On the bridge hypothesis in the glass transition of freestanding polymer films.

Freestanding thin polymer films with high molecular weights exhibit an anomalous decrease in the glass-transition temperature with film thickness. Specifically, in such materials, the measured glass-transition temperature evolves in an affine way with the film thickness, with a slope that weakly depends on the molecular weight. De Gennes proposed a sliding mechanism as the hypothetical dominant relaxation process in these systems, where stress kinks could propagate in a reptation-like fashion through so-called bridges, i.e. from one free interface to the other along the backbones of polymer macromolecules. Here, by considering the exact statistics of finite-sized random walks within a confined box, we investigate in details the bridge hypothesis. We show that the sliding mechanism cannot reproduce the basic features appearing in the experiments, and we exhibit the fundamental reasons behind such a fact.

Fate of articulation points and bredges in percolation.

We investigate the statistics of articulation points and bredges (bridge edges) in complex networks in which bonds are randomly removed in a percolation process. Because of the heterogeneous structure of a complex network, the probability of a node to be an articulation point or the probability of an edge to be a bredge will not be homogeneous across the network. We therefore analyze full distributions of articulation point probabilities as well as bredge probabilities, using a message-passing or cavity approach to the problem. Our methods allow us to obtain these distributions both for large single instances of networks and for ensembles of networks in the configuration model class in the thermodynamic limit, through a single unified approach. We also evaluate deconvolutions of these distributions according to degrees of the node or the degrees of both adjacent nodes in the case of bredges. We obtain closed form expressions for the large mean degree limit of Erdos-Rényi networks. Moreover, we reveal and are able to rationalize a significant amount of structure in the evolution of articulation point and bredge probabilities in response to random bond removal. We find that full distributions of articulation point and bredge probabilities in real networks and in their randomized counterparts may exhibit significant differences even where average articulation point and bredge probabilities do not. We argue that our results could be exploited in a variety of applications, including approaches to network dismantling or to vaccination and islanding strategies to prevent the spread of epidemics or of blackouts in process networks.

Statistical analysis of edges and bredges in configuration model networks.

A bredge (bridge-edge) in a network is an edge whose deletion would split the network component on which it resides into two separate components. Bredges are vulnerable links that play an important role in network collapse processes, which may result from node or link failures, attacks, or epidemics. Therefore, the abundance and properties of bredges affect the resilience of the network to these collapse scenarios. We present analytical results for the statistical properties of bredges in configuration model networks. Using a generating function approach based on the cavity method, we calculate the probability P[over ̂](e∈B) that a random edge e in a configuration model network with degree distribution P(k) is a bredge (B). We also calculate the joint degree distribution P[over ̂](k,k^{'}|B) of the end-nodes i and i^{'} of a random bredge. We examine the distinct properties of bredges on the giant component (GC) and on the finite tree components (FC) of the network. On the finite components all the edges are bredges and there are no degree-degree correlations. We calculate the probability P[over ̂](e∈B|GC) that a random edge on the giant component is a bredge. We also calculate the joint degree distribution P[over ̂](k,k^{'}|B,GC) of the end-nodes of bredges and the joint degree distribution P[over ̂](k,k^{'}|NB,GC) of the end-nodes of nonbredge edges on the giant component. Surprisingly, it is found that the degrees k and k^{'} of the end-nodes of bredges are correlated, while the degrees of the end-nodes of nonbredge edges are uncorrelated. We thus conclude that all the degree-degree correlations on the giant component are concentrated on the bredges. We calculate the covariance Γ(B,GC) of the joint degree distribution of end-nodes of bredges and show it is negative, namely bredges tend to connect high degree nodes to low degree nodes. We apply this analysis to ensembles of configuration model networks with degree distributions that follow a Poisson distribution (Erdos-Rényi networks), an exponential distribution and a power-law distribution (scale-free networks). The implications of these results are discussed in the context of common attack scenarios and network dismantling processes.

Distribution of shortest cycle lengths in random networks.

We present analytical results for the distribution of shortest cycle lengths (DSCL) in random networks. The approach is based on the relation between the DSCL and the distribution of shortest path lengths (DSPL). We apply this approach to configuration model networks, for which analytical results for the DSPL were obtained before. We first calculate the fraction of nodes in the network which reside on at least one cycle. Conditioning on being on a cycle, we provide the DSCL over ensembles of configuration model networks with degree distributions which follow a Poisson distribution (Erdos-Rényi network), degenerate distribution (random regular graph), and a power-law distribution (scale-free network). The mean and variance of the DSCL are calculated. The analytical results are found to be in very good agreement with the results of computer simulations.