Complex network growth model: Possible isomorphism between nonextensive statistical mechanics and random geometry

Tsallis, Constantino; Oliveira, Rute

doi:10.1063/5.0090864

Cited by 7 publications

(5 citation statements)

References 15 publications

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“…The connections between cities as well. In fact, the growth of virtually all asymptotically scale-free networks based on preferential attachment follow a q-statistical distribution of the number of degrees or, more generally speaking, of the site energies: see [92][93][94][95][96][97][98][99] and references therein. The definition of site (or local) energy is illustrated in figure 10, for a network with randomly weighted links.…”

Section: (G) Urbanism and Complex Networkmentioning

confidence: 99%

Non-additive entropies and statistical mechanics at the edge of chaos: a bridge between natural and social sciences

Tsallis

2023

Phil. Trans. R. Soc. A.

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The Boltzmann–Gibbs (BG) statistical mechanics constitutes one of the pillars of contemporary theoretical physics. It is constructed upon the other pillars—classical, quantum, relativistic mechanics and Maxwell equations for electromagnetism—and its foundations are grounded on the optimization of the BG (additive) entropic functional S BG = − k ∑ i p i ln ⁡ p i . Its use in the realm of classical mechanics is legitimate for vast classes of nonlinear dynamical systems under the assumption that the maximal Lyapunov exponent is positive (currently referred to as strong chaos ), and its validity has been experimentally verified in countless situations. It fails however when the maximal Lyapunov exponent vanishes (referred to as weak chaos ), which is virtually always the case with complex natural, artificial and social systems. To overcome this type of weakness of the BG theory, a generalization was proposed in 1988 grounded on the non-additive entropic functional S q = k ( ( 1 − ∑ i p i q ) / ( q − 1 ) ) ( q ∈ R ; S 1 = S BG ) . The index q and related ones are to be calculated, whenever mathematically tractable, from first principles and reflect the specific class of weak chaos. We review here the basics of this generalization and illustrate its validity with selected examples aiming to bridge natural and social sciences. This article is part of the theme issue ‘Thermodynamics 2.0: Bridging the natural and social sciences (Part 2)’.

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Section: (G) Urbanism and Complex Networkmentioning

confidence: 99%

Non-additive entropies and statistical mechanics at the edge of chaos: a bridge between natural and social sciences

Tsallis

2023

Phil. Trans. R. Soc. A.

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“…Rather unexpectedly a priori, some ubiquitous classes of growing networks-usually referred to as scale-free ones-are closely related [78][79][80][81][82][83][84][85][86] with various of the previous complex many-body systems. The relationship is neatly caused by the assumption of preferential attachment along the network growth.…”

Section: Asymptotically Scale-free Networkmentioning

confidence: 99%

Nonextensive Footprints in Dissipative and Conservative Dynamical Systems

et al. 2023

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Despite its centennial successes in describing physical systems at thermal equilibrium, Boltzmann–Gibbs (BG) statistical mechanics have exhibited, in the last several decades, several flaws in addressing out-of-equilibrium dynamics of many nonlinear complex systems. In such circumstances, it has been shown that an appropriate generalization of the BG theory, known as nonextensive statistical mechanics and based on nonadditive entropies, is able to satisfactorily handle wide classes of anomalous emerging features and violations of standard equilibrium prescriptions, such as ergodicity, mixing, breakdown of the symmetry of homogeneous occupancy of phase space, and related features. In the present study, we review various important results of nonextensive statistical mechanics for dissipative and conservative dynamical systems. In particular, we discuss applications to both discrete-time systems with a few degrees of freedom and continuous-time ones with many degrees of freedom, as well as to asymptotically scale-free networks and systems with diverse dimensionalities and ranges of interactions, of either classical or quantum nature.

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“…almost surely. * Proof 1: To compute the ratio in (2), recall that at time n = 0, we have one ball in the urn (this ball is of color c 1 ) and for each time instant n ≥ 1, we add ∆ n + 1 balls to the urn (∆ n of the color drawn and 1 of the new color c n+1 ). Hence the total number of balls in the urn at time t is given by 1 + t n=1 (∆ n + 1).…”

Section: The Modelmentioning

confidence: 99%

“…Preferential attachment graphs are an important class of randomly generated graphs which are often used to capture the "rich gets richer" phenomenon. This class of random graphs has been widely studied within the areas of statistical mechanics [1], [2], network science [3], probability theory [4], [5] and game theory [6]. One of the most popular model of a preferential attachment graph is the so-called Barabási-Albert model [7], which has since been modified in a variety of ways [8], [9], [10].…”

Section: Introductionmentioning

confidence: 99%

Modeling Network Contagion Via Interacting Finite Memory Pólya Urns

Singh

Alajaji

Gharesifard

2022

2022 IEEE International Symposium on Information Theory (ISIT)

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We introduce a novel preferential attachment model using the draw variables of a modified Pólya urn with an expanding number of colors, notably capable of modeling influential opinions (in terms of vertices of high degree) as the graph evolves. Similar to the Barabási-Albert model, the generated graph grows in size by one vertex at each time instance; in contrast however, each vertex of the graph is uniquely characterized by a color, which is represented by a ball color in the Pólya urn. More specifically at each time step, we draw a ball from the urn and return it to the urn along with a number (potentially time-varying and non-integer) of reinforcing balls of the same color; we also add another ball of a new color to the urn. We then construct an edge between the new vertex (corresponding to the new color) and the existing vertex whose color ball is drawn. Using color-coded vertices in conjunction with the time-varying reinforcing parameter allows for vertices added (born) later in the process to potentially attain a high degree in a way that is not captured in the Barabási-Albert model. We study the degree count of the vertices by analyzing the draw vectors of the underlying stochastic process. In particular, we establish the probability distribution of the random variable counting the number of draws of a given color which determines the degree of the vertex corresponding to that color in the graph. We further provide simulation results presenting a comparison between our model and the Barabási-Albert network.

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Complex network growth model: Possible isomorphism between nonextensive statistical mechanics and random geometry

Cited by 7 publications

References 15 publications

Non-additive entropies and statistical mechanics at the edge of chaos: a bridge between natural and social sciences

Non-additive entropies and statistical mechanics at the edge of chaos: a bridge between natural and social sciences

Nonextensive Footprints in Dissipative and Conservative Dynamical Systems

Modeling Network Contagion Via Interacting Finite Memory Pólya Urns

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