60 research outputs found
A weighted configuration model and inhomogeneous epidemics
A random graph model with prescribed degree distribution and degree dependent
edge weights is introduced. Each vertex is independently equipped with a random
number of half-edges and each half-edge is assigned an integer valued weight
according to a distribution that is allowed to depend on the degree of its
vertex. Half-edges with the same weight are then paired randomly to create
edges. An expression for the threshold for the appearance of a giant component
in the resulting graph is derived using results on multi-type branching
processes. The same technique also gives an expression for the basic
reproduction number for an epidemic on the graph where the probability that a
certain edge is used for transmission is a function of the edge weight. It is
demonstrated that, if vertices with large degree tend to have large (small)
weights on their edges and if the transmission probability increases with the
edge weight, then it is easier (harder) for the epidemic to take off compared
to a randomized epidemic with the same degree and weight distribution. A recipe
for calculating the probability of a large outbreak in the epidemic and the
size of such an outbreak is also given. Finally, the model is fitted to three
empirical weighted networks of importance for the spread of contagious diseases
and it is shown that can be substantially over- or underestimated if the
correlation between degree and weight is not taken into account
Pioneers of Influence Propagation in Social Networks
With the growing importance of corporate viral marketing campaigns on online
social networks, the interest in studies of influence propagation through
networks is higher than ever. In a viral marketing campaign, a firm initially
targets a small set of pioneers and hopes that they would influence a sizeable
fraction of the population by diffusion of influence through the network. In
general, any marketing campaign might fail to go viral in the first try. As
such, it would be useful to have some guide to evaluate the effectiveness of
the campaign and judge whether it is worthy of further resources, and in case
the campaign has potential, how to hit upon a good pioneer who can make the
campaign go viral. In this paper, we present a diffusion model developed by
enriching the generalized random graph (a.k.a. configuration model) to provide
insight into these questions. We offer the intuition behind the results on this
model, rigorously proved in Blaszczyszyn & Gaurav(2013), and illustrate them
here by taking examples of random networks having prototypical degree
distributions - Poisson degree distribution, which is commonly used as a kind
of benchmark, and Power Law degree distribution, which is normally used to
approximate the real-world networks. On these networks, the members are assumed
to have varying attitudes towards propagating the information. We analyze three
cases, in particular - (1) Bernoulli transmissions, when a member influences
each of its friend with probability p; (2) Node percolation, when a member
influences all its friends with probability p and none with probability 1-p;
(3) Coupon-collector transmissions, when a member randomly selects one of his
friends K times with replacement. We assume that the configuration model is the
closest approximation of a large online social network, when the information
available about the network is very limited. The key insight offered by this
study from a firm's perspective is regarding how to evaluate the effectiveness
of a marketing campaign and do cost-benefit analysis by collecting relevant
statistical data from the pioneers it selects. The campaign evaluation
criterion is informed by the observation that if the parameters of the
underlying network and the campaign effectiveness are such that the campaign
can indeed reach a significant fraction of the population, then the set of good
pioneers also forms a significant fraction of the population. Therefore, in
such a case, the firms can even adopt the naive strategy of repeatedly picking
and targeting some number of pioneers at random from the population. With this
strategy, the probability of them picking a good pioneer will increase
geometrically fast with the number of tries
Viral Marketing On Configuration Model
We consider propagation of influence on a Configuration Model, where each
vertex can be influenced by any of its neighbours but in its turn, it can only
influence a random subset of its neighbours. Our (enhanced) model is described
by the total degree of the typical vertex, representing the total number of its
neighbours and the transmitter degree, representing the number of neighbours it
is able to influence. We give a condition involving the joint distribution of
these two degrees, which if satisfied would allow with high probability the
influence to reach a non-negligible fraction of the vertices, called a big
(influenced) component, provided that the source vertex is chosen from a set of
good pioneers. We show that asymptotically the big component is essentially the
same, regardless of the good pioneer we choose, and we explicitly evaluate the
asymptotic relative size of this component. Finally, under some additional
technical assumption we calculate the relative size of the set of good
pioneers. The main technical tool employed is the "fluid limit" analysis of the
joint exploration of the configuration model and the propagation of the
influence up to the time when a big influenced component is completed. This
method was introduced in Janson & Luczak (2008) to study the giant component of
the configuration model. Using this approach we study also a reverse dynamic,
which traces all the possible sources of influence of a given vertex, and which
by a new "duality" relation allows to characterise the set of good pioneers
The front of the epidemic spread and first passage percolation
In this paper we establish a connection between epidemic models on random
networks with general infection times considered in Barbour and Reinert 2013
and first passage percolation. Using techniques developed in Bhamidi, van der
Hofstad, Hooghiemstra 2012, when each vertex has infinite contagious periods,
we extend results on the epidemic curve in Barbour Reinert 2013 from bounded
degree graphs to general sparse random graphs with degrees having finite third
moments as the number of vertices tends to infinity. We also study the epidemic
trail between the source and typical vertices in the graph. This connection to
first passage percolation can be also be used to study epidemic models with
general contagious periods as in Barbour Reinert 2013 without bounded degree
assumptions.Comment: 14 page
Critical behavior in inhomogeneous random graphs
We study the critical behavior of inhomogeneous random graphs where edges are
present independently but with unequal edge occupation probabilities. The edge
probabilities are moderated by vertex weights, and are such that the degree of
vertex i is close in distribution to a Poisson random variable with parameter
w_i, where w_i denotes the weight of vertex i. We choose the weights such that
the weight of a uniformly chosen vertex converges in distribution to a limiting
random variable W, in which case the proportion of vertices with degree k is
close to the probability that a Poisson random variable with random parameter W
takes the value k. We pay special attention to the power-law case, in which
P(W\geq k) is proportional to k^{-(\tau-1)} for some power-law exponent \tau>3,
a property which is then inherited by the asymptotic degree distribution.
We show that the critical behavior depends sensitively on the properties of
the asymptotic degree distribution moderated by the asymptotic weight
distribution W. Indeed, when P(W\geq k) \leq ck^{-(\tau-1)} for all k\geq 1 and
some \tau>4 and c>0, the largest critical connected component in a graph of
size n is of order n^{2/3}, as on the Erd\H{o}s-R\'enyi random graph. When,
instead, P(W\geq k)=ck^{-(\tau-1)}(1+o(1)) for k large and some \tau\in (3,4)
and c>0, the largest critical connected component is of the much smaller order
n^{(\tau-2)/(\tau-1)}.Comment: 26 page
The component sizes of a critical random graph with given degree sequence
Consider a critical random multigraph with vertices
constructed by the configuration model such that its vertex degrees are
independent random variables with the same distribution (criticality
means that the second moment of is finite and equals twice its first
moment). We specify the scaling limits of the ordered sequence of component
sizes of as tends to infinity in different cases. When
has finite third moment, the components sizes rescaled by
converge to the excursion lengths of a Brownian motion with parabolic drift
above past minima, whereas when is a power law distribution with exponent
, the components sizes rescaled by converge to the excursion lengths of a certain nontrivial
drifted process with independent increments above past minima. We deduce the
asymptotic behavior of the component sizes of a critical random simple graph
when has finite third moment.Comment: Published in at http://dx.doi.org/10.1214/13-AAP985 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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