use machine learning tools to further elucidate useful biological factors. It is well-known that genes operate as network systems; however, interpretable computational methods are still limited. In this work, we combine a set of network analysis tools to provide novel biological insights into how the apoptosis network is dysregulated in MM. We identify a set of genes associated with poor outcomes in MM using both univariate and network-based approaches. Genomic and clinical data, including RNA-sequencing (RNA-seq) and copy number alteration (CNA), for 659 subjects were obtained from the Multiple Myeloma Research Foundation's CoMMpass (IA19) database. The apoptosis gene set is defined using the apoptosis hallmark pathway provided by the gene set enrichment analysis tool. The interactions between the given list of genes are provided via MetaCore. Univariate gene modeling is done using a Cox proportional hazards model. To determine the relationship between genes, we used Ollivier-Ricci curvature (ORC), a measure of network robustness. The ORC value between two genes incorporates information about both the two genes in question and their local neighborhoods, a concept critical for understanding the role a neighborhood of genes has on a given connection. Once edge information is computed, we used a uniform manifold approximation and projection (UMAP) to reduce the complexity of the dataset. The resulting projection is clustered using k-means clustering, and survival analysis of the given clusters is done using the Kaplan-Meier (KM) method. To understand which features are most informative, we used a random forest to predict the cluster labels using the RNA-seq data. After five-fold cross-validation, feature importance was computed using random forest's permutation importance methodology. All statistical significance testing was corrected for multiple comparisons via the Benjamini-Hochberg false discovery rate method with an alpha set to 0.05. Using RNA-seq data of genes associated with apoptosis, we identified nineteen apoptosis genes associated with poor prognosis in MM. Some of these genes, such as ANXA1 and WEE1 are known to have an effect in MM, while others, such as AVPR1A have yet to be studied extensively for their role in MM. When examining CNA data associated with apoptosis, a similar pattern emerged - nineteen genes were associated with poor prognosis. However, between the two key gene sets found by RNA-seq and CNA, only FAS appeared in both lists. FAS, Fas Cell Surface Death Receptor, is critical for triggering apoptosis. Half of the 37 genes identified using RNA-seq and CNA data are directly connected to the TP53 gene. While TP53 mutations are not a marker of the presence of MM itself, TP53 mutations have been shown to be associated with poorer outcomes. After reducing the complexity of the Ollivier-Ricci curvature analysis of the RNA-seq apoptosis network using UMAPs, the data revealed five clusters with KM curves for progression-free survival were significant (p<0.001). To identify the genes most impactful for separating these clusters, we trained a random forest model on the RNA-seq - note, it was the edge robustness values that were clustered - and evaluated its performance for predicted class membership. The model had an overall accuracy of approximately 78% with F1 scores for each class ranging from 61% to 87%. The top three features associated with the model include CTNNB1, RELA, and LEF1. Of note, CTNNB1 and RELA are directly interacted with the TP53 gene. Because these three genes discriminate between all five clusters, their expression levels may play a mediating effect on MM outcomes. Further investigation with external datasets is needed to validate these results.

Robustness measurement of multiplex networks based on multiplex motifs

Multiplex network is a classic multilayer network model that can be used to model complex systems in the real world. Previous measures of network robustness mainly focused on the low-order characteristics of network topology, and the existence of motifs in the network is ignored. As simple building blocks of multiplex networks, multiplex motifs are indispensable tools to understand the structure of local networks. Therefore, from a mesoscopic perspective, we propose an index called normalized motif entropy to measure the robustness of multiplex networks under malicious attacks, and the effectiveness of our method is verified in real multiplex networks.

Measures and Optimization for Robustness and Vulnerability in Disconnected Networks

The function or performance of a network is strongly dependent on its robustness, quantifying the ability of the network to continue functioning under perturbations. While a wide variety of robustness metrics have been proposed, they have their respective limitations. In this paper, we propose to use the forest index as a measure of network robustness, which overcomes the deficiencies of existing metrics. Using such a measure as an optimization criterion, we propose and study the problem of breaking down a network by attacking some key edges. We show that the objective function of the problem is monotonic but not submodular, which impose more challenging on the problem. We thus resort to greedy algorithms extended for non-submodular functions by iteratively deleting the most promising edges. We first propose a simple greedy algorithm with a proved bound for the approximation ratio and cubic-time complexity. To confront the computation challenge for large networks, we further propose an improved nearly-linear time greedy algorithm, which significantly speeds up the process for edge selection but sacrifices little accuracy. Extensive experimental results for a large set of real-world networks verify the effectiveness and efficiency of our algorithms, demonstrating that our algorithms outperform several baseline schemes.

