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Role of dimensionality in preferential attachment growth in the Bianconi–Barabási model
Abstract
Scale-free networks are quite popular nowadays since many systems are well represented by such structures. In order to study these systems, several models were proposed. However, most of them do not take into account the node-to-node Euclidean distance, i.e. the geographical distance. In real networks, the distance between sites can be very relevant, e.g. those cases where it is intended to minimize costs. Within this scenario we studied the role of dimensionality d in the Bianconi–Barabási model with a preferential attachment growth involving Euclidean distances. The preferential attachment in this model follows the rule , where characterizes the fitness of the ith site and is randomly chosen within the interval. We verified that the degree distribution for dimensions are well fitted by , where is the q-exponential function naturally emerging within nonextensive statistical mechanics. We determine the index q and κ as functions of the quantities and d, and numerically verify that both present a universal behavior with respect to the scaled variable . The same behavior also has been displayed by the dynamical β exponent which characterizes the steadily growing number of links of a given site.
... The relationship between asymptotically scale-free d-dimensional geographic networks and nonextensive statistical mechanics started being explored in 2005 [28][29][30][31][32][33], where a preferential attachment index α A and a growth index α G were included. These studies showed that geographic networks exhibit three regimes: (a) 0 ≤ α A /d ≤ 1, corresponding to strongly long-range interactions, ...
... We verify that the energy distribution is completely independent from α G in all situations, as already discussed in earlier publications [28][29][30][31]33]. Consequently, we fix it to be α G = 1 in all our simulations. ...
... Also, we observe that p(ε) remains invariant when we fix α A /d and we modify the values of d = 1, 2, 3, 4, as shown in Fig. 2. This implies an universality property, namely that the energy distribution depends on the ratio α A /d and does not depend independently on α A and on d [29][30][31][32][33]. In consequence, we present our results by simply running d = 2. ...
Systems that consist of many localized constituents interacting with each other can be represented by complex networks. Consistently, network science has become highly popular in vast fields focusing on natural, artificial and social systems. We numerically analyze the growth of d -dimensional geographic networks (characterized by the index α G ⩾ 0; d = 1, 2, 3, 4) whose links are weighted through a predefined random probability distribution, namely P ( w ) ∝ e − | w − w c | / τ , w being the weight ( w c ⩾ 0; τ > 0). In this model, each site has an evolving degree k i and a local energy ε i ≡ ∑ j = 1 k i w i j / 2 ( i = 1, 2, …, N ) that depend on the weights of the links connected to it. Each newly arriving site links to one of the pre-existing ones through preferential attachment given by the probability Π i j ∝ ε i / d i j α A ( α A ⩾ 0 ) , where d ij is the Euclidean distance between the sites. Short- and long-range interactions respectively correspond to α A / d > 1 and 0 ⩽ α A / d ⩽ 1; α A / d → ∞ corresponds to interactions between close neighbors, and α A / d → 0 corresponds to infinitely-ranged interactions. The site energy distribution p ( ɛ ) corresponds to the usual degree distribution p ( k ) as the particular instance ( w c , τ ) = (1, 0). We numerically verify that the corresponding connectivity distribution p ( ɛ ) converges, when α A / d → ∞, to the weight distribution P ( w ) for infinitely wide distributions (i.e. τ → ∞, ∀ w c ) as well as for w c → 0, ∀ τ . Finally, we show that p ( ɛ ) is well approached by the q -exponential distribution e q − β q | ε − w c ′ | [ 0 ⩽ w c ′ ( w c , α A / d ) ⩽ w c ] , which optimizes the nonadditive entropy S q under simple constraints; q depends only on α A / d , thus exhibiting universality.
... The relationship between asymptotically scale-free d-dimensional geographic networks and nonextensive statistical mechanics started being explored in 2005 [27][28][29][30][31][32], where a preferential attachment index α A and a growth index α G were included. These studies showed that geographic networks exhibit three regimes: (a) 0 ≤ α A /d ≤ 1, corresponding to strongly long-range interactions, (b) 1 ≤ α A /d < ∼ 5, corresponding to moderately long-range interactions, and (c) α A /d > ∼ 5, corresponding to the BG-like regime, i.e., q 1 (short-range interactions). ...
... We verify that the energy distribution is completely independent from α G in all situations, as already discussed in earlier publications [27][28][29][30]32]. Consequently, we fix it to be α G = 1 in all our simulations. ...
... Also, we observe that p(ε) remains invariant when we fix α A /d and we modify the values of d = 1, 2, 3, 4, as shown in Fig. 2. This implies an universality property, namely that the energy distribution depends on the ratio α A /d and does not depend independently on α A and on d [28][29][30][31][32]. In consequence, we present our results by simply running d = 2. ...
Systems which consist of many localized constituents interacting with each other can be represented by complex networks. Consistently, network science has become highly popular in vast fields focusing on natural, artificial and social systems. We numerically analyze the growth of d-dimensional geographic networks (characterized by the index ; ) whose links are weighted through a predefined random probability distribution, namely , w being the weight . In this model, each site has an evolving degree and a local energy () that depend on the weights of the links connected to it. Each newly arriving site links to one of the pre-existing ones through preferential attachment given by the probability , where is the Euclidean distance between the sites. Short- and long-range interactions respectively correspond to and ; corresponds to interactions between close neighbors, and corresponds to infinitely-ranged interactions. The site energy distribution corresponds to the usual degree distribution p(k) as the particular instance . We numerically verify that the corresponding connectivity distribution converges, when , to the weight distribution P(w) for infinitely narrow distributions (i.e., ) as well as for .
... A , respectively. So, as α A increases-that means a decrease in the interaction range-, a topological phase transition occurs in the connectivity distribution [47,71]. The network changes from a completely heterogeneous network when α A = 0, which represent a long-range interaction regime, to an increasingly homogeneous network as α A tends to infinity, representing a shortrange interaction regime. ...
... It is possible to include euclidian distance in the homophilic and the fitness models, as investigated by Nunes et al. [71] and by Cinardi et al. [74], respectively. For instance, when we study social interaction in a city [69], the parameter η i can represent the influence of different places localized in a city. ...
