Title: Estimating Hitting Times Locally at Scale
Venue: NeurIPS 2025
Accepted: yes
Keywords: random walks

hitting times

local algorithms

graphs

effective resistance

markov chains
Abstract: Hitting times provide a fundamental measure of distance in random processes, quantifying the expected number of steps for a random walk starting at node $u$ to reach node $v$. They have broad applications across domains such as network centrality analysis, ranking and recommendation systems, and epidemiology.  

In this work, we develop local algorithms for estimating hitting times between a pair of vertices $u,v$ without accessing the full graph, overcoming scalability issues of prior global methods. Our first algorithm uses the key insight  that hitting time computations can be truncated at the meeting time of two independent random walks from $u$ and $v$. This leads to an efficient estimator analyzed via the Kronecker product graph and Markov Chain Chernoff bounds. We also present an algorithm extending the work of Peng et al. [2021] that introduces a novel adaptation of the spectral cutoff technique to account for the asymmetry of hitting times. This adaptation captures the directionality of the underlying random walk and requires non-trivial modifications to ensure accuracy and efficiency. In addition to the algorithmic upper bounds, we also provide tight asymptotic lower bounds. 

Finally, we reveal a connection between hitting time estimation and distribution testing, and we validate our algorithms using experiments on both real and synthetic data.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper considers the problem of estimating hitting times in a simple random walk on a graph. The paper puts forth three approaches. The first (Algorithm 1) estimates hitting times through meeting times. The second (Algorithm 3) is based on a truncation of a hitting time summation representation, given in Lemma 4.1. The third approach simply samples long random walks and returns the average hitting time. Experiments on real and synthetic data are included.\n\nSTRENGTHS AND WEAKNESSES:\nThis paper provides two new approaches for hitting times estimation, given in Algorithms 1 and 3. There were some issues with clarity. In particular, I felt that Lemma 4.1 should be moved earlier, so that Lemma 3.3. would not need a forward reference. \n\nMy biggest confusion is in Lemma 3.4. Why is there a single meeting time T? I thought there were multiple meeting times? As a result I am not understanding the definitions of A_i and B_i, and therefore I am not seeing how the display below Line 126 is the output of Algorithm 1. Furthermore, the stopping condition of Algorithm 1 refers to $\\chi$, but I don’t see $\\chi$ defined.\n\nAlso, Lemma 3.3 refers to W, but W is not defined until Lemma 4.1.\n\nThere were a few cases where notation was not defined: \n- Line 84: $\\chi_{uv}$\n- The set $\\chi$ is not defined in Algorithm 1, so I cannot interpret the failure condition.\n\nQUESTIONS:\nI did not follow the failure probability bound at the bottom of page 3. There is an intersection over \\sigma, but in the text above, \\sigma is defined to be a particular permutation. \n\nPlease address the point about Line 126.\n\nWhy does the Hoeffding bound apply in the line below 128? You are essentially conditioning on some events involving the A_i, B_i random variables, so you don’t have independence.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe work studies graphs as markov chains and formulate several methods to estimate hitting times using local state management of the graph random walks. Assuming steady state achievable markov chains on these graphs, they first relate chain mixing times to expected meeting times between 2 nodes/states on the graph. This helps them device parameters to decide on the estimated duration until which they run their algorithm (algorithm 1) and estimate hitting times based on several random walks they initiated from both the source and the target nodes. They also extend existing results on hitting times based on spectral values and tweak it into a local algorithm that runs on estimated local mixing time approximations. For all of these methods, authors provide algorithms, probabilistic error bounds, sample complexity bounds and run-time complexities. The focus was largely on the accuracy of the algorithm in achieving the true value in using fewer instances of random walks initiated by their algorithms. This was then backed by experiments on graphs where ground truth values of hitting times could be calculated.\n\nSTRENGTHS AND WEAKNESSES:\nStrength: 1. Rigorous theoretical analysis of several aspects of the proposed algorithm along with practical proof of concept.\n2. At first glance step line 9 of Algorithm 1 looks prohibitively expensive, especially for large graphs, but the running time grows at most $\\log^3n$, which is very promising for practice!\n\nWeakness: 1. While they prove tight bounds on the samples required for the algorithms, $l$ that is the number of processes that they start random walk for, are practically large for large graphs. While authors mention that one of the core motivations was to avoid using global algorithms that prohibit working with large graphs, the design choice of $l$ (which comes with excellent guarantees as well), is proportional to squared total edges in the graph as well as inverse square of the accuracy error. While the local approach allows them to not store all these walks, even storing the current state of these many walks incurs huge memory as well, albeit the memory requirements decline as the algo does away with intersected random walks.\n\nQUESTIONS:\n1. Algorithm 1 line 8, definition of $y_w$ should use $Y$ family of random walks to make sense.\n2. Algorithm 1 line 14 should be some another variable, maybe $I$ or $J$ instead of $\\chi$, right?\n3. Moving from last expression in line 117 to 118 first expression, how did the $\\pi_G(v)$ transform to $\\pi_G(u)$?\n\nLIMITATIONS:\n1. Extensions like algorithm 3 also face prohibitively large values of $r$, random walks with growth factors $\\frac{m^2}{\\epsilon^2}$. This is a crucial issue especially when one of the core assumptions is that the graph is connected and likely cannot be sparse!\n2. While identified by authors themselves in their diligent analysis, expecting $t_{\\text{mix}}$ as a one-shot input usually means running further sampling and approximation to estimate that first!", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-01"}

