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We present a framework to define a large class of neural networks for which, by construction, training by gradient flow provably reaches arbitrarily low loss when the number of parameters grows. Distinct from the fixed-space global…
We consider a variation of the supermarket model in which the servers can communicate with their neighbors and where the neighborhood relationships are described in terms of a suitable graph. Tasks with unit-exponential service time…
Random networks are a powerful tool in the analytical modeling of complex networks as they allow us to write approximate mathematical models for diverse properties and behaviors of networks. One notable shortcoming of these models is that…
The contact process and the slightly different susceptible-infected-susceptible model are studied on long-range connected networks in the presence of random transition rates by means of a strong disorder renormalization group method and…
We study the connectivity of random subgraphs of the $d$-dimensional Hamming graph $H(d, n)$, which is the Cartesian product of $d$ complete graphs on $n$ vertices. We sample the random subgraph with an i.i.d.\ Bernoulli bond percolation on…
We study a graph-theoretic property known as robustness, which plays a key role in certain classes of dynamics on networks (such as resilient consensus, contagion and bootstrap percolation). This property is stronger than other graph…
A wide array of random graph models have been postulated to understand properties of observed networks. Typically these models have a parameter $t$ and a critical time $t_c$ when a giant component emerges. It is conjectured that for a large…
We consider the contact process on a countable-infinite and connected graph of bounded degree. For this process started from the upper invariant measure, we prove certain uniform mixing properties under the assumption that the infection…
We prove central limit theorems (CLTs) for topological functionals of Bernoulli bond percolation on infinite graphs beyond the Euclidean lattice $\mathbb{Z}^{d}$. For quasi-transitive graphs of subexponential growth, we show that the number…
We present a detailed study of the evolution of the number of connected components in sub-critical multiplicative random graph processes. We consider a model where edges appear independently after an exponential time at rate equal to the…
Clustering $\unicode{x2013}$ the tendency for neighbors of nodes to be connected $\unicode{x2013}$ quantifies the coupling of a complex network to its latent metric space. In random geometric graphs, clustering undergoes a continuous phase…
In this paper, we formulate and investigate a generalized consensus algorithm which makes an attempt to unify distributed averaging and maximizing algorithms considered in the literature. Each node iteratively updates its state as a…
We consider the count of subgraphs with an arbitrary configuration of endpoints in the random-connection model based on a Poisson point process on ${\Bbb R}^d$. We present combinatorial expressions for the computation of the cumulants and…
A key task in the study of networked systems is to derive local and global properties that impact connectivity, synchronizability, and robustness; computing shortest paths or geodesics yields measures of network connectivity that can…
We theoretically study semi-supervised clustering in sparse graphs in the presence of pairwise constraints on the cluster assignments of nodes. We focus on bi-cluster graphs, and study the impact of semi-supervision for varying constraint…
We show that the canonical random-cluster measure associated to isoradial graphs is critical for all $q \geq 1$. Additionally, we prove that the phase transition of the model is of the same type on all isoradial graphs: continuous for $1…
Recently, it has been proposed that the natural connectivity can be used to efficiently characterise the robustness of complex networks. Natural connectivity quantifies the redundancy of alternative routes in a network by evaluating the…
We show that the contact process on a random $d$-regular graph initiated by a single infected vertex obeys the "cutoff phenomenon" in its supercritical phase. In particular, we prove that when the infection rate is larger than the critical…
We study analytic properties of graph product of finite groups with a hyperbolic defining graph. This is done by studying dynamics on the Bowditch compactification of the extension graph, or the crossing graph, of graph product. In…
Any infinite graph has site and bond percolation critical probabilities satisfying $p_c^{site}\geq p_c^{bond}$. The strict version of this inequality holds for many, but not all, infinite graphs. In this paper, the class of graphs for which…