Entropy and Heterogeneity Measures for Directed Graphs

Ye, Cheng; Wilson, Richard C.; Comin, César H.; Costa, Luciano da Fontoura; Hancock, Edwin R.

doi:10.1007/978-3-642-39140-8_15

Cited by 12 publications

(18 citation statements)

References 17 publications

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“…Hence we commence by computing the von Neumann entropy associated with each edge in a directed graph. An analysis, extending our own previously published work [8] shows that the entropy depends on the configuration of in and out-degrees of the two vertices defining a directed edge. This leads us to a four-dimensional characterization of directed graph structure, which depends on the distribution of entropy with the in and out-degrees of pairs of vertices connected by a directed edge.…”

Section: Contributionsupporting

confidence: 68%

Entropic Graph Embedding via Multivariate Degree Distributions

Wilson

Hancock

2014

Lecture Notes in Computer Science

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Although there are many existing alternative methods for using structural characterizations of undirected graphs for embedding, clustering and classification problems, there is relatively little literature aimed at dealing with such problems for directed graphs. In this paper we present a novel method for characterizing graph structure that can be used to embed directed graphs into a feature space. The method commences from a characterization based on the distribution of the von Neumann entropy of a directed graph with the in and out-degree configurations associated with directed edges. We start from a recently developed expression for the von Neumann entropy of a directed graph, which depends on vertex in-degree and out-degree statistics, and thus obtain a multivariate edge-based distribution of entropy. We show how this distribution can be encoded as a multi-dimensional histogram, which captures the structure of a directed graph and reflects its complexity. By performing principal components analysis on a sample of histograms, we embed populations of directed graphs into a low dimensional space. Finally, we undertake experiments on both artificial and real-world data to demonstrate that our directed graph embedding method is effective in distinguishing different types of directed graphs.

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Section: Contributionsupporting

confidence: 68%

Entropic Graph Embedding via Multivariate Degree Distributions

Wilson

Hancock

2014

Lecture Notes in Computer Science

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“…Such failure cases can be quantified using the approximate von Neumann entropy measure for directed graphs. Graph entropy is computed in terms of its in-degree and out-degree [41]. Consider a directed graph G(V, E) with a set of nodes denoted by V and the set of edges denoted by E. The adjacency matrix of such a graph is defined as, The maximum value of entropy for a directed graph is equal to 1− 1 |V | for a star graph where all the nodes in the graph have either incoming links or outgoing links but not both.…”

Section: Explanatory Modelmentioning

confidence: 99%

Face Phylogeny Tree Using Basis Functions

Banerjee

Ross

2020

IEEE Trans. Biom. Behav. Identity Sci.

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Photometric transformations, such as brightness and contrast adjustment, can be applied to a face image repeatedly creating a set of near-duplicate images. Identifying the original image from a set of such near-duplicates and deducing the relationship between them are important in the context of digital image forensics. This is commonly done by generating an image phylogeny treea hierarchical structure depicting the relationship between a set of near-duplicate images. In this work, we utilize three different families of basis functions to model pairwise relationships between near-duplicate images. The basis functions used in this work are orthogonal polynomials, wavelet basis functions and radial basis functions. We perform extensive experiments to assess the performance of the proposed method across three different modalities, namely, face, fingerprint and iris images; across different image phylogeny tree configurations; and across different types of photometric transformations. We also utilize the same basis functions to model geometric transformations and deep-learning based transformations. We also perform extensive analysis of each basis function with respect to its ability to model arbitrary transformations and to distinguish between the original and the transformed images. Finally, we utilize the concept of approximate von Neumann graph entropy to explain the success and failure cases of the proposed IPT generation algorithm. Experiments indicate that the proposed algorithm generalizes well across different scenarios thereby suggesting the merits of using basis functions to model the relationship between photometrically and geometrically modified images. !• Both the authors are with the

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“…(8) or Eq. (9), is always well defined, symmetric, negative definite and bounded, i.e., 0 ≤ D JS ≤ 1.…”

Section: A the Jensen-shannon Divergencementioning

confidence: 99%

“…This leads to a simple expression for the approximate entropy in terms of the degrees of adjacent vertices. Furthermore, to develop this work further, Ye et al [8] have shown how the von Neumann entropy for undirected graphs can be generalized to directed graphs. To this end, they commenced by using Chung's [9] definition of the normalized Laplacian on a directed graph.…”

Section: Introductionmentioning

confidence: 99%

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A novel entropy-based graph signature from the average mixing matrix

Bai

Rossi

Cui

et al. 2016

2016 23rd International Conference on Pattern Recognition (ICPR)

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Abstract-In this paper, we propose a novel entropic signature for graphs, where we probe the graphs by means of continuous-time quantum walks. More precisely, we characterise the structure of a graph through its average mixing matrix. The average mixing matrix is a doubly-stochastic matrix that encapsulates the time-averaged behaviour of a continuous-time quantum walk on the graph, i.e., the ij-th element of the average mixing matrix represents the time-averaged transition probability of a continuous-time quantum walk from the vertex vi to the vertex vj. With this matrix to hand, we can associate a probability distribution with each vertex of the graph. We define a novel entropic signature by concatenating the average Shannon entropy of these probability distributions with their JensenShannon divergence. We show that this new entropic measure can encaspulate the rich structural information of the graphs, thus allowing to discriminate between different structures. We explore the proposed entropic measure on several graph datasets abstracted from bioinformatics databases and we compare it with alternative entropic signatures in the literature. The experimental results demonstrate the effectiveness and efficiency of our method.

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Entropy and Heterogeneity Measures for Directed Graphs

Cited by 12 publications

References 17 publications

Entropic Graph Embedding via Multivariate Degree Distributions

Entropic Graph Embedding via Multivariate Degree Distributions

Face Phylogeny Tree Using Basis Functions

A novel entropy-based graph signature from the average mixing matrix

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