The sum of degrees in cliques

We investigate lower bounds on the average degree in r-cliques in graphs of order n and size greater than t(r,n), where t(r,n) is the size of the Turan graph on n vertices and r color classes. Continuing earlier research of Edwards and Faudree, we completely prove a conjecture of Bollobas and Erdoes from 1975.Comment: 10 page

Metric dimension for random graphs

The metric dimension of a graph $G$ is the minimum number of vertices in a subset $S$ of the vertex set of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $S$. In this paper we investigate the metric dimension of the random graph $G(n,p)$ for a wide range of probabilities $p=p(n)$

Knot Graphs

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph which is reducible by some finite sequence of these moves, to a graph with no edges is called a knot graph. We show that the class of knot graphs strictly contains the set of delta-wye graphs. We prove that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes

Modularity from Fluctuations in Random Graphs and Complex Networks

The mechanisms by which modularity emerges in complex networks are not well understood but recent reports have suggested that modularity may arise from evolutionary selection. We show that finding the modularity of a network is analogous to finding the ground-state energy of a spin system. Moreover, we demonstrate that, due to fluctuations, stochastic network models give rise to modular networks. Specifically, we show both numerically and analytically that random graphs and scale-free networks have modularity. We argue that this fact must be taken into consideration to define statistically-significant modularity in complex networks.Comment: 4 page

Robust Emergent Activity in Dynamical Networks

We study the evolution of a random weighted network with complex nonlinear dynamics at each node, whose activity may cease as a result of interactions with other nodes. Starting from a knowledge of the micro-level behaviour at each node, we develop a macroscopic description of the system in terms of the statistical features of the subnetwork of active nodes. We find the asymptotic characteristics of this subnetwork to be remarkably robust: the size of the active set is independent of the total number of nodes in the network, and the average degree of the active nodes is independent of both the network size and its connectivity. These results suggest that very different networks evolve to active subnetworks with the same characteristic features. This has strong implications for dynamical networks observed in the natural world, notably the existence of a characteristic range of links per species across ecological systems.Comment: 4 pages, 5 figure

Central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs

We consider the adjacency matrix $A$ of a large random graph and study fluctuations of the function $f_n(z,u)=\frac{1}{n}\sum_{k=1}^n\exp\{-uG_{kk}(z)\}$ with $G(z)=(z-iA)^{-1}$. We prove that the moments of fluctuations normalized by $n^{-1/2}$ in the limit $n\to\infty$ satisfy the Wick relations for the Gaussian random variables. This allows us to prove central limit theorem for $\hbox{Tr}G(z)$ and then extend the result on the linear eigenvalue statistics $\hbox{Tr}\phi(A)$ of any function $\phi:\mathbb{R}\to\mathbb{R}$ which increases, together with its first two derivatives, at infinity not faster than an exponential.Comment: 22 page

On the large N expansion in hyperbolic sigma-models

Invariant correlation functions for ${\rm SO}(1,N)$ hyperbolic sigma-models are investigated. The existence of a large $N$ asymptotic expansion is proven on finite lattices of dimension $d \geq 2$. The unique saddle point configuration is characterized by a negative gap vanishing at least like 1/V with the volume. Technical difficulties compared to the compact case are bypassed using horospherical coordinates and the matrix-tree theorem.Comment: 15 pages. Some changes in introduction and discussion; to appear in J. Math. Phy

Ising Model on Edge-Dual of Random Networks

We consider Ising model on edge-dual of uncorrelated random networks with arbitrary degree distribution. These networks have a finite clustering in the thermodynamic limit. High and low temperature expansions of Ising model on the edge-dual of random networks are derived. A detailed comparison of the critical behavior of Ising model on scale free random networks and their edge-dual is presented.Comment: 23 pages, 4 figures, 1 tabl

Numerical evaluation of the upper critical dimension of percolation in scale-free networks

We propose a numerical method to evaluate the upper critical dimension $d_c$ of random percolation clusters in Erd\H{o}s-R\'{e}nyi networks and in scale-free networks with degree distribution ${\cal P}(k) \sim k^{-\lambda}$, where $k$ is the degree of a node and $\lambda$ is the broadness of the degree distribution. Our results report the theoretical prediction, $d_c = 2(\lambda - 1)/(\lambda - 3)$ for scale-free networks with $3 < \lambda < 4$ and $d_c = 6$ for Erd\H{o}s-R\'{e}nyi networks and scale-free networks with $\lambda > 4$. When the removal of nodes is not random but targeted on removing the highest degree nodes we obtain $d_c = 6$ for all $\lambda > 2$. Our method also yields a better numerical evaluation of the critical percolation threshold, $p_c$, for scale-free networks. Our results suggest that the finite size effects increases when $\lambda$ approaches 3 from above.Comment: 10 pages, 5 figure