arXiv subjects

Nicola Apollonio

Publications and source records attributed to Nicola Apollonio.

14 recordsLinked to original sources

arXiv2026

A universal dichotomy for concentration in randomly colored graphs

Let $ζ$ be Euclidean norm of the degree sequence of a graph normalized by the graph size. We prove that when the vertices of a graph are randomly colored with $s$ colors such that the fraction of vertices in each color class is bounded away from zero, only two asymptotic regimes emerge. If $ζ=o(1)$, then the sizes of the subgraphs induced by the color classes concentrate around their expected values. If $ζ=Θ(1)$, then concentration depends on the color balance: for colorings with persisting imbalance, the total number $M$ of monochromatic edges stays bounded away from its mean with positive probability; otherwise, for vanishing imbalance, $M$ still concentrates. The same dichotomy holds for a broad class of randomly colored random graphs.

math.CO↗

arXiv2026

Posterior Bayesian Neural Networks with Dependent Weights

We consider fully connected and feedforward deep neural networks with dependent and possibly heavy-tailed weights, as introduced in [26], to address limitations of the standard Gaussian prior. It has been proved in [26] that, as the number of nodes in the hidden layers grows large, according to a sequential and ordered limit, the law of the output converges weakly to a Gaussian mixture. In this paper, we study the neural network through the lens of the posterior distribution with a Gaussian likelihood. If the random covariance matrix of the infinite-width limit is positive definite under the prior, we identify the posterior distribution of the output in the wide-width limit according to a sequential regime. Remarkably, we provide mild sufficient conditions to ensure the aforementioned invertibility of the random covariance matrix under the prior, thereby extending the results in [8]. Among our results, we present sufficient conditions on some model parameters (the activation function and the associated Lévy measures) which ensure that the sequential limits are independent of the order. We illustrate our findings with examples and numerical simulations.

stat.ML↗

arXiv2026

The Critical Patch Size Problem in Random Graphs

The problem of {\it critical patch size} -- a threshold condition for population persistence -- is investigated in the context of discrete habitats, modeled as graphs with a distinguished subset of vertices acting as sinks. These sinks impose boundary-like constraints analogous to Dirichlet conditions in continuous domains. The population proliferates locally at the vertices and diffuse across the network through the graph Laplacian. In the sinks the population cannot survive. The Dirichlet eigenvalue of the habitat is defined as the smallest eigenvalue of the principal submatrix of the Laplacian obtained by removing the rows and columns associated with sink vertices. This spectral parameter governs the habitat's viability: survival occurs when the Dirichlet eigenvalue of the habitat lies below a critical reaction-to-diffusion ratio. We study survival conditions for a sequence of random habitats built on binomial random graphs. We establish a law of large numbers for the corresponding sequence of Dirichlet eigenvalues and prove the emergence of a sharp threshold phenomenon: with high probability, a large random habitat is either viable or non-viable, depending on whether the reaction-to-diffusion ratio lies below or above this threshold. Our results provide the first general spectral theory for critical patch size on graphs, with implications for ecology, synthetic biology, and the modeling of processes on brain connectomes.

math.PR↗

arXiv2024

Functions that are uniquely maximized by sparse quasi-star graphs, and uniquely minimized by quasi-complete graphs

We show that for a certain class of convex functions $f$, including the exponential functions $x\mapsto e^{λx}$ with $λ>0$ a real number, and all the powers $x\mapsto x^β$, $x\geq 0$ and $β\geq 2$ a real number, with a unique small exception, if $(d_1,\ldots,d_n)$ ranges over the degree sequences of graphs with $n$ vertices and $m$ edges and $m\leq n-1$, then the maximum of $\sum_i f(d_i)$ is uniquely attained by the degree sequence of a quasi-star graph, namely, a graph consisting of a star plus possibly additional isolated vertices. This result significantly extends a similar result in [D.~Ismailescu, D.~Stefanica, Minimizer graphs for a class of extremal problems, J.~Graph Theory,~39~(4)~(2002)]. Dually, we show that for a certain class of concave functions $g$, including the negative exponential functions $x\mapsto 1-e^{-λx}$ with $λ>\ln(2)$ a real number, all the powers $x\mapsto x^α$, $x\geq 0$ and $0<α\leq \frac{1}{2}$ a real number, and the function $x\mapsto \frac{x}{x+1}$ for $x\geq 0$, if $(d_1,\ldots,d_n)$ ranges over the degree sequences of graphs with $n$ vertices and $m$ edges, then the minimum of $\sum_i g(d_i)$ is uniquely attained by the degree sequence of a quasi-complete graph, i.e., a graph consisting of a complete graph plus possibly an additional vertex connected to some but not all vertices of the complete graph, plus possibly isolated vertices. This result extends a similar result in the same paper.

math.CO↗

arXiv2023

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.

math.PR↗

arXiv2023

Network homophily via tail inequalities

Homophily is the principle whereby "similarity breeds connections". We give a quantitative formulation of this principle within networks. Given a network and a labeled partition of its vertices, the vector indexed by each class of the partition, whose entries are the number of edges 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 labeled partitions whose classes have the same cardinalities as the given one. In this perspective, the value of any homophily score $Θ$, namely a non decreasing real valued function in the sizes of subgraphs induced by the classes of the partition, evaluated at the observed outcome, can be thought of as the observed value of a random variable. Consequently, according to the score $Θ$, the input network is homophillic at the significance level $α$ whenever the one-sided tail probability of observing a value of $Θ$ at least as extreme as the observed one, is smaller than $α$. Since, as we show, even approximating $α$ is an NP-hard problem, we resort to classical tails inequality to bound $α$ from above. These upper bounds, obtained by specializing $Θ$, yield a class of quantifiers of network homophily. Computing the upper bounds 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, giving a meaningful, easy to compute class of indices for measuring network homophily. As demonstrated in real-world network applications, these indices are effective, reliable, and lead to new discoveries that could not be captured by the current state of the art.

