What exactly are the properties of scale-free and other networks?

The concept of scale-free networks has been widely applied across natural and physical sciences. Many claims are made about the properties of these networks, even though the concept of scale-free is often vaguely defined. We present tools and procedures to analyse the statistical properties of networks defined by arbitrary degree distributions and other constraints. Doing so reveals the highly likely properties, and some unrecognised richness, of scale-free networks, and casts doubt on some previously claimed properties being due to a scale-free characteristic.Comment: Preprint - submitted, 6 pages, 3 figure

Diophantine networks

We introduce a new class of deterministic networks by associating networks with Diophantine equations, thus relating network topology to algebraic properties. The network is formed by rep- resenting integers as vertices and by drawing cliques between M vertices every time that M dis- tinct integers satisfy the equation. We analyse the network generated by the Pythagorean equation x2+y2 = z2 showing that its degree distribution is well approximated by a power law with exponen- tial cut-o®. We also show that the properties of this network di®er considerably from the features of scale-free networks generated through preferential attachment. Remarkably we also recover a power law for the clustering coe±cient. We then study the network associated with the equation x2 + y2 = z showing that the degree distribution is consistent with a power-law for several decades of values of k and that, after having reached a minimum, the distribution begins rising again. The power law exponent, in this case, is given by ° » 4:5 We then analyse clustering and ageing and compare our results to the ones obtained in the Pythagorean case

Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)

Although the ``scale-free'' literature is large and growing, it gives neither a precise definition of scale-free graphs nor rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and verifiably false claims. In this paper, we propose a new, mathematically precise, and structural definition of the extent to which a graph is scale-free, and prove a series of results that recover many of the claimed properties while suggesting the potential for a rich and interesting theory. With this definition, scale-free (or its opposite, scale-rich) is closely related to other structural graph properties such as various notions of self-similarity (or respectively, self-dissimilarity). Scale-free graphs are also shown to be the likely outcome of random construction processes, consistent with the heuristic definitions implicit in existing random graph approaches. Our approach clarifies much of the confusion surrounding the sensational qualitative claims in the scale-free literature, and offers rigorous and quantitative alternatives.Comment: 44 pages, 16 figures. The primary version is to appear in Internet Mathematics (2005

Accelerated growth in outgoing links in evolving networks: deterministic vs. stochastic picture

In several real-world networks like the Internet, WWW etc., the number of links grow in time in a non-linear fashion. We consider growing networks in which the number of outgoing links is a non-linear function of time but new links between older nodes are forbidden. The attachments are made using a preferential attachment scheme. In the deterministic picture, the number of outgoing links $m(t)$ at any time $t$ is taken as $N(t)^\theta$ where $N(t)$ is the number of nodes present at that time. The continuum theory predicts a power law decay of the degree distribution: $P(k) \propto k^{-1-\frac{2} {1-\theta}}$, while the degree of the node introduced at time $t_i$ is given by $k(t_i,t) = t_i^{\theta}[ \frac {t}{t_i}]^{\frac {1+\theta}{2}}$ when the network is evolved till time $t$. Numerical results show a growth in the degree distribution for small $k$ values at any non-zero $\theta$. In the stochastic picture, $m(t)$ is a random variable. As long as $$is independent of time, the network shows a behaviour similar to the Barab\'asi-Albert (BA) model. Different results are obtained when$$ is time-dependent, e.g., when $m(t)$ follows a distribution $P(m) \propto m^{-\lambda}$. The behaviour of $P(k)$ changes significantly as $\lambda$ is varied: for $\lambda > 3$, the network has a scale-free distribution belonging to the BA class as predicted by the mean field theory, for smaller values of $\lambda$ it shows different behaviour. Characteristic features of the clustering coefficients in both models have also been discussed.Comment: Revised text, references added, to be published in PR