Color-Swap Models for Non-growing Scale-free Networks

Introduction Non-growing scale-free networks represent an interesting class of complex networks. Compared to growing networks, which have been studied in many different scenarios with great success [1, 2, 3], they aim at modeling situations where a complex topology different from purely random is emerging but no growth process is present. There are many examples, mainly in biology (see [4] and the references therein), where instead of growth one can observe a self-organizing stochastic process which moves some network connections to different nodes, and the number of nodes and edges stays statistically bounded within some interval. Although various aspects of the dynamic behavior of non-growing (equilibrium) networks were well studied [5] and several stochastic process candidates have been proposed, they provide the scale-free property only under special conditions. The discrepancies between real networks and most model designs can be summarized as follows: (i) Multigraph: the processes in [5, 4] allow multiple edges and/or self-loops between the nodes, whereas in most real networks this phenomena does not exist. (ii) Connection (edge) growth: the process defined in [6] will reach a scale-free state only in certain range of the node degree distribution and also the number of edges grows considerably before it stabilizes. (iii) Condensation: the networks go through a scale-free state but later condensate to a state where a small fraction of nodes posses all the edges [5]. (iv) Structural constrains: the stochastic process defined in [4] reveals very interesting aspects, however the scale-free character is a result of structural constrains and special initial conditions. We present an edge-colored graph concept aimed at developing processes, generating non-growing scale-free networks, which would lack the disadvantages of the processes known so far; in particular, we propose a processes ”Color-Swap” which avoids the above mentioned problems (i), (ii), (iv) and improves on the problem (iii). A more detailed discussion and extensive experimental results we made available in [7].