Entropy Bounds for Hierarchical Molecular Networks

Dehmer, Matthias; Borgert, Stephan; Emmert‐Streib, Frank

doi:10.1371/journal.pone.0003079

Cited by 38 publications

(37 citation statements)

References 47 publications

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“…However, hierarchy is a polysemous word, involving order, levels, inclusion, or control as possible descriptors (23), none of which captures either its complexity or the problem of its measure and origins. Although previous work using complex networks theory has quantitatively tackled the problem (4,(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34), some questions remain: Is hierarchy a widespread feature of complex systems organization? What types of hierarchies do exist?…”

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confidence: 99%

On the origins of hierarchy in complex networks

Corominas-Murtra

Goñi

Solé

et al. 2013

Proc. Natl. Acad. Sci. U.S.A.

148

147

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Hierarchy seems to pervade complexity in both living and artificial systems. Despite its relevance, no general theory that captures all features of hierarchy and its origins has been proposed yet. Here we present a formal approach resulting from the convergence of theoretical morphology and network theory that allows constructing a 3D morphospace of hierarchies and hence comparing the hierarchical organization of ecological, cellular, technological, and social networks. Embedded within large voids in the morphospace of all possible hierarchies, four major groups are identified. Two of them match the expected from random networks with similar connectivity, thus suggesting that nonadaptive factors are at work. Ecological and gene networks define the other two, indicating that their topological order is the result of functional constraints. These results are consistent with an exploration of the morphospace, using in silico evolved networks. modularity | evolution F ifty years ago (1), Herbert Simon defined complex systems as nested hierarchical networks of components organized as interconnected modules. Hierarchy seems a pervasive feature of the organization of natural and artificial systems (2, 3). The examples span from social interactions (4, 5), urban growth (6, 7), and allometric scaling (8) to cell function (9-13), development (14), ecosystem flows (15, 16), river networks (17), brain organization (18), and macroevolution (19,20). It also seems to pervade a coherent form of organization that allows reducing the costs associated to reliable information transmission (21) and to support efficient genetic and metabolic control in cellular networks (22). However, hierarchy is a polysemous word, involving order, levels, inclusion, or control as possible descriptors (23), none of which captures either its complexity or the problem of its measure and origins. Although previous work using complex networks theory has quantitatively tackled the problem (4, 24-34), some questions remain: Is hierarchy a widespread feature of complex systems organization? What types of hierarchies do exist? Are hierarchies the result of selection pressures or, conversely, do they arise as a by-product of structural constraints?A well-established concept where such questions are addressed involves the use of a morphospace (35-40), namely a phenotype space where a small set of quantitative traits can be defined as the axes. Here we take a step in this direction by combining morphospace and network theories, taking the intuitive idea of hierarchy as the starting point: a pattern of relations where there is no ambiguity in who controls whom with a pyramidal structure in which the few control the many. Formally, the picture of hierarchy matches a tree of relations (41), ideally represented by a directed graph. As shown in Fig. 1 A-D, the elements of the system are represented by nodes connected by arrows establishing the map of relations of who affects whom. Accordingly, a measure of hierarchy should account for the deviations from this ideal ...

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mentioning

confidence: 99%

On the origins of hierarchy in complex networks

Corominas-Murtra

Goñi

Solé

et al. 2013

Proc. Natl. Acad. Sci. U.S.A.

148

147

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“…Since the structural property of the complex networks plays a very significant role, it has been heavily studied [12][13][14][15]. Dehmer [16][17][18][19][20] introduced graph entropies based on information functionals which capture structural information and also studied their properties. He assigned a probability value to each individual node in a graph or a network.…”

Section: Introductionmentioning

confidence: 99%

New Upper Bound and Lower Bound for Degree-Based Network Entropy

Wang

2016

Symmetry

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Abstract:The degree-based network entropy which is inspired by Shannon's entropy concept becomes the information-theoretic quantity for measuring the structural information of graphs and complex networks. In this paper, we study some properties of the degree-based network entropy. Firstly we develop a refinement of Jensen's inequality. Next we present the new and more accurate upper bound and lower bound for the degree-based network entropy only using the order, the size, the maximum degree and minimum degree of a network. The bounds have desirable performance to restrict the entropy in different kinds of graphs. Finally, we show an application to structural complexity analysis of a computer network modeled by a connected graph.Keywords: Shannon's entropy; degree-based network entropy; Jensen's inequality; upper bound and lower bound of entropy; network structure

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“…For example, graph entropy measures have been used extensively to characterize the structure of graph-based systems in mathematical chemistry, biology [3] and in computer science-related areas, see [4]. The concept of graph entropy [5,6] introduced by Rashevsky [7] and Trucco [8] has been used to measure the structural complexity of graphs [3,9,10]. The entropy of a graph is an information-theoretic quantity that has been introduced by Mowshowitz [11].…”

Section: Introductionmentioning

confidence: 99%

A Note on Distance-based Graph Entropies

Chen

Dehmer

Shi

2014

Entropy

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135

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A variety of problems in, e.g., discrete mathematics, computer science, information theory, statistics, chemistry, biology, etc., deal with inferring and characterizing relational structures by using graph measures. In this sense, it has been proven that information-theoretic quantities representing graph entropies possess useful properties such as a meaningful structural interpretation and uniqueness. As classical work, many distance-based graph entropies, e.g., the ones due to Bonchev et al. and related quantities have been proposed and studied. Our contribution is to explore graph entropies that are based on a novel information functional, which is the number of vertices with distance k to a given vertex. In particular, we investigate some properties thereof leading to a better understanding of this new information-theoretic quantity.

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Entropy Bounds for Hierarchical Molecular Networks

Cited by 38 publications

References 47 publications

On the origins of hierarchy in complex networks

On the origins of hierarchy in complex networks

New Upper Bound and Lower Bound for Degree-Based Network Entropy

A Note on Distance-based Graph Entropies

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