CheiRank: Outgoing Influence in Networks

• CheiRank is a ranking method for directed networks computed from the reversed Google matrix, emphasizing nodes’ outgoing links to capture broadcasting and diffusion properties.
• Combining CheiRank with PageRank in a 2D ranking framework enables researchers to distinguish between popularity (incoming links) and communicative influence (outgoing links) across systems.
• Empirical studies demonstrate CheiRank's effectiveness in diverse domains, from identifying influential Twitter broadcasters to highlighting export-oriented roles in world trade.

CheiRank is a ranking on directed networks defined as the outgoing-link counterpart of PageRank. If PageRank measures how strongly a node is pointed to by important nodes, CheiRank measures how strongly a node points outward in the reversed network, and therefore captures communicativity, broadcasting, export orientation, diffusion, or source-likeness depending on domain (Ermann et al., 2011, Zhirov et al., 2010, Rollin et al., 8 Jul 2025). Together, PageRank and CheiRank place each node in a two-dimensional ranking space, usually represented by the PageRank–CheiRank plane $(K,K^*)$, which has been used for Wikipedia, citation networks, Twitter, Bitcoin transactions, world trade, world economic activities, business-process graphs, and biologically inspired Boolean networks (Zhirov et al., 2010, Ermann et al., 2011, Frahm et al., 2012, Ermann et al., 2017, Coquidé et al., 2019, Newby et al., 2023).

1. Definition and interpretive scope

CheiRank is obtained by applying the Google-matrix construction to the network with all link directions reversed. In the early two-dimensional-ranking literature, it was introduced as the complement to PageRank: PageRank measures popularity or authority through ingoing links, whereas CheiRank measures communicativity through outgoing links (Ermann et al., 2011, Abel et al., 2010). In the world trade literature, the inverse PageRank was explicitly named CheiRank to emphasize that it helps chercher information in a new way (Ermann et al., 2011).

The interpretation of CheiRank depends on the semantics of link direction. In Wikipedia and Twitter it measures communicative or broadcasting role; in citation networks it measures outgoing citation influence or communicativeness; in trade and economic-activity networks it is export-oriented or outgoing-flow importance; in Bitcoin it identifies important senders or seller-like users; in biologically inspired Boolean networks it is interpreted as source-likeness (Frahm et al., 2012, Frahm et al., 2013, Kandiah et al., 2015, Ermann et al., 2017, Coquidé et al., 2019, Newby et al., 2023).

Network class	PageRank interpretation	CheiRank interpretation
Wikipedia	popular or known node	communicative node
Citation networks	being cited by important papers	outgoing citation influence
World trade / WNEA	import-side prominence	export-side or outgoing influence
Bitcoin	receivers of flow	broadcasters / senders / outgoing-active users

This duality is central to the concept. The literature repeatedly treats CheiRank not as a replacement for PageRank but as an orthogonal ranking dimension that becomes informative precisely when incoming importance and outgoing influence do not coincide (Zhirov et al., 2010, Ermann et al., 2011).

2. Mathematical construction

For a directed network with adjacency matrix $A$, several papers use the convention $A_{ij}=1$ if node $j$ points to node $i$ (Zhirov et al., 2010, Rollin et al., 8 Jul 2025). The column-stochastic transition matrix is formed by normalizing outgoing links,

$$S_{ij}=\frac{A_{ij}}{k_{\rm out}(j)}, \qquad k_{\rm out}(j)=\sum_{i=1}^{N}A_{ij},$$

with dangling nodes replaced by $S_{ij}=1/N$ when $k_{\rm out}(j)=0$ (Rollin et al., 8 Jul 2025). The standard Google matrix is then

$G_{ij}=\alpha S_{ij}+(1-\alpha)\frac{1}{N},$

with $\alpha=0.85$ in much of the literature (Eom et al., 2013, Lages et al., 2015).

The PageRank vector $A$0 is the stationary distribution of this Markov process: $A$1 CheiRank is defined by reversing all links, constructing the Google matrix $A$2 of the reversed network, and taking its stationary vector: $A$3 Sorting $A$4 in decreasing order gives the PageRank index $A$5; sorting $A$6 gives the CheiRank index $A$7 (Frahm et al., 2013, Rollin et al., 8 Jul 2025).

