Are randomly grown graphs really random?

We analyze a minimal model of a growing network. At each time step, a new vertex is added; then, with probability delta, two vertices are chosen uniformly at random and joined by an undirected edge. This process is repeated for t time steps. In the limit of large t, the resulting graph displays surprisingly rich characteristics. In particular, a giant component emerges in an infinite-order phase transition at delta=1/8. At the transition, the average component size jumps discontinuously but remains finite. In contrast, a static random graph with the same degree distribution exhibits a second-order phase transition at delta=1/4, and the average component size diverges there. These dramatic differences between grown and static random graphs stem from a positive correlation between the degrees of connected vertices in the grown graph-older vertices tend to have higher degree, and to link with other high-degree vertices, merely by virtue of their age. We conclude that grown graphs, however randomly they are constructed, are fundamentally different from their static random graph counterparts.

Network robustness and fragility: percolation on random graphs.

Recent work on the Internet, social networks, and the power grid has addressed the resilience of these networks to either random or targeted deletion of network nodes or links. Such deletions include, for example, the failure of Internet routers or power transmission lines. Percolation models on random graphs provide a simple representation of this process but have typically been limited to graphs with Poisson degree distribution at their vertices. Such graphs are quite unlike real-world networks, which often possess power-law or other highly skewed degree distributions. In this paper we study percolation on graphs with completely general degree distribution, giving exact solutions for a variety of cases, including site percolation, bond percolation, and models in which occupation probabilities depend on vertex degree. We discuss the application of our theory to the understanding of network resilience.

Persistence of HIV-1 transcription in peripheral-blood mononuclear cells in patients receiving potent antiretroviral therapy.

BACKGROUND AND METHODS: Although potent antiretroviral therapy can control infection with human immunodeficiency virus type 1 (HIV-1), a long-lived reservoir of infectious virus persists in CD4+ T cells. We investigated this viral reservoir by measuring the levels of cell-associated viral DNA and messenger RNA (mRNA) that are essential for HIV-1 replication. Approximately every 6 months, we obtained samples of peripheral-blood mononuclear cells from five men with long-standing HIV-1 infection who had had undetectable levels of plasma HIV-1 RNA for 20 months or more during treatment with potent antiretroviral drugs. RESULTS: Before treatment, plasma levels of HIV-1 RNA correlated with the levels of cell-associated unintegrated HIV-1 DNA and unspliced viral mRNA. After treatment, plasma levels of HIV-1 RNA fell by more than 2.7 log to undetectable levels. The decrease in cell-associated integrated and unintegrated HIV-1 DNA and mRNA occurred in two phases. The first phase occurred during the initial 500 days of treatment and was characterized by substantial decreases in the levels of DNA and mRNA, but not to undetectable levels. The concentrations of cell-associated unintegrated viral DNA, integrated proviral DNA, and unspliced viral mRNA decreased by 1.25 to 1.46 log. The second phase occurred during the subsequent 300 days or more of treatment and was characterized by a plateau in the levels of HIV-1 DNA and unspliced mRNA. After an initial rapid decline, the ratio of unspliced to multiply spliced viral mRNA (a measure of active viral transcription) stabilized and remained greater than zero at each measurement. CONCLUSIONS: Despite treatment with potent antiretroviral drugs and the suppression of plasma HIV-1 RNA to undetectable levels for 20 months or more, HIV-1 transcription persists in peripheral-blood mononuclear cells. Unless the quasi-steady state levels of HIV DNA and mRNA eventually disappear with longer periods of therapy, these findings suggest that HIV-1 infection cannot be eradicated with current treatments.

Virus phenotype switching and disease progression in HIV-1 infection.

One of the phenotypic distinctions between different strains of human immunodeficiency virus type 1 (HIV-1) has to do with the ability to cause target cells to form large multinucleate bodies known as syncytia. There are two phenotypes according to this characterization: syncytium-inducing (SI) and non-syncytium-inducing (NSI). NSI strains are usually present throughout infection, while SI strains are typically seen at the beginning of the infection and near the onset of AIDS. The late emergence of SI strains is referred to as phenotype switching. In this paper we analyse the factors that lead to phenotype switching and contribute to the dynamics of disease progression. We show that a strong immune system selects for NSI strains while a weak immune system favours SI strains. The model explicitly accounts for the fact that CD4+ cells are both targets of HIV infection and crucial for activating immune responses against HIV In such a model, SI strains can emerge after a long and variable period of NSI dominated infection. Furthermore, versions of the model which do not explicitly account for HIV-specific, activated CD4+ cells do not exhibit phenotype switching, emphasizing the critical importance of this pool of cells.