Divergence measures have a long association with statistical inference, machine learning and information theory. The density power divergence and related measures have produced many useful (and popular) statistical procedures, which provide a good balance between model efficiency on one hand and outlier stability or robustness on the other. The logarithmic density power divergence, a particular logarithmic transform of the density power divergence, has also been very successful in producing efficient and stable inference procedures; in addition it has also led to significant demonstrated applications in information theory. The success of the minimum divergence procedures based on the density power divergence and the logarithmic density power divergence (which also go by the names$\beta $-divergence and$\gamma $-divergence, respectively) make it imperative and meaningful to look for other, similar divergences which may be obtained as transforms of the density power divergence in the same spirit. With this motivation we search for such transforms of the density power divergence, referred to herein as the functional density power divergence class. The present article characterizes this functional density power divergence class, and thus identifies the available divergence measures within this construct that may be explored further for possible applications in statistical inference, machine learning and information theory.

Many P2P-based storage systems use distributed indexing service for searching documents. There are two security issues when the nodes providing the index service are compromised by adversaries. First, the adversaries may delete the indexes or stop the program of indexing service, making the affected documents disappear in the search infrastructure. Second, the adversaries may leak the locations of the storage nodes hosting certain documents, making those nodes the target of DOS attacks. We propose a protocol called SCUBE which addresses these attacks by using secret-sharing based threshold cryptography and the concept of virtual addresses. Our results show that SCUBE performs appreciably well under different attack scenarios and incurs nominal overhead. A working prototype of SCUBE has also been implemented and tested on the Planetlab testbed.

Anonymous communication systems built on P2P infrastructures using anonymity forwarders are frequently affected by the churn problem, i.e. frequent joins and leaves of nodes. The problem unavoidably affects the quality of provided anonymity: The availability of anonymity forwarders will be decreased, which reduces the anonymity set; and the frequency of path reformation will increase, which increases the chance of successful intersection attacks. We propose an incentive-based P2P mechanism as an approach to providing reliable anonymity forwarding. It uses incentives to induce the peer nodes to provide anonymity forwarding as reliable service and to make stable and distributed forwarding decisions to minimize the frequency of path reformations. To support incentive, a payment system has been designed which meet the anonymity requirement and can handle typical scenarios of cheating and malicious attacks. To make sound forwarding decisions, we use game theory to carefully design the forwarding strategies used by the peer nodes. We have used event-driven simulations to evaluate the quality of anonymity provided by the mechanism under high churn and with the presence of malicious nodes. The results show that the quality of anonymity is maintained in those scenarios.

We study the resilience of structured peer-to-peer (P2P) networks in the presence of churn. Using the lifetime-based failure assumptions, we first show that a realistic churn model has an equivalent uniform failure model in the steady state. We then determine via percolation analysis and simulation the gap between the size of the connected component and the reachable component of a randomly picked node for symphony and chord. This gap represents the price of structured routing: the size of the set of nodes that are reachable by any unstructured routing method (e.g. broadcast on the unstructured overlay) from a randomly picked surviving node, but are not reachable using structured routing on the structured overlay. As an illustration, for 24- minute average node lifetime with 1-minute average node-search delay, the gap is around 12 thousand nodes or 1.2% in a chord network of around 1 million nodes. We finish by discussing potential techniques to mitigate the price of structured routing.