Water Distribution Networks (WDNs) are critical infrastructures that ensure a continuous supply of safe water to homes. In the face of challenges, like water scarcity, establishing resilient networks is imperative, especially in regions vulnerable to water crises. This study evaluates the resilience of network designs through graph theory, including its hydraulic feasibility using EPANET software, an aspect often overlooked. Novel mathematical algorithms, including Resilience by Design (RbD) and Resilience-strengthening (RS) algorithms, provide cost-effective and resilient network designs, even with budget constraints. A novel metric, Water Availability (WA), is introduced to offer a comprehensive measure of network resilience, thereby addressing ongoing discrepancies in resilience evaluation methods. Practical benefits are illustrated through a case study in which a resilient-by-design reclaimed water network is created, and an existing equivalent non-resilient network is improved. The resilient-by-design network demonstrates remarkably better results compared to the equivalent non-resilient design, including up to a 36 % reduction in the probability of service disruptions and a nearly 65 % decrease in the annual average unserved water due to service disruptions. These findings underscore the enormous advantages of a resilience-focused network design approach. When compared to the equivalent non-resilient design, the resilient-by-design network generated effectively safeguards up to a significant 91,700m³ of water from the impacts of water disruption events over a 50-year operational period. In addition, the resilient-by-design WDN solution incurs a subtle decrease in overall costs compared to consuming tap water from the drinking WDN baseline over a 50-year operational period. These findings highlight the cost-effectiveness of the approach, even offering financial benefits. This paper builds on our previous research by expanding its scope to include resilience considerations, providing algorithms that can be easily adapted from reclaimed to drinking WDNs. Ultimately, we contribute to the enhancement of water resource management and infrastructure planning in ever-evolving urban environments.

We propose an analytical approach to approximate the average two-Terminal reliability (ATT R) for graphs where a fraction of the nodes is removed. The approximation is based on the generating function of the network's degree distribution under random node removals and stochastic degree-based node removals. Through validation on synthetic graphs, including Erdos Renyi random graphs and Barabasi-Albert graphs, as well as four real-world networks from the Internet Topology Zoo, we observe that the analytical method effectively approximates the average two-Terminal reliability under random node removals for synthetic graphs. In the case of real-world graphs under random and stochastic degree-based node removals or synthetic graphs under stochastic degree-based node removals, the analytical ap-proximation yields reasonably accurate results when the fraction of removed nodes is small, specifically less than 10%, provided that the initial analytical approximation closely aligns with the real ATT R values.

Network controllability and its robustness has been widely studied. However, analytical methods to calculate network controllability with respect to node removals are currently lacking. This paper develops methods, based upon generating functions for the in- and out-degree distributions, to approximate the minimum number of driver nodes needed to control directed networks, during random and targeted node removals. By validating the proposed methods on synthetic and real-world networks, we show that our methods work very well in the case of random node removals and reasonably well in the case of targeted node removals, in particular for moderate fractions of attacked nodes.

Network controllability is a critical attribute of dynamic networked systems. Investigating methods to restore network controllability after network degradation is crucial for enhancing system resilience. In this study, we develop an analytical method based on degree distributions to estimate the minimum fraction of required driver nodes for network controllability under random node additions after the random removal of a subset of nodes. The outcomes of our method closely align with numerical simulation results for both synthetic and real-world networks. Additionally, we compare the efficacy of various node recovery strategies across directed Erdös-Rényi (ER) networks, swarm signaling networks (SSNs), and directed Barabàsi Albert (BA) networks. Our findings indicate that the most efficient recovery strategy for directed ER networks and SSNs is the greedy strategy, which considers node betweenness centrality. Similarly, for directed BA networks, the greedy strategy focusing on node degree centrality emerges as the most efficient. These strategies outperform recovery approaches based on degree centrality or betweenness centrality, as well as the strategy involving random node additions.

We introduce a Markov Modulated Process (MMP) to describe human mobility. We represent the mobility process as a time-varying graph, where a link specifies a connection between two nodes (humans) at any discrete time step. Each state of the Markov chain encodes a certain modification to the original graph. We show that our MMP model successfully captures the main features of a random mobility simulator, in which nodes moves in a square region. We apply our MMP model to human mobility, measured in a library.

A nearly universal trend in science today is the prominence of ever-increasing collaborative teams. Hence, identifying the relative credit due to each collaborator of published studies is of high significance. Although numerous methods have been employed to address this issue, allocating credit to all co-authors of new papers remains challenging. To address this cold-start issue, we introduce a credit allocation algorithm based on the co-citing network that captures the co-authors' shared credit of a multi-authored publication. Using the American Physical Society publication data, we validate the method by examining papers by Nobel laureates. Accordingly, we perform many experiments to demonstrate that the proposed method can be implemented on academic papers in any period after publication with a significantly higher degree of accuracy and robustness than the existing algorithms applied to new papers. This method enables us to explore the universal credit evolution pattern of scientific elites. Importantly, by testing the relation between an author's credit and authorship byline, we observe that the first authors of papers are currently assigned less credit than in the early days with respect to physics. With collaboration and a large team set to dominate the agenda of the current science system, our study provides a more effective method for allocating early credit to co-authors of a paper, which may be beneficial to various academic activities, including faculty hiring, funding, and promotion decisions.

The average fraction of infected nodes, in short the prevalence, of the Markovian ɛ-SIS (susceptible-infected-susceptible) process with small self-infection rate ɛ>0 exhibits, as a function of time, a typical "two-plateau" behavior, which was first discovered in the complete graph KN. Although the complete graph is often dismissed as an unacceptably simplistic approximation, its analytic tractability allows to unravel deeper details, that are surprisingly also observed in other graphs as demonstrated by simulations. The time-dependent mean-field approximation for KN performs only reasonably well for relatively large self-infection rates, but completely fails to mimic the typical Markovian ɛ-SIS process with small self-infection rates. While self-infections, particularly when their rate is small, are usually ignored, the interplay of nodal self-infection and spread over links may explain why absorbing processes are hardly observed in reality, even over long time intervals.