Fractional Graph Convolutional Networks (FGCN) for Semi-Supervised Learning
Authors: Yuzhou Chen
Yulia R. Gel
Konstantin Avrachenkov
Venue:  (2020)
Keywords: convolutional networks
node classification
Levy flight
graph-based semi-supervised learning
local graph topology

Abstract: Due to high utility in many applications, from social networks to blockchain to power grids, deep learning on non-Euclidean objects such as graphs and manifolds continues to gain an ever increasing interest. Most currently available techniques are based on the idea of performing  a convolution operation in the spectral domain with a suitably chosen nonlinear trainable filter and then approximating the filter with finite order polynomials. However, such polynomial approximation approaches tend to be both non-robust to changes in the graph structure and to capture primarily the global graph topology. In this paper we propose a new Fractional Generalized Graph Convolutional Networks (FGCN) method for semi-supervised learning, which casts the L\'evy Fights into random walks on graphs and, as a result, allows to more accurately account for the intrinsic graph topology and to substantially improve classification performance, especially for  heterogeneous graphs.

Under review as a conference paper at ICLR 2020
FRACTIONAL GRAPH CONVOLUTIONAL NETWORKS (FGCN) FOR SEMI-SUPERVISED LEARNING
Anonymous authors Paper under double-blind review
ABSTRACT
Due to high utility in many applications, from social networks to blockchain to power grids, deep learning on non-Euclidean objects such graphs and mani- folds continues to gain an ever increasing interest. Most currently available tech- niques are based on the idea of performing convolution operation in the spectral domain with a suitably chosen nonlinear trainable filter and then approximating the filter with finite order polynomials. However, such polynomial approxima- tion approaches tend to be both non-robust to changes in the graph structure and to capture primarily the global graph topology. In this paper we propose a new Fractional Generalized Graph Convolutional Networks (FGCN) method for semi- supervised learning, which casts the Lévy Flights into random walks on graphs and, as result, allows to more accurately account for the intrinsic graph topology and to substantially improve classification performance, especially for heteroge- neous graphs
INTRODUCTION
Adaptation of deep learning (DL) to graphs and other non-Euclidean objects has recently witnessed an ever increasing interest, leading to the new subfield of geometric deep learning. In particular, geometric deep learning is emerging direction in machine learning which aims at generalizing the concepts of deep learning for data in non-Euclidean spaces, e.g., graphs and manifolds, by bridging the gap between graph theory and deep neural networks (see discussion by Bronstein et al. 2017 Monti et al. 2017
Monti 2019) and references therein).
Many such DL approaches for non-Euclidian objects are based on the idea of performing a convo- lution operation in the spectral domain with a suitably chosen nonlinear trainable filter. As a result, node features are mapped into some Euclidian space. Next, graph filters approximated with various finite order polynomials, e.g., Chebyshev polynomials (i.e., the ChebNet model family of Defferrard et al.] 2016 Kipf & Welling 2017). Cayley transform (i.e., CayleyNer of Levie al., 2018) or the generalization of polynomaal filters in form of Auto-Regressive Moving Average (ARMA) models (Bianchi et al. 2019). However, deep learning approaches based on approxima- tion with finite order polynomials tend to be non-robust to even minor changes in the graph structure and largely disregard the local graph topology that often plays the critical role for learning on heterogeneous graphs. In contrast, as noted by Bianchi et al. (2019), one of the primary benefits of the ARMA filters over polynomial ones that ARMA filters are not computed in the Fourier space induced by graph Laplacian, and as a result, ARMA filters are local in the node space and enable to more flexibly and accurately capture the underlying graph topology.
In this paper we further advance this localized approach to DL on graphs and propose frac- tional generalized graph-based convolutional filter for semi-supervised learning, which casts the Lévy Flights into random walks on graphs. As result, Convolutional Networks (FGCN) method allows to more accurately account for the intrinsic local graph topology and to substantially improve classification performance, especially for heteroge- neous graphs.
Fractional Generalized Graph
The key contributions of our paper can be summarized as follows:

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we propose new fractional generalized graph-based convolutional filter for semi- supervised learning, which casts the Lévy Flights into random walks on graphs and, as result, provides more efficient exploration of the graph structure.
we develop new Fractional Generalized Graph Convolutional Networks (FGCN) method that substantially improves accuracy of node classification for graphs of smaller sizes and with lower number of node features.
the proposed architecture of FGCN uses two state-of-the-art operations - gated max- average pooling and residual block. The architecture has been shown to significantly im- prove the training convergence and model output stability.
RELATED WORK
Many earlier semi-supervised learning approaches on graphs, e.g., Gaussian mixture models, co- training, harmonic function, and label propagation, employed only the label information (i.e., la- beled instances) to train models based on the smoothness assumption over the labels (Zhu & Gold- berg 2009) and largely disregarded the graph structure. To improve performance, several learning methods graphs propose incorporate intrinsic ggraph-based" information by designing a classi- fying function via generalizing the normalized cut and adding smooth function with respect to the in- al.)
trinsic structure (Zhou et al. 2004 Zhou & Burges (007). recent approach of Avracheenkov (2012) proposes a generalized optimization framework through considering the above two methods as particular cases. However, one of the major limitations these graph- learning methods is disregarding important information contained in graph edges.
