Some Topological Considerations in Network Theory

Reza, F.M.

doi:10.1109/tct.1958.1086421

Cited by 22 publications

(8 citation statements)

References 25 publications

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“…where [NL] and [V] are given in Equations ( 10) and (11), respectively, and where × means matrix multiplication.…”

Section: Node-loop Methodsmentioning

confidence: 99%

“…All versions of the methods of solving problems related to a network of pipes are based on the mass and energy balance in the network at hand. The amount of gas flowing in and out of every node of the network and the pressure equilibrium in every loop or any closed path must be preserved, closely following the first and second Kirchhoff laws [11,12], keeping the network in a state of balance. To ensure such a balance, two approaches can be used, as follows.…”

Section: Introductionmentioning

confidence: 99%

“…Explanation of the two shown methods applied to the solution of the classical problem of flow distribution in looped pipe networks, improved Hardy Cross [5] versus nodeloop method [6]; • Detailed explanation of the correction of flow ∆Q in the Hardy Cross method [7] (with application to the correction of diameters ∆D during optimisation); • Sources and foundation of the idea on which the Hardy Cross method for pipe networks is based [8][9][10]; • Topological properties of pipe networks: number of pipes, nodes and loops and relations among them [11,81];…”

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confidence: 99%

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Two Iterative Methods for Sizing Pipe Diameters in Gas Distribution Networks with Loops

Brkić

2024

Computation

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Closed-loop pipe systems allow the possibility of the flow of gas from both directions across each route, ensuring supply continuity in the event of a failure at one point, but their main shortcoming is in the necessity to model them using iterative methods. Two iterative methods of determining the optimal pipe diameter in a gas distribution network with closed loops are described in this paper, offering the advantage of maintaining the gas velocity within specified technical limits, even during peak demand. They are based on the following: (1) a modified Hardy Cross method with the correction of the diameter in each iteration and (2) the node-loop method, which provides a new diameter directly in each iteration. The calculation of the optimal pipe diameter in such gas distribution networks relies on ensuring mass continuity at nodes, following the first Kirchhoff law, and concluding when the pressure drops in all the closed paths are algebraically balanced, adhering to the second Kirchhoff law for energy equilibrium. The presented optimisation is based on principles developed by Hardy Cross in the 1930s for the moment distribution analysis of statically indeterminate structures. The results are for steady-state conditions and for the highest possible estimated demand of gas, while the distributed gas is treated as a noncompressible fluid due to the relatively small drop in pressure in a typical network of pipes. There is no unique solution; instead, an infinite number of potential outcomes exist, alongside infinite combinations of pipe diameters for a given fixed flow pattern that can satisfy the first and second Kirchhoff laws in the given topology of the particular network at hand.

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“…where [NL] and [V] are given in Equations ( 10) and (11), respectively, and where × means matrix multiplication.…”

Section: Node-loop Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Two Iterative Methods for Sizing Pipe Diameters in Gas Distribution Networks with Loops

Brkić

2024

Computation

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“…[30] We emphasise, therefore, that, in the present work, there is certainly no intention on our part to attempt to supplant previous thorough and comprehensive treatments -such as, for example, the formal accounts of Bryant, [31,32] Slepian, [33] Chen [34] and Bondy & Murty [35] nor the extensive coverage in the electrical engineering literature, such as Refs. [36][37][38][39][40][41][42]. Rather, our emphasis here will be on giving a simple, easily comprehensible, exposition of the essence of the matter -illustrated, for convenience, by extended consideration of just one specific example -and on connecting Kirchhoff's work with late 20 th -century extensions that enable the spanning trees of a planar graph to be counted by considering its inner dual, [21,[43][44][45][46] as well as with 21 st -century developments concerning spanning-tree counts in graphs that may, in general, be non-planar.…”

Section: Introductionmentioning

confidence: 99%

What Kirchhoff Actually did Concerning Spanning Trees in Electrical Networks and its Relationship to Modern Graph-Theoretical Work

