Ranking cliques in higher-order complex networks

Abstract

Traditional network analysis focuses on the representation of complex systems with only pairwise interactions between nodes. However, the higher-order structure, which is beyond pairwise interactions, has a great influence on both network dynamics and function. Ranking cliques could help understand more emergent dynamical phenomena in large-scale complex networks with higher-order structures, regarding important issues, such as behavioral synchronization, dynamical evolution, and epidemic spreading. In this paper, motivated by multi-node interactions in a topological simplex, several higher-order centralities are proposed, namely, higher-order cycle (HOC) ratio, higher-order degree, higher-order H-index, and higher-order PageRank (HOP), to quantify and rank the importance of cliques. Experiments on both synthetic and real-world networks support that, compared with other traditional network metrics, the proposed higher-order centralities effectively reduce the dimension of a large-scale network and are more accurate in finding a set of vital nodes. Moreover, since the critical cliques ranked by the HOP and the HOC are scattered over a complex network, the HOP and the HOC outperform other metrics in ranking cliques that are vital in maintaining the network connectivity, thereby facilitating network dynamical synchronization and virus spread control in applications.

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RESEARCH ARTICLE | JU LY 18 2023
Ranking cliques in higher-order complex networks 
Special Collection: Disruption of Networks and System Dynamics
Yang Zhao ; Cong Li  ; Dinghua Shi ; Guanrong Chen ; Xiang Li
Chaos 33, 073139 (2023)
https://doi.org/10.1063/5.0147721
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Ranking cliques in higher-order complex
networks
Cite as: Chaos 33, 073139 (2023); doi: 10.1063/5.0147721
Submitted: 25 February 2023 ·Accepted: 20 June 2023 ·
Published Online: 18 July 2023
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Yang Zhao,1Cong Li,1,a)Dinghua Shi,2Guanrong Chen,3and Xiang Li4,5
AFFILIATIONS
1Adaptive Networks and Control Lab, Department of Electronic Engineering, School of Information Science and Technology,
Fudan University, Shanghai 200433, China
2Department of Mathematics, College of Science, Shanghai University, Shanghai 200444, China
3Department of Electrical Engineering, City University of Hong Kong, Hong Kong 999077, China
4Institute of Complex Networks and Intelligent Systems, Shanghai Research Institute for Intelligent Autonomous Systems,
Tongji University, Shanghai 201210, China
5State Key Laboratory of Intelligent Autonomous Systems, the Frontiers Science Center for Intelligent Autonomous Systems,
Tongji University, Shanghai 201210, China
Note: This paper is part of the Focus Issue on Disruption of Networks and System Dynamics.
a)Author to whom correspondence should be addressed: cong_li@fudan.edu.cn
ABSTRACT
Traditional network analysis focuses on the representation of complex systems with only pairwise interactions between nodes. However,
the higher-order structure, which is beyond pairwise interactions, has a great influence on both network dynamics and function. Ranking
cliques could help understand more emergent dynamical phenomena in large-scale complex networks with higher-order structures, regarding
important issues, such as behavioral synchronization, dynamical evolution, and epidemic spreading. In this paper, motivated by multi-node
interactions in a topological simplex, several higher-order centralities are proposed, namely, higher-order cycle (HOC) ratio, higher-order
degree, higher-order H-index, and higher-order PageRank (HOP), to quantify and rank the importance of cliques. Experiments on both
synthetic and real-world networks support that, compared with other traditional network metrics, the proposed higher-order centralities
effectively reduce the dimension of a large-scale network and are more accurate in finding a set of vital nodes. Moreover, since the critical
cliques ranked by the HOP and the HOC are scattered over a complex network, the HOP and the HOC outperform other metrics in ranking
cliques that are vital in maintaining the network connectivity, thereby facilitating network dynamical synchronization and virus spread control
in applications.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0147721
Numerous studies have shown that interactions in networks
are not limited to pairs of nodes, but often involve higher-
order cliques of nodes. To study higher-order interactions
among large cliques, new metrics and tools are needed. In
this paper, the higher-order cycle (HOC) ratio, higher-order
degree (HOD), higher-order H-index (HOH), and higher-order
PageRank (HOP) are introduced and used to rank cliques in
a large-scale network with higher-order structures. Extensive
experiments on both synthetic and real-world networks show that
these higher-order centralities could rank cliques in a more dis-
tinguished order compared to traditional lower-order metrics,
such as degree, cycle ratio, PageRank, and H-index, due to the
inherent properties. More importantly, the cliques ranked by the
proposed metrics could accurately reflect the functional impacts
of the network on structural robustness, behavioral synchroniza-
tion, and dynamics propagation.
I. INTRODUCTION
Many real-world networks exhibit scale-free properties, which
suggest that there is significant heterogeneity in the networks.1
Research on complex networks demonstrated that key nodes, which
play some particular roles in the networks, have a significant impact
on the network dynamics. In practice, ranking key nodes could
help manage many network tasks, such as better controlling diseases
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spreading by isolating the source of transmission,2–6predicting miss-
ing or future connected components in the network to infer the
network evolution, e.g., for social networks like Weibo and Twitter,7
formulating product marketing strategies for e-commerce,8opti-
mizing resource allocation to reduce cost budget,9and improving
the survivability of infrastructures to prevent catastrophic cascad-
ing failures.10 Therefore, ranking and selecting key nodes in pro-
cess initialization to improve network performances has become an
important issue in the studies of various complex networks.
Many effective methods have been developed11–23 based on net-
work properties to determine the ranking of nodes in a network.
Most of these methods, such as degree centrality,12 LocalRank,13
ClusterRank,14 and H-index,16 used local information of nodes as
a primary criterion for ranking nodes. The local centralities have
a low computational complexity; however, it might not be accu-
rate enough in ranking nodes. In the scenario of concerning traf-
fic flows, various path-based methods, such as the betweenness,17
eccentricity,18 and proximity,19 were proposed to assess the traffic
load shared by bridge nodes in the transmission of network flows.
