Spectral density of random graphs: convergence properties and application in model fitting

Santos, Suzana de Siqueira; Fujita, André; Matias, Catherine

doi:10.1093/comnet/cnab041

“…To estimate the spectral density (Santos et al, 2021 ), we use a Gaussian kernel regression with the Nadaraya-Watson estimator (Nadaraya, 1964 ; Watson, 1964 ). We set the bandwidth of the Gaussian kernel by (λ 1 − λ | V | )/number of bins, and the number of bins by using the Sturges criterion (Sturges, 1926 ).…”

Section: Methodsmentioning

confidence: 99%

Spectral density-based clustering algorithms for complex networks

Ramos

¹

,

Mourão-Miranda

²

,

Fujita

³

2023

Front. Neurosci.

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2

0

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IntroductionClustering is usually the first exploratory analysis step in empirical data. When the data set comprises graphs, the most common approaches focus on clustering its vertices. In this work, we are interested in grouping networks with similar connectivity structures together instead of grouping vertices of the graph. We could apply this approach to functional brain networks (FBNs) for identifying subgroups of people presenting similar functional connectivity, such as studying a mental disorder. The main problem is that real-world networks present natural fluctuations, which we should consider.MethodsIn this context, spectral density is an exciting feature because graphs generated by different models present distinct spectral densities, thus presenting different connectivity structures. We introduce two clustering methods: k-means for graphs of the same size and gCEM, a model-based approach for graphs of different sizes. We evaluated their performance in toy models. Finally, we applied them to FBNs of monkeys under anesthesia and a dataset of chemical compounds.ResultsWe show that our methods work well in both toy models and real-world data. They present good results for clustering graphs presenting different connectivity structures even when they present the same number of edges, vertices, and degree of centrality.DiscussionWe recommend using k-means-based clustering for graphs when graphs present the same number of vertices and the gCEM method when graphs present a different number of vertices.

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“…There is a huge literature on statistical models for ecological networks, see [132][133][134][135][136][137][138][139]. The variety of data available and the specificity of each ecosystem explain this vast corpus, which in itself would deserve an entire review.…”

Section: (A) Empirical Data and Real Interaction Networkmentioning

confidence: 99%

Complex systems in ecology: a guided tour with large Lotka–Volterra models and random matrices

Akjouj,

Barbier,

Clenet

et al. 2024

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Ecosystems represent archetypal complex dynamical systems, often modelled by coupled differential equations of the form d x i d t = x i ϕ i ( x 1 , … , x N ) , where N represents the number of species and x i , the abundance of species i . Among these families of coupled differential equations, Lotka–Volterra (LV) equations, corresponding to ϕ i ( x 1 , … , x N ) = r i − x i + ( Γ x ) i , play a privileged role, as the LV model represents an acceptable trade-off between complexity and tractability. Here, r i is the intrinsic growth of species i and Γ stands for the interaction matrix: Γ i j represents the effect of species j over species i . For large N , estimating matrix Γ is often an overwhelming task and an alternative is to draw Γ at random, parameterizing its statistical distribution by a limited number of model features. Dealing with large random matrices, we naturally rely on random matrix theory (RMT). The aim of this review article is to present an overview of the work at the junction of theoretical ecology and large RMT. It is intended to an interdisciplinary audience spanning theoretical ecology, complex systems, statistical physics and mathematical biology.

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“…Some other models to generate random graphs that may fit observed networks are represented in [4,96,97].…”

Section: Other Models Of Random Networkmentioning

confidence: 99%

“…In [97] such models like a deterministic block model, the configuration random graph model, d-regular random graph, a geometric random graph model apart from the ER, and the SBM described in Section 4.3 are presented as a list of most common models. In [4], much attention is devoted to spatial networks, e.g., the Spatially Embedded Random Networks and the Waxman model.…”

Section: Other Models Of Random Networkmentioning

confidence: 99%

“…Spectral density (i.e., the distribution of the graph adjacency matrix eigenvalues) and its kernel estimator are used in [97] to find the best graph model for the observed network in such a way that the divergence between the estimated spectral density and the known limiting spectral density (or one obtained using Monte Carlo) will be the smallest. A rich theory of the kernel estimators of probability density functions can be applied to spectral densities in a similar way.…”

Section: Other Models Of Random Networkmentioning

confidence: 99%

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Extreme Value Statistics for Evolving Random Networks

Markovich

¹

,

Vaičiulis

²

2023

Mathematics

0

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Our objective is to survey recent results concerning the evolution of random networks and related extreme value statistics, which are a subject of interest due to numerous applications. Our survey concerns the statistical methodology but not the structure of random networks. We focus on the problems arising in evolving networks mainly due to the heavy-tailed nature of node indices. Tail and extremal indices of the node influence characteristics like in-degrees, out-degrees, PageRanks, and Max-linear models arising in the evolving random networks are discussed. Related topics like preferential and clustering attachments, community detection, stationarity and dependence of graphs, information spreading, finding the most influential leading nodes and communities, and related methods are surveyed. This survey tries to propose possible solutions to unsolved problems, like testing the stationarity and dependence of random graphs using known results obtained for random sequences. We provide a discussion of unsolved or insufficiently developed problems like the distribution of triangle and circle counts in evolving networks, or the clustering attachment and the local dependence of the modularity, the impact of node or edge deletion at each step of evolution on extreme value statistics, among many others. Considering existing techniques of community detection, we pay attention to such related topics as coloring graphs and anomaly detection by machine learning algorithms based on extreme value theory. In order to understand how one can compute tail and extremal indices on random graphs, we provide a structured and comprehensive review of their estimators obtained for random sequences. Methods to calculate the PageRank and PageRank vector are shortly presented. This survey aims to provide a better understanding of the directions in which the study of random networks has been done and how extreme value analysis developed for random sequences can be applied to random networks.

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Spectral density of random graphs: convergence properties and application in model fitting

Cited by 4 publications

References 47 publications

Spectral density-based clustering algorithms for complex networks

Spectral density-based clustering algorithms for complex networks

Complex systems in ecology: a guided tour with large Lotka–Volterra models and random matrices

Extreme Value Statistics for Evolving Random Networks

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