Kirby¹,

Mallion²,

Pollak³

et al. 2016

Croat. Chem. Acta

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What Kirchhoff actually did concerning spanning trees in the course of his classic paper in the 1847 Annalen der Physik und Chemie has, to some extent, long been shrouded in myth in the literature of Graph Theory and Mathematical Chemistry. In this review, Kirchhoff's manipulation of the equations that arise from application of his two celebrated Laws of electrical circuits -formulated in the middle of the 19 th century -is related to 20 th -and 21 st -century work on the enumeration of spanning trees. It is shown that matrices encountered in an analysis of what Kirchhoff really did include (a) the Kirchhoff (Laplacian, Admittance) matrix, K, that features in the well-known Matrix Tree Theorem, (b) the matrix G encountered in the theorem of Gutman, Mallion & Essam (1983), applicable only to planar graphs, and (c) the analogous matrix M that arises in the Cycle Theorem (Kirby et al. 2004), a theorem that applies to graphs of any genus. It is concluded that Kirchhoff himself was not interested in counting spanning trees, and, accordingly, he did not explicitly do so. Nevertheless, it is shown how the modulus of the determinant of a certain matrix (here denoted by the label C') -associated with the linear equations arising from application of Kirchhoff's two Laws -is numerically equal to the number of spanning trees in the graph representing the connectivity of the electrical network being studied. Kirchhoff did, however, invoke the concept of spanning trees, introducing them in a complementary fashion by referring to the chords that must be removed from the original graph in order to form such trees. It is further emphasised that, in choosing the cycles in the network being studied, around which to apply his circuit Law, Kirchhoff explicitly selected what would now be called a 'Fundamental System of Cycles'.

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“…Simplicial models have further been studied from a geometric perspective [17], in terms of epidemic spreading [68], or in the context of extensions of random graph models [41,71,123]. Nevertheless, in contrast to graph-based methods, the analysis of higher-order interaction data using simplicial complexes is still nascent, even though the formal use of tools from algebraic topology for the analysis of networks was discussed already in the 1950s in the context of electrical network and circuit theory [105,107,108]. And Eckmann's seminal work introduced the ideas underpinning the Hodge Laplacian already in 1944 [45].…”

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confidence: 99%

Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian

Schaub,

Benson,

Horn

et al. 2018

Preprint

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Graphs modeling pairwise relationships between entities have become a dominant framework to study complex systems and data. Simplicial complexes extend this dyadic model of graphs to polyadic relationships and have emerged as a model for multi-node relationships occurring in many complex systems. For instance, biological interactions occur between sets of molecules, and communication systems include group messages and not only pairwise interactions. While the graph Laplacian and Laplacian dynamics have been intensely studied, corresponding notions of Laplacian dynamics beyond the node-space have so far remained largely unexplored for simplicial complexes. In particular, diffusion processes such as random walks and their relationship to the graph Laplacian, that underpin many methods such as centrality measures, flow-based rankings, community detection, and contagion models, lack a proper correspondence for general simplicial complexes Focusing on the coupling between edges, here we introduce a normalized Laplacian matrix for simplicial complexes and demonstrate its relationship to a random walk model on simplicial complexes as a foundational step towards translating many Laplacian-based analytics from graphs to simplicial complexes. Our key idea is to generalize the relationship between the normalized graph Laplacian and random walks on graphs by devising an appropriate normalization for the Hodge Laplacian, the analog of the graph Laplacian for simplicial complexes. We further discuss how this Hodge Laplacian gives rise to the Hodge decomposition, a decomposition of edge flows into intuitively interpretable components that are analogous to notions such as gradient flows or rotational flows from vector calculus. We demonstrate how these results can be leveraged for data analytics that extract information about the edge-space of a simplicial complex that complements and extends graph-based analysis. To illustrate the utility of these tools we derive spectral embeddings based on the Hodge Laplacian to examine trajectory data, and exemplify our ideas by an analysis of ocean drifters near Madagascar. We also present a generalization of personalized PageRank for the edge-space of simplicial complexes and apply it for the analysis of a book co-purchasing dataset.

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Some Topological Considerations in Network Theory

Cited by 22 publications

References 25 publications

Two Iterative Methods for Sizing Pipe Diameters in Gas Distribution Networks with Loops

Two Iterative Methods for Sizing Pipe Diameters in Gas Distribution Networks with Loops

What Kirchhoff Actually did Concerning Spanning Trees in Electrical Networks and its Relationship to Modern Graph-Theoretical Work

Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian

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