18 research outputs found
New Abilities and Limitations of Spectral Graph Bisection
Spectral based heuristics belong to well-known commonly used methods which determines provably minimal graph bisection or outputs "fail" when the optimality cannot be certified. In this paper we focus on Boppana\u27s algorithm which belongs to one of the most prominent methods of this type. It is well known that the algorithm works well in the random planted bisection model - the standard class of graphs for analysis minimum bisection and relevant problems. In 2001 Feige and Kilian posed the question if Boppana\u27s algorithm works well in the semirandom model by Blum and Spencer. In our paper we answer this question affirmatively. We show also that the algorithm achieves similar performance on graph classes which extend the semirandom model.
Since the behavior of Boppana\u27s algorithm on the semirandom graphs remained unknown, Feige and Kilian proposed a new semidefinite programming (SDP) based approach and proved that it works on this model. The relationship between the performance of the SDP based algorithm and Boppana\u27s approach was left as an open problem. In this paper we solve the problem in a complete way by proving that the bisection algorithm of Feige and Kilian provides exactly the same results as Boppana\u27s algorithm. As a consequence we get that Boppana\u27s algorithm achieves the optimal threshold for exact cluster recovery in the stochastic block model. On the other hand we prove some limitations of Boppana\u27s approach: we show that if the density difference on the parameters of the planted bisection model is too small then the algorithm fails with high probability in the model
Consistency Thresholds for the Planted Bisection Model
The planted bisection model is a random graph model in which the nodes are
divided into two equal-sized communities and then edges are added randomly in a
way that depends on the community membership. We establish necessary and
sufficient conditions for the asymptotic recoverability of the planted
bisection in this model. When the bisection is asymptotically recoverable, we
give an efficient algorithm that successfully recovers it. We also show that
the planted bisection is recoverable asymptotically if and only if with high
probability every node belongs to the same community as the majority of its
neighbors.
Our algorithm for finding the planted bisection runs in time almost linear in
the number of edges. It has three stages: spectral clustering to compute an
initial guess, a "replica" stage to get almost every vertex correct, and then
some simple local moves to finish the job. An independent work by Abbe,
Bandeira, and Hall establishes similar (slightly weaker) results but only in
the case of logarithmic average degree.Comment: latest version contains an erratum, addressing an error pointed out
by Jan van Waai
The minimum bisection in the planted bisection model
In the planted bisection model a random graph with
vertices is created by partitioning the vertices randomly into two classes of
equal size (up to ). Any two vertices that belong to the same class are
linked by an edge with probability and any two that belong to different
classes with probability independently. The planted bisection model
has been used extensively to benchmark graph partitioning algorithms. If
for numbers that remain fixed as
, then w.h.p. the ``planted'' bisection (the one used to construct
the graph) will not be a minimum bisection. In this paper we derive an
asymptotic formula for the minimum bisection width under the assumption that
for a certain constant
Clustering Partially Observed Graphs via Convex Optimization
This paper considers the problem of clustering a partially observed
unweighted graph---i.e., one where for some node pairs we know there is an edge
between them, for some others we know there is no edge, and for the remaining
we do not know whether or not there is an edge. We want to organize the nodes
into disjoint clusters so that there is relatively dense (observed)
connectivity within clusters, and sparse across clusters.
We take a novel yet natural approach to this problem, by focusing on finding
the clustering that minimizes the number of "disagreements"---i.e., the sum of
the number of (observed) missing edges within clusters, and (observed) present
edges across clusters. Our algorithm uses convex optimization; its basis is a
reduction of disagreement minimization to the problem of recovering an
(unknown) low-rank matrix and an (unknown) sparse matrix from their partially
observed sum. We evaluate the performance of our algorithm on the classical
Planted Partition/Stochastic Block Model. Our main theorem provides sufficient
conditions for the success of our algorithm as a function of the minimum
cluster size, edge density and observation probability; in particular, the
results characterize the tradeoff between the observation probability and the
edge density gap. When there are a constant number of clusters of equal size,
our results are optimal up to logarithmic factors.Comment: This is the final version published in Journal of Machine Learning
Research (JMLR). Partial results appeared in International Conference on
Machine Learning (ICML) 201
Asymptotic Mutual Information for the Two-Groups Stochastic Block Model
We develop an information-theoretic view of the stochastic block model, a
popular statistical model for the large-scale structure of complex networks. A
graph from such a model is generated by first assigning vertex labels at
random from a finite alphabet, and then connecting vertices with edge
probabilities depending on the labels of the endpoints. In the case of the
symmetric two-group model, we establish an explicit `single-letter'
characterization of the per-vertex mutual information between the vertex labels
and the graph.
The explicit expression of the mutual information is intimately related to
estimation-theoretic quantities, and --in particular-- reveals a phase
transition at the critical point for community detection. Below the critical
point the per-vertex mutual information is asymptotically the same as if edges
were independent. Correspondingly, no algorithm can estimate the partition
better than random guessing. Conversely, above the threshold, the per-vertex
mutual information is strictly smaller than the independent-edges upper bound.
In this regime there exists a procedure that estimates the vertex labels better
than random guessing.Comment: 41 pages, 3 pdf figure
Linear Programming and Community Detection
The problem of community detection with two equal-sized communities is
closely related to the minimum graph bisection problem over certain random
graph models. In the stochastic block model distribution over networks with
community structure, a well-known semidefinite programming (SDP) relaxation of
the minimum bisection problem recovers the underlying communities whenever
possible. Motivated by their superior scalability, we study the theoretical
performance of linear programming (LP) relaxations of the minimum bisection
problem for the same random models. We show that unlike the SDP relaxation that
undergoes a phase transition in the logarithmic average-degree regime, the LP
relaxation exhibits a transition from recovery to non-recovery in the linear
average-degree regime. We show that in the logarithmic average-degree regime,
the LP relaxation fails in recovering the planted bisection with high
probability.Comment: 35 pages, 3 figure