On measuring network robustness for weighted networks

Network robustness measures how well network structure is strong and healthy when it is under attack, such as vertices joining and leaving. It has been widely used in many applications, such as information diffusion, disease transmission, and network security. However, existing metrics, including node connectivity, edge connectivity, and graph expansion, can be suboptimal for measuring network robustness since they are inefficient to be computed and cannot directly apply to the weighted networks or disconnected networks. In this paper, we define the \({\mathcal {R}}\)-energy as a new robustness measurement for weighted networks based on the method of spectral analysis. \({\mathcal {R}}\)-energy can cope with disconnected networks and is efficient to compute with a time complexity of \(O(|V|+|E|)\), where V and E are sets of vertices and edges in the network, respectively. Our experiments illustrate the rationality and efficiency of computing \({\mathcal {R}}\)-energy: (1) Removal of high degree vertices reduces network robustness more than that of random or small degree vertices; (2) it takes as little as 120 s to compute for a network with about 6M vertices and 33M edges. We can further detect events occurring in a dynamic Twitter network with about 130K users and discover interesting weekly tweeting trends by tracking changes to \({\mathcal {R}}\)-energy.

Measuring Robustness and Coverage of Transportation Networks with Multiple Routes and Hubs

Robustness, or the resilience of a system to failures and attacks without changing its mode of operation, is an increasingly important characteristic of networks. Even though there are several measures of network robustness, they are not applicable to networks with several links between the same pair of nodes (multiline networks), such as transportation systems. A local index and a set of global indexes are introduced to capture the robustness of having multiple connections between the same nodes. These new indexes are designed to uncover the potential vulnerabilities of network components (i.e., hubs, terminals, and lines) to possible malfunctions that might not be well identified with traditional connectivity matrix–based methods. Results of these measures are compared with traditional network connectivity and robustness indexes for both simulated networks that vary topologically and are each composed of multiple lines that overlap in connectivity and a set of representative rail systems in the United States. A discussion of the applicability of these new multiline robustness measures for decision makers and transportation planners to prioritize the development, protection, or maintenance of network components is provided.

Robustness of Stakeholder Response Networks for Infrastructure System Protection

During disasters, maintaining the functionality of infrastructure systems requires effective and efficient stakeholder collaboration. Aiming at understanding stakeholder collaboration robustness in response to infrastructure interruptions, this research proposes a network-based approach that comprises two steps: (1) generating response networks through network analysis; and (2) indicating time evolution of stakeholder collaboration robustness through a measure of topological network robustness. A case study, which examines stakeholder collaboration on protecting transportation infrastructure in Texas during Hurricane Harvey, was conducted to demonstrate the feasibility and validity of the proposed approach. From the case study, several social processes were identified to facilitate decision making in future disaster response. In addition, the case study implies that the documentation and sharing of disaster response information at the local level require further improvements. Overall, this research promotes interorganizational collaboration in emergency management through creating a systematic, network-based approach for quantitatively analyzing time evolution of stakeholder collaboration robustness. The findings provide implementable support for stakeholders to advance their collaboration knowledge, thereby enhancing the overall response performance.

Quantifying the Robustness of Complex Networks with Heterogeneous Nodes

The robustness of a complex network measures its ability to withstand random or targeted attacks. Most network robustness measures operate under the assumption that the nodes in a network are homogeneous and abstract. However, most real-world networks consist of nodes that are heterogeneous in nature. In this work, we propose a robustness measure called fitness-incorporated average network efficiency, that attempts to capture the heterogeneity of nodes using the ‘fitness’ of nodes in measuring the robustness of a network. Further, we adopt the same measure to compare the robustness of networks with heterogeneous nodes under varying topologies, such as the scale-free topology or the Erdős–Rényi random topology. We apply the proposed robustness measure using a wireless sensor network simulator to show that it can be effectively used to measure the robustness of a network using a topological approach. We also apply the proposed robustness measure to two real-world networks; namely the CO2 exchange network and an air traffic network. We conclude that with the proposed measure, not only the topological structure, but also the fitness function and the fitness distribution among nodes, should be considered in evaluating the robustness of a complex network.

Modeling the criticality of a regional trucking network at the industry level: Evidence from the province of Ontario, Canada

Regional economic productivity and growth require a reliable, robust road transportation network to facilitate the movement of freight by trucks. Previous network robustness measures focus on the physical effects of network disruptions, such as travel time delays resulting from disabled links. However, the economic impacts associated with disruptions of freight flows must also be accounted for. This paper presents a methodology for generating industry-specific origin-destination freight demand matrices for the province of Ontario, Canada using truck GPS tracking data and the provincially mandated Commercial Vehicle Survey. Six mutually exclusive industry groups are defined, whose spatial interactions are expected to vary. A cost per shipment is estimated for each industry group and used to examine the value of goods carried on each network link for the 1 pm peak hour. Criticality is explored through the combined impacts on travel time due to link disruption and the value of goods carried on the respective links. As expected, a subset of links belonging to important trade corridors are highlighted as critical. Such characterization of the network provides an in-depth view of the economic activity and criticalities associated with regional trucking networks and can inform planning and mitigation measures to maximize network-wide benefits.