... for fitness and homophilic models, respectively. Nunes et al. [71] and Cinardi et al. [74] also shown a topological phase transition, as α A increases for fitness and homophilic models and no influence of the parameter α G in the pattern of the connectivity distribution. We obtained the same results for homophilic networks. ...
This article presents a brief overview of the main network models that use growth and preferential attachment. We start with the classical model proposed by Barabási and Albert: nodes are added to the network connecting preferably to nodes that are more connected. We also present models that consider more representative elements from social perspectives, such as the homophily between the nodes and the fitness that each node has, to build connections. Furthermore, we show a version of these models that includes Euclidean distance between the nodes as a preferential attachment component. Our objective is to study the fundamental properties of these networks, as distribution of connectivity, degree correlation, shortest path, cluster coefficient and how these characteristics are affected by the preferential attachment rules. In addition to the review, we also provided an application of these models using real-world networks.
... During the initial years, network science and nonextensive statistical mechanics were seen as completely different areas. But meaningful connections started in 2005 [16][17][18][19][20][21][22] . It is nowadays known that the degree distribution of asymptotically scale-free networks at the thermodynamic limit is of the form P(k) ∝ e −k/κ q , where the q-exponential function is defined as e z q ≡ [1 + (1 − q)z] ...
... We have in fact analyzed a large amount of typical cases in the space (d, α A , α G , w 0 , η) , and have systematically found the same results for d = 1, 2, 3 within the intervals (α A /d ∈ [0, 10]; α G ∈ [1, 10]; w 0 ∈ [0.5, 10]; η ∈ [0.5, 3]) . Similarly to previous works 16,[19][20][21] , p(ε) does not depend on α G ; also, it does not depend independently on d and α A , but only, remarkably, on the ratio α A /d ; p(ε) also depends on w 0 and η (see Fig. 2a-d). Because of these features, and without loss of generality, we have once for ever fixed α G = 1 , and d = 2 . ...
... This seemingly is the first time that, in a complex network, we identify a parameter which plays the role of an external parameter that we may fix at will, similarly to the manner in which we fix, in BG statistical mechanics, the temperature at which the thermally equilibrated system is placed. In all previous connections with random networks [16][17][18][19][20][21][22] , β q (sometimes noted 1/κ q ) was univocally related to q. A single value for β q for a given value of q is analogous to traditional critical points in BG statistics. ...
Boltzmann–Gibbs statistical mechanics applies satisfactorily to a plethora of systems. It fails however for complex systems generically involving nonlocal space–time entanglement. Its generalization based on nonadditive q -entropies adequately handles a wide class of such systems. We show here that scale-invariant networks belong to this class. We numerically study a d -dimensional geographically located network with weighted links and exhibit its ‘energy’ distribution per site at its quasi-stationary state. Our results strongly suggest a correspondence between the random geometric problem and a class of thermal problems within the generalised thermostatistics. The Boltzmann–Gibbs exponential factor is generically substituted by its q -generalisation, and is recovered in the q=1 q = 1 limit when the nonlocal effects fade away. The present connection should cross-fertilise experiments in both research areas.
... During the initial years, network science and nonextensive statistical mechanics were seen as completely different areas. But meaningful connections started in 2005 [16][17][18][19][20][21][22] . It is nowadays known that the degree distribution of asymptotically scale-free networks at the thermodynamic limit is of the form P(k) ∝ e −k/κ q , where the q-exponential function is defined as ...
... . Similarly to previous works 16,[19][20][21] , p(ε) does not depend on α G ; also, it does not depend independently on d and α A , but only, remarkably, on the ratio α A /d; p(ε) also depends on w 0 and η (see Fig. 2(a)-(d)). Because of these features, and without loss of generality, we have once for ever fixed α G = 1, and d = 2. ...
... This seemingly is the first time that, in a complex network, we identify a parameter which plays the role of an external parameter that we may fix at will, similarly to the manner in which we fix, in BG statistical mechanics, the temperature at which the thermally equilibrated system is placed. In all previous connections with random networks [16][17][18][19][20][21][22] , β q (sometimes noted 1/κ q ) was univocally related to q. A single value for β q for a given value of q is analogous to traditional critical points in BG statistics. ...
Boltzmann-Gibbs statistical mechanics applies satisfactorily to a plethora of systems. It fails however for complex systems generically involving strong space-time entanglement. Its generalization based on nonadditive q-entropies adequately handles a wide class of such systems. We show here that scale-invariant networks belong to this class. We numerically study a d-dimensional geographically located network with weighted links and exhibit its 'energy' distribution per site at its quasi-stationary state. Our results strongly suggest a correspondence between the random geometric problem and a class of thermal problems within the generalised thermostatistics. The Boltzmann-Gibbs exponential factor is generically substituted by its q-generalisation, and is recovered in the q=1 limit when the nonlocal effects fade away. The present connection should cross-fertilise experiments in both research areas.
... The unprecedented use of a nonadditive entropy (a conceptual possibility which, in one way or another, had already been considered historically [11][12][13]) in order to generalize the BG statistical mechanics, opened a door that has been being widely explored since 1988 [14,15]. It is for a wide class of anomalous situations, including analogous geometrical random systems such as asymptotically scale-free networks (see, for instance, [16][17][18] and references therein involving distance-dependent couplings of the type r −α a ; for the particular case of α A = 0, see [19][20][21][22]) that non-Boltzmannian entropies and related formalisms become useful. A neat explanation of the difference between entropic additivity and entropic extensivity will be provided below. ...
