Entropy rate of non-equilibrium growing networks (original) (raw)

Scale-free networks as entropy competition

Physical Review E, 2008

Complex networks describe several and different real-world systems consisting of a number of interacting elements. A very important characteristic of such networks is the degree distribution that strongly controls their behavior. Based on statistical mechanics, three classes of uncorrelated complex networks are identified here, depending on the role played by the connectivities amongst elements. In particular, by identifying the connectivities of a node with the number of its nearest neighbors, we show that the power law is the most probable degree distribution that both nodes and neighbors, in a reciprocal competition, assume when the respective entropy functions reach their maxima, under mutual constraint. As a result, we obtain scaling exponent values as a function of the structural characteristics of the whole network. Moreover, our approach sheds light on the exponential and Poissonian degree distributions, derived, respectively, when connectivities are thought of as degenerated connections or as half-edges. Thus, all three classes of degree distributions are derived, starting from a common principle and leading to a general and unified framework for investigating the network structure.

Characterizing the head of the degree distributions of growing networks

arXiv: Physics and Society, 2020

The analysis in this paper helps to explain the formation of growing networks with degree distributions that follow extended exponential or power-law tails. We present a generic model in which edge dynamics are driven by a continuous attachment of new nodes and a mixed attachment mechanism that triggers random or preferential attachment. Furthermore, reciprocal edges to newly added nodes are established according to a response mechanism. The proposed framework extends previous mixed attachment models by allowing the number of new edges to vary according to various discrete probability distributions, including Poisson, Binomial, Zeta, and Log-Series. We derive analytical expressions for the limit in-degree distribution that results from the mixed attachment and response mechanisms. Moreover, we describe the evolution of the dynamics of the cumulative in-degree distribution. Simulation results illustrate how the number of new edges and the process of reciprocity significantly impact t...

Characterizing the head of the degrees distributions of growing networks

2020

Dynamics of condensation in growing complex networks

Physical Review E, 2008

A condensation transition was predicted for growing technological networks evolving by preferential attachment and competing quality of their nodes, as described by the fitness model. When this condensation occurs a node acquires a finite fraction of all the links of the network. Earlier studies based on steady state degree distribution and on the mapping to Bose-Einstein condensation, were able to identify the critical point. Here we characterize the dynamics of condensation and we present evidence that below the condensation temperature there is a slow down of the dynamics and that a single node (not necessarily the best node in the network) emerges as the winner for very long times. The characteristic time t * at which this phenomenon occurs diverges both at the critical point and at T → 0 when new links are attached deterministically to the highest quality node of the network. PACS numbers: 89.75.Hc, 89.75.Da, 89.75.Fb

On the formation of structure in growing networks

Journal of Statistical Mechanics: Theory and Experiment, 2013

Based on the formation of triad junctions, the proposed mechanism generates networks that exhibit extended rather than single power law behavior. Triad formation guarantees strong neighborhood clustering and community-level characteristics as the network size grows to infinity. The asymptotic behavior is of interest in the study of directed networks in which (i) the formation of links cannot be described according to the principle of preferential attachment; (ii) the in-degree distribution fits a power law for nodes with a high degree and an exponential form otherwise; (iii) clustering properties emerge at multiple scales and depend on both the number of links that newly added nodes establish and the probability of forming triads; and (iv) groups of nodes form modules that feature less links to the rest of the nodes.

Correlations in Networks Associated to Preferential Growth

Physica Scripta, 2005

Combinations of random and preferential growth for both on-growing and stationary networks are studied and a hierarchical topology is observed. Thus for real world scale-free networks which do not exhibit hierarchical features preferential growth is probably not the main ingredient in the growth process. An example of such real world networks includes the protein-protein interaction network in yeast, which exhibits pronounced anti-hierarchical features.

A thermodynamic view of networks

Comptes Rendus Biologies, 2006

Networks can be described by the frequency distribution of the number of links associated with each node (the degree of the node). Of particular interest are the power law distributions, which give rise to the so-called scale-free networks, and the distributions of the form of the simplified canonical law (SCL) introduced by Mandelbrot, which give what we shall call the Mandelbrot networks. Many dynamical methods have been obtained for the construction of scale-free networks, but no dynamical construction of Mandelbrot networks has been demonstrated. Here we develop a systematic technique to obtain networks with any given distribution of the degrees of the nodes. This is done using a thermodynamic approach in which we maximise the entropy associated with degree distribution of the nodes of the network subject to certain constraints. These constraints can be chosen systematically to produce the desired network architecture. For large networks we therefore replace a dynamical approach to the stationary state by a thermodynamical viewpoint. We use the method to generate scale-free and Mandelbrot networks with arbitrarily chosen parameters. We emphasise that this approach opens the possibility of insights into a thermodynamics of networks by suggesting thermodynamic relations between macroscopic variables for networks. To cite this article: D.

Dynamical complexity as a proxy for the network degree distribution

Physical Review E, 2019

We explore the relation between the topological relevance of a node in a complex network and the individual dynamics it exhibits. When the system is weakly coupled, the effect of the coupling strength against the dynamical complexity of the nodes is found to be a function of their topological role, with nodes of higher degree displaying lower levels of complexity. We provide several examples of theoretical models of chaotic oscillators, pulse-coupled neurons and experimental networks of nonlinear electronic circuits evidencing such a hierarchical behavior. Importantly, our results imply that it is possible to infer the degree distribution of a network only from individual dynamical measurements.

Entropy distribution and condensation in random networks with a given degree distribution

Physical review. E, Statistical, nonlinear, and soft matter physics, 2014

The entropy of network ensembles characterizes the amount of information encoded in the network structure and can be used to quantify network complexity and the relevance of given structural properties observed in real network datasets with respect to a random hypothesis. In many real networks the degrees of individual nodes are not fixed but change in time, while their statistical properties, such as the degree distribution, are preserved. Here we characterize the distribution of entropy of random networks with given degree sequences, where each degree sequence is drawn randomly from a given degree distribution. We show that the leading term of the entropy of scale-free network ensembles depends only on the network size and average degree and that entropy is self-averaging, meaning that its relative variance vanishes in the thermodynamic limit. We also characterize large fluctuations of entropy that are fully determined by the average degree in the network. Finally, above a certain...

Entropy rate of non-equilibrium growing networks (original) (raw)

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