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Graph and Network Entropies
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Graph and Network Entropies

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Statistical Physics".

Deadline for manuscript submissions: closed (30 June 2018) | Viewed by 35710

Special Issue Editor

Prof. Dr. Ernesto Estrada

E-Mail Website
Guest Editor
Department of Mathematics & Statistics, University of Strathclyde, Glasgow G11XQ, UK
Interests: network sciences; walk entropies; algegraic graph theory; spectral methods; matrix functions

Special Issue Information

Dear Colleagues,

The use of graphs and networks to represent physical systems has become a major paradigm in modern science. Graphs are now ubiquitous to study quantum and molecular systems, macromolecules and their interactions, socio-economic and ecological systems, and infrastructural and technological systems, among others. The structural characterization of these systems is an important step in understanding their functioning, robustness and stability. From a mathematical and a physical point of view, the use of entropies and entropy-like measures to characterize graph/network structure is of paramount importance. This Special Issue focuses on original and new research results concerning the development and applications of entropies and entropy-like measures for studying graphs and networks. We welcome submissions addressing fundamental and methodological (mathematical, information, thermodynamics, statistical mechanics, and others) aspects of graph/networks entropies, applications of entropies to the study of structural and dynamical processes in graphs and networks in any area of applications, as well as those on more specific topics that illustrate the broad impact of entropy-based techniques in understanding the complexity of the systems represented by graphs and networks. We will consider computationally-oriented works when they give rise to a clear understanding of the structural and dynamical processes under consideration.

Prof. Dr. Ernesto Estrada
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • graph entropy
  • network entropy
  • entropy and network structure
  • entropy and quantum graphs
  • entropy and molecular graphs
  • applications of graph entropy
  • entropy and brain networks
  • entropy and biological networks
  • entropy and infrastructural networks
  • statistical mechanics of graphs/networks
  • graph thermodynamics

Published Papers (7 papers)

