Congestion phenomena on complex networks

arXiv:0808.0584v1 [physics.soc-ph] 5 Aug 2008 Congestion phenomena on complex networks Daniele De Martinoa, Luca Dall’Astab, Ginestra Bianconib, and Matteo Marsilib aInternational School for Advanced Studies SISSA and INFN, via Beirut 2-4,34014 Trieste, Italy, bThe Abdus Salam ICTP, Strada Costiera 11, 34014, Trieste, Italy (Dated: October 29, 2018) We define a minimal model of traffic flows in complex networks containing the most relevant features of real routing schemes, i.e. a trade–offstrategy between topological-based and traffic-based routing. The resulting collective behavior, obtained analytically for the ensemble of uncorrelated networks, is physically very rich and reproduces results recently observed in traffic simulations on scale-free networks. We find that traffic control is useless in homogeneous graphs but may improves global performance in inhomogeneous networks, enlarging the free-flow region in parameter space. Traffic control also introduces non-linear effects and, beyond a critical strength, may trigger the appearance of a congested phase in a discontinuous manner. PACS numbers: 02.50.Ey, 68.35.Rh, 89.20.Ff, 89.75.Fb The first identified Internet’s congestion collapse dates back to October 1986, when data troughput from Lawrence Berkeley National Laboratory to the Univer- sity of California in Berkeley dropped from 32 Kbps to 40 bps. After that initial event, traffic congestion con- tinued to threaten Internet’s practitioners, even after the implementation of congestion control algorithms able to recover the system in case of traffic overloads [1]. Com- puter scientists have also elaborated several schemes of congestion avoidance, that should prevent congestion to occur by keeping the system far from high levels of traffic [2]. Congestion avoidance and control are performed by continuously updating the dynamics of end-to-end flows in response to the variation of the load level in the net- work. Their functioning depend on the average round- trip-time (RTT) of the Acknowledgement signals (ACKs) used to exchange information between routers. For this reason, the observation of heterogeneous patterns in RTT time series has been often proposed as evidence of peri- ods of congestion [3]. Apart from these indirect measures, congestion events are difficult to monitor and study, so that a clear phenomenological picture is still missing. In spite of the wide interest in developing optimal routing algorithms, much less attention is devoted to explore the- oretically the dynamical mechanisms responsible of con- gestion. It is possible that a better comprehension of these mechanisms could help in understanding experi- mental data, in predicting congestion events and design- ing better routing protocols. The topological and dynamical properties of dis- tributed information systems, such as the Internet [4], pose theoretical challenges of a similar nature of those addressed in statistical physics. Therefore, understand- ing network congestion phenomena has become a subject of intense research in this field [5], in particular after the works by Takayasu and collaborators [6], in which the evidence of a phase transition from a free-flow regime to a congested phase depending on the load level was reported. In a recent work, Echenique et al. [7] have shown using numerical simulations that the nature of the congestion transition depends on the type of routing rules. They have adopted a routing scheme that could be considered a first approximation of realistic transmis- sion control protocol (TCP) routing: packets follow the shortest path between their source and destination, but small detours are admitted in order to avoid congested nodes. In case of purely topological routing (e.g. along the shortest paths) they found that the congested phase appears continuously, whereas the transition is discon- tinuous if some traffic-aware scheme is considered. The effect of routing rules on network performance has also been addressed in [8]. In this Letter, we put forward a minimal model of traf- fic that preserves all interesting features previously ob- served in simulations but is simple enough to be studied analytically. Both continuous and discontinuous phase transitions observed in [7] are reproduced, and their re- lation to microscopic packets dynamics is clarified. Let us consider a network of N nodes and let v(i) de- note the set of neighbors of node i. We describe traffic dy- namics as a continuous time stochastic process, in which packets are generated at each node i with a rate pi. Each node is endowed with a first-in first-out (FIFO) queue in which packets are stored waiting to be processed. Let ni be the number of packets in the queue of node i. If ni > 0, node i attempts to transmit packets at a rate ri, which represents bandwidth, to one of the neighbors j ∈v(i). We assume the following probabilistic routing protocol. First, the node j is chosen at random among the neighbors v(i) of i. Second, the fate of the packet being transmitted depe

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