Graph-based functional connectivity analysis at the level of individual voxels allows for spatially fine-grained characterization of functional networks in the human brain. However, with high-resolution data sets, such analyses can become infeasible due to the computational demands involved. Most previous studies investigating voxel-level functional connectivity graphs relied on Pearson’s *r* for estimating internodal functional connectivity [2, 8, 13, 15, 21–27, 29–31]. As demonstrated here, the tetrachoric correlation coefficient *r*
_{
t
} constitutes a time-efficient alternative to *r* as a measure of functional connectivity.

In order to reduce the computational costs associated with the analysis of voxel-level graphs, previous studies reduced the data’s spatial resolution [15, 26, 27], spatially restricted the graphical edges incorporated into the analysis [21], or utilized parallel computing [31]. In contrast, efficiency benefits from *r*
_{
t
}-based graph construction are achieved without sacrificing spatial resolution, disregarding graphical edges, or exploiting parallel computing. An open source software package containing the created programs is freely available for download [44]. Note that parallel versions of *r*- and *r*
_{
t
}-based graph construction have been implemented in addition to the sequential ones, thus providing additional efficiency gains that depend on the number of available processors. While this aspect is not the main focus of this article, as the resulting benefits (relative to sequential implementations) can be expected to be fairly independent of the choice of *r* versus *r*
_{
t
} as a measure of internodal functional connectivity, the parallel implementations are still included in the software package [44].

Even though the dichotomization procedure (a prerequisite to the computation of *r*
_{
t
}) entails discarding information in the time domain, important characteristics of the original data appear to be preserved. In applications to artificially generated as well as real fMRI data the new technique proved capable of closely reproducing results obtained in conventional ways. More specifically, the usefulness of the *r*
_{
t
}-based approach was assessed by comparison with *r* in estimating the correlation parameter *ρ* of bivariate normal populations of known properties. In this setting, both the bias and standard deviation were greater for *r*
_{
t
} than for *r*, but still reasonably small. Thus, *r*
_{
t
}-based correlation estimation yielded results closely resembling those obtained when using *r*. Beyond that, *r*- and *r*
_{
t
}-based graph construction and node degree computation were carried out for real fMRI data. A strong linear relationship was found between *r*- and *r*
_{
t
}-based correlations indicating that *r*
_{
t
}-based graphs closely resemble *r*-based graphs, since the graphs are derived from the correlation matrices. In line with this, the spatial distribution of node degrees was highly similar for *r*- and *r*
_{
t
}-based graphs and also in good correspondence with previous work [15, 27, 31].

As data mining approaches are currently gaining momentum in the neuroimaging community [36, 37, 49, 50], the amount of publicly available experimental data is steadily growing. Consequently, development and implementation of efficient exploratory methods, such as the one presented here, are necessary in order to take full advantage of this wealth of data, especially with respect to connectivity analyses [51]. Fast construction and subsequent analysis of graphs may thus open new avenues for applications, including those within a clinical setting, where the voxel-level approach may be of particular importance. It is worth noting in this context that voxel-level graph construction can operate at the original data resolution, thus avoiding the reduction of the analysis’ spatial sensitivity [4, 24]. For example, disease-related patterns, once identified, may serve as connectivity-based biomarkers that could aid, guide, or facilitate diagnostics and may increase prediction accuracy with respect to disease occurrence, recurrence, severity, or treatment outcome. Here, again, efficient methods are essential to facilitate assessment of individual patients within a narrow time frame [52]. If combined with efficient tools for subsequent analysis, the presented methods for fast graph construction may also be useful for online evaluation of functional connectivity in the context of real-time fMRI. This would allow for connectivity-based adaptation of experimental stimulation and interaction with the subject, for example, in task-based fMRI studies, or neurofeedback-based training. Taken together, we believe that there is a multitude of applications (be them experimental or clinical) that could benefit from the methods presented here, highlighting the growing importance of efficient tools for graph-based analysis of voxel-level connectivity.

### Limitations

As illustrated by the results, the accuracy of a correlation estimate naturally increases with the number of data points, i.e., the number of scans. Along the same lines, it has recently been shown that the reliability of functional homogeneity increases with scan duration [53]. For both correlation estimators, it is therefore recommended to avoid a low number of scans (caused, for example, by a short scan duration, or a long TR, or both). Since the deviation from the population correlation *ρ* is generally higher for *r*
_{
t
} than for *r*, a low number of scans will affect *r*
_{
t
} more severely than *r*.

The main focus of this work lies with the comparison of *r* and *r*
_{
t
} as functional connectivity estimators. To reduce the impact of preprocessing on the data’s correlation structure prior to this comparison, we limited the preprocessing of the fMRI data to a minimum. The effect of additional preprocessing steps, or a different preprocessing pipeline altogether, on the robustness of the proposed methods should be subject of future research. Unpublished results from our group indicate, however, that the comparability of *r* and *r*
_{
t
} remains essentially consistent.