arXiv subjects

Gadi Fibich

Publications and source records attributed to Gadi Fibich.

At least 19 recordsLinked to original sources

arXiv2025

A phase transition in the Susceptible-Infected model on hypernetworks

We derive the master equations for the Susceptible-Infected (SI) model on general hypernetworks with~$N$-body interactions. We solve these equations exactly for infinite~$d$-regular hypernetworks, and obtain an explicit solution for the expected infection level as a function of time. The solution shows that the epidemic spreads out to the entire population as~$t \to \infty$ if and only if the initial infection level exceeds a positive threshold value. This phase transition is a high-order interactions effect, which is absent with pairwise interactions.

physics.soc-ph↗

arXiv2025

Optimal promotions of new products on networks

We present a novel methodology for analyzing the optimal promotion in the Bass model for the spreading of new products on networks. For general networks with $M$ nodes, the optimal promotion is the solution of $2^M-1$ nonlinearly-coupled boundary-value problems. On structured networks, however, the number of equations can be reduced to a manageable size which is amendable to simulations and analysis. This enables us to gain insight into the effect of the network structure on optimal promotions. We find that the optimal advertising strategy decreases with time, whereas the optimal boosting of peer effects increases from zero and then decreases. In low-degree networks, it is optimal to prioritize advertising over boosting peer effects, but this relation is flipped in high-degree networks. When the planning horizon is finite, the optimal promotion continues until the last minute, as opposed to an infinite planning horizon where the optimal promotion decays to zero. Finally, promotions with short planning horizons can yield an order of magnitude higher increase of profits, compared to those with long planning horizons.

math.OC↗

arXiv2024

Funnel theorems for spreading on networks

We derive novel analytic tools for the Bass and SI models on networks for the spreading of innovations and epidemics on networks. We prove that the correlation between the nonadoption (noninfection) probabilities of $L \ge 2$ disjoint subsets of nodes $\{A_l\}_{l=1}^L$ is non-negative, find the necessary and sufficient condition that determines whether this correlation is positive or zero, and provide an upper bound for its magnitude. Using this result, we prove the funnel theorems, which provide lower and upper bounds for the difference between the non-adoption probability of a node and the product of its nonadoption probabilities on $L$ modified networks in which the node under consideration is only influenced by incoming edges from $A_l$ for $l=1, \dots, L$. The funnel theorems can be used, among other things, to explicitly compute the exact expected adoption/infection level on various types of networks, both with and without cycles.

physics.soc-ph↗

arXiv2024

Explicit solutions of the SI and Bass models on sparse Erd\H{o}s-R\'enyi and regular networks

We derive explicit expressions for the expected adoption and infection level in the Bass and SI models, respectively, on sparse Erd\H{o}s-R\'enyi networks and on $d$-regular networks. These expressions are soloutions of first-order ordinary differential equations, which are fairly easy to analyze. To prove that these expressions are exact, we show that the effect of cycles vanishes as the network size goes to infinity.

math.PR↗

arXiv2024

Explicit solutions of the Bass and SI models on hypernetworks

We analyze the Bass model and the Susceptible-Infected (SI) model on hypergraphs with 3-body interactions. We derive the master equations for general hypernetworks, and use them to obtain explicit expressions for the expected adoption/infection level on infinite complete hypernetworks, infinite Erdős-Rényi hypernetworks, and on infinite hyperlines. These expressions are exact, as they are derived without making any approximation.

physics.soc-ph↗

arXiv2024

Monotone convergence of spreading processes on networks

We analyze the Bass and SI models for the spreading of innovations and epidemics, respectively, on homogeneous complete networks, circular networks, and heterogeneous complete networks with two homogeneous groups. We allow the network parameters to be time dependent, which is a prerequisite for the analysis of optimal strategies on networks. Using a novel top-down analysis of the master equations, we present a simple proof for the monotone convergence of these models to their respective infinite-population limits. This leads to explicit expressions for the expected adoption or infection level in the Bass and SI models, respectively, on infinite homogeneous complete and circular networks, and on heterogeneous complete networks with two homogeneous groups with time-dependent parameters.

math.CA↗

arXiv2024

Boundary Effects in the Diffusion of New Products on Cartesian Networks

We analyze the effect of boundaries in the discrete Bass model on D-dimensional Cartesian networks. In 2D, this model describes the diffusion of new products that spread primarily by spatial peer effects, such as residential photovoltaic solar systems. We show analytically that nodes (residential units) that are located near the boundary are less likely to adopt than centrally-located ones. This boundary effect is local, and decays exponentially with the distance from the boundary. At the aggregate level, boundary effects reduce the overall adoption level. The magnitude of this reduction scales as~$\frac{1}{M^{1/D}}$, where~$M$ is the number of nodes. Our analysis is supported by empirical evidence on the effect of boundaries on the adoption of solar.