semi-supervised
To address this disadvantage, Defferrard et al. (2016) propose a formulation of convolutional neural SaaadvaattgeeDeeffrraaddd
networks (CNN) based on spectral graph theory ChebNet, which employs approximation via finite order polynomials and, in particular, is based the Chebyshev expansion for fast filtering instead of the expensive digen-decomposition. Graph Convolutional Networks (GCN) of Kipf & Welling (017) simplifies ChebNet and further addresses the gradient vanishing problem and reduces the number of optimization. Other related approaches to graph learning with deep neural networks in- clude, for instance, mixture model networks (MoNet) (Monti et al. (017). graph attention networks (GAT) (Veličković et al. 2017). graph convolutional recurrent networks (Seo et al. 2018). dual graph convolutional networks (Zhuang & Ma 2018), FastGCN (Chen et al. 2018). and simplified version of GCN (Wu et al. (019))
To extend the success of GCN on undirected graphs to directed graphs, MotifNet of Monti et al. (2018) replaces the normalized Laplacian with the motif Laplacian in a multivariate polynomial filter, where the motifs information can help capture the network structure. Finally, the most recent approach Bianchi al. (2019) provides more flexible responses than GCN by using parallel and periodic concatenations of the convolutional kernel via the ARMA filter. As a result, the ARMA approach which is applicable to both directed and undirected networks allows to more accurately incorporate the underlying local graph structure into the graph learning process.
METHODOLOGY
Given a graph structure G (V, WJ. with V the set of nodes (N |V)) and & S V x V the set of edges between any two nodes. Let square matrix W represents the adjacency matrix of graph whose entries { isijsN are the edge weight function such that each edge eij E E has weight wij. We call W the symmetric adjacency matrix (wij * 0 if there is an edge between vertex vi and vertex v;) if we assume that graph G is undirected. In reality, however, undirected graph is simplified representation of complex networks, we also consider directed graph (in which case W T # W) and we can casily transform the symmtriet W to the symmetric one through W' = (WT + W)/2.
We take N x Q feature matrix X as the input to remi-supervised learning algorithm where Q is the number of different node features. Suppose we would like to classify N data points into K classes (communities), define N x K label matrix Y as:
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1, if vertex labelled as class k,
Yx
(I)
O, otherwise.
Here we refer to each column Y. of matrix Y as labeling function. Also define an N x K matrix F and call its columns F.x classification functions.
3.1 GRAPH SIGNAL PROCESSING
In graph signal processing, real-symmetric matrix has N real eigenvalues and its N real eigen- vectors form an orthonormal basis. Given the symmetric adjacency matrix W of a graph, let D be the degree matrix where d;; = Σド= Wij and denote L = U AU the Standard Laplacian matrix, where A diag(Ao, Av- and U [ ................................ - 1] is the matrix of eigenvectors.
In the following, we will revisit three popular supervise learning methods graph-based semi- supervised learning, fractional graph-based semi-supervised learning, and graph convolutional net- works and gain new insights for improving their modeling capabilities.
Graph-based supervisd learning. Graph-based semi-supervised learning (G-SSL) has ceived much attention as an alternative approach to the population paradigm of supervised learning in recent years. G-SSL develops generalized optimization framework, which has three particular cases (i) the Standard Laplacian (SL); (ii) Normalized Laplacian (NL); (iii) PageRank (PR). The optimization formulation with the following expression:
min DD (( (F- ((FFF+Y.... +
(2)
where y is a regularization parameter. The minimization of the first term in Eq. corresponds to the idea that if two nodes are close in graph with respect to some metric, they should belong the same class; and by minimizing the second term we would like to bring the classification function F& as close possible to the labelling function Y.,. The Eq. 2q.2aam allows us to obtain the Standard Laplacian based formulation (a = 1), the Normalized Laplacian Tormulation (a = }), and PageRank formulation (a 0). The objective of the generalized optimization framework for G-SSL a convex function and the corresponding classification function:
F+ (1 a) (I aD aD- - Y
(3)
where a
2 and for h: = 1,...,.. K. Σ+μ
Tuning the parameter a on the power of degree matrix D, we can obtain three mentioned above particular semi-supervised learning methods:
SL method (a = 1): F& =1 a) (1 Y.
NL. method (o = 1): F.A (1 - =11-001-aDDWWD
Y
PR method (a 0): F. (1 (1- a) (1-a)(1-aW))) (1-aWD-I)-
From above formulations, the classification function F, i.e., the result of random walk process which provides the connection the probabilistic interpretation G-SSL. The parameter a controls the strength of the ground truth label matrix Y in the generalized optimization framework.