These methods are able to rank network nodes with the largest
information load capacity, but their computational complexity is
high. There are also numerous effective centrality-based iterative
optimization techniques, which emphasize that the importance of
a node is influenced by both its neighbors and itself, which is
known as the mutual enhancement effect.20 Typical centralities with
iterative refinements include eigenvector centrality,21 PageRank,22
LeaderRank,23 and so on.24,25
However, the aforementioned methods for studying the impor-
tance of nodes have their limitations prohibiting them from study-
ing the ubiquitous higher-order interactions in large-scale complex
networks.26–32 Higher-order interactions are not only common but
also important impacting the network dynamics and function. For
example, it was found that the emergence of higher-order cliques
can fundamentally alter the propagation dynamics of networks.30,31
By studying the topological signals on different higher-order struc-
tures, it was found33 that signals defined on boundary couplings
between nodes and edges lead to explosive topological synchro-
nization. For two-hyperlink interactions, it was found34 that higher-
order interactions could significantly optimize the synchronization
performances of networks. Moreover, a simplicial contagion model
was proposed30 to study the impact of higher-order interactions on
infectious disease processes and found that the higher-order inter-
actions cause abrupt transition at the epidemic threshold leading to
the emergence of bistable regions in the parameter space.
The complexity of the structure limits the importance of indi-
vidual nodes. To take into account the clique structures that are
represented by higher-order motifs and subgraphs to rank cliques in
a network, we introduce the centralities of higher-order cycle (HOC)
ratio, higher-order degree (HOD), higher-order H-index (HOH),
and higher-order PageRank (HOP). Since only the cliques in the net-
work are considered, our method could reduce the dimensionality of
large-scale networks and also reduce the complexity of the computa-
tional algorithm. Our experiments on both synthetic and real-world
networks show that such higher-order metrics are inherently dif-
ferent from lower-order ones. The higher-order centralities have a
much more distinguishable ranking of cliques than the traditional
metrics centralities, i.e., degree, cycle ratio, PageRank, and H-index.
Moreover, we find a general connection between the utility of dif-
ferent higher-order metrics and the network topology. Last but not
least, our extensive experiments on structural robustness, behav-
ioral synchronization, and dynamical propagation clearly demon-
strate that the proposed metrics discover critical cliques, which the
lower-order metrics cannot identify.
II. LOWER-ORDER CENTRALITY METRICS
Lower-order centrality metrics quantify node properties in net-
works. Here, some wildly used lower-order centrality metrics are
reviewed.
A complex network is represented as G=(V,E), which con-
sists of a set of Nnodes V={1, 2, ...,N}and a set of Medges
E=eij|i6= j, where eij is the link between nodes iand j. Network
Gcan be represented by an N×Nadjacency matrix A, consisting of
elements aij =1 if eij exists, or aij =0 otherwise. Degree kiof node
iis defined as the number of links via node i, and ki=Pj∈Vaij.
The Laplacian matrix of Gis an N×Nmatrix L=D−A, where
D=diag(k1,k2,...,kN)is the degree matrix. Figure 1(a) illustrates
a general network with V={1, 2, ..., 9}and E={e12,e13 ,...,e89}.
Definition 1 (Degree centrality): For node iwith degree
ki=Pj∈Vaij, its normalized degree centrality Diis defined as
Di=ki
N. (1)
For instance, the normalized degree centrality of node 7 shown
in Fig. 1(a) is 1/3.
Definition 2 (H-index, Ref. 35): For node iwith degree ki,
the degrees of its neighbors (j1,j2,...,jki)are kj1,kj2,...,kjki. The
first-order H-index h(1)
iof node iis the maximum number of its
neighbors whose degrees are not less than h(1)
i, defined by
h(1)
i=Hh(0)
j1,h(0)
j2,...,h(0)
jki, (2)
where h(0)
j1,h(0)
j2,...,h(0)
jki=kj1,kj2,...,kjki.
The first-order H-index of node 7 shown in Fig. 1(a) is 3.
FIG. 1. Schematic diagram of the networks with nine nodes. (a) A general net-
work Gwith 9 nodes and 17 edges. (b) A simplicial network G3with 9 0-cliques, 17
1-cliques, 10 2-cliques, and 1 3-clique. Light blue and dark blue indicate 2-cliques
and 3-cliques, respectively.
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Definition 3 (PageRank): The adjacency matrix Ais normal-
ized to obtain the transition probability matrix P, where the element
Pij =aij/Piaij . The basic PageRank at time tis defined by
x(t)=PTx(t−1), (3)
where x∈RNand xiis the PageRank value of the ith node in G.
Considering that there might be a group of interconnected pages
without links to others, the PageRank values remain the same within
the group and will not be changed. To avoid this from happen-
ing, a damping factor sis introduced to obtain a revised PageRank
formula, as follows:
x(t)=sPTx(t−1)+u(1−s)
N, (4)
where u∈RNis a vector with every entry equal to 1, and each ele-
ment of x(0)is 1/N,s∈(0, 1)is a damping factor, which reduces to
the original PageRank for the network when s=1.
Moreover, the LeaderRank could also solve this problem by
adding a ground node connected to all other nodes.23 Since our
study focuses on ranking cliques, we do not describe this algorithm
in detail. As shown in Fig. 1(a), the transition probability matrix
P=diag k1,k2,...,k9−1A.
Definition 4 (Cycle ratio, Ref. 36): The smallest cycle con-
taining node iis the shortest girth starting and ending with node i.
The cycle ratio of node iis defined by
ri=
N
X
j=1
cij/cjj , (5)
where cij is the total number of smallest cycles containing both node
iand node jif i6= j; if i=j,cii is the total number of smallest cycles
containing node i.
As shown in Fig. 1(a), node 1 has three associated cycles, i.e.,
(1, 2, 3), (1, 2, 4), and (1, 3, 4); thus, c11 =3. Node 2 has three associ-
ated cycles, i.e., (1, 2, 3), (1, 2, 4), and (2, 3, 4); thus, c22 =3. Node 3
has four associated cycles, i.e., (1, 2, 3), (1, 3, 4), (2, 3, 4), and (3, 4,
5); thus, c33 =4. Node 4 has six associated cycles, i.e., (1, 2, 4), (1,
3, 4), (2, 3, 4), (3, 4, 5), (4, 5, 6), and (4, 6, 9); thus, c44 =6. More-
over, c12,c13 , and c14 are equal to 2. Thus, the cycle ratio of node 1 is
r1=P9
j=1c1j/cjj =2.5.
III. HIGHER-ORDER CENTRALITY METRICS
Higher-order structures of networks could be characterized by
simplex.32 A simplex is a complete subgraph of a certain order in a
network, also called a clique in graph theory. Specifically, a p-clique
is composed of p+1 completely connected nodes, where all subsets
of the p+1 nodes are contained in the p-clique. A subset with Nsub
nodes is called the Nsub-face of the p-clique and all p-faces are called
the boundaries of the p-clique.37 Therefore, we define the networks
composed of cliques as simplicial networks.