comments: SUMMARY:
This paper introduces local algorithms for estimating the hitting time $H_G(u,v)$ between two vertices.  The central idea is simple: consider the meeting time of two independent random walks originating at different vertices, $u$ and $v$.  Because the infinite series that defines a hitting time collapses once the walks intersect, the resulting estimator is unbiased, comes with explicit additive‑error guarantees, and provably uses nearly the minimum possible number of samples.

Beyond this meeting‑time approach, the authors adapt the spectral‑cutoff method of Peng et al. to the asymmetric case.  They provide matching lower bounds that justify why straightforward walk sampling cannot achieve the same accuracy, and they reinterpret hitting‑time estimation as a property testing problem, swapping global mixing‑time parameters for quantities that can be measured locally.

STRENGTHS AND WEAKNESSES:
Strengths:

The results are backed up with strong theory: Provides additive‑error guarantees, a near‑optimal lower bound, an asymmetric spectral‑cutoff analysis, and a reduction to a property testing problem.

Weakness:

1) It would strengthen the paper to include experiments on graphs of at least 100k nodes (and ideally larger) to demonstrate scalability.

2) Adding empirical comparisons against common heuristics, such as personalized or approximate PageRank, would help illustrate the method’s practical relevance.

QUESTIONS:
Please refer to the weakness section

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper is on estimating the hitting time in an undirected graph. Classic algorithms that compute the hitting time run in O(n^3) time. This paper gives a random walk based algorithm that estimates the hitting time. The core idea is to use (what I would call) a \"bidirectional estimator\". To estimate the hitting time from u to v, one runs random walks from both u and v, look at the walks that \"collide\", and scale that to get an estimate. The difficult part is usually in bounding the concentration and proving that the estimate converges with a few samples.\n\nEssentially, when the mixing time is small (much smaller than the hitting time), then the bidirectional estimates converge rapidly. This can be shown using matrix Chernoff bounds.\n\nThe running time of this estimator is expressed in terms of the mixing time and the stationary distribution. In the worst case, the running time significantly worse than O(n^3). If the graph is an expander (mixing time = O(\\log n)), then the running time is better than the worst cast (though still worse than n^3?). If the degree distribution is skewed, then I think there is a significant advantage.\n\nThe result also gives empirical experiments showing that the algorithm works well in practice, which I think is nice.\n\nSTRENGTHS AND WEAKNESSES:\nGetting local/sublinear graph algorithms is a fantastic topic and needs more focus. I think the hitting time is an excellent testbed for such algorithm. The bidirectional estimator has shown much promise in theory and practice, and it's great to see more applications of this technique.\n\nThe major weakness is a lack of an explanation of when this algorithm works. Theorem 3.1 and Corollary 3.1 should be accompanied with worst-case running times (in terms of n), an explanation of the rapidly mixing case, and possible of the skewed degree distribution case (so that \\pi(v) can be much larger than 1/n). I believe that the latter is the reason why the algorithm works well in practice.\n\nQUESTIONS:\n1. As said earlier: please give an explanation of when the algorithm is better than n^3, and the impact of skewed pi(v) values.\n2. There is a lot of literature on similar estimators used for PageRank, PersonalizedPageRank, and estimating the stationary distribution. I would recommend citing some of these papers, and the papers that they cite.\n - FAST-PPR: Scaling Personalized PageRank Estimation for Large Graphs, Peter Lofgren, Siddhartha Banerjee, Ashish Goel, and C. Seshadhri, KDD 2014.\n - Personalized PageRank Estimation and Search: A Bidirectional Approach. Peter Lofgren, Siddhartha Banerjee, Ashish Goel. WSDM 2016\n - Fast Bidirectional Probability Estimation in Markov Models. Siddhartha Banerjee, Peter Lofgren. NIPS 2015\n - Estimating Single-Node PageRank in O ̃ (min{dt,√m}) Time. Hanzhi Wang and Zhewei Wei, VLDB 2023\n - Efficient Algorithms for Personalized PageRank Computation: A Survey. Mingji Yang, Hanzhi Wang, Zhewei Wei, Sibo Wang, and Ji-Rong Wen. TKDE 2024\n - Revisiting Local PageRank Estimation on Undirected Graphs: Simple and Optimal. Hanzhi Wang. KDD 2024\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-03"}