An official website of the United States government
Abstract Community detection is considered for a stochastic block model graph of n vertices, with K vertices in the planted community, edge probability p for pairs of vertices both in the community, and edge probability q for other pairs of vertices. The main focus of the paper is on weak recovery of the community based on the graph G , with o ( K ) misclassified vertices on average, in the sublinear regime n 1- o (1) ≤ K ≤ o ( n ). A critical parameter is the effective signal-to-noise ratio λ = K 2 ( p - q ) 2 / (( n - K ) q ), with λ = 1 corresponding to the Kesten–Stigum threshold. We show that a belief propagation (BP) algorithm achieves weak recovery if λ > 1 / e, beyond the Kesten–Stigum threshold by a factor of 1 / e. The BP algorithm only needs to run for log * n + O (1) iterations, with the total time complexity O (| E |log * n ), where log * n is the iterated logarithm of n . Conversely, if λ ≤ 1 / e, no local algorithm can asymptotically outperform trivial random guessing. Furthermore, a linear message-passing algorithm that corresponds to applying a power iteration to the nonbacktracking matrix of the graph is shown to attain weak recovery if and only if λ > 1. In addition, the BP algorithm can be combined with a linear-time voting procedure to achieve the information limit of exact recovery (correctly classify all vertices with high probability) for all K ≥ ( n / log n ) (ρ BP + o (1)), where ρ BP is a function of p / q .
more »
« less
Hamiltonicity of random graphs in the stochastic block model
We study the Hamiltonicity of the following model of a random graph. Suppose that we partition $[n]$ into $$V_1,V_2,\ldots,V_k$$ and add edge $$\{x,y\}$$ to our graph with probability $$p$$ if there exists $$i$$ such that $$x,y\in V_i$$. Otherwise, we add the edge with probability $$q$$. We denote this model by $$\G(n, p,q)$$ and give tight results for Hamiltonicity, including a critical window analysis, under various conditions.
more »
« less
- Award ID(s):
- 1952285
- PAR ID:
- 10320464
- Date Published:
- Journal Name:
- SIAM journal on discrete mathematics
- Volume:
- 35
- ISSN:
- 1095-7146
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
The planted densest subgraph detection problem refers to the task of testing whether in a given (random) graph there is a subgraph that is unusually dense. Specifically, we observe an undirected and unweighted graph on n vertices. Under the null hypothesis, the graph is a realization of an Erdös-R{\'e}nyi graph with edge probability (or, density) q. Under the alternative, there is a subgraph on k vertices with edge probability p>q. The statistical as well as the computational barriers of this problem are well-understood for a wide range of the edge parameters p and q. In this paper, we consider a natural variant of the above problem, where one can only observe a relatively small part of the graph using adaptive edge queries. For this model, we determine the number of queries necessary and sufficient (accompanied with a quasi-polynomial optimal algorithm) for detecting the presence of the planted subgraph. We also propose a polynomial-time algorithm which is able to detect the planted subgraph, albeit with more queries compared to the above lower bound. We conjecture that in the leftover regime, no polynomial-time algorithms exist. Our results resolve two open questions posed in the past literature.more » « less
-
Chakrabarti, Amit; Swamy, Chaitanya (Ed.)We consider the problem of sampling from the ferromagnetic Potts and random-cluster models on a general family of random graphs via the Glauber dynamics for the random-cluster model. The random-cluster model is parametrized by an edge probability p ∈ (0,1) and a cluster weight q > 0. We establish that for every q ≥ 1, the random-cluster Glauber dynamics mixes in optimal Θ(nlog n) steps on n-vertex random graphs having a prescribed degree sequence with bounded average branching γ throughout the full high-temperature uniqueness regime p < p_u(q,γ). The family of random graph models we consider includes the Erdős-Rényi random graph G(n,γ/n), and so we provide the first polynomial-time sampling algorithm for the ferromagnetic Potts model on Erdős-Rényi random graphs for the full tree uniqueness regime. We accompany our results with mixing time lower bounds (exponential in the largest degree) for the Potts Glauber dynamics, in the same settings where our Θ(n log n) bounds for the random-cluster Glauber dynamics apply. This reveals a novel and significant computational advantage of random-cluster based algorithms for sampling from the Potts model at high temperatures.more » « less
-
We study the problem of efficiently estimating the effect of an intervention on a single variable using observational samples. Our goal is to give algorithms with polynomial time and sample complexity in a non-parametric setting. Tian and Pearl (AAAI ’02) have exactly characterized the class of causal graphs for which causal effects of atomic interventions can be identified from observational data. We make their result quantitative. Suppose 𝒫 is a causal model on a set V of n observable variables with respect to a given causal graph G, and let do(x) be an identifiable intervention on a variable X. We show that assuming that G has bounded in-degree and bounded c-components (k) and that the observational distribution satisfies a strong positivity condition: (i) [Evaluation] There is an algorithm that outputs with probability 2/3 an evaluator for a distribution P^ that satisfies TV(P(V | do(x)), P^(V)) < eps using m=O (n/eps^2) samples from P and O(mn) time. The evaluator can return in O(n) time the probability P^(v) for any assignment v to V. (ii) [Sampling] There is an algorithm that outputs with probability 2/3 a sampler for a distribution P^ that satisfies TV(P(V | do(x)), P^(V)) < eps using m=O (n/eps^2) samples from P and O(mn) time. The sampler returns an iid sample from P^ with probability 1 in O(n) time. We extend our techniques to estimate P(Y | do(x)) for a subset Y of variables of interest. We also show lower bounds for the sample complexity, demonstrating that our sample complexity has optimal dependence on the parameters n and eps, as well as if k=1 on the strong positivity parameter.more » « less
-
An edge‐ordered graph is a graph with a linear ordering of its edges. Two edge‐ordered graphs areequivalentif there is an isomorphism between them preserving the edge‐ordering. Theedge‐ordered Ramsey number redge(H; q) of an edge‐ordered graphHis the smallestNsuch that there exists an edge‐ordered graphGonNvertices such that, for everyq‐coloring of the edges ofG, there is a monochromatic subgraph ofGequivalent toH. Recently, Balko and Vizer proved thatredge(H; q) exists, but their proof gave enormous upper bounds on these numbers. We give a new proof with a much better bound, showing there exists a constantcsuch thatfor every edge‐ordered graphHonnvertices. We also prove a polynomial bound for the edge‐ordered Ramsey number of graphs of bounded degeneracy. Finally, we prove a strengthening for graphs where every edge has a label and the labels do not necessarily have an ordering.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1137/19M1296069
