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Dynamic Processes on Complex Networks

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

Deadline for manuscript submissions: closed (31 July 2020) | Viewed by 31199

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Special Issue Editor

Prof. Bosiljka Tadic

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1. Department of Theoretical Physics, Jozef Stefan Institute, P.O. Box 3000, SI-1001 Ljubljana, Slovenia
2. Complexity Science Hub, 1080 Vienna, Austria
Interests: critical phenomena in disordered systems; nonlinear dynamics; collective behavior in online social systems; modeling of structure and dynamics of complex networks; agent-based models of web users; brain networks; self-assembly processes and nanonetworks

Special Issue Information

Due to the COVID19 epidemic, we are prepared to discuss the extension of the submission date with each author individually. Besides, given the current worldwide interest in COVID19 data analysis and modelling, contributions on this topic are highly welcome.

Dear Colleagues,

The emergence of new functional properties at a larger scale, as a paradigm of complexity, is based on the apparent interdependence between nonlinear dynamics and the underlying network structure. This Special Issue welcomes contributions highlighting current trends in structural and dynamical perspectives of complex systems. The goal is to unravel the impact of specific structural properties (described by directed weighted mono- and bipartite networks, hyperbolic graphs, and graphs with simplicial complexes architecture, as well as networks inferred from the empirical data such as brain graphs and online social networks) in the emergence of collective dynamics. On the other hand, some aspects of the dynamics are sensibly combined with or modified by a particular structure to stimulate cooperative behavior. These dynamical features can be sought in traffic and diffusion of various contents (information, knowledge, emotion, trust), dissemination of social conducts (opinions, rumors, riots, computer viruses, diseases), as well as synchronization processes in the original brain networks, spin kinetics, and avalanching dynamics in complex materials. A further goal is to understand and describe the complexity of the interface between structure and dynamics, based on temporal and geometric interpretation of time series, information topology and entropy, and on various concepts of multiscale processes. We look forward to submissions, both research articles and short reviews, addressing theoretical developments and numerical modeling, as well as empirical data analysis.

Prof. Bosiljka Tadic
Guest Editor

Manuscript Submission Information

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Keywords

avalanches in complex systems
diffusion of contents
directed bipartite networks
entropy
hyperbolic graphs
information topology
multiscale processes
online social dynamics
simplicial complexes architecture
spin dynamics in complex materials
spreading of behaviors
synchronization on brain graphs
time–series graphs
traffic on networks

Published Papers (10 papers)

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Research

21 pages, 469 KiB

Open AccessArticle

Modelling Excess Mortality in Covid-19-Like Epidemics

by Zdzislaw Burda

Entropy 2020, 22(11), 1236; https://doi.org/10.3390/e22111236 - 30 Oct 2020

Cited by 15 | Viewed by 2917

Abstract

We develop an agent-based model to assess the cumulative number of deaths during hypothetical Covid-19-like epidemics for various non-pharmaceutical intervention strategies. The model simulates three interrelated stochastic processes: epidemic spreading, availability of respiratory ventilators and changes in death statistics. We consider local and non-local modes of disease transmission. The first simulates transmission through social contacts in the vicinity of the place of residence while the second through social contacts in public places: schools, hospitals, airports, etc., where many people meet, who live in remote geographic locations. Epidemic spreading is modelled as a discrete-time stochastic process on random geometric networks. We use the Monte–Carlo method in the simulations. The following assumptions are made. The basic reproduction number is

R_{0} = 2.5

and the infectious period lasts approximately ten days. Infections lead to severe acute respiratory syndrome in about one percent of cases, which are likely to lead to respiratory default and death, unless the patient receives an appropriate medical treatment. The healthcare system capacity is simulated by the availability of respiratory ventilators or intensive care beds. Some parameters of the model, like mortality rates or the number of respiratory ventilators per

100,000

inhabitants, are chosen to simulate the real values for the USA and Poland. In the simulations we compare ‘do-nothing’ strategy with mitigation strategies based on social distancing and reducing social mixing. We study epidemics in the pre-vacine era, where immunity is obtained only by infection. The model applies only to epidemics for which reinfections are rare and can be neglected. The results of the simulations show that strategies that slow the development of an epidemic too much in the early stages do not significantly reduce the overall number of deaths in the long term, but increase the duration of the epidemic. In particular, a hybrid strategy where lockdown is held for some time and is then completely released, is inefficient. Full article

(This article belongs to the Special Issue Dynamic Processes on Complex Networks)

