arXiv subjects

Markus Brede

Publications and source records attributed to Markus Brede.

29 records · Page 2Linked to original sources

arXiv2014

A k-deformed Model of Growing Complex Networks with Fitness

The Barabási-Bianconi (BB) fitness model can be solved by a mapping between the original network growth model to an idealized bosonic gas. The well-known transition to Bose-Einstein condensation in the latter then corresponds to the emergence of "super-hubs" in the network model. Motivated by the preservation of the scale-free property, thermodynamic stability and self-duality, we generalize the original extensive mapping of the BB fitness model by using the nonextensive Kaniadakis k-distribution. Through numerical simulation and mean-field calculations we show that deviations from extensivity do not compromise qualitative features of the phase transition. Analysis of the critical temperature yields a monotonically decreasing dependence on the nonextensive parameter k.

cond-mat.stat-mech↗

arXiv2011

Growth and Optimality in Network Evolution

In this paper we investigate networks whose evolution is governed by the interaction of a random assembly process and an optimization process. In the first process, new nodes are added one at a time and form connections to randomly selected old nodes. In between node additions, the network is rewired to minimize its pathlength. For timescales, at which neither the assembly nor the optimization processes are dominant, we find a rich variety of complex networks with power law tails in the degree distributions. These networks also exhibit non-trivial clustering, a hierarchical organization and interesting degree mixing patterns.

cond-mat.dis-nn↗

arXiv2010

The synchronization transition in correlated oscillator populations

The synchronization transition of correlated ensembles of coupled Kuramoto oscillators on sparse random networks is investigated. Extensive numerical simulations show that correlations between the native frequencies of adjacent oscillators on the network systematically shift the critical point as well as the critical exponents characterizing the transition. Negative correlations imply an onset of synchronization for smaller coupling, whereas positive correlations shift the critical coupling towards larger interaction strengths. For negatively correlated oscillators the transition still exhibits critical behaviour similar to the all-to-all coupled Kuramoto system, while positive correlations change the universality class of the transition depending on the correlation strength. Crucially, the paper demonstrates that the synchronization behaviour is not only determined by the coupling architecture, but is also strongly influenced by the oscillator placement on the coupling network.

cond-mat.stat-mech↗

arXiv2010

Coordinated and Uncoordinated Optimization of Networks

In this paper we consider spatial networks that realize a balance between an infrastructure cost (the cost of wire needed to connect the network in space) and communication efficiency, measured by average shortest pathlength. A global optimization procedure yields network topologies in which this balance is optimized. These are compared with network topologies generated by a competitive process in which each node strives to optimize its own cost-communication balance. Three phases are observed in globally optimal configurations for different cost-communication trade-offs: (i) regular small worlds, (ii) star-like networks and (iii) trees with a centre of interconnected hubs. In the latter regime, i.e. for very expensive wire, power laws in the link length distributions $P(w)\propto w^{-α}$ are found, which can be explained by a hierarchical organization of the networks. In contrast, in the local optimization process the presence of sharp transitions between different network regimes depends on the dimension of the underlying space. Whereas for $d=\infty$ sharp transitions between fully connected networks, regular small worlds and highly cliquish periphery-core networks are found, for $d=1$ sharp transitions are absent and the power law behaviour in the link length distribution persists over a much wider range of link cost parameters. The measured power law exponents are in agreement with the hypothesis that the locally optimized networks consist of multiple overlapping sub-optimal hierarchical trees.

cond-mat.dis-nn↗

arXiv2010

Optimal Synchronization in Space

In this Rapid Communication we investigate spatially constrained networks that realize optimal synchronization properties. After arguing that spatial constraints can be imposed by limiting the amount of `wire' available to connect nodes distributed in space, we use numerical optimization methods to construct networks that realize different trade-offs between optimal synchronization and spatial constraints. Over a large range of parameters such optimal networks are found to have a link length distribution characterized by power law tails $P(l)\propto l^{-α}$, with exponents $α$ increasing as the networks become more constrained in space. It is also shown that the optimal networks, which constitute a particular type of small world network, are characterized by the presence of nodes of distinctly larger than average degree around which long distance links are centred.

