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Sample Return Capsules (SRCs) entering Earth's atmosphere at hypervelocity from interplanetary space are a valuable resource for studying meteor phenomena. The 24 September 2023 arrival of the OSIRIS-REx (Origins, Spectral Interpretation, Resource Identification, and Security-Regolith Explorer) SRC provided an unprecedented chance for geophysical observations of a well-characterized source with known parameters, including timing and trajectory. A collaborative effort involving researchers from 16 institutions executed a carefully planned geophysical observational campaign at strategically chosen locations, deploying over 400 ground-based sensors encompassing infrasound, seismic, distributed acoustic sensing (DAS), and GPS technologies. Additionally, balloons equipped with infrasound sensors were launched to capture signals at higher altitudes. This campaign (the largest of its kind so far) yielded a wealth of invaluable data anticipated to fuel scientific inquiry for years to come. The success of the observational campaign is evidenced by the near-universal detection of signals across instruments, both proximal and distal. This paper presents a comprehensive overview of the collective scientific effort, field deployment, and preliminary findings. The early findings have the potential to inform future space missions and terrestrial campaigns, contributing to our understanding of meteoroid interactions with planetary atmospheres. Furthermore, the dataset collected during this campaign will improve entry and propagation models as well as augment the study of atmospheric dynamics and shock phenomena generated by meteoroids and similar sources., Comment: 87 pages, 14 figures

Heliotropes are passive solar hot air balloons that are capable of achieving nearly level flight within the lower stratosphere for several hours. These inexpensive flight platforms enable stratospheric sensing with high-cadence enabled by the low cost to manufacture, but their performance has not yet been assessed systematically. During July to September of 2021, 29 heliotropes were successfully launched from Oklahoma and achieved float altitude as part of the Balloon-based Acoustic Seismology Study (BASS). All of the heliotrope envelopes were nearly identical with only minor variations to the flight line throughout the campaign. Flight data collected during this campaign comprise a large sample to characterize the typical heliotrope flight behavior during launch, ascent, float, and descent. Each flight stage is characterized, dependence on various parameters is quantified, and a discussion of nominal and anomalous flights is provided.

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 $\zeta_\star\in\big((d-1)/d,1\big)$. We prove that this exponent $\zeta_\star$ 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., Comment: 67 pages, 1 figure

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 $\lambda/f(d_u, d_v)$ for some positive function $f$. In the case $f(d_u, d_v)=\max(d_u, d_v)^\mu$ for some $\mu>0$, 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 $\mu$ (at $1/2$) and the degree distribution $D$ of the GW tree. - When $\mu\ge 1$, the process goes extinct for all distributions $D$ for all sufficiently small $\lambda>0$; - When $\mu\in(1/2, 1)$, and the tail of $D$ weakly follows a power law with tail-exponent less than $1-\mu$, the process survives globally but not locally for all $\lambda$ small enough; - When $\mu\in(1/2, 1)$, and $\mathbb{E}[D^{1-\mu}]<\infty$, the process goes extinct almost surely, for all $\lambda$ small enough; - When $\mu<1/2$, and $D$ is heavier then stretched exponential with stretch-exponent $1-2\mu$, the process survives (locally) with positive probability for all $\lambda>0$. We also study the product case $f(x,y)=(xy)^\mu$. In that case, the situation for $\mu < 1/2$ is the same as the one described above, but $\mu\ge 1/2$ always leads to a subcritical contact process for small enough $\lambda>0$ 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., Comment: 71 pages, 3 figures

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 ($|x-y|^{\eta_0+o(1)}$ for an explicit $\eta_0<1$), and linear ($\Theta(|x-y|)$) 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., Comment: 51 pages. To be published in Electronic Journal of Probability. Companion paper: arXiv:2309.11840

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 $(W_xW_y)^\mu$, 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., $\mu$ 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., Comment: 82 pages. Companion paper: arXiv:2309.11880

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 $\zeta\in(0,1)$ 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 \[ \mathbb{P}(k< |\mathcal{C}(0)|<\infty)=\exp\big(-\Theta(k^{\zeta})\big), \qquad \text{as }k\to\infty. \] The value of $\zeta$ undergoes several phase transitions with respect to three main model parameters: the Euclidean dimension $d$, the power-law tail exponent $\tau$ of the degree distribution and a long-range parameter $\alpha$ 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: $\zeta$ in this regime depends both on $\tau,\alpha$. We also prove that the second-largest component in a box of volume $n$ is of size $\Theta((\log n)^{1/\zeta})$ 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., Comment: 74 pages, 3 figures. Fixed typos, to appear in Annals of Probability

