Searcharxiv⌕ Search

arXiv subjects

Pan Peng

Publications and source records attributed to Pan Peng.

At least 37 records · Page 2Linked to original sources
arXiv2024

Sublinear-Time Opinion Estimation in the Friedkin--Johnsen Model

Online social networks are ubiquitous parts of modern societies and the discussions that take place in these networks impact people's opinions on diverse topics, such as politics or vaccination. One of the most popular models to formally describe this opinion formation process is the Friedkin--Johnsen (FJ) model, which allows to define measures, such as the polarization and the disagreement of a network. Recently, Xu, Bao and Zhang (WebConf'21) showed that all opinions and relevant measures in the FJ model can be approximated in near-linear time. However, their algorithm requires the entire network and the opinions of all nodes as input. Given the sheer size of online social networks and increasing data-access limitations, obtaining the entirety of this data might, however, be unrealistic in practice. In this paper, we show that node opinions and all relevant measures, like polarization and disagreement, can be efficiently approximated in time that is sublinear in the size of the network. Particularly, our algorithms only require query-access to the network and do not have to preprocess the graph. Furthermore, we use a connection between FJ opinion dynamics and personalized PageRank, and show that in $d$-regular graphs, we can deterministically approximate each node's opinion by only looking at a constant-size neighborhood, independently of the network size. We also experimentally validate that our estimation algorithms perform well in practice.

cs.SI↗
arXiv2024

A Differentially Private Clustering Algorithm for Well-Clustered Graphs

We study differentially private (DP) algorithms for recovering clusters in well-clustered graphs, which are graphs whose vertex set can be partitioned into a small number of sets, each inducing a subgraph of high inner conductance and small outer conductance. Such graphs have widespread application as a benchmark in the theoretical analysis of spectral clustering. We provide an efficient ($ε$,$δ$)-DP algorithm tailored specifically for such graphs. Our algorithm draws inspiration from the recent work of Chen et al., who developed DP algorithms for recovery of stochastic block models in cases where the graph comprises exactly two nearly-balanced clusters. Our algorithm works for well-clustered graphs with $k$ nearly-balanced clusters, and the misclassification ratio almost matches the one of the best-known non-private algorithms. We conduct experimental evaluations on datasets with known ground truth clusters to substantiate the prowess of our algorithm. We also show that any (pure) $ε$-DP algorithm would result in substantial error.

cs.DS↗
arXiv2023

A Sublinear-Time Spectral Clustering Oracle with Improved Preprocessing Time

We address the problem of designing a sublinear-time spectral clustering oracle for graphs that exhibit strong clusterability. Such graphs contain $k$ latent clusters, each characterized by a large inner conductance (at least $φ$) and a small outer conductance (at most $\varepsilon$). Our aim is to preprocess the graph to enable clustering membership queries, with the key requirement that both preprocessing and query answering should be performed in sublinear time, and the resulting partition should be consistent with a $k$-partition that is close to the ground-truth clustering. Previous oracles have relied on either a $\textrm{poly}(k)\log n$ gap between inner and outer conductances or exponential (in $k/\varepsilon$) preprocessing time. Our algorithm relaxes these assumptions, albeit at the cost of a slightly higher misclassification ratio. We also show that our clustering oracle is robust against a few random edge deletions. To validate our theoretical bounds, we conducted experiments on synthetic networks.

cs.DS↗
arXiv2023

Recovering Unbalanced Communities in the Stochastic Block Model With Application to Clustering with a Faulty Oracle

The stochastic block model (SBM) is a fundamental model for studying graph clustering or community detection in networks. It has received great attention in the last decade and the balanced case, i.e., assuming all clusters have large size, has been well studied. However, our understanding of SBM with unbalanced communities (arguably, more relevant in practice) is still limited. In this paper, we provide a simple SVD-based algorithm for recovering the communities in the SBM with communities of varying sizes. We improve upon a result of Ailon, Chen and Xu [ICML 2013; JMLR 2015] by removing the assumption that there is a large interval such that the sizes of clusters do not fall in, and also remove the dependency of the size of the recoverable clusters on the number of underlying clusters. We further complement our theoretical improvements with experimental comparisons. Under the planted clique conjecture, the size of the clusters that can be recovered by our algorithm is nearly optimal (up to poly-logarithmic factors) when the probability parameters are constant. As a byproduct, we obtain an efficient clustering algorithm with sublinear query complexity in a faulty oracle model, which is capable of detecting all clusters larger than $\tildeΩ({\sqrt{n}})$, even in the presence of $Ω(n)$ small clusters in the graph. In contrast, previous efficient algorithms that use a sublinear number of queries are incapable of recovering any large clusters if there are more than $\tildeΩ(n^{2/5})$ small clusters.

