374 research outputs found
Dynamical Systems on Networks: A Tutorial
We give a tutorial for the study of dynamical systems on networks. We focus
especially on "simple" situations that are tractable analytically, because they
can be very insightful and provide useful springboards for the study of more
complicated scenarios. We briefly motivate why examining dynamical systems on
networks is interesting and important, and we then give several fascinating
examples and discuss some theoretical results. We also briefly discuss
dynamical systems on dynamical (i.e., time-dependent) networks, overview
software implementations, and give an outlook on the field.Comment: 39 pages, 1 figure, submitted, more examples and discussion than
original version, some reorganization and also more pointers to interesting
direction
Long ties accelerate noisy threshold-based contagions
Network structure can affect when and how widely new ideas, products, and
behaviors are adopted. In widely-used models of biological contagion,
interventions that randomly rewire edges (generally making them "longer")
accelerate spread. However, there are other models relevant to social
contagion, such as those motivated by myopic best-response in games with
strategic complements, in which an individual's behavior is described by a
threshold number of adopting neighbors above which adoption occurs (i.e.,
complex contagions). Recent work has argued that highly clustered, rather than
random, networks facilitate spread of these complex contagions. Here we show
that minor modifications to this model, which make it more realistic, reverse
this result: we allow very rare below-threshold adoption, i.e., rarely adoption
occurs when there is only one adopting neighbor. To model the trade-off between
long and short edges we consider networks that are the union of cycle-power-
graphs and random graphs on nodes. Allowing adoptions below threshold to
occur with order probability along some "short" cycle edges is
enough to ensure that random rewiring accelerates spread. Simulations
illustrate the robustness of these results to other commonly-posited models for
noisy best-response behavior. Hypothetical interventions that randomly rewire
existing edges or add random edges (versus adding "short", triad-closing edges)
in hundreds of empirical social networks reduce time to spread. This revised
conclusion suggests that those wanting to increase spread should induce
formation of long ties, rather than triad-closing ties. More generally, this
highlights the importance of noise in game-theoretic analyses of behavior
The Majority Illusion in Social Networks
Social behaviors are often contagious, spreading through a population as
individuals imitate the decisions and choices of others. A variety of global
phenomena, from innovation adoption to the emergence of social norms and
political movements, arise as a result of people following a simple local rule,
such as copy what others are doing. However, individuals often lack global
knowledge of the behaviors of others and must estimate them from the
observations of their friends' behaviors. In some cases, the structure of the
underlying social network can dramatically skew an individual's local
observations, making a behavior appear far more common locally than it is
globally. We trace the origins of this phenomenon, which we call "the majority
illusion," to the friendship paradox in social networks. As a result of this
paradox, a behavior that is globally rare may be systematically overrepresented
in the local neighborhoods of many people, i.e., among their friends. Thus, the
"majority illusion" may facilitate the spread of social contagions in networks
and also explain why systematic biases in social perceptions, for example, of
risky behavior, arise. Using synthetic and real-world networks, we explore how
the "majority illusion" depends on network structure and develop a statistical
model to calculate its magnitude in a network
Contagions in Random Networks with Overlapping Communities
We consider a threshold epidemic model on a clustered random graph with
overlapping communities. In other words, our epidemic model is such that an
individual becomes infected as soon as the proportion of her infected neighbors
exceeds the threshold q of the epidemic. In our random graph model, each
individual can belong to several communities. The distributions for the
community sizes and the number of communities an individual belongs to are
arbitrary.
We consider the case where the epidemic starts from a single individual, and
we prove a phase transition (when the parameter q of the model varies) for the
appearance of a cascade, i.e. when the epidemic can be propagated to an
infinite part of the population. More precisely, we show that our epidemic is
entirely described by a multi-type (and alternating) branching process, and
then we apply Sevastyanov's theorem about the phase transition of multi-type
Galton-Watson branching processes. In addition, we compute the entries of the
matrix whose largest eigenvalue gives the phase transition.Comment: Minor modifications for the second version: added comments (end of
Section 3.2, beginning of Section 5.3); moved remark (end of Section 3.1,
beginning of Section 4.1); corrected typos; changed titl
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