Statistical Characterizers of Transport in a Communication Network

arXiv:0810.4058v2 [physics.soc-ph] 31 Aug 2009 Statistical Characterizers of Transport in Communication Networks Satyam Mukherjee and Neelima Gupte Department of Physics, Indian Institute of Technology Madras, India. Gautam Mukherjee Bidhan Chandra College, Asansol 713304, Dt. Burdwan, West Bengal, India. We identify the statistical characterizers of congestion and decongestion for message transport in model communication lattices. These turn out to be the travel time distributions, which are Gaussian in the congested phase, and log-normal in the decongested phase. Our results are demon- strated for two dimensional lattices, such the Waxman graph, and for lattices with local clustering and geographic separations, gradient connections, as well as for a 1 −d ring lattice with random assortative connections. The behavior of the distribution identifies the congested and decongested phase correctly for these distinct network topologies and decongestion strategies. The waiting time distributions of the systems also show identical signatures of the congested and decongested phases. PACS numbers: 89.75.Hc Investigations of traffic flows on substrates of various topologies have been a topic of recent research interest. [1]. Congestion effects can occur in real networks like telephone networks, computer networks and the Inter- net due to various factors like capacity, band-width and network topology [2]. These lead to deterioration of the service quality experienced by users due to an increase in network load. Statistical characterizers which can iden- tify the state of the network, whether congested or de- congested, can be of practical utility. In this paper, we identify statistical characterizers which carry the signa- ture of the state of congestion or decongestion of the network. The statistical characterizer which carries the signa- ture of the congested or decongested phase, is identified to be the travel time distribution of the messages. The travel time distribution has been studied earlier in the context of vehicular traffic [3], server traffic [4] and the Internet [5]. Hence the travel time distribution can be re- garded as an useful statistical characterizer of transport. In our model networks, the travel time is defined to be the time required for a message to travel from source to tar- get, including the time spent waiting at congested hubs. This distribution turns out to be normal or Gaussian in the congested phase, and log-normal in the decongested phase. We demonstrate that the travel time distribution is able to identify correctly the congested/decongested state in the case of two dimensional model networks, such as the Waxman topology network, a popular model for In- ternet topology[7], as well as for a network with local clustering[8], and its variants with gradient connections [9]. The same characterizer is able to distinguish be- tween the congested and decongested phases in a net- work with a one dimensional ring geometry. Thus, the travel time distribution is a robust characterizer of the congested/decongested phase. We first consider models based on 2 −d lattices. We note that communication networks based on two- dimensional lattices have been considered earlier in the context of search algorithms [10] and of network traffic with routers and hosts [11, 12] and have been observed to reproduce realistic features of Internet traffic. The first network based on a 2 −d geometry is the Waxman graph [7], which incorporates the distance de- pendence in link formation which is characteristic of real world networks [13] and has been widely used to model the topology of intra-domain networks [14]. We consider the case where the Waxman graphs are generated on a rectangular coordinate grid of side L with the probability P(a, b) of an edge from node a to node b given by P(a, b) = β exp(−d αM ) (1) where the parameters 0 < α, β < 1, d is the Euclidean distance from a to b and M = √ 2 × L is the maximum distance between any two nodes [7]. Large values of β result in graphs with larger link densities and small values of α increase the density of short links as compared to the longer ones. A topology similar to Waxman graphs is generated by selecting randomly a predetermined number Nw of nodes in the 2 −d lattice for generating the edges. Additionally, each node of the lattice has a connection to its nearest neighbors ( See Fig.1(a)). The second network that we study is a model which incorporates local clustering and geographic separations developed in Ref.[8]. As shown in Fig.1(b), this network consists of a 2 −d lattice with nodes and hubs, where the hubs are randomly located on the lattice, and are connected to all nodes inside their given area of influence [15]. A given number of messages Nm are allowed to travel on these lattices between fixed source target pairs by a distance based routing algorithm by which a node which holds a message looks for a hub in the direction of the target which is nearest to itself, and routes

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