0. Introduction

Lauren Ross (2021) argues that topological explanation and causal explanation are not exclusive, and that some explanations can be both. Ross suggests a general three-part framework for scientific explanations, including (1) explanandum, (2) explanans, and the (3) dependency relation between the explanandum and the explanans (Woodward, 2003; Reutlinger, 2016; Ross, 2021). Depending on whether the dependency relation is an empirical or mathematical one, topological explanations can be either causal or non-causal, respectively. I end somewhat dissatisfied with Ross, but suggest a positive direction.

Here is the roadmap: first, I will give a standard summary of topological explanations.1 Next, I will introduce a contested case of causal topology: the bow-tie structure of immune biological pathways. After discussing Ross' view about why this is a causal topology, I turn to the virtues and vices of her account. I ask "what is really doing the work in causal topological cases," and "what is the dependency relation between the explanandum and the explanans." Finally, I turn to a discussion about what role graphs and network representations play for Ross (2021), contrasting this with Bechtel (2019).

1. Topological Explanations

Recently, philosophers of science have focused their attention on topological explanations (e.g., Huneman, 2010) in contrast to causal (e.g., Woodward, 2003) or mechanistic explanations (e.g., Craver, 2007). Topological explanations appeal to topological properties of a system, thought to be mathematical or formal properties of a system.

For an example of a topological explanation, consider the classic network theory problem, the seven bridges of Konigsberg (brought to philosophical relevance by Pincock, 2007). The question is whether one can visit each island and bank by crossing each bridge once, and only once (n.d.). Euler provided a topological explanation as to why this is not possible. First, represent the banks and islands as nodes connected by edges representing each of the seven bridges. Then, consider the degree of each node: the number of edges connected to a node. Euler proved that in order to visit each node in a system once and only once, (a) every node in the system must be connected to each other, and (b) that either zero or two nodes must have an odd degree (Euler, 1956; Ross, 2021). The Konigsberg system fails both (a) and (b), and thus cannot be traversed in an "Eulerian path" (Euler, 1956).

[Figure 1]

To characterize topological explanation, we see that the explanandum—the fact that you can't cross each bridge exactly once—follows by deduction from the explanans, the system's topological properties not fulfilling conditions (a) and (b) (Huneman 2010). This is similar to a covering law explanation, though not shown by deduction through natural laws. Rather, it is a necessary consequence of the mathematical topological properties. This is thought to be a non-causal explanation as it abstracts away from the causal information of the system. The causal happenings of the bridges, islands, and banks realize the topological structure of nodes and edges; many real-world systems might have the same topology and thus not be Eulerian traversable. Furthermore, these topological properties are abstract, mathematical structures and therefore non-temporal (Huneman 2010).2 Considering that temporality is necessary for causality, the consensus is that these topological explanations are non-causal (Ross 2021).

Topological explanations have gained traction in fields such as biology, neuroscience, and ecology. A prime example are findings of many small-world networks in systems of interest (Bechtel, 2019). Small-world networks are networks in which most nodes are not connected to one another, but most nodes are connected by a small number of edges.3 Being small-world is an abstract mathematical property of a system, and systems with this property have proved less susceptible to random node deletion than other network structures. Neural networks of C. elegans, for example, are small-world networks (Watts and Strogratz, 1998), and some have suggested it explains why brain functions are robust despite random cell death (Behrens and Sporns, 2011; Craver, 2016).

However, while we may grant that some scientific explanations are genuinely non-causal, scientists use increasingly complex networks to explain biological systems (Fornito et al., 2016). As Ross (2021) argues, while others have assumed a clean division between causal and topological explanations, some networks can encode causal information, thereby blurring the line between the two. One borderline case is the bow-tie structure of the human immune system.

2. Bow Ties and CD4+ T Cells

Human immunodeficiency virus (HIV) binds to CD4+ T-lymphocyte immune cells, thereby depleting the body of CD4+ T cells and leading to immune system failure (Jones, 2014). To explain the vulnerability of the human immune system to HIV, Kitano and Oda (2006) appeal to the structure of the human immune system's biological pathway (Jones, 2014; Kitano and Oda, 2006). They characterize this biological pathway as a directed network that displays the functional relationships between cells in the immune system (Jones, 2014). This reveals a bow-tie structure, in which the components of the system converge on the CD4+ T cell and then "fan out" (Ross, 2021; see fig. 2).

[Figure 2]

The CD4+ T-cell rests at the center of the bow-tie, and, if disrupted, disrupts the 'right-hand' side of the bow-tie. This is because the biological pathway passes through the CD4+ T cell. Therefore, intervention by the HIV virus thus disrupts the pathway. If disrupted, the signals from the 'left-hand' side of the bow tie do not propagate to the 'right-hand' side. The explanation of this system's fragility is explained by the bow-tie structure, specifically the "location of the CD4+ T cells in this structure" (Jones 2014: 1139). If the CD4+ T cell were located at another point in the structure, or the system had another topology, then it might not be as fragile to HIV binding (Jones, 2014; Ross, 2021).

