arXiv subjects

Júlia Komjáthy

Publications and source records attributed to Júlia Komjáthy.

At least 19 recordsLinked to original sources

arXiv2026

From local giants to locality in long-range percolation

We prove the analogue of Schramm's locality conjecture for long-range percolation on transitive graphs of polynomial growth with $α\in (0,2)$. In this setting, we also prove the joint continuity of the percolation probability $θ$ with respect to three parameters: the underlying graph with respect to the local topology, the connectivity kernel, and the percolation parameter $β$ for all values of $β\in \mathbf{R}_+$, including the critical parameter $β_c$. We also prove a number of results related to the supercritical sharpness of long-range percolation: the long-range order decay of the distribution of finite clusters, the truncation problem, the anchored isoperimetric dimension and the transience of the infinite percolation cluster, and the smoothness of the percolation characters. We obtain these results from proving the local existence-and-uniqueness of the linear-sized (giant) cluster. As an immediate corollary of the local existence-and-uniqueness of the giant we obtain the law of large numbers, which answers a special case of a question of Nekrashevych and Pete \cite[Question 1.3]{nekrashevych_scale-invariant_2011}. The main technical contribution is the construction of a renormalisation scheme combining iteratively merged Voronoi tiles with scale-invariant nets, related to the scale-invariant groups of Benjamini.

math.PR↗

arXiv2026

Degree-dependent and distance-dependent contact rates interpolate between explosive, exponential and polynomial epidemic growth

It is a fundamental question in epidemiology to estimate, model and predict the growth rate of a pandemic. Analogously, analysing the diffusion of innovation, (fake) news, memes, and rumours is of key importance in the social sciences. The resulting epidemic growth curves can be classified according to their growth rates. These have been found to range from exponential to both faster super-exponential curves and slower subexponential or polynomial curves. Previous research has lacked a unified explanatory framework capable of accommodating super-exponential, (stretched) exponential, and polynomial growth patterns within the same contact network. In this paper we propose a simple agent-based network model that can capture all these phases. We provide such a framework by modelling how transmission rates depend on spatial distance and on individuals' numbers of contacts. By comparing the growth rate of spreading processes with or without degree-dependent and/or distance-dependent contact rates through data-driven and synthetic simulations on real and modelled networks with underlying geometry, we find evidence that even a 'sublinear presence' of these causes may cause a significant slow down of the growth rate on the same underlying network. We find that the growth rate is governed by a combination of three factors: geometry, the prevalence of weak ties, and superspreaders. We confirm our results with rigorous proofs in a theoretical model, using a spatial multiscale-argument in long-range heterogeneous first passage percolation. Our results give a plausible explanation of why the consecutive waves of a single pandemic can differ in their growth even if their spreading mechanisms are similar.

math.PR↗

arXiv2025

Large deviations of the giant in supercritical kernel-based spatial random graphs

We study cluster sizes in supercritical $d$-dimensional inhomogeneous percolation models with long-range edges -- such as long-range percolation -- and/or heavy-tailed degree distributions -- such as geometric inhomogeneous random graphs and the age-dependent random connection model. Our focus is on large deviations of the size of the largest cluster in the graph restricted to a finite box as its volume tends to infinity. Compared to nearest-neighbor Bernoulli bond percolation on $\mathbb{Z}^d$, we show that long edges can increase the exponent of the polynomial speed of the lower tail from $(d-1)/d$ to any $ζ_\star\in\big((d-1)/d,1\big)$. We prove that this exponent $ζ_\star$ also governs the size of the second-largest cluster, and the distribution of the size of the cluster containing the origin $\mathcal{C}(0)$. For the upper tail of large deviations, we prove that its speed is logarithmic for models with power-law degree distributions. We express the rate function via the generating function of $|\mathcal{C}(0)|$. The upper tail in degree-homogeneous models decays much faster: the speed in long-range percolation is linear.

