Biased random walks on complex networks: the role of local navigation rules

We study the biased random walk process in random uncorrelated networks with arbitrary degree distributions. In our model, the bias is defined by the preferential transition probability, which, in recent years, has been commonly used to study efficiency of different routing protocols in communication networks. We derive exact expressions for the stationary occupation probability, and for the mean transit time between two nodes. The effect of the cyclic search on transit times is also explored. Results presented in this paper give the basis for theoretical treatment of the transport-related problems on complex networks, including quantitative estimation of the critical value of the packet generation rate.Comment: 5 pages (Phys. Rev style), 3 Figure

Thermodynamic forces, flows, and Onsager coefficients in complex networks

We present Onsager formalism applied to random networks with arbitrary degree distribution. Using the well-known methods of non-equilibrium thermodynamics we identify thermodynamic forces and their conjugated flows induced in networks as a result of single node degree perturbation. The forces and the flows can be understood as a response of the system to events, such as random removal of nodes or intentional attacks on them. Finally, we show that cross effects (such as thermodiffusion, or thermoelectric phenomena), in which one force may not only give rise to its own corresponding flow, but to many other flows, can be observed also in complex networks.Comment: 4 pages, 2 figure

Percolation in the classical blockmodel

Classical blockmodel is known as the simplest among models of networks with community structure. The model can be also seen as an extremely simply example of interconnected networks. For this reason, it is surprising that the percolation transition in the classical blockmodel has not been examined so far, although the phenomenon has been studied in a variety of much more complicated models of interconnected and multiplex networks. In this paper we derive the self-consistent equation for the size the global percolation cluster in the classical blockmodel. We also find the condition for percolation threshold which characterizes the emergence of the giant component. We show that the discussed percolation phenomenon may cause unexpected problems in a simple optimization process of the multilevel network construction. Numerical simulations confirm the correctness of our theoretical derivations.Comment: 7 pages, 6 figure

A novel configuration model for random graphs with given degree sequence

Recently, random graphs in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices have attracted much attention. Here, we present a specific realization of a class of random network models in which the connection probability between two vertices (i,j) is a specific function of degrees ki and kj. In the framework of the configuration model of random graphs, we find analytical expressions for the degree correlation and clustering as a function of the variance of the desired degree distribution. The expressions obtained are checked by means of numerical simulations. Possible applications of our model are discussed.Comment: 7 pages, 3 figure

Theoretical approach and impact of correlations on the critical packet generation rate in traffic dynamics on complex networks

Using the formalism of the biased random walk in random uncorrelated networks with arbitrary degree distributions, we develop theoretical approach to the critical packet generation rate in traffic based on routing strategy with local information. We explain microscopic origins of the transition from the flow to the jammed phase and discuss how the node neighbourhood topology affects the transport capacity in uncorrelated and correlated networks.Comment: 6 pages, 5 figure

Generalized Shortest Path Kernel on Graphs

We consider the problem of classifying graphs using graph kernels. We define a new graph kernel, called the generalized shortest path kernel, based on the number and length of shortest paths between nodes. For our example classification problem, we consider the task of classifying random graphs from two well-known families, by the number of clusters they contain. We verify empirically that the generalized shortest path kernel outperforms the original shortest path kernel on a number of datasets. We give a theoretical analysis for explaining our experimental results. In particular, we estimate distributions of the expected feature vectors for the shortest path kernel and the generalized shortest path kernel, and we show some evidence explaining why our graph kernel outperforms the shortest path kernel for our graph classification problem.Comment: Short version presented at Discovery Science 2015 in Banf

Average path length in random networks

Analytic solution for the average path length in a large class of random graphs is found. We apply the approach to classical random graphs of Erd\"{o}s and R\'{e}nyi (ER) and to scale-free networks of Barab\'{a}si and Albert (BA). In both cases our results confirm previous observations: small world behavior in classical random graphs $l_{ER} \sim \ln N$ and ultra small world effect characterizing scale-free BA networks $l_{BA} \sim \ln N/\ln\ln N$. In the case of scale-free random graphs with power law degree distributions we observed the saturation of the average path length in the limit of $N\to\infty$ for systems with the scaling exponent $2< \alpha <3$ and the small-world behaviour for systems with $\alpha>3$.Comment: 4 pages, 2 figures, changed conten

A minimal model for congestion phenomena on complex networks

We study a minimal model of traffic flows in complex networks, simple enough to get analytical results, but with a very rich phenomenology, presenting continuous, discontinuous as well as hybrid phase transitions between a free-flow phase and a congested phase, critical points and different scaling behaviors in the system size. It consists of random walkers on a queueing network with one-range repulsion, where particles can be destroyed only if they can move. We focus on the dependence on the topology as well as on the level of traffic control. We are able to obtain transition curves and phase diagrams at analytical level for the ensemble of uncorrelated networks and numerically for single instances. We find that traffic control improves global performance, enlarging the free-flow region in parameter space only in heterogeneous networks. Traffic control introduces non-linear effects and, beyond a critical strength, may trigger the appearance of a congested phase in a discontinuous manner. The model also reproduces the cross-over in the scaling of traffic fluctuations empirically observed in the Internet, and moreover, a conserved version can reproduce qualitatively some stylized facts of traffic in transportation networks

Transition from fractal to non-fractal scalings in growing scale-free networks

Real networks can be classified into two categories: fractal networks and non-fractal networks. Here we introduce a unifying model for the two types of networks. Our model network is governed by a parameter $q$. We obtain the topological properties of the network including the degree distribution, average path length, diameter, fractal dimensions, and betweenness centrality distribution, which are controlled by parameter $q$. Interestingly, we show that by adjusting $q$, the networks undergo a transition from fractal to non-fractal scalings, and exhibit a crossover from `large' to small worlds at the same time. Our research may shed some light on understanding the evolution and relationships of fractal and non-fractal networks.Comment: 7 pages, 3 figures, definitive version accepted for publication in EPJ