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Simultaneous Dimensionality and Complexity Model Selection for Spectral Graph Clustering [article]

Congyuan Yang, Carey E. Priebe, Youngser Park, David J. Marchette
2020 arXiv   pre-print
The first contribution is a probabilistic model which approximates the distribution of the extended spectral embedding of a graph.  ...  We illustrate our method via application to a collection of brain graphs.  ...  In this paper, we propose a novel simultaneous dimensionality and complexity model selection framework for spectral graph clustering.  ... 
arXiv:1904.02926v4 fatcat:fru3b7fcivf35l6lxbhcox4ffy

Spectral properties of complex unit gain graphs

Nathan Reff
2012 Linear Algebra and its Applications  
We extend some fundamental concepts from spectral graph theory to complex unit gain graphs. We define the adjacency, incidence and Laplacian matrices, and study each of them.  ...  A complex unit gain graph is a graph where each orientation of an edge is given a complex unit, which is the inverse of the complex unit assigned to the opposite orientation.  ...  A T-gain graph (or complex unit gain graph) is a graph with the additional structure that each orientation of an edge is given a complex unit, called a gain, which is the inverse of the complex E-mail  ... 
doi:10.1016/j.laa.2011.10.021 fatcat:5uzim2j4lfbg3efs2d2tl7vxpy

Graph spectral characterization of the XY model on complex networks

Paul Expert, Sarah de Nigris, Taro Takaguchi, Renaud Lambiotte
2017 Physical review. E  
This work opens new avenues to analyse and characterise dynamics on complex networks using temporal Graph Signal Analysis.  ...  We then use the temporal Graph Signal Transform technique to decompose the time series of the spins on the eigenbasis of the Laplacian.  ...  detection error probability behavior § Relations to the phase and phase transition phenomenon § In empty graph and chain graph, : ; (<) never reduces to 0. § In complete graph and lattice graph, there  ... 
doi:10.1103/physreve.96.012312 pmid:29347091 fatcat:bizxpowtcrg3zmvhfoq2totviy

CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters [article]

Ron Levie, Federico Monti, Xavier Bresson, Michael M. Bronstein
2018 arXiv   pre-print
The core ingredient of our model is a new class of parametric rational complex functions (Cayley polynomials) allowing to efficiently compute spectral filters on graphs that specialize on frequency bands  ...  In this paper, we introduce a new spectral domain convolutional architecture for deep learning on graphs.  ...  In this paper we focus on spectral graph CNNs.  ... 
arXiv:1705.07664v2 fatcat:kgu74rfizzcurkg5ovq45vqi6i

Spectral Complexity of Directed Graphs and Application to Structural Decomposition [article]

Igor Mezić, Vladimir A. Fonoberov, Maria Fonoberova, Tuhin Sahai
2018 arXiv   pre-print
We introduce a new measure of complexity (called spectral complexity) for directed graphs. We start with splitting of the directed graph into its recurrent and non-recurrent parts.  ...  We show that the total complexity of the graph can then be defined in terms of the spectral complexity, complexities of individual components and edge weights.  ...  The following fact on the graph with least spectral complexity is obvious: Fact: The graph with K disconnected nodes has complexity 0.  ... 
arXiv:1808.06004v2 fatcat:xa2vyhsa6ba4vjj66xzufydwfy

Graph product multilayer networks: spectral properties and applications

Hiroki Sayama
2017 Journal of Complex Networks  
This paper aims to establish theoretical foundations of graph product multilayer networks (GPMNs), a family of multilayer networks that can be obtained as a graph product of two or more factor networks  ...  SPECTRAL PROPERTIES OF NONSIMPLE GPMNS In this paper, we aim to extend the definitions of GPMNs to make them capable of capturing a greater variety of complex networks.  ...  SIMPLE GRAPH PRODUCT MULTILAYER NETWORKS AND THEIR SPECTRAL PROPERTIES We define graph product multilayer networks (GPMNs) as a particular family of multilayer networks that can be obtained by applying  ... 
doi:10.1093/comnet/cnx042 fatcat:kruvevvcafhhpehdy6svqhcrhu

First-principles multiway spectral partitioning of graphs

M. A. Riolo, M. E. J. Newman
2014 Journal of Complex Networks  
We consider the minimum-cut partitioning of a graph into more than two parts using spectral methods.  ...  defined by the leading eigenvectors of the graph Laplacian.  ...  SPECTRAL BISECTION The term spectral bisection refers to the special case in which we partition a graph into exactly k = 2 parts.  ... 
doi:10.1093/comnet/cnt021 fatcat:mdbiql3ry5d45mttkbtsbcpqiu

Higher rank graphs from cube complexes and their spectral theory [article]

Nadia S. Larsen, Alina Vdovina
2021 arXiv   pre-print
We introduce Ramanujan k-graphs satisfying optimal spectral gap property, and show explicitly how to construct such k-graphs.  ...  Guided by geometric insight, we obtain several new series of k-graphs using cube complexes covered by Cartesian products of trees, for k ≥ 3.  ...  Spectral theory of k-graphs.  ... 
arXiv:2111.09120v1 fatcat:ea3ygubh4fdbnhdcbbc3tj4s4m