math.PR↗

arXiv2024

Cluster-size decay in supercritical kernel-based spatial random graphs

We consider a large class of spatially-embedded random graphs that includes among others long-range percolation, continuum scale-free percolation and the age-dependent random connection model. We assume that the model is supercritical: there is an infinite component. We identify the stretch-exponent $ζ\in(0,1)$ of the decay of the cluster-size distribution. That is, with $|\mathcal{C}(0)|$ denoting the number of vertices in the component of the vertex at $0\in \mathbb{R}^d$, we prove \[ \mathbb{P}(k< |\mathcal{C}(0)|<\infty)=\exp\big(-Θ(k^ζ)\big), \qquad \text{as }k\to\infty. \] The value of $ζ$ undergoes several phase transitions with respect to three main model parameters: the Euclidean dimension $d$, the power-law tail exponent $τ$ of the degree distribution and a long-range parameter $α$ governing the presence of long edges in Euclidean space. In this paper we present the proof for the region in the phase diagram where the model is a generalization of continuum scale-free percolation and/or hyperbolic random graphs: $ζ$ in this regime depends both on $τ,α$. We also prove that the second-largest component in a box of volume $n$ is of size $Θ((\log n)^{1/ζ})$ with high probability. We develop a deterministic algorithm, the cover expansion, as new methodology. This algorithm enables us to prevent too large components that may be de-localized or locally dense in space.

math.PR↗

arXiv2024

Polynomial growth in degree-dependent first passage percolation on spatial random graphs

In this paper we study a version of (non-Markovian) first passage percolation on graphs, where the transmission time between two connected vertices is non-iid, but increases by a penalty factor polynomial in their expected degrees. Based on the exponent of the penalty-polynomial, this makes it increasingly harder to transmit to and from high-degree vertices. This choice is motivated by awareness or time-limitations. For the iid part of the transmission times we allow any nonnegative distribution with regularly varying behaviour at $0$. For the underlying graph models we choose spatial random graphs that have power-law degree distributions, so that the effect of the penalisation becomes visible: (finite and infinite) Geometric Inhomogeneous Random Graphs, and Scale-Free Percolation. In these spatial models, the connection probability between two vertices depends on their spatial distance and on their expected degrees. We prove that upon increasing the penalty exponent, the transmission time between two far away vertices $x,y$ sweeps through four universal phases even for a single underlying graph: explosive (tight transmission times), polylogarithmic, polynomial but sublinear ($|x-y|^{η_0+o(1)}$ for an explicit $η_0<1$), and linear ($Θ(|x-y|)$) in their Euclidean distance. Further, none of these phases are restricted to phase boundaries, and those are non-trivial in the main model parameters: the tail of the degree-distribution, a long-range parameter, and the exponent of regular variation of the iid part of the transmission times. In this paper we present proofs of lower bounds for the latter two phases and the upper bound for the linear phase. These complement the matching upper bounds for the polynomial regime in our companion paper.

math.PR↗

arXiv2024

Cluster-size decay in supercritical long-range percolation

We study the cluster-size distribution of supercritical long-range percolation on $\mathbb{Z}^d$, where two vertices $x,y\in\mathbb{Z}^d$ are connected by an edge with probability $\mathrm{p}(\|x-y\|):=p\min(1,β\|x-y\|)^{-dα}$ for parameters $p\in(0, 1]$, $α>1$, and $β>0$. We show that when $α>1+1/d$, and either $β$ or $p$ is sufficiently large, the probability that the origin is in a finite cluster of size at least $k$ decays as $\exp\big(-Θ(k^{(d-1)/d})\big)$. This corresponds to classical results for nearest-neighbor Bernoulli percolation on $\mathbb{Z}^d$, but is in contrast to long-range percolation with $α<1+1/d$, when the exponent of the stretched exponential decay changes to $2-α$. This result, together with our accompanying paper, establishes the phase diagram of long-range percolation with respect to cluster-size decay. Our proofs rely on combinatorial methods that show that large delocalized components are unlikely to occur. As a side result we determine the asymptotic growth of the second-largest connected component when the graph is restricted to a finite box.