Fractional graph-based semi-supervised learning. To improve the classification performance (in particular, fuzzy graphs and unbalanced labeled data) of G-SSL, fractional graph-based semi- supervised learning (De Nigris et al., (017)) embeds Lévy Flights into random walks on graphs by constructing from powers of the Laplacian matrix, i.e., the L operator. This operation can be used to generate different transition probabilities (i.c., corresponding to stochastic adjacency matrix) based on different y values. Intuitively, embedding Lévy Flights into random walks allows for better capturing mixing properties (i.e., dependence) in the data. Based on a fractional Laplacian matrix, 0 Y 1, the anomalous (fractional) diffusion processes networks can be constructed from the spectra data and eigenvectors the Laplacian matrix. The fractional powers of allows Lévy random walks with long-range navigation network. For example, the long-range transitions
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on network directly move node and node with the transition probability m through
random walker, where m is an element in the fractional transition matrix M(h) (see the graph within dotted circle in Figure []. The transition probability (1) between any two nodes whose geodesic distance is not infinite can summarized as follow:
m mm = 6w (L2) uu k((2)
(4)
where due represents the Kronecker delta, k((2) denotes the fractional degree of the node and x(()) (LD) Eq. provides transition probabilities for the Lévy Flights. Unlike the standard random walk, the Lévy Flights jump immediately makes Lévy Flights a very good exploratory process. There is a price to pay for this: the typically sparse transition probability matrix becomes non-sparse. We mitigate non-sparsity by taking reasonable number of principal singular eigenvectors or limiting the number of terms in the Taylor expansion. Through replacing the operator with L UTA the new optimization formulation leaves us with the following expression:
several hops in graph. This feature
min D2 D.- F. + U - Y.x) Y 220-111200 (F.x - Y++)
(5)
where (Dy)ss = (L))
Let 0 y 1, the closed form solution for Eq. 5 can be obtain fellow:
F& = a) - a for k = =,,,.... I.....K. Therefore, we can conclude three particular fractional semi-supervised learning mehtods like G-SSL:
W W, Y.s,
Fractional SL method (a 1): F. (1 - a) (I aD, aD,'W,, Yx -1 Fractional NL method (o = 1): F = - a)) (1 aI- Wr Y.x
Fractional PR method 0): F.x (1 a) (1 DD Y.k
3.2 PROPOSED FRACTIONAL GENERALIZED GRAPH CONVOLUTIONAL NETWORKS FOR SEMI-SUPERVISED NODE CLASSIFICATION
Although both G-SSL and fractional G-SSL achieve comparable and consistent (low variance) per- formance some datasets, e.g., Les Miserables, Wikipedia-math, and MNIST, these approaches consider only the given adjacency matrix W and the label matrix without using the feature ma- trix X. Such limitation is crucial especially when dealing with datasets that not only exhibit sophisticated topological graph structure but also provide node feature information, such as cita- tion, biological, financial, and power grid networks. To address this drawback, through using the feature matrix X instead of the label matrix Y and encoding the graph structure by using neural net- work framework, many graph-based neural networks methods, e.g., graph convolutional networks (GCN), have been recently proposed and have been shown to demonstrate very impressive advances in semi-supervised learning performance graphs. Next, we Generalized Graph Convolutional Networks (FGCN) for supirvieeeeeeed node classification.
discussing the new Fractional
Fractional Generalized Graph Convolutional Networks (FGCN), The key idea behind our proposed method is Fractional Generalized Sisma-based based (FGS) filter gras(a, = (1
a) (I - (W-)) the expression of the FGS filter and this has the advantage of avoiding the inverse computation. We have the following expression:
a) a Let us insert Taylor series expansion into
gras(a,o,n) = (1 -a)(1+
(aI))++...) [+ +aī)² + (aī)² + (aL)3 +...) = (1 - a) Σ (at)
(7)
1=0
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where 0
< 1,0 v 1,0 Y 1. We then obtain the general classification function by
multiplying the feature matrix X:
X = gFas 0, 7)
- a)
X
oLx the second item
+...).
(8)
the first item
the third item
where, recursively,
(x);= x +i((X')- (x')= x, 11=1,
(9)
Convolutional layer. During the training process, the convolutional model need to train the trainable parameters (W, b) of the graph filter which can scan the given input feature matrix into a series of feature maps with neurons. Thereby, we provide an implementation of Eq, al a FGS convolutional layer:
H = - (1- a)
Σ aL)) H(w
8
(10)
where H[++1) is the hidden layer output matrix of activations in the (-th layer and 17 = X, (r-) is adopted activation function, and W is the trainable weight in the /-th layer. Besides, we borrow the concept of the parallel system in reliability theory to improve the consistency of proposed method. A parallel system is a configuration that the entire system works as long as not all involved components in the system fail. Hence, the parallel system structure is more robust against noisy inputs, compared to single system structure.
Lemma 1. Let X be the output matrix P1 from a pooling layer. Let (1, finite population such that each unit i, U is associated with an output matrix x
N} be FGS i
1,..., N. Then
S N
Var(X
(11)
N
n'
where $2 =
(x(0
Pas XFas) /(N -).
Suppose there are n components in parallel system, with the probability of non-failure p(t) (where i= 1.....,.,. n) in parallel system, then the reliability of this parallel system PPS can be obtained with the following expression:
P =1
I
P
= ) (()) x (1 (1- (1-P()) x x (1-p))))
p(7)).
(12)
According to Lemma the parallel system allows for enhancing stability and reducing estimation variance up order of (i.e., Var(XFGs) O(S2 /n)).. In this way, establish both theoretical and practical guarantees for our proposed model to reach stable over a large set of hyperparameters, small datasets, and noisy labels based this parallel implementation.
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(t) HI---
12
max
OLW Co)
oft
ELU
ELU
2 (a Were!
Gieee Pootin
x
Figure I: Illustration Fractional Generalized Graph Convolutional Network model. The input the feature matrix X and the graph within dotted circle represents embedding Lévy Flights into random walks on graph (where L is the Laplacian matrix to power 7). FGCN architecture consists of three main components: (i) FGS convolutional layer with parallel structure; (ii) gated max-average pooling layer; (iii) activation block for residual learning.