... Also worth mentioning are selected entropic applications beyond BG in other areas of knowledge: complex networks [16][17][18]; economics [149][150][151][152][153][154][155][156]; geophysics (earthquakes, atmosphere) [157][158][159][160][161][162][163][164][165][166]; general and quantum chemistry [139,[167][168][169][170][171]; hydrology and engineering (water engineering [172] and materials engineering [173,174]); power grids [175]; the environment [176]; medicine [177][178][179]; biology [180,181]; computational processing of medical images (microcalcifications in mammograms [182] and magnetic resonance for multiple sclerosis [183]) and time series (e.g., ECG in coronary disease [184] and EEG in epilepsy [185,186]); train delays [187]; citations of scientific publications and scientometrics [188,189]; global optimization techniques [190][191][192]; ecology [193][194][195]; cognitive science [196][197][198]; mathematics (functions [199], uniqueness theorems and related axiomatic approaches [200][201][202][203][204][205], central limit theorems, and generalized Fourier transform [206][207][208][209][210][211][212][213][214]); probabilistic models [215][216][217]; information geometry [218,219]. ...
... We see here that both the thermal and the geometrical model exhibit the interesting α/d scaling. The same happens for the α-Heisenberg inertial ferromagnet [100], the α-Fermi-Pasta-Ulam model [101][102][103][104], and other asymptotically scale-free networks [17,18], thus exhibiting the ubiquity of this grounding scaling law. Before proceeding, let us clarify why the statistical mechanical description of scale-free networks appears as a particular instance of q-statistics. ...
The pillars of contemporary theoretical physics are classical mechanics, Maxwell electromagnetism, relativity, quantum mechanics, and Boltzmann–Gibbs (BG) statistical mechanics –including its connection with thermodynamics. The BG theory describes amazingly well the thermal equilibrium of a plethora of so-called simple systems. However, BG statistical mechanics and its basic additive entropy SBG started, in recent decades, to exhibit failures or inadequacies in an increasing number of complex systems. The emergence of such intriguing features became apparent in quantum systems as well, such as black holes and other area-law-like scenarios for the von Neumann entropy. In a different arena, the efficiency of the Shannon entropy—as the BG functional is currently called in engineering and communication theory—started to be perceived as not necessarily optimal in the processing of images (e.g., medical ones) and time series (e.g., economic ones). Such is the case in the presence of generic long-range space correlations, long memory, sub-exponential sensitivity to the initial conditions (hence vanishing largest Lyapunov exponents), and similar features. Finally, we witnessed, during the last two decades, an explosion of asymptotically scale-free complex networks. This wide range of important systems eventually gave support, since 1988, to the generalization of the BG theory. Nonadditive entropies generalizing the BG one and their consequences have been introduced and intensively studied worldwide. The present review focuses on these concepts and their predictions, verifications, and applications in physics and elsewhere. Some selected examples (in quantum information, high- and low-energy physics, low-dimensional nonlinear dynamical systems, earthquakes, turbulence, long-range interacting systems, and scale-free networks) illustrate successful applications. The grounding thermodynamical framework is briefly described as well.
... and the q-Gaussian exp q [−βx 2 ] appear naturally by extremizing S q under appropriate constraints [27][28][29]. The generalized thermostatistics based on S q frequently applies when assumptions underlying the BG thermostatistics are not fulfilled (like, e.g., mixing and ergodicity) [14,15,[18][19][20][29][30][31][32][33][34]. ...
... We explore how higher values of d modify previous d = 1 results for the velocity distribution; moreover, we show how the energy distribution changes from the celebrated exponential Boltzmann weight to a distribution well described by a q-exponential, as the system goes from the short-to the long-range regime. We have also verified a universal scaling law of these distributions governed by the α/d ratio, similarly to what was found in other complex systems [18,19]. ...
... Intriguingly, these values of q do not attain unit around α/d = 1, but rather at some higher value, close to α/d = 2. This fact has also been observed in recent simulations of other models with power-law decay of interactions: (i) a one-dimensional quantum Ising ferromagnet [36]; (ii) a Fermi-Pasta-Ulam-like one-dimensional Hamiltonian with a quartic coupling constant decaying with the distance between oscillators [14,20,37]; (iii) scale-free complex networks [18,19]. Similarly to the present investigation, in these previous works three distinct regimes were found, namely, a non-BG long-range interacting regime (0 ≤ α/d ≤ 1), a non-BG shortrange one (1 < α/d ≤ a c ), and the standard BG short-range regime (α/d > a c ); for some classical Hamiltonians, a c ≈ 2, whereas, for complex networks, a c ≈ 5. ...
The dynamics and thermostatistics of a classical inertial XY model, characterized by long-range interactions, are investigated on d-dimensional lattices (d=1,2, and 3), through molecular dynamics. The interactions between rotators decay with the distance like~ (), where and respectively correspond to the nearest-neighbor and infinite-range interactions. We verify that the momenta probability distributions are Maxwellians in the short-range regime, whereas q-Gaussians emerge in the long-range regime. Moreover, in this latter regime, the individual energy probability distributions are characterized by long tails, corresponding to q-exponential functions. The present investigation strongly indicates that, in the long-range regime, central properties fall out of the scope of Boltzmann-Gibbs statistical mechanics, depending on d and through the ratio .
... Complex networks constitute a powerful tool for the description of many natural, artificial and social systems. In the last decades hundreds of models have been proposed to describe them [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Of particular importance, and representative of many such systems, are the so-called asymptotically-scale free networks, or simply scale-free networks. ...
... A possible extension of this model is the well-known Bianconi-Barabási one [6], where now the fitness parameter is (introduced and chosen) uniformly between zero and one. It turned out that these are particular cases of more general models [9,14,15], where the dependence on the distance of the growing mechanism and therefore the role of dimensionality of the system is introduced. The nodes are placed in a specific geographical position (for dimension d = 1, 2 and 3) based on an isotropic distribution; then the topology of the network is dictated by the degree, the fitness and distances between nodes. ...