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 $\mathrm{p}(\|x-y\|):=p\min(1,\beta\|x-y\|)^{-d\alpha}$ for parameters $p\in(0, 1]$, $\alpha>1$, and $\beta>0$. We show that when $\alpha>1+1/d$, and either $\beta$ or $p$ is sufficiently large, the probability that the origin is in a finite cluster of size at least $k$ decays as $\exp\big(-\Theta(k^{(d-1)/d})\big)$. This corresponds to classical results for nearest-neighbor Bernoulli percolation on $\mathbb{Z}^d$, but is in contrast to long-range percolation with $\alpha<1+1/d$, when the exponent of the stretched exponential decay changes to $2-\alpha$. 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., Comment: 36 pages, minor revision: to appear in Electronic Journal of Probability

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., Comment: 35 pages, 4 figures

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., Comment: 29 pages, 6 figures

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., Comment: 17 pages text, 27 figures of which 22 in the appendix

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)^\mu for all \mu >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 \tau > 1. For \tau < 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-\tau)/(2\mu) 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-\tau)/(2\mu), 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.

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 $\tau\in(2,3)$, 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 $4\frac{\log\log(t)-\log(\log(t'/t)\vee1)}{|\log(\tau-2)|}\vee 2$ 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 $L$., Comment: 42 pages, 4 figures. Revised version with corrected typos and more elaborate proofs. Includes correction of an error in Theorem 2.5 that required a shift of indices in the summation

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.

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., Comment: Revised version, results are unchanged. 30 pages, 1 figure. To appear in Random Structures and Algorithms

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., Comment: 45 pages, 8 figures

Random intersection graphs model networks with communities, assuming an underlying bipartite structure of groups and individuals, where these groups may overlap. Group memberships are generated through the bipartite configuration model. Conditionally on the group memberships, the classical random intersection graph is obtained by connecting individuals when they are together in at least one group. We generalize this definition, allowing for arbitrary community structures within the groups. In our new model, groups might overlap and they have their own internal structure described by a graph, the classical setting corresponding to groups being complete graphs. Our model turns out to be tractable. We analyze the overlapping structure of the communities, derive the asymptotic degree distribution and the local clustering coefficient. These proofs rely on local weak convergence, which also implies that subgraph counts converge. We further exploit the connection to the bipartite configuration model, for which we also prove local weak convergence, and which is interesting in its own right., Comment: 34 pages, 12 figures

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 Komj\'athy and Simon, Song-Havlin-Makse model, spherically symmetric trees, and supercritical Galton-Watson trees., Comment: 26 pages, 6 figures

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, $n=\Theta(\mathrm{e}^{R/2})$ 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 $\tau\in(2,3)$ (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., Comment: 49 pages, 4 figures

In this paper we study first-passage percolation in the configuration model with empirical degree distribution that follows a power-law with exponent $\tau \in (2,3)$. We assign independent and identically distributed (i.i.d.)\ weights to the edges of the graph. We investigate the weighted distance (the length of the shortest weighted path) between two uniformly chosen vertices, called typical distances. When the underlying age-dependent branching process approximating the local neighborhoods of vertices is found to produce infinitely many individuals in finite time -- called explosive branching process -- Baroni, Hofstad and the second author showed that typical distances converge in distribution to a bounded random variable. The order of magnitude of typical distances remained open for the $\tau\in (2,3)$ case when the underlying branching process is not explosive. We close this gap by determining the first order of magnitude of typical distances in this regime for arbitrary, not necessary continuous edge-weight distributions that produce a non-explosive age-dependent branching process with infinite mean power-law offspring distributions. This sequence tends to infinity with the amount of vertices, and, by choosing an appropriate weight distribution, can be tuned to be any growing function that is $O(\log\log n)$, where $n$ is the number of vertices in the graph. We show that the result remains valid for the the erased configuration model as well, where we delete loops and any second and further edges between two vertices., Comment: 24 pages

