789 research outputs found
Seeded Graph Matching: Efficient Algorithms and Theoretical Guarantees
In this paper, a new information theoretic framework for graph matching is
introduced. Using this framework, the graph isomorphism and seeded graph
matching problems are studied. The maximum degree algorithm for graph
isomorphism is analyzed and sufficient conditions for successful matching are
rederived using type analysis. Furthermore, a new seeded matching algorithm
with polynomial time complexity is introduced. The algorithm uses `typicality
matching' and techniques from point-to-point communications for reliable
matching. Assuming an Erdos-Renyi model on the correlated graph pair, it is
shown that successful matching is guaranteed when the number of seeds grows
logarithmically with the number of vertices in the graphs. The logarithmic
coefficient is shown to be inversely proportional to the mutual information
between the edge variables in the two graphs
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
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