Non-autonomous dynamic network model involving growth and decay

Quantifying growth model in large scale complex networks has been analyzed using various distributions. Networks which are of ample interest in current scenario tend to follow the exponential distribution in their degree characteristics. This preferential attachment model only reflected the growth mechanism of the system. Birth and death are two indistinguishable phenomena of nature, thus losses of links and nodes in a network are bound to affect the distribution. Whether it be social network or biological reactions or body growth or it be popularity of actor or popularity of music genre all goes through three phases of growth, saturation and finally tends to decay. In this paper, we study this effect and come up with a more general “Gamma Distribution” model for these systems. A truly astonishing characteristic of these systems is their non-autonomy. The degree distribution parameter shows time dependency and follows an `inverted bathtub' curve. Initially when the birth rate exceeds the death rate of links, the number of links increases, which lowers the number of hubs to number of links ratio. Thus, the distribution parameter (γ for scale-free) increases. When both rates are same, flat portion of bathtub curve is obtained. Finally, when the death of links dominates, three possibilities are there: When the number of hubs remains almost unchanged, decreasing portion of `inverted bathtub curve' is observed. When hubs also start decaying, constant parameter value persists or creates spike in the curve on abrupt hub deaths. These sudden spikes can occur anytime during the life period of a network.