Definition 5 (Simplicial network): A simplicial network
Gn=(C0,C1,...,Cn)is a network of cliques, where nis the high-
est order of its cliques and C0(node), C1(edge),...,Cn(n-clique)
is the set of 0-cliques, 1-cliques, ...,n-cliques in the network,
respectively. Let NCnbe the number of n-cliques, and Sp,q(p<q)be
the NCp×NCqincidence matrix between a p-clique and a q-clique,
where the element sp,q=1 means that the p-clique is a subset of the
q-clique, or 0 otherwise. An=ST
0,nS0,nis the NCn×NCnadjacency
matrix of the n-clique, where the element of anIJ =xif the n-cliques
Iand Jhave xnodes in common, or anIJ =0 otherwise.
Figure 1(b) illustrates a general simplicial network
G3=(C0,C1,C2,C3)with C0={1, 2, ..., 9},C1={e12,e13 ,...,e89},
C2={(3, 4, 5),(4, 5, 6),...,(6, 8, 9)}, and C3={(1, 2, 3, 4)}.
It is first necessary to find all the cliques in the network
according to a certain criterion by Algorithm 1.28 Then, the follow-
ing proposed higher-order centralities can be applied to rank the
cliques.
Definition 6 (Higher-Order Degree—HOD): The HOD of
any n-clique is the number of non-zero elements in the correspond-
ing row of An, except for the diagonal elements. For the n-clique I,
the HOD kIis computed by
kI=
NCn
X
J=1,J6=I
1(anIJ 6=0). (6)
In other words, the kIis the number of n-cliques, which have at least
one common node with n-clique I. For instance, in Fig. 1(b), the
HOD of the 2-clique (6, 8, 9) is 4, which characterizes the number
of 2-clique neighbors, i.e., (6, 7, 8), (6, 7, 5), (6, 5, 4), and (6, 4, 9). In
contrast to the upper and lower-adjacencies,38 HOD considers the
relationship between structures of the same order.
Definition 7 (Higher-Order H-index—HOH): The HOH of
an n-clique Iis the maximum value H, such that there exists at least
Hneighbors of the n-clique Iwith the HODs no less than H. For the
n-clique Iwith kIneighbors J1,J2...JkI, its HOH hIis defined by
h(1)
I=Hh(0)
J1,h(0)
J2,...,h(0)
JkI, (7)
where h(0)
J1,h(0)
J2,...,h(0)
JkI=kJ1,kJ2,...,kJkI.
As shown in Fig. 1(b), the 2-clique (3, 4, 5) has seven higher-
order neighbors, with HODs 8, 8, 6, 6, 6, 5, and 4, respectively. Thus,
the HOH of (3, 4, 5) is 5.
Definition 8 (Higher-Order PageRank—HOP): Similarly to
PageRank, the element of the nth-order transition matrix Pnas
PnIJ =k−1
I∗anIJ if I 6= J,
0if I =J.(8)
The nth-order stationary distribution P∞
nis defined by
P∞
n=lim
t→∞ Pt
n. (9)
Further, the steady-state value of PageRank for each clique is
defined by
xn(t)=s(P∞
n)Txn(0)+u(1−s)
NCn
, (10)
where xn∈RNCn,xIis the PageRank value of the Ith clique in Gn,
and each element of xn(0)is 1/NCn.
Definition 9 (Higher-Order Cycle Ratio—HOC): An-cycle
is a closed higher-order structure, where each n-clique of the
structure could start from one of its n-face, passes through cer-
tain n-cliques, and end to its other n−1n-faces. The size of a
n-cycle equals the number of n-cliques it contains. Note that an
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n-clique Icould belong to more than one n-cycles. The small-
est n-cycle of n-clique Iis the smallest size of n-cycle which
contains n-clique I. The rule for forming the n-cycle could be
interpreted as a boundary operation for n-cliques. The set of bound-
aries of an n-clique Iis expressed with wI. Thus, the boundary
operation of m n-cliques is defined by w1+w2+ · · · + wm
:=(w1∪w2∪,...,∪wm)−Sm
I,J,I6=J(wI∩wJ). The m n-cliques form
an-cycle, when the result of boundary operation is an empty
set, or when the number of elements in the boundary operation
is not greater than m. A set of smallest n-cycles csn could be
found by Algorithm 2. We give an example of the smallest n-cycle
in Appendix A.
Let Ncsn be the size of smallest n-cycles, and Tnbe the
NCn×Ncsn smallest n-cycle incidence matrix of n-cliques and small-
est n-cycles, where the element tn=1 means that the n-clique is a
subset of n-cycle, or 0 otherwise. The NCn×NCnn-cycle number
matrix cnof the network is calculated by TnT0
n, where cnIJ is the
number of smallest cycles containing n-cliques Iand Jwhen I6= J;
if I=J,cnII is the number of smallest cycles containing the n-clique
I. The HOC of the n-clique Iis defined by
rI=
NCn
X
J=1
cnIJ /cnJJ . (11)
As shown in Fig. 1(b), there are only two smallest 2-cycles,
which are (1, 2, 3, 4) and (4, 5, 6, 7, 8, 9). The c2of the network
is T2T0
2, where the T2is
(1,2,3,4) (4,5,6,7,8,9)








(1,2,3) 1 0
(1,2,4) 1 0
.
.
..
.
..
.
.
(4,6,9) 0 1
.
Therefore, the HOC of each 2-clique can be calculated by
Eq. (11). For instance, r(1,2,3)=1/1+1/1+1/1+1/1=4.
Through the definition of higher-order centralities, we find that
the HOD and HOH of n-clique are calculated in a similar way by the
incidence matrix between the 0-cliques and the n-cliques, and the
calculation of the HOH requires the calculation of the HOD first. In
addition, the HOC and HOP of n-clique are both calculated based
TABLE I. Time complexity analysis of various metrics to search the ranking of
components in a network. Here, D, C, H, and P represent degree, cycle ratio, H-index,
and PageRank, respectively.
Metrics HOD HOC HOH
Complexity O(N2
Cn)O(NCn+logNCn)O(N2
CnlogNCn)
(Cn+1)
Metrics HOP D C
Complexity O(t(ε)N2
Cn)O(NE)O((n+e)(c+1))
Metrics H P
Complexity O(N2logN)O(t(ε)N2)
TABLE II. Notations.
