In a 2015 symposium on big data and measurement, Maggie Penn and I argued that social choice theory is central to the analysis of complex data precisely because any reduction of high-dimensional data into a usable measure involves aggregation, and aggregation involves choices about what to preserve and what to discard.1 We used the Florentine marriage network to illustrate how different centrality indices — degree, betweenness, closeness — each make different and irreconcilable choices about what information to retain. We noted, in a footnote, that even drawing a network requires choices to be made, citing the Fruchterman-Reingold algorithm as a ubiquitous example.
That footnote deserves more than a footnote. What follows is the argument we didn’t make in 2015.
Running a physics simulation
When a political scientist visualizes a network, she is running a physics simulation. This is not a metaphor. The standard algorithms used to draw networks — Fruchterman-Reingold, ForceAtlas2, and their relatives, implemented in packages like Gephi and igraph — model edges as springs and nodes as mutually repulsive charged particles. The algorithm releases this simulated physical system from a random initial configuration and records where everything lands when it reaches mechanical equilibrium. The picture you see is a snapshot of a steady state.
This would be unproblematic if the steady state encoded something meaningful about the underlying social structure. In one limited sense it does: densely connected nodes tend to cluster together visually, and nodes with no connections tend to drift apart. Rough community structure is often visible. The developers of ForceAtlas2 describe this as the algorithm’s ambition — to turn structural proximities into visual proximities.
What is less appreciated, including apparently by many of the algorithm’s users, is what the steady state does not encode. The coordinates of any individual node are not data. They are the output of an optimization that depends on the random initialization of positions, on software parameter choices, and on the full composition of the graph. The ForceAtlas2 documentation states this plainly: “The result cannot be read as a Cartesian projection. The position of a node cannot be interpreted on its own.”
The implication is immediate. When a researcher looks at a network picture and says “Group A sits closer to Group B than to Group C, suggesting A and B are more aligned” — or worse, compares two pictures from different time periods and says “A and B have moved closer together, suggesting convergence” — she is making claims about ink. The x and y coordinates carry no warranted inferential content about the groups they represent. This is not a minor caveat. It is a fundamental misidentification of what the picture is.
We can state the problem precisely. Say that a layout algorithm satisfies Network Embedding Independence of Irrelevant Alternatives (NEIIA) if, for any two graphs that agree on the edge between nodes \(x\) and \(y\) and on all edges incident to \(x\) or \(y\), the algorithm places \(x\) and \(y\) in the same relative position in both graphs. Force-directed algorithms fail NEIIA by construction. The equilibrium position of every node is determined by a global optimization — repulsion terms involve every pair of nodes, and attraction terms propagate through the entire graph. There is no sense in which any node’s position is computed locally. The picture of \(x\) and \(y\) is always a function of the full graph.2
The most famous network in political science
Consider the Florentine marriage network, assembled by Padgett and Ansell from fifteenth-century records of intermarriage among elite Florentine families.3 It is probably the most cited network dataset in political science and sociology. The central finding — that the Medici’s structural position in the network, not their wealth or ideology, explains their political rise — is a landmark result reproduced in textbooks and review articles for thirty years.
Notice what Padgett and Ansell actually computed. Their measure of choice was betweenness centrality: the number of shortest paths between other pairs of families that pass through the Medici. This quantity is a property of the combinatorial graph — who is connected to whom — not of any geometric rendering. The Medici’s betweenness centrality is 47.5 (normalized: 0.522), and it is the same whether the network is drawn vertically or horizontally, clockwise or counterclockwise, by Gephi or by hand on a napkin. As Patty and Penn (2015) discuss at length, the celebrated result survives any redrawing because it was never about the drawing.
Below is the Padgett-Ansell dataset as an interactive simulation. Try pressing “Re-run layout” several times before reading further.
The network you just saw rearranged is the same sixteen families connected by the same twenty edges. The Medici’s betweenness centrality remained 47.5 through every run. What changed was a random seed — a number fed to the algorithm’s initialization routine that determines where each node starts before the simulation runs. There is no sense in which one picture is more correct than another. They are all equally arbitrary representations of the same combinatorial object.
A hierarchy of what is actually identified
Before proceeding to the specific distortions that force-directed layouts introduce, it is worth establishing a hierarchy. Not all network statistics are equally vulnerable. The 2015 paper with Maggie proved a theorem, due originally to van den Brink and Gilles (2003), that organizes this hierarchy cleanly.
The theorem states that the only centrality index satisfying both Positive Responsiveness — a node’s rank never decreases when it gains links — and Independence of Irrelevant Edges (IIE) — the relative ranking of nodes \(i\) and \(j\) is invariant to edges not incident to either — is degree centrality. Every other widely used centrality measure necessarily violates at least one of these conditions.