math.PR↗

arXiv2024

Four universal growth regimes in degree-dependent first passage percolation on spatial random graphs I

One-dependent first passage percolation is a spreading process on a graph where the transmission time through each edge depends on the direct surroundings of the edge. In particular, the classical iid transmission time $L_{xy}$ is multiplied by $(W_xW_y)^μ$, a polynomial of the expected degrees $W_x, W_y$ of the endpoints of the edge $xy$, which we call the penalty function. Beyond the Markov case, we also allow any distribution for $L_{xy}$ with regularly varying distribution near $0$. We then run this process on three spatial scale-free random graph models: finite and infinite Geometric Inhomogeneous Random Graphs, and Scale-Free Percolation. In these spatial models, the connection probability between two vertices depends on their spatial distance and on their expected degrees. We show that as the penalty-function, i.e., $μ$ increases, the transmission time between two far away vertices sweeps through four universal phases: explosive (with tight transmission times), polylogarithmic, polynomial but strictly sublinear, and linear in the Euclidean distance. The strictly polynomial growth phase here is a new phenomenon that so far was extremely rare in spatial graph models. The four growth phases are highly robust in the model parameters and are not restricted to phase boundaries. Further, the transition points between the phases depend non-trivially on the main model parameters: the tail of the degree distribution, a long-range parameter governing the presence of long edges, and the behaviour of the distribution $L$ near $0$. In this paper we develop new methods to prove the upper bounds in all sub-explosive phases. Our companion paper complements these results by providing matching lower bounds in the polynomial and linear regimes.

math.PR↗

arXiv2023

Degree-penalized contact processes

We study degree-penalized contact processes on Galton-Watson trees (GW) and the configuration model. The model we consider is a modification of the usual contact process on a graph. In particular, each vertex can be either infected or healthy. When infected, each vertex heals at rate one. Also, when infected, a vertex $v$ with degree $d_v$ infects its neighboring vertex $u$ with degree $d_u$ with rate $λ/f(d_u, d_v)$ for some positive function $f$. In the case $f(d_u, d_v)=\max(d_u, d_v)^μ$ for some $μ>0$, the infection is slowed down to and from high degree vertices. This is in line with arguments used in social network science: people with many contacts do not have the time to infect their neighbors at the same rate as people with fewer contacts. We show that new phase transitions occur in terms of the parameter $μ$ (at $1/2$) and the degree distribution $D$ of the GW tree. - When $μ\ge 1$, the process goes extinct for all distributions $D$ for all sufficiently small $λ>0$; - When $μ\in(1/2, 1)$, and the tail of $D$ weakly follows a power law with tail-exponent less than $1-μ$, the process survives globally but not locally for all $λ$ small enough; - When $μ\in(1/2, 1)$, and $\mathbb{E}[D^{1-μ}]<\infty$, the process goes extinct almost surely, for all $λ$ small enough; - When $μ<1/2$, and $D$ is heavier then stretched exponential with stretch-exponent $1-2μ$, the process survives (locally) with positive probability for all $λ>0$. We also study the product case $f(x,y)=(xy)^μ$. In that case, the situation for $μ< 1/2$ is the same as the one described above, but $μ\ge 1/2$ always leads to a subcritical contact process for small enough $λ>0$ on all graphs. Furthermore, for finite random graphs with prescribed degree sequences, we establish the corresponding phase transitions in terms of the length of survival.

