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Optimization and scale-freeness for complex networks

### Abstract

Complex networks are mapped to a model of boxes and balls where the balls are distinguishable. It is shown that the scale-free size distribution of boxes maximizes the information associated with the boxes *provided* configurations including boxes containing a finite fraction of the total amount of balls are excluded. It is conjectured that for a connected network with only links between different nodes, the nodes with a finite fraction of links are effectively suppressed. It is hence suggested that for such networks the scale-free node-size distribution maximizes the information encoded on the nodes. The noise associated with the size distributions is also obtained from a maximum entropy principle. Finally, explicit predictions from our least bias approach are found to be borne out by metabolic networks.

© 2007 American Institute of Physics

Received 19 December 2006
Accepted 05 March 2007
Published online 28 June 2007

Lead Paragraph:
Networks galore! Representations of real complex systems in terms of networks range over all science from social science, economics, and the internet, to physics, chemistry, biology, and much more. Basically, a network is a representation of who or what is connected to or influences whom or what. It takes the form of some irregular cobweb where the parts (the who or what) are connected by links. The parts are called nodes, so the representation is in terms of nodes and links (see Fig. 1). One feature of a network is the number of links that are attached to a node: a network can be characterized by ; i.e., the number of nodes with links attached to them. In many real networks, one finds that this distribution over sizes is very broad and power law like; i.e., . Why is this and what does it imply? This is still to a large extent an open question. Here we address this question using the maximum entropy principle. The predictive value of this principle is greatest when it fails! For example, when it was found that the measured blackbody radiation did have a smaller entropy than the one predicted from Maxwell's equations and classical statistical mechanics, the maximum entropy principle (had it been known at the time) would immediately tell you that a physical constraint was lacking in the theory. This, alas, turned out to be Planck's constant. We use the same reasoning here: we first find that the maximum entropy for an unconstrained network does have a larger entropy than the broad distribution found in real systems. So there is a missing constraint! We argue that this missing constraint is the advantage (in many cases) of maximizing the information encoded on the nodes. Thus, we are suggesting that the real advantage is not to maximize the total number of possibilities but rather to maximize the information encoded on the nodes. Our “least bias” approach gives explicit predictions for real networks, which can be tested. We demonstrate that metabolic networks (a network representation of the metabolism in a cell) are likely to be a maximum informationnetwork.

Acknowledgments:
This work was supported by the Swedish Research Council through Contract No. 50412501.

Article outline:

I. INTRODUCTION
II. RANDOMNESS AND STATES
III. BALLS IN BOXES
IV. BOX INFORMATION AND SCALE-FREENESS
V. SCALE-FREENESS AND NOISE
VI. CONSEQUENCES FOR DIRECTED NETWORKS
VII. CONCLUDING REMARKS

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2007-06-28

2016-10-21

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