Project-Gang:Small world networks structure

Inria / Raweb 2009
Presentation of the Project Gang

Section: New Results

Small world networks structure

Participants : Pierre Fraigniaud, Cyril Gavoille [ CNRS LABRI, University of Bordeaux, France ] , George Giakkoupis [ CNRS LIAFA, University of Paris Diderot, France ] , Adrian Kosowski [ CNRS LABRI, University of Bordeaux, France ] , Zvi Lotker [ University of Tel Aviv, Israel ] .

Small world routing

In [12] , we analyze decentralized routing in small-world networks that combine a wide variation in node degrees with a notion of spatial embedding. Specifically, we consider a variation of Kleinberg's augmented-lattice model (STOC 2000), where the number of long-range contacts for each node is drawn from a power-law distribution. This model is motivated by the experimental observation that many “real-world” networks have power-law degrees. In such networks, the exponent $\alpha$ of the power law is typically between 2 and 3. We prove that, in our model, for this range of values, 2< $\alpha$ <3 , the expected number of steps of greedy routing from any source to any target is $Im1 ${O(log^{\#945 -1}n)}$$ steps. This bound is tight in a strong sense. Indeed, we prove that the expected number of steps of greedy routing for a uniformly-random pair of source–target nodes is $Im2 ${\#937 (log^{\#945 -1}n)}$$ steps. We also show that, for $\alpha$ <2 or $\alpha$ $\ge$ 3 , greedy routing performs in $\upper_theta$ (log²n) expected steps, and for $\alpha$ = 2 , $Im3 ${\#920 (log^{1+\#1013 }n)}$$ expected steps are required, where 1/3 $\le$ $\epsilon$ $\le$ 1/2 . To the best of our knowledge, these results are the first to formally quantifying the effect of the power-law degree distribution on the navigability of small worlds. Moreover, they show that this effect is significant. In particular, when $\alpha$ approaches 2 from above, the expected number of steps of greedy routing in the augmented lattice with power-law degrees approaches the square-root of the expected number of steps of greedy routing in the augmented lattice with fixed degrees , although both networks have the same average degree .

Rumor spreading

Gossip protocols are communication protocols in which, periodically, every node of a network exchanges information with some other node chosen according to some (randomized) strategy. These protocols have recently found various types of applications for the management of distributed systems. Spatial gossip protocols are gossip protocols that use the underlying spatial structure of the network, in particular for achieving the "closest-first" property. This latter property states that the closer a node is to the source of a message the more likely it is to receive this message within a prescribed amount of time. Spatial gossip protocols find many applications, including the propagation of alarms in sensor networks, and the location of resources in P2P networks. In [6] , we design a sub-linear spatial gossip protocol for arbitrary graphs metric. More specifically, we prove that, for any graph metric with maximum degree $\upper_delta$ , for any source s and any ball centered at s with size b , new information is spread from s to all nodes in the ball within $Im4 ${O({(\sqrt {blogb}~loglogb+\#916 )}logb)}$$ rounds, with high probability. Moreover, when applied to general metrics with uniform density, the same protocol achieves a propagation time of O(log²bloglogb) rounds.

An edge-Markovian process with birth-rate p and death-rate q generates sequences of graphs (G₀, G₁, G₂, ...) with the same node set [n] such that G_t is obtained from G_t-1 as follows: if $Im5 ${e\#8713 E(G_{t-1})}$$ then e $\in$ E(G_t) with probability p , and if e $\in$ E(G_t-1) then $Im6 ${e\#8713 E(G_t)}$$ with probability q . Clementi et al. (PODC 2008) analyzed thoroughly information dissemination in such dynamic graphs, by establishing bounds on their flooding time — flooding is the basic mechanism in which every node becoming aware of an information at step t forwards this information to all its neighbors at all forthcoming steps t^'>t . In [5] , we establish tight bounds on the complexity of flooding for all possible birth rates and death rates, completing the previous results by Clementi et al. Moreover, we note that despite its many advantages in term of simplicity and robustness, flooding suffers from its high bandwidth consumption. Hence we al! so show that flooding in dynamic graphs can be implemented in a more parsimonious manner, so that to save bandwidth, yet preserving efficiency in term of simplicity and completion time. For a positive integer k , we say that the flooding protocol is k -active if each node forwards an information only during the k time steps immediately following the step at which the node receives that information for the first time. We define the reachability threshold for the flooding protocol as the smallest integer k such that, for any source s $\in$ [n] , the k -active flooding protocol from s completes (i.e., reaches all nodes), and we establish tight bounds for this parameter. We show that, for a large spectrum of parameters p and q , the reachability threshold is by several orders of magnitude smaller than the flooding time. In particular, we show that it is even constant whenever the ratio p/(p + q) exceeds logn/n . Moreover, we also show that being active for a number of steps equal to the reachability threshold (up to a multiplicative constant) allows the flooding protocol to complete in optimal time, i.e., in asymptotically the sa! me number of steps as when being perpetually active. These results demonstrate that flooding can be implemented in a practical and efficient manner in dynamic graphs. The main ingredient in the proofs of our results is a reduction lemma enabling to overcome the time dependencies in edge-Markovian dynamic graphs.

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