Peer-To-Peer Networks

Abstract

Peer-To-Peer (P2P) networks are distributed systems that lack any hierarchical organization or centralized control. Peers form self-organizing networks that are overlayed on the Internet Protocol (IP) networks, offering a mix of various features such as robust wide-area routing architecture, efficient search of data items, selection of nearby peers, redundant storage, permanence, hierarchical naming, trust and authentication, anonymity, massive scalability and fault tolerance. These systems go beyond services offered by client-server systems by having symmetry in roles where a client may also be a server. It allows access to its resources by other systems and supports resource-sharing, which requires fault-tolerance, self-organization and massive scalability properties. Unlike Grid systems, P2P networks do not arise from the collaboration between established and connected groups of systems and without a more reliable set of resources to share. The core operation in a P2P network is the efficient location of data items. In this chapter, we will look at two widely-known P2P networks, Chord and Freenet.

Chord [3] is a distributed lookup protocol that addresses the problem of efficiently locating the node that stores a particular data item in a structured P2P application. Given a key, it maps the key onto a node. A key is associated with each data item and key/data item pair is stored at the node to which the key maps. Chord is designed to adapt efficiently as nodes join and leave the network dynamically. Freenet [1] is an unstructured P2P network application that allows the publication, replication and retrieval of data while protecting the anonymity of both the authors and the readers. It operates as a network of identical nodes that collectively pool their storage space to store data files and cooperate to route requests to the most likely physical location of data. The files are referred to in a location-independent manner, and are dynamically replicated in locations near requestors and deleted from locations where there is no interest. It is infeasible to discover the true origin or destination of a file passing through the network, and difficult for a node operator to be held responsible for the actual physical contents of her node.

Problems

In this exercise, the task is to evaluate a decentralized search algorithm on a network where the edges are created according to a hierarchical tree structure. The leaves of the tree will form the nodes of the network and the edge probabilities between two nodes depends on their proximity in the underlying tree structure.

P2P networks can be organized in a tree hierarchy, where the root is the main software application and the second level contains the different countries. The third level represents the different states and the fourth level is the different cities. There could be several more levels depending on the size and structure of the P2P network. Nevertheless, the final level are the clients.

Consider a situation where client A wants a file that is located in client B. If A cannot access B directly, A may connect to a node C which is, for instance, in the same city and ask C to access the file instead. If A does not have access to any node in the same city as B, it may try to access a node in the same state. In general, A will attempt to connect to the node “closest” to B.

In this problem, there are two networks: one is the observed network, i.e, the edge between P2P clients and the other is the hierarchical tree structure that is used to generated the edges in the observed network.

For this exercise, we will use a complete, perfectly balanced b-ary tree T (each node has b children and $b\ge 2$), and a network whose nodes are the leaves of T. For any pair of network nodes v and w, h(v, w) denotes the distance between the nodes and is defined as the height of the subtree L(v, w) of T rooted at the lowest common ancestor of v and w. The distance captures the intuition that clients in the same city are more likely to be connected than, for example, in the same state.

To model this intuition, generate a random network on the leaf nodes where for a node v, the probability distribution of node v creating an edge to any other node w is given by Eq. 6.1

$$\begin{aligned} p_{v}(w) = \frac{1}{Z}b^{-h(v,w)} \end{aligned}$$

(6.1)

where $Z = \sum _{w \ne v}b^{-h(v,w)}$ is a normalizing constant.

Next, set some parameter k and ensure that every node v has exactly k outgoing edges, using the following procedure. For each node v, sample a random node w according to $p_{v}$ and create edge (v, w) in the network. Continue this until v has exactly k neighbours. Equivalently, after an edge is added from v to w, set $p_{v}(w)$ to 0 and renormalize with a new Z to ensure that $\sum _{w} p(w) = 1$. This results in a k-regular directed network.

Now experimentally investigate a more general case where the edge probability is proportional to $b^{-\alpha h(v,w)}$. Here $\alpha > 0$ is a parameter in our experiments.

Consider a network with the setting $h(T) = 10$, $b = 2$, $k = 5$, and a given $\alpha$, i.e, the network consists of all the leaves in a binary tree of height 10; the out degree of each node is 5. Given $\alpha$, create edges according to the distribution described above.

46

Create random networks for $\alpha = 0.1, 0.2, \ldots , 10$. For each of these networks, sample 1000 unique random (s, t) pairs $(s \ne t)$. Then do a decentralized search starting from s as follows. Assuming that the current node is s, pick its neighbour u with smallest h(u, t) (break ties arbitrarily). If $u = t$, the search succeeds. If $h(s, t) > h(u, t)$, set s to u and repeat. If $h(s, t) \le h(u, t)$, the search fails.

For each $\alpha$, pick 1000 pairs of nodes and compute the average path length for the searches that succeeded. Then draw a plot of the average path length as a function of $\alpha$. Also, plot the search success probability as a function of $\alpha$.