The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences $(G_n)$ of graphs in terms of a limiting object that may be represented by a symmetric function $W$ on $[0, 1]^2$ , i.e., a kernel or graphon. In this context it is natural to wish to relate specific properties of the sequence to specific properties of the kernel. Here we show that the kernel is monotone (i.e., increasing in both variables) if and only if the sequence satisfies a “quasimonotonicity” property defined by a certain functional tending to zero. As a tool we prove an inequality relating the cut and $L^1$ norms of kernels of the form $W_1 −W_2$ with $W_1$ and $W_2$ monotone that may be of interest in its own right; no such inequality holds for general kernels.

Stochastic Kronecker graphs supply a parsimonious model for large sparse real-world graphs. They can specify the distribution of a large random graph using only three or four parameters. Those parameters have, however, proved difficult to choose in specific applications. This article looks at method-of-moments estimators that are computationally much simpler than maximum likelihood. The estimators are fast, and in our examples, they typically yield Kronecker parameters with expected feature counts closer to a given graph than we get from KronFit. The improvement is especially prominent for the number of triangles in the graph.

In this paper, we study some important statistics of the random graph $H^{(t)}_{a ,k}$ in the Buckley-Osthus model, where $t$ is the number of nodes, $kt$ is the number of edges (so that $k \in \mathbb{N}$), and $a \gt 0$ is the so-called initial attractiveness of a node. This model is a modification of the well-known Bollobás-Riordan model. First, we find a new asymptotic formula for the expectation of the number $R(d, t)$ of nodes of a given degree $d$ in a graph in this model. Such a formula is known for $a \in \mathbb{N}$ and $d \le t^{1/100(a+1)}$ . Both restrictions are unsatisfactory from theoretical and practical points of view. We completely remove them. Then we calculate the covariances between any two quantities $R(d_1 , t)$ and $R(d_2 , t)$, and using the second-moment method we show that $R(d, t)$ is tightly concentrated around its mean for all possible values of $d$ and $t$. Furthermore, we study a more complicated statistic of the web graph: $X(d_1, d_2 , t)$ is the total number of edges between nodes whose degrees are equal to $d_1$ and $d_2$ respectively. We also find an asymptotic formula for the expectation of $X(d_1, d_2 , t)$ and prove a tight concentration result. Again, we do not impose any substantial restrictions on the values of $d_1 , d_2$ , and $t$.

Quantitative security analysis of networked computer systems has been an open problem in computer security for decades. Recently, a promising approach was proposed in Li et al., which, however, made some strong assumptions including the exponential distribution of, and the independence among, the relevant random variables. In this paper, we substantially weaken these assumptions while offering, in addition to the same types of analytical results as in Li et al., methods for obtaining the desired security quantities in practice. Moreover, we investigate the problem from a higher-level abstraction, which also leads to both analytical results and practical methods for obtaining the desired security quantities. These should represent a significant step toward ultimately solving the problem of quantitative security analysis of networked computer systems.