Citation Network Analysis Explained

A citation network is a directed graph where a count of in-edges is the field’s default measure of importance. That count is a one-hop-deep summary of a structure that is many hops deep, and the two disagree in a way you can derive exactly rather than assert.

What the edges mean

An edge runs from the citing paper to the cited paper. Three properties of this graph shape every method applied to it:

• It is almost acyclic. A paper can only cite work that already exists, so edges point backwards in time. Cycles occur only through preprints, simultaneous publication and self-citation between versions — rare enough that many algorithms assume a DAG and rare enough that the exceptions break them.
• Out-degree is bounded and in-degree is not. A paper has a reference list of a few dozen entries, fixed at publication and never updated. Its citation count grows without limit for decades. The graph is fundamentally asymmetric in a way the web graph is not.
• Edges carry no sign. A citation refuting a paper and a citation building on it are the same edge. Every measure below inherits this, and it is the reason a retracted paper can retain a high score.

What a citation count measures

In-degree is cheap, transparent and genuinely informative, and it has known distortions that are not opinions: citation rates differ by an order of magnitude between fields, so a raw count compares nothing across disciplines; counts accumulate with age, so a five-year-old paper is compared against a one-year-old paper on unequal terms; review articles attract citations far out of proportion to their originality; and self-citation inflates counts without any external judgement.

The deeper limitation is that in-degree treats every citing paper as equal. One citation from a foundational work and one from an uncited preprint contribute the same 1. Network centrality is the attempt to fix exactly that, and the fix is recursive: a paper is important if important papers cite it.

Where counting and centrality invert

Here is a nine-paper network built to make the disagreement exact. Arrows point from citing to cited.

A → P1        F → D        D → P2
B → P1        F → E        E → P2
C → P1        G → D
              G → E

citation counts (in-degree)
  P1 = 3   (from A, B, C — none of which is itself cited)
  P2 = 2   (from D and E — each of which is cited twice)
  D  = 2   E = 2
  A, B, C, F, G = 0

By citation count P1 beats P2, three to two. Now run PageRank with the standard damping factor of 0.85, which Page, Brin, Motwani and Winograd introduced in 1998. P1 and P2 have no outgoing edges, so their rank mass is redistributed uniformly across all N = 9 nodes in the usual way; call the resulting baseline every node receives b. Then:

nodes with no in-links (A, B, C, F, G)
  PR = b

D and E:  cited by F and G, each of out-degree 2
  PR(D) = b + 0.85 · ( b/2 + b/2 ) = b(1 + 0.85) = 1.85 b
  PR(E) = 1.85 b

P1: cited by A, B, C, each of out-degree 1
  PR(P1) = b + 0.85 · ( b + b + b ) = b(1 + 2.55) = 3.55 b

P2: cited by D and E, each of out-degree 1
  PR(P2) = b + 0.85 · ( 1.85b + 1.85b ) = b(1 + 3.145) = 4.145 b

normalise: total = 5b + 2(1.85b) + 3.55b + 4.145b = 16.395 b = 1
           b = 0.0610

  PR(P1) = 3.55  × 0.0610 = 0.2165
  PR(P2) = 4.145 × 0.0610 = 0.2528
  PR(D)  = PR(E) = 0.1128
  PR(A)  = PR(B) = PR(C) = PR(F) = PR(G) = 0.0610

P2, with two citations, outranks P1, with three. The reason is entirely local and entirely explicable: P1’s three citing papers are themselves uncited, so each passes only the baseline mass every node gets for free. P2’s two citing papers have each been cited twice, so each arrives carrying 1.85 times the baseline. Weighting by the standing of the citer, rather than counting citers, reverses the order.

It is also worth seeing what the damping factor is doing to that result. At d = 0.85 each citer passes 85% of its own rank onward and every node keeps a floor. Lower d and the floor dominates, every node converges toward 1/N and the ranking flattens toward the citation count; raise d toward 1 and the recursion is allowed to run further, so differences several hops upstream matter more and the inversion above widens. Damping is not a nuisance constant — it is the parameter that sets how many generations of citation the score is allowed to see.

This is not a toy artefact. It is the phenomenon Chen, Xie, Maslov and Redner reported in 2007 when they applied PageRank to a physics citation network: a small number of papers with unremarkable citation counts scored far above their rank by count, and inspecting them found influential work whose impact ran through a few important descendants rather than through volume.

Time makes this graph unlike the web

PageRank was designed for a graph where links are added and removed continuously in both directions. Citations are not: a paper’s out-edges are frozen at publication, and its in-edges accumulate forever. That asymmetry produces a systematic bias toward old papers, because rank flows backwards in time and there has been more time for it to arrive.

The standard correction is to make the random surfer prefer recent work — CiteRank and related schemes apply an exponentially decaying restart distribution weighted by paper age, so the walk restarts on new papers and old ones must keep earning their rank rather than accumulating it. The mechanics of a restart distribution are the same personalised random walk covered on random walk algorithms on graphs.

Two other structures are specific to this graph and worth knowing. Co-citation joins two papers when a third cites both, and bibliographic coupling joins two papers when they cite the same third. They point in opposite directions in time: coupling is fixed at publication and available immediately, co-citation grows and reflects how a field later came to read the work. Both are projections of the bipartite paper-reference graph, exactly the projection operation worked through on fraud ring detection, and both are what the field’s cluster maps are built from.

What none of these numbers support

Every measure on this page describes a graph. None of them measures quality, and the gap between those two statements is where citation analysis does damage.

The h-index, introduced by Hirsch in PNAS in 2005 as the largest h for which an author has h papers with at least h citations each, is bounded by an author’s total output and so penalises short careers and rewards long ones regardless of impact. Journal-level averages are worse: a journal’s citation distribution is heavily skewed, so its mean describes almost none of its papers, and using it to judge an individual paper is a category error the San Francisco Declaration on Research Assessment (2012) was written specifically to oppose.

The defensible uses are the aggregate ones: mapping a field’s structure, tracing how an idea propagated, identifying communities and bridging work, finding literature you would not have found by keyword. Those are questions about the shape of a network, which is what a network measure can answer.