IEEE Computational Intelligence Magazine - August 2019 - 49

F1 Score

K Value
(e)

K Value
(f)

0.2

NMI/Cond.

NMI/Cond.

0.4

0.6
0.4
0.2

K Value
(j)

K Value
(k)

0.9
0.8
0.7
0.6
0.5
0.4

5
10
15
50
100

ComeE (Conductance)

5
10
13
50
100

K Value
(i)

0.6

40
30
20
10
5
10
40
50
100

40
30

ComE (NMI)
NMI/Cond.

5
10
40
50
100

20

K Value
(h)

50

0.8

0.9
0.8

60

5
10
13
50
100

5
10
15
50
100

20

5

NMI/Cond.

K Value
(g)

10

0

5

0.2
3

NMI/Cond.
5
10
39
50
100

0.4

70

K Value
(d)

1

0.6

80

K Value
(c)
ComeE+ (Conductance)

0.8

NMI/Cond.

0.9
0.88
0.86
0.84
0.82
0.8

70

90

60

K Value
(b)
ComE+ (NMI)

ComeE (Micro-F1)

5
10
19
50
100

K Value
(a)

10

3

91

90
80
70
60
50

F1 Score

F1 Score

F1 Score

92

5
10
39
50
100

F1 Score

93
40
35
30
25

ComE (Macro-F1)

F1 Score

ComeE+ (Micro-F1)

5
10
19
50
100

ComE+ (Marco-F1)

K Value
(l)

FIGURE 3 Impact of parameter K. The smaller variance of the performance when K $ K l suggests that ComE+ is more stable with respect to
ComE. (a) BlogCatalog, (b) DBLP, (c) Wikipedia, (d) Rochester, (e) Mich, (f) Amherst, (g) BlogCatalog, (h) DBLP, (i) Wikipedia, (j) Rochester,
(k) Mich and (I) Amherst.

30 M
Loss

20 M
Loss

over-segmentation appears to be useful
in modeling this behavior. Finally, as
expected, the performance of ComE+
are more robust than ComE concerning
both community detection and node
classification especially when K $ K l,
validating the Bayesian inference process
used to handle the uncertainty in the
number of the community.

10 M
0

10 M
0

0 1 2 3 4 5 6 7 8 9
Iterations
(a)

D. Convergence and efficiency

As final experiments, we compare the
convergence and efficiency of both
models. We record the value of the loss
functions at the end of every iteration.
As shown in Fig. 4, the loss of both
ComE and ComE+ converge quickly
within 2-3 iterations.
To demonstrate the efficiency of our
models, we test them on all the six datasets at different scales. More precisely, for
each dataset, we generate four subgraphs
in which we keep 25%, 50%, 75% and
100% of the total number of nodes and
edges. It has to be noted that, to speed
up the computation time of those
experiments, we set d = 2 and g = 5.
The diagram in Fig. 5 shows the processing time of ComE and ComE+ in

20 M

BlogCatalog
Rochester

0 1 2 3 4 5 6 7 8 9
Iterations
(b)
DBLP
Mich

Wikipedia
Amherst

FIGURE 4 Model convergence. (a) ComE+ and (b) ComE.

different datasets. Clearly, the processing
time of our algorithms is linear to the
graph size (i.e., ;V ; and ; E ; ). This validates our complexity analysis at the end
of Sec. IV.
VI. Conclusion

In this paper, we studied the important
(yet largely under-explored) problem of
embedding communities on graphs. We
have investigated the existence of a
closed loop among community embedding, community detection and node

embedding that preserve a communityaware higher-order proximity. More in
detail, we extend the ComE algorithm
to achieve such closed loop in a Bayesian inference setting. The proposed
ComE+ algorithm can better handle
the uncertainty related to the unknown
number of communities.We also designed
an efficient iterative inference algorithm
for ComE+, which can still retain a low
complexity of O ( ;V ; +; E ; ). We evaluated our model on seven real-world datasets and with multiple application tasks.

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IEEE Computational Intelligence Magazine - August 2019

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