Jones takes this as a non-causal topological explanation (Jones, 2014). His argument is as follows: (1) The biological pathway realizes a bowtie structure, which is identified as topological. (2) CD4+ T cells are located at the core and (3) are non-redundant components of the core; furthermore, (4) all bow-tie structures are susceptible to the deletion of non-redundant nodes at the core (Jones, 2014). Therefore, by appealing to topological properties, the immune system is vulnerable to disruptions of non-redundant core components, such as HIV's targeting of CD4+ T cells (Jones, 2014). This is by deduction from topological properties, much like a covering-law explanation (Huneman, 2018).

However, it is apparent that Jones' (2014) and Kitano and Oda's (2006) network representation of the pathway involves more than mere nodes and edges. It also involves directed edges, as indicated by the "arrows" or "arcs" between components. Graphs with directed edges, like the bow-tie graph, are called directed graphs. Ross (2021) argues that directed edges are often explicitly taken to encode causal information (see Palumbo et al. 2006). In the case of the bow tie structure, the directed arrows represent "interactions" between cells which "activate" CD4+ T cells, which in turn "activate" various downstream effects, something that Jones explicitly admits (Jones, 2014: 1139). So, Ross takes issue with (1): that this bowtie structure is best understood as non-causal.

3. Ross' Causal Topological Explanations

Ross' argument and account of causal versus non-causal topological explanation requires that the network representations represent causal information (Ross, 2021). In this case, specifically, the directed edges represent causal interactions. Granting this, Ross suggests that bow-tie structures are causal topologies or "causal patterns" (Ross, 2021). The possibility of this depends on our understanding of abstraction. While we might abstract away from the bridges of Konigsberg a purely mathematical network, in other cases, we might not abstract away the entirety of the system's causes (Ross, 2021).4 We can abstract away to multiply-realizable causal patterns, examples of which include linear, branching, and cyclic pathways (Ross, 2021). These patterns, such as bow-ties, commonly occur in biology and other systems, such as freight yards (Niss et al., 2018; Tieri et al., 2010). A helpful comparison is between these networks and "wiring topology." Wiring diagrams are diagrams of electric circuits, capturing how electricity travels through them (Tun et al. 2006, p. 5; see Figure 3).

[Figure 3]

Recalling that topological explanations are explanations that appeal to topological properties of a system, and granting that there are causal topologies, what does a causal topological explanation look like? Here, Ross gives a general three-part framework for scientific explanations, including (1) explanandum, (2) explanans, and the (3) dependency relation between the explanandum and the explanans (Woodward, 2003; Reutlinger, 2016; Ross, 2021). As dependency relations can be empirical or mathematical, and this marks the distinction between causal and non-causal explanations, topological explanations with empirical dependence relations are causal topological explanations.5

It is good to clarify what a "dependency relation" is. Dependency relations are those that show how the explanans "makes a difference" to the explanandum (Ross, 2021). We can set a mathematical dependency in contrast with the empirical dependency involved in causal explanations (Woodward, 2003). In the interventionist framework, Y is caused by X if an ideal intervention on X, holding background conditions B fixed, would lead to changes in Y, and this relationship is invariant across a range of such interventions (Woodward, 2003). For example, in establishing that a virus causes the common cold, we hold all other variables fixed and intervene on the virus to see if it produces changes in symptoms of the common cold. Seeing that "wiggling" one variable induces a change in the other, under ideal circumstances, establishes cause. But importantly, we may ask how the cold depends on the virus. In causal cases, this is not established mathematically but empirically, through investigation of the world (Ross, 2021).

Applying this three-component framework to the bridges of Konigsberg, the explanandum is set by our question: "Can you visit each island and bank by crossing each bridge once, and only once?" This is whether there is an Eulerian path. The explanans involves topological properties of the system, such as node degree and conditions (a) and (b). The dependency relation is mathematical: the explanandum follows mathematically from the explanans once we see the system fails (a) and (b), demonstrating the impossibility of an Eulerian path for any such system. The availability of an Eulerian path depends mathematically on the topological properties of the network without empirical investigation (Ross, 2021).

This is in contrast to the bow-tie: an example of causal topological explanation. Granting that the bow-tie encodes causal information, this involves causal topological properties in the explanans. However, how does this relate to the explanandum? Unfortunately, Ross leaves out an important—perhaps the most important—piece of this story. To complete her account, the dependency relation between the causal topology of the bow-tie must relate empirically to the "higher-level explanatory target of interest," or the structure realized by the biological pathway (Ross, 2021: 9817). However, she does not flesh out how this relation is empirical rather than mathematical. We only have "good reason" to suppose that this relation is causal (Ross, 2021).

To sum up: explanations are topological when the explanans include topological properties. And the dependency relation between explanans and explanandum is either empirical or mathematical (Woodward, 2003; Reutlinger, 2016), which allows for causal topological explanations.

It is good to step back and appreciate the virtues of Ross' account. First, while we may be tempted to compare topological explanations to covering law explanations (Hempel, 1965), Ross accommodates topological explanations under her three-part account. And this is in wide agreement with Huneman and Jones in non-causal topological explanations: all parties are committed to topological properties involved in the explanans from which the explanandum is derived.

