The Sum-over-Forests Density Index: Identifying Dense Regions in a Graph.

This work introduces a novel nonparametric density index defined on graphs, the Sum-over-Forests (SoF) density index. It is based on a clear and intuitive idea: high-density regions in a graph are characterized by the fact that they contain a large amount of low-cost trees with high outdegrees while low-density regions contain few ones. Therefore, a Boltzmann probability distribution on the countable set of forests in the graph is defined so that large (high-cost) forests occur with a low probability while short (low-cost) forests occur with a high probability. Then, the SoF density index of a node is defined as the expected outdegree of this node on the set of forests, thus providing a measure of density around that node. Following the matrix-forest theorem and a statistical physics framework, it is shown that the SoF density index can be easily computed in closed form through a simple matrix inversion. Experiments on artificial and real datasets show that the proposed index performs well on finding dense regions, for graphs of various origins.

The sum-over-paths covariance kernel: a novel covariance measure between nodes of a directed graph.

This work introduces a link-based covariance measure between the nodes of a weighted directed graph, where a cost is associated with each arc. To this end, a probability distribution on the (usually infinite) countable set of paths through the graph is defined by minimizing the total expected cost between all pairs of nodes while fixing the total relative entropy spread in the graph. This results in a Boltzmann distribution on the set of paths such that long (high-cost) paths occur with a low probability while short (low-cost) paths occur with a high probability. The sum-over-paths (SoP) covariance measure between nodes is then defined according to this probability distribution: two nodes are considered as highly correlated if they often co-occur together on the same--preferably short--paths. The resulting covariance matrix between nodes (say n nodes in total) is a Gram matrix and therefore defines a valid kernel on the graph. It is obtained by inverting an n\times n matrix depending on the costs assigned to the arcs. In the same spirit, a betweenness score is also defined, measuring the expected number of times a node occurs on a path. The proposed measures could be used for various graph mining tasks such as computing betweenness centrality, semi-supervised classification of nodes, visualization, etc., as shown in Section 7.

PSA doubling time as a predictive factor on repeat biopsy for detection of prostate cancer.

OBJECTIVE: Detection of prostate cancer needs a biopsy of the prostate. Suspecting cancer from an increase in prostate-specific antigen (PSA) has a high negative rate at an initial prostate biopsy. Cases with negative initial biopsy may be the candidates of subsequent biopsy. For lowering unnecessary repeat biopsy, the use of predictive factors before a repeat biopsy is applied for indication. METHODS: Seventy-seven cases with negative initial prostate biopsy received a repeat biopsy and factors for the detection of cancer were examined. RESULTS: PSA doubling time distinguished a part of cancer cases. Its sensitivity of 30, 50 and 70 months was 36.6%, 30.4% and 10%, respectively. Cancer case did not show PSA doubling time of >100 months in general. Values of PSA transition zone density, %Free/total PSA and PSA velocity were similar between cancer and no cancer cases. CONCLUSIONS: PSA doubling time was one of the predictive factors for the detection of prostate cancer and was valuable for avoiding unnecessary repeat biopsy in some cases.

Outcome of patients with hormone-refractory prostate cancer: prognostic significance of prostate-specific antigen-doubling time and nadir prostate-specific antigen.

OBJECTIVE: Most patients with advanced prostate cancer after prostate-specific antigen (PSA) relapse following maximum androgen blockade rapidly progress to death. The present study was aimed to predict the survival of these serious patients after PSA relapse. METHODS: Sixty-eight patients with M1b and 20 patients with T3b, who relapsed and died of cancer within a short period, were studied. PSA-doubling time (PSA-DT) at PSA relapse influenced the outcome after PSA relapse [hazard ratio (CI): 2.000 (1.283-3.226)]; thus, on the basis of the median values of PSA-DT (>2 months) and additionally nadir PSA in previous treatment (2 months and nadir PSA of