cs.LG↗
arXiv2023

Massively Parallel Algorithms for the Stochastic Block Model

Learning the community structure of a large-scale graph is a fundamental problem in machine learning, computer science and statistics. We study the problem of exactly recovering the communities in a graph generated from the Stochastic Block Model (SBM) in the Massively Parallel Computation (MPC) model. Specifically, given $kn$ vertices that are partitioned into $k$ equal-sized clusters (i.e., each has size $n$), a graph on these $kn$ vertices is randomly generated such that each pair of vertices is connected with probability~$p$ if they are in the same cluster and with probability $q$ if not, where $p > q > 0$. We give MPC algorithms for the SBM in the (very general) \emph{$s$-space MPC model}, where each machine has memory $s=Ω(\log n)$. Under the condition that $\frac{p-q}{\sqrt{p}}\geq \tildeΩ(k^{\frac12}n^{-\frac12+\frac{1}{2(r-1)}})$ for any integer $r\in [3,O(\log n)]$, our first algorithm exactly recovers all the $k$ clusters in $O(kr\log_s n)$ rounds using $\tilde{O}(m)$ total space, or in $O(r\log_s n)$ rounds using $\tilde{O}(km)$ total space. If $\frac{p-q}{\sqrt{p}}\geq \tildeΩ(k^{\frac34}n^{-\frac14})$, our second algorithm achieves $O(\log_s n)$ rounds and $\tilde{O}(m)$ total space complexity. Both algorithms significantly improve upon a recent result of Cohen-Addad et al. [PODC'22], who gave algorithms that only work in the \emph{sublinear space MPC model}, where each machine has local memory~$s=O(n^δ)$ for some constant $δ>0$, with a much stronger condition on $p,q,k$. Our algorithms are based on collecting the $r$-step neighborhood of each vertex and comparing the difference of some statistical information generated from the local neighborhoods for each pair of vertices. To implement the clustering algorithms in parallel, we present efficient approaches for implementing some basic graph operations in the $s$-space MPC model.

cs.DS↗
arXiv2023

Effective Resistances in Non-Expander Graphs

Effective resistances are ubiquitous in graph algorithms and network analysis. In this work, we study sublinear time algorithms to approximate the effective resistance of an adjacent pair $s$ and $t$. We consider the classical adjacency list model for local algorithms. While recent works have provided sublinear time algorithms for expander graphs, we prove several lower bounds for general graphs of $n$ vertices and $m$ edges: 1.It needs $Ω(n)$ queries to obtain $1.01$-approximations of the effective resistance of an adjacent pair $s$ and $t$, even for graphs of degree at most 3 except $s$ and $t$. 2.For graphs of degree at most $d$ and any parameter $\ell$, it needs $Ω(m/\ell)$ queries to obtain $c \cdot \min\{d, \ell\}$-approximations where $c>0$ is a universal constant. Moreover, we supplement the first lower bound by providing a sublinear time $(1+ε)$-approximation algorithm for graphs of degree 2 except the pair $s$ and $t$. One of our technical ingredients is to bound the expansion of a graph in terms of the smallest non-trivial eigenvalue of its Laplacian matrix after removing edges. We discover a new lower bound on the eigenvalues of perturbed graphs (resp. perturbed matrices) by incorporating the effective resistance of the removed edge (resp. the leverage scores of the removed rows), which may be of independent interest.

cs.DS↗
arXiv2023

An Optimal Separation between Two Property Testing Models for Bounded Degree Directed Graphs