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21 pages, 1231 KiB

Open AccessArticle

Filtering Statistics on Networks

by G. J. Baxter, R. A. da Costa, S. N. Dorogovtsev and J. F. F. Mendes

Entropy 2020, 22(10), 1149; https://doi.org/10.3390/e22101149 - 13 Oct 2020

Viewed by 2307

Abstract

Compression, filtering, and cryptography, as well as the sampling of complex systems, can be seen as processing information. A large initial configuration or input space is nontrivially mapped to a smaller set of output or final states. We explored the statistics of filtering of simple patterns on a number of deterministic and random graphs as a tractable example of such information processing in complex systems. In this problem, multiple inputs map to the same output, and the statistics of filtering is represented by the distribution of this degeneracy. For a few simple filter patterns on a ring, we obtained an exact solution of the problem and numerically described more difficult filter setups. For each of the filter patterns and networks, we found three key numbers that essentially describe the statistics of filtering and compared them for different networks. Our results for networks with diverse architectures are essentially determined by two factors: whether the graphs structure is deterministic or random and the vertex degree. We find that filtering in random graphs produces much richer statistics than in deterministic graphs, reflecting the greater complexity of such graphs. Increasing the graph’s degree reduces this statistical richness, while being at its maximum at the smallest degree not equal to two. A filter pattern with a strong dependence on the neighbourhood of a node is much more sensitive to these effects. Full article

(This article belongs to the Special Issue Dynamic Processes on Complex Networks)

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18 pages, 484 KiB

Open AccessArticle

Modeling User Reputation in Online Social Networks: The Role of Costs, Benefits, and Reciprocity

by Frank Schweitzer, Pavlin Mavrodiev, Adrian M. Seufert and David Garcia

Entropy 2020, 22(10), 1073; https://doi.org/10.3390/e22101073 - 24 Sep 2020

Cited by 7 | Viewed by 2298

Abstract

We analyze an agent-based model to estimate how the costs and benefits of users in an online social network (OSN) impact the robustness of the OSN. Benefits are measured in terms of relative reputation that users receive from their followers. They can be increased by direct and indirect reciprocity in following each other, which leads to a core-periphery structure of the OSN. Costs relate to the effort to login, to maintain the profile, etc. and are assumed as constant for all users. The robustness of the OSN depends on the entry and exit of users over time. Intuitively, one would expect that higher costs lead to more users leaving and hence to a less robust OSN. We demonstrate that an optimal cost level exists, which maximizes both the performance of the OSN, measured by means of the long-term average benefit of its users, and the robustness of the OSN, measured by means of the lifetime of the core of the OSN. Our mathematical and computational analyses unfold how changes in the cost level impact reciprocity and subsequently the core-periphery structure of the OSN, to explain the optimal cost level. Full article

(This article belongs to the Special Issue Dynamic Processes on Complex Networks)

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15 pages, 982 KiB

Open AccessArticle

A Veritable Zoology of Successive Phase Transitions in the Asymmetric q-Voter Model on Multiplex Networks

by Anna Chmiel, Julian Sienkiewicz, Agata Fronczak and Piotr Fronczak

Entropy 2020, 22(9), 1018; https://doi.org/10.3390/e22091018 - 11 Sep 2020

Cited by 13 | Viewed by 2282

Abstract

We analyze a nonlinear q-voter model with stochastic noise, interpreted in the social context as independence, on a duplex network. The size of the lobby q (i.e., the pressure group) is a crucial parameter that changes the behavior of the system. The q-voter model has been applied on multiplex networks, and it has been shown that the character of the phase transition depends on the number of levels in the multiplex network as well as on the value of q. The primary aim of this study is to examine phase transition character in the case when on each level of the network the lobby size is different, resulting in two parameters

q_{1}

and

q_{2}

. In a system of a duplex clique (i.e., two fully overlapped complete graphs) we find evidence of successive phase transitions when a continuous phase transition is followed by a discontinuous one or two consecutive discontinuous phase transitions appear, depending on the parameter. When analyzing this system, we even encounter mixed-order (or hybrid) phase transition. The observation of successive phase transitions is a new quantity in binary state opinion formation models and we show that our analytical considerations are fully supported by Monte-Carlo simulations. Full article

(This article belongs to the Special Issue Dynamic Processes on Complex Networks)

