Balazs Kozma

Consensus formation on adaptive networks

The structure of a network can significantly influence the properties of the dynamical processes that take place on them. While many studies have been paid to this influence, much less attention has been devoted to the interplay and feedback mechanisms between dynamical processes and network topology on adaptive networks. Adaptive rewiring of links can happen in real life systems such as acquaintance networks, where people are more likely to maintain a social connection if their views and values are similar. In our study, we consider different variants of a model for consensus formation. Our investigations reveal that the adaptation of the network topology fosters cluster formation by enhancing communication between agents of similar opinion, although it also promotes the division of these clusters. The temporal behavior is also strongly affected by adaptivity: while, on static networks, it is influenced by percolation properties, on adaptive networks, both the early and late time evolutions of the system are determined by the rewiring process. The investigation of a variant of the model reveals that the scenarios of transitions between consensus and polarized states are more robust on adaptive networks.

Who's talking first? Consensus or lack thereof in coevolving opinion formation models.

We investigate different opinion formation models on adaptive network topologies. Depending on the dynamical process, rewiring can either (i) lead to the elimination of interactions between agents in different states, and accelerate the convergence to a consensus state or break the network in noninteracting groups or (ii), counterintuitively, favor the existence of diverse interacting groups for exponentially long times. The mean-field analysis allows us to elucidate the mechanisms at play. Strikingly, allowing the interacting agents to bear more than one opinion at the same time drastically changes the models behavior and leads to fast consensus.

Diffusion Processes on Power-Law Small-World Networks

We consider diffusion processes on power-law small-world networks in different dimensions. In one dimension, we find a rich phase diagram, with different transient and recurrent phases, including a critical line with continuously varying exponents. The results were obtained using self-consistent perturbation theory and can also be understood in terms of a scaling theory, which provides a general framework for understanding processes on small-world networks with different distributions of long-range links.

Scaling in small-world resistor networks

We study the effective resistance of small-world resistor networks. Utilizing recent analytic results for the propagator of the Edwards–Wilkinson process on small-world networks, we obtain the asymptotic behavior of the disorder-averaged two-point resistance in the large system-size limit. We find that the small-world structure suppresses large network resistances: both the average resistance and its standard deviation approaches a finite value in the large system-size limit for any non-zero density of random links. We also consider a scenario where the link conductance decays as a power of the length of the random links, l −α . In this case we find that the average effective system resistance diverges for any non-zero value of α.

Fisher Waves and the Velocity of Front Propagation in a Two-Species Invasion Model with Preemptive Competition

We consider an individual-based two-dimensional spatial model with nearest-neighbor preemptive competition to study front propagation between an invader and a resident species. In particular, we investigate the asymptotic front velocity and compare it with mean-field predictions.

Stochastic Growth in a Small World

We considered the Edwards-Wilkinson model on a small-world network. We studied the finite-size behavior of the surface width by performing exact numerical diagonalization for the underlying coupling matrix. We found that the spectrum exhibits a gap or a pseudo-gap, which is responsible for a finite width in the thermodynamic limit for an arbitrarily weak but nonzero magnitude of the random interactions.

Processes on annealed and quenched power-law small-world networks

We considered diffusion driven processes on power-law small-world networks: random walk on randomly folded polymers and surface growth related to synchronization problems. We found a rich phase diagram, with different transient and recurrent phases. The calculations were done in two limiting cases: the annealed case, when the rearrangement of the random links is fast (the configuration of the polymer changes fast) and the quenched case, when the link rearrangement is slow (the polymer configuration is static) compared to the motion of the random walker. In the quenched case, the random links introduced in small-world networks often lead to mean-filed coupling (i.e., the random links can be treated in an annealed fashion) but in some systems mean-field predictions break down, such as for diffusion in one dimension. This break-down can be understood treating the random links perturbatively where the mean field prediction appears as the lowest order term of a naive perturbation expansion. Our results were obtained using self-consisten perturbation theory. Numerical results will also be shown as a confirmation of the theory.