![● Applicable for the elements having multiple membership
● Applicable for Large Datasets: ideally O(N), runtime up to O(N2
)
Families with the accuracy metrics satisfying our requirements:
● Pair Counting Based Metrics: Omega Index [Collins,1988]
● Cluster Matching Based Metrics: Average F1 score [Yang,2013]
● Information Theory Based Metrics: Generalized NMI
[Esquivel,2012]
Problem: accuracy values interpretability and the metric selection. 17](https://image.slidesharecdn.com/daoc-stableclusteringoflargenetworks-191217085430/85/DAOC-Stable-Clustering-of-Large-Networks-17-320.jpg)
![F1a is defined as the average of the weighted F1 scores of a) the best
matching ground-truth clusters to the formed clusters and b) the best
matching formed clusters to the ground-truth clusters:
20
,
F1 - F1-measure
[Rijsbergen, 1974]](https://image.slidesharecdn.com/daoc-stableclusteringoflargenetworks-191217085430/85/DAOC-Stable-Clustering-of-Large-Networks-20-320.jpg)
![NMI is Mutual Information I(C’:C) normalized by the max or mean
value of the unconditional entropy H of the clusterings C’, C:
25
,
,
GNMI
[Esquivel,2012]
uses a stochastic
process to
compute MI.](https://image.slidesharecdn.com/daoc-stableclusteringoflargenetworks-191217085430/85/DAOC-Stable-Clustering-of-Large-Networks-25-320.jpg)
DAOC: Stable Clustering of Large Networks
- 1. Artem Lutov, Mourad Khayati and Philippe Cudré-Mauroux eXascale Infolab, University of Fribourg, Switzerland https://github.com/eXascaleInfolab/daoc https://bit.ly/daoc-slides
- 2. Stability includes both robustness and determinism in our paper: ● Robustness means that the forming clusters should evolve gracefully and without any surges on minor perturbations (i.e., some changes in the links or nodes) of the input network (i.e., graph). ● Determinism represents both non-stochastic and input order-independent (i.e., reshuffling-resistant) results. 2
- 3. Stable (i.e., robust and deterministic) clustering of large networks: DAOC - Deterministic and Agglomerative Overlapping Clustering ● Mutual Maximal Gain, to ensure robustness while being capable of identifying micro-scale clusters ● Overlap Decomposition, to identify fine-grained clusters in a deterministic way, while capturing multiple optima even for the algorithms whose optimization function is not supported overlaps 3
- 4. Human perception-adapted Taxonomy construction for large Evolving Networks by Incremental Clustering ● Stable ● Fully-automatic ● Browsable ● Large ● Multi-viewpoint ● Narrow (7 ± 2 rule) 4 ● Robust + Determ. ● Parameter-free ● Hierarchical ● Near-linear runtime ● Overlapping ● Fine-grained
- 5. Human perception-adapted Taxonomy construction for large Evolving Networks by Incremental Clustering ● Stable ● Fully-automatic ● Browsable ● Large ● Multi-viewpoint ● Narrow (7 ± 2 rule) 5 ● Robust + Determ. ● Parameter-free ● Hierarchical ● Near-linear runtime ● Overlapping ● Fine-grained Louvain
- 7. Decomposition of a node of degree d=3 into K=3 fragments: 7 OD constraints:
- 8. The following algorithms are evaluated on synthetic (multiple instances of each category) and real-world networks using open-source benchmarking framework, Clubmark: 8 * the feature is partially available, parameters tuning might be required for specific cases ◦ the feature is not supported by the original implementation of the algorithm
- 9. 9 F1h (average value and deviation) for subsequent perturbations (link removals) of a synthetic network. Stable algorithms (among non- stochastic and ensemble) are outlined with a bold line and expected to have a gracefully decreasing F1h without surges.
