Local Search in Unstructured Networks

We review a number of message-passing algorithms that can be used to search through power-law networks. Most of these algorithms are meant to be improvements for peer-to-peer file sharing systems, and some may also shed some light on how unstructured social networks with certain topologies might function relatively efficiently with local information. Like the networks that they are designed for, these algorithms are completely decentralized, and they exploit the power-law link distribution in the node degree. We demonstrate that some of these search algorithms can work well on real Gnutella networks, scale sub-linearly with the number of nodes, and may help reduce the network search traffic that tends to cripple such networks.Comment: v2 includes minor revisions: corrections to Fig. 8's caption and references. 23 pages, 10 figures, a review of local search strategies in unstructured networks, a contribution to `Handbook of Graphs and Networks: From the Genome to the Internet', eds. S. Bornholdt and H.G. Schuster (Wiley-VCH, Berlin, 2002), to be publishe

The Routing of Complex Contagion in Kleinberg's Small-World Networks

In Kleinberg's small-world network model, strong ties are modeled as deterministic edges in the underlying base grid and weak ties are modeled as random edges connecting remote nodes. The probability of connecting a node $u$ with node $v$ through a weak tie is proportional to $1/|uv|^\alpha$, where $|uv|$ is the grid distance between $u$ and $v$ and $\alpha\ge 0$ is the parameter of the model. Complex contagion refers to the propagation mechanism in a network where each node is activated only after $k \ge 2$ neighbors of the node are activated. In this paper, we propose the concept of routing of complex contagion (or complex routing), where we can activate one node at one time step with the goal of activating the targeted node in the end. We consider decentralized routing scheme where only the weak ties from the activated nodes are revealed. We study the routing time of complex contagion and compare the result with simple routing and complex diffusion (the diffusion of complex contagion, where all nodes that could be activated are activated immediately in the same step with the goal of activating all nodes in the end). We show that for decentralized complex routing, the routing time is lower bounded by a polynomial in $n$ (the number of nodes in the network) for all range of $\alpha$ both in expectation and with high probability (in particular, $\Omega(n^{\frac{1}{\alpha+2}})$ for $\alpha \le 2$ and $\Omega(n^{\frac{\alpha}{2(\alpha+2)}})$ for $\alpha > 2$ in expectation), while the routing time of simple contagion has polylogarithmic upper bound when $\alpha = 2$. Our results indicate that complex routing is harder than complex diffusion and the routing time of complex contagion differs exponentially compared to simple contagion at sweetspot.Comment: Conference version will appear in COCOON 201

Paul Baran, Network Theory, and the Past, Present, and Future of Internet

Paul Baran’s seminal 1964 article “On Distributed Communications Networks” that first proposed packet switching also advanced an underappreciated vision of network architecture: a lattice-like, distributed network, in which each node of the Internet would be homogeneous and equal in status to all other nodes. Scholars who have subsequently embraced the concept of a lattice-like network approach have largely overlooked the extent to which it is both inconsistent with network theory (associated with the work of Duncan Watts and Albert-László Barabási), which emphasizes the importance of short cuts and hubs in enabling networks to scale, and the actual way, the Internet initially deployed, which relied on a three-tiered, hierarchical architecture that was actually what Baran called a decentralized network. However, empirical studies reveal that the Internet’s architecture is changing: it is in the process of becoming flatter and less hierarchical, as large content providers build extensive wide area networks and undersea cables to connect directly to last-mile networks. This change is making the network more centralized rather than becoming more distributed. As a result, this article suggests that the standard reference model that places backbones at the center of the architecture should be replaced with a radically different vision: a stack of centralized star networks, each centered on one of the leading content providers

Mixing navigation on networks

In this Letter, we proposed a mixing navigation mechanism, which interpolates between random-walk and shortest-path protocol. The navigation efficiency can be remarkably enhanced via a few routers. Some advanced strategies are also designed: For non-geographical scale-free networks, the targeted strategy with a tiny fraction of routers can guarantee an efficient navigation with low and stable delivery time almost independent of network size. For geographical localized networks, the clustering strategy can simultaneously increase the efficiency and reduce the communication cost. The present mixing navigation mechanism is of significance especially for information organization of wireless sensor networks and distributed autonomous robotic systems.Comment: 4 pages, and 7 figure

Neighbor selection and hitting probability in small-world graphs

Small-world graphs, which combine randomized and structured elements, are seen as prevalent in nature. Jon Kleinberg showed that in some graphs of this type it is possible to route, or navigate, between vertices in few steps even with very little knowledge of the graph itself. In an attempt to understand how such graphs arise we introduce a different criterion for graphs to be navigable in this sense, relating the neighbor selection of a vertex to the hitting probability of routed walks. In several models starting from both discrete and continuous settings, this can be shown to lead to graphs with the desired properties. It also leads directly to an evolutionary model for the creation of similar graphs by the stepwise rewiring of the edges, and we conjecture, supported by simulations, that these too are navigable.Comment: Published in at http://dx.doi.org/10.1214/07-AAP499 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Dynamics of Rumor Spreading in Complex Networks

We derive the mean-field equations characterizing the dynamics of a rumor process that takes place on top of complex heterogeneous networks. These equations are solved numerically by means of a stochastic approach. First, we present analytical and Monte Carlo calculations for homogeneous networks and compare the results with those obtained by the numerical method. Then, we study the spreading process in detail for random scale-free networks. The time profiles for several quantities are numerically computed, which allow us to distinguish among different variants of rumor spreading algorithms. Our conclusions are directed to possible applications in replicated database maintenance, peer to peer communication networks and social spreading phenomena.Comment: Final version to appear in PR

The missing links in the BGP-based AS connectivity maps

PAM2003 - The Passive and Active Measurement Workshop(http://www.pam2003.org), San Diego, USA, April 2003PAM2003 - The Passive and Active Measurement Workshop(http://www.pam2003.org), San Diego, USA, April 2003PAM2003 - The Passive and Active Measurement Workshop(http://www.pam2003.org), San Diego, USA, April 2003A number of recent studies of the Internet topology at the autonomous systems level (AS graph) are based on the BGP-based AS connectivity maps (original maps). The so-called extended maps use additional data sources and contain more complete pictures of the AS graph. In this paper, we compare an original map, an extended map and a synthetic map generated by the Barabasi-Albert model. We examine the recently reported rich-club phenomenon, alternative routing paths and attack tolerance. We point out that the majority of the missing links of the original maps are the connecting links between rich nodes (nodes with large numbers of links) of the extended maps. We show that the missing links are relevant because links between rich nodes can be crucial for the network structure

Task Routing for Prediction Tasks

We study principles and methods for task routing that aim to harness people’s abilities to jointly contribute to a task and to route tasks to others who can provide further contributions. In the particular context of prediction tasks, the goal is to efficiently obtain accurate probability assessments for an event of interest. We introduce routing scoring rules for promoting collaborative behavior, that bring truthfully contributing information and optimally routing tasks into a Perfect Bayesian Equilibrium under common knowledge about agents’ abilities. However, for networks where agents only have local knowledge about other agents’ abilities, optimal routing requires complex reasoning over the history and future routing decisions of users outside of local neighborhoods. Avoiding this, we introduce a class of local routing rules that isolate simple routing decisions in equilibrium, while still promoting effective routing decisions. We present simulation results that show that following routing decisions induced by local routing rules lead to efficient information aggregation.Engineering and Applied Science