Heat diffusion: Thermodynamic depth complexity of networks

Escolano, Francisco; Hancock, Edwin R.; Lozano, Miguel Ángel

doi:10.1103/physreve.85.036206

Cited by 63 publications

(68 citation statements)

References 39 publications

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“…Sun, Ovsjanikov and Guibas [11] histogram the elements of the heat kernel trace to compute the heat kernel signature, and use this for shape recognition. In a recent paper, Escolano et al [4] introduced an alternative technique based on the analysis of the heat flow on a graph. Heat flow is derived from the heat kernel, which is the solution of the heat diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

“…Since the dynamic system is non-stationary and non-ergodic, it makes sense neither to characterize it using its steady state behaviour (since this does exist) nor its phase transitions (as is the case in the heat flow method). Instead we turn to the Fourier transform as a natural way of providing a frequency domain characterization of the time evolution of the complex wave equation, and use this instead of the heat flow trace [4].…”

Section: Introductionmentioning

confidence: 99%

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Bragg Diffraction Patterns as Graph Characteristics

Escolano

Hancock

2018

Lecture Notes in Computer Science

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Abstract. In this paper we establish a link between diffraction theory and graph characterization through the Schrödinger operator. This provides a natural way of characterizing wave propagation on a graph. In order to do so, we compute the spatio-temporal Fourier transform of the operator and then pack its spherical representation in a point of a Stiefel manifold. We show that when the temporal interval of analysis is set according to quantum efficiency principles the proposed approach outperforms the alternatives in graph discrimination.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bragg Diffraction Patterns as Graph Characteristics

Escolano

Hancock

2018

Lecture Notes in Computer Science

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“…In fact, thermodynamic depth has proved to provide a powerful means of characterizing a graph in terms of statistical complexity [17]. Recently, the normalized Laplacian spectrum has been shown to provide a complexity level characterization via definition of the von Neumann entropy (or quantum entropy) associated with a density matrix [9,18].…”

Section: A Related Literaturementioning

confidence: 99%

Approximate von Neumann entropy for directed graphs

et al. 2014

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In this paper, we develop an entropy measure for assessing the structural complexity of directed graphs. Although there are many existing alternative measures for quantifying the structural properties of undirected graphs, there are relatively few corresponding measures for directed graphs. To fill this gap in the literature, we explore an alternative technique that is applicable to directed graphs. We commence by using Chung's generalization of the Laplacian of a directed graph to extend the computation of von Neumann entropy from undirected to directed graphs. We provide a simplified form of the entropy which can be expressed in terms of simple node in-degree and out-degree statistics. Moreover, we find approximate forms of the von Neumann entropy that apply to both weakly and strongly directed graphs, and that can be used to characterize network structure. We illustrate the usefulness of these simplified entropy forms defined in this paper on both artificial and real-world data sets, including structures from protein databases and high energy physics theory citation networks.

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“…In this way some of the strengths of both the randomness and statistical approaches to complexity can be combined. One approach that takes an important step in this direction is thermodynamic depth complexity [9]. Here a Birckhoff-vonNeumann polytope is fitted to the heat kernel of a graph.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, by using the logical or thermodynamic depth of a network, the details of inhomogeneous degree structure can be problem. One powerful techniques here is to use a variant of the Kologomorov-Chaitin [4,5] complexity to measure how many operations are need to transform a graph into a canonical form (see [9] for a review of network complexity).…”

Section: Introductionmentioning

confidence: 99%

Graph Entropy from Closed Walk and Cycle Functionals

Aziz

Hancock

Wilson

2016

Lecture Notes in Computer Science

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Abstract. This paper presents an informational functional that can be used to characterise the entropy of a graph or network structure, using closed random walks and cycles. The work commences from Dehmer's information functional, that characterises networks at the vertex level, and extends this to structures which capture the correlation of vertices, using walk and cycle structures. The resulting entropies are applied to synthetic networks and to network time series. Here they prove effective in discriminating between different types of network structure, and detecting changes in the structure of networks with time.

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Heat diffusion: Thermodynamic depth complexity of networks

Cited by 63 publications

References 39 publications

Bragg Diffraction Patterns as Graph Characteristics

Bragg Diffraction Patterns as Graph Characteristics

Approximate von Neumann entropy for directed graphs

Graph Entropy from Closed Walk and Cycle Functionals

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