Complex networks created by aggregation

We study aggregation as a mechanism for the creation of complex networks. In this evolution process vertices merge together, which increases the number of highly connected hubs. We study a range of complex network architectures produced by the aggregation. Fat-tailed (in particular, scale-free) distributions of connections are obtained both for networks with a finite number of vertices and growing networks. We observe a strong variation of a network structure with growing density of connections and find the phase transition of the condensation of edges. Finally, we demonstrate the importance of structural correlations in these networks.Comment: 12 pages, 13 figure

Accelerated growth in outgoing links in evolving networks: deterministic vs. stochastic picture

In several real-world networks like the Internet, WWW etc., the number of links grow in time in a non-linear fashion. We consider growing networks in which the number of outgoing links is a non-linear function of time but new links between older nodes are forbidden. The attachments are made using a preferential attachment scheme. In the deterministic picture, the number of outgoing links $m(t)$ at any time $t$ is taken as $N(t)^\theta$ where $N(t)$ is the number of nodes present at that time. The continuum theory predicts a power law decay of the degree distribution: $P(k) \propto k^{-1-\frac{2} {1-\theta}}$, while the degree of the node introduced at time $t_i$ is given by $k(t_i,t) = t_i^{\theta}[ \frac {t}{t_i}]^{\frac {1+\theta}{2}}$ when the network is evolved till time $t$. Numerical results show a growth in the degree distribution for small $k$ values at any non-zero $\theta$. In the stochastic picture, $m(t)$ is a random variable. As long as $$is independent of time, the network shows a behaviour similar to the Barab\'asi-Albert (BA) model. Different results are obtained when$$ is time-dependent, e.g., when $m(t)$ follows a distribution $P(m) \propto m^{-\lambda}$. The behaviour of $P(k)$ changes significantly as $\lambda$ is varied: for $\lambda > 3$, the network has a scale-free distribution belonging to the BA class as predicted by the mean field theory, for smaller values of $\lambda$ it shows different behaviour. Characteristic features of the clustering coefficients in both models have also been discussed.Comment: Revised text, references added, to be published in PR

Emergence of Clusters in Growing Networks with Aging

We study numerically a model of nonequilibrium networks where nodes and links are added at each time step with aging of nodes and connectivity- and age-dependent attachment of links. By varying the effects of age in the attachment probability we find, with numerical simulations and scaling arguments, that a giant cluster emerges at a first-order critical point and that the problem is in the universality class of one dimensional percolation. This transition is followed by a change in the giant cluster's topology from tree-like to quasi-linear, as inferred from measurements of the average shortest-path length, which scales logarithmically with system size in one phase and linearly in the other.Comment: 8 pages, 6 figures, accepted for publication in JSTA

Phase Transition with the Berezinskii--Kosterlitz--Thouless Singularity in the Ising Model on a Growing Network

We consider the ferromagnetic Ising model on a highly inhomogeneous network created by a growth process. We find that the phase transition in this system is characterised by the Berezinskii--Kosterlitz--Thouless singularity, although critical fluctuations are absent, and the mean-field description is exact. Below this infinite order transition, the magnetization behaves as $exp(-const/\sqrt{T_c-T})$. We show that the critical point separates the phase with the power-law distribution of the linear response to a local field and the phase where this distribution rapidly decreases. We suggest that this phase transition occurs in a wide range of cooperative models with a strong infinite-range inhomogeneity. {\em Note added}.--After this paper had been published, we have learnt that the infinite order phase transition in the effective model we arrived at was discovered by O. Costin, R.D. Costin and C.P. Grunfeld in 1990. This phase transition was considered in the papers: [1] O. Costin, R.D. Costin and C.P. Grunfeld, J. Stat. Phys. 59, 1531 (1990); [2] O. Costin and R.D. Costin, J. Stat. Phys. 64, 193 (1991); [3] M. Bundaru and C.P. Grunfeld, J. Phys. A 32, 875 (1999); [4] S. Romano, Mod. Phys. Lett. B 9, 1447 (1995). We would like to note that Costin, Costin and Grunfeld treated this model as a one-dimensional inhomogeneous system. We have arrived at the same model as a one-replica ansatz for a random growing network where expected to find a phase transition of this sort based on earlier results for random networks (see the text). We have also obtained the distribution of the linear response to a local field, which characterises correlations in this system. We thank O. Costin and S. Romano for indicating these publications of 90s.Comment: 5 pages, 2 figures. We have added a note indicating that the infinite order phase transition in the effective model we arrived at was discovered in the work: O. Costin, R.D. Costin and C.P. Grunfeld, J. Stat. Phys. 59, 1531 (1990). Appropriate references to the papers of 90s have been adde

Effect of the accelerating growth of communications networks on their structure

Motivated by data on the evolution of the Internet and World Wide Web we consider scenarios of self-organization of the nonlinearly growing networks into free-scale structures. We find that the accelerating growth of the networks establishes their structure. For the growing networks with preferential linking and increasing density of links, two scenarios are possible. In one of them, the value of the exponent $\gamma$ of the connectivity distribution is between 3/2 and 2. In the other, $\gamma>2$ and the distribution is necessarily non-stationary.Comment: 4 pages revtex, 3 figure

Belief-propagation algorithm and the Ising model on networks with arbitrary distributions of motifs

We generalize the belief-propagation algorithm to sparse random networks with arbitrary distributions of motifs (triangles, loops, etc.). Each vertex in these networks belongs to a given set of motifs (generalization of the configuration model). These networks can be treated as sparse uncorrelated hypergraphs in which hyperedges represent motifs. Here a hypergraph is a generalization of a graph, where a hyperedge can connect any number of vertices. These uncorrelated hypergraphs are tree-like (hypertrees), which crucially simplify the problem and allow us to apply the belief-propagation algorithm to these loopy networks with arbitrary motifs. As natural examples, we consider motifs in the form of finite loops and cliques. We apply the belief-propagation algorithm to the ferromagnetic Ising model on the resulting random networks. We obtain an exact solution of this model on networks with finite loops or cliques as motifs. We find an exact critical temperature of the ferromagnetic phase transition and demonstrate that with increasing the clustering coefficient and the loop size, the critical temperature increases compared to ordinary tree-like complex networks. Our solution also gives the birth point of the giant connected component in these loopy networks.Comment: 9 pages, 4 figure

Log-Networks

We introduce a growing network model in which a new node attaches to a randomly-selected node, as well as to all ancestors of the target node. This mechanism produces a sparse, ultra-small network where the average node degree grows logarithmically with network size while the network diameter equals 2. We determine basic geometrical network properties, such as the size dependence of the number of links and the in- and out-degree distributions. We also compare our predictions with real networks where the node degree also grows slowly with time -- the Internet and the citation network of all Physical Review papers.Comment: 7 pages, 6 figures, 2-column revtex4 format. Version 2: minor changes in response to referee comments and to another proofreading; final version for PR

Correlated electrons systems on the Apollonian network

Strongly correlated electrons on an Apollonian network are studied using the Hubbard model. Ground-state and thermodynamic properties, including specific heat, magnetic susceptibility, spin-spin correlation function, double occupancy and one-electron transfer, are evaluated applying direct diagonalization and quantum Monte Carlo. The results support several types of magnetic behavior. In the strong-coupling limit, the quantum anisotropic spin 1/2 Heisenberg model is used and the phase diagram is discussed using the renormalization group method. For ferromagnetic coupling, we always observe the existence of long-range order. For antiferromagnetic coupling, we find a paramagnetic phase for all finite temperatures.Comment: 7 pages, 8 figure