History-Guided Sampling (HiGS)

• History-Guided Sampling (HiGS) is a method that uses the historical trajectory of random walks to reduce burn-in times and improve estimator efficiency in network sampling.
• Its variants, CNRW and GNRW, use history-dependent neighbor selection to avoid redundant traversals and maintain unbiased stationary distributions.
• Empirical evaluations demonstrate that HiGS methods lower query costs and reduce RMSE compared to classical random walk methods in social network sampling.

History-Guided Sampling (HiGS) refers to a class of sampling methodologies designed to accelerate and improve the statistical efficiency of random walks and related algorithms by leveraging the history of the sampling process. In the context of online social networks—where access is often limited to neighborhood queries via restrictive API interfaces—HiGS specifically addresses the challenge of reducing the “burn-in” period associated with classical Markov Chain Monte Carlo (MCMC) random walks, thereby producing more representative samples with fewer queries. The seminal work "Leveraging History for Faster Sampling of Online Social Networks" (Zhou et al., 2015) introduced HiGS as a higher-ordered Markov chain derived from the trajectory history of the walk, resulting in both Circulated Neighbors Random Walk (CNRW) and Groupby Neighbors Random Walk (GNRW) algorithms. These techniques maintain the stationary distribution of simple random walks (SRW) but achieve lower estimator variance and improved efficiency.

1. Classical Random Walks and Burn-in Problems

In online social network sampling, algorithms typically navigate a node-limited interface by issuing neighborhood queries. SRW—Simple Random Walk—is the prevailing method: at each step, the sampler transitions from the current node to a randomly chosen neighbor. The stationary distribution $\pi(v)$ in undirected graphs is given by $k_v / (2|E|)$, where $k_v$ is the degree of node $v$ and $|E|$ is the number of edges. However, SRW exhibits several limitations:

• Burn-in Period: A substantial number of transitions is necessary before the chain reaches stationarity, incurring high query costs.
• Local Mixing: The memoryless nature of SRW leads to repeated traversals within local clusters, delaying mixing and causing inefficiencies in aggregate estimate tasks (e.g., computing average or total properties).

2. Higher-Ordered Markov Chains via History Guidance

HiGS mitigates these drawbacks by constructing a higher-ordered (history-dependent) Markov chain. The principle is to condition the next move not solely on the current node, but also on the edge or group from which it was reached. Key elements include:

• Path Block: Defined as the segment of a walk between repeated traversals of a given edge $(u \rightarrow v)$, used to delineate windows of history-dependence.
• Sampling Without Replacement: Upon traversing into node $v$ from node $u$, the algorithm selects the next neighbor from $N(v)$ (the set of neighbors) avoiding those previously selected via this same incoming edge until all are covered, then resets.
• Stationarity Maintenance: Despite introducing history, the stationary distribution $\pi(v)$ remains unchanged compared to SRW, ensuring unbiased aggregate estimates.

3. Algorithmic Innovations: CNRW and GNRW

The paper formalizes HiGS as two distinct algorithms:

Algorithm	History Guidance Mechanism	Summary Impact
CNRW (Circulated)	Tracks neighbor selection per edge $(u,v)$; circulates without replacement; resets after all neighbors chosen	Avoids repeated edge transitions, accelerates exploration
GNRW (Groupby)	Partitions neighbors into attribute groups; circulates among groups and within groups without replacement	Forces alternation across attribute strata, enhances global mixing

• CNRW: For each incoming edge $(u,v)$, maintains a record $b(u,v)$; selects unvisited neighbors from $N(v)$ until depleted, only then resets.
• GNRW: For attributes (age, degree, location), partitions $N(v)$ into groups; at each visit, samples new group (unvisited), then within group, neighbor—ensures attribute-level stratification.
• Mixing Speed/Variance Reduction: By preventing repeated traversals/local trapping and promoting attribute diversity, both algorithms empirically and theoretically achieve lower variance in estimators and reduced burn-in query requirements.

4. Theoretical Properties and Estimator Efficiency

The stationary distribution under both CNRW/GNRW matches that of SRW:

$\pi(v) = \frac{k_v}{2|E|}$

The asymptotic variance of any aggregate estimator $\hat{A}_n$ (e.g., average neighbor attribute) is reduced due to less local correlation in the sample trajectory. This makes the estimators more statistically efficient: for a fixed query budget, HiGS-based methods yield aggregate estimates with smaller mean squared error and bias compared to SRW or non-backtracking random walks.

5. Empirical Evaluation and Query Savings

Extensive experimentation on public and synthetic datasets (Google Plus, Yelp, Facebook, barbell graphs, etc.) demonstrates:

• HiGS methods consistently reach stationary sampling distributions much faster than SRW, requiring fewer queries (API calls).
• Aggregate estimation error (bias or RMSE) converges more rapidly with HiGS, especially in graphs with low conductance or pronounced community structures.
• In some cases, both CNRW and GNRW outperform advanced alternatives (e.g., non-backtracking walks) at identical query levels.

6. Practical Applications and Tuning

HiGS is particularly suited for environments with restrictive API rate limits and limited graph topology access. It enables rapid estimation of global aggregates (e.g., mean friend count) and conditional aggregates (e.g., averages by geographic region) using only feasible neighbor queries. GNRW provides tunability for attribute-specific analyses: by defining groups according to the analyst’s interest (age, influence, location, etc.), mixing can be accelerated in desired subspaces, further improving estimation accuracy.

7. Extensions and Future Work

Prospective developments for HiGS include:

• Extending grouping methods to dynamic, attribute-driven group definitions or adaptive grouping based on feedback during sampling.
• Generalizing history-guided sampling beyond SRW/GNRW, including to non-backtracking and other sophisticated random walk variants.
• Investigating scalability and query cost minimization for extremely large or evolving online social graphs, using further history-dependent mechanisms.

History-Guided Sampling (HiGS) thus establishes a rigorous framework for leveraging walk history in network sampling, delivering lower variance and higher efficiency while retaining unbiased stationary distributions—a significant advance for social network analytics in constrained-access scenarios.