Queue-length synchronization in a communication networks

arXiv:0809.0417v3 [physics.soc-ph] 20 May 2009 Queue Length Synchronization in a Communication Network Satyam Mukherjee∗and Neelima Gupte† Department of Physics, Indian Institute of Technology Madras, Chennai - 600036, India. (Dated: November 2, 2018) We study synchronization in the context of network traffic on a 2−d communication network with local clustering and geographic separations. The network consists of nodes and randomly distributed hubs where the top five hubs ranked according to their coefficient of betweenness centrality (CBC) are connected by random assortative and gradient mechanisms. For multiple message traffic, messages can trap at the high CBC hubs, and congestion can build up on the network with long queues at the congested hubs. The queue lengths are seen to synchronize in the congested phase. Both complete and phase synchronization is seen, between pairs of hubs. In the decongested phase, the pairs start clearing, and synchronization is lost. A cascading master-slave relation is seen between the hubs, with the slower hubs (which are slow to decongest) driving the faster ones. These are usually the hubs of high CBC. Similar results are seen for traffic of constant density. Total synchronization between the hubs of high CBC is also seen in the congested regime. Similar behavior is seen for traffic on a network constructed using the Waxman random topology generator. We also demonstrate the existence of phase synchronization in real Internet traffic data. PACS numbers: 89.75.Hc I. INTRODUCTION The phenomenon of synchronization has been studied in contexts ranging from the synchronization of clocks and the flashing of fire-flies [1] to synchronization in os- cillator networks [2] and in complex networks [3]. Syn- chronized states have been seen in the context of traf- fic flows as well [4], and investigations of traffic flow on substrates of various geometries have been the focus of recent research interest [5, 6, 7, 8]. The synchronization of processes at the nodes, or hubs, of complex networks can have serious consequences for the performance of the network [9]. In the case of communication networks, the performance of the networks is assessed in terms of their efficiency at packet delivery. Such networks can show a congestion-decongestion transition [10]. We note that an intimate connection between congestion and synchro- nization effects has been seen in the case of real networks [11, 12]. The aim of this paper is to study the interplay of con- gestion and synchronization effects on each other, and examine their effect on the efficiency of the network for packet delivery in the context of two model networks based on two dimensional grids. The first network con- sists of nodes and hubs, with the hubs being connected by random assortative or gradient connections[13]. In the case of the second network, in addition to nearest neigh- bour connections between nodes, the nodes are connected probabilistically to other nodes, with the probability of a connection between nodes being dependent on the Eu- clidean distance between them[14]. Such networks are called Waxman networks and are popular models of in- ∗Electronic address: mukherjee@physics.iitm.ac.in †Electronic address: gupte@physics.iitm.ac.in ternet topology[15]. Synchronisation effects are observed in the congested phase of both these model networks. In addition to these two networks, we also discuss synchro- nisation effects seen in actual internet data. We first study synchronization behavior in a two di- mensional communication network of nodes and hubs. Such networks have been considered earlier in the con- text of search algorithms [16] and of network traffic with routers and hosts [17, 18, 19]. Despite the regular 2 −d geometry such models have shown log-normal distribu- tion in latency times as seen in Internet dynamics [20]. The lattice consists of two types of nodes, the regular or ordinary nodes, which are connected to each of their nearest neighbors, and the hubs, which are connected to all the nodes in a given area of influence, and are randomly distributed in the lattice. Thus, the network represents a model with local clustering and geographi- cal separations [21, 22]. Congestion effects are seen on this network when a large number of messages travel be- tween multiple sources and targets due to various factors like capacity, band-width and network topology [23]. De- congestion strategies, which involve the manipulation of factors like capacity and connectivity have been set up for these networks. Effective connectivity strategies have fo- cused on setting up random assortative[24], or gradient connections[25] between hubs of high betweenness cen- trality. We introduce the ideas of phase synchronization and complete synchronization in the context of the queue lengths at the hubs. The queue at a given hub is de- fined to be the number of messages which have the hub as a temporary target. During multiple message transfer, when many messages run simultaneously

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