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Scale-free networks are abundant in nature and society, describing such diverse systems as the world wide web, the web of human sexual contacts, or the chemical network of a cell. All models used to generate a scale-free topology are…
Although two-dimensional periodic structures have functioned as the primary platform for exploring topological phenomena, recent advances have substantially expanded this research boundary to include more intricate, aperiodic structures:…
We introduce a new class of deterministic networks by associating networks with Diophantine equations, thus relating network topology to algebraic properties. The network is formed by representing integers as vertices and by drawing cliques…
Complex systems, ranging from soft materials to wireless communication, are often organised as random geometric networks in which nodes and edges evenly fill up the volume of some space. Studying such networks is difficult because they…
Networks of living neurons exhibit an avalanche mode of activity, experimentally found in organotypic cultures. Moreover, experimental studies of morphology indicate that neurons develop a network of small-world-like connections, with the…
In this paper, by both simulations and theoretical predictions we study two and three node (or degree) correlations in random Apollonian network (RAN), which have small-world and scale-free topologies. Using the rate equation approach under…
In this paper we study the toughness of Random Apollonian Networks (RANs), a random graph model which generates planar graphs with power-law properties. We consider their important characteristics: every RAN is a uniquely representable…
Let $r$ and $d$ be positive integers with $r<d$. Consider a random $d$-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it $d$ newly created offspring. Let ${\mathcal…
It is known that many networks modeling real-life complex systems are small-word (large local clustering and small diameter) and scale-free (power law of the degree distribution), and very often they are also hierarchical. Although most of…
We investigate the differences between scale-free recursive nets constructed by a synchronous, deterministic updating rule (e.g., Apollonian nets), versus an asynchronous, random sequential updating rule (e.g., random Apollonian nets). We…
Several interesting approaches have been reported in the literature on complex networks, random walks, and hierarchy of graphs. While many of these works perform random walks on stable, fixed networks, in the present work we address the…
We consider the model of complex hyperbranched polymer structures formed on the basis of scale-free graphs, where functionalities (degrees) $k$ of nodes obey a power law decaying probability $p(k)\sim{k^{-\alpha}}$. Such polymer topologies…
In this paper, we study the distribution of distances in random Apollonian network structures (RANS), a family of graphs which has a one-to-one correspondence with planar ternary trees. Using multivariate generating functions that express…
This study introduces an algorithm that generates undirected graphs with three main characteristics of real-world networks: scale-freeness, short distances between nodes (small-world phenomenon), and large clustering coefficients. The main…
Evolutionary game theory is employed to study topological conditions of scale-free networks for the evolution of cooperation. We show that Apollonian Networks (ANs) are perfect scale-free networks, on which cooperation can spread to all…
We consider Gallai's graph Modular Decomposition theory for network analytics. On the one hand, by arguing that this is a choice tool for understanding structural and functional similarities among nodes in a network. On the other, by…
It has been shown that many networks associated with complex systems are small-world (they have both a large local clustering coefficient and a small diameter) and they are also scale-free (the degrees are distributed according to a power…
The Apollonian networks display the remarkable power-law and small-world properties as observed in most realistic networked systems. Their dual graphs are extended Tower of Hanoi graphs, which are obtained from the Tower of Hanoi graphs by…
In this work we analyze basic properties of Random Apollonian Networks \cite{zhang,zhou}, a popular stochastic model which generates planar graphs with power law properties. Specifically, let $k$ be a constant and $\Delta_1 \geq \Delta_2…
Extensive studies have been done to understand the principles behind architectures of real networks. Recently, evidences for hierarchical organization in many real networks have also been reported. Here, we present a new hierarchical model…