Notations Definition
CnSet of n-cliques
Tn
Smallest n-cycle incidence matrix of n-cliques
and smallest n-cycles
AnAdjacency matrix of the n-clique
cnCycle number matrix of the n-clique
N(NCn)Number of nodes (n-cliques)
ki(kI) Degree of node i(clique I)
h(n)
i(h(n)
I)The n-order H-index of node i(clique I)
x(t) (xn(t)) PageRank of nodes (n-cliques)
ri(rI) Cycle ratio of node i(clique I)
g(i) (g(I))
Number of nodes (cliques) in the maximum
connected subgraph after the corresponding i
(I) is removed
µ(n)
1(L−C)
Minimum non-zero eigenvalue of the n-order
principal submatrix after deleting C pinned
n-cliques
µCuring rate
β(β1) Infection rate among nodes (cliques)
on the incidence matrix of (n−1)-cliques and n-cliques. HOP is
a more global metric compared to HOC. Hence, in the ranking of
importance under general networks, the results of HOD are similar
to HOH and the results of HOC are similar to HOP.
Table I shows the computation complexity analysis of various
metrics to search the ranking of components in a network, where
t(ε) is the number of iterations, and it is related to the threshold
value εfor convergence. Table II summarizes the main notation and
variables used.
ALGORITHM 1. Nearest-neighbor method to find all cliques in a network.
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ALGORITHM 2. Find the smallest n-cycle of the network.
IV. PERFORMANCE MEASURES
The performances of the above-defined higher-order metrics
in ranking cliques are tested, subject to three representative dynam-
ical processes, i.e., percolation,39 synchronization,34,40 and epidemic
spreading.6,41
A. Robustness evaluation of networks
Analyzing the robustness is essential to rank the influential
components of the considered networks. Given a network, one node
in a ranked clique is randomly removed at each time step. The size of
the largest component of the remaining network is then calculated
to assess the significance of the clique in maintaining the network
connectivity. The network robustness is defined as
R=1
NCn
NCn
X
I=1
g(I), (12)
where g(I)is the number of cliques in the maximum connected
subgraph after the corresponding n-clique Iis removed.
B. Synchronization evaluation of networks
The pinning control aims to pin a small number of network
nodes to stabilize a complex dynamic network onto some homoge-
nous stationary states.42 To evaluate the accuracy of the clique rank-
ing, the effect of pinning the corresponding structures on network
FIG. 2. A form of virus transmission where a 2-clique is the basic unit.
synchronization is measured. Consider a Kuramoto-type model
with Ninteracting phase oscillators described by34
˙
θi=f(θi)+K1
N
X
l1=1
a(1)
il1ψ(1)θi,θl1
+K2
N
X
l1=1
N
X
l2=1
a(2)
il1l2ψ(2)θi,θl1,θl2+ · · ·
+KD
N
X
l1=1
N
X
l2=1
...
N
X
lD=1
a(D)
il1...lDψ(D)θi,θl1,...,θlD, (13)
where θiis the state variable of the ith oscillator, the function f
describes the self-dynamics of the ith oscillator, Kdare the cou-
pling constant of the dth-order interaction, where d=1, 2, ...,D;
D<N,ψ(d)is the internal coupling function of the dth-order inter-
action, a(d)
il1...ldare the dth-order adjacency tensors containing oscil-
lator i, with the element of a(d)
il1...ld=1 if the dth interaction between
oscillators i,l1,l2, . . . , and ldexists or a(d)
il1...ld=0 otherwise.
The master stability equation depending on the generalized
Laplacians40,43 is given by
L(d)
ij =dk(d)
iδij −k(d)
ij , (14)
TABLE III. Network topological characteristics of six real-world networks. Here, hkiis
the mean degree and Cis the clustering coefficient.
Network Node Edge Triangle Tetrahedron Chki
C. elegans 297 2148 3 241 2 010 0.29 14.46
NS 379 914 918 630 0.73 4.82
US-grid 4941 6594 651 90 0.08 2.67
US-air 332 2126 12 184 60 173 0.62 12.81
Email 1133 5451 5 343 3 419 0.22 9.62
Jazz 198 2742 17 899 78 442 0.61 27.69
Chaos 33, 073139 (2023); doi: 10.1063/5.0147721 33, 073139-5
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FIG. 3. Degree distribution in (a) Email and (c) C. elegans. HOD distribution in (b) Email and (d) C. elegans.
where δij is the Kronecker function, with δij =1 if i=j, or δij =0
otherwise, in which k(d)
iis the generalized dth-order degree of node
i, given by
k(d)
i=1
d!
N
X
l1=1
...
N
X
ld=1
a(d)
il1...ld, (15)
and k(d)
ij is the generalized dth-order degree between the nodes iand
j, given by
k(d)
ij =1
(d−1)!
N
X
l2=1
...
N
X
ld=1
a(d)
ijl2...ld. (16)
The synchronization capability of the network is positively
correlated with the eigen-ratio (the minimum non-zero eigenvalue
divided by the maximum eigenvalue) of the Laplacian matrix L
described in the method.43 The same criterion is extended to the
higher-order setting.34 As a result, cliques are pinned one by one
according to the importance ranking to modify the synchroniz-
ability of the network, which is measured by the reciprocal of
the minimum non-zero eigenvalue of the principal submatrix,36,44,45
namely, µ(n)
1(L−C). Here, Cis the number of pinned cliques, L−Cis
obtained by removing the rows and columns corresponding to the C
pinned cliques from the generalized Laplacian matrix, and µ(n)
1(L−C)
is the smallest non-zero eigenvalue of L−C, where ndenotes the
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FIG. 4. Visualization of ranking the cliques produced by HOC, HOP, HOD, and HOH in the US-grid. The color of the node indicates its importance, which decreases in order
of the color bar from top to bottom.
order of the pinned cliques. A smaller value of µ(n)
1(L−C)indicates
a lower synchronizability of the network. Analogous to the clique
percolation, the pinning efficiency of the n-clique is defined as
PEn=1
NCn
NCn
X
C=1
1
µ(n)
1(L−C), (17)
where 1/NCnensures that the PEnvalues of various networks with
different sizes can be compared.
C. Epidemic dynamic of networks
Now, cliques are used as the fundamental units of a network to
study virus transmission control over large-scale complex networks
with higher-order structures.
We here consider the classical susceptible-infected-susceptible
(SIS) model,41 where a susceptible node can be infected by an
infected neighbor with probability β∈[0, 1], and each infected
node cures to be an susceptible node with probability µ∈[0, 1].
In addition to the direct transmission with probability βof the SIS
component, an indirect transmission with probability β1brought
about by higher-order structures of the networks is considered. It
could be understood as the interaction between two groups of nodes
in a real-world network.
