arXiv subjects

G. J. Rodgers

Publications and source records attributed to G. J. Rodgers.

At least 19 recordsLinked to original sources

arXiv2013

Kinetics of node splitting in evolving complex networks

We introduce a collection of complex networks generated by a combination of preferential attachment and a previously unexamined process of "splitting" nodes of degree $k$ into $k$ nodes of degree 1. Four networks are considered, each evolves at each time step by either preferential attachment, with probability $p$, or splitting with probability $1-p$. Two methods of attachment are considered; first, attachment of an edge between a newly created node and existing node in the network, and secondly by attachment of an edge between two existing nodes. Splitting is also considered in two separate ways; first by selecting each node with equal probability and secondly, selecting the node with probability proportional to its degree. Exact solutions for the degree distributions are found and scale-free structure is exhibited in those networks where the candidates for splitting are chosen with uniform probability, those that are chosen preferentially are distributed with a power law with exponential cut-off.

physics.soc-ph↗

arXiv2010

Modelling conflicts with cluster dynamics on networks

We introduce cluster dynamical models of conflicts in which only the largest cluster can be involved in an action. This mimics the situations in which an attack is planned by a central body, and the largest attack force is used. We study the model in its annealed random graph version, on a fixed network, and on a network evolving through the actions. The sizes of actions are distributed with a power-law tail, however, the exponent is non-universal and depends on the frequency of actions and sparseness of the available connections between units. Allowing the network reconstruction over time in a self-organized manner, e.g., by adding the links based on previous liaisons between units, we find that the power-law exponent depends on the evolution time of the network. Its lower limit is given by the universal value 5/2, derived analytically for the case of random fragmentation processes. In the temporal patterns behind the size of actions we find long-range correlations in the time series of number of clusters and non-trivial distribution of time that a unit waits between two actions. In the case of an evolving network the distribution develops a power-law tail, indicating that through the repeated actions, the system develops internal structure which is not just more effective in terms of the size of events, but also has a full hierarchy of units.

physics.soc-ph↗

arXiv2008

Differences between normal and shuffled texts: structural properties of weighted networks

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.

physics.data-an↗

arXiv2008

Spectral Density of Complex Networks with a Finite Mean Degree

In order to clarify the statistical features of complex networks, the spectral density of adjacency matrices has often been investigated. Adopting a static model introduced by Goh, Kahng and Kim, we analyse the spectral density of complex scale free networks. For that purpose, we utilize the replica method and effective medium approximation (EMA) in statistical mechanics. As a result, we identify a new integral equation which determines the asymptotic spectral density of scale free networks with a finite mean degree $p$. In the limit $p \to \infty$, known asymptotic formulae are rederived. Moreover, the $1/p$ corrections to known results are analytically calculated by a perturbative method.

cond-mat.stat-mech↗

arXiv2008

Local Information Based Algorithms for Packet Transport in Complex Networks

We introduce four algorithms for packet transport in complex networks. These algorithms use deterministic rules which depend, in different ways, on the degree of the node, the number of packets posted down each edge, the mean delivery time of packets sent down each edge to each destination and the time since an edge last transmitted a packet. On scale-free networks all our algorithms are considerably more efficient and can handle a larger load than the random walk algorithm. We consider in detail various attributes of our algorithms, for instance we show that an algorithm that bases its decisions on the mean delivery time jams unless it incorporates information about the degree of the destination node.

cs.NI↗

arXiv2007

The Network of Commuters in London

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.

physics.soc-ph↗

arXiv2007

Diophantine Networks

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.

physics.soc-ph↗

arXiv2007

Multi-directed Eulerian growing networks

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.

physics.soc-ph↗

arXiv2006

Transport on Complex Networks: Flow, Jamming and Optimization

Many transport processes on networks depend crucially on the underlying network geometry, although the exact relationship between the structure of the network and the properties of transport processes remain elusive. In this paper we address this question by using numerical models in which both structure and dynamics are controlled systematically. We consider the traffic of information packets that include driving, searching and queuing. We present the results of extensive simulations on two classes of networks; a correlated cyclic scale-free network and an uncorrelated homogeneous weakly clustered network. By measuring different dynamical variables in the free flow regime we show how the global statistical properties of the transport are related to the temporal fluctuations at individual nodes (the traffic noise) and the links (the traffic flow). We then demonstrate that these two network classes appear as representative topologies for optimal traffic flow in the regimes of low density and high density traffic, respectively. We also determine statistical indicators of the pre-jamming regime on different network geometries and discuss the role of queuing and dynamical betweenness for the traffic congestion. The transition to the jammed traffic regime at a critical posting rate on different network topologies is studied as a phase transition with an appropriate order parameter. We also address several open theoretical problems related to the network dynamics.

physics.soc-ph↗

arXiv2006

Network properties of written human language

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.

physics.soc-ph↗

arXiv2005

Spin Glass Phase Transition on Scale-Free Networks

We study the Ising spin glass model on scale-free networks generated by the static model using the replica method. Based on the replica-symmetric solution, we derive the phase diagram consisting of the paramagnetic (P), ferromagnetic (F), and spin glass (SG) phases as well as the Almeida-Thouless line as functions of the degree exponent $λ$, the mean degree $K$, and the fraction of ferromagnetic interactions $r$. To reflect the inhomogeneity of vertices, we modify the magnetization $m$ and the spin glass order parameter $q$ with vertex-weights. The transition temperature $T_c$ ($T_g$) between the P-F (P-SG) phases and the critical behaviors of the order parameters are found analytically. When $2 < λ< 3$, $T_c$ and $T_g$ are infinite, and the system is in the F phase or the mixed phase for $r > 1/2$, while it is in the SG phase at $r=1/2$. $m$ and $q$ decay as power-laws with increasing temperature with different $λ$-dependent exponents. When $λ> 3$, the $T_c$ and $T_g$ are finite and related to the percolation threshold. The critical exponents associated with $m$ and $q$ depend on $λ$ for $3 < λ< 5$ ($3 < λ< 4$) at the P-F (P-SG) boundary.

