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Register a new userCommunication Bottlenecks in Scale-Free Networks
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We consider the effects of network topology on the optimality of packet routing quantified by $gamma_c$, the rate of packet insertion beyond which congestion and queue growth occurs. The key result of this paper is to show that for any network, there exists an absolute upper bound, expressed in terms of vertex separators, for the scaling of $gamma_c$ with network size $N$, irrespective of the routing algorithm used. We then derive an estimate to this upper bound for scale-free networks, and introduce a novel static routing protocol which is superior to shortest path routing under intense packet insertion rates.
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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.
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.
We give an intuitive though general explanation of the finite-size effect in scale-free networks in terms of the degree distribution of the starting network. This result clarifies the relevance of the starting network in the final degree distribution. We use two different approaches: the deterministic mean-field approximation used by Barabasi and Albert (but taking into account the nodes of the starting network), and the probability distribution of the degree of each node, which considers the stochastic process. Numerical simulations show that the accuracy of the predictions of the mean-field approximation depend on the contribution of the dispersion in the final distribution. The results in terms of the probability distribution of the degree of each node are very accurate when compared to numerical simulations. The analysis of the standard deviation of the degree distribution allows us to assess the influence of the starting core when fitting the model to real data.
Networks with a scale-free degree distribution are widely thought to promote cooperation in various games. Herein, by studying the well-known prisoners dilemma game, we demonstrate that this need not necessarily be true. For the very same degree sequence and degree distribution, we present a variety of possible behaviour. We reassess the perceived importance of hubs in a network towards the maintenance of cooperation. We also reevaluate the dependence of cooperation on network clustering and assortativity.
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.
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