1,242 research outputs found
Robustness of Randomized Rumour Spreading
In this work we consider three well-studied broadcast protocols: Push, Pull
and Push&Pull. A key property of all these models, which is also an important
reason for their popularity, is that they are presumed to be very robust, since
they are simple, randomized, and, crucially, do not utilize explicitly the
global structure of the underlying graph. While sporadic results exist, there
has been no systematic theoretical treatment quantifying the robustness of
these models. Here we investigate this question with respect to two orthogonal
aspects: (adversarial) modifications of the underlying graph and message
transmission failures.
We explore in particular the following notion of Local Resilience: beginning
with a graph, we investigate up to which fraction of the edges an adversary has
to be allowed to delete at each vertex, so that the protocols need
significantly more rounds to broadcast the information. Our main findings
establish a separation among the three models. It turns out that Pull is robust
with respect to all parameters that we consider. On the other hand, Push may
slow down significantly, even if the adversary is allowed to modify the degrees
of the vertices by an arbitrarily small positive fraction only. Finally,
Push&Pull is robust when no message transmission failures are considered,
otherwise it may be slowed down.
On the technical side, we develop two novel methods for the analysis of
randomized rumour spreading protocols. First, we exploit the notion of
self-bounding functions to facilitate significantly the round-based analysis:
we show that for any graph the variance of the growth of informed vertices is
bounded by its expectation, so that concentration results follow immediately.
Second, in order to control adversarial modifications of the graph we make use
of a powerful tool from extremal graph theory, namely Szemer\`edi's Regularity
Lemma.Comment: version 2: more thorough literature revie
Push is Fast on Sparse Random Graphs
We consider the classical push broadcast process on a large class of sparse
random multigraphs that includes random power law graphs and multigraphs. Our
analysis shows that for every , whp rounds are
sufficient to inform all but an -fraction of the vertices.
It is not hard to see that, e.g. for random power law graphs, the push
process needs whp rounds to inform all vertices. Fountoulakis,
Panagiotou and Sauerwald proved that for random graphs that have power law
degree sequences with , the push-pull protocol needs
to inform all but vertices whp. Our result demonstrates that,
for such random graphs, the pull mechanism does not (asymptotically) improve
the running time. This is surprising as it is known that, on random power law
graphs with , push-pull is exponentially faster than pull
Gossip vs. Markov Chains, and Randomness-Efficient Rumor Spreading
We study gossip algorithms for the rumor spreading problem which asks one
node to deliver a rumor to all nodes in an unknown network. We present the
first protocol for any expander graph with nodes such that, the
protocol informs every node in rounds with high probability, and
uses random bits in total. The runtime of our protocol is
tight, and the randomness requirement of random bits almost
matches the lower bound of random bits for dense graphs. We
further show that, for many graph families, polylogarithmic number of random
bits in total suffice to spread the rumor in rounds.
These results together give us an almost complete understanding of the
randomness requirement of this fundamental gossip process.
Our analysis relies on unexpectedly tight connections among gossip processes,
Markov chains, and branching programs. First, we establish a connection between
rumor spreading processes and Markov chains, which is used to approximate the
rumor spreading time by the mixing time of Markov chains. Second, we show a
reduction from rumor spreading processes to branching programs, and this
reduction provides a general framework to derandomize gossip processes. In
addition to designing rumor spreading protocols, these novel techniques may
have applications in studying parallel and multiple random walks, and
randomness complexity of distributed algorithms.Comment: 41 pages, 1 figure. arXiv admin note: substantial text overlap with
arXiv:1304.135
On the push&pull protocol for rumour spreading
The asynchronous push&pull protocol, a randomized distributed algorithm for
spreading a rumour in a graph , works as follows. Independent Poisson clocks
of rate 1 are associated with the vertices of . Initially, one vertex of
knows the rumour. Whenever the clock of a vertex rings, it calls a random
neighbour : if knows the rumour and does not, then tells the
rumour (a push operation), and if does not know the rumour and knows
it, tells the rumour (a pull operation). The average spread time of
is the expected time it takes for all vertices to know the rumour, and the
guaranteed spread time of is the smallest time such that with
probability at least , after time all vertices know the rumour. The
synchronous variant of this protocol, in which each clock rings precisely at
times , has been studied extensively. We prove the following results
for any -vertex graph: In either version, the average spread time is at most
linear even if only the pull operation is used, and the guaranteed spread time
is within a logarithmic factor of the average spread time, so it is . In the asynchronous version, both the average and guaranteed spread times
are . We give examples of graphs illustrating that these bounds
are best possible up to constant factors. We also prove theoretical
relationships between the guaranteed spread times in the two versions. Firstly,
in all graphs the guaranteed spread time in the asynchronous version is within
an factor of that in the synchronous version, and this is tight.
Next, we find examples of graphs whose asynchronous spread times are
logarithmic, but the synchronous versions are polynomially large. Finally, we
show for any graph that the ratio of the synchronous spread time to the
asynchronous spread time is .Comment: 25 page
Understanding the spreading power of all nodes in a network: a continuous-time perspective
Centrality measures such as the degree, k-shell, or eigenvalue centrality can
identify a network's most influential nodes, but are rarely usefully accurate
in quantifying the spreading power of the vast majority of nodes which are not
highly influential. The spreading power of all network nodes is better
explained by considering, from a continuous-time epidemiological perspective,
the distribution of the force of infection each node generates. The resulting
metric, the \textit{expected force}, accurately quantifies node spreading power
under all primary epidemiological models across a wide range of archetypical
human contact networks. When node power is low, influence is a function of
neighbor degree. As power increases, a node's own degree becomes more
important. The strength of this relationship is modulated by network structure,
being more pronounced in narrow, dense networks typical of social networking
and weakening in broader, looser association networks such as the Internet. The
expected force can be computed independently for individual nodes, making it
applicable for networks whose adjacency matrix is dynamic, not well specified,
or overwhelmingly large
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