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We analyze the distribution of PageRank on a directed configuration model and show that as the size of the graph grows to infinity it can be closely approximated by the PageRank of the root node of an appropriately constructed tree. This…
The brain's structural and functional systems, protein-protein interaction, and gene networks are examples of biological systems that share some features of complex networks, such as highly connected nodes, modularity, and small-world…
Google's PageRank method was developed to evaluate the importance of web-pages via their link structure. The mathematics of PageRank, however, are entirely general and apply to any graph or network in any domain. Thus, PageRank is now…
Background: Recent models of genome-proteome evolution have shown that some of the key traits displayed by the global structure of cellular networks might be a natural result of a duplication-diversification (DD) process. One of the…
Distance matrices are matrices whose elements are the relative distances between points located on a certain manifold. In all cases considered here all their eigenvalues except one are non-positive. When the points are uncorrelated and…
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…
The Internet is composed of networks, called Autonomous Systems (or, ASes), interconnected to each other, thus forming a large graph. While both the AS-graph is known and there is a multitude of data available for the ASes (i.e., node…
The conception of multi-alphabetical genetics is represented. Matrix forms of the representation of the multi-level system of molecular-genetic alphabets have revealed algebraic properties of this system. These properties are connected with…
Geometric constraints impact the formation of a broad range of spatial networks, from amino acid chains folding to proteins structures to rearranging particle aggregates. How the network of interactions dynamically self-organizes in such…
We propose a general approach to the description of spectra of complex networks. For the spectra of networks with uncorrelated vertices (and a local tree-like structure), exact equations are derived. These equations are generalized to the…
Systems which consist of many localized constituents interacting with each other can be represented by complex networks. Consistently, network science has become highly popular in vast fields focusing on natural, artificial and social…
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…
Scale-free networks are one of the most famous examples of emergent behavior and are ubiquitous in social systems, especially online social media in which users can follow each other. By analyzing the interactions of multiple generative…
Semantic networks qualify the meaning of an edge relating any two vertices. Determining which vertices are most "central" in a semantic network is difficult because one relationship type may be deemed subjectively more important than…
Knowledge about the graph structure of the Web is important for understanding this complex socio-technical system and for devising proper policies supporting its future development. Knowledge about the differences between clean and…
An essential quantity to ensure evolvability of populations is the navigability of the genotype space. Navigability relies on the existence of sufficiently large genotype networks, that is ensembles of sequences with the same phenotype that…
We investigate the searchability of complex systems in terms of their interconnectedness. Associating searchability with the number and size of branch points along the paths between the nodes, we find that scale-free networks are relatively…
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…
Random walks on simple graphs in connection with electrical resistor networks lead to the definition of Markov chains with transition probability matrix in terms of electrical conductances. We extend this definition to an effective…
The eigenvalues of matrices representing the structure of large-scale complex networks present a wide range of applications, from the analysis of dynamical processes taking place in the network to spectral techniques aiming to rank the…