Pooling layer. When implementing the form of pooling operation to aggregate information from the outputs parallel FGS convolutional layer, instead of using some popular pooling functions such max and average pooling, we apply the state-of-art pooling operation gated max-average pooling (Lee et al. 2016) to capture the local and global information from all the nodes and graph structure. The rationale behind the gated max-average pooling. that considers "responsive" strategy (i.e., improving translation invariance and scale invariance via considering input in each gating mask) based on the mixed max-average pooling equation. We have:
Sgate( (xFas) = a (WT X Xras) m
(X (X + (1 (W Xras)) favg (X Xras),
(13)
where W is the trainable weight matrix, XFGS is the output matrix from the parallel FGS convolu- tional layer after concatenation operation.
Residual building block. Inspired by residual learning, apply residual block by adding the skip connection after the pooling layer. One of the advantages of the residual learning is the identity mapping which provides direct path for propagating information. When using the residual building block, we adopt similar scheme in He et al. (2016) to deal with the output of the pooling layer. Let H(x) be an underlying mapping and cast it as H(x) mapping defined by H(x) I. In other words, it is easier to optimize the residual mapping F(x) than optimizing the direct mapping Hxx and helps to avoid the gradient vanishing problem during training. To this end, we use an exponential linear unit (EIJU) in direct mapping and place rectified linear unit (Rel.U) after addition our model.
F(x) + x. where F(x) is the residual
4 EXPERIMENTS
4.1 DATASETS
Undirected networks. Cora-ML (this Cora dataset consists of Machine Learning papers), Cite- Seer and PubMed are three standard citation networks benchmark datasets used for semi-supervised learning evaluation. In these citation networks, nodes represent publications, edges denote citation, the input feature matrix are bag of words and label matrix contain the class label of publication. We use the same data format GCN, i..,, 20 labels per class in each citation network.
Directed networks. We evaluate our method on four directed networks Cora, IEEE 118-bus sys- tem (IEEE bus), Texas 2000-bus system (TX bus), and South Carolina 500-bus system (SC bus). Unlike the small subset of the Cora-ML dataset (above), Cora extracted from original dataset which contains more than 50, 000 computer science research papers, where nodes represent scien- tific papers and directed edges between node u and node e (u - v) represent citation of paper u in paper u. In addition citation networks, also test our method power grid networks. For

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IEEE 118-bus system, we consider setweighted-dreetee graph as model for the IEEE 118- bus system where nodes represent units such as generators, loads and buses, and edges represent the transmission lines. The input features of power grid network are generator active power upper bound (PMAX) and real power demand (PD) obtained from MATPOWER case struct. For Texas 2000-bus system and South Carolina 500-bus system, treat them as weighted directed power grid networks. In particular, we use total line charging susceptance (BR B) as edge weight and input features are: (i) real power demand (PD); (ii) reactive power demand (QD); (ii) voltage magnitude (VM); (iv) voltage angle (VA); (v) base voltage (BASE_KV) VV). On directed networks, we test them separately from two scenarios: (i) on Cora, the number of features are decomposed into a low dimension (130 components) based on principal component analysis (PCA), the dataset is trained with 10% label rate and evaluated in another 10% fraction whose labels are withheld; (ii) on IEEE 118-bus system, Texas 2000-bus system, and South Carolina 500-bus system, 10% label rate for training set, 20% for validation and 70% for test sets. The statistics of data we used in the experimental section are summarized in [Tble
Table 1: Dataset statistics.
Dataset
Vertices
Edges
Features
Classes
Label rate
Train/Valid/Test
Cora-ML
2,708
5,429
1,433
7
0.052
140 140/500/ 1,000
CiteSeer
3,327
4,732
3,703
6
0.036
120 500 1,000
PubMed
19,717
44.338
500
3
0.003
60 / 500 / 1,000
Cora
19,793
65,311
8,710
70
0,100
1,979 1,979 / 1,979
IEEE Bus
118
182
2
3
0.100
11 /5 770
TX Bus
2,000
2,668
5
3
0.100
200 400 1,400
SC Bus
500
584
5
0.100 3
50/ / 100 / 350
4.2 SETUP AND BASELINES
Baselines. In this setting, on undirected networks, we compare FGCN with the state-of-the art semi-supervised classification approaches including (i) using the label matrix as input: label propagation (LP) (Zhu & Ghahramani] 2002): (ii) using the feature matrix as input: DeepWalk (DW) (Perozzi et al., 2014), graph attention networks (GAT), GNN with ChebNet polynomials filter (ChebNet), GCN, GNNs with convolutional ARMA filters (ARMA), Graph Markov Neural Networks (GMNN) (Qu et al. 2019), and Large-Scale Learnable Graph Convolutional Networks (LGCNs) (Gao al. 2018). In addition, on undirected networks, metric motif adjacency matrix instead of Normalized Laplacian matrix in multivariate polynomial filters), ChebNet, GCN, ARMA, GMNN, and LGCNSs as the benchmarks.
MotifNet (using sym-
Training setting details. The training is done by using Adam optimizer with learning rate Iri = 0.01 for undirected networks and lrz 1: 0.001} for directed networks. To prevent the model from overfitting, we consider both adding dropout layer before two graph convolutional layers and kernel regularizers (f2) in each layer. For undirected networks: we follow the experimental setup in Kipf & Welling (2017) to set the parameters of baselines (GAT, ChebNet, GCN, ARMA, GMNN, and LGCNs) by using two graph convolutional layers with 16 hidden units, l regularization term with coefficient 5 x 10-4, and dropout probability Pdrop. of 0.5 (for ARMA, except for number of hidden units, the hyperparameters setting are significantly different from others). For directed networks: we consider the MotifNet setting with using dropout rate Pdrap. of 0.5, E2 regularization term with coefficient 0.001, and the degree of multivariate polynomials k = 4 for MotifNet: for other three baselines (i.e., ChebNet, GCN, ARMA, GMNN, and LGCNs), we also use two-layer network but with 16 hidden units, learning rate Ir. with 0.001, 0.5 dropout rate and & regularization weight of 5 x 10-4. 4 In the following, Table 2 displays the best hyperparameter configurations of FGCN for each dataset by using standard grid search mechanism (the optimal kernel regularization weight & always equal to 5 x 10-4).
