We study stationary epidemic processes in scale-free networks with local-awareness behavior adopted by only susceptible, infected, or all nodes. find that, while the size susceptible-aware and all-aware models scales linearly network size, scaling becomes sublinear infected-aware model. Hence, fewer aware nodes may reduce more effectively; a phenomenon reminiscent of Braess's paradox. present numerical theoretical analysis highlight role influential their disassortativity to raise awareness....

We investigate local-density dependent Markov processes on a class of large graphs sampled from graphon, where the transition rates vertices are influenced by states their neighbors. show that as average degree converges to infinity (faster than given threshold), evolution process in transient regime solution set non-local integro-partial differential equations depending limit graphon. also provide rigorous derivation for epidemic threshold case Susceptible–Infected–Susceptible (SIS) such graphons.

We provide error bounds for the N-intertwined mean-field approximation (NIMFA) local density-dependent Markov population processes with a well-distributed underlying network structure showing NIMFA being accurate when typical vertex has many neighbors. The result justifies some of most common approximations used in epidemiology, statistical physics and opinion dynamics literature under certain conditions. allow interactions between more than 2 individuals, an hypergraph accordingly.

Abstract We study Markov population processes on large graphs, with the local state transition rates of a single vertex being linear function its neighborhood. A simple way to approximate such is by system ODEs called homogeneous mean-field approximation (HMFA). Our main result showing that HMFA guaranteed be graph limit stochastic dynamics finite time horizon if and only graph-sequence quasi-random. An explicit error bound given it $$\frac{1}{\sqrt{N}}$$ <mml:math...

In physical networks, like the brain or metamaterials, we often observe local bundles, corresponding to locally aligned link configurations. Here introduce a minimal model for bundle formation, modeling networks as non-equilibrium packings of hard-core 3D elongated links. We show that growth is logarithmic in time, stark contrast with algebraic behavior lower dimensional random packing models. Equally important, find this slow kinetics metastable, allowing us analytically predict an due...

We study Markov processes on weighted directed hypergraphs where the state of at most one vertex can change a time. Our setting is general enough to include simplicial epidemic processes, multilayered networks or even dynamics edges graph. results are twofold. Firstly, we prove concentration bounds for number vertices in certain under mild assumptions. imply that empirical averages subpopulations diverging but possibly sublinear size well concentrated around their mean. In case undirected...

We study stationary epidemic processes in scale-free networks with local awareness behavior adopted by only susceptible, infected, or all nodes. find that while the size susceptible-aware and all-aware scenarios scales linearly network size, scaling becomes sublinear infected-aware scenario, suggesting fewer aware nodes may reduce more effectively. explain this paradox via numerical theoretical analysis, highlight role of influential their disassortativity to raise scenarios.

Abstract We study SIR‐type epidemics (susceptible‐infected‐resistant) on graphs in two scenarios: (i) when the initial infections start from a well‐connected central region and (ii) are distributed uniformly. Previously, Ódor et al. demonstrated few random graph models that expectation of total number undergoes switchover phenomenon; is more dangerous for small infection rates, while large uniform seeding expected to infect nodes. rigorously prove this claim under mild, deterministic...

We study Markov population processes on large graphs, with the local state transition rates of a single vertex being linear function its neighborhood. A simple way to approximate such is by system ODEs called homogeneous mean-field approximation (HMFA). Our main result showing that HMFA guaranteed be graph limit stochastic dynamics finite time horizon if and only graph-sequence quasi-random. Explicit error bound given order $\frac{1}{\sqrt{N}}$ plus largest discrepancy graph. For Erd\H{o}s...

We study SIR type epidemics on graphs in two scenarios: (i) when the initial infections start from a well connected central region, (ii) are distributed uniformly. Previously, \'Odor et al. demonstrated few random graph models that expectation of total number undergoes switchover phenomenon; region is more dangerous for small infection rates, while large uniform seeding expected to infect nodes. rigorously prove this claim under mild, deterministic assumptions underlying graph. If we further...

We are investigating deterministic SIS dynamics on large networks starting from only a few infected individuals. Under mild assumptions we show that any two epidemic curves -- the same network and with parameters almost identical up to time translation when initial conditions small enough, regardless of how infections distributed at beginning. The limit object an infinite past infinitesimally prevalence is identified as nontrivial eternal solution connecting disease free state endemic...

We investigate local-density dependent Markov processes on a class of large graphs sampled from graphon, where the transition rates vertices are influenced by states their neighbors. show that as average degree converges to infinity, evolution process in transient regime solution set non-local integro-partial differential equations. also provide rigorous derivation for epidemic threshold case Susceptible-Infected-Susceptible (SIS) such graphons.

We study the asymptotic behaviour of Markov processes on large weighted Erdos-Renyi graphs where transition rates vertices are only influenced by state their neighbours and corresponding weight edges. find ratio being in a certain will converge to solution differential equation obtained from mean field approximation if graph is dense enough, namely, average degree at least order $N^{\frac{1}{2}+\epsilon}$. Proof for convergence probability transient regime shown.

We provide error bounds for the N-intertwined mean-field approximation (NIMFA) local density-dependent Markov population processes with a well-distributed underlying network structure showing NIMFA being accurate when typical vertex has many neighbors. The result justifies some of most common approximations used in epidemiology, statistical physics and opinion dynamics literature under certain conditions. allow interactions between more than 2 individuals, an hypergraph accordingly.

We are investigating deterministic SIS dynamics on large networks starting from only a few infected individuals. Under mild assumptions we show that any two epidemic curves - the same network and with parameters almost identical up to time translation when initial conditions small enough regardless of how infections distributed at beginning. The limit object an infinite past infinitesimal prevalence is identified as nontrivial eternal solution connecting disease free state endemic...

We provide error bounds for the N-intertwined mean-field approximation (NIMFA) local density-dependent Markov population processes with a well-distributed underlying network structure showing NIMFA being accurate when typical vertex has many neighbors. The result justifies some of most common approximations used in epidemiological modeling literature under certain conditions. allow interactions between more than 2 individuals, and an hypergraph accordingly.