We consider a generalised d-dimensional model for asymptotically-
scale-free geographical networks. Central to many networks of this kind, when
considering their growth in time, is the attachment rule, i.e. the probability
that a new node is attached to one (or more) preexistent nodes. In order to
be more realistic, a fitness parameter ηi ∈ [0, 1] for each node i of the network
is also taken into account to reflect the ability of the nodes to attract new
ones. Our d-dimensional model takes into account the geographical distances
between nodes, with dierent probability distribution for η which sensibly
modifies the growth dynamics. The preferential attachment rule is assumed to
be Πi ∝ kiηir−αA where ki is the connectivity of the ith pre-existing site and ij
αA characterizes the importance of the euclidean distance r for the network
growth. For special values of the parameters, this model recovers respectivelythe Bianconi–Barabási and the Barabási–Albert ones. The present generalisedmodel is asymptotically scale-free in all cases, and its degree distribution is verywell fitted with q-exponential distributions, which optimizes the nonadditive −k/κ entropy Sq, given by p(k) ∝ eq1/(q−1)≡ 1/[1 + (q − 1)k/κ] , with (q, κ) depending uniquely only on the ratio αA/d and the fitness distribution. Hence this model constitutes a realization of asymptotically-scale-free geographical networks within nonextensive statistical mechanics, where k plays the role of energy and κ plays the role of temperature. General scaling laws are also found for q as a function of the parameters of the model.
... Complex networks constitute a powerful tool for the description of many natural, artificial and social systems. In the last decades hundreds of models have been proposed to describe them [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Of particular importance, and representative of many such systems, are the so-called asymptotically-scale free networks, or simply scale-free networks. ...
... A possible extension of this model is the well-known Bianconi-Barabási one [6], where now the fitness parameter is (introduced and chosen) uniformly between zero and one. It turned out that these are particular cases of more general models [9,14,15], where the dependence on the distance of the growing mechanism and therefore the role of dimensionality of the system is introduced. The nodes are placed in a specific geographical position (for dimension d = 1, 2 and 3) based on an isotropic distribution; then the topology of the network is dictated by the degree, the fitness and distances between nodes. ...
We consider a generalised d-dimensional model for asymptotically-scale-free geographical networks. Central to many networks of this kind, when considering their growth in time, is the attachment rule, i.e. the probability that a new node is attached to one (or more) preexistent nodes. In order to be more realistic, a fitness parameter for each node i of the network is also taken into account to reflect the ability of the nodes to attract new ones. Our d-dimensional model takes into account the geographical distances between nodes, with different probability distribution for which sensibly modifies the growth dynamics. The preferential attachment rule is assumed to be where is the connectivity of the ith pre-existing site and characterizes the importance of the euclidean distance r for the network growth. For special values of the parameters, this model recovers respectively the Bianconi-Barab\'{a}si and the Barab\'{a}si-Albert ones. The present generalised model is asymptotically scale-free in all cases, and its degree distribution is very well fitted with q-exponential distributions, which optimise the nonadditive entropy , given by , with depending uniquely only on the ratio and the fitness distribution. Hence this model constitutes a realization of asymptotically-scale-free geographical networks within nonextensive statistical mechanics, where k plays the role of energy and plays the role of temperature. General scaling laws are also found for q as a function of the parameters of the model.
... The same departure from the BG predictions is observed for the time-averaged energy distributions; in this case, instead of the usual BG exponential, q-exponential probability distributions are obtained, with q ≡ q E (α/d) [34]. Similar universality features have been recently obtained in quite different contexts, such as a generalized Fermi-Pasta-Ulam model with long-range interactions [35][36][37][38][39], complex networks with preferential attachment growth [40,41], and a system of particles under overdamped motion interacting repulsively with power-law interactions [42]. ...
... The present numerical analysis is consistent with the theoretical prevision of a change of behavior at α/d = 1, in agreement with recent results obtained in other long-range-interaction models, such as a generalized Fermi-Pasta-Ulam model [35][36][37][38][39], complex networks with preferential attachment growth [40,41], and a system of particles under overdamped motion interacting repulsively [42]. In future works we intend to cover the entire α/d ≥ 0 region in order to clarify further this point, and particularly, to investigate why some of the Heisenberg results found herein sensibly differ from those available in the literature for the XY and Fermi-Pasta-Ulam models. ...
We numerically study the first-principle dynamics and thermostatistics of a d-dimensional classical inertial Heisenberg ferromagnetic model (d=1,2,3) with interactions decaying with the distance rij as 1/rijα (α≥0), where the limit α=0 (α→∞) corresponds to infinite-range (nearest-neighbour) interactions, and the ratio α/d>1 (0≤α/d≤1) characterizes the short-ranged (long-ranged) regime. By means of first-principle molecular dynamics we study: (i) The scaling with the system size N of the maximum Lyapunov exponent λ in the form λ∼N−κ, where κ(α/d) depends only on the ratio α/d; (ii) The time-averaged single-particle angular momenta probability distributions for a typical case in the long-range regime 0≤α/d≤1 (which turns out to be well fitted by q-Gaussians), and (iii) The time-averaged single-particle energies probability distributions for a typical case in the long-range regime 0≤α/d≤1 (which turns out to be well fitted by q-exponentials). Through the Lyapunov exponents we observe an intriguing, and possibly size-dependent, persistence of the non-Boltzmannian behavior even in the α/d>1 regime. The universality that we observe for the probability distributions with regard to the ratio α/d makes this model similar to the α-XY and α-Fermi-Pasta-Ulam Hamiltonian models as well as to asymptotically scale-invariant growing networks.
... 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. ...
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 SBG=−k∑ipilnpi. 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 Sq=k((1−∑ipiq)/(q−1)) (q∈R; S1=SBG). 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)’.
... 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. ...
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.
... Fan and Chen [47] developed a Multi Local World (MLW) model to describe the Internet structure with better statistical performance. The Bianconi-Barabási model is another extension of the BA model, with it being assumed that nodes with higher fitness have a greater probability of acquiring new links [48]. ...
Given they are two critical infrastructure areas, the security of electricity and gas networks is highly important due to potential multifaceted social and economic impacts. Unexpected errors or sabotage can lead to blackouts, causing a significant loss for the public, businesses, and governments. Climate change and an increasing number of consequent natural disasters (e.g., bushfires and floods) are other emerging network resilience challenges. In this paper, we used network science to examine the topological resilience of national energy networks with two case studies of Australian gas and electricity networks. To measure the fragility and resilience of these energy networks, we assessed various topological features and theories of percolation. We found that both networks follow the degree distribution of power-law and the characteristics of a scale-free network. Then, using these models, we conducted node and edge removal experiments. The analysis identified the most critical nodes that can trigger cascading failure within the network upon a fault. The analysis results can be used by the network operators to improve network resilience through various mitigation strategies implemented on the identified critical nodes.