This construction has weighted and personalized variants. In Bitcoin, the transition matrix is built from normalized transaction volume, and the same reversal procedure produces CheiRank on the inverted transaction network (Ermann et al., 2017). In multilingual Wikipedia with clickstreams and pageviews, the Google matrix is generalized to

$A$8

so that clickstream counts modify $A$9 and pageviews bias the teleportation vector $A_{ij}=1$0 (Coquidé et al., 2020). In world economic activities, the Google matrices $A_{ij}=1$1 and $A_{ij}=1$2 use personalization vectors chosen by the Google Personalized Vector Method so that countries are treated democratically while sector weights remain proportional to exchange volumes (Kandiah et al., 2015).

3. Two-dimensional ranking and correlation structure

Because each node has both $A_{ij}=1$3 and $A_{ij}=1$4, CheiRank is normally studied jointly with PageRank in the PageRank–CheiRank plane $A_{ij}=1$5 (Zhirov et al., 2010, Frahm et al., 2013). The node density in this plane is described by

$A_{ij}=1$6

and a standard global correlation measure is

$A_{ij}=1$7

(Ermann et al., 2011, Frahm et al., 2013).

Reported values of $A_{ij}=1$8 vary strongly across network classes.

Network	Reported $A_{ij}=1$9	Interpretation
English Wikipedia	$j$0	strong positive correlation (Zhirov et al., 2010)
Twitter	$j$1	very large overlap of popular and communicative elite (Frahm et al., 2012)
Bitcoin, later quarters	about $j$2	very strong coupling between receiving and sending activity (Ermann et al., 2017)
Physical Review citation network	$j$3 for full CNPR; $j$4 without Rev. Mod. Phys.	small and negative correlation (Frahm et al., 2013)
Business process graph	$j$5	almost no correlation (Abel et al., 2010)
Linux kernel and gene regulation	$j$6 or slightly negative	near-independence (Ermann et al., 2011)

These values show that CheiRank is useful precisely because PageRank–CheiRank relations are network dependent. In Wikipedia, the density is concentrated along a band and the correlator is positive (Zhirov et al., 2010). In the Physical Review citation network, the correlation is weak and negative, reflecting the near-triangular time ordering of citations and the structural difference between the original and reversed graphs (Frahm et al., 2013). In Bitcoin, the density concentrates near the diagonal $j$7, which the authors interpret as users trying to maintain a rough balance of inflows and outflows (Ermann et al., 2017).

The two-dimensional framework also motivates combined rankings. Several papers define 2DRank by scanning expanding square ribs in the $j$8 plane (Zhirov et al., 2010, Eom et al., 2013), while other studies use a combined criterion such as $j$9 or $i$0 (Eom et al., 2014, Coquidé et al., 2020). In either form, 2DRank favors nodes that are simultaneously strong in incoming and outgoing importance.

4. Empirical behavior across major network families

In Wikipedia, CheiRank highlights highly communicative pages with many outgoing links, often including portals, lists, and pages rich in outward references (Zhirov et al., 2010). This produces category-dependent contrasts with PageRank. For personalities, PageRank selection is dominated by politicians, whereas 2DRank gives more accent on personalities of arts; for universities, outgoing links to alumni and related institutions can strongly affect CheiRank positions (Eom et al., 2013). In multilingual Wikipedia, CheiRank lists of persons are much more culture-specific than PageRank lists: the average overlap of top persons across editions is about $i$1 for PageRank, $i$2 for 2DRank, and about $i$3 for CheiRank (Eom et al., 2013).

In citation networks, CheiRank measures outgoing citation influence rather than incoming citation prestige (Frahm et al., 2013). In the Physical Review network, PageRank is strongly localized on a small number of highly cited papers, whereas CheiRank is much more spread out, with a much larger inverse participation ratio (Frahm et al., 2013). CheiRank tends to emphasize papers with long bibliographies, especially review-like articles and papers in Rev. Mod. Phys. (Frahm et al., 2013). The philosopher study on Wikipedia likewise uses CheiRank as diffusion or communicative reach, and reports that philosopher pages are generally more cited than citing, tending to occupy the region $i$4 and $i$5 (Rollin et al., 8 Jul 2025).

