I thank my co-authors for allowing me to use material from our joint papers. Some of results in this thesis have already been previously published in the following papers:
[L1] Yang Liu, Xi Wang, and Jürgen Kurths. “Optimization of targeted node set in complex networks under percolation and selection”. In: Physical Review E 98.1 (2018), p.
012313.
[L2] Yang Liu, Xi Wang, and Jürgen Kurths. “Framework of evolutionary algorithm for investigation of influential nodes in complex networks”. In: IEEE Transactions on Evolutionary Computation 23.6 (2019), pp. 1049–1063.
[L3] Yang Liu and Jürgen Kurths. “Effects of network robustness on explosive synchroniza-tion”. In: Physical Review E 100.1 (2019), p. 012312.
Additional research work not covered by the material in this thesis can be found in:
• Yong Zhao, Xiaoyan Sun, Yang Liu and Jürgen Kurths, “Phase synchronization dynamics of coupled neurons with coupling phase in the electromagnetic field”. In:
Nonlinear Dynamics 93.3 (2018), pp. 1315–1324.
• Xiujing Han, Yang Liu, Qinsheng Bi and Jürgen Kurths, “Frequency-truncation fast-slow analysis for parametrically and externally excited systems with two slow incommensurate excitation frequencies”. In: Communications in Nonlinear Science and Numerical Simulation 72 (2019), pp. 16–25.
• Jingfang Fan, Jun Meng, Yang Liu, Abbas Ali Saberi, Jürgen Kurths and Jan Nagler,
“Universal gap scaling in percolation”. In: Nature Physics 16.4 (2020), pp. 455–461.
Complex Network Theory and Network 2
Percolation
In this chapter we present underlying concepts regarding complex networks, attributes and measures of nodes and networks, percolation theory, and explosive percolation.
2.1 Complex Network Theory
2.1.1 Networks
Anetwork(graph1)G(N,M)consists of a number of nodes (vertices) tied by a group of edges (links) whereN and Mare accordingly the node set and the edge set. Letn=|N | andm=|M|be the corresponding number of nodes and edges, respectively. Then we also refer to a network asG(n,m)which indicates a networkGconstructed withnnodes andm edges (see Fig. 2.1 for an example). In general, the node (see Table A.1) could be an agent in a multi-agent system, an interaction in a road network, or an airport in the global airline network. Such node could also have some properties, like the size, security, and location of an airport. The edge could be a road connecting two interactions, an airline between two airports, or friendships among individuals. In practice, different scenarios could share a similar fundamental structure such as following similar macroscopic characteristics, even though their node and edge have different meanings. Besides, the structure of a network might also be far complicated, e.g., node and edge could have weights, or connections between two nodes could be multiple (see Fig. 2.1). Among them, the most fundamental and important structure in both network science and graph theory is the simple network [2, 41]. A simple network is an undirected and unweighted network without self-loops (Fig.
2.1). In this thesis, we will mainly consider simple networks and refer to a simple network as a network if there is no special explanation.
1The difference between network and graph is from the difference of network science and graph theory. They are actually interchangeable. Thus, we view network and graph as the same thing throughout this thesis, even though there are some subtle differences [2].
Simple Network
Node Edge
Node {individual, protein, airport, …}
Edge {i
{Friendship, interaction, airline, …}
Weighted nodes and edges
Multi-, self-, undirected, and directed edges 1
2
3 4
5 6
Figure 2.1: Example of nodes, edges, and networks. A simple network is an unweighted and undirected network without self-loops.
2.1.2 Adjacency matrix
The core to represent a network is to find an appropriate way which can fully capture those interactions among nodes. For example, we can firstly label each node from the simple network in Fig. 2.1, and then employ the following two arrays to represent that network,
adj = [2, 3, 4, 5, 1, 3, 6, 1, 2, 4, 1, 3, 1, 6, 2, 5], idx= [0, 4, 7, 10, 12, 14, 16].
In this manner, one can obtain the nodes connecting to node i through2 adj[idx[i] +1 : idx[i+1]], or get an edgeeij where3 j=adj[idx[i] +1]. This representation is actually very useful in storing the network or for some calculation running on the network.
However, a better such way for mathematical calculations is theadjacency matrixthat could efficiently represent a network by the aid of a matrix. For a simple network like the one in Fig. 2.1, the element of the adjacency matrix follows
Aij =
{︄1, if nodeiandjare connected to each other,
0, otherwise. (2.1)
Thus, the corresponding adjacency matrixAis written as
A=
0 1 1 1 1 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0
.
