Abstract:
In this work we consider the topological analysis of symbolic formal systems in the framework of network theory. In particular we analyse the network extracted by Principia Mathematica of B. Russell and A.N. Whitehead, where the vertices are the statements and two statements are connected with a directed link if one statement is used to demonstrate the other one. We compare the obtained network with other directed acyclic graphs, such as a scientific citation network and a stochastic model. We also introduce a novel topological ordering for directed acyclic graphs and we discuss its properties in respect to the classical one. The main result is the observation that formal systems of knowledge topologically behave similarly to self-organised systems.

Abstract:
We study the directed and weighted network in which the wards of London are vertices and two vertices are connected whenever there is at least one person commuting to work from a ward to another. Remarkably the in-strength and in-degree distribution tail is a power law with exponent around -2, while the out-strength and out-degree distribution tail is exponential. We propose a simple square lattice model to explain the observed empirical behaviour.

Abstract:
We introduce and analyze a model of a multi-directed Eulerian network, that is a directed and weighted network where a path exists that passes through all the edges of the network once and only once. Networks of this type can be used to describe information networks such as human language or DNA chains. We are able to calculate the strength and degree distribution in this network and find that they both exhibit a power law with an exponent between 2 and 3. We then analyze the behavior of the accelerated version of the model and find that the strength distribution has a double slope power law behavior. Finally we introduce a non-Eulerian version of the model and find that the statistical topological properties remain unchanged. Our analytical results are compared with numerical simulations.

Abstract:
In this paper we deal with the structural properties of weighted networks. Starting from an empirical analysis of a linguistic network, we analyse the differences between the statistical properties of a real and a shuffled network and we show that the scale free degree distribution and the scale free weight distribution are induced by the scale free strength distribution, that is Zipf's law. We test the result on a scientific collaboration network, that is a social network, and we define a measure, the vertex selectivity, that can easily distinguish a real network from a shuffled network. We prove, via an ad-hoc stochastic growing network with second order correlations, that this measure can effectively capture the correlations within the topology of the network.

Abstract:
We investigate the nature of written human language within the framework of complex network theory. In particular, we analyse the topology of Orwell's \textit{1984} focusing on the local properties of the network, such as the properties of the nearest neighbors and the clustering coefficient. We find a composite power law behavior for both the average nearest neighbor's degree and average clustering coefficient as a function of the vertex degree. This implies the existence of different functional classes of vertices. Furthermore we find that the second order vertex correlations are an essential component of the network architecture. To model our empirical results we extend a previously introduced model for language due to Dorogovtsev and Mendes. We propose an accelerated growing network model that contains three growth mechanisms: linear preferential attachment, local preferential attachment and the random growth of a pre-determined small finite subset of initial vertices. We find that with these elementary stochastic rules we are able to produce a network showing syntactic-like structures.

Abstract:
In this paper we analyse the street network of London both in its primary and dual representation. To understand its properties, we consider three idealised models based on a grid, a static random planar graph and a growing random planar graph. Comparing the models and the street network, we find that the streets of London form a self-organising system whose growth is characterised by a strict interaction between the metrical and informational space. In particular, a principle of least effort appears to create a balance between the physical and the mental effort required to navigate the city.

Abstract:
We test the recently introduced radiation model against the gravity model for the system composed of England and Wales, both for commuting patterns and for public transportation flows. The analysis is performed both at macroscopic scales, i.e. at the national scale, and at microscopic scales, i.e. at the city level. It is shown that the thermodynamic limit assumption for the original radiation model significantly underestimates the commuting flows for large cities. We then generalize the radiation model, introducing the correct normalisation factor for finite systems. We show that even if the gravity model has a better overall performance the parameter-free radiation model gives competitive results, especially for large scales.

Abstract:
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 exponential cut-off. We also show that the properties of this network differ considerably from the features of scale-free networks generated through preferential attachment. Remarkably we also recover a power law for the clustering coefficient.

Abstract:
We introduce a general method to infer the directional information flow between populations whose elements are described by n-dimensional vectors of symbolic attributes. The method is based on the Jensen-Shannon divergence and on the Shannon entropy and has a wide range of application. We show here the results of two applications: first extracting the network of genetic flow between the meadows of the seagrass Poseidonia Oceanica, where the meadow elements are specified by sets of microsatellite markers, then we extract the semantic flow network from a set of Wikipedia pages, showing the semantic channels between different areas of knowledge.

Abstract:
Street networks are important infrastructural transportation systems that cover a great part of the planet. It is now widely accepted that transportation properties of street networks are better understood in the interplay between the street network itself and the so called \textit{information} or \textit{dual network}, which embeds the topology of the street network navigation system. In this work, we present a novel robustness analysis, based on the interaction between the primal and the dual transportation layer for two large metropolis, London and Chicago, thus considering the structural differences to intentional attacks for \textit{self-organized} and planned cities. We elaborate the results through an accurate closeness centrality analysis in the Euclidean space and in the relationship between primal and dual space. Interestingly enough, we find that even if the considered planar graphs display very distinct properties, the information space induce them to converge toward systems which are similar in terms of transportation properties.