math.ST↗

arXiv2023

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.

math.PR↗

arXiv2023

Second-order moments of the size of randomly induced subgraphs of given order

For a graph $G$ and a positive integer $c$, let $M_c(G)$ be the size of a subgraph of $G$ induced by a randomly sampled subset of $c$ vertices. Second-order moments of $M_c(G)$ encode part of the structure of $G$. We use this fact, coupled to classical moment inequalities, to prove graph theoretical results, to give combinatorial identities, to bound the size of the $c$-densest subgraph from below and the size of the $c$-sparsest subgraph from above, and to provide bounds for approximate enumeration of trivial subgraphs.

math.CO↗

arXiv2022

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.

math.CO↗

arXiv2022

A novel method for assessing and measuring homophily in networks through second-order statistics

We present a new method for assessing and measuring homophily in networks whose nodes have categorical attributes, namely when the nodes of networks come partitioned into classes (colors). We probe this method in two different classes of networks: i) protein-protein interaction (PPI) networks, where nodes correspond to proteins, partitioned according to their functional role, and edges represent functional interactions between proteins ii) Pokec on-line social network, where nodes correspond to users, partitioned according to their age, and edges respresent friendship between users. Similarly to other classical and well consolidated approaches, our method compares the relative edge density of the subgraphs induced by each class with the corresponding expected relative edge density under a null model. The novelty of our approach consists in prescribing an endogenous null model, namely, the sample space of the null model is built on the input network itself. This allows us to give exact explicit expression for the z-score of the relative edge density of each class as well as other related statistics. The z-scores directly quantify the statistical significance of the observed homophily via Tchebycheff inequality. The expression of each z-score is entered by the network structure through basic combinatorial invariant such as the number of subgraphs with two spanning edges. Each z-score is computed in O(n + m) time for a network with n nodes and m edges. This leads to an overall efficient computational method for assesing homophily. We complement the analysis of homophily/heterophily by considering z-scores of the number of isolated nodes in the subgraphs induced by each class, that are computed in O(nm) time. Theoretical results are then exploited to show that, as expected, both the analyzed network classes are significantly homophilic with respect to the considered node properties.

cs.DM↗

arXiv2021

A New Characterization of Path Graphs

Path graphs are intersection graphs of paths in a tree.~In this paper we give a "6\ good characterization" of path graphs, namely, we prove that path graph membership is in $NP\cap CoNP$ without resorting to existing polynomial time algorithms. The characterization is given in terms of the collection of the \emph{attachedness graphs} of a graph, a novel device to deal with the connected components of a graph after the removal of clique separators. On the one hand, the characterization refines and simplifies the characterization of path graphs due to Monma and Wei [C.L.~Monma,~and~V.K.~Wei, Intersection {G}raphs of {P}aths in a {T}ree, J. Combin. Theory Ser. B, 41:2 (1986) 141--181], which we build on, by reducing a constrained vertex coloring problem defined on the \emph{attachedness graphs} to a vertex 2-coloring problem on the same graphs. On the other hand, the characterization allows us to exhibit two exhaustive lists of obstructions to path graph membership in the form of minimal forbidden induced/partial 2-edge colored subgraphs in each of the \emph{attachedness graphs}.

cs.DM↗

arXiv2015

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 repeatedly adding either pending 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 prove 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, to characterize self-dual outer-planar graphs and, finally, to provide a new class of polynomially-solvable instances for the integer multi commodity flow of maximum value.

cs.DM↗

arXiv2015

On Computing the Galois Lattice of Bipartite Distance Hereditary Graphs

The class of Bipartite Distance Hereditary (BDH) graphs is the intersection between bipartite domino-free and chordal bipartite graphs.\ Graphs in both the latter classes have linearly many maximal bicliques, implying the existence of polynomial-time algorithms for computing the associated Galois lattice.\ Such a lattice can indeed be built in $O(m\times n)$ worst case-time for a domino-free graph with $m$ edges and $n$ vertices.\ In this paper we give a sharp estimate on the number of the maximal bicliques of BDH graphs and exploit such result to give an $O(m)$ worst case time algorithm for computing the Galois lattice of BDH graphs. By relying on the fact that neighborhoods of vertices of BDH graphs can be realized as directed paths in a arborescence, we give an $O(n)$ worst-case space and time encoding of both the input graph and its Galois lattice, provided that the reverse of a Bandelt and Mulder building sequence is given.

cs.DM↗

arXiv2014

On the Galois Lattice of Bipartite Distance Hereditary Graphs

We give a complete characterization of bipartite graphs having tree-like Galois lattices. We prove that the poset obtained by deleting bottom and top elements from the Galois lattice of a bipartite graph is tree-like if and only if the graph is a Bipartite Distance Hereditary graph. By relying on the interplay between bipartite distance hereditary graphs and series-parallel graphs, we show that the lattice can be realized as the containment relation among directed paths in an arborescence. Moreover, a compact encoding of Bipartite Distance Hereditary graphs is proposed, that allows optimal time computation of neighborhood intersections and maximal bicliques.

cs.DM↗