In Fig. 2, red (solid) and white (hollow) nodes represent
infected and susceptible individuals, respectively. The dashed line
indicates that the existence of an overlap between two cliques will
affect the propagation even if two nodes belonging to two different
cliques share no common edges. For instance, despite that there is
no substantial connection between individuals aand d, the cliques
formed by their overlap will lead to the situation that individual dis
infected by individual awith probability β1.
V. EXPERIMENTAL RESULTS
In this section, experimental results are presented and the effect
of each higher-order metric is compared to the existing ones, includ-
ing degree, cycle ratio, PageRank, and H-index, to demonstrate the
performances of the higher-order metrics in ranking cliques on both
synthetic and real-world networks.
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FIG. 5. Performances of eight metrics on percolation in real-world networks: (a) Jazz, (b) C. elegans, (c) Email, (d) NS, (e) US-air, and (f) US-grid, respectively. The yaxis
represents the measurement of network robustness after clique (node) removal and the xaxis denotes the ratio of the removed cliques (nodes).
A. Data settings
In the experiments here, six real-world networks46 are sim-
ulated. Each node in a network represents an entity, each edge
represents the first-order relationship between two nodes, each tri-
angle represents the second-order relationship among three nodes,
and so on. Without loss of generality, all such relationships are
viewed as cliques in the network. The parameters of these real-world
networks are shown in Table III.
B. Structural analysis
In addition to the basic components (namely, nodes, edges),
higher-order ones (especially, triangles, tetrahedrons) are now taken
into consideration. Although our method could theoretically rank
the importance of n-cliques in the network, in this paper, we focus
on the importance of 2-cliques, which are the most fundamental
higher-order units in the network. By analyzing the distributions of
degrees (of nodes) and HODs (of 2-cliques) of the C. elegans and
Email datasets shown in Fig. 3, we discover that the information
in HOD distribution obscures the heterogeneity among nodes. This
enables us to rank cliques, thereby helping to identify critical cliques
especially when the importance of a clique is not prominent.
Moreover, using the US-grid dataset, the top 15% of the cliques
ranked by each higher-order metric are visualized in Fig. 4. We find
that the critical cliques ranked by the HOP and the HOC are scat-
tered over the network, yielding the possibility of selecting fewer
nodes (cliques) to cover a large area. In this way, critical cliques can
be clearly ranked.
C. Performance comparison
To quantify the ability of metrics for ranking cliques, consider
the importance of cliques as their ability to maintain the network
connectivity, facilitate network synchronization, and control virus
propagation.
1. Robustness analysis of real-world networks
We first study the application of cliques to network percolation
dynamics. The cliques are ranked for each metric in the descending
order, and the ones in the top places are deleted first. Obviously, a
smaller Rmeans a quicker collapse and thus a better performance.
Recall that the network robustness under the lower-order
metrics is defined as39
R=1
N
N
X
n=1
g(n), (18)
where g(n)represents the size of the maximal connected subgraph
after deleting the corresponding node n.
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Figure 5 shows that, as the nodes are removed from the network
one by one, the robustness of these real-world networks continues to
decline. In particular, both HOC and HOP curves lie at the bottom
of the others in all the six real-world networks; that is, when the same
number of nodes are removed, HOC and HOP metrics show signifi-
cant impacts on the robustness of the networks. In most cases, when
the proportion of removed cliques increases under the HOP, HOC,
HOD, and HOH, the collapse rate of the network slows down. How-
ever, under the metrics, including PageRank, cycle ratio, degree, and
h-index, the collapse rate of the network accelerates with the propor-
tion of removed nodes. This clearly indicates that, in most real-world
networks, the percolation rate for nodes gradually accelerates while
the percolation rate for cliques decelerates. Moreover, except for the
NS (a network of coauthorships) and US-grid, the higher-order met-
rics perform much better than lower-order ones, in the sense that a
few cycles formed in the NS and US-grid result in the collapse speed
of the nodes could keep up with the cliques. Hence, we need fur-
ther explore the universal characteristics of networks in which the
proposed higher-order metrics are practicable (see Sec. V C 4).
2. Synchronization analysis of real-world networks
A commonly used synchronizability measure, the eigen-ratio,
is used to verify the effectiveness of higher-order metrics for ranking
cliques with comparison to that using lower-order metrics of degree,
H-index, PageRank, and cycle ratio. The network synchronizability
under the lower-order metrics is expressed as36
PE =1
Qmax
Qmax
X
Q=1
1
µ(1)
1(L−Q), (19)
where µ(1)
1(L−Q)is the minimum non-zero eigenvalue of the Lapla-
cian matrix Lafter deleting the rows and columns corresponding to
the Qpinned nodes of the network.
Figure 6 shows how the network synchronizability decreases
with the increasing proportion pof the pinned nodes (cliques). Since
critical structures have a significant impact on the synchronization
of the network, we pin the top 30% cliques in the network. We find
that when pis larger than 30%, it does not affect the results and
conclusions. In the figure, a faster decay corresponds to a better
synchronization performance. Table IV lists the pinning efficiency
of eight metrics, in which the bold black color is utilized to high-
light the best performance of higher-order and lower-order metrics,
respectively.
From the experiment results, one can observe that the effi-
ciency of HOP and HOC in ranking cliques is better than HOD,
and HOD is better than HOH, in the networks of C. elegans, Email,
and Jazz, whereas, unlike higher-order metrics, the ranking of the
FIG. 6. Performances of the eight metrics on synchronization in real-world networks, (a) C. elegans, (b) Email, (c) Jazz, (d) NS, (e) US-air, and (f) US-grid, respectively. The y
axis represents the network synchronization after pinning a fraction of cliques (nodes) and the xaxis denotes the ratio of pinned cliques (nodes).
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TABLE IV. Pinning efficiency (PE2and PE) of the eight metrics on the simulated
real-world networks. Here, D, C, H, and P represent degree, cycle ratio, H-index, and
PageRank, respectively.
Network HOD HOC HOH HOP D C H P
C. elegans 0.54 0.53 0.54 0.52 0.67 0.74 0.80 0.66
NS 60.53 23.29 70.09 13.27 15.68 12.87 26.25 30.19
US-grid 16.28 26.94 18.72 9.45 0.41 0.98 1.02 1.46
US-air 2.66 1.86 2.66 2.35 7.65 3.65 8.51 7.14
Email 3.14 2.22 3.15 1.76 2.89 3.06 2.86 2.79
Jazz 1.29 1.20 1.81 1.17 1.79 1.75 1.82 1.79
effects of the lower-order metrics is inconsistent for these real-world
networks. For instance, PageRank has the best performance on C.
elegans and the worst performance on NS. Furthermore, for the
results on networks of Email, NS, and Jazz shown in Fig. 6, the
values of µ1L−Qin different lower-order metrics are very close
to each other under the same ratio of pinned components. Thus,
the effect of lower-order metrics is not clearly distinguished, while
the differences of higher-order metrics could be well distinguished
for all simulated real-world networks. Last, the US-grid network is
sparse with very few cliques, resulting in the worst performance of
the HOC.
