Related papers: Small Cycles in Small Worlds
I start by reviewing some basic properties of random graphs. I then consider the role of random walks in complex networks and show how they may be used to explain why so many long tailed distributions are found in real data sets. The key…
We investigate equilibrium properties of small world networks, in which both connectivity and spin variables are dynamic, using replicated transfer matrices within the replica symmetric approximation. Population dynamics techniques allow us…
Consider the following routing problem in the context of a large scale network $G$, with particular interest paid to power law networks, although our results do not assume a particular degree distribution. A small number of nodes want to…
Various molecular interaction networks have been claimed to follow power-law decay for their global connectivity distribution. It has been proposed that there may be underlying generative models that explain this heavy-tailed behavior by…
We investigate several geometric models of network which simultaneously have some nice global properties, that the small diameter property, the small-community phenomenon, which is defined to capture the common experience that (almost)…
Although the analysis of loops is not so much because of the complications, it has already been found that heuristically enhancing loops decreases the variance of degree distributions for improving the robustness of connectivity. While many…
The metabolic networks are very well characterized for a large set of organisms, a unique case in within the large-scale biological networks. For this reason they provide a a very interesting framework for the construction of analytically…
Subgraphs and cycles are often used to characterize the local properties of complex networks. Here we show that the subgraph structure of real networks is highly time dependent: as the network grows, the density of some subgraphs remains…
Metabolic networks have two properties that are generally regarded as unrelated: One, they have metabolic reactions whose single knockout is lethal for the organism, and two, they have correlated sets of reactions forming functional…
In this article, we explicitly derive the limiting degree distribution of the shortest path tree from a single source on various random network models with edge weights. We determine the asymptotics of the degree distribution for large…
Networks observed in the real world often have many short loops. This violates the tree-like assumption that underpins the majority of random graph models and most of the methods used for their analysis. In this paper we sketch possible…
Small-world (SW) networks have been identified in many different fields. Topological coefficients like the clustering coefficient and the characteristic path length have been used in the past for a qualitative characterization of these…
Watts and Strogatz [Nature 393, 440 (1998)] have recently introduced a model for disordered networks and reported that, even for very small values of the disorder $p$ in the links, the network behaves as a small-world. Here, we test the…
The asymptotic behaviour of dynamical processes in networks can be expressed as a function of spectral properties of the corresponding adjacency and Laplacian matrices. Although many theoretical results are known for the spectra of…
Empirical studies on the spatial structures in several real transport networks reveal that the distance distribution in these networks obeys power law. To discuss the influence of the power-law exponent on the network's structure and…
Topological properties of "scale-free" networks are investigated by determining their spectral dimensions $d_S$, which reflect a diffusion process in the corresponding graphs. Data bases for citation networks and metabolic networks together…
Temporal networks have been increasingly used to model a diversity of systems that evolve in time; for example human contact structures over which dynamic processes such as epidemics take place. A fundamental aspect of real-life networks is…
Different network models have been suggested for the topology underlying complex interactions in natural systems. These models are aimed at replicating specific statistical features encountered in real-world networks. However, it is rarely…
Small-worlds represent efficient communication networks that obey two distinguishing characteristics: a high clustering coefficient together with a small characteristic path length. This paper focuses on an interesting paradox, that…
In this thesis, we have studied the large scale structure and system level dynamics of certain biological networks using tools from graph theory, computational biology and dynamical systems. We study the structure and dynamics of large…