High Dimensional Apollonian Networks

In this manuscript, we provide a concise review of the concept of metric dimension for both deterministic as well as random graphs. Algorithms to approximate this quantity, as well as potential applications, are also reviewed. This work has been partially funded by the NSF IIS grant 1836914.

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 law). Moreover, these networks are very often hierarchical, as they describe the modularity of the systems that are modeled. Most of the studies for complex networks are based on stochastic methods. However, a deterministic method, with an e...

We study the growth of random networks under a constraint that the diameter, defined as the average shortest path length between all nodes, remains approximately constant. We show that if the graph maintains the form of its degree distribution then that distribution must be approximately scale-free with an exponent between 2 and 3. The diameter constraint can be interpreted as an environmental selection pressure that may help explain the scale-free nature of graphs for which ...

We introduce a simple one-parameter network growth algorithm which is able to reproduce a wide variety of realistic network structures but without having to invoke any global information about node degrees such as preferential-attachment probabilities. Scale-free networks arise at the transition point between quasi-random and quasi-ordered networks. We provide a detailed formalism which accurately describes the entire network range, including this critical point. Our formalis...

Various real-life networks of current interest are simultaneously scale-free and modular. Here we study analytically the average distance in a class of deterministically growing scale-free modular networks. By virtue of the recursive relations derived from the self-similar structure of the networks, we compute rigorously this important quantity, obtaining an explicit closed-form solution, which recovers the previous result and is corroborated by extensive numerical calculatio...

We present a statistical mechanics approach for the description of complex networks. We first define an energy and an entropy associated to a degree distribution which have a geometrical interpretation. Next we evaluate the distribution which extremize the free energy of the network. We find two important limiting cases: a scale-free degree distribution and a finite-scale degree distribution. The size of the space of allowed simple networks given these distribution is evaluat...

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 stochastic, that is they create networks in which the nodes appear to be randomly connected to each other. Here we propose a simple model that generates scale-free networks in a deterministic fashion. We solve exactly the model, showing that the ...

We study the distribution function for minimal paths in small-world networks. Using properties of this distribution function, we derive analytic results which greatly simplify the numerical calculation of the average minimal distance, $\bar{\ell}$, and its variance, $\sigma^2$. We also discuss the scaling properties of the distribution function. Finally, we study the limit of large system sizes and obtain some analytic results.

The metric dimension of a graph $G$ is the minimum number of vertices in a subset $S$ of the vertex set of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $S$. In this paper we investigate the metric dimension of the random graph $G(n,p)$ for a wide range of probabilities $p=p(n)$.

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 between M vertices every time that M distinct integers satisfy the equation. We analyse the network generated by the Pythagorean equation $x^{2}+y^{2}= z^{2}$ showing that its degree distribution is well approximated by a power law with ex...