3. Epidemic transmission analysis of real-world
networks
To measure the efficiency of ranking cliques by each metrics,
we employ the SIS model to simulate the influence of ranking cliques
on viral transmission. We sequentially select the ranked cliques from
the network for immunization. Parameter pof the network is given
by the proportion of infected nodes when the number of infected
nodes does not change over time. High propagation thresholds and
low pvalues correspond to good performances.
Without loss of generality, the recovery rate µis set to 1 and
parameter βis re-scaled to λ=βhki
µ, where hkiis the average degree
of the network. Assuming that the path of indirect propagation is
twice as long as that of direct propagation, we set β1=0.5β. Ini-
tially, we select the top 10% cliques for immunization and randomly
infect 10% of the susceptible nodes from the network. We perform
all the simulations for 100 network realizations and then consider
their averages.
For comparison, we use the numerical sum of lower-order met-
rics to rank cliques. Figure 7 shows that HOC and HOP have better
performances, which is manifested with the curve represented by
FIG. 7. Impact of each metric on the spread of infectious diseases in real-world networks: (a) C. elegans, (b) US-air, (c) NS, (d) Jazz, (e) US-grid, and (f) Email, respectively.
The yaxis indicates the proportion of infected nodes in steady state after node immunization and the xaxis denotes the scaling of the infection probability.
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TABLE V. Network topological characteristics of six synthetic networks. Here, hkiis
the mean degree and Cis the clustering coefficient.
Network Nodes Edge Triangle Tetrahedron Chki
BA(500-2-0) 500 1975 1 206 85 0.30 7.9
BA(500-2-1) 500 1925 1 491 168 0.41 7.7
BA(500-4-0) 500 3750 3 647 718 0.18 15
BA(500-4-1) 500 3750 5 012 1 905 0.30 15
ER(500-0.05) 500 2425 2 290 842 0.53 9.7
ER(500-0.15) 500 6700 14 173 10 996 0.37 26.8
HOP that has a high propagation threshold, and with the curve
represented by HOC that has a low steady-state infection ratio, espe-
cially in C. elegans, US-air, NS, and US-grid datasets. However,
HOD and HOH cannot achieve better results than lower-order met-
rics. The reason is that all edge relations connected to any triangle
are totally considered in low-order metrics, thereby yielding better
performances than HOD and HOH, which ignore the spread of the
virus on lower-order relationships.
4. Performances in synthetic networks
Considering that the structures of these real-world networks
are idiosyncratic and cannot represent general characteristics of net-
works, we choose synthetic BA and ER simplicial networks with
different parameters to explore the relationship between the effect
of the proposed metrics and the network topologies. Table V and
Appendix B show the parameters and generation methods of these
synthetic networks, respectively.
Similarly, we first study the degree and HOD distributions of
the synthetic networks. Figure 8 shows that there are very few nodes
of large degree, but cliques with high HODs occupy a large pro-
portion in the BA and ER simplicial networks. In particular, the
preferential rule in the BA simplicial networks leads to a much larger
FIG. 8. Degree distribution in synthetic networks: (a) BA(500, 2, 0), (c) BA(500,2, 1), (e) BA(500, 4, 0), (g) BA(500, 4, 1), (i) ER(500, 0.05), and (k) ER(500, 0.15),
respectively. HOD distribution in synthetic networks: (b) BA(500, 2, 0), (d) BA(500, 2, 1), (f) BA(500, 4, 0), (h) BA(500, 4, 1), (j) ER(500, 0.05), and (l) ER(500, 0.15),
respectively.
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range of HOD values as compared to the networks generated by
non-preferential rules. Therefore, in BA and ER simplicial networks,
identifying vital nodes is relatively easy, while it is difficult to do
so for critical cliques. Moreover, the degree distribution of nodes
and the HOD distribution of triangles are transformed into skewed
normal distributions in ER simplicial networks.
Next, we compare the influences of the cliques ranking deter-
mined by the proposed metrics on the robustness, synchronization,
and propagation dynamics in these synthetic networks. Experimen-
tal parameter settings are consistent with the previous ones. The
main results show that in BA simplicial networks, all conclusions
are consistent with the previous ones, that is, the performances of
our proposed metrics satisfy the ranking of HOP >HOC >HOD
>HOH, where “>” means “better than.” However, in ER simplicial
networks, HOC loses its advantage even if the networks have numer-
ous cycles. The reason is that the small circumferences of the cycles
formed in ER simplicial networks lead to many equally important
cycles. Detailed results about the BA and ER simplicial networks are
presented in Appendixes C–E.
VI. CONCLUSION AND DISCUSSION
Traditional node importance ranking only take into account
edge-based relationships, ignoring higher-order cliques represented
by motifs and subgraphs between some key nodes in a complex net-
work. In this paper, we propose higher-order cycle (HOC) ratio,
higher-order degree (HOD), higher-order H-index (HOH), and
higher-order PageRank (HOP) to rank cliques in complex networks.
Experiments on both synthetic and real-world networks show that
the HOC and HOP outperform the other metrics in ranking crit-
ical cliques that are important in maintaining network connectiv-
ity, facilitating network synchronization, and controlling epidemic
spreading.
Since only the clique structures in networks are considered, our
proposed metrics could reduce the dimensionality of large-scale net-
works and optimize the complexity of the associated algorithms.
However, this is also the reason why HOH and HOD are not as
effective as lower-order metrics in some other considerations. The
problems encountered in node importance ranking have been thor-
oughly discussed before.47 Here, we focus on two concerned issues:
(i) it is difficult to find an index that best quantifies the importance of
nodes in all possible measures, and (ii) most known methods were
essentially designed to rank individual vital nodes instead of a set
of vital nodes, while the latter is more relevant to many real-world
applications such as group infection-testing and immunization. In
the process of ranking cliques, we find that, in terms of network
robustness, synchronization, and propagation, almost all simulation
results satisfy the ranking of HOP >HOC >HOD >HOH, where
“>” means “better than.” By selecting one representative node from
each group, we could get a set of vital nodes of certain importance,
which could be beneficial to such applications as product marketing,
vaccine injection, etc.
