Small-World Internet Topologies: Possible Causes and Implications on Scalability of End-System Multicast

Recent work has shown the prevalence of small-world phenomena [28] in many networks. Small-world graphs exhibit a high degree of clustering, yet have typically short path lengths between arbitrary vertices. Internet AS-level graphs have been shown to exhibit small-world behaviors [9]. In this paper, we show that both Internet AS-level and router-level graphs exhibit small-world behavior. We attribute such behavior to two possible causes–namely the high variability of vertex degree distributions (which were found to follow approximately a power law [15]) and the preference of vertices to have local connections. We show that both factors contribute with different relative degrees to the small-world behavior of AS-level and router-level topologies. Our findings underscore the inefficacy of the Barabasi-Albert model [6] in explaining the growth process of the Internet, and provide a basis for more promising approaches to the development of Internet topology generators. We present such a generator and show the resemblance of the synthetic graphs it generates to real Internet AS-level and router-level graphs. Using these graphs, we have examined how small-world behaviors affect the scalability of end-system multicast. Our findings indicate that lower variability of vertex degree and stronger preference for local connectivity in small-world graphs results in slower network neighborhood expansion, and in longer average path length between two arbitrary vertices, which in turn results in better scaling of end system multicast.

Social Networks and Small World phenomena

Social Networking tools like Facebook yield recognisable small world phenomena, that is particular kinds of social graphs that facilitate particular kinds of interaction and information exchange.

Small-world effects in lattice stochastic diffusion search

Stochastic Diffusion Search is an efficient probabilistic bestfit search technique, capable of transformation invariant pattern matching. Although inherently parallel in operation it is difficult to implement efficiently in hardware as it requires full inter-agent connectivity. This paper describes a lattice implementation, which, while qualitatively retaining the properties of the original algorithm, restricts connectivity, enabling simpler implementation on parallel hardware. Diffusion times are examined for different network topologies, ranging from ordered lattices, over small-world networks to random graphs.

Spectra of modular and small-world matrices

We compute spectra of symmetric random matrices describing graphs with general modular structure and arbitrary inter- and intra-module degree distributions, subject only to the constraint of finite mean connectivities. We also evaluate spectra of a certain class of small-world matrices generated from random graphs by introducing shortcuts via additional random connectivity components. Both adjacency matrices and the associated graph Laplacians are investigated. For the Laplacians, we find Lifshitz-type singular behaviour of the spectral density in a localized region of small |?| values. In the case of modular networks, we can identify contributions of local densities of state from individual modules. For small-world networks, we find that the introduction of short cuts can lead to the creation of satellite bands outside the central band of extended states, exhibiting only localized states in the band gaps. Results for the ensemble in the thermodynamic limit are in excellent agreement with those obtained via a cavity approach for large finite single instances, and with direct diagonalization results.

Collective dynamics and control of a 3-D small-world network with time delays

The paper presents a detailed analysis on the collective dynamics and delayed state feedback control of a three-dimensional delayed small-world network. The trivial equilibrium of the model is first investigated, showing that the uncontrolled model exhibits complicated unbounded behavior. Then three control strategies, namely a position feedback control, a velocity feedback control, and a hybrid control combined velocity with acceleration feedback, are then introduced to stabilize this unstable system. It is shown in these three control schemes that only the hybrid control can easily stabilize the 3-D network system. And with properly chosen delay and gain in the delayed feedback path, the hybrid controlled model may have stable equilibrium, or periodic solutions resulting from the Hopf bifurcation, or complex stranger attractor from the period-doubling bifurcation. Moreover, the direction of Hopf bifurcation and stability of the bifurcation periodic solutions are analyzed. The results are further extended to any "d" dimensional network. It shows that to stabilize a "d" dimensional delayed small-world network, at least a "d – 1" order completed differential feedback is needed. This work provides a constructive suggestion for the high dimensional delayed systems.

Efficiency Dynamics on Two Coupled Small-World Networks

We investigate the effect of clusters in complex networks on efficiency dynamics by studying a simple efficiency model in two coupled small-world networks. It is shown that the critical network randomness corresponding to transition from a stagnant phase to a growing one decreases to zero as the connection strength of clusters increases. It is also shown for fixed randomness that the state of clusters transits from a stagnant phase to a growing one as the connection strength of clusters increases. This work can be useful for understanding the critical transition appearing in many dynamic processes on the cluster networks.

Small World Folksonomies: Clustering in Tri-Partite Hypergraphs

Many recent Web 2.0 resource sharing applications can be subsumed under the "folksonomy" moniker. Regardless of the type of resource shared, all of these share a common structure describing the assignment of tags to resources by users. In this report, we generalize the notions of clustering and characteristic path length which play a major role in the current research on networks, where they are used to describe the small-world effects on many observable network datasets. To that end, we show that the notion of clustering has two facets which are not equivalent in the generalized setting. The new measures are evaluated on two large-scale folksonomy datasets from resource sharing systems on the web.

TU Graz: Course: 707.000 Web Science and Web Technology: Lecture 2: Small World Problem

We will discuss several examples and research efforts related to the small world problem and set the ground for our discussion of network theory and social network analysis. Readings: An Experimental Study of the Small World Problem, J. Travers and S. Milgram Sociometry 32 425-443 (1969) [Protected Access] Optional: The Strength of Weak Ties, M.S. Granovetter The American Journal of Sociology 78 1360--1380 (1973) [Protected Access] Optional: Worldwide Buzz: Planetary-Scale Views on an Instant-Messaging Network, J. Leskovec and E. Horvitz MSR-TR-2006-186. Microsoft Research, June 2007. [Web Link, the most recent and comprehensive study on the subject!] Originally from: http://kmi.tugraz.at/staff/markus/courses/SS2008/707.000_web-science/

Lethal Closeness: The Evolution of a Small-World Drug Trafficking Network

The evolution of the drug trafficking network –so-called– ‘Cartel del Norte del Valle’, is studied using network analysis methods. We found that the average length between any pair of its members was bounded by 4 –an attribute of smallworld networks. In this tightly connected network, informational shocks induce fear and the unleashing of searches of threatening nodes, using available paths. Lethal violence ensues in clusters of increasing sizes that fragment the network, without compromising, however, the survival of the largest component, which proved to be resilient to massive violence. In spite of a success from the point of view of head counting, the US’ socialization program for drug traffickers did not effectively change the cyclical dynamics of the drug dealing business: war survivors took over what was left from the old network initiating a new cycle of business and violence.

Emergence of a small-world functional network in cultured neurons

The functional networks of cultured neurons exhibit complex network properties similar to those found in vivo. Starting from random seeding, cultures undergo significant reorganization during the initial period in vitro, yet despite providing an ideal platform for observing developmental changes in neuronal connectivity, little is known about how a complex functional network evolves from isolated neurons. In the present study, evolution of functional connectivity was estimated from correlations of spontaneous activity. Network properties were quantified using complex measures from graph theory and used to compare cultures at different stages of development during the first 5 weeks in vitro. Networks obtained from young cultures (14 days in vitro) exhibited a random topology, which evolved to a small-world topology during maturation. The topology change was accompanied by an increased presence of highly connected areas (hubs) and network efficiency increased with age. The small-world topology balances integration of network areas with segregation of specialized processing units. The emergence of such network structure in cultured neurons, despite a lack of external input, points to complex intrinsic biological mechanisms. Moreover, the functional network of cultures at mature ages is efficient and highly suited to complex processing tasks.