Network Growth with Feedback

📝 Abstract

Existing models of network growth typically have one or two parameters or strategies which are fixed for all times. We introduce a general framework where feedback on the current state of a network is used to dynamically alter the values of such parameters. A specific model is analyzed where limited resources are shared amongst arriving nodes, all vying to connect close to the root. We show that tunable feedback leads to growth of larger, more efficient networks. Exact results show that linear scaling of resources with system size yields crossover to a trivial condensed state, which can be considerably delayed with sublinear scaling.

💡 Analysis

Existing models of network growth typically have one or two parameters or strategies which are fixed for all times. We introduce a general framework where feedback on the current state of a network is used to dynamically alter the values of such parameters. A specific model is analyzed where limited resources are shared amongst arriving nodes, all vying to connect close to the root. We show that tunable feedback leads to growth of larger, more efficient networks. Exact results show that linear scaling of resources with system size yields crossover to a trivial condensed state, which can be considerably delayed with sublinear scaling.

📄 Content

Network Growth with Feedback Raissa M. D’Souza1, 2 and Soumen Roy1, 2 1Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA 95616 2Center for Computational Science and Engineering, University of California, Davis, CA 95616 Existing models of network growth typically have one or two parameters or strategies which are fixed for all times. We introduce a general framework where feedback on the current state of a network is used to dynamically alter the values of such parameters. A specific model is analyzed where limited resources are shared amongst arriving nodes, all vying to connect close to the root. We show that tunable feedback leads to growth of larger, more efficient networks. Exact results show that linear scaling of resources with system size yields crossover to a trivial condensed state, which can be considerably delayed with sublinear scaling. PACS numbers: 64.60.aq, 02.70.Hm, 89.75.Fb, 89.70.-a The prevalence and importance of network structures in physical, biological and social systems is becoming widely recognized. Current research on network growth focuses on models which reproduce aspects of real-world networks, in particular the broad range of node degrees typically ob- served [1, 2, 3, 4, 5, 6, 7]. These simple and elegant models have just one or two free parameters, or strategies which are specified initially and remain unaltered even as the network grows to a massive size, starting from a few seed nodes. Yet, the functionality and performance required of a small network may be radically different from that of a large network. Thus, it is natural that the param- eters of the growth strategy should change over time as the network grows. The mechanisms underlying these dis- tinct growth models can be generally classified as growth via either preferential attachment [1, 3, 4], copying [2, 6], or optimization [5, 7]. In preferential attachment models, the extent of the preference (i.e., the connection kernel) could be altered, tuning properties of the resulting degree distribution [3, 8]. In copying models, the probability of successfully copying links could be changed, thus affecting degree distribution. In optimization models, the explicit parameter values of the optimization function could be al- tered, leading to a range of interesting behaviors [5, 7, 9]. In this rapid communication, we introduce a framework where information on the current state of a network pro- vides feedback to the system allowing it to dynamically alter and self-tune the parameter values throughout the growth process. It combines local optimization models of growth [7, 9] with measures of efficient information flow in a network [10]. We show that with feedback, one can grow larger and more efficient network structures in less time. This framework can be applied to many systems exhibit- ing a hierarchical “chain of command” structure. Simple examples are business enterprises, armed forces, etc., with the “CEO” or the commander in chief respectively being the root node of the hierarchy. Such a structure has also recently been found in the organization of genetic regu- latory networks [11]. More generally, hierarchy appears to be a central organizing principle of complex networks, providing insight into structures such as food webs, bio- chemical and social networks [12]. We are interested in growth of hierarchical networks where information flow is essential to the network’s func- tion. Two basic considerations are: (i) ensuring a smooth “flow” of commands or information throughout the struc- ture, and, (ii) addition of new nodes subject to con- straints on resources. More explicitly, only some fraction, 0 < c ≤1, of existing resources can be dedicated to opti- mizing new growth. The remaining portion of the system is involved with performing some task (e.g., information processing, regulation, transport and routing), crucial to the sustenance and function of the organization. We show herein that how the resources allocated for growth scale with system size N directly impacts the resulting net- work structure. Moreover, we show that incorporating feedback leads to flatter hierarchies on which information flows more efficiently, providing a quantitative underpin- ning to previous case studies of individual organizations where this is found in practice [13]. We consider a simple growth model incorporating (i) and (ii). It is a discrete time process starting from a sin- gle root node. Let G(t) denote the network at time t and N(t) the number of nodes. At each time-step, an integer number of new nodes, λ(t) ≥1, arrive and must con- nect to the existing network. In accord with (ii), the frac- tion, 0 < c ≤1, of resources dedicated to optimizing new growth must be shared equally by all λ(t) arriving nodes. Thus, each arriving node sees only k(t) = [c/λ(t)][N(t)]α randomly chosen candidate parent nodes, where α deter- mines the scaling of resources and system size, e.g., α = 1 is linear scaling. It then cho

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