The Truth about Power Laws: Theory and Reality

Using a simple model with link removals as well as link additions, we show that an evolving network is scale free with a degree exponent in the range of (2, 4]. We then establish a relation between the network evolution and a set of non-homogeneous birth-and-death processes, and, with which, we capture the process by which the network connectivity evolves. We develop an effective algorithm to compute the network degree distribution accurately. Comparing analytical and numeric...

In this paper we describe the emergence of scale-free degree distributions from statistical mechanics principles. We define an energy associated to a degree sequence as the logarithm of the number of indistinguishable simple networks it is possible to draw given the degree sequence. Keeping fixed the total number of nodes and links, we show that the energy of scale-free distribution is much higher than the energy associated to the degree sequence of regular random graphs. Thi...

Various important and useful quantities or measures that characterize the topological network structure are usually investigated for a network, then they are averaged over the samples. In this paper, we propose an explicit representation by the beforehand averaged adjacency matrix over samples of growing networks as a new general framework for investigating the characteristic quantities. It is applied to some network models, and shows a good approximation of degree distributi...

Scale-free power law structure describes complex networks derived from a wide range of real world processes. The extensive literature focuses almost exclusively on networks with power law exponent strictly larger than 2, which can be explained by constant vertex growth and preferential attachment. The complementary scale-free behavior in the range between 1 and 2 has been mostly neglected as atypical because there is no known generating mechanism to explain how networks with ...

Approaches from statistical physics are applied to investigate the structure of network models whose growth rules mimic aspects of the evolution of the world-wide web. We first determine the degree distribution of a growing network in which nodes are introduced one at a time and attach to an earlier node of degree k with rate A_ksim k^gamma. Very different behaviors arise for gamma<1, gamma=1, and gamma>1. We also analyze the degree distribution of a heterogeneous network, th...

It was experimentally observed that the majority of real-world networks follow power law degree distribution. The aim of this paper is to study the algorithmic complexity of such "typical" networks. The contribution of this work is twofold. First, we define a deterministic condition for checking whether a graph has a power law degree distribution and experimentally validate it on real-world networks. This definition allows us to derive interesting properties of power law ne...

Zipf's power law is a general empirical regularity found in many natural and social systems. A recently developed theory predicts that Zipf's law corresponds to systems that are growing according to a maximally sustainable path in the presence of random proportional growth, stochastic birth and death processes. We report a detailed empirical analysis of a burgeoning network of social groups, in which all ingredients needed for Zipf's law to apply are verifiable and verified. ...

The probability distribution of number of ties of an individual in a social network follows a scale-free power-law. However, how this distribution arises has not been conclusively demonstrated in direct analyses of people's actions in social networks. Here, we perform a causal inference analysis and find an underlying cause for this phenomenon. Our analysis indicates that heavy-tailed degree distribution is causally determined by similarly skewed distribution of human activit...

Real-world networks are generally claimed to be scale-free, meaning that the degree distributions follow the classical power-law, at least asymptotically. Yet, closer observation shows that the classical power-law distribution is often inadequate to meet the data characteristics due to the existence of a clearly identifiable non-linearity in the entire degree distribution in the log-log scale. The present paper proposes a new variant of the popular heavy-tailed Lomax distribu...

Power-law networks such as the Internet, terrorist cells, species relationships, and cellular metabolic interactions are susceptible to node failures, yet maintaining network connectivity is essential for network functionality. Disconnection of the network leads to fragmentation and, in some cases, collapse of the underlying system. However, the influences of the topology of networks on their ability to withstand node failures are poorly understood. Based on a study of the re...