A weighted configuration model and inhomogeneous epidemics

A random graph model with prescribed degree distribution and degree dependent edge weights is introduced. Each vertex is independently equipped with a random number of half-edges and each half-edge is assigned an integer valued weight according to a distribution that is allowed to depend on the degree of its vertex. Half-edges with the same weight are then paired randomly to create edges. An expression for the threshold for the appearance of a giant component in the resulting graph is derived using results on multi-type branching processes. The same technique also gives an expression for the basic reproduction number for an epidemic on the graph where the probability that a certain edge is used for transmission is a function of the edge weight. It is demonstrated that, if vertices with large degree tend to have large (small) weights on their edges and if the transmission probability increases with the edge weight, then it is easier (harder) for the epidemic to take off compared to a randomized epidemic with the same degree and weight distribution. A recipe for calculating the probability of a large outbreak in the epidemic and the size of such an outbreak is also given. Finally, the model is fitted to three empirical weighted networks of importance for the spread of contagious diseases and it is shown that $R_0$ can be substantially over- or underestimated if the correlation between degree and weight is not taken into account

The Future of Gwydir: Community Engagement 2015-16

In mid-2015 Gwydir Shire Council engaged the Centre for Local Government at the University of Technology Sydney (UTS:CLG) to conduct workshops with Council, a deliberative panel, and a community survey exploring the financial sustainability of Council and future service delivery in the local area

Percolation in invariant Poisson graphs with i.i.d. degrees

Let each point of a homogeneous Poisson process in R^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution mu on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution mu. Leaving aside degenerate cases, we prove that for any mu there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme that is a natural extension of Gale-Shapley stable marriage, we give sufficient conditions on mu for the absence and presence of infinite components

Non-equilibrium mean-field theories on scale-free networks

Many non-equilibrium processes on scale-free networks present anomalous critical behavior that is not explained by standard mean-field theories. We propose a systematic method to derive stochastic equations for mean-field order parameters that implicitly account for the degree heterogeneity. The method is used to correctly predict the dynamical critical behavior of some binary spin models and reaction-diffusion processes. The validity of our non-equilibrium theory is furtherly supported by showing its relation with the generalized Landau theory of equilibrium critical phenomena on networks.Comment: 4 pages, no figures, major changes in the structure of the pape

Percolation transition and distribution of connected components in generalized random network ensembles

In this work, we study the percolation transition and large deviation properties of generalized canonical network ensembles. This new type of random networks might have a very rich complex structure, including high heterogeneous degree sequences, non-trivial community structure or specific spatial dependence of the link probability for networks embedded in a metric space. We find the cluster distribution of the networks in these ensembles by mapping the problem to a fully connected Potts model with heterogeneous couplings. We show that the nature of the Potts model phase transition, linked to the birth of a giant component, has a crossover from second to first order when the number of critical colors $q_c = 2$ in all the networks under study. These results shed light on the properties of dynamical processes defined on these network ensembles.Comment: 27 pages, 15 figure

Synchronization in Scale Free networks: The role of finite size effects

Synchronization problems in complex networks are very often studied by researchers due to its many applications to various fields such as neurobiology, e-commerce and completion of tasks. In particular, Scale Free networks with degree distribution $P(k)\sim k^{-\lambda}$, are widely used in research since they are ubiquitous in nature and other real systems. In this paper we focus on the surface relaxation growth model in Scale Free networks with $2.5< \lambda <3$, and study the scaling behavior of the fluctuations, in the steady state, with the system size $N$. We find a novel behavior of the fluctuations characterized by a crossover between two regimes at a value of $N=N^*$ that depends on $\lambda$: a logarithmic regime, found in previous research, and a constant regime. We propose a function that describes this crossover, which is in very good agreement with the simulations. We also find that, for a system size above $N^{*}$, the fluctuations decrease with $\lambda$, which means that the synchronization of the system improves as $\lambda$ increases. We explain this crossover analyzing the role of the network's heterogeneity produced by the system size $N$ and the exponent of the degree distribution.Comment: 9 pages and 5 figures. Accepted in Europhysics Letter