In social and information networks, CheiRank often isolates nodes that control dissemination. In Twitter, PageRank and CheiRank are both power-law-like, with $i$6 for PageRank and $i$7 for CheiRank, and top PageRank users are exceptionally strongly interconnected (Frahm et al., 2012). The CheiRank interpretation there is communicative activity, and the large $i$8 supports the notion of a tightly connected social-network elite (Frahm et al., 2012). In the Ising-PageRank model of opinion formation, elite nodes chosen by top PageRank, CheiRank, or 2DRank can significantly shift the final vote even if the elite fraction is very small; CheiRank elites represent highly outgoing and communicative nodes and can be especially influential depending on network structure (Frahm et al., 2018).

In transaction and economic networks, CheiRank acquires an explicitly flow-based meaning. For Bitcoin, PageRank identifies important receivers and CheiRank important senders; the dimensionless balance

$i$9

distinguishes seller-like from buyer-like users (Coquidé et al., 2019). For world trade and world economic activities, CheiRank is export-oriented and PageRank import-oriented, yielding country balances such as

$$S_{ij}=\frac{A_{ij}}{k_{\rm out}(j)}, \qquad k_{\rm out}(j)=\sum_{i=1}^{N}A_{ij},$$0

(Kandiah et al., 2015, Coquidé et al., 2022). In the world trade network, countries with $$S_{ij}=\frac{A_{ij}}{k_{\rm out}(j)}, \qquad k_{\rm out}(j)=\sum_{i=1}^{N}A_{ij},$$1 are interpreted as successful traders because their export rank is better than their import rank (Ermann et al., 2011). In the COVID-19 trade study, the PageRank–CheiRank product balance

$$S_{ij}=\frac{A_{ij}}{k_{\rm out}(j)}, \qquad k_{\rm out}(j)=\sum_{i=1}^{N}A_{ij},$$2

reveals structural export-oriented and import-oriented product groups and shows that 2020 involved a major rewiring of trade flows (Coquidé et al., 2022).

5. Variants, extensions, and associated methods

CheiRank has been extended beyond the unweighted, uniformly teleported Google matrix. The most direct generalization is weighted CheiRank, where link reversal is applied after link weights have been defined from transactions, trade volumes, or clickstream counts (Ermann et al., 2017, Coquidé et al., 2020). In the WikiClick and WikiClick Plus View models, clickstream counts replace unit hyperlink weights, pageviews bias teleportation, and CheiRank changes much more drastically than PageRank; in the English network, CheiRank overlap between standard and weighted methods is low, and under the pageview-biased model the exact overlap is essentially $$S_{ij}=\frac{A_{ij}}{k_{\rm out}(j)}, \qquad k_{\rm out}(j)=\sum_{i=1}^{N}A_{ij},$$3 across compared methods by $$S_{ij}=\frac{A_{ij}}{k_{\rm out}(j)}, \qquad k_{\rm out}(j)=\sum_{i=1}^{N}A_{ij},$$4 (Coquidé et al., 2020). The paper interprets the resulting CheiRank as identifying entry points of Wikipedia rather than merely list pages (Coquidé et al., 2020).

A second line of development combines CheiRank with reduced-network methods. The reduced Google matrix algorithm constructs a small matrix for a selected node set while preserving both direct and indirect interactions mediated by the full network (Coquidé et al., 2019, Coquidé et al., 2018). In Bitcoin, the reduced matrix is decomposed as

$$S_{ij}=\frac{A_{ij}}{k_{\rm out}(j)}, \qquad k_{\rm out}(j)=\sum_{i=1}^{N}A_{ij},$$5