nlin.AO↗

arXiv2009

Optimal synchronization on strongly connected directed networks

In this paper we construct and analyse strongly connected sparse directed networks with an enhanced propensity for synchronization (PFS). Two types of PFS-enhanced networks are considered: (i) an eigenratio minimizing ensemble with non-vanishing complex parts of the spectrum and (ii) a class of networks with real spectrum but slightly larger eigenratios than (i). We relate the superior PFSs to a strongly skewed out-degree distribution, the density of double links and a hierarchical periphery-core organization. Ensembles (i) and (ii) are found to differ in the density of double links and the particular organization of the core and the periphery-core linkage.

nlin.PS↗

arXiv2008

Synchrony-optimized Networks of Non-identical Kuramoto Oscillators

In this letter we discuss a method for generating synchrony-optimized coupling architectures of Kuramoto oscillators with a heterogeneous distribution of native frequencies. The method allows us to relate the properties of the coupling network to its synchronizability. These relations were previously only established from a linear stability analysis of the identical oscillator case. We further demonstrate that the heterogeneity in the oscillator population produces heterogeneity in the optimal coupling network as well. Two rules for enhancing the synchronizability of a given network by a suitable placement of oscillators are given: (i) native frequencies of adjacent oscillators must be anti-correlated and (ii) frequency magnitudes should positively correlate with the degree of the node they are placed at.

nlin.PS↗

arXiv2008

Synchronization on directed small worlds: feed forward loops and cycles

In this paper we investigate the influence of directed motifs on the synchronization properties of Kuramoto oscillators on directed networks. Building different types of sparse directed small world networks similar to the Watts and Strogatz procedure we establish that feed forward loops favour synchronization on directed networks. The paper highlights the importance of local network characteristics for synchronization.

nlin.AO↗

arXiv2005

Assortative mixing by degree makes a network more unstable

We investigate the role of degree correlation among nodes on the stability of complex networks, by studying spectral properties of randomly weighted matrices constructed from directed Erdös-Rényi and scale-free random graph models. We focus on the behaviour of the largest real part of the eigenvalues, $λ_\text{max}$, that governs the growth rate of perturbations about an equilibrium (and hence, determines stability). We find that assortative mixing by degree, where nodes with many links connect preferentially to other nodes with many links, reduces the stability of networks. In particular, for sparse scale-free networks with $N$ nodes, $λ_\text{max}$ scales as $N^α$ for highly assortative networks, while for disassortative graphs, $λ_\text{max}$ scales logarithmically with $N$. This difference may be a possible reason for the prevalence of disassortative networks in nature.

cond-mat.stat-mech↗

arXiv2004

Growing Networks with Enhanced Resilience to Perturbation

Scale-free (SF) networks and small world networks have been found to occur in very diverse contexts. It is this striking universality which makes one look for widely applicable mechanisms which lead to the formation of such networks. In this letter we propose a new mechanism for the construction of SF networks: Evolving networks as interaction networks of systems which are distinguished by their stability if perturbed out of equilibrium. Stability is measured by the largest real part of any eigenvalue of a matrix associated with the graph. We extend the model to weighted directed networks and report power law behaviour of the link strength distribution of the weighted graphs in the SF regime. The model we propose for the first time relates SF networks to stability properties of the underlying dynamical system.

cond-mat.dis-nn↗

arXiv2001

The Architecture of Idiotypic Networks: Percolation and Scaling Behaviour

We investigate a model where idiotypes (characterizing B-lymphocytes and antibodies of an immune system) and anti-idiotypes are represented by complementary bitstrings of a given length d allowing for a number of mismatches (matching rules). In this model, the vertices of the hypercube in dimension d represent the potential repertoire of idiotypes. A random set of (with probability p) occupied vertices corresponds to the expressed repertoire of idiotypes at a given moment. Vertices of this set linked by the above matching rules build random clusters. We give a structural and statistical characterisation of these clusters - or in other words - of the architecture of the idiotypic network. Increasing the probability p one finds at a critical p a percolation transition where for the first time a large connected graph occures with probability one. Increasing p further, there is a second transition above which the repertoire is complete in the sense that any newly introduced idiotype finds a complementary anti-idiotype. We introduce structural characteristics such as the mass distributions and the fragmentation rate for random clusters, and determine the scaling behaviour of the cluster size distribution near the percolation transition, including finite size corrections. We find that slightly above the percolation transition the large connected cluster (the central part of the idiotypic network) consists typically of one highly connected part and a number of weakly connected constituents and coexists with a number of small, isolated clusters. This is in accordance with the picture of a central and a peripheral part of the idiotypic network and gives some support to idealized architectures of the central part used in recent dynamical mean field models.

physics.bio-ph↗