... We show that these equations gain a foundation when they are involved with the proposal of non-extensive statistical mechanics [31]. Nowadays, the Tsallis statistic [31] and nonlinear diffusion have assumed important roles in the application of more subtle problems in thermodynamics, such as black holes [21,143], generalised forms of H-theorem [19,20,144], financial market [145,146], and many other systems [94,147,148]. Here, it is worth mentioning to the reader that from the point of view of the H theorem [11] the porous media equations may imply the Tsallis entropy. ...
This review article aims to stress and reunite some of the analytic formalism of the anomalous diffusive processes that have succeeded in their description. Also, it has the objective to discuss which of the new directions they have taken nowadays. The discussion is started by a brief historical report that starts with the studies of thermal machines and combines in theories such as the statistical mechanics of Boltzmann-Gibbs and the Brownian Movement. In this scenario, in the twentieth century, a series of experiments were reported that were not described by the usual model of diffusion. Such experiments paved the way for deeper investigation into anomalous diffusion. These processes are very abundant in physics, and the mechanisms for them to occur are diverse. For this reason, there are many possible ways of modelling the diffusive processes. This article discusses three analytic approaches to investigate anomalous diffusion: fractional diffusion equation, nonlinear diffusion equation and Langevin equation in the presence of fractional, coloured or multiplicative noises. All these formalisms presented different degrees of complexity and for this reason, they have succeeded in describing anomalous diffusion phenomena.
We numerically investigate a geographical d-dimensional (d=1,2,3,4) Bianconi–Barabási-like model, characterized by preferential attachment growth mechanisms influenced by Euclidean distances and weighted edges. The weights of the edges follow a predetermined random probability distribution. This model is implemented through a straightforward energy-driven dynamics and exhibits the distribution of ’energy’ per site in its quasi-stationary state. Across all networks generated by this model, we observe q-exponential energy distributions over the entire parameter space, which exhibits that this model belongs to the realm of nonadditive q-entropies. Additionally, the time evolution of the site energies, characterized by the dynamic exponent, is analyzed.
In this paper we introduce a new type of preferential attachment network, the growth of which is based on the eigenvector centrality. In this network, the agents attach most probably to the nodes with larger eigenvector centrality which represents that the agent has stronger connections. A new network is presented, namely a dandelion network, which shares some properties of star-like structure and also a hierarchical network. We show that this network, having hub-and-spoke topology is not generally scale free, and shows essential differences with respect to the Barabási–Albert preferential attachment model. Most importantly, there is a super hub agent in the system (identified by a pronounced peak in the spectrum), and the other agents are classified in terms of the distance to this super-hub. We explore a plenty of statistical centralities like the nodes degree, the betweenness and the eigenvector centrality, along with various measures of structure like the community and hierarchical structures, and the clustering coefficient. Global measures like the shortest path statistics and the self-similarity are also examined.
The Beck-Cohen superstatistics became an important theory in the scenario of complex systems because it generates distributions representing regions of a nonequilibrium system, characterized by different temperatures T≡β−1, leading to a probability distribution f(β). In superstatistics, some classes have been most frequently considered for f(β), like χ2, χ2 inverse, and log-normal ones. Herein we investigate the superstatistics resulting from a χη2 distribution through a modification of the usual χ2 by introducing a real index η (0<η≤1). In this way, one covers two common and relevant distributions as particular cases, proportional to the q-exponential (eq−βx=[1−(1−q)βx]11−q) and the stretched exponential (e−(βx)η). Furthermore, an associated generalized entropic form is found. Since these two particular-case distributions have been frequently found in the literature, we expect that the present results should be applicable to a wide range of classes of complex systems.
A classical α−XY inertial model, consisting of N two-component rotators and characterized by interactions decaying with the distance rij as 1/rijα (α≥0) is studied through first-principle molecular-dynamics simulations on d-dimensional lattices of linear size L (N≡Ld and d=1,2,3). The limits α=0 and α→∞ correspond to infinite-range and nearest-neighbor interactions, respectively, whereas the ratio α/d>1 (0≤α/d≤1) is associated with short-range (long-range) interactions. By analyzing the time evolution of the kinetic temperature T(t) in the long-range-interaction regime, one finds a quasi-stationary state (QSS) characterized by a temperature TQSS; for fixed N and after a sufficiently long time, a crossover to a second plateau occurs, corresponding to the Boltzmann-Gibbs temperature TBG (as predicted within the BG theory), with TBG>TQSS. It is shown that the QSS duration (tQSS) depends on N, α, and d, although the dependence on α appears only through the ratio α/d; in fact, tQSS decreases with α/d and increases with both N and d. Considering a fixed energy value, a scaling for tQSS is proposed, namely, tQSS∝NA(α/d)e−B(N)(α/d)2, analogous to a recent analysis carried out for the classical α-Heisenberg inertial model. It is shown that the exponent A(α/d) and the coefficient B(N) present universal behavior (within error bars), comparing the XY and Heisenberg cases. The present results should be useful for other long-range systems, very common in nature, like those characterized by gravitational and Coulomb forces.
We develop a framework based on discrete-time Markov chains (MCs) to model the Brazilian air transport market networks. Our results suggest that in this market, economic activity explains the centralities of the airports and the players' choices. Periods of economic prosperity affect the air transport networks and they move from a hub-and-spoke topology towards a mixed point-to-point and hub-and-spoke topology. Furthermore, we use the MC transition and stationary probabilities to build geographical subgraphs that reveal (1) the most important routes, (2) the stability of these routes, (3) the most important flights, and (4) the airlines' specific market niches.