Extremal Results on Vertex and Link Residual Closeness

Robustness of the network topology is a key aspect in the design of computer networks. Vertex (Link, respectively) residual closeness is a new graph-theoretic concept defined as a measure of network robustness due to the failure of individual vertices (links, respectively). In this paper, we identify the trees and unicyclic graphs with the first a few smallest vertex residual closeness, and determine the graphs that minimize or maximize the vertex (link, respectively) residual closeness over some classes of graphs.

Serotonin signaling modulates aging-associated metabolic network integrity in response to nutrient choice in Drosophila melanogaster

Aging arises from complex interactions among multiple biochemical products. Systems-level analyses of biological networks may provide insights into the causes and consequences of aging that evade single-gene studies. We have previously found that dietary choice is sufficient to modulate aging in the vinegar fly, Drosophila melanogaster. Here we show that nutrient choice influenced several measures of metabolic network integrity, including connectivity, community structure, and robustness. Importantly, these effects are mediated by serotonin signaling, as a mutation in serotonin receptor 2A (5-HT2A) eliminated the effects of nutrient choice. Changes in network structure were associated with organism resilience and increased susceptibility to genetic perturbation. Our data suggest that the behavioral or perceptual consequences of exposure to individual macronutrients, involving serotonin signaling through 5-HT2A, qualitatively change the state of metabolic networks throughout the organism from one that is highly connected and robust to one that is fragmented, fragile, and vulnerable to perturbations.

Robustness of Brain Structural Networks Is Affected in Cognitively Impaired MS Patients.

The robustness of brain structural networks, estimated from diffusion MRI data, may be relevant to cognition. We investigate whether measures of network robustness, such as Ollivier-Ricci curvature, can explain cognitive impairment in multiple sclerosis (MS). We assessed whether local (i.e., cortical area) and/or global (i.e., whole brain) robustness, differs between cognitively impaired (MSCI) and non-impaired (MSNI) MS patients. Fifty patients, with Expanded Disability Status Scale mean (m): 3.2, disease duration m: 12 years, and age m: 40 years, were enrolled. Cognitive impairment scores were estimated from the Minimal Assessment of Cognitive Function in Multiple Sclerosis. Images were obtained in a 3T MRI using a diffusion protocol with a 2 min acquisition time. Brain structural networks were created using 333 cortical areas. Local and global robustness was estimated for each individual, and comparisons were performed between MSCI and MSNI patients. 31 MSCI and 10 MSNI patients were included in the analyses. Brain structural network robustness and centrality showed significant correlations with cognitive impairment. Measures of network robustness and centrality identified specific cortical areas relevant to MS-related cognitive impairment. These measures can be obtained on clinical scanners and are succinct yet accurate potential biomarkers of cognitive impairment.

A comparative analysis of link removal strategies in real complex weighted networks

In this report we offer the widest comparison of links removal (attack) strategies efficacy in impairing the robustness of six real-world complex weighted networks. We test eleven different link removal strategies by computing their impact on network robustness by means of using three different measures, i.e. the largest connected cluster (LCC), the efficiency (Eff) and the total flow (TF). We find that, in most of cases, the removal strategy based on the binary betweenness centrality of the links is the most efficient to disrupt the LCC. The link removal strategies based on binary-topological network features are less efficient in decreasing the weighted measures of the network robustness (e.g. Eff and TF). Removing highest weight links first is the best strategy to decrease the efficiency (Eff) in most of the networks. Last, we found that the removal of a very small fraction of links connecting higher strength nodes or of highest weight does not affect the LCC but it determines a rapid collapse of the network efficiency Eff and the total flow TF. This last outcome raises the importance of both to adopt weighted measures of network robustness and to focus the analyses on network response to few link removals.

Estimating Robustness Through Kirchhoff Index in Mesh Graphs

The Kirchhoff index is a new measure of network robustness. In this paper, we study the robustness of n × m mesh graphes (denoted by M n×m ) by determining the most important edges and the least important edges. In other words, we aim to find the edges (denoted by edge max ) which have the biggest impact on the Kirchhoff index after the edge is deleted and the edges (denoted by edge min ) which have the least impact on Kirchhoff index after the edge is deleted. The distributions of edge max and edge min are fully characterized. Consequently, we propose a new strategy called modified resistance distance strategy to locate edge max and edge min of M n×m . The applicability and rationality of the modified resistance distance strategy in mesh graphs is proved by comparing with other known strategies, such as the semi-random strategy, the degree product strategy and the resistance distance strategy. Moreover, the modified resistance distance strategy is still applicable in mesh graphs when we use the algebraic connectivity as the measure of graph robustness.