math.PR↗

arXiv2021

Distance evolutions in growing preferential attachment graphs

We study the evolution of the graph distance and weighted distance between two fixed vertices in dynamically growing random graph models. More precisely, we consider preferential attachment models with power-law exponent $τ\in(2,3)$, sample two vertices $u_t,v_t$ uniformly at random when the graph has $t$ vertices, and study the evolution of the graph distance between these two fixed vertices as the surrounding graph grows. This yields a discrete-time stochastic process in $t'\geq t$, called the distance evolution. We show that there is a tight strip around the function $4\frac{\log\log(t)-\log(\log(t'/t)\vee1)}{|\log(τ-2)|}\vee 2$ that the distance evolution never leaves with high probability as $t$ tends to infinity. We extend our results to weighted distances, where every edge is equipped with an i.i.d. copy of a non-negative random variable $L$.

math.PR↗

arXiv2021

Sharp bound on the threshold metric dimension of trees

The threshold-$k$ metric dimension ($\mathrm{Tmd}_k$) of a graph is the minimum number of sensors -- a subset of the vertex set -- needed to uniquely identify any vertex in the graph, solely based on its distances from the sensors, when the measuring radius of a sensor is $k$. We give a sharp lower bound on the $\mathrm{Tmd}_k$ of trees, depending only on the number of vertices $n$ and the measuring radius $k$. This sharp lower bound grows linearly in $n$ with leading coefficient $3/(k^2+4k+3+\mathbf{1}\{k\equiv 1\pmod 3\})$, disproving earlier conjectures by Tillquist et al. in arXiv:2106.14314 that suspected $n/(\lfloor k^2/4\rfloor +2k)$ as main order term. We provide a construction for the largest possible trees with a given $\mathrm{Tmd}_k$ value. The proof that our optimal construction cannot be improved relies on edge-rewiring procedures of arbitrary (suboptimal) trees with arbitrary resolving sets, which reveal the structure of how small subsets of sensors measure and resolve certain areas in the tree that we call the attraction of those sensors. The notion of `attraction of sensors' might be useful in other contexts beyond trees to solve related problems. We also provide an improved lower bound on the $\mathrm{Tmd}_k$ of arbitrary trees that takes into account the structural properties of the tree, in particular, the number and length of simple paths of degree-two vertices terminating in leaf vertices. This bound complements arXiv:2106.14314, where only trees without degree-two vertices were considered, except the simple case of a single path.

math.CO↗

arXiv2021

Switchover phenomenon induced by epidemic seeding on geometric networks

It is a fundamental question in disease modelling how the initial seeding of an epidemic, spreading over a network, determines its final outcome. Research in this topic has primarily concentrated on finding the seed configuration which infects the most individuals. Although these optimal configurations give insight into how the initial state affects the outcome of an epidemic, they are unlikely to occur in real life. In this paper we identify two important seeding scenarios, both motivated by historical data, that reveal a new complex phenomenon. In one scenario, the seeds are concentrated on the central nodes of a network, while in the second, they are spread uniformly in the population. Comparing the final size of the epidemic started from these two initial conditions through data-driven and synthetic simulations on real and modelled geometric metapopulation networks, we find evidence for a switchover phenomenon: When the basic reproduction number $R_0$ is close to its critical value, more individuals become infected in the first seeding scenario, but for larger values of $R_0$, the second scenario is more dangerous. We find that the switchover phenomenon is amplified by the geometric nature of the underlying network, and confirm our results via mathematically rigorous proofs, by mapping the network epidemic processes to bond percolation. Our results expand on the previous finding that in case of a single seed, the first scenario is always more dangerous, and further our understanding why the sizes of consecutive waves can differ even if their epidemic characters are similar.

physics.soc-ph↗

arXiv2021

The metric dimension of critical Galton-Watson trees and linear preferential attachment trees

