Cantelli's bounds for generalized tail inequalities in Euclidean spaces

Let $X$ be a centered random vector in a finite dimensional real inner product space $\mathcal{E}$. For a subset $C$ of the ambient vector space $V$ of $\mathcal{E}$ and $x,\,y\in V$, write $x\preceq_C y$ if $y-x\in C$. When $C$ is a closed convex cone in $\mathcal{E}$, then $\preceq_C$ is a pre-order on $V$, whereas if $C$ is a proper cone in $\mathcal{E}$, then $\preceq_C$ is actually a partial order on $V$. In this paper we give sharp Cantelli's type inequalities for generalized tail probabilities like $\text{Pr}\{X\succeq_C b\}$ for $b\in V$. These inequalities are obtained by ``scalarizing'' $X\succeq_C b$ via cone duality and then by minimizing the classical univariate Cantelli's bound over the scalarized inequalities

A Superclass of Edge-Path-Tree graphs with few cliques

Edge-Path-Tree graphs are intersection graphs of Edge-Path-Tree matrices that is matrices whose columns are incidence vectors of edge-sets of paths in a given tree. Edge-Path-Tree graphs have polynomially many cliques as proved in [4] and [7]. Therefore, the problem of finding a clique of maximum weight in these graphs is solvable in strongly polynomial time. In this paper we extend this result to a proper superclass of Edge-Path-Tree graphs. Each graph in the class is defined as the intersection graph of a matrix with no submatrix in a set W of seven small forbidden submatrices. By forbidding an eighth small matrix, our result specializes to Edge-Path-Tree graph

Two New Characterizations of Path Graphs

Path graphs are intersection graphs of paths in a tree. We start from the characterization of path graphs by Monma and Wei [C.L.~Monma,~and~V.K.~Wei, Intersection Graphs of Paths in a Tree, J. Combin. Theory Ser. B, 41:2 (1986) 141--181] and we reduce it to some 2-colorings subproblems, obtaining the first characterization that directly leads to a polynomial recognition algorithm. Then we introduce the collection of the attachedness graphs of a graph and we exhibit a list of minimal forbidden 2-edge colored subgraphs in each of the attachedness graph.Comment: 18 pages, 6 figure

Network homophily via multi-dimensional extensions of Cantelli's inequality

Homophily is the principle whereby "similarity breeds connections". We give a quantitative formulation of this principle within networks. We say that a network is homophillic with respect to a given labeled partition of its vertices, when the classes of the partition induce subgraphs that are significantly denser than what we expect under a random labeled partition into classes maintaining the same cardinalities (type). This is the recently introduced \emph{random coloring model} for network homophily. In this perspective, the vector whose entries are the sizes of the subgraphs induced by the corresponding classes, is viewed as the observed outcome of the random vector described by picking labeled partitions at random among partitions with the same type.\,Consequently, the input network is homophillic at the significance level $\alpha$ whenever the one-sided tail probability of observing an outcome at least as extreme as the observed one, is smaller than $\alpha$. Clearly, $\alpha$ can also be thought of as a quantifier of homophily in the scale $[0,1]$. Since, as we show, even approximating this tail probability is an NP-hard problem, we resort multidimensional extensions of classical Cantelli's inequality to bound $\alpha$ from above. This upper bound is the homophily index we propose. It requires the knowledge of the covariance matrix of the random vector, which was not previously known within the random coloring model. In this paper we close this gap by computing the covariance matrix of subgraph sizes under the random coloring model. Interestingly, the matrix depends on the input partition only through its type and on the network only through its degrees. Furthermore all the covariances have the same sign and this sign is a graph invariant. Plugging this structure into Cantelli's bound yields a meaningful, easy to compute index for measuring network homophily

A tight relation between series--parallel graphs and bipartite distance hereditary graphs

Bandelt and Mulder’s structural characterization of bipartite distance hereditary graphs asserts that such graphs can be built inductively starting from a single vertex and by re17 peatedly adding either pendant vertices or twins (i.e., vertices with the same neighborhood as an existing one). Dirac and Duffin’s structural characterization of 2–connected series–parallel graphs asserts that such graphs can be built inductively starting from a single edge by adding either edges in series or in parallel. In this paper we give an elementary proof that the two constructions are the same construction when bipartite graphs are viewed as the fundamental graphs of a graphic matroid. We then apply the result to re-prove known results concerning bipartite distance hereditary graphs and series–parallel graphs and to provide a new class of polynomially-solvable instances for the integer multi-commodity flow of maximum valu

Polynomial algorithms for partitioning a tree into single-center subtrees to minimize flat service costs

This paper deals with the following graph partitioning problem. Consider a connected graph with n nodes, p of which are centers, while the remaining ones are units. For each unit-center pair there is a fixed service cost and the goal is to find a partition into connected components such that each component contains only one center and the total service cost is minimum. This problem is known to be NP-hard on general graphs, and here we show that it remains such even if the service cost is monotone and the graph is bipartite. However, in this paper we derive some polynomial time algorithms for trees. For this class of graphs we provide several reformulations of the problem as integer linear programs proving the integrality of the corresponding polyhedra. As a consequence, the tree partitioning problem can be solved in polynomial time either by linear programming or by suitable convex nondifferentiable optimization algorithms. Moreover, we develop a dynamic programming algorithm, whose recursion is based on sequences of minimum weight closure problems, which solves the problem on trees in O(np) time

Normal approximation of Random Gaussian Neural Networks

In this paper we provide explicit upper bounds on some distances between the (law of the) output of a random Gaussian NN and (the law of) a random Gaussian vector. Our results concern both shallow random Gaussian neural networks with univariate output and fully connected and deep random Gaussian neural networks, with a rather general activation function. The upper bounds show how the widths of the layers, the activation functions and other architecture parameters affect the Gaussian approximation of the ouput. Our techniques, relying on Stein's method and integration by parts formulas for the Gaussian law, yield estimates on distances which are indeed integral probability metrics, and include the total variation and the convex distances. These latter metrics are defined by testing against indicator functions of suitable measurable sets, and so allow for accurate estimates of the probability that the output is localized in some region of the space. Such estimates have a significant interest both from a practitioner's and a theorist's perspective

A human-neutral large carnivore? No patterns in the body mass of gray wolves across a gradient of anthropization

The gray wolf (Canis lupus) expanded its distribution in Europe over the last few decades. To better understand the extent to which wolves could re-occupy their historical range, it is important to test if anthropization can affect their fitness-related traits. After having accounted for ecologically relevant confounders, we assessed how anthropization influenced i) the growth of wolves during their first year of age (n = 53), ii) sexual dimorphism between male and female adult wolves (n = 121), in a sample of individuals that had been found dead in Italy between 1999 and 2021. Wolves in anthropized areas have a smaller overall variation in their body mass, during their first year of age. Because they already have slightly higher body weight at 3–5 months, possibly due to the availability of human-derived food sources. The difference in the body weight of adult females and males slightly increases with anthropization. However, this happens because of an increase in the body mass of males only, possibly due to sex-specific differences in dispersal and/or to “dispersal phenotypes”. Anthropization in Italy does not seem to have any clear, nor large, effect on the body mass of wolves. As body mass is in turn linked to important processes, like survival and reproduction, our findings indicates that wolves could potentially re-occupy most of their historical range in Europe, as anthropized landscapes do not seem to constrain such of an important life-history trait. Wolf management could therefore be needed across vast spatial scales and in anthropized areas prone to social conflicts