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Table Hyperparameters setting of FGCN
Dataset
Pdrop.
1/
[1]
7
TFGS
;
Cora-ML
0.75
32
0.6
0.50
1.000
7
{5:2]
ELU + ReLU
CiteSeer
0.75
32
0.6
0.53
1.000
{3:2]
ELU ELU
PubMed
0.00
32
0.6
0.50
1.000
7
{5:2]
ELU + ReLU
Cora
0.00
128
0,4
1.00
1.000
15
{1:1]
ELU+ ELU
IEEE Bus
0.00
128
0.1
0.20
0.004
40
{7:2)
ELU ELU
TX Bus
0.00
128
0.3
0.10
0.100
14
[5:2]
ELU ELU
SC Bus
0.00
128
0.2
0.10
0.100
20
{5:2]
ELU ELU
Th denotes the number of hidden units in the first layer; #CD and #C represents the number statis-
tically independent parallel components in the first layer and the second layer respectively, RFas denotes the number of items Taylor series polynomial expansion of FGS filter; of (RB represents the type of activation
functions in residual block structure.
RESULTS
Performance comparison of semi-supervised node classification. Table 3 reports the average accuracy delivered by FGCN and competing methods for fixed train/valid/test sizes (see Table I The best performance for each dataset is marked in bold.
Table that FGCN outperforms all competing approaches all data sets, except for CiteSeer (FGCN delivers the fouth best accuracy result), PubMed (FGCN delivers the second best accuracy result), Cora (FGCN delivers the third best accuracy result after the LGCNs approach ference between FGCN and LGCNs for Cora is less than 0.05%). The improvement gain of FGCN over the next most accurate method ranges from 0.75% (for undirected Cora-ML over GMNN) to 4.21% (for directed IEEE 118-Bus to undirected and directed networks (i.e., ChebNet, GCN, ARMA, GMNN, and LGCNs) tend to deliver noticeably lower accuracy results for a directed networks (especially weighted-direeted networks), while the new FGCN method yields a more stable performance across both directed and undirected networks. In turn, CiteSeer (unweighted-undircttee))) GMNN and LGCNs outperform our FGCN model delivers marginally
the dif-
GMNN). Remarkably, methods that applicable both
FGCN to 1.7%. For the PubMed wwwwweghhed-undiretteeee better performance results than LGCNs, and GMNN has better performance than FGCN. Finally, both GMNN and LGCNs outperform FGCN on CORA curacy results are very close. Based on the obtain results, the new FGCN approach tends be the most competitive and, hence, preferred node classification method for sparser networks with higher label rates.
but the delivered ac-
Furthermore, the IEEE 118-Bus data set is the smallest among the considered data (see Table I1. and we might expect to observe lower accuracy results for this data set due to a limited training set. However, the accuracy yielded by FGCN is among the highest ones across all data sets.
Table 3: Comparison of average accuracy (%) of semi- semi-supervised classification approaches for both undirected networks and directed networks.
Undirect networks
Directed networks
Method
Cora-ML
CiteSeer
PubMed
Cora
IEEE Bus
Bus
SC Bus
LP
68.70
46.32
65.92
DW
67.20
43.27
65.33
MotifNet
60.00
65.75
82.00
95.18
GAT
83.11
70.85
78.56
ChebNet
81.45
70.23
78.40
58.93
60.00
80.04
94.13
GCN
81.50
71.11
79.00
57.75
52.86
73.36
90.33
ARMA
82.80
72.30
78.80
58.99
70.55
81.20
94.33
GMNN
83.72
73.10
81.80
61.20
78.88
86.21
96.57
L.CNS
83.35
73.08
79.51
60.72
71.43
85.57
95.14
FGCN (ours)
84.35
71.89
79.56
60.70
82.20
87.74
97.61
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Evaluation of FGCN-specifiee parameters. During grid search over three parameters (i.e., a, 0, and 7), we find that: (i) the regularization parameter a which used to specify the relative importance of graph in clustering strongly relates to the probability of initial conditions for random walks when the self-refreshing process works, and it strongly influences the network's ' generalization ability and node classification performance for all datasets; (ii) the free unifying parameter o provides enough flexibility to construct a canonical formulation of different graph-based smii-superiieed methods. Table indicates that the optimal value of a depends on both the types of networks (undirected and directed) and label rate not on the size of network; (iii) the fractional power parameter substantially impacts the accuracy of node classification for the small datasets (see e.g.. Figure 2 however, no similarly strong influence is found in the larger datasets.