This gives us three tiers. Node positions in a force-directed layout sit at the bottom: sensitive to the random seed, the algorithm, the parameter choices, and every node and edge in the graph. They carry essentially no invariant information about the underlying network. Betweenness centrality — Padgett and Ansell’s measure — sits in the middle: invariant to the drawing, but sensitive to network composition. Adding or removing families anywhere can change who ranks highest. This is the source of its power, capturing global structural position, but it comes with a sensitivity that degree centrality avoids. Degree centrality sits at the top: invariant to the drawing and satisfying IIE, so the ranking of the Medici against the Guadagni depends only on their own connections. This robustness is also a limitation — degree centrality cannot see past immediate neighbors, which is precisely why Padgett and Ansell needed betweenness.
The practical lesson is not “use degree centrality for everything.” It is: be precise about which tier your inferential claims are drawing from. A claim about where a node sits on a page is drawing from the bottom tier. A claim about betweenness centrality is drawing from the middle. A claim about degree is drawing from the top. Most published dynamic network visualization is presenting pictures as if they were evidence from the top tier when they are evidence from the bottom.
The layout dictator
Now press “Remove Medici.”
The Medici are connected to six other families — Acciaiuoli, Albizzi, Barbadori, Ridolfi, Salviati, and Tornabuoni. In the spring-embedding equilibrium, this makes them a gravitational center: every family connected to them is pulled toward the Medici’s position, and therefore toward every other family connected to them. The apparent proximity of, say, the Tornabuoni and the Acciaiuoli in any given rendering is partly a function of their both being Medici clients — not of any direct relationship between them.
With the Medici removed, the network fragments. Most dramatically, the Pazzi and Salviati become a completely isolated dyad — cut off from the rest of Florence entirely. Their prior apparent proximity to the Albizzi, Guadagni, and Strozzi clusters was purely a function of the Medici hub pulling everything together. This is particularly striking because the Pazzi were one of the wealthiest families in the dataset — roughly 72,500 florins, comparable to the Medici themselves — and listed in Padgett and Ansell’s data as party “Medici,” meaning active political supporters. Yet structurally they were marginal: degree 1, hanging off a single marriage tie to Salviati. The visualization with the Medici present obscures this completely.
This is the layout dictator property: a sufficiently well-connected node controls the geometric positions of all other nodes, regardless of the direct relationships among those others. Its influence on the picture is proportional to its degree. The organizations most likely to function as layout dictators in advocacy networks are the ones that are substantively most important: major civil rights organizations, large labor federations, pan-ethnic legal defense funds. They are important precisely because they are highly connected. Dropping them to “clean up” the picture is not methodological hygiene. It is the systematic removal of the most powerful actors from the analysis.
The stretched spring
Now press “Add hypothetical Pazzi–Strozzi edge” and watch what happens to the clusters from their current positions before the simulation re-settles.
No marriage alliance between the Pazzi and Strozzi is recorded in the historical data — this is a hypothetical. But the mechanic it demonstrates is real and operates on every long edge in every network picture ever drawn. Adding a single spring between two nodes that the rest of the graph has been holding far apart introduces a force that is stretched — it wants to contract to its natural length, but the intra-cluster springs resist. The equilibrium shifts: not just those two nodes, but every node in each cluster drifts toward the other community, because every node is coupled to its neighbors through its own springs.
Consider the visual signal this produces. The new edge appears as a long dashed line — suggesting, to the eye, a peripheral or weak connection between distant groups. But mechanically it is doing maximum work. A stretched spring exerts more force than a relaxed one. The visual impression — “this edge is long, so it must be minor” — is exactly backwards from the mechanical reality: this edge is long precisely because it is fighting hardest against the intra-cluster springs, displacing every node in both communities in the process.
The magnitude of displacement depends on the internal density of the clusters being connected. A dense, tightly coupled community resists displacement more than a sparse, loosely coupled one. The algorithm therefore has systematically different error properties in different parts of the graph. Dense, insular communities are geometrically stable. Open, cross-cutting communities are geometrically volatile. The visualization rewards insularity with geometric stability. A community that actively builds cross-cutting coalitions will have its apparent position contaminated by every such coalition it forms. A community that never reaches across the divide will appear as a coherent, well-defined cluster. This is not a finding about the political landscape. It is a property of the physics.
The dynamic case
Both distortions compound severely in dynamic network analysis — the increasingly common practice of showing a network evolving across time periods and interpreting changes in node positions as evidence of political change.
When new nodes enter the graph at \(t_2\), the algorithm re-optimizes globally. Nodes that have not changed a single edge can move dramatically on the canvas, purely because the energy landscape shifted with the arrival of new actors. A single new marriage alliance between the Pazzi and Strozzi clusters, appearing at \(t_2\), will make every family in both clusters appear to converge — regardless of whether anything else changed. The researcher who interprets this as political rapprochement is reading an artifact of one stretched spring, amplified through the coupling network of each cluster’s internal edges, further confounded by whatever new families entered the dataset in the same period.