It is worth noting that the performances of our proposed met-
rics are related to the topology of the underlying network. In this
paper, we generate different types of synthetic simplicial networks to
demonstrate the effects of the cliques determined by various metrics
on network robustness, synchronicity, and propagation dynamics.
We find that in BA simplicial networks, the proposed higher-order
centralities are applicable to large-scale networks with rich cycle
structures, whereas a large proportion of equally critical cliques
in ER simplicial networks lead to poor performance of the HOC
(see Appendixes C–E).
An obvious inefficiency of higher-order metrics is that they
may not be directly applicable to tree or tree-like networks. Even
for normal networks, a fraction of nodes may not be associated with
any clique. Thus, the influences of these nodes may be different but
they all are assigned the same importance value of zero. Therefore,
a straightforward and effective solution is to combine higher-order
metrics with lower-order metrics in an appropriate manner. For
example, a mixed index could be r∗
(1,2,3)=r(1,2,3)+ξ(k1+k2+k3),
with ξbeing a tunable parameter, so that all cliques that are not in
higher-order structures could be ranked by their degrees. Combin-
ing different higher- and lower-order metrics has good potential to
get more accurate results, which is a good topic for future research.
This paper focuses on the importance of ranking cliques. Con-
sidering that there are many cliques of equal importance in the above
ranking, one might perform a secondary importance ranking on
such cliques. If the importance of a pair of cliques is the same, lower-
order metrics (one-to-one correspondence between higher-order
metrics and lower-order metrics, for example, HOC corresponds
to cycle ratio) could be introduced to reorder their importance.
Indeed, we have found that good performances could be achieved
after appropriate metrics fusion, especially for HOC and HOH
(see Appendix F).
All in all, we believe our present work can serve as a fundamen-
tal framework for studying higher-order networks, and our findings
can shed some new lights on the control, prediction, and protection
of various large-scale complex networks.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science
Foundation of China (NNSFC) (Grant Nos. 62173095 and
71731004), the Natural Science Foundation of Shanghai (Grant No.
21ZR1404700), the Shanghai Municipal Science and Technology
Major Project (No. 2021SHZDZX0100), and partly supported by the
Fundamental Research Funds for the Central Universities.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Yang Zhao: Conceptualization (equal); Data curation (equal); For-
mal analysis (equal); Investigation (equal); Methodology (equal);
Software (equal); Writing – original draft (equal); Writing – review
& editing (equal). Cong Li: Conceptualization (equal); Formal anal-
ysis (equal); Funding acquisition (equal); Methodology (equal);
Project administration (equal); Supervision (equal); Writing – orig-
inal draft (equal); Writing – review & editing (equal). Dinghua
Shi: Conceptualization (equal); Formal analysis (equal); Methodol-
ogy (equal); Validation (equal); Writing – review & editing (equal).
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Guanrong Chen: Formal analysis (equal); Methodology (equal);
Validation (equal); Writing – review & editing (equal). Xiang Li:
Formal analysis (equal); Funding acquisition (equal); Resources
(equal); Supervision (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are openly
available in https://doi.org/10.1609/aaai.v29i1.9277, Ref. 46.
APPENDIX A: EXAMPLES OF SMALLEST
n
-CYCLE
As shown in Fig. 9, the 2-faces of the 2-clique (2, 3, 4)are (2, 3),
(2, 4), and (3, 4)in simplicial network G3. The 2-clique (2, 3, 4)starts
from its 2-face (2, 4), passes through 2-cliques (1, 2, 4),(1, 2, 3), and
ends to its 2-face (2, 3)to form 2-cycle (1, 2, 3, 4). The 2-clique
(2, 3, 4)is also the basic structure that forms 2-cycles (2, 3, 4, 5, 6, 9)
and (2, 3, 4, 5, 6, 7, 8, 9). According to Definition 9, the smallest
2-cycle based on 2-clique (2,3,4) is (1,2,3,4).
Next, we utilize the boundary operation to verify that the set
{1, 2, 3, 4}is a 2-cycle of 2-clique (2, 3, 4). The set {1, 2, 3, 4}has
four associated 2-cliques, i.e., (1, 2, 3),(1, 2, 4),(1, 3, 4), and (2, 3, 4).
From Definition 9, the set of boundaries of 2-clique (1, 2, 3)is
w(1,2,3)= {(1, 2),(1, 3),(2, 3)}, thus, the result of the boundary oper-
ation is w(1,2,3)+w(1,2,4)+w(1,3,4)+w(2,3,4):= {(1, 2),(1, 3),(2, 3)}
∪ {(1, 2),(1, 4),(2, 4)} ∪ {(1, 3),(1, 4),(3, 4)} ∪ {(2, 3),(2, 4),(3, 4)}
− {(1, 2)∪(1, 3)∪(2, 3)∪(2, 4)∪(3, 4)∪(1, 4)} = ∅. Hence, the
set {1, 2, 3, 4}is the smallest 2-cycle of 2-clique (2, 3, 4).
APPENDIX B: MODEL GENERATING RULES FOR
SIMPLICIAL NETWORKS
The generation rule of BA simplicial networks,48 BA(n,m,w),
is shown in Fig. 10, as follows:
1. Start with a triangle or multiple connected triangles.
2. At each time step, add one node to the existing network and
connect it to mselected edges from the network so as to form
FIG. 9. A simplicial network G3with 9 0-cliques, 17 1-cliques, 11 2-cliques, and
1 3-clique. Light blue and dark blue indicate 2-cliques and 3-cliques, respectively.
new triangles. There are two ways to generate new triangles: one
is a non-preferential connection mechanism (w=0), where the
node randomly selects existing edges from the network to form
new triangles; the other is a preferential connection mechanism
(w=1), where the probability of electing an edge for the node
to form a triangle is proportional to its HOD.
3. Repeat step 2 until the number of added nodes reaches a pre-
desired number n.
Similarly, the generation rule of ER simplicial networks,
ER(n,η), is as follows:
1. Start with Nnon-adjacent triangles consisting of nnodes.
2. Select two disjoint cliques and add a triangle to connect them at
random with probability η.
3. Repeat step 2 until all pairs of cliques in the network are selected
once and once only.