We investigate the weighted scale-free percolation (SFPW) model on $\mathbb Z^d$. In the SFPW model, the vertices of $\mathbb Z^d$ are assigned i.i.d. weights $(W_x)_{x\in \mathbb Z^d}$, following a power-law distribution with tail exponent $\tau>1$. Conditioned on the collection of weights, the edges $(x,y)_{x, y \in \mathbb Z^d}$ are present independently with probability that a Poisson random variable with parameter $\lambda W_x W_y/(\|x-y\|)^\alpha$ is at least one, for some $\alpha, \lambda>0$, and where $\|\cdot \|$ denotes the Euclidean distance. After the graph is constructed this way, we assign independent and identically distributed (i.i.d.) random edge lengths from distribution $L$ to all existing edges in the graph. The focus of the paper is to determine when is the obtained model \emph{explosive}, that is, when it is possible to reach infinitely many vertices in finite time from a vertex. We show that explosion happens precisely for those edge length distributions that produce explosive branching processes with infinite mean power law offspring distributions. For non-explosive edge-length distributions, when $\gamma \in (1,2)$, we characterise the asymptotic behaviour of the time it takes to reach the first vertex that is graph distance $n$ away. For $\gamma>2$, we show that the number of vertices reachable by time $t$ from the origin grows at most exponentially, thus explosion is never possible. For the non-explosive edge-length distributions, when $\gamma \in (1,2)$, we further determine the first order asymptotics of distances when $\gamma\in (1,2)$. As a corollary we obtain a sharp upper and lower bound for graph distances, closing a gap between a lower and upper bound on graph distances in Deijfen Hofstad `13 when $\gamma \in(1,2), \tau>2$., Comment: 40 pages. We fixed some typos, and shortened the paper

Multifractal characteristics of the Internet traffic have been discovered and discussed in several research papers so far. However, the origin of this phenomenon is still not fully understood. It has been proven that the congestion control mechanism of the Internet transport protocol, i.e., the mechanism of TCP Reno can generate multifractal traffic properties. Nonetheless, TCP Reno does not exist in today's network any longer, surprisingly traffic multifractality has still been observed. In this paper we give the theoretical proof that TCP CUBIC, which is the default TCP version in the Linux world, can generate multifractal traffic. We give the multifractal spectrum of TCP CUBIC traffic and compare it with the multifractal spectrum of TCP Reno traffic. Moreover, we present the multifractal spectrum for a more general model, where TCP CUBIC and TCP Reno are special cases. Our results also show that TCP CUBIC produces less bursty traffic than TCP Reno., Comment: 42 pages, 9 figures

In this paper we study typical distances in the configuration model, when the degrees have asymptotically infinite variance. We assume that the empirical degree distribution follows a power law with exponent $\tau\in (2,3)$, up to value $n^{\beta_n}$ for some $\beta_n\gg (\log n)^{-\gamma}$ and $\gamma\in(0,1)$. This assumption is satisfied for power law i.i.d. degrees, and also includes truncated power-law distributions where the (possibly exponential) truncation happens at $n^{\beta_n}$. We show that the graph distance between two uniformly chosen vertices centers around $2 \log \log (n^{\beta_n}) / |\log (\tau-2)| + 1/(\beta_n(3-\tau))$, with tight fluctuations. Thus, the graph is an \emph{ultrasmall world} whenever $1/\beta_n=o(\log\log n)$. We determine the distribution of the fluctuations around this value, in particular we prove that these are non-converging tight random variables that show $\log \log$-periodicity. We describe the topology and number of shortest paths: We show that the number of shortest paths is of order $n^{f_n\beta_n}$, where $f_n \in (0,1)$ is a random variable that oscillates with $n$. The two end-segments of any shortest path have length $\log \log (n^{\beta_n}) / |\log (\tau-2)|$+tight, and the total degree is increasing towards the middle of the path on these segments. The connecting middle segment has length $1/(\beta_n(3-\tau))$+tight, and it contains only vertices with degree at least of order $n^{(1-f_n)\beta_n}$, thus all the degrees on this segment are comparable to the maximal degree. Our theorems also apply when instead of truncating the degrees, we start with a configuration model and we remove every vertex with degree at least $n^{\beta_n}$, and the edges attached to these vertices. This sheds light on the attack vulnerability of the configuration model with infinite variance degrees., Comment: 36 pages, 1 figure

We prove results for first-passage percolation on the configuration model with i.i.d. degrees having finite mean, infinite variance and i.i.d. weights with strictly positive support of the form Y=a+X, where a is a positive constant. We prove that the weight of the optimal path has tight fluctuations around the asymptotical mean of the graph-distance if and only if the following condition holds: the random variable X is such that the continuous-time branching process describing first-passage percolation exploration in the same graph with excess edge weight X has a positive probability to reach infinitely many individuals in a finite time. This shows that almost shortest paths in the graph-distance proliferate, in the sense that there are even ones having tight total excess edge weight for various edge-weight distributions., Comment: 19 pages

