**Chapter 1: Introduction**

**3.4. Centrality and Network Positions**

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center is an actual patron. Administrative rank levels only assist but do not conclude the case. A weak leader with bare influence in the party’s leadership might be at vice-state level only because he is awarded a trivial vice-chairmanship in the National People’s Congress or National Committee of CPPCC. A seasoned and influential leader may stop at full-provincial level in the ladder of administrative ranking, but his connections to the core of party leadership make him resourceful and capable of sustaining his own faction. The administrative rank level is not conclusive with regard to the actual influence of a cadre amongst the regime elites. Here we are in the position of introducing relational statistics in network analysis to help us quantitatively determine whether the possible patrons with the highest rank level indeed occupy the most central positions in their circles.

### 3.4. Centrality and Network Positions

Centrality is the measurement of relative positions of nodes in a network. The ego-centered perspective of centrality measures the positions of individual nodes in the whole network. People who are central in the network have better positions to access information and better opportunities to spread information. (de Nooy, Mrvar &

Batagelj, 2005) In the context of factional politics, a factional patron is expected to occupy a relatively central position and thus has higher centrality score than his followers.

There are various measures of centrality in the field of network analysis. Degree is the most straightforward measure, which simply counts the number of lines incident with a node (de Nooy, Mrvar & Batagelj, 2005). In a network of an ideal-typed faction, the degree of each faction follower is 1, and the degree of the patron equals to the number of his followers. Unfortunately, the reality is never this neat. As we choose to count colleague relations, almost all cadres certainly have ties with others who are not in the

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same factions with them. As a result, a low-ranking follower may score higher in degree than his patron does.

Closeness and betweenness are also commonly used. Closeness is a measure of how long it takes to spread information from one individual node to others in the network.

(Newman, 2005) It is calculated as the inverse of the farness of a node: the sum of distances between the node and all others. Betweenness is the number of shortest paths (geodesics) from all nodes to all others that pass through the node in question (Freeman, 1977; Ulrik, 2008). Closeness and betweenness, though, may mislead our analysis for two reasons. First, they do not distinguish between nodes that are immediately adjacent to a given node and nodes distant from the given node. As we are interested in local structure of the network, the nodes with direct ties with a given node should have more weight in the algorithm that calculates centrality. Second, neither closeness nor betweenness distinguishes central nodes from peripheral nodes.

In our case, relations with central nodes – the more powerful cadres – are more important than relations with peripheral ones. We need an algorithm that weight relations according to the relative importance of adjacent nodes.

All things considered, this thesis adopts eigenvector centrality. Eigenvector centrality, like degree centrality, counts the number of nodes adjacent to a given node. Moreover, it weights each of the adjacent nodes by their own centrality. The algorithm of eigenvector centrality is expressed as follows:

𝑒_{𝑖} = 𝜆 ∑ 𝑥_{𝑖𝑗}𝑒_{𝑗}

𝑗

𝑒_{𝑖} is the centrality of the node in question; 𝑒_{𝑗} is the centrality of each node adjacent
to node 𝑖; 𝜆 is a proportionality constant called the eigenvalue. (Borgatti, Everett &

Johnson, 2013)^{37}

37 For detailed discussions of various measures of centrality, see Chapter 3 and 6 in de Nooy, Mrvar & Batagelj

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By eigenvector centrality, the centrality of a given node is primarily determined by its ties with adjacent nodes, which rules out the noise of distant nodes irrelevant to local structure. Eigenvector algorithm weights different nodes by their own centrality. It reflects the political reality that a cadre’s political influence is primarily a result of his connections with the powerful ones. In this thesis, we apply a variant algorithm of eigenvector centrality, PageRank. It is originally a system to rank web pages developed by the founders of Google (Brin & Page, 1998). PageRank is not the only algorithm that Google uses but it still plays a central role in its web search.

The working principle of PageRank is to count how many web pages are linked to a given page whose relative importance we want to calculate. The more incoming links a given web page has, the more probable that a “random surfer” will visit this page. It resembles a voting system in which web pages “vote” for each other using links as ballots. But the suffrage is not equal. The more important a web page is, the more weight its “ballots” carry. (Brin & Page, 1998) In short, PageRank counts the number and significance of links to a given web page to calculate its importance. The algorithm is as follows:

𝑃𝑅_{𝑖} = (1 − 𝑑) + 𝑑 ∑𝑃𝑅_{𝑗}
𝐶_{𝑗}

𝑗

Page 𝑖 is the given web page we are interested in. 𝑗 stands for all pages that are
linked to page 𝑖, and 𝐶_{𝑗} is the number of links going out from page 𝑗. 𝑑 is a
damping factor usually set at 0.85. (Brin & Page, 1998)

In the context of our analysis, PageRank operates as a “voting” system in which all tigers count “ballots” for each other by their colleague relations. The more ties in number and more important ties in quality one tiger has, the higher his PageRank

(2005), and Chapter 10 in Borgatti, Everett & Johnson (2013).

