105 research outputs found
Rumor Spreading on Random Regular Graphs and Expanders
Broadcasting algorithms are important building blocks of distributed systems.
In this work we investigate the typical performance of the classical and
well-studied push model. Assume that initially one node in a given network
holds some piece of information. In each round, every one of the informed nodes
chooses independently a neighbor uniformly at random and transmits the message
to it.
In this paper we consider random networks where each vertex has degree d,
which is at least 3, i.e., the underlying graph is drawn uniformly at random
from the set of all d-regular graphs with n vertices. We show that with
probability 1 - o(1) the push model broadcasts the message to all nodes within
(1 + o(1))C_d ln n rounds, where C_d = 1/ ln(2(1-1/d)) - 1/(d ln(1 - 1/d)). In
particular, we can characterize precisely the effect of the node degree to the
typical broadcast time of the push model. Moreover, we consider pseudo-random
regular networks, where we assume that the degree of each node is very large.
There we show that the broadcast time is (1+o(1))C ln n with probability 1 -
o(1), where C= 1/ ln 2 + 1, is the limit of C_d as d grows.Comment: 18 page
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
Low Randomness Rumor Spreading via Hashing
International audienceWe consider the classical rumor spreading problem, where a piece of information must be disseminated from a single node to all n nodes of a given network. We devise two simple push-based protocols, in which nodes choose the neighbor they send the information to in each round using pairwise independent hash functions, or a pseudo-random generator, respectively. For several well-studied topologies our algorithms use exponentially fewer random bits than previous protocols. For example, in complete graphs, expanders, and random graphs only a polylogarithmic number of random bits are needed in total to spread the rumor in O(log n) rounds with high probability. Previous explicit algorithms require Omega(n) random bits to achieve the same round complexity. For complete graphs, the amount of randomness used by our hashing-based algorithm is within an O(log n)-factor of the theoretical minimum determined by [Giakkoupis and Woelfel, 2011]
Spread of Information and Diseases via Random Walks in Sparse Graphs
We consider a natural network diffusion process, modeling the spread of information or infectious diseases. Multiple mobile agents perform independent simple random walks on an n-vertex connected graph G. The number of agents is linear in n and the walks start from the stationary distribution. Initially, a single vertex has a piece of information (or a virus). An agent becomes informed (or infected) the first time it visits some vertex with the information (or virus); thereafter, the agent informs (infects) all vertices it visits. Giakkoupis et al. (PODC'19) have shown that the spreading time, i.e., the time before all vertices are informed, is asymptotically and w.h.p. the same as in the well-studied randomized rumor spreading process, on any d-regular graph with d=Ω(logn). The case of sub-logarithmic degree was left open, and is the main focus of this paper. First, we observe that the equivalence shown by Giakkoupis et al. does not hold for small d: We give an example of a 3-regular graph with logarithmic diameter for which the expected spreading time is Ω(log^2n/loglogn), whereas randomized rumor spreading is completed in time Î(logn), w.h.p. Next, we show a general upper bound of O~(dâ
diam(G)+log^3n/d), w.h.p., for the spreading time on any d-regular graph. We also provide a version of the bound based on the average degree, for non-regular graphs. Next, we give tight analyses for specific graph families. We show that the spreading time is O(logn), w.h.p., for constant-degree regular expanders. For the binary tree, we show an upper bound of O(lognâ
loglogn), w.h.p., and prove that this is tight, by giving a matching lower bound for the cover time of the tree by n random walks. Finally, we show a bound of O(diam(G)), w.h.p., for k-dimensional grids, by adapting a technique by Kesten and Sidoravicius.Supported in part by ANR Project PAMELA (ANR16-CE23-0016-01).
Gates Cambridge Scholarship programme.
Supported by the ERC Grant `Dynamic Marchâ
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, and every node is only allowed to call one neighbor in each round. In this work we introduce two fundamentally new techniques in studying the rumor spreading problem:
First, we establish a new connection between the rumor spreading process in an arbitrary graph and certain Markov chains. While most previous work analyzed the rumor spreading time in general graphs by studying the rate of the number of (un-)informed nodes after every round, we show that the mixing time of a certain Markov chain suffices to bound the rumor spreading time in an arbitrary graph.
Second, we construct a reduction from rumor spreading processes to branching programs. This reduction gives us a general framework to derandomize the rumor spreading and other gossip processes. In particular, we show that, for any n-vertex expander graph, there is a protocol which informs every node in O(log n) rounds with high probability, and uses O (log n · log log n) random bits in total. The runtime of our protocol is tight, and the randomness requirement of O (log n· log log n) random bits almost matches the lower bound of Ω(log n) random bits. We further show that, for many graph families (defined with respect to the expansion and the degree), O (poly log n) random bits in total suffice for fast rumor spreading. These results give us an almost complete understanding of the role of randomness in the rumor spreading process, which was extensively studied over the past years
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
Quasirandom Rumor Spreading
We propose and analyze a quasirandom analogue of the classical push model for disseminating information in networks (ârandomized rumor spreadingâ). In the classical model, in each round, each informed vertex chooses a neighbor at random and informs it, if it was not informed before. It is known that this simple protocol succeeds in spreading a rumor from one vertex to all others within
O
(log
n
) rounds on complete graphs, hypercubes, random regular graphs, ErdĆs-RĂ©nyi random graphs, and Ramanujan graphs with probability 1 â
o
(1). In the quasirandom model, we assume that each vertex has a (cyclic) list of its neighbors. Once informed, it starts at a random position on the list, but from then on informs its neighbors in the order of the list. Surprisingly, irrespective of the orders of the lists, the above-mentioned bounds still hold. In some cases, even better bounds than for the classical model can be shown.
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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
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