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18 pages, 4714 KiB

Open AccessArticle

Towards a More Realistic Citation Model: The Key Role of Research Team Sizes

by Staša Milojević

Entropy 2020, 22(8), 875; https://doi.org/10.3390/e22080875 - 10 Aug 2020

Cited by 7 | Viewed by 2694

Abstract

We propose a new citation model which builds on the existing models that explicitly or implicitly include “direct” and “indirect” (learning about a cited paper’s existence from references in another paper) citation mechanisms. Our model departs from the usual, unrealistic assumption of uniform probability of direct citation, in which initial differences in citation arise purely randomly. Instead, we demonstrate that a two-mechanism model in which the probability of direct citation is proportional to the number of authors on a paper (team size) is able to reproduce the empirical citation distributions of articles published in the field of astronomy remarkably well, and at different points in time. Interpretation of our model is that the intrinsic citation capacity, and hence the initial visibility of a paper, will be enhanced when more people are intimately familiar with some work, favoring papers from larger teams. While the intrinsic citation capacity cannot depend only on the team size, our model demonstrates that it must be to some degree correlated with it, and distributed in a similar way, i.e., having a power-law tail. Consequently, our team-size model qualitatively explains the existence of a correlation between the number of citations and the number of authors on a paper. Full article

(This article belongs to the Special Issue Dynamic Processes on Complex Networks)

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24 pages, 1754 KiB

Open AccessFeature PaperArticle

Evolution of Cooperation in the Presence of Higher-Order Interactions: From Networks to Hypergraphs

by Giulio Burgio, Joan T. Matamalas, Sergio Gómez and Alex Arenas

Entropy 2020, 22(7), 744; https://doi.org/10.3390/e22070744 - 06 Jul 2020

Cited by 32 | Viewed by 4379

Abstract

Many real systems are strongly characterized by collective cooperative phenomena whose existence and properties still need a satisfactory explanation. Coherently with their collective nature, they call for new and more accurate descriptions going beyond pairwise models, such as graphs, in which all the interactions are considered as involving only two individuals at a time. Hypergraphs respond to this need, providing a mathematical representation of a system allowing from pairs to larger groups. In this work, through the use of different hypergraphs, we study how group interactions influence the evolution of cooperation in a structured population, by analyzing the evolutionary dynamics of the public goods game. Here we show that, likewise to network reciprocity, group interactions also promote cooperation. More importantly, by means of an invasion analysis in which the conditions for a strategy to survive are studied, we show how, in heterogeneously-structured populations, reciprocity among players is expected to grow with the increasing of the order of the interactions. This is due to the heterogeneity of connections and, particularly, to the presence of individuals standing out as hubs in the population. Our analysis represents a first step towards the study of evolutionary dynamics through higher-order interactions, and gives insights into why cooperation in heterogeneous higher-order structures is enhanced. Lastly, it also gives clues about the co-existence of cooperative and non-cooperative behaviors related to the structural properties of the interaction patterns. Full article

(This article belongs to the Special Issue Dynamic Processes on Complex Networks)

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21 pages, 784 KiB

Open AccessArticle

Power-Law Distributions of Dynamic Cascade Failures in Power-Grid Models

by Géza Ódor and Bálint Hartmann

Entropy 2020, 22(6), 666; https://doi.org/10.3390/e22060666 - 16 Jun 2020

Cited by 9 | Viewed by 3115

Abstract

Power-law distributed cascade failures are well known in power-grid systems. Understanding this phenomena has been done by various DC threshold models, self-tuned at their critical point. Here, we attempt to describe it using an AC threshold model, with a second-order Kuramoto type equation of motion of the power-flow. We have focused on the exploration of network heterogeneity effects, starting from homogeneous two-dimensional (2D) square lattices to the US power-grid, possessing identical nodes and links, to a realistic electric power-grid obtained from the Hungarian electrical database. The last one exhibits node dependent parameters, topologically marginally on the verge of robust networks. We show that too weak quenched heterogeneity, coming solely from the probabilistic self-frequencies of nodes (2D square lattice), is not sufficient for finding power-law distributed cascades. On the other hand, too strong heterogeneity destroys the synchronization of the system. We found agreement with the empirically observed power-law failure size distributions on the US grid, as well as on the Hungarian networks near the synchronization transition point. We have also investigated the consequence of replacing the usual Gaussian self-frequencies to exponential distributed ones, describing renewable energy sources. We found a drop in the steady state synchronization averages, but the cascade size distribution, both for the US and Hungarian systems, remained insensitive and have kept the universal tails, being characterized by the exponent

τ ≃ 1.8

. We have also investigated the effect of an instantaneous feedback mechanism in case of the Hungarian power-grid. Full article

(This article belongs to the Special Issue Dynamic Processes on Complex Networks)

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15 pages, 3589 KiB

Open AccessArticle

Cooperation on Interdependent Networks by Means of Migration and Stochastic Imitation

by Sayantan Nag Chowdhury, Srilena Kundu, Maja Duh, Matjaž Perc and Dibakar Ghosh