The area of networks is very interdisciplinary and exhibits many applications in several fields of science. Nevertheless, there are few studies focusing on geographically located d-dimensional networks. In this paper, we study scaling properties of a wide class of d-dimensional geographically located networks which grow with preferential attachment involving Euclidean distances through r −α A ij (αA ≥ 0). We have numerically analyzed the time evolution of the connectivity of sites, the average shortest path, the degree distribution entropy, and the average clustering coefficient, for d = 1, 2, 3, 4, and typical values of αA. Remarkably enough, virtually all the curves can be made to collapse as functions of the scaled variable αA/d. These observations confirm the existence of three regimes. The first one occurs in the interval αA/d ∈ [0, 1]; it is non-Boltzmannian with very-long-range interactions in the sense that the degree distribution is a q-exponential with q constant and above unity. The critical value αA/d = 1 that emerges in many of these properties is replaced by αA/d = 1/2 for the β-exponent which characterizes the time evolution of the connectivity of sites. The second regime is still non-Boltzmannian, now with moderately long-range interactions, and reflects in an index q monotonically decreasing with αA/d increasing from its critical value to a characteristic value αA/d ≃ 5. Finally, the third regime is Boltzmannian (with q = 1), and corresponds to short-range interactions.
The dynamics and thermostatistics of a classical inertial XY model, characterized by long-range interactions, are investigated on d-dimensional lattices (d = 1, 2, and 3), through molecular dynamics. The interactions between rotators decay with the distance r ij like, where α → ∞ and respectively correspond to the nearest-neighbor and infinite-range interactions. We verify that the momenta probability distributions are Maxwellians in the short-range regime, whereas q-Gaussians emerge in the long-range regime. Moreover, in this latter regime, the individual energy probability distributions are characterized by long tails, corresponding to q-exponential functions. The present investigation strongly indicates that, in the long-range regime, central properties fall out of the scope of Boltzmann-Gibbs statistical mechanics, depending on d and α through the ratio .
We present a study of social networks based on the analysis of Brazilian and Portuguese family names (surnames). We construct networks whose nodes are names of families and whose edges represent parental relations between two families. From these networks we extract the connectivity distribution, clustering coefficient, shortest path and centrality. We find that the connectivity distribution follows an approximate power law. We associate the number of hubs, centrality and entropy to the degree of miscegenation in the societies in both countries. Our results show that Portuguese society has a higher miscegenation degree than Brazilian society. All networks analyzed lead to approximate inverse square power laws in the degree distribution. We conclude that the thermodynamic limit is reached for small networks (3 or 4 thousand nodes). The assortative mixing of all networks is negative, showing that the more connected vertices are connected to vertices with lower connectivity. Finally, the network of surnames presents some small world characteristics.
Deep connections are known to exist between scale-free networks and non-Gibbsian statistics. For example, typical degree distributions at the thermodynamical limit are of the form , where the q-exponential form optimizes the nonadditive entropy Sq (which, for q → 1, recovers the Boltzmann-Gibbs entropy). We introduce and study here d-dimensional geographically-located networks which grow with preferential attachment involving Euclidean distances through . Revealing the connection with q-statistics, we numerically verify (for d = 1, 2, 3 and 4) that the q-exponential degree distributions exhibit, for both q and k, universal dependences on the ratio αA/d. Moreover, the q = 1 limit is rapidly achieved by increasing αA/d to infinity.
Network geometry is attracting increasing attention because it has a wide
range of applications, ranging from data mining to routing protocols in the
Internet. At the same time advances in the understanding of the geometrical
properties of networks are essential for further progress in quantum gravity.
In network geometry, simplicial complexes describing the interaction between
two or more nodes, play a spacial role. In fact these structures can be used to
discretize a geometrical d dimensional space, and for this reason they have
been already widely used in quantum gravity. Here we introduce the Network
Geometry with Flavor (NGF) describing simplicial complexes defined
in arbitrary dimension d and evolving by a non-equilibrium dynamics. The NGF
can generate discrete geometries of different nature, ranging from chains and
higher dimensional manifolds to scale-free networks with small-world
properties, scale-free degree distribution and non-trivial community structure.
The thermodynamic properties of NGF reveal that NGF obeys a generalized area
law opening a new scenario for formulating its coarse-grained limit. The
structure of NGF is strongly dependent on the dimensionality d.Here we find
that, for NGF with dimension , generalizing growing complex networks, it
is not necessary to have an explicit preferential attachment rule to generate
scale-free topologies. We also show that NGF admits a quantum mechanical
description in terms of associated quantum network states. Interestingly the
NGF remains fully classical but its statistical properties reveal the relation
to its quantum mechanical description. In fact the -dimensional faces
of the NGF have generalized degrees that follow either the Fermi-Dirac,
Boltzmann or Bose-Einstein statistics depending on the flavor s and the
dimensions d and .
The product of the differential production cross-section and the branching fraction of the decay is measured as a function of the beauty hadron transverse momentum, pT, and rapidity, y. The kinematic region of the measurements is pT < 20 GeV/c and 2.0 < y < 4.5. The measurements use a data sample corresponding to an integrated luminosity of 3fb⁻¹ collected by the LHCb detector in pp collisions at centre-of-mass energies in 2011 and in 2012. Based on previous LHCb results of the fragmentation fraction ratio the branching fraction of the decay is measured to be
where the first uncertainty is statistical, the second is systematic, the third is due to the uncertainty on the branching fraction of the decay B̅⁰ → J/ψK̅*(892)⁰, and the fourth is due to the knowledge of . The sum of the asymmetries in the production and decay between and is also measured as a function of pT and y. The previously published branching fraction of , relative to that of , is updated. The branching fractions of are determined.
The rate at which nodes in a network
increase their connectivity depends on their fitness to compete
for links. For example, in social networks some individuals
acquire more social links than others, or on the www some webpages
attract considerably more links than others. We find that this
competition for links translates into multiscaling, i.e. a fitness-dependent
dynamic exponent, allowing fitter nodes to overcome the
more connected but less fit ones. Uncovering this
fitter-gets-richer phenomenon can help us understand in
quantitative terms the evolution of many competitive systems in
nature and society.
