Uniformly Most-Reliable Graphs and Antiholes

Rela, Guillermo; Robledo, Franco; Romero, Pablo

doi:10.1007/978-3-030-37599-7_36

Cited by 2 publications

(3 citation statements)

References 16 publications

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“…In this paper, we formally proved that the K 4,4 is UMRG. To the best of our knowledge, this is the second non-trivial 4-regular UMRG in the literature (since K 5 is 4-regular) after H = C 7 proved in Rela et al (2019).…”

Section: Ifmentioning

confidence: 83%

“…Rela et al (2018) proved that Petersen graph is UMRG. Furthermore, the only non-trivial 4-regular UMRG known so far is the complement of a cycle with seven nodes H = C 7 also known in the literature as an odd-antihole (Rela et al, 2019). A complementary research is focused on UMRGs under node-failures, and several bipartite and multipartite graphs are UMRGs.…”

Section: Recent Progressmentioning

confidence: 99%

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Finding uniformly most reliable graphs by counting trivial cuts

Canale¹,

Rela²,

Robledo³

et al. 2020

Preprint

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In network design, the all-terminal reliability maximization is of paramount importance. In this classical setting, we assume a simple graph with perfect nodes but independent edge failures with identical probability ρ. The goal is to communicate n terminals using e edges, in such a way that the connectedness probability of the resulting random graph is maximum. A graph with n nodes and e edges that meets the maximum reliability property for all ρ ∈ (0, 1) is called uniformly most-reliable (n, e)-graph (UMRG). The discovery of these graphs is a challenging problem that involves an interplay between extremal graph theory and computational optimization. Recent works confirm the existence of special cubic UMRGs, such as Wagner, Petersen and Yutsis graphs, and a 4-regular graph H = C7. In a foundational work in the field, Boesch. et. al. state with no proof that the bipartite complete graph K4,4 is UMRG. In this paper, we revisit the breakthroughs in the theory of UMRG. A simple methodology to determine UMRGs based on counting trivial cuts is presented. Finally, we test this methodology to mathematically prove that the complete bipartite graph K4,4 is UMRG.

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Section: Ifmentioning

confidence: 83%

Section: Recent Progressmentioning

confidence: 99%

Finding uniformly most reliable graphs by counting trivial cuts

Canale¹,

Rela²,

Robledo³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Let us consider the elementary setting of a simple graph with a fixed number of units and edges. If every edge can fail independently with a given probability, a uniformly most reliable graph has to maximize the number of spanning trees, i. e. be τ -optimal [72]. In this sense we could extend the Hebbian principle from pairs of neurons to assemblies: What robustly wires together, fires together.…”

Section: Analytical Avalanche Size Distributions In Small or Structur...mentioning

confidence: 99%

A rigorous stochastic theory for spike pattern formation in recurrent neural networks with arbitrary connection topologies

Schünemann¹,

Ernst²,

Kesseböhmer³

2022

Preprint

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Cortical networks exhibit synchronized activity which often occurs in spontaneous events in the form of spike avalanches. Since synchronization has been causally linked to central aspects of brain function such as selective signal processing and integration of stimulus information, participating in an avalanche is a form of a transient synchrony which temporarily creates neural assemblies and hence might especially be useful for implementing flexible information processing. For understanding how assembly formation supports neural computation, it is therefore essential to establish a comprehensive theory of how network structure and dynamics interact to generate specific avalanche patterns and sequences. Here we derive exact avalanche distributions for a finite network of recurrently coupled spiking neurons with arbitrary non-negative interaction weights, which is made possible by formally mapping the model dynamics to a linear, random dynamical system on the N -torus and by exploiting self-similarities inherent in the phase space. We introduce the notion of relative unique ergodicity and show that this property is guaranteed if the system is driven by a time-invariant Bernoulli process. This approach allows us not only to provide closed-form analytical expressions for avalanche size, but also to determine the detailed set(s) of units firing in an avalanche (i.e., the avalanche assembly). The underlying dependence between network structure and dynamics is made transparent by expressing the distribution of avalanche assemblies in terms of the induced graph Laplacian. We explore analytical consequences of this dependence and provide illustrating examples.In summary, our framework provides a major extension of previous analytical work which was restricted to regularly coupled or discrete state networks in the infinite network limit. For systems with a sufficiently homogeneous or translationally invariant coupling topology, we make an explicit link to critical states and the existence of scale-free distributions.

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Uniformly Most-Reliable Graphs and Antiholes

Cited by 2 publications

References 16 publications

Finding uniformly most reliable graphs by counting trivial cuts

Finding uniformly most reliable graphs by counting trivial cuts

A rigorous stochastic theory for spike pattern formation in recurrent neural networks with arbitrary connection topologies

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