The metric dimension of a graph $G$ is the minimal size of a subset $R$ of vertices of $G$ that, upon reporting their graph distance from a distingished (source) vertex $v^\star$, enable unique identification of the source vertex $v^\star$ among all possible vertices of $G$. In this paper we show a Law of Large Numbers (LLN) for the metric dimension of some classes of trees: critical Galton-Watson trees conditioned to have size $n$, and growing general linear preferential attachment trees. The former class includes uniform random trees, the latter class includes Yule-trees (also called random recursive trees), $m$-ary increasing trees, binary search trees, and positive linear preferential attachment trees. In all these cases, we are able to identify the limiting constant in the LLN explicitly. Our result relies on the insight that the metric dimension can be related to subtree properties, and hence we can make use of the powerful fringe-tree literature developed by Aldous and Janson et al.

math.PR↗

arXiv2020

Not all interventions are equal for the height of the second peak

In this paper we conduct a simulation study of the spread of an epidemic like COVID-19 with temporary immunity on finite spatial and non-spatial network models. In particular, we assume that an epidemic spreads stochastically on a scale-free network and that each infected individual in the network gains a temporary immunity after its infectious period is over. After the temporary immunity period is over, the individual becomes susceptible to the virus again. When the underlying contact network is embedded in Euclidean geometry, we model three different intervention strategies that aim to control the spread of the epidemic: social distancing, restrictions on travel, and restrictions on maximal number of social contacts per node. Our first finding is that on a finite network, a long enough average immunity period leads to extinction of the pandemic after the first peak, analogous to the concept of "herd immunity". For each model, there is a critical average immunity length $L_c$ above which this happens. Our second finding is that all three interventions manage to flatten the first peak (the travel restrictions most efficiently), as well as decrease the critical immunity length $L_c$, but elongate the epidemic. However, when the average immunity length $L$ is shorter than $L_c$, the price for the flattened first peak is often a high second peak: for limiting the maximal number of contacts, the second peak can be as high as 1/3 of the first peak, and twice as high as it would be without intervention. Thirdly, interventions introduce oscillations into the system and the time to reach equilibrium is, for almost all scenarios, much longer. We conclude that network-based epidemic models can show a variety of behaviors that are not captured by the continuous compartmental models.

physics.soc-ph↗

arXiv2020

Weighted distances in scale-free preferential attachment models

We study three preferential attachment models where the parameters are such that the asymptotic degree distribution has infinite variance. Every edge is equipped with a non-negative i.i.d. weight. We study the weighted distance between two vertices chosen uniformly at random, the typical weighted distance, and the number of edges on this path, the typical hopcount. We prove that there are precisely two universality classes of weight distributions, called the explosive and conservative class. In the explosive class, we show that the typical weighted distance converges in distribution to the sum of two i.i.d. finite random variables. In the conservative class, we prove that the typical weighted distance tends to infinity, and we give an explicit expression for the main growth term, as well as for the hopcount. Under a mild assumption on the weight distribution the fluctuations around the main term are tight.

math.PR↗

arXiv2020

Stopping explosion by penalising transmission to hubs in scale-free spatial random graphs

We study the spread of information in finite and infinite inhomogeneous spatial random graphs. We assume that each edge has a transmission cost that is a product of an i.i.d. random variable L and a penalty factor: edges between vertices of expected degrees w_1 and w_2 are penalised by a factor of (w_1w_2)^μfor all μ>0. We study this process for scale-free percolation, for (finite and infinite) Geometric Inhomogeneous Random Graphs, and for Hyperbolic Random Graphs, all with power law degree distributions with exponent τ> 1. For τ< 3, we find a threshold behaviour, depending on how fast the cumulative distribution function of L decays at zero. If it decays at most polynomially with exponent smaller than (3-τ)/(2μ) then explosion happens, i.e., with positive probability we can reach infinitely many vertices with finite cost (for the infinite models), or reach a linear fraction of all vertices with bounded costs (for the finite models). On the other hand, if the cdf of L decays at zero at least polynomially with exponent larger than (3-τ)/(2μ), then no explosion happens. This behaviour is arguably a better representation of information spreading processes in social networks than the case without penalising factor, in which explosion always happens unless the cdf of L is doubly exponentially flat around zero. Finally, we extend the results to other penalty functions, including arbitrary polynomials in w_1 and w_2. In some cases the interesting phenomenon occurs that the model changes behaviour (from explosive to conservative and vice versa) when we reverse the role of w_1 and w_2. Intuitively, this could corresponds to reversing the flow of information: gathering information might take much longer than sending it out.