We introduce a model which consists in a planar network which grows by adding nodes at a distance r from the pre-existing barycenter. Each new node position is randomly located through the distribution law P(r) ∝1/rγ with γ> 1. The new node j is linked to only one pre-existing node according to the probability law
P(i ↔j) ∝ηi ki / rijαA (1 ≤i ≪ j; αA ≥0); ki is the number of links of the ith node, ηi is its fitness (or quality factor), and rij is the distance. We consider in the present paper two models for ηi. In one of them, the single fitness model (SFM [D. J. B. Soares, C. Tsallis, A. M. Mariz and L. R. da Silva, Europhys. Letters 70 (2005), 70.]), we consider ηi = 1 ∀i. In the other one, the uniformly distributed fitness model (UDFM), ηi is chosen to be uniformly distributed within the interval (0,1]. We have determined numerically the degree distribution P(k). This distribution appears to be well fitted with P(k) = P(1) k-λ eq-(k-1) / κ with q ≥1, where eqx ≡[1+(1 q)x]1/(1-q) (e1x = ex) is the q-exponential function naturally emerging within Tsallis nonextensive statistical mechanics. We determine, for both models,
the entropic index q as a function of αA. Additionally, we determine the average topological (or chemical) distance within the network, and the time evolution of
the average number of links ≪ki >. We obtain that, asymptotically, ≪ki >∝(t/i)β, (i coincides with the input-time of the ith node) and β(αA) to both cases.
Phenomenological Tsallis fits to the CMS, ATLAS, and ALICE transverse
momentum spectra of hadrons for pp collisions at LHC were recently found to
extend over a large range of the transverse momentum. We investigate whether
the few degrees of freedom in the Tsallis parametrization may arise from the
relativistic parton-parton hard-scattering and related processes. The effects
of the multiple hard-scattering and parton showering processes on the power law
are discussed. We find empirically that whereas the transverse spectra of both
hadrons and jets exhibit power-law behavior of 1/pT^n at high pT, the power
indices n for hadrons are systematically greater than those for jets, for which
n~4-5.
The predictions from a non-extensive self-consistent theory recently proposed
are investigated. Transverse momentum () distribution for several hadrons
obtained in p+p collisions are analyzed to verify if there are evidence for a
limiting effective temperature and a limiting entropic factor. In addition, the
hadron-mass spectrum proposed in that theory is confronted with available data.
It turns out that all -distributions and the mass spectrum obtained in
the theory are in good agreement with experiment with constant effective
temperature and constant entropic factor. The results confirm that the
non-extensive statistics plays an important role in the description of the
termodynamics of hadronic systems, and also that the self-consistent principle
holds for energies as high as those achieved in the LHC. A discussion on the
best -distirbution formula for fitting experimental data is presented.
With the use of a quantity normally scaled in multifractals, a generalized form is postulated for entropy, namelyS
q
k [1 –
i=1
W
p
i
q
]/(q-1), whereq characterizes the generalization andp
i are the probabilities associated withW (microscopic) configurations (W). The main properties associated with this entropy are established, particularly those corresponding to the microcanonical and canonical ensembles. The Boltzmann-Gibbs statistics is recovered as theq1 limit.
Complex systems are very often organized under the form of networks where
nodes and edges are embedded in space. Transportation and mobility networks,
Internet, mobile phone networks, power grids, social and contact networks,
neural networks, are all examples where space is relevant and where topology
alone does not contain all the information. Characterizing and understanding
the structure and the evolution of spatial networks is thus crucial for many
different fields ranging from urbanism to epidemiology. An important
consequence of space on networks is that there is a cost associated to the
length of edges which in turn has dramatic effects on the topological structure
of these networks. We will expose thoroughly the current state of our
understanding of how the spatial constraints affect the structure and
properties of these networks. We will review the most recent empirical
observations and the most important models of spatial networks. We will also
discuss various processes which take place on these spatial networks, such as
phase transitions, random walks, synchronization, navigation, resilience, and
disease spread.
Systems as diverse as genetic networks or the world wide web are best
described as networks with complex topology. A common property of many large
networks is that the vertex connectivities follow a scale-free power-law
distribution. This feature is found to be a consequence of the two generic
mechanisms that networks expand continuously by the addition of new vertices,
and new vertices attach preferentially to already well connected sites. A model
based on these two ingredients reproduces the observed stationary scale-free
distributions, indicating that the development of large networks is governed by
robust self-organizing phenomena that go beyond the particulars of the
individual systems.
We demonstrated experimentally that the momentum distribution of cold atoms in dissipative optical lattices is a Tsallis distribution. The parameters of the distribution can be continuously varied by changing the parameters of the optical potential. In particular, by changing the depth of the optical lattice, it is possible to change the momentum distribution from Gaussian, at deep potentials, to a power-law tail distribution at shallow optical potentials.
Anomalous diffusion and non-Gaussian statistics are detected experimentally in a two-dimensional driven-dissipative system. A single-layer dusty plasma suspension with a Yukawa interaction and frictional dissipation is heated with laser radiation pressure to yield a structure with liquid ordering. Analyzing the time series for mean-square displacement, superdiffusion is detected at a low but statistically significant level over a wide range of temperatures. The probability distribution function fits a Tsallis distribution, yielding q, a measure of nonextensivity for non-Gaussian statistics.
