Uniformly optimally reliable graphs: A survey

Romero, Pablo

doi:10.1002/net.22085

Cited by 8 publications

(3 citation statements)

References 80 publications

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“…Here, collective functionality assures redundancy, while independence maintains modularity. We need more comprehensive complex network analyses quantifying these benefits (e.g., Romero, 2021), to guide topological design for DWSs that are affordable and robust under both normal and contingency conditions.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Mesoscale Modeling of Distributed Water Systems Enables Policy Search

Zhou

Dueñas‐Osorio

Doss‐Gollin

et al. 2023

Water Resources Research

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It is widely acknowledged that distributed water systems (DWSs), which integrate distributed water supply and treatment with existing centralized infrastructure, can mitigate challenges to water security from extreme events, climate change, and aged infrastructure. However, it is unclear which are beneficial DWS configurations, i.e., where and at what scale to implement distributed water supply. We develop a mesoscale representation model that approximates DWSs with reduced backbone networks to enable efficient system emulation while preserving key physical realism. Moreover, system emulation allows us to build a multiobjective optimization model for computational policy search that addresses energy utilization and economic impacts. We demonstrate our models on a hypothetical DWS with distributed direct potable reuse (DPR) based on the City of Houston's water and wastewater infrastructure. The backbone DWS with greater than 92% $92\%$ link and node reductions achieves satisfactory approximation of global flows and water pressures, to enable configuration optimization analysis. Results from the optimization model reveal case‐specific as well as general opportunities, constraints, and their interactions for DPR allocation. Implementing DPR can be beneficial in areas with high energy intensities of water distribution, considerable local water demands, and commensurate wastewater reuse capacities. The mesoscale modeling approach and the multiobjective optimization model developed in this study can serve as practical decision‐support tools for stakeholders to search for alternative DWS options in urban settings.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Mesoscale Modeling of Distributed Water Systems Enables Policy Search

Zhou

Dueñas‐Osorio

Doss‐Gollin

et al. 2023

Water Resources Research

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“…The study of similar notions of optimal graphs has occupied a number of researchers in other areas of reliability (such as all-terminal and two-terminal), where only partial results are known about the existence or non-existence of such graphs. The reader can consult a recent survey of uniformly optimally reliable graphs in [8] as well as practical guidelines in [4]. A study of the most-reliable multigraphs can be found in [2,6,7].…”

Section: Optimal Graphsmentioning

confidence: 99%

On the split reliability of graphs

Brown,

McMullin

2023

Networks

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A common model of robustness of a graph against random failures has all vertices operational, but the edges independently operational with probability p$$ p $$. One can ask for the probability that all vertices can communicate (all‐terminal reliability) or that two specific vertices (or terminals) can communicate with each other (two‐terminal reliability). A relatively new measure is split reliability, where for two fixed vertices s$$ s $$ and t$$ t $$, we consider the probability that every vertex communicates with one of s$$ s $$ or t$$ t $$, but not both. In this article, we explore the existence for fixed numbers n≥2$$ n\ge 2 $$ and m≥nprefix−1$$ m\ge n-1 $$ of an optimal connected false(n,mfalse)$$ \left(n,m\right) $$‐graph Gn,m$$ {G}_{n,m} $$ for split reliability, that is, a connected graph with n$$ n $$ vertices and m$$ m $$ edges for which for any other such graph H$$ H $$, the split reliability of Gn,m$$ {G}_{n,m} $$ is at least as large as that of H$$ H $$, for all values of p∈false[0,1false]$$ p\in \left[0,1\right] $$. Unlike the similar problems for all‐terminal and two‐terminal reliability, where only partial results are known, we completely solve the issue for split reliability, where we show that there is an optimal false(n,mfalse)$$ \left(n,m\right) $$‐graph for split reliability if and only if n≤3$$ n\le 3 $$, m=nprefix−1$$ m=n-1 $$, or n=m=4$$ n=m=4 $$.

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“…The topic of networks, and more specifically the reliability of networks, has been extensively studied. (See References [4,9,10] for recent survey papers on the topic.) These networks are modeled using graphs where the vertices of the graph represent the nodes of the network, and the edges of the graph represent node connections.…”

Section: Introductionmentioning

confidence: 99%

More reliable graphs are not always stronger

Graves

2022

Networks

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A graph G$$ G $$ is stronger than a graph H$$ H $$ if G$$ G $$ has at least as many connected spanning subgraphs of size k$$ k $$ as H$$ H $$ for any positive integer k$$ k $$. Counting the number of connected spanning subgraphs of fixed size allows us to compute the reliability of a graph. Formally, the reliability polynomial of a graph is the probability that the graph is connected when each edge is deleted independently with the same fixed probability. A graph G$$ G $$ is uniformly more reliable than H$$ H $$ if its reliability polynomial is greater than or equal to the reliability polynomial of H$$ H $$ for all probabilities. As a direct consequence of the definition, a sufficient condition for G$$ G $$ to be uniformly more reliable than H$$ H $$ is for G$$ G $$ to be stronger than H$$ H $$. In this paper, we show that the sufficient condition is not necessary by providing an example of two infinite families of graphs, Gk$$ {G}_k $$ and Hk$$ {H}_k $$, such that Gk$$ {G}_k $$ is uniformly more reliable than Hk$$ {H}_k $$ but is not stronger than Hk$$ {H}_k $$.

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Uniformly optimally reliable graphs: A survey

Cited by 8 publications

References 80 publications

Mesoscale Modeling of Distributed Water Systems Enables Policy Search

Mesoscale Modeling of Distributed Water Systems Enables Policy Search

On the split reliability of graphs

More reliable graphs are not always stronger

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