Sensitivity analysis. In the sensitivity analysis setting, we have the ability to analyze the sen- sitivity of the node classification accuracy to variation from three FGCN-specific parameters E {0.1,... 1), a E 0., .... 1}. and 7 E {0.001, 0.01, 0.1, 1}. In this case, we only show the results from sensitively analysis for FGCN model on IEEE 1 118-bus dataset. First, we perform the parameter learning experiments on four scenarios with a fixed parameter 7. Figure 2 shows that the accuracy substantially decreases when is larger than 0.8, especially in y equals to 0.001 and 0.01 (see Figure Za 2b). Setting 7 = {0.1, 1}. we observe that the classification accuracy nearly monotonic decreases while increasing a. Additionally, there little difference of classifica- tion accuracy between = 0.001 and ? = 0.01 and FGCN generally gives consistent and higher accuracy when the a parameter is within the range of {0.1, 0.2, 0.3, 0.4). We then explore the vari- ation of accuracy based on tuning parameter within the range of [0.001, 0.002, ... 0.01] (setting {0, 0.1, 1 at the same time), however, is hard to obtain the optimal (o, 4) combination through gathering finite experimental results (100 runs) since some of the results are very close. Therefore, we run the following experiments to demonstrate the impact evaluation y:
Step I: Setting y E {0.001, 0.002, ... 0.13).
Step 2: For each y (i.e., fix the y value in each experiment), we our proposed model 100 times separately for a from 0 to I by 0.01; Then we can obtain the 100 average accuracies for each y under fixed y, that is, ALACo==,ACC---. ... AACo=1] where = 1,....10 ,10.
Step 3: Fitting the Gaussian distribution to (AACo=O: ACC,=0.01;"' "AACo (see
Figure 3b).
Similar to y, we can fit the Gaussian distribution to (AAC, ACC,y ... o.. by fixing o, where j 1. 11 (see Figure Ba
to
From Figure 3 we find that there exist larger difference between the shapes of approximate Gaussian distributions by fixing the parameter a than fixing the parameter y which means the parameter more important factor in FGCN approach for small datasets.
6 CONCLUSION
In this paper we have proposed a new Fractional Generalized Graph Convolutional Networks (FGCN) method to semi-supervised learning on graphs that enables to better capture the intrin- sic local graph topology. The key idea behind our new approach is a new fractional generalized braph-based convolutional filter which casts the Lévy Flights into random walks on graphs and, as result, allows for more efficient, accurate and robust exploration of the local graph structure which often plays the key role in graph learning performance, especially for heterogeneous graphs.
Our numerical studies have indicated that the new FGCN method tends to outperform competing deep learning approaches on both unweighted-directt
and unweighted-undired considered data sets, except of Cora for which the difference between the best result delivered by the GMNN and FGCN is less than 0.85%. The gain in learning accuracy of FGCN over the next best competitor ranges from 0.75% to 4.21%, and the highest gain has been achieved for the IEEE 118-Bus data set which is the smallest among the considered data sets. Furthermore, in contrast to the competing approaches, FGCN tends to deliver a more stable performance across directed and undirected networks regardless of the label rate.
graphs in all

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Under review as a conference paper at ICLR 2020
00
0.2
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10
(a) 7 0.001
(h) 0.01
0.8
Ausinge accuumcy
(c) 0.1
(d) 7 1.0
Figure 2: Accuracy FGCN for PageRank, Normalized Laplacian, and Standard Laplacian, which
depends on the function of and fixes the fractional power (0.001 (a), 0.01 (b), 0.1 (c), 1.0 (d)
012
0112
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0444
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Averae sccumnee--/110000) VCI00000 1.00
Averaee accuragy o 010.
(a) Density of accuracy depends on fixing
(b) Density of accuracy depends on fixing
Figure 3: Generalized Gaussian density accuracy FGCN: (a) the solid curves represent three particular scenarios in FGCN, where the red curve denotes the PR-based method (o = 0), green curve denotes the NL-based method (a 0.5), and blue curve denotes the SL-based method (a 1); the dotted curves represent common scenarios in FGCN model, that is, the generalized parameter falls within the range of (0, 0.5) U (0.5, 1). (b) the red, green, blue dotted curve represents the scenario when setting the fractional parameter equals 0.001. 0.005, and 0.01, respectively; The solid curves represent scenarios where in the range of (0.001, 0.005) U (0.005, 0.01).
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Under review as a conference paper at ICLR 2020
In the future plan to advance the proposed FGCN technique to learning multilayer networks and to enhance graph learning process with topological information on the underlying deep neural network.
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Filippo Maria Bianchi, Daniele Grattarola, Lorenzo Livi, and Cesare Alippi. Graph neural networks with convolutional arma filters. arXiv preprint arXiv: 1eti:999010133 2019.
M. M. Bronstein, J. Bruna, Y. LeCun, A. Szlam, and P. Vandergheynst. Geometric deep learning: going beyond euclidean data. IEEE Signal Processing Magazine, 34(4):18442 2017.
J. Chen, T. Ma, and C. Xiao. Fastgen: fast learning with graph convolutional networks via impor- tance sampling. arXiv preprint /801 2018.
S. De Nigris, F. Bautista, P. Abry, K. Avrachenkov, and P. Gonçalves. Fractional graph-based semi- supervised learning. In 25th EUSIPCO, pp. 356-360, 2017.
M. Defferrard, X. Bresson, and P. Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. In NIPS, pp. 3844-3852, 2016.
Hongyang Gao, Zhengyang Wang, and Shuiwang Ji. Large-scale learnable graph convolutional networks. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 1416-1424. ACM, 2018
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R. Levie, F. Monti, X. Bresson, and M. M. Bronstein. Cayleynets: Graph convolutional neural networks with complex rational spectral filters. IEEE Transactions on Signal Processing, 67(1): 97-109, 2018.
Monti, D. Boscaini, Masci, E. Rodola, J. Svoboda, and M. M. Bronstein. Geometric deep learning on graphs and manifolds using mixture model cnns. In CVRPR, pp. 5115-5124, 2017.