2Here we follow the rules in computer programme but with a slight difference. That is, for instance, considering node 3 in Fig. 2.1,idx[3] =7 (not 10),idx[3] +1 :idx[3+1], i.e., 8 : 10, denotes[8, 9, 10], and thus we getadj[idx[i] +1 :idx[i+1]] = [1, 2, 4]. Note that we will use these symbols and rules through the whole thesis.
3We also useeij∈ Mto denote an edge starting from nodeiand ending at nodejthrough the whole thesis.
2.1 Complex Network Theory
Apparently, here Ais symmetrical because the network that we considered is an undirected network. For a directed network, Aij = 1 indicates that there is an edge starting from nodei and ending at nodej.
2.1.3 Essentials
1 2
3
4 5
6 Γ(1) ={2,3,4,5}
k1= 4, k2= 3 hki= 4+3+3+2+2+2
6 = 16/6
p2= 3/6, p3= 2/6, p4= 1/6
Figure 2.2:Examples for the nearest neighbor setΓ(i), node degreeki, average degree⟨k⟩and degree distribution pk.
2.1.3.1 Nearest neighbor
Thenearest neighborsetΓ(i)of a node iis a node set which contains all nodes directly connecting to nodei, that is (Fig. 2.2),
Γ(i) ={j|Aji =1,∀j∈ N }. (2.2) 2.1.3.2 Degree, average degree, degree distribution
The degree ki (see Fig. 2.2 as an example) of node i is the sum of weights on edges associated with all nodes in its nearest neighborsΓ(i). For a simple network where weights of both node and edge are fixed to 1, the degree ofiis equivalent to the number of nodes in Γ(i), i.e.,
ki =
∑
j∈Γ(i)
Aji =|Γ(i)|. (2.3)
Obviously, the number of edges m = 12∑i∈Nki. Besides, one can also obtain ki through idx[i+1]−idx[i].
Theaverage (mean) degree⟨k⟩is defined as
⟨k⟩= 1 n
∑
i∈N
ki = 2m
n , (2.4)
which corresponds to the first moment of the degree sequence.
Further, thedegree distribution pkrepresents the probability that a randomly chosen node has degreek. For a specific network, sincepk has to follow∑kpk =1, it is usually defined as
pk = #nodes with degreek
n , (2.5)
where # means ‘the number of’. With this, the average degree can also be obtained through
⟨k⟩=
∑
k
pk. (2.6)
2.1.3.3 Walks, paths, connected components, and cycles
1 2
3
4 5
6 7
Figure 2.3: Examples of walks, paths, components and cycles. Node 6 can reach 3 through one of three walks[6, 5, 4, 3],[6, 7, 5, 4, 3]or[6, 5, 4, 5, 4, 3], in which[6, 5, 4, 3]and[6, 7, 5, 4, 3]are both path and[6, 5, 4, 3]is the shortest path. Besides, none of 1, 2, and 3 can be reached by each other.
Thus, the network contains three connected components covered by independent gray shadows.
An example of a cycle could be[6, 5, 7, 6].
In graph theory, for a given network G(N,M), awalkfrom nodei∈ N to nodej∈ N is asequence4 S starting from iand ending at j, in which S comprises nodes from G and satisfieseuv ∈ Mifu=S[l]andv= S[l+1]for∀S[l]̸= j(see Fig. 2.3 as an example5). It is worth noting that both nodes and edges inSare repeatable. The length of the walk fromi tojis the number of edges inS, or|S| −1 where|S|denotes the number of elements in S6. One can also easily obtain how many walks with length|S| −1 exist between any pair of nodes as
A|S|−1,
where theith-row-jth-column value is the number of walks with length|S| −1 fromitoj.
A pathfrom i to j is a distinct sequence7 S starting from i and ending at j, in which repetitions of nodes and edges are not allowed. Ashortest pathis a path with minimum|S|, and the corresponding lengthdij can be obtained through
dij =min
S |S| −1,S[1] =i,S[|S|] = j. (2.7) Note that it is possible to have multiple shortest paths between two nodes. Thediameter dmax of a network is the longest shortest path among all node pairs,
dmax=maxdij,∀i̸=j. (2.8)
Further theaverage shortest path⟨d⟩is defined as
⟨d⟩= 1
n(n−1)
∑
i,j∈N,i̸=j
dij. (2.9)
Sometimes, it is also possible that nodeicannot reach node jthrough existing edges, i.e., there is no path fromitoj. For such a case, we denote the length of the shortest path betweeniand jwithdij =d∞= ∞. The meaning of this might be, for instance, that a virus
4Sequence is a well-defined collection of ordered objects where repetitions are allowed.
5Through the whole thesis, we use[...]instead of regular representation(...)to denote a sequence.
6View occurrences of the same node as different elements if there is.
7Distinct sequence means that the elements in the sequence are distinct, i.e., repetitions are not allowed.