Navigability is a Robust Property

The Small World phenomenon has inspired researchers across a number of fields. A breakthrough in its understanding was made by Kleinberg who introduced Rank Based Augmentation (RBA): add to each vertex independently an arc to a random destination selected from a carefully crafted probability distribution. Kleinberg proved that RBA makes many networks navigable, i.e., it allows greedy routing to successfully deliver messages between any two vertices in a polylogarithmic number of steps. We prove that navigability is an inherent property of many random networks, arising without coordination, or even independence assumptions

Navigability of temporal networks in hyperbolic space

Information routing is one of the main tasks in many complex networks with a communication function. Maps produced by embedding the networks in hyperbolic space can assist this task enabling the implementation of efficient navigation strategies. However, only static maps have been considered so far, while navigation in more realistic situations, where the network structure may vary in time, remain largely unexplored. Here, we analyze the navigability of real networks by using greedy routing in hyperbolic space, where the nodes are subject to a stochastic activation-inactivation dynamics. We find that such dynamics enhances navigability with respect to the static case. Interestingly, there exists an optimal intermediate activation value, which ensures the best trade-off between the increase in the number of successful paths and a limited growth of their length. Contrary to expectations, the enhanced navigability is robust even when the most connected nodes inactivate with very high probability. Finally, our results indicate that some real networks are ultranavigable and remain highly navigable even if the network structure is extremely unsteady. These findings have important implications for the design and evaluation of efficient routing protocols that account for the temporal nature of real complex networks.Comment: 10 pages, 4 figures. Includes Supplemental Informatio

Network Information Flow in Small World Networks

Recent results from statistical physics show that large classes of complex networks, both man-made and of natural origin, are characterized by high clustering properties yet strikingly short path lengths between pairs of nodes. This class of networks are said to have a small-world topology. In the context of communication networks, navigable small-world topologies, i.e. those which admit efficient distributed routing algorithms, are deemed particularly effective, for example in resource discovery tasks and peer-to-peer applications. Breaking with the traditional approach to small-world topologies that privileges graph parameters pertaining to connectivity, and intrigued by the fundamental limits of communication in networks that exploit this type of topology, we investigate the capacity of these networks from the perspective of network information flow. Our contribution includes upper and lower bounds for the capacity of standard and navigable small-world models, and the somewhat surprising result that, with high probability, random rewiring does not alter the capacity of a small-world network.Comment: 23 pages, 8 fitures, submitted to the IEEE Transactions on Information Theory, November 200

Hyperbolic Geometry of Complex Networks

We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry. Conversely, we show that if a network has some metric structure, and if the network degree distribution is heterogeneous, then the network has an effective hyperbolic geometry underneath. We then establish a mapping between our geometric framework and statistical mechanics of complex networks. This mapping interprets edges in a network as non-interacting fermions whose energies are hyperbolic distances between nodes, while the auxiliary fields coupled to edges are linear functions of these energies or distances. The geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures. Finally, we show that targeted transport processes without global topology knowledge, made possible by our geometric framework, are maximally efficient, according to all efficiency measures, in networks with strongest heterogeneity and clustering, and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure

Finding the most navigable path in road networks

Input to the Most Navigable Path (MNP) problem consists of the following: (a) a road network represented as a directed graph, where each edge is associated with numeric attributes of cost and “navigability score” values; (b) a source and a destination and; (c) a budget value which denotes the maximum permissible cost of the solution. Given the input, MNP aims to determine a path between the source and the destination which maximizes the navigability score while constraining its cost to be within the given budget value. The problem can be modeled as the arc orienteering problem which is known to be NP-hard. The current state-of-the-art for this problem may generate paths having loops, and its adaptation for MNP that yields simple paths, was found to be inefficient. In this paper, we propose five novel algorithms for the MNP problem. Our algorithms first compute a seed path from the source to the destination, and then modify the seed path to improve its navigability. We explore two approaches to compute the seed path. For modification of the seed path, we explore different Dynamic Programming based approaches. We also propose an indexing structure for the MNP problem which helps in reducing the running time of some of our algorithms. Our experimental results indicate that the proposed solutions yield comparable or better solutions while being orders of magnitude faster than the current state-of-the-art for large real road networks