Computational and analytical studies of the Randić index in Erdös–Rényi models

Martinez-Martinez, C. T.; Méndez-Bermúdez, J. A.; Rodrı́guez, José M.; Sigarreta, José M.

doi:10.1016/j.amc.2020.125137

Cited by 35 publications

(51 citation statements)

References 39 publications

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“…The Randić Index of a graph has been introduced in 1976 by Milan Randić [57] to study molecular structures of organic chemistry compounds. Even though it is behind the aims of our investigation, from a mathematical viewpoint we highlight, as this index is still widely investigated for its properties as, for instance, in [62][63][64].…”

Section: Randić Indexmentioning

confidence: 99%

Testing Graph Robustness Indexes for EEG Analysis in Alzheimer’s Disease Diagnosis

et al. 2021

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Alzheimer’s Disease (AD) is an incurable neurodegenerative disorder which mainly affects older adults. An early diagnosis is essential because medical treatments can slow down the progression of the disease only if provided during the first stage, called Mild Cognitive Impairment (MCI). Starting from the study of electroencephalografic signals, brain functional connectivity analyses can be performed with the support of the graph theory. In particular, the purpose of this work is to verify the performances of three indexes, typically adopted to evaluate the graph robustness, in order to estimate the functional connectivity for three groups of subjects: healthy controls and people affected by dementia at two different stages (MCI and AD). The results obtained by the Connection Density Index, the Randić Index, and a normalized version of the Kirchhoff Index revealed a higher robustness in the brain networks of healthy people, followed by MCI and, finally, by AD patients, consistent with the hallmarks of Alzheimer’s disease. The statistical analysis showed that there is a significant difference between controls and AD for all three indexes. Finally, all three indexes were compared, revealing that the the Randić Index outperformed the other two indexes. These preliminary outcomes will be exploited to address further in-depth and time-expensive analyses for improving the diagnosis of Alzheimer’s disease.

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Section: Randić Indexmentioning

confidence: 99%

Testing Graph Robustness Indexes for EEG Analysis in Alzheimer’s Disease Diagnosis

et al. 2021

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“…For recent exceptions see Refs. [25,43,44], where the average Randić index has been used to probe the percolation transition in Erdös-Rényi graphs and RGGs.…”

Section: B Randić Connectivity Indexmentioning

confidence: 99%

Non-uniform random graphs on the plane: A scaling study

Martinez-Martinez,

Mendez-Bermudez,

Rodrigues

et al. 2021

Preprint

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We consider random geometric graphs on the plane characterized by a non-uniform density of vertices. In particular, we introduce a graph model where n vertices are independently distributed in the unit disc with positions, in polar coordinates (l, θ), obeying the probability density functions ρ(l) and ρ(θ). Here we choose ρ(l) as a normal distribution with zero mean and variance σ ∈ (0, ∞) and ρ(θ) as an uniform distribution in the interval θ ∈ [0, 2π). Then, two vertices are connected by an edge if their Euclidian distance is less or equal than the connection radius ℓ. We characterize the topological properties of this random graph model, which depends on the parameter set (n, σ, ℓ), by the use of the average degree k and the number of non-isolated vertices V×; while we approach their spectral properties with two measures on the graph adjacency matrix: the ratio of consecutive eigenvalue spacings r and the Shannon entropy S of eigenvectors. First we propose a heuristic expression for k(n, σ, ℓ) . Then, we look for the scaling properties of the normalized average measure X (where X stands for V×, r and S) over graph ensembles. We demonstrate that the scaling parameter of V× = V× /n is indeed k ; with V× ≈ 1 − exp(− k ). Meanwhile, the scaling parameter of both r and S is proportional to n −γ k with γ ≈ 0.16.

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“…This statistical approach, well known in random matrix theory studies, has been recently applied to random networks by means of topological indices, see e.g. [18][19][20]. Moreover, it has been shown that average topological indices may serve as complexity measures equivalent to standard random matrix theory measures [21,22].…”

Section: Computational Study Of Sp α (G) On Random Networkmentioning

confidence: 99%

Stolarsky-Puebla index

Méndez-Bermúdez¹,

Aguilar-Sanchez²,

Blaya³

et al. 2021

Preprint

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We introduce a degree-based variable topological index inspired on the Stolarsky mean (known as the generalization of the logarithmic mean). We name this new index as the Stolarsky-Puebla index:Here, uv denotes the edge of the network G connecting the vertices u and v, du is the degree of the vertex u, and α ∈ R\{0, 1}. Indeed, for given values of α, the Stolarsky-Puebla index reproduces well-known topological indices such as the reciprocal Randic index, the first Zagreb index, and several mean Sombor indices. Moreover, we apply these indices to random networks and demonstrate that SPα(G) , normalized to the order of the network, scale with the corresponding average degree d .

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Computational and analytical studies of the Randić index in Erdös–Rényi models

Cited by 35 publications

References 39 publications

Testing Graph Robustness Indexes for EEG Analysis in Alzheimer’s Disease Diagnosis

Testing Graph Robustness Indexes for EEG Analysis in Alzheimer’s Disease Diagnosis

Non-uniform random graphs on the plane: A scaling study

Stolarsky-Puebla index

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