Communication in Networks with Hierarchical Branching

Abstract

We present a simple model of communication in networks with hierarchical branching. We analyze the behavior of the model from the viewpoint of critical systems under different situations. For certain values of the parameters, a continuous phase transition between a sparse and a congested regime is observed and accurately described by an order parameter and the power spectra. At the critical point the behavior of the model is totally independent of the number of hierarchical levels. Also scaling properties are observed when the size of the system varies. The presence of noise in the communication is shown to break the transition. The analytical results are a useful guide to forecasting the main features of real networks. DOI: 10.1103/PhysRevLett.86.3196 PACS numbers: 89.20.Ff, 05.70.Jk, 64.60.-i Nowadays, many challenging questions have arisen concerning the behavior of complex technological, economical, and social systems [1]. In particular, computer simulations of agents and their interactions (agent-based modeling) have become a widely used tool in our current understanding of their macroscopic behavior [2]. Especially interesting is the study of hierarchical branching in networks because it seems to be the basic structure underlying complex organizational systems. Our interest is focused on the behavior of hierarchical structures formed by elements (or agents) that interact with each other via communication processes. This framework is especially adequate to study, e.g., Internet flow [3][4][5][6][7], traffic networks [8], river networks [9], and even communication flows in organizations [10].In this Letter, we propose and study a very simple model of communication. The model includes only the basic ingredients present in a communication process between two elements: (i) information packets to be transmitted (delivered) and (ii) communication channels to transmit the packets. Despite its simplicity, the model reproduces the main characteristics of the flow of information packets in a network, and is general enough to allow the study of communication processes in many conditions: for example, different capabilities of agents to transmit packets, and/or heterogeneity in the communication channels (miscommunication, exogenous effects, etc.) represented by introducing disorder. We observe three different behaviors depending on the capability of agents to transmit packets. In particular, for a certain capability, we observe a continuous phase transition between a sparse and a congested regime when the number of packets to deliver reaches a critical value. Near the transition point signs of criticality arise, we find large fluctuations, critical slowing down, and power law behavior of power spectrum of the amount of information flowing in the network, in agreement with reported empirical data [5]. We provide a mean-field estimation of the critical point in good agreement with simulation results and we define analytically an order parameter to characterize the behavior of the system.The model is defined in the f...

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“…Each agent has a certain capability that decreases as the number of packets to deliver increases. The transition from a free phase to a congested phase has been studied for different network architectures in [16,18], whereas in [21] the cost of maintaining communication channels was considered. Finally in [22] we have attacked the problem of network optimization for fixed number of links and nodes.…”

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