We revisit the relation between two fundamental property testing models for bounded-degree directed graphs: the bidirectional model in which the algorithms are allowed to query both the outgoing edges and incoming edges of a vertex, and the unidirectional model in which only queries to the outgoing edges are allowed. Czumaj, Peng and Sohler [STOC 2016] showed that for directed graphs with both maximum indegree and maximum outdegree upper bounded by $d$, any property that can be tested with query complexity $O_{\varepsilon,d}(1)$ in the bidirectional model can be tested with $n^{1-Ω_{\varepsilon,d}(1)}$ queries in the unidirectional model. In particular, if the proximity parameter $\varepsilon$ approaches $0$, then the query complexity of the transformed tester in the unidirectional model approaches $n$. It was left open if this transformation can be further improved or there exists any property that exhibits such an extreme separation. We prove that testing subgraph-freeness in which the subgraph contains $k$ source components, requires $Ω(n^{1-\frac{1}{k}})$ queries in the unidirectional model. This directly gives the first explicit properties that exhibit an $O_{\varepsilon,d}(1)$ vs $Ω(n^{1-f(\varepsilon,d)})$ separation of the query complexities between the bidirectional model and unidirectional model, where $f(\varepsilon,d)$ is a function that approaches $0$ as $\varepsilon$ approaches $0$. Furthermore, our lower bound also resolves a conjecture by Hellweg and Sohler [ESA 2012] on the query complexity of testing $k$-star-freeness.

cs.DS↗
arXiv2023

On Testability of First-Order Properties in Bounded-Degree Graphs and Connections to Proximity-Oblivious Testing

We study property testing of properties that are definable in first-order logic (FO) in the bounded-degree graph and relational structure models. We show that any FO property that is defined by a formula with quantifier prefix $\exists^*\forall^*$ is testable (i.e., testable with constant query complexity), while there exists an FO property that is expressible by a formula with quantifier prefix $\forall^*\exists^*$ that is not testable. In the dense graph model, a similar picture is long known (Alon, Fischer, Krivelevich, Szegedy, Combinatorica 2000), despite the very different nature of the two models. In particular, we obtain our lower bound by an FO formula that defines a class of bounded-degree expanders, based on zig-zag products of graphs. We expect this to be of independent interest. We then use our class of FO definable bounded-degree expanders to answer a long-standing open problem for proximity-oblivious testers (POTs). POTs are a class of particularly simple testing algorithms, where a basic test is performed a number of times that may depend on the proximity parameter, but the basic test itself is independent of the proximity parameter. In their seminal work, Goldreich and Ron [STOC 2009; SICOMP 2011] show that the graph properties that are constant-query proximity-oblivious testable in the bounded-degree model are precisely the properties that can be expressed as a generalised subgraph freeness (GSF) property that satisfies the non-propagation condition. It is left open whether the non-propagation condition is necessary. We give a negative answer by showing that our property is a GSF property which is propagating. Hence in particular, our property does not admit a POT. For this result we establish a new connection between FO properties and GSF-local properties via neighbourhood profiles.

cs.LO↗
arXiv2022

Sublinear-Time Algorithms for Max Cut, Max E2Lin$(q)$, and Unique Label Cover on Expanders

We show sublinear-time algorithms for Max Cut and Max E2Lin$(q)$ on expanders in the adjacency list model that distinguishes instances with the optimal value more than $1-\varepsilon$ from those with the optimal value less than $1-ρ$ for $ρ\gg \varepsilon$. The time complexities for Max Cut and Max $2$Lin$(q)$ are $\widetilde{O}(\frac{1}{ϕ^2ρ} \cdot m^{1/2+O(\varepsilon/(ϕ^2ρ))})$ and $\widetilde{O}(\mathrm{poly}(\frac{q}{ϕρ})\cdot {(mq)}^{1/2+O(q^6\varepsilon/ϕ^2ρ^2)})$, respectively, where $m$ is the number of edges in the underlying graph and $ϕ$ is its conductance. Then, we show a sublinear-time algorithm for Unique Label Cover on expanders with $ϕ\gg ε$ in the bounded-degree model. The time complexity of our algorithm is $\widetilde{O}_d(2^{q^{O(1)}\cdotϕ^{1/q}\cdot \varepsilon^{-1/2}}\cdot n^{1/2+q^{O(q)}\cdot \varepsilon^{4^{1.5-q}}\cdot ϕ^{-2}})$, where $n$ is the number of variables. We complement these algorithmic results by showing that testing $3$-colorability requires $Ω(n)$ queries even on expanders.