- 10. 10
- 11. 11
- 12. DAOC is a novel clustering algorithm design for a stable (both robust and deterministic) clustering of large networks aiming to construct human perception-adapted taxonomies without any manual tuning. ● DAOC is 25% more accurate on average than state-of-the-art non-stochastic clustering algorithms being on par with the most accurate existing (including stochastic) clustering algorithms ● DAOC is one the least memory consuming and fastest state-of- the-art clustering algorithms being applicable to large networks 12
- 14. 14
- 15. 1515 Racing cars Overlapping Clusters Clusters on Various Resolutions Blue cars Jeeps Cars Racing cars Bikes Racing & blue cars Bikes
- 16. Matching the clusterings (unordered sets of elements) even with the elements having a single membership may yield multiple best matches: => Strictclusterslabeling is not always possible and undesirable. Many dedicated accuracy metrics are designed but few of them are applicable for the elements with multiplemembership. 16 Produced Ground-truth Dark or Cyan? Yellow
- 17. ● Applicable for the elements having multiple membership ● Applicable for Large Datasets: ideally O(N), runtime up to O(N2 ) Families with the accuracy metrics satisfying our requirements: ● Pair Counting Based Metrics: Omega Index [Collins,1988] ● Cluster Matching Based Metrics: Average F1 score [Yang,2013] ● Information Theory Based Metrics: Generalized NMI [Esquivel,2012] Problem: accuracy values interpretability and the metric selection. 17
- 18. Omega Index (𝛀) counts the number of pairs of elements occurring in exactly the same number of clusters as in the number of categories and adjusted to the expected number of such pairs: 18 , , C’ - ground-truth (categories) C - produced cls.
- 19. Soft Omega Index take into account pairs present in different number of clusters by normalizing smaller number of occurrences of each pair of elements in all clusters of one clustering by the larger number of occurrences in another clustering: 19 ,
- 20. F1a is defined as the average of the weighted F1 scores of a) the best matching ground-truth clusters to the formed clusters and b) the best matching formed clusters to the ground-truth clusters: 20 , F1 - F1-measure [Rijsbergen, 1974]
- 21. F1h uses harmonic instead of the arithm. mean to address F1a ≳ 0.5 for the clusters produced from all combinations of the nodes (F1C‘,C = 1 since for each category there exists the exactly matching cluster, F1C,C’ →0 since majority of the clusters have low similarity to the categories): 21 , for the contribution m of the nodes:
- 22. F1p is the harmonic mean of the average over each clustering of the best local probabilities (f1 ➞ pprob) for each cluster: 22
- 23. Purpose: O(N(|C’| + |C|)) ➞ O(N) Cluster mbs # Member nodes, const cont #Members contrib, const counter # Contribs counter Counter orig # Originating cluster ctr # Raw counter, <= mbs C 23 .. ... a for a in g2.mbs: for c in cls(C.a): cc = c.counter; if cc.orig != g2: cc.ctr=0; cc.orig=g2 cc.ctr += 1 / |C.a| if ovp else 1 fmatch(cc.ctr, c.cont, g2.cont) g2 c1 c3 C’
- 24. SNAP DBLP (Nodes: 317,080 Edges: 1,049,866 Clusters: 13,477) ground-truth vs clustering by the Louvain. Evaluation on Intel Xeon E5-2620 (32 logical CPUs) @ 2.10 GHz, apps compiled using GCC 5.4 with -O3 flag. 24
- 25. NMI is Mutual Information I(C’:C) normalized by the max or mean value of the unconditional entropy H of the clusterings C’, C: 25 , , GNMI [Esquivel,2012] uses a stochastic process to compute MI.
- 26. (Soft) 𝛀 MF1 GNMI 26 O(N2 ), performs purely for the multi-resolution clusterings. Evaluates the best-matching clusters only (unfair advantage for the larger clusters). Biased to the number of clusters, non-deterministic results, the convergence is not guaranteed in the stochastic implementation. values are not affected by the number of clusters. O(N), F1p satisfiers more formal constraints than others. Highly parallelized, evaluates full matches, well-grounded theoretically.