APPENDIX C: ROBUSTNESS ANALYSIS OF SYNTHETIC
NETWORKS
In the BA simplicial networks shown in Fig. 11, one can see that
among the higher-order metrics, HOP performs best, HOC is close
to the best, and HOH is the worst. It is consistent with the exper-
imental results on the simulated real-world networks. Moreover,
when the preferential connection mechanism is adopted, as shown
in Figs. 11(b) and 11(d), the collapse speed of the cliques in the ini-
tial phase is greatly increased while the collapse speed of the nodes
remains constant. However, different phenomena emerge from the
ER simplicial networks shown in Figs. 11(e) and 11(f), where HOC
loses its advantage in ranking cliques even if the number of cycles in
a network is huge. Moreover, one can see that in the ER simplicial
networks, most curves representing the lower-order metrics drop to
zero before the others, indicating that the collapse speed of nodes
could catch up with the collapse speed of cliques.
FIG. 10. Process of generating BA simplicial network.
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FIG. 11. Performances of the eight metrics on percolation for synthetic networks: (a) BA(500, 2, 0), (b) BA(500, 2, 1), (c) BA(500, 4, 0), (d) BA(500, 4, 1), (e) ER(500, 0.05),
and (f) ER(500, 0.15), respectively. The yaxis represents the measurement of network robustness after cliques (node) removal and the xaxis denotes the ratio of removed
cliques (nodes).
FIG. 12. Performances of the eight metrics on synchronization for synthetic networks: (a) BA(500, 2, 0), (b) BA(500, 2, 1), (c) BA(500, 4, 0), (d) BA(500,4, 1),
(e) ER(500, 0.05), and (f) ER(500, 0.15), respectively. The yaxis represents the network synchronization after pinning a fraction of cliques (nodes) and the xaxis denotes
the ratio of pinned cliques (nodes).
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TABLE VI. Pinning efficiency (PE2and PE) of the eight metrics on synthetic networks. Here, D, C, H, and P represent degree, cycle ratio, H-index, and PageRank, respectively.
Network HOD HOC HOH HOP D C H P
BA(500-2-0) 0.55 0.48 0.60 0.47 0.85 0.83 0.84 0.87
BA(500-2-1) 0.34 0.32 0.35 0.31 0.58 0.30 0.53 0.59
BA(500-4-0) 0.19 0.18 0.19 0.17 0.58 0.60 0.57 0.59
BA(500-4-1) 0.14 0.13 0.14 0.13 0.30 0.29 0.28 0.30
ER(500-0.05) 1.26 1.41 1.34 1.20 1.72 1.72 2.05 1.57
ER(500-0.15) 1.05 1.23 1.06 1.03 0.16 0.17 0.28 0.17
APPENDIX D: SYNCHRONIZATION ANALYSIS OF
SYNTHETIC NETWORKS
Figure 12 shows the effect of each proposed metric on network
synchronization. Table VI lists the pinning efficiency of the metrics
for synthetic networks, in which the bold black color is utilized
to highlight the best performances of higher-order and lower-
order metrics, respectively. Overall, among the higher-order met-
rics, HOC and HOP outperform HOC and HOD in ranking cliques
that are efficient in pinning control. However, unlike higher-order
metrics, the ranking of the effects of the lower-order metrics is
inconsistent for these networks.
APPENDIX E: EPIDEMIC DYNAMIC ANALYSIS OF
SYNTHETIC NETWORKS
Figure 13 shows the impact of the proposed metrics on the
spread of infectious diseases for synthetic simplicial networks. The
curve of HOP is at the bottom of all the curves; that is, when
immunizing the same proportion of cliques, HOP has a significant
effect on the transmission threshold as well as the steady-state infec-
tion proportion. In addition, the performance of HOC gradually
improves in BA simplicial networks when the number of cliques in
the network gradually increases, while the opposite is true in classical
ER networks.
FIG. 13. Impact of each proposed metric on the spread of infectious diseases on synthetic networks: (a) BA(500, 2, 0), (b) BA(500,2, 1), (c) BA(500, 4, 0), (d) BA(500, 4, 1),
(e) ER(500, 0.05), and (f) ER(500, 0.15), respectively. The yaxis indicates the proportion of infected nodes in steady state after node immunization and the xaxis denotes
the scaling of infection probability.
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FIG. 14. Effects of eight fusion metrics on the robustness of synthetic networks: (a) BA(500, 2, 0), (b) BA(500, 2, 1), (c) BA(500,4, 0), (d) BA(500, 4, 1), (e) ER(500, 0.05),
and (f) ER(500, 0.15), respectively. The yaxis represents the measurement of network robustness after cliques (nodes) removal and the xaxis denotes the ratio of removed
cliques (nodes).
In summary, HOP shows advantages in most cases and HOC
significantly affects networks with prominent heterogeneity. In
addition, the importance ranking of cliques described by HOC has
little overlap with the cliques described by several other metrics. It
suggests that HOC can provide novel insights beyond the known
metrics.
APPENDIX F: FUSING HIGHER AND LOWER METRICS
Considering that there are many cliques of equal importance
in complex networks, we perform a secondary importance ranking
on the cliques. If cliques are of equal importance, low-order met-
rics are introduced to reorder them. We find that better perfor-
mance could be achieved after such index fusion, especially HOC
and HOH. Figure 14 shows the effects of fused metrics on the
robustness of synthetic networks, and a comparison of the met-
rics is shown in Table VII, where the bolded font color indi-
cates the metrics with the fastest decline in robustness. In gen-
eral, our results with a secondary ranking cannot achieve a sig-
nificant improvement, implying that our results after the primary
ranking already accurately represent the importance ranking of
cliques.
TABLE VII. Robustness of eight metrics for synthetic networks. Here, MD, MC, MH, and MP represent the fusion of higher-order metrics with the corresponding lower-order
metrics, respectively.
Network HOD HOC HOH HOP MD MC MH MP
BA(500-2-0) 50.3 44.957.1 24.4 50.2 32.855.5 24.4
BA(500-2-1) 16.4 11.124.9 5.1 16.4 9.623.4 5.1
BA(500-4-0) 46.6 40.154.9 26.8 46.5 38.654.5 26.8
BA(500-4-1) 23.9 12.4 26.67.4 23.9 12.38 25.97.2
ER(500-0.05) 91.2 123.199.3 53.1 90.9 119.398.2 53.1
ER(500-0.15) 66.9 139.2 104.939.9 66.8 139.2 104.039.9
Chaos 33, 073139 (2023); doi: 10.1063/5.0147721 33, 073139-16
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Chaos ARTICLE pubs.aip.org/aip/cha
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Chaos 33, 073139 (2023); doi: 10.1063/5.0147721 33, 073139-17
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