cs.DS↗
arXiv2022

Sublinear-Time Clustering Oracle for Signed Graphs

Social networks are often modeled using signed graphs, where vertices correspond to users and edges have a sign that indicates whether an interaction between users was positive or negative. The arising signed graphs typically contain a clear community structure in the sense that the graph can be partitioned into a small number of polarized communities, each defining a sparse cut and indivisible into smaller polarized sub-communities. We provide a local clustering oracle for signed graphs with such a clear community structure, that can answer membership queries, i.e., "Given a vertex $v$, which community does $v$ belong to?", in sublinear time by reading only a small portion of the graph. Formally, when the graph has bounded maximum degree and the number of communities is at most $O(\log n)$, then with $\tilde{O}(\sqrt{n}\operatorname{poly}(1/\varepsilon))$ preprocessing time, our oracle can answer each membership query in $\tilde{O}(\sqrt{n}\operatorname{poly}(1/\varepsilon))$ time, and it correctly classifies a $(1-\varepsilon)$-fraction of vertices w.r.t. a set of hidden planted ground-truth communities. Our oracle is desirable in applications where the clustering information is needed for only a small number of vertices. Previously, such local clustering oracles were only known for unsigned graphs; our generalization to signed graphs requires a number of new ideas and gives a novel spectral analysis of the behavior of random walks with signs. We evaluate our algorithm for constructing such an oracle and answering membership queries on both synthetic and real-world datasets, validating its performance in practice.

cs.DS↗
arXiv2022

Approximately Counting Subgraphs in Data Streams

Estimating the number of subgraphs in data streams is a fundamental problem that has received great attention in the past decade. In this paper, we give improved streaming algorithms for approximately counting the number of occurrences of an arbitrary subgraph $H$, denoted $\# H$, when the input graph $G$ is represented as a stream of $m$ edges. To obtain our algorithms, we provide a generic transformation that converts constant-round sublinear-time graph algorithms in the query access model to constant-pass sublinear-space graph streaming algorithms. Using this transformation, we obtain the following results. 1. We give a $3$-pass turnstile streaming algorithm for $(1\pm ε)$-approximating $\# H$ in $\tilde{O}(\frac{m^{ρ(H)}}{ε^2\cdot \# H})$ space, where $ρ(H)$ is the fractional edge-cover of $H$. This improves upon and generalizes a result of McGregor et al. [PODS 2016], who gave a $3$-pass insertion-only streaming algorithm for $(1\pm ε)$-approximating the number $\# T$ of triangles in $\tilde{O}(\frac{m^{3/2}}{ε^2\cdot \# T})$ space if the algorithm is given additional oracle access to the degrees. 2. We provide a constant-pass streaming algorithm for $(1\pm ε)$-approximating $\# K_r$ in $\tilde{O}(\frac{mλ^{r-2}}{ε^2\cdot \# K_r})$ space for any $r\geq 3$, in a graph $G$ with degeneracy $λ$, where $K_r$ is a clique on $r$ vertices. This resolves a conjecture by Bera and Seshadhri [PODS 2020]. More generally, our reduction relates the adaptivity of a query algorithm to the pass complexity of a corresponding streaming algorithm, and it is applicable to all algorithms in standard sublinear-time graph query models, e.g., the (augmented) general model.