There is no standard diagnostic for disentangling genuine relational change from layout artifact in dynamic network visualizations. In most published work, no attempt is made to do so. This is not a minor technical oversight. It means that a substantial literature on political polarization, coalition evolution, and inter-group cooperation over time is drawing inferences from a sequence of pictures whose spatial arrangement is determined partly by the social process of interest and partly — perhaps dominantly — by the composition of the graph and the software defaults.
What is actually identified
The hierarchy established above suggests a constructive conclusion, not merely a critique. Padgett and Ansell got the Florentine analysis right for the right reasons: they used the drawing as an illustration and the centrality measure as the finding. Their figures 2a and 2b in the original paper are not force-directed layouts at all. They are hand-drawn blockmodel diagrams, with the positions of blocks chosen by the analysts to communicate structure that had already been established algebraically. The Medici are at the center of the diagram because betweenness centrality put them there. Nobody looks at those figures and asks why the Medici block is positioned where it is — because the question doesn’t arise. The drawing illustrates a finding; it does not make one.
This discipline is worth recovering. Any procedure that takes a network and produces positions in \(\mathbb{R}^2\) must make assumptions somewhere. Force-directed layouts hide those assumptions in physics metaphors and software defaults, presenting the output as if it were data rather than as the consequence of choices. Alternative approaches — latent space models, for instance — produce node positions that are estimates of parameters in a generative model of edge formation, which carries genuine inferential content.4 But even these make assumptions about the latent space: its dimensionality, its distance function, its link function. The assumptions don’t disappear — they migrate. The appropriate discipline is not to find the assumption-free method, because there isn’t one. It is to be explicit about what your method assumes and whether those assumptions are defensible given your theory of the social process.
The picture is a heuristic for community structure, and in that limited role it is useful. It is not a map of social space. The distance between two nodes in a Gephi rendering is not the distance between the groups they represent. And the distance between the same two nodes across two Gephi renderings from different time periods is not a measure of how their relationship changed. What is identified — genuinely, robustly, independently of how the picture is drawn — is the combinatorial structure of the graph: who is connected to whom, how many steps separate them, how much of the network’s traffic flows through them. Padgett and Ansell measured that. Thirty years of follow-on work largely hasn’t. The Pazzi were wealthy, politically aligned with the Medici, and structurally peripheral. No force-directed layout will tell you that. The betweenness calculation will.
With that, I leave you with this.
Notes
1 Patty, John W. and Elizabeth Maggie Penn. “Analyzing Big Data: Social Choice and Measurement.” PS: Political Science & Politics 48(1): 95–101, 2015. The argument is developed at greater length in our book Social Choice and Legitimacy: The Possibilities of Impossibility (Cambridge University Press, 2014), and extended to fairness and machine learning in Patty and Penn, “Measuring Fairness, Inequality, and Big Data: Social Choice Since Arrow,” Annual Review of Political Science 22: 435–460, 2019.
2 This failure is structurally parallel to the Independence of Irrelevant Edges (IIE) condition defined in Patty and Penn (2015) for centrality indices, and to Arrow’s Independence of Irrelevant Alternatives in social choice theory. IIE asks whether the relative ranking of nodes \(i\) and \(j\) can be computed without reference to edges not incident to either. Arrow’s IIA asks whether the social ranking of \(x\) versus \(y\) can be computed without reference to where \(z\) sits in individual ballots. NEIIA asks whether the apparent spatial relationship between \(x\) and \(y\) can be computed without reference to the rest of the graph. All three are conditions of local computability, and all three are violated by the natural holistic procedures in their respective domains.
3 Padgett, John F. and Christopher K. Ansell. “Robust Action and the Rise of the Medici, 1400–1434.” American Journal of Sociology 98(6): 1259–1319, 1993. The dataset used here is the standard 16-family, 20-edge marriage network distributed with R’s statnet and igraph packages. Note that this is the marriage network only; Padgett and Ansell also analyzed business, patronage, and friendship networks separately. The Pazzi’s single marriage tie is to Salviati; there is no direct Pazzi–Medici marriage edge in the data, a fact with some historical resonance given subsequent events.
4 The foundational treatment of latent space models for networks is Hoff, Peter D., Adrian E. Raftery, and Mark S. Handcock. “Latent Space Approaches to Social Network Analysis.” Journal of the American Statistical Association 97(460): 1090–1098, 2002. These models assume that tie probability is a decreasing function of distance in an unobserved latent space, producing positions that are genuine statistical estimates rather than fixed points of an aesthetic energy function. The positions remain unidentified up to rotation, reflection, and translation — only relative distances carry information — but uncertainty is properly quantified. The gap between what force-directed layouts are and what latent space models are is almost never acknowledged when researchers present network pictures.
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