We explore the possibility to interpret as a 'gas' the dynamical self-organized scale-free network recently introduced by Kim et al (2005). The role of 'momentum' of individual nodes is played by the degree of the node, the 'configuration space' (metric defining distance between nodes) being determined by the dynamically evolving adjacency matrix. In a constant-size network process, 'inelastic' interactions occur between pairs of nodes, which are realized by the merger of a pair of two nodes into one. The resulting node possesses the union of all links of the previously separate nodes. We consider chemostat conditions, i.e., for each merger there will be a newly created node which is then linked to the existing network randomly. We also introduce an interaction 'potential' (node-merging probability) which decays with distance d_ij as 1/d_ij^alpha; alpha >= 0). We numerically exhibit that this system exhibits nonextensive statistics in the degree distribution, and calculate how the entropic index q depends on alpha. The particular cases alpha=0 and alpha to infinity recover the two models introduced by Kim et al. Comment: 7 pages, 5 figures
We introduce a two-dimensional growth model where every new site is located, at a distance r from the barycenter of the pre-existing graph, according to the probability law , and is attached to (only) one pre-existing site with a probability ; is the number of links of the site of the pre-existing graph, and its distance to the new site). Then we numerically determine that the probability distribution for a site to have k links is asymptotically given, for all values of , by , where is the function naturally emerging within nonextensive statistical mechanics. The entropic index is numerically given (at least for not too large) by , and the characteristic number of links by . The particular case belongs to the same universality class to which the Barabasi-Albert model belongs. In addition to this, we have numerically studied the rate at which the average number of links increases with the scaled time t/i; asymptotically, , the exponent being close to for , and zero otherwise. The present results reinforce the conjecture that the microscopic dynamics of nonextensive systems typically build (for instance, in Gibbs -space for Hamiltonian systems) a scale-free network.
Three models of growing random networks with fitness dependent growth rates are analysed using the rate equations for the distribution of their connectivities. In the first model (A), a network is built by connecting incoming nodes to nodes of connectivity k and random additive fitness , with rate . For we find the connectivity distribution is power law with exponent . In the second model (B), the network is built by connecting nodes to nodes of connectivity k, random additive fitness and random multiplicative fitness with rate . This model also has a power law connectivity distribution, but with an exponent which depends on the multiplicative fitness at each node. In the third model (C), a directed graph is considered and is built by the addition of nodes and the creation of links. A node with fitness , i incoming links and j outgoing links gains a new incoming link with rate , and a new outgoing link with rate . The distributions of the number of incoming and outgoing links both scale as power laws, with inverse logarithmic corrections. Comment: 6 pages and 1 figure, submitted to publication
A great variety of complex phenomena in many scientific fields exhibit power-law behavior, reflecting a hierarchical or fractal structure. Many of these phenomena seem to be susceptible to description using approaches drawn from thermodynamics or statistical mechanics, particularly approaches involving the maximization of entropy and of Boltzmann-Gibbs statistical mechanics and standard laws in a natural way. The book addresses the interdisciplinary applications of these ideas, and also on various phenomena that could possibly be quantitatively describable in terms of these ideas.
The product of the A(b)(0) ((B) over bar (0)) differential production cross-section and the branching fraction of the decay A(b)(0)-> J/psi pK(-) ((B) over bar (0)-> J/psi p (K) over bar*(892)(0)) is measured as a function of the beauty hadron transverse momentum, p(T), and rapidity, y. The kinematic region of the measurements is p(T) J/psi pK(-) is measured to be B(A(b)(0)-> J/psi pK(-))=(3.17 +/- 0.04 +/- 0.07 +/- 0.34(-0.28)(+0.45))x10(-4) where the first uncertainty is statistical, the second is systematic, the third is due to the uncertainty on the branching fraction of the decay (B) over bar (0)-> J/psi p (K) over bar*(892)(0), and the fourth is due to the knowledge of f(Ab0)/f(d). The sum of the asymmetries in the production and decay between A(b)(0) and (A) over bar (0)(b) is also measured as a function of p(T) and y. The previously published branching fraction of A(b)(0)-> J/psi p pi(-), relative to that of A(b)(0)-> J/psi pK(-), is updated. The branching fractions of A(b)(0)-> P-c(+)(-> J/psi p)K- are determined.
The scientific study of networks, including computer networks, social networks, and biological networks, has received an enormous amount of interest in the last few years. The rise of the Internet and the wide availability of inexpensive computers have made it possible to gather and analyze network data on a large scale, and the development of a variety of new theoretical tools has allowed us to extract new knowledge from many different kinds of networks. The study of networks is broadly interdisciplinary and important developments have occurred in many fields, including mathematics, physics, computer and information sciences, biology, and the social sciences. This book brings together the most important breakthroughs in each of these fields and presents them in a coherent fashion, highlighting the strong interconnections between work in different areas. Subjects covered include the measurement and structure of networks in many branches of science, methods for analyzing network data, including methods developed in physics, statistics, and sociology, the fundamentals of graph theory, computer algorithms, and spectral methods, mathematical models of networks, including random graph models and generative models, and theories of dynamical processes taking place on networks.
The PHENIX experiment at the Relativistic Heavy Ion Collider has measured omega meson production via leptonic and hadronic decay channels in p+p, d+Au, Cu+Cu, and Au+Au collisions at sNN = 200 GeV. The invariant transverse momentum spectra measured in different decay modes give consistent results. Measurements in the hadronic decay channel in Cu+Cu and Au+Au collisions show that omega production has a suppression pattern at high transverse momentum, similar to that of pi0 and eta in central collisions, but no suppression is observed in peripheral collisions. The nuclear modification factors, RAA, are consistent in Cu+Cu and Au+Au collisions at similar numbers of participant nucleons.
Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems-be they neurons, power stations or lasers-will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks.
Inspired by empirical studies of networked systems such as the Internet,
social networks, and biological networks, researchers have in recent years
developed a variety of techniques and models to help us understand or predict
the behavior of these systems. Here we review developments in this field,
including such concepts as the small-world effect, degree distributions,
clustering, network correlations, random graph models, models of network growth
and preferential attachment, and dynamical processes taking place on networks.
- M Gell-Mann
- C Tsallis
Gell-Mann M and Tsallis C 2004 Nonextensive Entropy-Interdisciplinary Applications
(New York: Oxford University Press)
- C Tsallis
Tsallis C 2009 Introduction to Nonextensive Statistical Mechanics-Approaching a Complex World
(New York: Springer)
- S Brito
- L R Da Silva
- C Tsallis
Brito S, da Silva L R and Tsallis C 2016 Role of dimensionality in complex networks Sci. Rep. 6 27992
- Gell-Mann M