In this paper we initiate the theory of Crump-Mode-Jagers branching processes (BP) in the setting where no Malthusian parameter exist, i.e., the process grows faster than exponential. A Crump-Mode-Jagers BP is a branching process (in continuous time) where arbitrary dependencies are allowed between the birth-times of the children of a single individual in the population. It is however assumed that these reproduction processes are i.i.d. point processes for different individuals. This paper focuses on determining whether this branching process explodes, that is, the process reaches infinitely many individuals in finite time. We develop comparison techniques between reproduction processes. We study special cases in terms of explosivity such as age-dependent BPs, and epidemic models with contagious intervals. For this, we superimpose a random contagious interval $[I, C]$ on every individual in the BP and keep only the children with birth-times that fall in this interval of the parent. We show that the distribution of the end $C$ of the contagious interval does not matter in terms of explosion, while the distribution of $I$ does: the epidemic explodes if and only if the two age-dependent BPs with the original birth-times and birth-times $I$ explode. We finish studying some pathological examples such as birth-time distributions that are singular to the Lebesque-measure yet they produce an explosive BP with arbitrary power-law offspring distributions., Comment: 40 pages

The Newman-Watts model is given by taking a cycle graph of n vertices and then adding each possible edge $(i,j), |i-j|\neq 1 \mod n$ with probability $\rho/n$ for some $\rho>0$ constant. In this paper we add i.i.d. exponential edge weights to this graph, and investigate typical distances in the corresponding random metric space given by the least weight paths between vertices. We show that typical distances grow as $\frac1\lambda \log n$ for a $\lambda>0$ and determine the distribution of smaller order terms in terms of limits of branching process random variables. We prove that the number of edges along the shortest weight path follows a Central Limit Theorem, and show that in a corresponding epidemic spread model the fraction of infected vertices follows a deterministic curve with a random shift., Comment: 29 pages, 4 figures

These are the notes for the minicourse on Markov chains delivered at the Saint Petersburg Summer School, June 2012. The main emphasis is on methods for estimating mixing times (for finite chains) and escape rates (for infinite chains). Lamplighter groups are key examples in both topics and the Varopolous-Carne long range estimate is useful in both settings., Comment: 28 pages, 1 figure

We prove non-universality results for first-passage percolation on the configuration model with i.i.d. degrees having infinite variance. We focus on the weight of the optimal path between two uniform vertices. Depending on the properties of the weight distribution, we use an example-based approach and show that rather different behaviors are possible. When the weights are a.s. larger than a constant, the weight and number of edges in the graph grow proportionally to loglog(n), as for the graph distances. On the other hand, when the continuous-time branching process describing the first passage percolation exploration through the graph reaches infinitely many vertices in finite time, the weight converges to the sum of two i.i.d. random variables representing the explosion times of the processes started from the two sources. This non-universality is in sharp contrast to the setting where the degree sequence has a finite variance (see Bhamidi, Hofstad and Hooghiemstra arXiv: 1210.6839)., Comment: 18 pages, 0 figures

We study competition of two spreading colors starting from single sources on the configuration model with i.i.d. degrees following a power-law distribution with exponent $\tau\in (2,3)$. In this model two colors spread with a fixed and equal speed on the unweighted random graph. We analyse how many vertices the two colors paint eventually. We show that coexistence sensitively depends on the initial local neighborhoods of the source vertices: if these neighborhoods are `dissimilar enough', then there is no coexistence, and the `loser' color paints a polynomial fraction of the vertices with a random exponent. If the local neighborhoods of the starting vertices are `similar enough', then there is coexistence, i.e., both colors paint a strictly positive proportion of vertices. We give a quantitative characterization of `similar' local neighborhoods: two random variables describing the double exponential growth of local neighborhoods of the source vertices must be within a factor $\tau-2$ of each other. Both of the two outcomes happen with positive probability with asymptotic value that is explicitly computable. This picture reinforces the common belief that location is an important feature in advertising. This paper is a follow-up of the similarly named paper that handles the case when the speeds of the two colors are not equal. There, we have shown that the faster color paints almost all vertices, while the slower color paints only a random sub-polynomial fraction of the vertices., Comment: 62 pages, 11 figures