Monti, K. Otness, and M. M. Bronstein. Motifnet: motif-based graph convolutional network for directed graphs. In 2018 IEEE DSW, pp. 225-228, 2018.
et al. Monti. Geometric deep learning, 2019. attp://geonsriicdeepleennngg
Andrzej Nowak and Kevin R Collins. Reliability of structures. CRC Press, 2012.
Omer Ozturk and Narayanaswamy Balakrishnan. Constructing quantile confidence intervals using extended simple random sample in finite populations. Statistics, pp. 1-15, 2019.
B. Perozzi, R. Al-Rfou, and S. Skiena. Deepwalk: Online learning of social representations. In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 701-710. ACM, 2014.
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Meng Qu, Yoshua Bengio, and Jian Tang. Gmnn: Graph markov neural networks. arXiv preprint 1rXiv:190S.0621,, 2019.
Charles T Scott and Michael Köhl. Sampling with partial replacement and stratification. Forest Science, 40(1):30 46, 1994.
Y. Seo, M. Defferrard, P. Vandergheynst, and X. Bresson. Structured sequence modeling with graph convolutional recurrent networks. In International Conference on Neural Information Processing, pp. 362-373. Springer, 2018.
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A APPENDIX
Proof of Lemma 1: The parallel structure (Henley Kumamoto, 1981 constructed in a sim- ilar spirit as bagging of features in random forest and other ensemble learning methods. The key rationale behind the parallel structure
reduce variance and increase stability.
Let us first consider I Suppose that T is random sample drawn without replacement from finite population = -1, 2,.,,...NN according to sampling design p(.). Each unit i, i ,,...,N FGS. i.e. the i-th output matrix from FGS convolution for new feature
of 4 is associated with (() matrix X,, Probability of choosing sample Pr(T v) = p(T) > for all E U and + 0. Let Z be the indicator variable such that
1, ==
X, is in the sample 0, otherwise.
Z4
(14)
Hence, probability that the unit i, 1...., N is selected, is given by म; = F(Zz), and proba- bility that units and j. i,j = 1,..., N are selected simultancously is vij EyZ,Z,); π. and Tij are called first order inclusion probability and second order inclusion probability, respectively.
Since in the current paper, we consider simple random sampling design of units x FGS without replacement from U, n; = n/N and Tij = n I)/N(N -1-
Let Xras EiEn (1) F Ei=1 x Pts/n. Hence, given (x FGS...... that
x (N) ) we find PGS)
N E[Xaas = Σ
x(i)
N
x
N
x((
N
(
[2]]
FGS
FGS n
FGS
Pas
PU Fas+
n
N
N
1=1
i=
12

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Under review as a conference paper at ICLR 2020
where XFGS is be the output matrix P1 from pooling layer (see Figure 1 which implies that FGS is an unbiased estimator of population mean yas In turn.
1 V
Var(X
Σ
Zi
Z
n²"
FaS
N
1
Cov
ZiX ZiX (t)
Σ
Zj 2
n
FGS1
FGS
1=1
j=1
N
N
N
(x
Var (2z)
x
FGS PGS Cov (Zi, Zz)
x(1
n
FGS)
1=1
f=1
N
N Σ
N
72
12
(1
12
n
[X
n²
N
N
FGS)
PGS
PGS
N
-
N
N
1=1 jzz
N
N
2
N
12
1
(1
N
n
N(N -
1
(N 1)
(x
rGs)²
FGS
Σ
x
PGS
+
(x
FGS
i
N
27
(1 N
1
(x FGS)
Σ
x
12
N(N -
Fas
= =(1
n1
S
N
(15)
FGS - xxas )² /(N-1) 11) and the factor (1-n/N) is called the finite popu- lation correction (FPC). Hence, the proposed PS structure decreases an original estimation variance S by order of n. Note that we can consider other sampling designs p(-). including, for example, weighted sampling and p-extended simple random sampling with replacement (see discussion by
where 52 = EE x FFss=
Wong & Easton 1980, Scott& & Köhl 1994 Ozturk & Balakrishnan 2019 and references therein).
The stability result of the parallel structure (Nowak Collins 2012) follows from verbatim appli- cation of the probability bound on the intersection of independent events.
B
APPENDIX
The general idea of graph-based semi-supervised learning (G-SSL) is based on two widely used optimization frameworks. The first formulation, the Standard Laplacian based formulation (Zhou & Burges 2007) follows:
N de P..
min
Wij
F..
P31
Y3
(16)
E=1
where dus is (i, i) -eleemen in degree matrix D and w.y represents the edge weight for edge 6g in ad- jacency matrix W. For the second formulation, the Normalized Laplacian based formulation Zhou
et al. 2004), is as follows:
Fr. du
N |..
min
Wij
Y.11
(17)
d
The following lemma (Avrachenkov et al. 2012) asserts that the generalized optimization frame- work, i.e., G-SSL, which has particular cases the two above mentioned formulations:
13

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Under review as a conference paper at ICLR 2020
Lemma 2. Let denote alternative parameter the power of degree matrix D whose entries are the degrees dui and let I a S 1. Then
N
N
N P. - Ys.11
min
Wij 1 P.
-d
+ p Σ 2o-
The classification functions for the generalized semi-supervised learning given by
F = WD Y...
