Global clustering coefficient in scale-free networks

In this paper, we analyze the behavior of the global clustering coefficient in scale free graphs. We are especially interested in the case of degree distribution with an infinite variance, since such degree distribution is usually observed in real-world networks of diverse nature. There are two common definitions of the clustering coefficient of a graph: global clustering and average local clustering. It is widely believed that in real networks both clustering coefficients tend to some positive constant as the networks grow. There are several models for which the average local clustering coefficient tends to a positive constant. On the other hand, there are no models of scale-free networks with an infinite variance of degree distribution and with a constant global clustering. In this paper we prove that if the degree distribution obeys the power law with an infinite variance, then the global clustering coefficient tends to zero with high probability as the size of a graph grows

Transforming Graphs for Enhanced Attribute Clustering: An Innovative Graph Transformer-Based Method

Graph Representation Learning (GRL) is an influential methodology, enabling a more profound understanding of graph-structured data and aiding graph clustering, a critical task across various domains. The recent incursion of attention mechanisms, originally an artifact of Natural Language Processing (NLP), into the realm of graph learning has spearheaded a notable shift in research trends. Consequently, Graph Attention Networks (GATs) and Graph Attention Auto-Encoders have emerged as preferred tools for graph clustering tasks. Yet, these methods primarily employ a local attention mechanism, thereby curbing their capacity to apprehend the intricate global dependencies between nodes within graphs. Addressing these impediments, this study introduces an innovative method known as the Graph Transformer Auto-Encoder for Graph Clustering (GTAGC). By melding the Graph Auto-Encoder with the Graph Transformer, GTAGC is adept at capturing global dependencies between nodes. This integration amplifies the graph representation and surmounts the constraints posed by the local attention mechanism. The architecture of GTAGC encompasses graph embedding, integration of the Graph Transformer within the autoencoder structure, and a clustering component. It strategically alternates between graph embedding and clustering, thereby tailoring the Graph Transformer for clustering tasks, whilst preserving the graph's global structural information. Through extensive experimentation on diverse benchmark datasets, GTAGC has exhibited superior performance against existing state-of-the-art graph clustering methodologies

ENHANCING RESILIENCE OF COMPLEX NETWORKS: WASHINGTON D.C. URBAN RAIL TRANSIT AS A CASE STUDY

According to the United Nation’s Department of Economic and Social Affairs Population Division, 66% of the world’s population will reside in urban areas by 2050; a boost from 30 % in 1950. Urbanization has indeed triumphed and its speed has brought innovation and economic growth. Its synergies within infrastructure systems are undeniable and have increased the demand for such systems. However, urbanization is one reason infrastructure systems are knocked out of equilibrium and show complex dynamical behavior. Most infrastructure systems have been designed without planning for this magnitude of potential demographic changes; thus redesigns are long overdue. Also, climate change looms. Resource scarcity and host of other factors leave their impacts; all pose some incidence of perturbation in the state of the infrastructure system. These perturbations can affect the system’s resilience, which is a defining property of each system for remaining functional in the midst of disruption from an adverse event. Therefore, it is essential to develop appropriate metrics and methods to enhance the resilience of infrastructures at the network level. Such enhancements are critical for sustainable infrastructure development that is capable of performing satisfactorily through intentional and/or stochastic disruptions. A resilience evaluation of a network typically entails assessing vulnerability and robustness as well as identifying strategies to increasing network efficiency and performance and offering recovery strategies ideally taken in a cost-effective manner. This dissertation uses complex network theory (CNT) as the theoretic basis to enhance the resilience of large-scale infrastructure networks, such as urban rail transit systems. Urban rail transit infrastructures are heterogeneous, complex systems consisting of a large number of interacting nodes and links, which can imitate a network paradigm. Any adverse event leading to a disruption in the interaction and connectivity of network components would dramatically affect the safety and wellbeing of commuters, as well as the direct and indirect costs associated with performance loss. Therefore, enhancing their resilience is necessary. Using the Washington D.C. Urban rail transit as a case study, this dissertation develops a methodology to analyze network topology, compute its efficiency, vulnerability, and robustness in addition to provide a unified metric for assessing the network resilience. The steps of methodology are applied to two models of weighted and unweighted networks. For the weighted model two novel algorithms are proposed to capture the general pattern of ridership in the network, and to reflect the weights on assessing network efficiency, respectively. This dissertation then proposes an effective strategy to increase the network resilience prior to a disruptive event, e.g., a natural disaster, by adding several loop lines in the network for topological enhancement. As such, adding a loop line can create redundancy to the vulnerable components and improve network resilience. Expanding on this, the dissertation offers comparative recovery strategies and cost model in the case of disruption. An effective recovery strategy must demonstrate rapid optimal restoration of a disrupted system performance while minimizing recovery costs. In summary, the systematic methodology described above, assesses and enhances the network resilience. The initial results rank the most vulnerable and robust components of the network. The algorithms developed throughout the study advance the weighted network analysis state of art. The topological enhancement strategy offered basis to justify capital improvement. Post failure recovery analysis and the cost model serves to inform decision makers in identifying best recover strategies with special attention not only to restoring performance of a system but also on reducing associated failure and recovery costs. The use of the methodology proposed in this dissertation may lead to significant societal benefits by reducing the risk of catastrophic failures, providing references for mitigation of disruption due to adverse events, and offering resilience- based strategies, and related pursuits