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scores, and thus the more central this tiger is. A patron in a central position is expectedly linked to many other tigers who are mostly his followers, and thus he shall score high in PageRank. A factional follower, in the periphery of a factional structure, has fewer links and thus lower score in PageRank. A faction follower’s PageRank will not be overstated by his link to his patron because the “ballot” from the patron is weighted by all the patron’s outgoing links. All tigers, be they patrons, followers or non-factional tigers, have redundant links that are not factional ties, but the influence of this redundancy is neutralized because each linkage is weighted in the equation. If the PageRank of a low-ranking tiger is boosted by his links to multiple “big tigers”

*(大老虎, da laohu), he is probably not a member of any faction in the clientelist *
definition and thus we need not worry about his PageRank score being higher than a
factional patron. With the help of PageRank score, we can see whether the
highest-ranking cadre in a circle does occupy a central position and thus is an actual
patron.

**Table 3-1: Top 10 Tigers in PageRank Score **

Name Rank Level^{1} Remark

No. 1 Zhou Yongkang FS Standing Member of Politburo

No. 2 Ling Jihua VS National Leader

No. 3 Su Rong VS National Leader

No. 4 Li Dongsheng FP Tied to Zhou Yongkang

No. 5 Shen Weichen FP Highest-Ranking in Shanxi

No. 6 Liu Zheng VMR

No. 7 Bai Enpei FP Highest-Ranking in Yunnan

No. 8 Liang Bin VP Tied to Shanxi

No. 9 Ji Wenlin VP Tied to Zhou Yongkang

No. 10 Xu Caihou VS Vice-Chairman of CMC

Note:

1. FS: full-state level; VS: vice-state level; VMR: vice-military-regional level; FP: full-provincial level;

VP: vice-provincial level.

Table 3-1 shows the top 10 tigers with the highest PageRank score. The four national

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leaders all make into this top-10 list. The only possible patron below vice-state level, Bai Enpei, is also on the list. 9 senior corrupt cadres out of these top 10 are either potential patrons of factional groups or part of the circles we point out in the previous section. The PageRank measurement preliminarily reflects political reality because the most powerful cadres in our common sense do have the highest PageRank scores.

**Figure 3-3: PageRank of Circles**

Figure 3-3 presents the average PageRank of all 104 tigers, the average PageRank of each circle and the PageRank of the four possible patrons. The overall average score is 0.010, and the average PageRank of each circle is slightly higher than this overall average. Zhou Yongkang Circle has the highest circle average at 0.014, followed by Shanxi Circle at 0.013. All of the other three circles have an average of 0.012 in PageRank score. It is suggested that the network positions of circle tigers are not significantly more central to the non-circle tigers.

0.014

Zhou Yongkang Circle Su Rong Circle Bai Enpei Circle Xu Caihou Circle Shanxi Circle All

Average PageRank Possible Patron's PageRank

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As factional theories expect, all hub tigers at the highest rank level in circles as the possible patrons have much higher PageRank scores than the average number of their respective circle. The intra-circle central positions of these four seasoned politicians are not only visually inspected but also quantitatively confirmed by centrality. Zhou Yongkang has the PageRank score of 0.037, higher than any tiger among the 104. Xu Caihou, the possible patron with the smallest potential factional group, scores 0.017 in PageRank. The main body of Shanxi Circle is a clique and the circle does not have an obvious possible patron. But its highest-ranking tiger, Shen Weichen, also scores the highest PageRank in the circle at 0.020.

As a summary of this section, the PageRank statistics confirm that centrality measurement is a powerful tool to quantify how powerful a tiger is among the whole network as PageRank ranking coincides wth tigers’ administrative rank levels. All of the possible patrons pass the test of local centrality as they all score significantly higher than average circle members.

### 3.5. Summary

In this chapter, we diagram the colleague network of senior corrupt cadres for the identification of potential factional groups and further discussion of factional politics in play. The first two sections present the coding principles of the tigers’ career tracks and their colleague relations. We acquire the resumes of the 104 tigers from open sources and code their overlapping of service as into colleague ties. In Section 3.3 we transform the relational data into a network and identify 5 circles as potential factional groups. 4 circles are centered on either state-level leaders (Zhou Yongkang, Su Rong, and Xu Caihou) or a seasoned politician with exceptional seniority (Bai Enpei).

Shanxi Circle is the only one without a possible patron. There is 1 national leader, Ling Jihua, without a circle around. In Section 3.4 we use PageRank algorithm to

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calculate centrality scores of the 104 tigers and find centrality a rather faithful quantitative indicator of these cadres’ relative importance. As factional theories expect, all the possible patrons of identified circles have higher centrality scores than their respective circle subordinates.