Proof: The objective functtinn of the generalized semi-supervised learning framework (i.e., Lemma can be rewritten in the following matrix form:
Q(F) 2 Σ D
Fxx Σ (Fx -xx)" D20-1 120 -
Given the first order optimality condition Q 0, we have
2F (D- + D0~1 + 2 (Fxx Yx) 120-1 0.
Multiplying the above expression from the right hand side by 20+1 leads
TF (D (L+ 1.7 DT*) + 2p (F. Y )² = 0.
Then, substituting = and rearranging the terms yields
F (2T nー-1 (W + w²) D + nJ) 1 = 0. D
Since the resulting adjacency matrix W is a symmetric matrix (see transformation 3 the main body of the paper through replacing with = 2/(2 + n)), we obtain
F = I [II - 2D - WD + +pl)
which concludes the proof.
14


Reviews:
[{"review_id": "SyxdoagL5B", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper provides a new model for semi-supervised node classification in directed and undirected graphs. It is based on a novel fractional filter for graph conv networks, which generalizes several previously employed graph semi-supervised learning frameworks, by introducing a fractional hyperparameter (sigma in the paper), using fractional powers of the Laplace operator.\n\nThe relevant previous work in the area seems to be cited, and the paper appropriately embedded in the previous work\n\nEmpirically, the method outperforms several established baseline models (classical and neural) on standard datasets in the node classification task. in particular one with a low number of labeled nodes. A sensitivity analysis is performed to assess the impact of the hyperparameters of the FGCN. \n\nI believe the experimental results could justify an accept, but I would not claim I am an expert in semi-supervised learning on graphs.\n\n\nQuestions:\n\nHow can the architecture be extended to handle edge types?\n\nHow were the hyperparameters of the baselines tuned?\n\nSmall things:\nCould you provide more context around the reliability in parallel systems (eq 11)? It is not clear how this relates to the rest of the paper.\n\nPlease add a citation for gated max average pooling.\n", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I did not assess the derivations or theory."}, "extra_text": {}}, {"review_id": "ByeG3YPpFB", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes a fractional graph convolutional networks for semi-supervised learning. The proposed method used a classification function of a fractional graph semi-supervised learning (GSSL) [De Nigris et al., 2017] as a graph filter. In addition, the authors adopt a parallel system and weighted combination of max and average pool. Experimental results show that the proposed method (FGCN) shows the best accuracy compared to other recent graph-based neural networks for all datasets except one.\n\nThe key approach of the proposed method is to apply a classification function (equation (3)) obtained by solving a GSSL problem to graph convolutional networks. However, this idea is too incremental and applying the classification function to graph filter is very trivial. This works also combines the fractional GSSL with a parallel system and weighted pool. But, it is not clear which contribution actually improves the results. Moreover, the intuition of the fractional approach is not clear too, e.g., how the optimization (equation (2)) is derived?, and some explanations are unnatural to demonstrate the methodology, e.g., equation (4). For these reasons, this paper is under the bar of acceptance.\n\nMain concerns:\n1. What is the intuition of the optimization of GSSL? How is it obtained? And among all fractional methods (e.g., SL, NL, and PR), which one doe achieve the best performance?\n\n2. The FGS filter in equation (7) is the sum of infinite terms. However, in practical, it is impossible to compute the infinite terms. Does this approximate the sum of finite terms? If does, what is the number of truncation?\n\n3. The authors mention that they establish a theoretical guarantee of the parallel system. But, I could not find any theoretical results. It would be better to include the analysis in the paper.\n\n\nMinor concerns:\n1. In page 3, please edit “forulation” -> “formulation”\n2. In equation (8), I think “X+\\alpha \\tilde{L} X” should change to “\\tilde{L} X”\n", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "Bke3S1BTYS", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper presents a fractional generalized graph convolutional networks for semi-supervised learning. The authors design a new graph convolutional filter based on Levy Flights, and propose new feature propagation rules on graphs. Experimental results on multiple graph datasets are reported and discussed.\n\nPros.\n1. This paper presents a nice overview of three popular semi-supervised learning methods in Section 3.1, and presents insights regarding these models.\n2. A new graph convolutional filter is proposed. The motivation is clear, and the technical details are easy to follow.\n3. Experiments on five benchmark datasets are conducted. Both undirected and directed graphs are used in experiments.\n\nCons.\n1. The proposed method contains three major components: parallel FGS convolution, pooling, and residual block. Although some justifications are provided for such designs, it is difficult to justify the role of each component. In other words, it is unclear whether the performance gain is from the parallel structure, or the residual block. What would be the model performance without parallel structures? Also, given that FGCN has quite a few layers, the motivation of using residual blocks should be carefully justified. \n2. Many recent methods on graph neural networks are not discussed or included as baselines, such as [a-b].\n[a] GMNN: Graph Markov Neural Networks, ICML 2019\n[b] Large-Scale Learnable Graph Convolutional Networks, KDD 2018\n[c] SPAGAN: Shortest Path Graph Attention Network, IJCAI 2019  \n\n\n-------------------------------------------------\nThe response from authors addressed many of my concerns. The rating has been updated.", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a fractional graph convolutional networks for semi-supervised learning, using a classification function repurposed from previous work, as well as parallelization and weighted combinations of pooling function. This leads to good results on several tasks.
Reviewers had concerns about the part played by each piece, the lack of comparison to recent related work, and asked for better explanation of the rationale of the method and more experimental details. Authors provided explanations and details, and a more thorough set of comparison to other work, showing better performance in some but not all cases.
However, concerns that the proposed innovations are too incremental remain.
Therefore, we cannot recommend acceptance.
Decision: Reject