where $$S_{ij}=\frac{A_{ij}}{k_{\rm out}(j)}, \qquad k_{\rm out}(j)=\sum_{i=1}^{N}A_{ij},$$6 captures direct links, $$S_{ij}=\frac{A_{ij}}{k_{\rm out}(j)}, \qquad k_{\rm out}(j)=\sum_{i=1}^{N}A_{ij},$$7 is close to the PageRank distribution, and $$S_{ij}=\frac{A_{ij}}{k_{\rm out}(j)}, \qquad k_{\rm out}(j)=\sum_{i=1}^{N}A_{ij},$$8 contains nontrivial indirect pathways; this helps explain why top PageRank and CheiRank users collapse rapidly under contagion (Coquidé et al., 2019). In the Presocratic-philosopher study, the same framework reveals hidden links such as $$S_{ij}=\frac{A_{ij}}{k_{\rm out}(j)}, \qquad k_{\rm out}(j)=\sum_{i=1}^{N}A_{ij},$$9 and $S_{ij}=1/N$0 that are not necessarily direct hyperlinks (Rollin et al., 8 Jul 2025).

CheiRank is also associated with localized or source-specific propagation measures. In the Physical Review citation network, ImpactRank is introduced through

$S_{ij}=1/N$1

or equivalently through a personalized Google matrix, to study the influence domain of a specific article (Frahm et al., 2013). That work explicitly distinguishes the two notions: CheiRank is a global ranking of nodes in the reversed citation network, whereas ImpactRank is a localized propagation measure centered on one article (Frahm et al., 2013). In biologically inspired Boolean networks, CheiRank is used as one of three structural propagation metrics for ranking feedback vertex set subsets, together with PRINCE propagation and modified PRINCE propagation, and contributes to the propagation intersection metric used to predict attractor control strength (Newby et al., 2023).

6. Significance, limitations, and recurrent misconceptions

A recurrent misconception is that CheiRank is merely the out-degree ordering. The literature does state that CheiRank is, on average, proportional to the number of outgoing links (Ermann et al., 2011, Frahm et al., 2012, Lages et al., 2015). However, its definition is the stationary vector of the reversed-network Google matrix, and empirical studies show that weights, damping, teleportation, and indirect pathways can substantially alter the ranking (Ermann et al., 2017, Coquidé et al., 2020, Kandiah et al., 2015). In trade networks, CheiRank can elevate countries with broad, diversified export networks over countries with large but concentrated export totals (Ermann et al., 2011). In Bitcoin, CheiRank participates in a balance-based contagion model in which top PageRank and CheiRank users can fail rapidly as a tightly coupled core, even away from the critical threshold $S_{ij}=1/N$2 (Coquidé et al., 2019).

A second misconception is that PageRank and CheiRank usually agree. They can agree strongly, as in mature Bitcoin or Twitter, but they can also be weakly correlated or negatively correlated, as in business-process graphs, Linux and gene-regulation networks, or the Physical Review citation network (Ermann et al., 2017, Frahm et al., 2012, Abel et al., 2010, Ermann et al., 2011, Frahm et al., 2013). This suggests that outgoing influence and incoming prestige are network-specific structural roles rather than interchangeable views of the same property.

CheiRank is also more sensitive to editable or rapidly changing outward structure. In Wikipedia, outgoing links are easier for editors to modify than ingoing links, and CheiRank positions fluctuate more than PageRank positions for personalities and universities (Eom et al., 2013). In multilingual person rankings, the very low cross-edition overlap of top CheiRank persons makes CheiRank less suitable than PageRank or 2DRank for identifying robust global heroes (Eom et al., 2013). In specialized encyclopedias such as SEP and IEP, CheiRank can reveal a sharper separation between central and outgoing articles than in Wikipedia; for example, in SEP Aristotle is $S_{ij}=1/N$3 but $S_{ij}=1/N$4, whereas John Dewey has $S_{ij}=1/N$5 but $S_{ij}=1/N$6 (Rollin et al., 8 Jul 2025).

Within Google-matrix research, the lasting significance of CheiRank is that it turns one-dimensional authority ranking into a two-dimensional analysis of directed flow. Across encyclopedic networks, citation systems, transaction graphs, trade networks, and control problems, it provides a mathematically uniform description of outward connectivity while remaining semantically adaptable to the domain under study (Zhirov et al., 2010, Ermann et al., 2011, Frahm et al., 2013, Kandiah et al., 2015).