High Dimensional Apollonian Networks

Experimentally observed complex networks are often scale-free, small-world and have unexpectedly large number of small cycles. Apollonian network is one notable example of a model network respecting simultaneously having all three of these properties. This network is constructed by a deterministic procedure of consequentially splitting a triangle into smaller and smaller triangles. Here we present a similar construction based on consequential splitting of tetragons and other ...

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 the assumption of continuous degree, we first give the analytical solution for two node correlations, expressed by average nearest-neighbor degree (ANND). Then, we revisit the degree distribution of RAN using rate equation method and get t...

In a recursive way and by including a parameter, we introduce a family of deterministic scale-free networks. The resulting networks exhibit small-world effects. We calculate the exact results for the degree exponent, the clustering coefficient and the diameter. The major points of our results indicate: the degree exponent can be adjusted; the clustering coefficient of each individual vertex is inversely proportional to its degree and the average clustering coefficient of all ...

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 all information on distances, and singularity analysis for evaluating the coefficients of these functions, we describe the distribution of distances to an outermost vertex, and show that the average value of the distance between any pair of ...

We introduce a family of complex networks that interpolates between the Apollonian network and its binary version, recently introduced in [Phys. Rev. E \textbf{107}, 024305 (2023)], via random removal of nodes. The dilution process allows the clustering coefficient to vary from $C=0.828$ to $C=0$ while maintaining the behavior of average path length and other relevant quantities as in the deterministic Apollonian network. Robustness against the random deletion of nodes is als...

The network of contacts in space-filling disk packings, such as the Apollonian packing, are examined. These networks provide an interesting example of spatial scale-free networks, where the topology reflects the broad distribution of disk areas. A wide variety of topological and spatial properties of these systems are characterized. Their potential as models for networks of connected minima on energy landscapes is discussed.

Many real life networks present an average path length logarithmic with the number of nodes and a degree distribution which follows a power law. Often these networks have also a modular and self-similar structure and, in some cases - usually associated with topological restrictions- their clustering is low and they are almost planar. In this paper we introduce a family of graphs which share all these properties and are defined by two parameters. As their construction is deter...

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 idea is to perform random walks across the network and, at each iteration, add special edges with a decreasing probability to link more distant nodes, following a specific probability distribution. A key advantage of our algorithm is its si...

Probabilistic networks display a wide range of high average clustering coefficients independent of the number of nodes in the network. In particular, the local clustering coefficient decreases with the degree of the subtending node in a complicated manner not explained by any current models. While a number of hypotheses have been proposed to explain some of these observed properties, there are no solvable models that explain them all. We propose a novel growth model for both ...

Networks representing many complex systems in nature and society share some common structural properties like heterogeneous degree distributions and strong clustering. Recent research on network geometry has shown that those real networks can be adequately modeled as random geometric graphs in hyperbolic spaces. In this paper, we present a computer program to generate such graphs. Besides real-world-like networks, the program can generate random graphs from other well-known g...