G. J. Rodgers

Degree distributions of growing networks

The in-degree and out-degree distributions of a growing network model are determined. The in-degree is the number of incoming links to a given node (and vice versa for out-degree). The network is built by (i) creation of new nodes which each immediately attach to a preexisting node, and (ii) creation of new links between preexisting nodes. This process naturally generates correlated in-degree and out-degree distributions. When the node and link creation rates are linear functions of node degree, these distributions exhibit distinct power-law forms. By tuning the parameters in these rates to reasonable values, exponents which agree with those of the web graph are obtained.

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 time series exhibits antipersistence due to the regulatory role of the superstructure associated with two hub nodes in the network. We discuss how the superstructure affects 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.

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 ...

Growing random networks with fitness

Three models of growing random networks with fitness-dependent growth rates are analysed using the rate equations for the distribution of their connectivities. In the first model (A), a network is built by connecting incoming nodes to nodes of connectivity k and random additive fitness η, with rate (k−1)+η. For η>0 we find the connectivity distribution is power law with exponent γ=〈η〉+2. In the second model (B), the network is built by connecting nodes to nodes of connectivity k, random additive fitness η and random multiplicative fitness ζ with rate ζ(k−1)+η. This model also has a power law connectivity distribution, but with an exponent which depends on the multiplicative fitness at each node. In the third model (C), a directed graph is considered and is built by the addition of nodes and the creation of links. A node with fitness (α,β), i incoming links and j outgoing links gains a new incoming link with rate α(i+1), and a new outgoing link with rate β(j+1). The distributions of the number of incoming and outgoing links both scale as power laws, with inverse logarithmic corrections.

The Hamming Distance in the Minority Game

We investigate different versions of the minority game, a toy model for agents buying and selling a commodity. The Hamming distance between the strategies used by agents to make decisions is introduced as an analytical tool to determine several properties of these models. The success rate of the agents in an adaptive version of the game is compared with the rate from a stochastic version. It is shown numerically and analytically that the adaptive process is inefficient, increasing the success rate of the unused strategies while decreasing the success rate of the strategies used by the agents. The agents do not do as well as if they were forced to use only one strategy permanently. A version of the game in which the agents strategies evolve is also analysed using the notion of distance. The agents evolve into a state in which they are all using one strategy, which is again the state that yields the maximum success rate.

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 networks activity bursts. The packet transit time distribution shows a power-law dependence with average transit time increasing with network size. This implies dynamic queuing on the network, in which many interacting queues are mutually driven by temporally correlated packet streams.

Eigenvalue spectra of complex networks

We examine the eigenvalue spectrum, ρ(μ), of the adjacency matrix of a random scale-free network with an average of p edges per vertex using the replica method. We show how in the dense limit, when p → ∞, one can obtain two relatively simple coupled equations whose solution yields ρ(μ) for an arbitrary complex network. For scale-free graphs, with degree distribution exponent λ, we obtain an exact expression for the eigenvalue spectrum when λ = 3 and show that ρ(μ) ~ 1/μ2λ−1 for large μ. In the limit λ → ∞ we recover known results for the Erdos–Renyi random graph.

Density of states of sparse random matrices

A supersymmetric formalism is used to derive a set of equations giving the density of states of any real, symmetric, sparse random matrix as a function of the distribution of non-zero elements and the mean number of non-zero elements per row, p. In the matrix where the non-zero elements take the values +or-1 with equal probability the equations are solved as p to infinity recovering results obtained previously with the replica method. As E to 0 the density of states rho (E) behaves as 1/E(ln(E))2. The more general case where +or-1 occur with unequal probabilities is also considered.

Tsallis Entropy and the transition to scaling in fragmentation

By using the maximum entropy principle with Tsallis entropy we obtain a fragment size distribution function which undergoes a transition to scaling. This distribution function reduces to those obtained by other authors using Shannon entropy. The treatment is easily generalisable to any process of fractioning with suitable constraints.

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 analyze the differences between the statistical properties of a real and a shuffled network. We show that the scale-free degree distribution and the scale-free weight distribution are induced by the scale-free strength distribution, that is Zipfs 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 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.