Entropy 2020, 22(4), 485; https://doi.org/10.3390/e22040485 - 23 Apr 2020

Cited by 53 | Viewed by 3531

Abstract

Evolutionary game theory in the realm of network science appeals to a lot of research communities, as it constitutes a popular theoretical framework for studying the evolution of cooperation in social dilemmas. Recent research has shown that cooperation is markedly more resistant in interdependent networks, where traditional network reciprocity can be further enhanced due to various forms of interdependence between different network layers. However, the role of mobility in interdependent networks is yet to gain its well-deserved attention. Here we consider an interdependent network model, where individuals in each layer follow different evolutionary games, and where each player is considered as a mobile agent that can move locally inside its own layer to improve its fitness. Probabilistically, we also consider an imitation possibility from a neighbor on the other layer. We show that, by considering migration and stochastic imitation, further fascinating gateways to cooperation on interdependent networks can be observed. Notably, cooperation can be promoted on both layers, even if cooperation without interdependence would be improbable on one of the layers due to adverse conditions. Our results provide a rationale for engineering better social systems at the interface of networks and human decision making under testing dilemmas. Full article

(This article belongs to the Special Issue Dynamic Processes on Complex Networks)

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24 pages, 905 KiB

Open AccessFeature PaperArticle

On the Structure of the World Economy: An Absorbing Markov Chain Approach

by Olivera Kostoska, Viktor Stojkoski and Ljupco Kocarev

Entropy 2020, 22(4), 482; https://doi.org/10.3390/e22040482 - 23 Apr 2020

Cited by 6 | Viewed by 4444

Abstract

The expansion of global production networks has raised many important questions about the interdependence among countries and how future changes in the world economy are likely to affect the countries’ positioning in global value chains. We are approaching the structure and lengths of value chains from a completely different perspective than has been available so far. By assigning a random endogenous variable to a network linkage representing the number of intermediate sales/purchases before absorption (final use or value added), the discrete-time absorbing Markov chains proposed here shed new light on the world input/output networks. The variance of this variable can help assess the risk when shaping the chain length and optimize the level of production. Contrary to what might be expected simply on the basis of comparative advantage, the results reveal that both the input and output chains exhibit the same quasi-stationary product distribution. Put differently, the expected proportion of time spent in a state before absorption is invariant to changes of the network type. Finally, the several global metrics proposed here, including the probability distribution of global value added/final output, provide guidance for policy makers when estimating the resilience of world trading system and forecasting the macroeconomic developments. Full article

(This article belongs to the Special Issue Dynamic Processes on Complex Networks)

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15 pages, 8211 KiB

Open AccessArticle

Magnetisation Processes in Geometrically Frustrated Spin Networks with Self-Assembled Cliques

by Bosiljka Tadić, Miroslav Andjelković, Milovan Šuvakov and Geoff J. Rodgers

Entropy 2020, 22(3), 336; https://doi.org/10.3390/e22030336 - 14 Mar 2020

Cited by 7 | Viewed by 2606

Abstract

Functional designs of nanostructured materials seek to exploit the potential of complex morphologies and disorder. In this context, the spin dynamics in disordered antiferromagnetic materials present a significant challenge due to induced geometric frustration. Here we analyse the processes of magnetisation reversal driven by an external field in generalised spin networks with higher-order connectivity and antiferromagnetic defects. Using the model in (Tadić et al. Arxiv:1912.02433), we grow nanonetworks with geometrically constrained self-assemblies of simplexes (cliques) of a given size n, and with probability p each simplex possesses a defect edge affecting its binding, leading to a tree-like pattern of defects. The Ising spins are attached to vertices and have ferromagnetic interactions, while antiferromagnetic couplings apply between pairs of spins along each defect edge. Thus, a defect edge induces

n - 2

frustrated triangles per n-clique participating in a larger-scale complex. We determine several topological, entropic, and graph-theoretic measures to characterise the structures of these assemblies. Further, we show how the sizes of simplexes building the aggregates with a given pattern of defects affects the magnetisation curves, the length of the domain walls and the shape of the hysteresis loop. The hysteresis shows a sequence of plateaus of fractional magnetisation and multiscale fluctuations in the passage between them. For fully antiferromagnetic interactions, the loop splits into two parts only in mono-disperse assemblies of cliques consisting of an odd number of vertices n. At the same time, remnant magnetisation occurs when n is even, and in poly-disperse assemblies of cliques in the range

n \in [2, 10]

. These results shed light on spin dynamics in complex nanomagnetic assemblies in which geometric frustration arises in the interplay of higher-order connectivity and antiferromagnetic interactions. Full article

(This article belongs to the Special Issue Dynamic Processes on Complex Networks)

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