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Research

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15 pages, 1298 KiB  
Article
Thermodynamic Analysis of Time Evolving Networks
by Cheng Ye, Richard C. Wilson, Luca Rossi, Andrea Torsello and Edwin R. Hancock
Entropy 2018, 20(10), 759; https://doi.org/10.3390/e20100759 - 02 Oct 2018
Cited by 7 | Viewed by 4578
Abstract
The problem of how to represent networks, and from this representation, derive succinct characterizations of network structure and in particular how this structure evolves with time, is of central importance in complex network analysis. This paper tackles the problem by proposing a thermodynamic [...] Read more.
The problem of how to represent networks, and from this representation, derive succinct characterizations of network structure and in particular how this structure evolves with time, is of central importance in complex network analysis. This paper tackles the problem by proposing a thermodynamic framework to represent the structure of time-varying complex networks. More importantly, such a framework provides a powerful tool for better understanding the network time evolution. Specifically, the method uses a recently-developed approximation of the network von Neumann entropy and interprets it as the thermodynamic entropy for networks. With an appropriately-defined internal energy in hand, the temperature between networks at consecutive time points can be readily derived, which is computed as the ratio of change of entropy and change in energy. It is critical to emphasize that one of the main advantages of the proposed method is that all these thermodynamic variables can be computed in terms of simple network statistics, such as network size and degree statistics. To demonstrate the usefulness of the thermodynamic framework, the paper uses real-world network data, which are extracted from time-evolving complex systems in the financial and biological domains. The experimental results successfully illustrate that critical events, including abrupt changes and distinct periods in the evolution of complex networks, can be effectively characterized. Full article
(This article belongs to the Special Issue Graph and Network Entropies)
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24 pages, 2022 KiB  
Article
Optimal Placement of Pressure Gauges for Water Distribution Networks Using Entropy Theory Based on Pressure Dependent Hydraulic Simulation
by Do Guen Yoo, Dong Eil Chang, Yang Ho Song and Jung Ho Lee
Entropy 2018, 20(8), 576; https://doi.org/10.3390/e20080576 - 04 Aug 2018
Cited by 6 | Viewed by 3138
Abstract
This study proposed a pressure driven entropy method (PDEM) that determines a priority order of pressure gauge locations, which enables the impact of abnormal condition (e.g., pipe failures) to be quantitatively identified in water distribution networks (WDNs). The method developed utilizes the entropy [...] Read more.
This study proposed a pressure driven entropy method (PDEM) that determines a priority order of pressure gauge locations, which enables the impact of abnormal condition (e.g., pipe failures) to be quantitatively identified in water distribution networks (WDNs). The method developed utilizes the entropy method from information theory and pressure driven analysis (PDA), which is the latest hydraulic analysis method. The conventional hydraulic approach has problems in determining the locations of pressure gauges, attributable to unrealistic results under abnormal conditions (e.g., negative pressure). The proposed method was applied to two benchmark pipe networks and one real pipe network. The priority order for optimal locations was produced, and the result was compared to existing approach. The results of the conventional method show that the pressure reduction difference of each node became so excessive, which resulted in a distorted distribution. However, with the method developed, which considers the connectivity of a system and the influence among nodes based on PDA and entropy method results, pressure gauges can be more realistically and reasonably located. Full article
(This article belongs to the Special Issue Graph and Network Entropies)
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16 pages, 17206 KiB  
Article
Virtual Network Embedding Based on Graph Entropy
by Jingjing Zhang, Chenggui Zhao, Honggang Wu, Minghui Lin and Ren Duan
Entropy 2018, 20(5), 315; https://doi.org/10.3390/e20050315 - 25 Apr 2018
Cited by 4 | Viewed by 3468
Abstract
For embedding virtual networks into a large scale substrate network, a massive amount of time is needed to search the resource space even if the scale of the virtual network is small. The complexity of searching the candidate resource will be reduced if [...] Read more.
For embedding virtual networks into a large scale substrate network, a massive amount of time is needed to search the resource space even if the scale of the virtual network is small. The complexity of searching the candidate resource will be reduced if candidates in substrate network can be located in a group of particularly matched areas, in which the resource distribution and communication structure of the substrate network exhibit a maximal similarity with the objective virtual network. This work proposes to discover the optimally suitable resource in a substrate network corresponding to the objective virtual network through comparison of their graph entropies. Aiming for this, the substrate network is divided into substructures referring to the importance of nodes in it, and the entropies of these substructures are calculated. The virtual network will be embedded preferentially into the substructure with the closest entropy if the substrate resource satisfies the demand of the virtual network. The experimental results validate that the efficiency of virtual network embedding can be improved through our proposal. Simultaneously, the quality of embedding has been guaranteed without significant degradation. Full article
(This article belongs to the Special Issue Graph and Network Entropies)
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15 pages, 18706 KiB  
Article
Network Entropy for the Sequence Analysis of Functional Connectivity Graphs of the Brain
by Chi Zhang, Fengyu Cong, Tuomo Kujala, Wenya Liu, Jia Liu, Tiina Parviainen and Tapani Ristaniemi
Entropy 2018, 20(5), 311; https://doi.org/10.3390/e20050311 - 25 Apr 2018
Cited by 19 | Viewed by 5562
Abstract
Dynamic representation of functional brain networks involved in the sequence analysis of functional connectivity graphs of the brain (FCGB) gains advances in uncovering evolved interaction mechanisms. However, most of the networks, even the event-related ones, are highly heterogeneous due to spurious interactions, which [...] Read more.
Dynamic representation of functional brain networks involved in the sequence analysis of functional connectivity graphs of the brain (FCGB) gains advances in uncovering evolved interaction mechanisms. However, most of the networks, even the event-related ones, are highly heterogeneous due to spurious interactions, which bring challenges to revealing the change patterns of interactive information in the complex dynamic process. In this paper, we propose a network entropy (NE) method to measure connectivity uncertainty of FCGB sequences to alleviate the spurious interaction problem in dynamic network analysis to realize associations with different events during a complex cognitive task. The proposed dynamic analysis approach calculated the adjacency matrices from ongoing electroencephalpgram (EEG) in a sliding time-window to form the FCGB sequences. The probability distribution of Shannon entropy was replaced by the connection sequence distribution to measure the uncertainty of FCGB constituting NE. Without averaging, we used time frequency transform of the NE of FCGB sequences to analyze the event-related changes in oscillatory activity in the single-trial traces during the complex cognitive process of driving. Finally, the results of a verification experiment showed that the NE of the FCGB sequences has a certain time-locked performance for different events related to driver fatigue in a prolonged driving task. The time errors between the extracted time of high-power NE and the recorded time of event occurrence were distributed within the range [−30 s, 30 s] and 90.1% of the time errors were distributed within the range [−10 s, 10 s]. The high correlation (r = 0.99997, p < 0.001) between the timing characteristics of the two types of signals indicates that the NE can reflect the actual dynamic interaction states of brain. Thus, the method may have potential implications for cognitive studies and for the detection of physiological states. Full article
(This article belongs to the Special Issue Graph and Network Entropies)
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12 pages, 809 KiB  
Article
Distance Entropy Cartography Characterises Centrality in Complex Networks
by Massimo Stella and Manlio De Domenico
Entropy 2018, 20(4), 268; https://doi.org/10.3390/e20040268 - 11 Apr 2018
Cited by 27 | Viewed by 5581
Abstract
We introduce distance entropy as a measure of homogeneity in the distribution of path lengths between a given node and its neighbours in a complex network. Distance entropy defines a new centrality measure whose properties are investigated for a variety of synthetic network [...] Read more.
We introduce distance entropy as a measure of homogeneity in the distribution of path lengths between a given node and its neighbours in a complex network. Distance entropy defines a new centrality measure whose properties are investigated for a variety of synthetic network models. By coupling distance entropy information with closeness centrality, we introduce a network cartography which allows one to reduce the degeneracy of ranking based on closeness alone. We apply this methodology to the empirical multiplex lexical network encoding the linguistic relationships known to English speaking toddlers. We show that the distance entropy cartography better predicts how children learn words compared to closeness centrality. Our results highlight the importance of distance entropy for gaining insights from distance patterns in complex networks. Full article
(This article belongs to the Special Issue Graph and Network Entropies)
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17 pages, 681 KiB  
Article
Sparse Power-Law Network Model for Reliable Statistical Predictions Based on Sampled Data
by Alexander P. Kartun-Giles, Dmitri Krioukov, James P. Gleeson, Yamir Moreno and Ginestra Bianconi
Entropy 2018, 20(4), 257; https://doi.org/10.3390/e20040257 - 07 Apr 2018
Cited by 4 | Viewed by 5129
Abstract
A projective network model is a model that enables predictions to be made based on a subsample of the network data, with the predictions remaining unchanged if a larger sample is taken into consideration. An exchangeable model is a model that does not [...] Read more.
A projective network model is a model that enables predictions to be made based on a subsample of the network data, with the predictions remaining unchanged if a larger sample is taken into consideration. An exchangeable model is a model that does not depend on the order in which nodes are sampled. Despite a large variety of non-equilibrium (growing) and equilibrium (static) sparse complex network models that are widely used in network science, how to reconcile sparseness (constant average degree) with the desired statistical properties of projectivity and exchangeability is currently an outstanding scientific problem. Here we propose a network process with hidden variables which is projective and can generate sparse power-law networks. Despite the model not being exchangeable, it can be closely related to exchangeable uncorrelated networks as indicated by its information theory characterization and its network entropy. The use of the proposed network process as a null model is here tested on real data, indicating that the model offers a promising avenue for statistical network modelling. Full article
(This article belongs to the Special Issue Graph and Network Entropies)
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Review

