arXiv subjects

Mark Dickison

Publications and source records attributed to Mark Dickison.

4 recordsLinked to original sources

arXiv2012

Preferential Attachment in the Interaction between Dynamically Generated Interdependent Networks

We generalize the scale-free network model of Barabàsi and Albert [Science 286, 509 (1999)] by proposing a class of stochastic models for scale-free interdependent networks in which interdependent nodes are not randomly connected but rather are connected via preferential attachment (PA). Each network grows through the continuous addition of new nodes, and new nodes in each network attach preferentially and simultaneously to (a) well-connected nodes within the same network and (b) well-connected nodes in other networks. We present analytic solutions for the power-law exponents as functions of the number of links both between networks and within networks. We show that a cross-clustering coefficient vs. size of network $N$ follows a power law. We illustrate the models using selected examples from the Internet and finance.

physics.soc-ph↗

arXiv2009

Dynamic networks and directed percolation

We introduce a model for dynamic networks, where the links or the strengths of the links change over time. We solve the model by mapping dynamic networks to the problem of directed percolation, where the direction corresponds to the evolution of the network in time. We show that the dynamic network undergoes a percolation phase transition at a critical concentration $p_c$, which decreases with the rate $r$ at which the network links are changed. The behavior near criticality is universal and independent of $r$. We find fundamental network laws are changed. (i) For Erdős-Rényi networks we find that the size of the giant component at criticality scales with the network size $N$ for all values of $r$, rather than as $N^{2/3}$. (ii) In the presence of a broad distribution of disorder, the optimal path length between two nodes in a dynamic network scales as $N^{1/2}$, compared to $N^{1/3}$ in a static network.

cond-mat.dis-nn↗

arXiv2005

Critical behavior and Griffiths effects in the disordered contact process

We study the nonequilibrium phase transition in the one-dimensional contact process with quenched spatial disorder by means of large-scale Monte-Carlo simulations for times up to $10^9$ and system sizes up to $10^7$ sites. In agreement with recent predictions of an infinite-randomness fixed point, our simulations demonstrate activated (exponential) dynamical scaling at the critical point. The critical behavior turns out to be universal, even for weak disorder. However, the approach to this asymptotic behavior is extremely slow, with crossover times of the order of $10^4$ or larger. In the Griffiths region between the clean and the dirty critical points, we find power-law dynamical behavior with continuously varying exponents. We discuss the generality of our findings and relate them to a broader theory of rare region effects at phase transitions with quenched disorder.

cond-mat.stat-mech↗

arXiv2005

Monte-Carlo simulations of the smeared phase transition in a contact process with extended defects

We study the nonequilibrium phase transition in a contact process with extended quenched defects by means of Monte-Carlo simulations. We find that the spatial disorder correlations dramatically increase the effects of the impurities. As a result, the sharp phase transition is completely destroyed by smearing. This is caused by effects similar to but stronger than the usual Griffiths phenomena, viz., rare strongly coupled spatial regions can undergo the phase transition independently from the bulk system. We determine both the stationary density in the vicinity of the smeared transition and its time evolution, and we compare the simulation results to a recent theory based on extremal statistics.

cond-mat.stat-mech↗