The Large Scale Curvature of Networks

Understanding key structural properties of large scale networks are crucial for analyzing and optimizing their performance, and improving their reliability and security. Here we show that these networks possess a previously unnoticed feature, global curvature, which we argue has a major impact on core congestion: the load at the core of a network with N nodes scales as N^2 as compared to N^1.5 for a flat network. We substantiate this claim through analysis of a collection of real data networks across the globe as measured and documented by previous researchers.Comment: 4 pages, 5 figure

Optimization of transport protocols with path-length constraints in complex networks

We propose a protocol optimization technique that is applicable to both weighted or unweighted graphs. Our aim is to explore by how much a small variation around the Shortest Path or Optimal Path protocols can enhance protocol performance. Such an optimization strategy can be necessary because even though some protocols can achieve very high traffic tolerance levels, this is commonly done by enlarging the path-lengths, which may jeopardize scalability. We use ideas borrowed from Extremal Optimization to guide our algorithm, which proves to be an effective technique. Our method exploits the degeneracy of the paths or their close-weight alternatives, which significantly improves the scalability of the protocols in comparison to Shortest Paths or Optimal Paths protocols, keeping at the same time almost intact the length or weight of the paths. This characteristic ensures that the optimized routing protocols are composed of paths that are quick to traverse, avoiding negative effects in data communication due to path-length increases that can become specially relevant when information losses are present.Comment: 8 pages, 8 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

Internet's Critical Path Horizon

Internet is known to display a highly heterogeneous structure and complex fluctuations in its traffic dynamics. Congestion seems to be an inevitable result of user's behavior coupled to the network dynamics and it effects should be minimized by choosing appropriate routing strategies. But what are the requirements of routing depth in order to optimize the traffic flow? In this paper we analyse the behavior of Internet traffic with a topologically realistic spatial structure as described in a previous study (S-H. Yook et al. ,Proc. Natl Acad. Sci. USA, {\bf 99} (2002) 13382). The model involves self-regulation of packet generation and different levels of routing depth. It is shown that it reproduces the relevant key, statistical features of Internet's traffic. Moreover, we also report the existence of a critical path horizon defining a transition from low-efficient traffic to highly efficient flow. This transition is actually a direct consequence of the web's small world architecture exploited by the routing algorithm. Once routing tables reach the network diameter, the traffic experiences a sudden transition from a low-efficient to a highly-efficient behavior. It is conjectured that routing policies might have spontaneously reached such a compromise in a distributed manner. Internet would thus be operating close to such critical path horizon.Comment: 8 pages, 8 figures. To appear in European Journal of Physics B (2004

Transitions in spatial networks

Networks embedded in space can display all sorts of transitions when their structure is modified. The nature of these transitions (and in some cases crossovers) can differ from the usual appearance of a giant component as observed for the Erdos-Renyi graph, and spatial networks display a large variety of behaviors. We will discuss here some (mostly recent) results about topological transitions, `localization' transitions seen in the shortest paths pattern, and also about the effect of congestion and fluctuations on the structure of optimal networks. The importance of spatial networks in real-world applications makes these transitions very relevant and this review is meant as a step towards a deeper understanding of the effect of space on network structures.Comment: Corrected version and updated list of reference