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Register a new userTrading interactions for topology in scale-free networks
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Scale-free networks with topology-dependent interactions are studied. It is shown that the universality classes of critical behavior, which conventionally depend only on topology, can also be explored by tuning the interactions. A mapping, $gamma = (gamma - mu)/(1-mu)$, describes how a shift of the standard exponent $gamma$ of the degree distribution $P(q)$ can absorb the effect of degree-dependent pair interactions $J_{ij} propto (q_iq_j)^{-mu}$. Replica technique, cavity method and Monte Carlo simulation support the physical picture suggested by Landau theory for the critical exponents and by the Bethe-Peierls approximation for the critical temperature. The equivalence of topology and interaction holds for equilibrium and non-equilibrium systems, and is illustrated with interdisciplinary applications.
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We investigate analytically and numerically the critical line in undirected random Boolean networks with arbitrary degree distributions, including scale-free topology of connections $P(k)sim k^{-gamma}$. We show that in infinite scale-free networks the transition between frozen and chaotic phase occurs for $3<gamma < 3.5$. The observation is interesting for two reasons. First, since most of critical phenomena in scale-free networks reveal their non-trivial character for $gamma<3$, the position of the critical line in Kauffman model seems to be an important exception from the rule. Second, since gene regulatory networks are characterized by scale-free topology with $gamma<3$, the observation that in finite-size networks the mentioned transition moves towards smaller $gamma$ is an argument for Kauffman model as a good starting point to model real systems. We also explain that the unattainability of the critical line in numerical simulations of classical random graphs is due to percolation phenomena.
A complete understanding of real networks requires us to understand the consequences of the uneven interaction strengths between a systems components. Here we use the minimum spanning tree (MST) to explore the effect of weight assignment and network topology on the organization of complex networks. We find that if the weight distribution is correlated with the network topology, the MSTs are either scale-free or exponential. In contrast, when the correlations between weights and topology are absent, the MST degree distribution is a power-law and independent of the weight distribution. These results offer a systematic way to explore the impact of weak links on the structure and integrity of complex networks.
The studies based on $A+A rightarrow emptyset$ and $A+Brightarrow emptyset$ diffusion-annihilation processes have so far been studied on weighted uncorrelated scale-free networks and fractal scale-free networks. In the previous reports, it is widely accepted that the segregation of particles in the processes is introduced by the fractal structure. In this paper, we study these processes on a family of weighted scale-free networks with identical degree sequence. We find that the depletion zone and segregation are essentially caused by the disassortative mixing, namely, high-degree nodes tend to connect with low-degree nodes. Their influence on the processes is governed by the correlation between the weight and degree. Our finding suggests both the weight and degree distribution dont suffice to characterize the diffusion-annihilation processes on weighted scale-free networks.
Networks with fat-tailed degree distributions are omnipresent across many scientific disciplines. Such systems are characterized by so-called hubs, specific nodes with high numbers of connections to other nodes. By this property, they are expected to be key to the collective network behavior, e.g., in Ising models on such complex topologies. This applies in particular to the transition into a globally ordered network state, which thereby proceeds in a hierarchical fashion, and with a non-trivial local structure. Standard mean-field theory of Ising models on scale-free networks underrates the presence of the hubs, while nevertheless providing remarkably reliable estimates for the onset of global order. Here, we expose that a spurious self-feedback effect, inherent to mean-field theory, underlies this apparent paradox. More specifically, we demonstrate that higher order interaction effects precisely cancel the self-feedback on the hubs, and we expose the importance of hubs for the distinct onset of local versus global order in the network. Due to the generic nature of our arguments, we expect the mechanism that we uncover for the archetypal case of Ising networks of the Barabasi-Albert type to be also relevant for other systems with a strongly hierarchical underlying network structure.
In the reaction-diffusion process $A+B to varnothing$ on random scale-free (SF) networks with the degree exponent $gamma$, the particle density decays with time in a power law with an exponent $alpha$ when initial densities of each species are the same. The exponent $alpha$ is $alpha > 1$ for $2 < gamma < 3$ and $alpha=1$ for $gamma ge 3$. Here, we examine the reaction process on fractal SF networks, finding that $alpha < 1$ even for $2 < gamma < 3$. This slowly decaying behavior originates from the segregation effect: Fractal SF networks contain local hubs, which are repulsive to each other. Those hubs attract particles and accelerate the reaction, and then create domains containing the same species of particles. It follows that the reaction takes place at the non-hub boundaries between those domains and thus the particle density decays slowly. Since many real SF networks are fractal, the segregation effect has to be taken into account in the reaction kinetics among heterogeneous particles.
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