Adaptive Dynamics of Realistic Small-World Networks

Continuing in the steps of Jon Kleinberg's and others celebrated work on decentralized search in small-world networks, we conduct an experimental analysis of a dynamic algorithm that produces small-world networks. We find that the algorithm adapts robustly to a wide variety of situations in realistic geographic networks with synthetic test data and with real world data, even when vertices are uneven and non-homogeneously distributed. We investigate the same algorithm in the case where some vertices are more popular destinations for searches than others, for example obeying power-laws. We find that the algorithm adapts and adjusts the networks according to the distributions, leading to improved performance. The ability of the dynamic process to adapt and create small worlds in such diverse settings suggests a possible mechanism by which such networks appear in nature

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

A Framework for Web Object Self-Preservation

We propose and develop a framework based on emergent behavior principles for the long-term preservation of digital data using the web infrastructure. We present the development of the framework called unsupervised small-world (USW) which is at the nexus of emergent behavior, graph theory, and digital preservation. The USW algorithm creates graph based structures on the Web used for preservation of web objects (WOs). Emergent behavior activities, based on Craig Reynolds’ “boids” concept, are used to preserve WOs without the need for a central archiving authority. Graph theory is extended by developing an algorithm that incrementally creates small-world graphs. Graph theory provides a foundation to discuss the vulnerability of graphs to different types of failures and attack profiles. Investigation into the robustness and resilience of USW graphs lead to the development of a metric to quantify the effect of damage inflicted on a graph. The metric remains valid whether the graph is connected or not. Different USW preservation policies are explored within a simulation environment where preservation copies have to be spread across hosts. Spreading the copies across hosts helps to ensure that copies will remain available even when there is a concerted effort to remove all copies of a USW component. A moderately aggressive preservation policy is the most effective at making the best use of host and network resources. Our efforts are directed at answering the following research questions: 1. Can web objects (WOs) be constructed to outlive the people and institutions that created them? We have developed, analyzed, tested through simulations, and developed a reference implementation of the unsupervised small-world (USW) algorithm that we believe will create a connected network of WOs based on the web infrastructure (WI) that will outlive the people and institutions that created the WOs. The USW graph will outlive its creators by being robust and continuing to operate when some of its WOs are lost, and it is resilient and will recover when some of its WOs are lost. 2. Can we leverage aspects of naturally occurring networks and group behavior for preservation? We used Reynolds’ tenets for “boids” to guide our analysis and development of the USW algorithm. The USW algorithm allows a WO to “explore” a portion of the USW graph before making connections to members of the graph and before making preservation copies across the “discovered” graph. Analysis and simulation show that the USW graph has an average path length (L(G)) and clustering coefficient (C(G)) values comparable to small-world graphs. A high C(G) is important because it reflects how likely it is that a WO will be able spread copies to other domains, thereby increasing its likelihood of long term survival. A short L(G) is important because it means that a WO will not have to look too far to identify new candidate preservation domains, if needed. Small-world graphs occur in nature and are thus believed to be robust and resilient. The USW algorithms use these small-world graph characteristics to spread preservation copies across as many hosts as needed and possible. USW graph creation, damage, repair and preservation has been developed and tested in a simulation and reference implementation

Towards Small World Emergence

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