physics.soc-ph↗

arXiv2023

Universal Bounds for Spreading on Networks

Spreading (diffusion) of innovations is a stochastic process on social networks. When the key driving mechanism is peer effects (word of mouth), the rate at which the aggregate adoption level increases with time depends strongly on the network structure. In many applications, however, the network structure is unknown. To estimate the aggregate adoption level for such innovations, we show that the two networks that correspond to the slowest and fastest adoption levels are a homogeneous two-node network and a homogeneous infinite complete network, respectively. Solving the stochastic Bass model on these two networks yields explicit lower and upper bounds for the adoption level on any network. These bounds are tight, and they also hold for the individual adoption probabilities of nodes. The gap between the lower and upper bounds increases monotonically with the ratio of the rates of internal and external influences.

math.PR↗

arXiv2023

Funnel Theorems for Spreading on Networks

We derive novel analytic tools for the discrete Bass model, which models the diffusion of new products on networks. We prove that the probability that any two nodes adopt by time t, is greater than or equal to the product of the probabilities that each of the two nodes adopts by time t. We introduce the notion of an "influential node", and use it to determine whether the above inequality is strict or an equality. We then use the above inequality to prove the "funnel inequality", which relates the adoption probability of a node to the product of its adoption probability on two sub-networks. We introduce the notion of a "funnel node", and use it to determine whether the funnel inequality is strict or an equality. The above analytic tools can be exptended to epidemiological models on networks. We then use the funnel theorems to derive a new inequality for diffusion on circles and a new explicit expression for the adoption probabilities of nodes on two-sided line, and to prove that the adoption level on one-sided lines is strictly slower than on anisotropic two-sided lines, and that the adoption level on multi-dimensional Cartesian networks is bounded from below by that on one-dimensional networks.

math.PR↗

arXiv2022

Compartmental limit of discrete Bass models on networks

We introduce a new method for proving the convergence and the rate of convergence of discrete Bass models on various networks to their respective compartmental Bass models, as the population size $M$ becomes infinite. In this method, the full set of master equations is reduced to a smaller system of equations, which is closed and exact. The reduced finite system is embedded into an infinite system, and the convergence of that system to the infinite limit system is proved using standard ODE estimates. Finally, an ansatz provides an exact closure of the infinite limit system, which reduces that system to the compartmental model. Using this method, we show that when the network is complete and homogeneous, the discrete Bass model converges to the original 1969 compartmental Bass model, at the rate of $1/M$. When the network is circular, however, the compartmental limit is different, and the rate of convergence is exponential in $M$. In the case of a heterogeneous network that consists of $K$ homogeneous groups, the limit is given by a heterogeneous compartmental Bass model, and the rate of convergence is $1/M$. Using this compartmental model, we show that when the heterogeneity in the external and internal influence parameters among the $K$ groups is positively monotonically related, heterogeneity slows down the diffusion.

math.CA↗

arXiv2022

Diffusion of new products with heterogeneous consumers

Does a new product spread faster among heterogeneous or homogeneous consumers? We analyze this question using the stochastic discrete Bass model, in which consumers may differ in their individual external influence rates $\{p_j \}$ and in their individual internal influence rates $\{ q_j \}$. When the network is complete and the heterogeneity is only manifested in $\{p_j \}$ or only in $\{ q_j \}$, it always slows down the diffusion, compared to the corresponding homogeneous network. When, however, consumers are heterogeneous in both $\{p_j\}$ and $\{q_{j}\}$, heterogeneity slows down the diffusion in some cases, but accelerates it in others. Moreover, the dominance between the heterogeneous and homogeneous adoption levels is global in time in some cases, but changes with time in others. Perhaps surprisingly, global dominance between two networks is not always preserved under "additive transformations", such as adding an identical node to both networks. When the network is not complete, the effect of heterogeneity depends also on its spatial distribution within the network.

math.OC↗

arXiv2021

Exact description of SIR-Bass epidemics on 1D lattices

This paper is devoted to the study of a stochastic epidemiological model which is a variant of the SIR model to which we add an extra factor in the transition rate from susceptible to infected accounting for the inflow of infection due to immigration or environmental sources of infection. This factor yields the formation of new clusters of infections, without having to specify a priori and explicitly their date and place of appearance.We establish an {exact deterministic description} for such stochastic processes on 1D lattices (finite lines, semi-infinite lines, infinite lines, circles) by showing that the probability of infection at a given point in space and time can be obtained as the solution of a deterministic ODE system on the lattice. Our results allow stochastic initial conditions and arbitrary spatio-temporal heterogeneities on the parameters.We then apply our results to some concrete situations and obtain useful qualitative results and explicit formulae on the macroscopic dynamics and also the local temporal behavior of each individual. In particular, we provide a fine analysis of some aspects of cluster formation through the study of {patient-zero problems} and the effects of {time-varying point sources}.Finally, we show that the space-discrete model gives rise to new space-continuous models, which are either ODEs or PDEs, depending on the rescaling regime assumed on the parameters.

