Section: Scientific Foundations
Recent years have brought tremendous progress along the peer-to-peer paradigm allowing large scale decentralized application to be practically deployed. The main achievement of this trend is certainly efficient content distribution through the BitTorrent protocol  . The power of peer-to-peer content distribution is to rely on the upload capacity of the node interested in receiving the content. This allows to scale to very large number of participants. The main breakthrough of BitTorrent resides in its “tit for tat” strategy inspired from game theory to give incentive to cooperate. For that purpose, a peer preferentially uploads preferentially offering best reciprocity. This kind of preferences induces an interesting graph structure with ordered neighborhoods. Understanding the dynamic behavior of such affinity graphs is an important for stabilizing the performance of such protocols.
A second major achievement of the peer-to-peer paradigm concerns indexing with distributed hashtables  ,  ,  . The idea behind these proposals is to organize the peers into a structure close to well known graphs with low diameter such as hight dimension torus, hypercube or de Bruijn graph. Efficient routing to the node storing a given key is then guaranteed. This academic work has lead to practical basic indexing facilities by introducing redundancy in the structure  . This is typically the kind of approach we want to promote: from known efficient theoretical solutions to practical working protocols. We have contributed to this trend by introducing de Bruijn based solutions  ,  with redundancy in the contact graph to resist node churn.
Small World Phenomenon
Popularized emerging properties include degree distributions observed to be power law in many networks or clustering coefficient observed to be high in social networks or low average distances. This last point gave the denomination of “small worlds” for this type of networks. Some work  ,  try to give models that give raise to such statistical properties. In that line, numerous results such as  try to derive efficient algorithms based only on these statistical properties. This particular approach tends to concentrate load on nodes with high degrees and may not be suited for applications where nodes have similar capacities. Other interesting work  try to explain this statistical observation forms an inherent optimization problem operating when constructing the network.
On the other hand, in its seminal paper  , Kleinberg focuses on the algorithmic aspects of such social networks and shows how adding random links to a torus can produce efficient greedy routing. This result has been extend to more general classes of graphs  ,  such as bounded growth metric graphs  . However, this augmentation process is not always feasible  . Such theoretical work is particularly interesting for overlay networks where this augmenting process simply consists in opening additional connections.
Bounded growth and doubling metrics generalize Euclidean metrics. A metric has bounded growth if the size of any ball increases by a factor not larger than 2d when its radius is doubled  . A metric is doubling if any set of diameter D can be covered with at most 2d sets of diameter D/2  . In both cases, the smallest acceptable value of d is called the dimension of the metric. The metric of any d dimensional Euclidean space has bounded growth dimension O(d) . Any bounded growth metric of dimension d has doubling dimension O(d) . The doubling metric is the most general and has the additional property of being inherited by subspaces: the metric induced by a doubling metric on a subset of nodes is also doubling. For example, sampling nodes in an Euclidean space always results in a doubling metric.
As networks are embedded in our usual three dimensional space, it is legitimate to think than some network metrics may be modeled through doubling metrics. Recent results thus investigate network problems for the restricted classes of graphs with bounded growth or doubling metric  ,  ,  ,  . However, the doubling nature of large networks such as the Internet has still not be tested.
Bounded Width Classes of Graphs
Many graph parameters such as treewidth, branchwidth, rankwidth, cutwidth, cliquewidth ...have been introduced in recent years  ,  to measure the structure of a given graph. These parameters are of course NP-complete to compute, but when it can be proved that for a given class of graphs the parameter is bounded by a constant then it can be proved using graph grammars (see Courcelle's fundamental work) that most of the optimization problems on this class are polynomially tractable, and sometimes we know the existence of a linear algorithm (but the hidden constant can be very high !)
The most famous parameter, namely the treewidth captures the distance of the graph to a tree, and therefore when the treewidth is small a dynamic programming approach can be used  .
Despite some promising results, applications of these notions has still to be done for networks, in a practical perspective.