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15 pages, 457 KiB  
Review
A Review of Graph and Network Complexity from an Algorithmic Information Perspective
by Hector Zenil, Narsis A. Kiani and Jesper Tegnér
Entropy 2018, 20(8), 551; https://doi.org/10.3390/e20080551 - 25 Jul 2018
Cited by 43 | Viewed by 7101
Abstract
Information-theoretic-based measures have been useful in quantifying network complexity. Here we briefly survey and contrast (algorithmic) information-theoretic methods which have been used to characterize graphs and networks. We illustrate the strengths and limitations of Shannon’s entropy, lossless compressibility and algorithmic complexity when used [...] Read more.
Information-theoretic-based measures have been useful in quantifying network complexity. Here we briefly survey and contrast (algorithmic) information-theoretic methods which have been used to characterize graphs and networks. We illustrate the strengths and limitations of Shannon’s entropy, lossless compressibility and algorithmic complexity when used to identify aspects and properties of complex networks. We review the fragility of computable measures on the one hand and the invariant properties of algorithmic measures on the other demonstrating how current approaches to algorithmic complexity are misguided and suffer of similar limitations than traditional statistical approaches such as Shannon entropy. Finally, we review some current definitions of algorithmic complexity which are used in analyzing labelled and unlabelled graphs. This analysis opens up several new opportunities to advance beyond traditional measures. Full article
(This article belongs to the Special Issue Graph and Network Entropies)
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