cs.DS↗
arXiv2021

Constant-Time Dynamic Weight Approximation for Minimum Spanning Forest

We give two fully dynamic algorithms that maintain a $(1+\varepsilon)$-approximation of the weight $M$ of a minimum spanning forest (MSF) of an $n$-node graph $G$ with edges weights in $[1,W]$, for any $\varepsilon>0$. (1) Our deterministic algorithm takes $O({W^2 \log W}/{\varepsilon^3})$ worst-case update time, which is $O(1)$ if both $W$ and $\varepsilon$ are constants. Note that there is a lower bound by Patrascu and Demaine (SIAM J. Comput. 2006) which shows that it takes $Ω(\log n)$ time per operation to maintain the exact weight of an MSF that holds even in the unweighted case, i.e. for $W=1$. We further show that any deterministic data structure that dynamically maintains the $(1+\varepsilon)$-approximate weight of an MSF requires super constant time per operation, if $W\geq (\log n)^{ω_n(1)}$. (2) Our randomized (Monte-Carlo style) algorithm works with high probability and runs in worst-case $O(\log W/ \varepsilon^{4})$ update time if $W= O({(m^*)^{1/6}}/{\log^{2/3} n})$, where $m^*$ is the minimum number of edges in the graph throughout all the updates. It works even against an adaptive adversary. This implies a randomized algorithm with worst-case $o(\log n)$ update time, whenever $W=\min\{O((m^*)^{1/6}/\log^{2/3} n), 2^{o({\log n})}\}$ and $\varepsilon$ is constant. We complement this result by showing that for any constant $\varepsilon,α>0$ and $W=n^α$, any (randomized) data structure that dynamically maintains the weight of an MSF of a graph $G$ with edge weights in $[1,W]$ and $W = Ω(\varepsilon m^*)$ within a multiplicative factor of $(1+\varepsilon)$ takes $Ω(\log n)$ time per operation.

cs.DS↗
arXiv2021

Towards a Query-Optimal and Time-Efficient Algorithm for Clustering with a Faulty Oracle

Motivated by applications in crowdsourced entity resolution in database, signed edge prediction in social networks and correlation clustering, Mazumdar and Saha [NIPS 2017] proposed an elegant theoretical model for studying clustering with a faulty oracle. In this model, given a set of $n$ items which belong to $k$ unknown groups (or clusters), our goal is to recover the clusters by asking pairwise queries to an oracle. This oracle can answer the query that ``do items $u$ and $v$ belong to the same cluster?''. However, the answer to each pairwise query errs with probability $\varepsilon$, for some $\varepsilon\in(0,\frac12)$. Mazumdar and Saha provided two algorithms under this model: one algorithm is query-optimal while time-inefficient (i.e., running in quasi-polynomial time), the other is time efficient (i.e., in polynomial time) while query-suboptimal. Larsen, Mitzenmacher and Tsourakakis [WWW 2020] then gave a new time-efficient algorithm for the special case of $2$ clusters, which is query-optimal if the bias $δ:=1-2\varepsilon$ of the model is large. It was left as an open question whether one can obtain a query-optimal, time-efficient algorithm for the general case of $k$ clusters and other regimes of $δ$. In this paper, we make progress on the above question and provide a time-efficient algorithm with nearly-optimal query complexity (up to a factor of $O(\log^2 n)$) for all constant $k$ and any $δ$ in the regime when information-theoretic recovery is possible. Our algorithm is built on a connection to the stochastic block model.

cs.LG↗
arXiv2021

Local Algorithms for Estimating Effective Resistance

Effective resistance is an important metric that measures the similarity of two vertices in a graph. It has found applications in graph clustering, recommendation systems and network reliability, among others. In spite of the importance of the effective resistances, we still lack efficient algorithms to exactly compute or approximate them on massive graphs. In this work, we design several \emph{local algorithms} for estimating effective resistances, which are algorithms that only read a small portion of the input while still having provable performance guarantees. To illustrate, our main algorithm approximates the effective resistance between any vertex pair $s,t$ with an arbitrarily small additive error $\varepsilon$ in time $O(\mathrm{poly}(\log n/\varepsilon))$, whenever the underlying graph has bounded mixing time. We perform an extensive empirical study on several benchmark datasets, validating the performance of our algorithms.

cs.DS↗
arXiv2021

Time Complexity Analysis of Randomized Search Heuristics for the Dynamic Graph Coloring Problem

We contribute to the theoretical understanding of randomized search heuristics for dynamic problems. We consider the classical vertex coloring problem on graphs and investigate the dynamic setting where edges are added to the current graph. We then analyze the expected time for randomized search heuristics to recompute high quality solutions. The (1+1)~Evolutionary Algorithm and RLS operate in a setting where the number of colors is bounded and we are minimizing the number of conflicts. Iterated local search algorithms use an unbounded color palette and aim to use the smallest colors and, consequently, the smallest number of colors. We identify classes of bipartite graphs where reoptimization is as hard as or even harder than optimization from scratch, i.e., starting with a random initialization. Even adding a single edge can lead to hard symmetry problems. However, graph classes that are hard for one algorithm turn out to be easy for others. In most cases our bounds show that reoptimization is faster than optimizing from scratch. We further show that tailoring mutation operators to parts of the graph where changes have occurred can significantly reduce the expected reoptimization time. In most settings the expected reoptimization time for such tailored algorithms is linear in the number of added edges. However, tailored algorithms cannot prevent exponential times in settings where the original algorithm is inefficient.

