A bootstrap percolation process on a graph G is an infection process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round every uninfected node which has at least r infected neighbours becomes infected
and remains so forever. The parameter r > 1 is fixed. We consider this process in the case where the underlying graph is an inhomogeneous random graph whose kernel is of rank 1. Assuming that initially every vertex is infected independently with probability p > 0, we provide a law of large numbers for the number of vertices that will have been infected by the end of the process. We also focus on a special case of such random graphs which exhibit a power-law degree distribution with exponent in (2,3). The first two authors have shown the existence of a critical function a_c(n) such that a_c(n)=o(n) with the following property. Let n be the number of vertices of the underlying random graph and let a(n) be the number of the vertices that are initially infected. Assume that a set of a(n) vertices is chosen randomly and becomes externally infected. If a(n) << a_c(n), then the process does not evolve at all, with high probability as n grows, whereas if a(n)>> a_c(n), then with high probability the final set of infected vertices is linear. Using the techniques of the previous theorem, we give the precise asymptotic fraction of vertices which will be eventually infected when a(n) >> a_c (n) but a(n) = o(n). Note that this corresponds to the case where p approaches 0.
Consider a random regular graph with degree $d$ and of size $n$. Assign to each edge an i.i.d. exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weigh
t paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed $d geq 3$, we show that the longest of these shortest-weight paths has about $hat{alpha}log n$ edges where $hat{alpha}$ is the unique solution of the equation $alpha log(frac{d-2}{d-1}alpha) - alpha = frac{d-3}{d-2}$, for $alpha > frac{d-1}{d-2}$.
In this paper we study the impact of random exponential edge weights on the distances in a random graph and, in particular, on its diameter. Our main result consists of a precise asymptotic expression for the maximal weight of the shortest weight pat
hs between all vertices (the weighted diameter) of sparse random graphs, when the edge weights are i.i.d. exponential random variables.
Propagation of balance-sheet or cash-flow insolvency across financial institutions may be modeled as a cascade process on a network representing their mutual exposures. We derive rigorous asymptotic results for the magnitude of contagion in a large f
inancial network and give an analytical expression for the asymptotic fraction of defaults, in terms of network characteristics. Our results extend previous studies on contagion in random graphs to inhomogeneous directed graphs with a given degree sequence and arbitrary distribution of weights. We introduce a criterion for the resilience of a large financial network to the insolvency of a small group of financial institutions and quantify how contagion amplifies small shocks to the network. Our results emphasize the role played by contagious links and show that institutions which contribute most to network instability in case of default have both large connectivity and a large fraction of contagious links. The asymptotic results show good agreement with simulations for networks with realistic sizes.
A bootstrap percolation process on a graph $G$ is an infection process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round each uninfected node which has at least $r$ infected neighbours becomes infect
ed and remains so forever. The parameter $rgeq 2$ is fixed. Such processes have been used as models for the spread of ideas or trends within a network of individuals. We analyse bootstrap percolation process in the case where the underlying graph is an inhomogeneous random graph, which exhibits a power-law degree distribution, and initially there are $a(n)$ randomly infected nodes. The main focus of this paper is the number of vertices that will have been infected by the end of the process. The main result of this work is that if the degree sequence of the random graph follows a power law with exponent $beta$, where $2 < beta < 3$, then a sublinear number of initially infected vertices is enough to spread the infection over a linear fraction of the nodes of the random graph, with high probability. More specifically, we determine explicitly a critical function $a_c(n)$ such that $a_c(n)=o(n)$ with the following property. Assuming that $n$ is the number of vertices of the underlying random graph, if $a(n) ll a_c(n)$, then the process does not evolve at all, with high probability as $n$ grows, whereas if $a(n)gg a_c(n)$, then there is a constant $eps>0$ such that, with high probability, the final set of infected vertices has size at least $eps n$. It turns out that when the maximum degree is $o(n^{1/(beta -1)})$, then $a_c(n)$ depends also on $r$. But when the maximum degree is $Theta (n^{1/(beta -1)})$, then $a_c (n)=n^{beta -2 over beta -1}$.
We report a novel technique to passively create strong secondary flows at moderate to high flow rates in microchannels, accurately control them and finally, due to their deterministic nature, program them into microfluidic platforms. Based on the flo
w conditions and due to the presence of the pillars in the channel, the flow streamlines will lose their fore-aft symmetry. As a result of this broken symmetry the fluid is pushed away from the pillar at the center of the channel (i.e. central z-plane). As the flow needs to maintain conservation of mass, the fluid will laterally travel in the opposite direction near the top and bottom walls. Therefore, a NET secondary flow will be created in the channel cross-section which is depicted in this video. The main platform is a simple straight channel with posts (i.e. cylindrical pillars - although other pillar cross-sections should also function) placed along the channel. Channel measures were 200 mumtimes50 mum, with pillars of 100 mum in diameter. Positioning the pillars in different locations within the cross-section of the channel will result in induction of different secondary flow patterns, which can be carefully engineered. The longitudinal spacing of the pillars is another design parameter (600 mum spacing was used for this video). The device works over a wide range (moderate to high) flow rates. We used 150 muL/min in this experiment. The device has 3 inlets where a dye stream is co-flowed between two water streams. In this video, one can see the effect of the net secondary flow created by inertia in the microchannel by visualizing the cross-section of the fluorescently labeled stream. Confocal images are sequentially taken at the inlet and after 49 consecutive pillars.