Spectral Lower Bounds on the I/O Complexity of Computation Graphs [article]

Saachi Jain, Matei Zaharia
2020 arXiv   pre-print
Executions of complex computations can be formalized as an evaluation order over the underlying computation graph.  ...  This spectral bound is not only efficiently computable by power iteration, but can also be computed in closed form for graphs with known spectra.  ...  This spectral bound is efficiently computable and can be applied to arbitrarily large and complex graphs. For graphs with known spectra, this bound can also be computed in closed form.  ... 
arXiv:1909.09791v2 fatcat:if7i3g3norbhrlupbujmzzakbu

Spectral Complexity of Directed Graphs and Application to Structural Decomposition

Igor Mezić, Vladimir A. Fonoberov, Maria Fonoberova, Tuhin Sahai
2019 Complexity  
We introduce a new measure of complexity (called spectral complexity) for directed graphs. We start with splitting of the directed graph into its recurrent and nonrecurrent parts.  ...  We show that the total complexity of the graph can then be defined in terms of the spectral complexity, complexities of individual components, and edge weights.  ...  The following fact on the graph with least spectral complexity is obvious: Fact. The graph with disconnected nodes has spectral complexity 0.  ... 
doi:10.1155/2019/9610826 fatcat:hn32fl37dnd2dlzklqofolm35a

Spectral Sufficient Conditions on Pancyclic Graphs

Guidong Yu, Tao Yu, Xiangwei Xia, Huan Xu, Shaohui Wang
2021 Complexity  
in terms of the spectral radius and the signless Laplacian spectral radius of the graph.  ...  Because the spectrum of graphs is convenient to be calculated, in this study, we try to use the spectral theory of graphs to study this problem and give some sufficient conditions for a graph to be pancyclic  ...  Laplacian spectral radius conditions for a graph to be pancyclic.  ... 
doi:10.1155/2021/3630245 fatcat:reiksb2q3rfk5lrbk3uhrdne2a

A Manifold-Based Dimension Reduction Algorithm Framework for Noisy Data Using Graph Sampling and Spectral Graph

Tao Yang, Dongmei Fu, Jintao Meng, Marcin Mrugalski
2020 Complexity  
Subsequently, the specific range of localities is determined using graph spectral analysis, and the density within each local range is estimated to obtain the distribution parameters.  ...  The proposed framework follows the idea of the localization of manifolds and uses graph sampling to determine some local anchor points from the given data.  ...  A spectral graph is a special type of spectral analysis. e spectral analysis itself is based on the frequency domain where a signal is characterized by its spectral coefficient or spectral energy.  ... 
doi:10.1155/2020/8954341 fatcat:27hfyl6qrze7hotdssmqnalxre

A graph complexity measure based on the spectral analysis of the Laplace operator [article]

Diego M. Mateos and Federico Morana and Hugo Aimar
2021 arXiv   pre-print
In this work we introduce a concept of complexity for undirected graphs in terms of the spectral analysis of the Laplacian operator defined by the incidence matrix of the graph.  ...  Second, complexity of complementary graphs coincide.  ...  In order to introduce our definition of spectral complexity of a graph, let us set Z to denote the null graph, i.e w ij = 0 for every i, j = 1, ..., n and F the complete graph i.e w ij = 1 for every i  ... 
arXiv:2109.06706v1 fatcat:mh53alg7v5b4vipcjbl3kpp36a

The Limit Spectral Graph in the Semi-Classical Approximation for the Sturm-Liouville Problem With a Complex Polynomial Potential [article]

A. A. Shkalikov, S. N. Tumanov
2016 arXiv   pre-print
It is shown that at large parameter values, the eigenvalues are concentrated along the so-called limit spectral graph; the curves forming this graph are classified.  ...  The limit distribution of the discrete spectrum of the Sturm-Liouville problem with complex-valued polynomial potential on an interval, on a half-axis, and on the entire axis is studied.  ...  On the right: Limit spectral graph (red). Example 2. Limit spectral set (red); examples of singular and critical curves which are not essential singular and not essential critical (black).  ... 
arXiv:1603.08905v2 fatcat:i4yzgx23pbhblcyypnqzi2i2ee

Low-Complexity Factor Graph Receivers for Spectrally Efficient MIMO-IDMA

C. Novak, F. Hlawatsch, G. Matz
2008 2008 IEEE International Conference on Communications  
Gaussian approximations for certain messages propagated through the factor graph lead to a complexity that scales only linearly with the number of users.  ...  In this paper, we consider a MIMO-IDMA system with increased spectral efficiency due to the use of higher-order symbol constellations.  ...  The complex transmit symbol x [n] ∈ S with the one-to-one symbol mapping χ. We will refer to c III. FACTOR GRAPH AND MESSAGES We next derive a factor graph for an iterative MIMO-IDMA receiver.  ... 
doi:10.1109/icc.2008.151 dblp:conf/icc/NovakHM08 fatcat:rlnoxorkevexxf6zxuw6tslqrq
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