Modularity and anti-modularity in networks with arbitrary degree distribution

Networks describing the interaction of the elements that constitute a complex system grow and develop via a number of different mechanisms, such as the addition and deletion of nodes, the addition and deletion of edges, as well as the duplication or fusion of nodes. While each of these mechanisms can have a different cause depending on whether the network is biological, technological, or social, their impact on the network's structure, as well as its local and global properties, is similar. This allows us to study how each of these mechanisms affects networks either alone or together with the other processes, and how they shape the characteristics that have been observed. We study how a network's growth parameters impact the distribution of edges in the network, how they affect a network's modularity, and point out that some parameters will give rise to networks that have the opposite tendency, namely to display anti-modularity. Within the model we are describing, we can search the space of possible networks for parameter sets that generate networks that are very similar to well-known and well-studied examples, such as the brain of a worm, and the network of interactions of the proteins in baker's yeast.Comment: 23 pages. 13 figures, 1 table. Includes Supplementary tex

Random networks with sublinear preferential attachment: The giant component

We study a dynamical random network model in which at every construction step a new vertex is introduced and attached to every existing vertex independently with a probability proportional to a concave function f of its current degree. We give a criterion for the existence of a giant component, which is both necessary and sufficient, and which becomes explicit when f is linear. Otherwise it allows the derivation of explicit necessary and sufficient conditions, which are often fairly close. We give an explicit criterion to decide whether the giant component is robust under random removal of edges. We also determine asymptotically the size of the giant component and the empirical distribution of component sizes in terms of the survival probability and size distribution of a multitype branching random walk associated with f.Comment: Published in at http://dx.doi.org/10.1214/11-AOP697 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The phase transition in the uniformly grown random graph has infinite order

The aim of this paper is to study the emergence of the giant component in the uniformly grown random graph G(n)(c), 0 < c < 1,the graph on the set [n]= {1, 2,...,n} in which each possible edge ij is present with probability c/max {i,j}, independently of all other edges. Equivalently, we may start with the random graph G(n)(1) with vertex set [n], where each vertex j is joined to each "earlier" vertex i < j with probability 1/j, independently of all other choices. The graph G(n)(c) is formed by the open bonds in the bond percolation on G(n)(1) in which a bond is open with probability c. The model G(n)(c) is the finite version of a model proposed by Dubins in 1984, and is also closely related to a random graph process defined by Callaway, Hopcroft, Kleinberg, Newman, and Strogatz [Phys Rev E 64 (2001), 041902]. Results of Kalikow and Weiss [Israel J Math 62 (1988), 257-268] and Shepp [Israel J Math 67 (1989), 23-33] imply that the percolation threshold is at c = 1/4. The main result of this paper is that for c = 1/4 + epsilon,epsilon > 0, the giant component in G(n)(c) has order exp n. In particular, the phase transition in the bond percolation on G(n)(1) has infinite order. Using nonrigorous methods, Dorogovtsev, Mendes, and Samukhin [Phys Rev E 64 (2001), 066110] showed that an even more precise result is likely to hold. (C) 2004 Wiley Periodicals, Inc

Susceptibility in inhomogeneous random graphs

We study the susceptibility, i.e., the mean size of the component containing a random vertex, in a general model of inhomogeneous random graphs. This is one of the fundamental quantities associated to (percolation) phase transitions; in practice one of its main uses is that it often gives a way of determining the critical point by solving certain linear equations. Here we relate the susceptibility of suitable random graphs to a quantity associated to the corresponding branching process, and study both quantities in various natural examples.Comment: 51 page

Statistical Inference on Stochastic Graphs

This thesis considers modelling and applications of random graph processes. A brief review on contemporary random graph models and a general Birth-Death model with relevant maximum likelihood inference procedure are provided in chapter one. The main result in this thesis is the construction of an epidemic model by embedding a competing hazard model within a stochastic graph process (chapter 2). This model includes both individual characteristics and the population connectivity pattern in analyzing the infection propagation. The dynamic outdegrees and indegrees, estimated by the model, provide insight into important epidemiological concepts such as the reproductive number. A dynamic reproductive number based on the disease graph process is developed and applied in several simulated and actual epidemic outbreaks. In addition, graph-based statistical measures are proposed to quantify the effect of individual characteristics on the disease propagation. The epidemic model is applied to two real outbreaks: the 2001 foot-and-mouth epidemic in the United Kingdom (chapter 3) and the 1861 measles outbreak in Hagelloch, Germany (chapter 4). Both applications provide valuable insight into the behaviour of infectious disease propagation with di erent connectivity patterns and human interventions