Spectral Partitioning for Node Criticality

Finding critical nodes in a network is a significant task, highly relevant to network vulnerability and security. We consider the node criticality problem as an algebraic connectivity minimization problem where the objective is to choose nodes which minimize the algebraic connectivity of the resulting network. Previous suboptimal solutions of the problem suffer from the computational complexity associated with the implementation of a maximization consensus algorithm. In this work, we use spectral partitioning concepts introduced by Fiedler, to propose a new suboptimal solution which significantly reduces the implementation complexity. Our approach, combined with recently proposed distributed Fiedler vector calculation algorithms enable each node to decide by itself whether it is a critical node. If a single node is required then the maximization algorithm is applied on a restricted set of nodes within the network. We derive a lower bound for the achievable algebraic connectivity when nodes are removed from the network and we show through simulations that our approach leads to algebraic connectivity values close to this lower bound. Similar behaviour is exhibited by other approaches at the expense, however, of a higher implementation complexity

A network approach for power grid robustness against cascading failures

Cascading failures are one of the main reasons for blackouts in electrical power grids. Stable power supply requires a robust design of the power grid topology. Currently, the impact of the grid structure on the grid robustness is mainly assessed by purely topological metrics, that fail to capture the fundamental properties of the electrical power grids such as power flow allocation according to Kirchhoff's laws. This paper deploys the effective graph resistance as a metric to relate the topology of a grid to its robustness against cascading failures. Specifically, the effective graph resistance is deployed as a metric for network expansions (by means of transmission line additions) of an existing power grid. Four strategies based on network properties are investigated to optimize the effective graph resistance, accordingly to improve the robustness, of a given power grid at a low computational complexity. Experimental results suggest the existence of Braess's paradox in power grids: bringing an additional line into the system occasionally results in decrease of the grid robustness. This paper further investigates the impact of the topology on the Braess's paradox, and identifies specific sub-structures whose existence results in Braess's paradox. Careful assessment of the design and expansion choices of grid topologies incorporating the insights provided by this paper optimizes the robustness of a power grid, while avoiding the Braess's paradox in the system.Comment: 7 pages, 13 figures conferenc

Similarity-Aware Spectral Sparsification by Edge Filtering

In recent years, spectral graph sparsification techniques that can compute ultra-sparse graph proxies have been extensively studied for accelerating various numerical and graph-related applications. Prior nearly-linear-time spectral sparsification methods first extract low-stretch spanning tree from the original graph to form the backbone of the sparsifier, and then recover small portions of spectrally-critical off-tree edges to the spanning tree to significantly improve the approximation quality. However, it is not clear how many off-tree edges should be recovered for achieving a desired spectral similarity level within the sparsifier. Motivated by recent graph signal processing techniques, this paper proposes a similarity-aware spectral graph sparsification framework that leverages efficient spectral off-tree edge embedding and filtering schemes to construct spectral sparsifiers with guaranteed spectral similarity (relative condition number) level. An iterative graph densification scheme is introduced to facilitate efficient and effective filtering of off-tree edges for highly ill-conditioned problems. The proposed method has been validated using various kinds of graphs obtained from public domain sparse matrix collections relevant to VLSI CAD, finite element analysis, as well as social and data networks frequently studied in many machine learning and data mining applications

Considerations about multistep community detection

The problem and implications of community detection in networks have raised a huge attention, for its important applications in both natural and social sciences. A number of algorithms has been developed to solve this problem, addressing either speed optimization or the quality of the partitions calculated. In this paper we propose a multi-step procedure bridging the fastest, but less accurate algorithms (coarse clustering), with the slowest, most effective ones (refinement). By adopting heuristic ranking of the nodes, and classifying a fraction of them as `critical', a refinement step can be restricted to this subset of the network, thus saving computational time. Preliminary numerical results are discussed, showing improvement of the final partition.Comment: 12 page

Critical Node Identifcation for accessing network vulnerability, a necessary consideration

Timely identification of critical nodes is crucial for assessing network vulnerability and survivability. This thesis presents two new approaches for the identification of critical nodes in a network with the first being an intuition based approach and the second being build on a mathematical framework. The first approach which is referred to as the Combined Banzhaf & Diversity Index (CBDI) uses a newly devised diversity metric, that uses the variability of a node’s attributes relative to its neighbours and the Banzhaf power index which characterizes the degree of participation of a node in forming the shortest path route. The Banzhaf power index is inspired from the theory of voting games in game theory whereas, the diversity index is inspired from the analysis and understanding of the influence of the average path length of a network on its performance. This thesis also presents a new approach for evaluating this average path length metric of a network with reduced computational complexity and proposes a new mechanism for reducing the average path length of a network for relatively larger network structures. The proposed average path length reduction mechanism is tested for a wireless sensor network and the results compared for multiple existing approaches. It has been observed using simulations that, the proposed average path length reduction mechanism outperforms existing approaches by reducing the average path length to a greater extent and with a simpler hardware requirement. The second approach proposed in this thesis for the identification of critical nodes is build on a mathematical framework and it is based on suboptimal solutions of two optimization problems, namely the algebraic connectivity minimization problem and a min-max network utility problem. The former attempts to address the topological as- pect of node criticality whereas, the latter attempts to address its connection-oriented nature. The suboptimal solution of the algebraic connectivity minimization problem is obtained through spectral partitioning considerations. This approach leads to a distributed solution which is computationally less expensive than other approaches that exist in the literature and is near optimal, in the sense that it is shown through simulations to approximate a lower bound which is obtained analytically. Despite the generality of the proposed approaches, this thesis evaluates their performance on a wireless ad hoc network and demonstrates through extensive simulations that the proposed solutions are able to choose more critical nodes relative to other approaches, as it is observed that when these nodes are removed they lead to the highest degrada- tion in network performance in terms of the achieved network throughput, the average network delay, the average network jitter and the number of dropped packets

Combined Banzhaf & Diversity Index (CBDI) for critical node detection

Critical node discovery plays a vital role in assessing the vulnerability of a computer network to malicious attacks and failures and provides a useful tool with which one can greatly improve network security and reliability. In this paper, we propose a new metric to characterize the criticality of a node in an arbitrary computer network which we refer to as the Combined Banzhaf & Diversity Index (CBDI). The metric utilizes a diversity index which is based on the variability of a node׳s attributes relative to its neighbours and the Banzhaf power index which characterizes the degree of participation of a node in forming shortest paths. The Banzhaf power index is inspired from the theory of voting games in game theory. The proposed metric is evaluated using analysis and simulations. The criticality of nodes in a network is assessed based on the degradation in network performance achieved when these nodes are removed. We use several performance metrics to evaluate network performance including the algebraic connectivity which is a spectral metric characterizing the connectivity robustness of the network. Extensive simulations in a number of network topologies indicate that the proposed CBDI index chooses more critical nodes which, when removed, degrade network performance to a greater extent than if critical nodes based on other criticality metrics were removed