Scalable Peer-to-Peer Indexing with Constant State

We present a distributed indexing scheme for peer to peer networks. Past work on distributed indexing traded off fast search times with non-constant degree topologies or network-unfriendly behavior such as flooding. In contrast, the scheme we present optimizes all three of these performance measures. That is, we provide logarithmic round searches while maintaining connections to a fixed number of peers and avoiding network flooding. In comparison to the well known scheme Chord, we provide competitive constant factors. Finally, we observe that arbitrary linear speedups are possible and discuss both a general brute force approach and specific economical optimizations

Bootstrap Percolation on Complex Networks

We consider bootstrap percolation on uncorrelated complex networks. We obtain the phase diagram for this process with respect to two parameters: $f$, the fraction of vertices initially activated, and $p$, the fraction of undamaged vertices in the graph. We observe two transitions: the giant active component appears continuously at a first threshold. There may also be a second, discontinuous, hybrid transition at a higher threshold. Avalanches of activations increase in size as this second critical point is approached, finally diverging at this threshold. We describe the existence of a special critical point at which this second transition first appears. In networks with degree distributions whose second moment diverges (but whose first moment does not), we find a qualitatively different behavior. In this case the giant active component appears for any $f>0$ and $p>0$, and the discontinuous transition is absent. This means that the giant active component is robust to damage, and also is very easily activated. We also formulate a generalized bootstrap process in which each vertex can have an arbitrary threshold.Comment: 9 pages, 3 figure

Clustering Phase Transitions and Hysteresis: Pitfalls in Constructing Network Ensembles

Ensembles of networks are used as null models in many applications. However, simple null models often show much less clustering than their real-world counterparts. In this paper, we study a model where clustering is enhanced by means of a fugacity term as in the Strauss (or "triangle") model, but where the degree sequence is strictly preserved -- thus maintaining the quenched heterogeneity of nodes found in the original degree sequence. Similar models had been proposed previously in [R. Milo et al., Science 298, 824 (2002)]. We find that our model exhibits phase transitions as the fugacity is changed. For regular graphs (identical degrees for all nodes) with degree k > 2 we find a single first order transition. For all non-regular networks that we studied (including Erdos - Renyi and scale-free networks) we find multiple jumps resembling first order transitions, together with strong hysteresis. The latter transitions are driven by the sudden emergence of "cluster cores": groups of highly interconnected nodes with higher than average degrees. To study these cluster cores visually, we introduce q-clique adjacency plots. We find that these cluster cores constitute distinct communities which emerge spontaneously from the triangle generating process. Finally, we point out that cluster cores produce pitfalls when using the present (and similar) models as null models for strongly clustered networks, due to the very strong hysteresis which effectively leads to broken ergodicity on realistic time scales.Comment: 13 pages, 11 figure