31 research outputs found

### Connectivity of inhomogeneous random graphs

We find conditions for the connectivity of inhomogeneous random graphs with
intermediate density. Our results generalize the classical result for G(n, p),
when p = c log n/n. We draw n independent points X_i from a general
distribution on a separable metric space, and let their indices form the vertex
set of a graph. An edge (i,j) is added with probability min(1, \K(X_i,X_j) log
n/n), where \K \ge 0 is a fixed kernel. We show that, under reasonably weak
assumptions, the connectivity threshold of the model can be determined.Comment: 13 pages. To appear in Random Structures and Algorithm

### THE DIAMETER OF INHOMOGENEOUS RANDOM GRAPHS

International audienceIn this paper we study the diameter of inhomogeneous random graphs G(n, κ, p) that are induced by irreducible kernels κ. The kernels we consider act on separable metric spaces and are almost everywhere continuous. We generalize results known for the Erd˝ os-Rényi model G(n, p) for several ranges of p. We find upper and lower bounds for the diameter of G(n, κ, p) in terms of the expansion factor and two explicit constants that depend on the behavior of the kernel over partitions of the metric space

### The Bulk and the Extremes of Minimal Spanning Acycles and Persistence Diagrams of Random Complexes

Frieze showed that the expected weight of the minimum spanning tree (MST) of
the uniformly weighted graph converges to $\zeta(3)$. Recently, this result was
extended to a uniformly weighted simplicial complex, where the role of the MST
is played by its higher-dimensional analogue -- the Minimum Spanning Acycle
(MSA). In this work, we go beyond and look at the histogram of the weights in
this random MSA -- both in the bulk and in the extremes. In particular, we
focus on the `incomplete' setting, where one has access only to a fraction of
the potential face weights. Our first result is that the empirical distribution
of the MSA weights asymptotically converges to a measure based on the shadow --
the complement of graph components in higher dimensions. As far as we know,
this result is the first to explore the connection between the MSA weights and
the shadow. Our second result is that the extremal weights converge to an
inhomogeneous Poisson point process. A interesting consequence of our two
results is that we can also state the distribution of the death times in the
persistence diagram corresponding to the above weighted complex, a result of
interest in applied topology.Comment: 15 pages, 5 figures, Corrected Typo

### Opinion dynamics on directed complex networks

We propose and analyze a mathematical model for the evolution of opinions on
directed complex networks. Our model generalizes the popular DeGroot and
Friedkin-Johnsen models by allowing vertices to have attributes that may
influence the opinion dynamics. We start by establishing sufficient conditions
for the existence of a stationary opinion distribution on any fixed graph, and
then provide an increasingly detailed characterization of its behavior by
considering a sequence of directed random graphs having a local weak limit. Our
most explicit results are obtained for graph sequences whose local weak limit
is a marked Galton-Watson tree, in which case our model can be used to explain
a variety of phenomena, e.g., conditions under which consensus can be achieved,
mechanisms in which opinions can become polarized, and the effect of disruptive
stubborn agents on the formation of opinions

### Stochastic recursions on directed random graphs

For a directed graph $G(V_n, E_n)$ on the vertices $V_n = \{1,2, \dots, n\}$,
we study the distribution of a Markov chain $\{ {\bf R}^{(k)}: k \geq 0\}$ on
$\mathbb{R}^n$ such that the $i$th component of ${\bf R}^{(k)}$, denoted
$R_i^{(k)}$, corresponds to the value of the process on vertex $i$ at time $k$.
We focus on processes $\{ {\bf R}^{(k)}: k \geq 0\}$ where the value of
$R_i^{(k+1)}$ depends only on the values $\{ R_j^{(k)}: j \to i\}$ of its
inbound neighbors, and possibly on vertex attributes. We then show that,
provided $G(V_n, E_n)$ converges in the local weak sense to a marked
Galton-Watson process, the dynamics of the process for a uniformly chosen
vertex in $V_n$ can be coupled, for any fixed $k$, to a process $\{
\mathcal{R}_\emptyset^{(r)}: 0 \leq r \leq k\}$ constructed on the limiting
marked Galton-Watson tree. Moreover, we derive sufficient conditions under
which $\mathcal{R}^{(k)}_\emptyset$ converges, as $k \to \infty$, to a random
variable $\mathcal{R}^*$ that can be characterized in terms of the attracting
endogenous solution to a branching distributional fixed-point equation. Our
framework can also be applied to processes $\{ {\bf R}^{(k)}: k \geq 0\}$ whose
only source of randomness comes from the realization of the graph $G(V_n,
E_n)$