math.AP↗

arXiv2020

Loss of Physical Reversibility in Reversible Systems

A dynamical system is said to be reversible if, given an output, the input can always be recovered in a well-posed manner. Nevertheless, we argue that reversible systems that have a time-reversal symmetry, such as the Nonlinear Schr{ö}dinger equation and the $ϕ^4$ equation can become "physically irreversible". By this, we mean that realistically-small experimental errors in measuring the output can lead to dramatic differences between the recovered input and the original one. The loss of reversibility reveals a natural "arrow of time", reminiscent of the thermodynamic one, which is the direction in which the radiation is emitted outward. Our results are relevant to imaging and reversal applications in nonlinear optics.

math.AP↗

arXiv2019

Density Estimation in Uncertainty Propagation Problems Using a Surrogate Model

The effect of uncertainties and noise on a quantity of interest (model output) is often better described by its probability density function (PDF) than by its moments. Although density estimation is a common task, the adequacy of approximation methods (surrogate models) for density estimation has not been analyzed before in the uncertainty-quantification (UQ) literature. In this paper, we first show that standard surrogate models (such as generalized polynomial chaos), which are highly accurate for moment estimation, might completely fail to approximate the PDF, even for one-dimensional noise. This is because density estimation requires that the surrogate model accurately approximates the gradient of the quantity of interest, and not just the quantity of interest itself. Hence, we develop a novel spline-based algorithm for density-estimation whose convergence rate in $L^q$ is polynomial in the sampling resolution. This convergence rate is better than that of standard statistical density-estimation methods (such as histograms and kernel density estimators) at dimensions $1 \leq d\leq \frac{5}{2}m$, where $m$ is the spline order. Furthermore, we obtain the convergence rate for density estimation with any surrogate model that approximates the quantity of interest and its gradient in $L^{\infty}$. Finally, we demonstrate our algorithm for problems in nonlinear optics and fluid dynamics.

math.NA↗

arXiv2019

Boundary Effects in the Discrete Bass Model

To study the effect of boundaries on diffusion of new products, we introduce two novel analytic tools: The indifference principle, which enables us to explicitly compute the aggregate diffusion on various networks, and the dominance principle, which enables us to rank the diffusion on different networks. Using these principles, we prove our main result that on a finite line, one-sided diffusion (i.e., when each consumer can only be influenced by her left neighbor) is strictly slower than two-sided diffusion (i.e., when each consumer can be influenced by her left and right neighbor). This is different from the periodic case of diffusion on a circle, where one-sided and two-sided diffusion are identical. We observe numerically similar results in higher dimensions.

cs.SI↗

arXiv2018

Loss of Polarization in Collapsing Beams

We show theoretically and demonstrate experimentally that collapsing elliptically-polarized laser beams experience a nonlinear ellipse rotation that is highly sensitive to small fluctuations in the input power. For arbitrarily small fluctuations in the input power and after a sufficiently large propagation distance, the polarization angle becomes uniformly distributed in [0, 2$π$] from shot-to-shot. We term this novel phenomenon 'loss of polarization'. We perform experiments in fused-silica glass, nitrogen gas and water, and observe a significant increase in the fluctuations of the output polarization angle for elliptically-polarized femtosecond pulses as the power is increased beyond the critical power for self-focusing. We also show numerically and confirm experimentally that this effect is more prominent in the anomalous group-velocity dispersion (GVD) regime compared to the normal-GVD regime due to the extended lengths of the filaments for the former. Such effects could play an important role in intense-field light-matter interactions in which elliptically-polarized pulses are utilized.

physics.optics↗

arXiv2017

Diffusion of new products with recovering consumers

We consider the diffusion of new products in the discrete Bass-SIR model, in which consumers who adopt the product can later "recover" and stop influencing their peers to adopt the product. To gain insight into the effect of the social network structure on the diffusion, we focus on two extreme cases. In the "most-connected" configuration where all consumers are inter-connected (complete network), averaging over all consumers leads to an aggregate model, which combines the Bass model for diffusion of new products with the SIR model for epidemics. In the "least-connected" configuration where consumers are arranged on a circle and each consumer can only be influenced by his left neighbor (one-sided 1D network), averaging over all consumers leads to a different aggregate model which is linear, and can be solved explicitly. We conjecture that for any other network, the diffusion is bounded from below and from above by that on a one-sided 1D network and on a complete network, respectively. When consumers are arranged on a circle and each consumer can be influenced by his left and right neighbors (two-sided 1D network), the diffusion is strictly faster than on a one-sided 1D network. This is different from the case of non-recovering adopters, where the diffusion on one-sided and on two-sided 1D networks is identical. We also propose a nonlinear model for recoveries, and show that consumers' heterogeneity has a negligible effect on the aggregate diffusion.

physics.soc-ph↗

arXiv2017

Loss of phase and universality of stochastic interactions between laser beams

We show that all laser beams gradually lose their initial phase information in nonlinear propagation. Therefore, if two beams travel a sufficiently long distance before interacting, it is not possible to predict whether they would intersect in- or out-of-phase. Hence, if the underlying propagation model is non-integrable, deterministic predictions and control of the interaction outcome become impossible. Because the relative phase between the two beams becomes uniformly distributed in $[0,2π]$, however, the statistics of the interaction outcome are universal, and can be efficiently computed using a polynomial-chaos approach, even when the distributions of the noise sources are unknown.

physics.optics↗