On the formation of structure in growing networks

2013 American Control Conference, 2013

Based on the formation of triad junctions, the proposed mechanism generates growing networks that exhibit extended power law behavior and strong neighborhood clustering. The asymptotic behavior of both properties is of interest in the study of 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; and (iii) the degree of clustering depends on both the number of links that newly added nodes establish and the probability of forming triads.

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

Most of the complex social, technological, and biological networks have a significant community structure. Therefore the community structure of complex networks has to be considered as a universal property, together with the much explored small-world and scale-free properties of these networks. Despite the large interest in characterizing the community structures of real networks, not enough attention has been devoted to the detection of universal mechanisms able to spontaneously generate networks with communities. Triadic closure is a natural mechanism to make new connections, especially in social networks. Here we show that models of network growth based on simple triadic closure naturally lead to the emergence of community structure, together with fat-tailed distributions of node degree and high clustering coefficients. Communities emerge from the initial stochastic heterogeneity in the concentration of links, followed by a cycle of growth and fragmentation. Communities are the m...

Physical Review E, 2001

In this work we propose an idealized model for competitive cluster growth in complex networks. Each cluster can be thought of as a fraction of a community that shares some common opinion. Our results show that the cluster size distribution depends on the particular choice for the topology of the network of contacts among the agents. As an application, we show that the cluster size distributions obtained when the growth process is performed on hierarchical networks, e.g., the Apollonian network, have a scaling form similar to what has been observed for the distribution of a number of votes in an electoral process. We suggest that this similarity may be due to the fact that social networks involved in the electoral process may also possess an underlining hierarchical structure.

International Journal of Modern Physics C, 2007

In this paper we provide numerical evidence of the richer behavior of the connectivity degrees in heterogeneous preferential attachment networks in comparison to their homogeneous counterparts. We analyze the degree distribution in the threshold model, a preferential attachment model where the affinity between node states biases the attachment probabilities of links. We show that the degree densities exhibit a power-law multiscaling which points to a signature of heterogeneity in preferential attachment networks. This translates into a power-law scaling in the degree distribution, whose exponent depends on the specific form of heterogeneity in the attachment mechanism.

Nature Physics, 2012

Many complex systems reveal a small-world topology 1,2 which allows simultaneously for local and global efficiency in the interaction between system constituents 3-5. Here, we show that strong interactions in complex systems, quantified by a high link weight, support high network traffic across clustered neighborhoods 1,6. For brain, gene, social, and language networks, we found a local integrative weight organization in which strong links preferentially occur between nodes with overlapping neighbourhoods with the consequence that globally the clustering is robust to removal of the weakest links. We identify local learning rules that establish integrative networks and improve network traffic in response to past traffic failures. Our findings identify a general organization for complex systems that strikes a balance between efficient local and global communication in their strong interactions, while allowing for robust, exploratory development of weak interactions. Networks as diverse as those linking scientific collaborations and those connecting the U.S. electrical power grid are characterized by small-world topology 1,2,7. In the brain, this topology captures the organization of neural connections at different spatial scales and in various species 8-12 , including the structure of spontaneous neural activity characterized by neuronal avalanches 13-15. In these sparse networks, most nodes are only separated by a few links and nodes are highly clustered, that is, neighboring nodes are very likely to be connected themselves, quantified by a high clustering coefficient, C 7,16,17. This enables complex systems to simultaneously achieve both global and local efficiency in the interactions of their components 3-5 .

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...

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...

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. * [email protected]

Nature Physics, 2011

Complex networks appear in almost every aspect of science and technology. Although most results in the field have been obtained by analysing isolated networks, many real-world networks do in fact interact with and depend on other networks. The set of extensive results for the limiting case of non-interacting networks holds only to the extent that ignoring the presence of other networks can be justified. Recently, an analytical framework for studying the percolation properties of interacting networks has been developed. Here we review this framework and the results obtained so far for connectivity properties of 'networks of networks' formed by interdependent random networks. T he interdisciplinary field of network science has attracted a great deal of attention in recent years 1-30. This development is based on the enormous number of data that are now routinely being collected, modelled and analysed, concerning social 31-39 , economic 14,36,40,41 , technological 40,42-48 and biological 9,13,49,50 systems. The investigation and growing understanding of this extraordinary volume of data will enable us to make the infrastructures we use in everyday life more efficient and more robust. The original model of networks, random graph theory, was developed in the 1960s by ErdAEs and Rényi, and is based on the assumption that every pair of nodes is randomly connected with the same probability, leading to a Poisson degree distribution. In parallel, in physics, lattice networks, where each node has exactly the same number of links, have been studied to model physical systems. Although graph theory is a well-established tool in the mathematics and computer science literature, it cannot describe well modern, real-life networks. Indeed, the pioneering 1999 observation by Barabasi 2 , that many real networks do not follow the ErdAEs-Rényi model but that organizational principles naturally arise in most systems, led to an overwhelming accumulation of supporting data, new models and computational and analytical results, and to the emergence of a new science, that of complex networks. Complex networks are usually non-homogeneous structures that in many cases obey a power-law form in their degree (that is, number of links per node) distribution. These systems are called scale-free networks. Real networks that can be approximated as scale-free networks include the Internet 3 , the World Wide Web 4 , social networks 31-39 representing the relations between individuals, infrastructure networks such as those of airlines 51 , networks in biology 9,13,49,50 , in particular networks of proteinprotein interactions 10 , gene regulation and biochemical pathways, and networks in physics, such as polymer networks or the potentialenergy-landscape network. The discovery of scale-free networks led to a re-evaluation of the basic properties of networks, such as their robustness, which exhibit a drastically different character than those of ErdAEs-Rényi networks. For example, whereas homogeneous ErdAEs-Rényi networks are extremely vulnerable to random failures, heterogeneous scale-free networks are remarkably robust 4,5. A great part of our current knowledge on networks is based on ideas borrowed from statistical physics, such as percolation theory, fractals and scaling analysis. An important property of these infrastructures is their stability, and it is thus important that we understand and quantify their robustness in terms of node and