The effect of network structure on phase transitions in queuing networks

Recently, De Martino et al have presented a general framework for the study of transportation phenomena on complex networks. One of their most significant achievements was a deeper understanding of the phase transition from the uncongested to the congested phase at a critical traffic load. In this paper, we also study phase transition in transportation networks using a discrete time random walk model. Our aim is to establish a direct connection between the structure of the graph and the value of the critical traffic load. Applying spectral graph theory, we show that the original results of De Martino et al showing that the critical loading depends only on the degree sequence of the graph -- suggesting that different graphs with the same degree sequence have the same critical loading if all other circumstances are fixed -- is valid only if the graph is dense enough. For sparse graphs, higher order corrections, related to the local structure of the network, appear.Comment: 12 pages, 7 figure

An Analytical Model for Wireless Mesh Networks with Collision-Free TDMA and Finite Queues

Wireless mesh networks are a promising technology for connecting sensors and actuators with high flexibility and low investment costs. In industrial applications, however, reliability is essential. Therefore, two time-slotted medium access methods, DSME and TSCH, were added to the IEEE 802.15.4 standard. They allow collision-free communication in multi-hop networks and provide channel hopping for mitigating external interferences. The slot schedule used in these networks is of high importance for the network performance. This paper supports the development of efficient schedules by providing an analytical model for the assessment of such schedules, focused on TSCH. A Markov chain model for the finite queue on every node is introduced that takes the slot distribution into account. The models of all nodes are interconnected to calculate network metrics such as packet delivery ratio, end-to-end delay and throughput. An evaluation compares the model with a simulation of the Orchestra schedule. The model is applied to Orchestra as well as to two simple distributed scheduling algorithms to demonstrate the importance of traffic-awareness for achieving high throughput.Comment: 17 pages, 14 figure

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

A critical look at power law modelling of the Internet

This paper takes a critical look at the usefulness of power law models of the Internet. The twin focuses of the paper are Internet traffic and topology generation. The aim of the paper is twofold. Firstly it summarises the state of the art in power law modelling particularly giving attention to existing open research questions. Secondly it provides insight into the failings of such models and where progress needs to be made for power law research to feed through to actual improvements in network performance.Comment: To appear Computer Communication

Preferential Behaviour and Scaling in Diffusive Dynamics on Networks

We study the fluctuation properties and return-time statistics on inhomogeneous scale-free networks using packets moving with two different dynamical rules; random diffusion and locally navigated diffusive motion with preferred edges. Scaling in the fluctuations occurs when the dispersion of a quantity at each node or edge increases like the its mean to the power $\mu$. We show that the occurrence of scaling in the fluctuations of both the number of packets passing nodes and the number flowing along edges is related to preferential behaviour in either the topology (in the case of nodes) or in the dynamics (in case the of edges). Within our model the absence of any preference leads to the absence of scaling, and when scaling occurs it is non-universal; for random diffusion the number of packets passing a node scales with an exponent $\mu$ which increases continuously with increased acquisition time window from $\mu =1/2$ at small windows, to $\mu =1$ at long time windows; In the preferentially navigated diffusive motion, busy nodes and edges have exponent $\mu =1$, in contrast to less busy parts of the network, where an exponent $\mu =1/2$ is found. Broad distributions of the return times at nodes and edges illustrate that the basis of the observed scaling is the cooperative behaviour between groups of nodes or edges. These conclusions are relevant for a large class of diffusive dynamics on networks, including packet transport with local navigation rules