cs.NE↗
arXiv2021

GSF-locality is not sufficient for proximity-oblivious testing

In Property Testing, proximity-oblivious testers (POTs) form a class of particularly simple testing algorithms, where a basic test is performed a number of times that may depend on the proximity parameter, but the basic test itself is independent of the proximity parameter. In their seminal work, Goldreich and Ron [STOC 2009; SICOMP 2011] show that the graph properties that allow constant-query proximity-oblivious testing in the bounded-degree model are precisely the properties that can be expressed as a generalised subgraph freeness (GSF) property that satisfies the non-propagation condition. It is left open whether the non-propagation condition is necessary. Indeed, calling properties expressible as a generalised subgraph freeness property GSF-local properties, they ask whether all GSF-local properties are non-propagating. We give a negative answer by exhibiting a property of graphs that is GSF-local and propagating. Hence in particular, our property does not admit a POT, despite being GSF-local. We prove our result by exploiting a recent work of the authors which constructed a first-order (FO) property that is not testable [SODA 2021], and a new connection between FO properties and GSF-local properties via neighbourhood profiles.

cs.CC↗
arXiv2021

Sampling Arbitrary Subgraphs Exactly Uniformly in Sublinear Time

We present a simple sublinear-time algorithm for sampling an arbitrary subgraph $H$ \emph{exactly uniformly} from a graph $G$ with $m$ edges, to which the algorithm has access by performing the following types of queries: (1) degree queries, (2) neighbor queries, (3) pair queries and (4) edge sampling queries. The query complexity and running time of our algorithm are $\tilde{O}(\min\{m, \frac{m^{ρ(H)}}{\# H}\})$ and $\tilde{O}(\frac{m^{ρ(H)}}{\# H})$, respectively, where $ρ(H)$ is the fractional edge-cover of $H$ and $\# H$ is the number of copies of $H$ in $G$. For any clique on $r$ vertices, i.e., $H=K_r$, our algorithm is almost optimal as any algorithm that samples an $H$ from any distribution that has $Ω(1)$ total probability mass on the set of all copies of $H$ must perform $Ω(\min\{m, \frac{m^{ρ(H)}}{\# H\cdot (cr)^r}\})$ queries. Together with the query and time complexities of the $(1\pm \varepsilon)$-approximation algorithm for the number of subgraphs $H$ by Assadi, Kapralov and Khanna [ITCS 2018] and the lower bound by Eden and Rosenbaum [APPROX 2018] for approximately counting cliques, our results suggest that in our query model, approximately counting cliques is "equivalent to" exactly uniformly sampling cliques, in the sense that the query and time complexities of exactly uniform sampling and randomized approximate counting are within a polylogarithmic factor of each other. This stands in interesting contrast to an analogous relation between approximate counting and almost uniformly sampling for self-reducible problems in the polynomial-time regime by Jerrum, Valiant and Vazirani [TCS 1986].

cs.DS↗
arXiv2021

On Testability of First-Order Properties in Bounded-Degree Graphs

We study property testing of properties that are definable in first-order logic (FO) in the bounded-degree graph and relational structure models. We show that any FO property that is defined by a formula with quantifier prefix $\exists^*\forall^*$ is testable (i.e., testable with constant query complexity), while there exists an FO property that is expressible by a formula with quantifier prefix $\forall^*\exists^*$ that is not testable. In the dense graph model, a similar picture is long known (Alon, Fischer, Krivelevich, Szegedy, Combinatorica 2000), despite the very different nature of the two models. In particular, we obtain our lower bound by a first-order formula that defines a class of bounded-degree expanders, based on zig-zag products of graphs. We expect this to be of independent interest. We then prove testability of some first-order properties that speak about isomorphism types of neighbourhoods, including testability of $1$-neighbourhood-freeness, and $r$-neighbourhood-freeness under a mild assumption on the degrees.

cs.LO↗