math.PR↗

arXiv2019

Phase transition in random intersection graphs with communities

The `random intersection graph with communities' models networks with communities, assuming an underlying bipartite structure of groups and individuals. Each group has its own internal structure described by a (small) graph, while groups may overlap. The group memberships are generated by a bipartite configuration model. The model generalizes the classical random intersection graph model that is included as the special case where each community is a complete graph (or clique). The `random intersection graph with communities' is analytically tractable. We prove a phase transition in the size of the largest connected component based on the choice of model parameters. Further, we prove that percolation on our model produces a graph within the same family, and that percolation also undergoes a phase transition. Our proofs rely on the connection to the bipartite configuration model, however, with the arbitrary structure of the groups, it is not completely straightforward to translate results on the group structure into results on the graph. Our related results on the bipartite configuration model are not only instrumental to the study of the random intersection graph with communities, but are also of independent interest, and shed light on interesting differences from the unipartite case.

math.PR↗

arXiv2019

Explosion in weighted Hyperbolic Random Graphs and Geometric Inhomogeneous Random Graphs

In this paper we study weighted distances in scale-free spatial network models: hyperbolic random graphs (HRG), geometric inhomogeneous random graphs (GIRG) and scale-free percolation (SFP). In HRGs, $n=Θ(\mathrm{e}^{R/2})$ vertices are sampled independently from the hyperbolic disk with radius $R$ and two vertices are connected either when they are within hyperbolic distance $R$, or independently with a probability depending on the hyperbolic distance. In GIRGs and SFP, each vertex is given an independent weight and location from an underlying measured metric space and $\mathbb{Z}^d$, respectively, and two vertices are connected independently with a probability that is a function of their distance and weights. We assign i.i.d. weights to the edges of the random graphs and study the weighted distance between two uniformly chosen vertices. In SFP, we study the weighted distance from the origin of vertex-sequences with norm tending to infinity. In particular, we study the case when the parameters are so that the degree distribution in the graph follows a power law with exponent $τ\in(2,3)$ (infinite variance), and the edge-weight distribution is such that it produces an explosive age-dependent branching process with power-law offspring distribution. We show that in all three models, typical distances within the giant/infinite component converge in distribution, solving an open question in [Explosion and distances in scale-free percolation (2017)]. The main tools of our proof are to couple the models to infinite versions, to follow the shortest paths to infinity and to connect these paths using weight-dependent percolation on the graphs: delete edges attached to vertices with higher weight with higher probability. We realise this using the edge-weights: only short edges connected to high weight vertices will stay, yielding arbitrarily short upper bounds for the connections.

math.PR↗

arXiv2019

Transfinite fractal dimension of trees and hierarchical scale-free graphs

In this paper, we introduce a new concept: the transfinite fractal dimension of graph sequences motivated by the notion of fractality of complex networks proposed by Song et al. We show that the definition of fractality cannot be applied to networks with `tree-like' structure and exponential growth rate of neighborhoods. However, we show that the definition of fractal dimension could be modified in a way that takes into account the exponential growth, and with the modified definition, the fractal dimension becomes a proper parameter of graph sequences. We find that this parameter is related to the growth rate of trees. We also generalize the concept of box dimension further and introduce the transfinite Cesaro fractal dimension. Using rigorous proofs we determine the optimal box-covering and transfinite fractal dimension of various models: the hierarchical graph sequence model introduced by Komjáthy and Simon, Song-Havlin-Makse model, spherically symmetric trees, and supercritical Galton-Watson trees.

math.CO↗