cond-mat.stat-mech↗

arXiv2004

Traffic on complex networks: Towards understanding global statistical properties from microscopic density fluctuations

We study the microscopic time fluctuations of traffic-load and the global statistical properties of a dense traffic of particles on scale-free cyclic graphs. For a wide range of driving rates $R$ the traffic is stationary and the load timeseries exhibit anti-persistence due to the regulatory role of the super-structure associated with two hub nodes in the network. We discuss how the super-structure effects the functioning of the network at high traffic density and at the jamming threshold. The degree of correlations systematically decreases with increasing traffic density and eventually disappears when approaching a jamming density $R_c$. Already before jamming we observe qualitative changes in the global network-load distributions and the particle queuing-times. These changes are related to the occurrence of temporary crises in which the network-load increases dramatically, and then slowly falls back to a value characterizing free-flow.

cond-mat.stat-mech↗

arXiv2003

Self-organized Model for Modular Complex Networks : Division and Independence

We introduce a minimal network model which generates a modular structure in a self-organized way. To this end, we modify the Barabasi-Albert model into the one evolving under the principle of division and independence as well as growth and preferential attachment (PA). A newly added vertex chooses one of the modules composed of existing vertices, and attaches edges to vertices belonging to that module following the PA rule. When the module size reaches a proper size, the module is divided into two, and a new module is created. The karate club network studied by Zachary is a prototypical example. We find that the model can reproduce successfully the behavior of the hierarchical clustering coefficient of a vertex with degree k, C(k), in good agreement with empirical measurements of real world networks.

cond-mat.stat-mech↗

arXiv2003

Bose-Einstein condensation in random directed networks

We consider the phenomenon of Bose-Einstein condensation in a random growing directed network. The network grows by the addition of vertices and edges. At each time step the network gains a vertex with probabilty $p$ and an edge with probability $1-p$. The new vertex has a fitness $(a,b)$ with probability $f(a,b)$. A vertex with fitness $(a,b)$, in-degree $i$ and out-degree $j$ gains a new incoming edge with rate $a(i+1)$ and an outgoing edge with rate $b(j+1)$. The Bose-Einstein condensation occurs as a function of fitness distribution $f(a,b)$.

cond-mat.stat-mech↗

arXiv2002

Packet Transport on Scale Free Networks

We introduce a model of information packet transport on networks in which the packets are posted by a given rate and move in parallel according to a local search algorithm. By performing a number of simulations we investigate the major kinetic properties of the transport as a function of the network geometry, the packet input rate and the buffer size. We find long-range correlations in the power spectra of arriving packet density and the network's activity bursts. The packet transit time distribution shows a power-law dependence with average transit time increasing with network size. This implies dynamic queueing on the network, in which many interacting queues are mutually driven by temporally correlated packet streams.

cond-mat.stat-mech↗

arXiv2002

A non extensive approach for DNA breaking by ionizing radiation

Tsallis entropy and a maximum entropy principle allows to reproduce experimental data of DNA double strand breaking by electron and neutron radiation. Analytic results for the probability of finding a DNA segment of length l are obtained reproducing quite well the fragment distribution function experimentally obtained.

cond-mat↗

arXiv2002

A model for the size distribution of customer groups and businesses

We present a generalization of the dynamical model of information transmission and herd behavior proposed by Eguiluz and Zimmermann. A characteristic size of group of agents $s_{0}$ is introduced. The fragmentation and coagulation rates of groups of agents are assumed to depend on the size of the group. We present results of numerical simulations and mean field analysis. It is found that the size distribution of groups of agents $n_{s}$ exhibits two distinct scaling behavior depending on $s \leq s_{0}$ or $s > s_{0}$. For $s \leq s_{0}$, $n_{s} \sim s^{-(5/2 + δ)}$, while for $s > s_{0}$, $n_{s} \sim s^{-(5/2 -δ)}$, where $δ$ is a model parameter representing the sensitivity of the fragmentation and coagulation rates to the size of the group. Our model thus gives a tunable exponent for the size distribution together with two scaling regimes separated by a characteristic size $s_{0}$. Suitably interpreted, our model can be used to represent the formation of groups of customers for certain products produced by manufacturers. This, in turn, leads to a distribution in the size of businesses. The characteristic size $s_{0}$, in this context, represents the size of a business for which the customer group becomes too large to be kept happy but too small for the business to become a brand name.

cond-mat.stat-mech↗

arXiv2001

A Herding Model with Preferential Attachment and Fragmentation

We introduce and solve a model that mimics the herding effect in financial markets when groups of agents share information. The number of agents in the model is growing and at each time step either (i) with probability $p$ an incoming agent joins an existing group, or (ii) with probability $1-p$ a group is fragmented into individual agents. The group size distribution is found to be power-law with an exponent that depends continuously on $p$. A number of variants of our basic model are discussed. Comparisons are made between these models and other models of herding and random growing networks.

cond-mat.dis-nn↗