A novel measure of edge and vertex centrality for assessing robustness in complex networks

In this work, we propose a novel robustness measure for networks, which we refer to as Effective Resistance Centrality of a vertex (or an edge), defined as the relative drop of the Kirchhoff index due to deletion of this vertex (edge) from the network. Indeed, we provide a local robustness measure, able to catch which is the effect of either a specific vertex or a specific edge on the network robustness. The validness of this new measure is illustrated on some typical graphs and on a wide variety of well-known model networks. Furthermore, we analyse the topology of the US domestic flight connections. In particular, we investigate the role that airports play in maintaining the structure of the entire network.

Designing Robust Air Transportation Networks via Minimizing Total Effective Resistance

Designing a robust air transportation network is an ongoing research effort that seeks to improve the extent of a network being connected against failures and attacks. The total effective resistance can be a promising measure for network robustness as demonstrated by case studies conducted in this paper. To enhance the robustness of air transportation networks, we consider to solve a flight route selection problem in which a set of routes is chosen from a candidate route set to minimize the utility function defined by the total effective resistance. Since it is an integer nonlinear programming problem, to balance the tradeoff between optimality performance and computational efficiency, two methods that implement the total effective resistance are developed to suit different network scales. For small/medium-scale networks, we develop an interior-point method based on convex relaxation and duality gap, which achieves a near optimality performance within acceptable time. For large-scale networks, we develop an accelerated greedy algorithm based on proved monotone submodularity, which can substantially reduce the computation time in deriving a good solution with a guaranteed optimality gap. Their optimality performance and computational efficiency are verified and compared through numerical results. Moreover, three case studies from real-world examples are also performed to demonstrate the application of the proposed methods for different network scales.

Kirchhoffian indices for weighted digraphs

The resistance indices, namely the Kirchhoff index and its generalisations, have undergone intense critical scrutiny in recent years. Based on random walks, we derive three Kirchhoffian indices for strongly connected and weighted digraphs. These indices are expressed in terms of (i) hitting times and (ii) the trace and eigenvalues of suitable matrices associated to the graph, namely the asymmetric Laplacian, the diagonally scaled Laplacian and their Moore–Penrose inverses. The appropriateness of the generalised Kirchhoff index as a measure of network robustness is discussed, providing an alternative interpretation which is supported by an empirical application to the World Trade Network.

Network robustness and residual closeness

A central issue in the analysis of complex networks is the assessment of their robustness and vulnerability. A variety of measures have been proposed in the literature to quantify the robustness of networks and a number of graph-theoretic parameters have been used to derive formulas for calculating network reliability. In this paper, we study the vulnerability of interconnection networks to the failure of individual nodes, using a graph-theoretic concept of residual closeness as a measure of network robustness which provides a much fuller characterization of the network.

Robustness of Regular Caterpillars

Robustness of the network topology is a key aspect in the design of computer networks. Vertex residual closeness is a new graph-theoretic concept defined as a measure of network robustness. In this model, edges are perfectly reliable and the vertices fail independently of each other. In this paper, vertex residual closeness of paths and regular caterpillars are calculated by giving an insight of how to evaluate the vertex residual closeness of path-like graphs.

A comparative study of network robustness measures

The robustness is an important functionality of networks because it manifests the ability of networks to resist failures or attacks. Many robustness measures have been proposed from different aspects, which provide us various ways to evaluate the network robustness. However, whether these measures can properly evaluate the network robustness and which aspects of network robustness these measures can evaluate are still open questions. Therefore, in this paper, a thorough introduction over attacks and robustness measures is first given, and then nine widely used robustness measures are comparatively studied. To validate whether a robustness measure can evaluate the network robustness properly, the sensitivity of robustness measures is first studied on both initial and optimized networks. Then, the performance of robustness measures in guiding the optimization process is studied, where both the optimization process and the obtained optimized networks are studied. The experimental results show that, first, the robustness measures are more sensitive to the changes in initial networks than to those in optimized networks; second, an optimized network may not be useful in practical situations because some useful functionalities, such as the shortest path length and communication efficiency, are sacrificed too much to improve the robustness; third, the robustness of networks in terms of closely correlated robustness measures can often be improved together. These results indicate that it is not wise to just apply the optimized networks obtained by optimizing over one certain robustness measure into practical situations. Practical requirements should be considered, and optimizing over two or more Received February 22, 2016; accepted September 29, 2016 E-mail: neouma@163.com suitable robustnessmeasures simultaneously is also a promising way.