Furthermore, Ross maintains a pluralism about the kinds of explanations involved in science. Some mechanists, such as Craver (2007), argue that topological explanations do not constitute a bona fide explanation, and that mechanisms ultimately do the explanatory work. But Ross' three-component account of explanation, at least prima facie, proffers a monism for the structure of scientific explanations (see Ruetlinger, 2016). But, as no specific kind of explanation has priority over another, this accommodates an explanatory pluralism.

Now, let us turn to vices. A notable gap in this essay is that the notion of a "causal topology" is not strongly defended. Ross really supports her claim that there can be "causal topologies" by citing scientists who discuss the notion. Furthermore, if we grant her this notion, then it seems obvious that the distinction between causal and topological explanation is blurred. But causal topology might be considered a strange thing. For example, if it is both causal and abstract, how might we intervene upon it? Woodward's (2003) interventionist framework concerns interactions between local variables (Xs and Ys). Therefore, there is no room for intervening on topologies or structures. So, Ross owes us an explanation.

However, causal topology might have some intuitive appeal. Therefore, I would like to move on to some other objections. But first, I would like to discuss more about what we consider topological properties.

4. What Are Topological Properties?

I think that more turns on this than is explicit in Ross' discussion. This being said, Ross, Jones, and Huneman are all in agreement about the extension of "topological properties." This is what adds force to Ross' argument that there are causal topological explanations, as bow-ties are topological structures that figure in the explanans for all parties. But thus far, we have been working with a lax notion of "topological properties," where the "topology of a graph defines how the links between system elements are organized" (Fornito et al., 2016: 6). We may, however, desire some more nuance about what counts as a topological property.

Interestingly, putting their quote in context, the graphs that Fornito et al. refer to are classic examples of networks, explicitly Euler's discussion of the bridges of Konigsberg, which deal with nodes, edges, paths, and degrees. But it is important to realize that directed graph theory is an extension of graph theory, although still in the family of graph theory (Fornito et al. 2016). So what justifies this extension?

The simple and intuitive answer is that directed edges can be mathematically formalized. And it is (partly) in virtue of this that they fall under the domain of "topological." If they weren't, there would be no reason they would fall under the extension. So, I suggest this as a relevant necessary property of topological properties.

To be perfectly explicit about the mathematics and formalisms involved, we may introduce a directed graph formally as such:

A directed graph is an ordered pair G = (V, A) where:

With this formulation out of the way, we can also formalize a particularly important notion: a path. A path is a sequence of nodes connected by directed edges. For example, in Figure 4, the path from (b) to (d) may be represented in sentential graph theory notation as such:

(b, c), (c, a), (a, d).

Each ordered pair is a directed edge between a node that we can "hop" between (Jones, 2014). Explicitly, the directed edges are formal asymmetries. This contrasts with the Konigsberg case, where the non-directed edges are symmetric, thus making it an extension of standard graph theory.

As shown in Figure 4, there are several ways to represent directed graphs and directed edges. For example, we can represent this information mathematically in an incidence matrix. This is to further show that the graph encodes mathematical information.

[Figure 4]

There is nothing about these formal asymmetries that requires us to interpret it causally. This is, of course, a point that Ross acknowledges. For example, directed arrows could represent priority relations or hierarchies (Ross, 2021). This is not to suggest that directed arrows and graphs are not often used in causal analysis (Glymour, 1998). The question at hand is what role these formal properties play in explanation. And now with the notions of asymmetry and paths in our hands, I will turn again to Ross' discussion of bow-ties.

5. What Is Really Doing the Explanatory Work?

Returning to the bow-ties, Ross asks, "How do we know that the causal information encoded in the graph is really doing the explanatory work?" (Ross, 2021: 9810). She argues that without the causal information, this cannot be so. I will argue in this section that Ross' argument is not convincing.

For her argument, Ross has us consider removing all causal information from the bow-tie topology by replacing its directed edges with non-directed ones. She argues that this non-causal topology is insufficient to explain the fragility of the biological pathway, as the degree of the central node alone cannot account for the system's fragility. Consider a node downstream or upstream of the node that has a higher degree than the central node (a node other than the CD4+ T-cell). That node's high degree does not make it the system's weak point. Rather, what explains the system's fragility is not the central node's degree but its location in the directed network. It is situated between all incoming signals and all outgoing effects, functioning as a bottleneck (Ross, 2021). And identifying something as a bottleneck in this sense, Ross argues, requires that we know the direction of the signal, downstream to upstream. Therefore, the non-causal non-directed topology is insufficient, and the causal information encoded in the directed edges is doing genuine explanatory work. She ends by saying, "to the extent that these bow tie explanations depend on identifying causal bottlenecks, they cannot be provided with network topologies that abstract from causal information" (Ross, 2021: 9811).

Ross' argument here is odd. Removing the causal information from the graph does require replacing directed edges with non-directed ones. Yet, this simultaneously removes the relevant formal asymmetries from the topology. Ross equates removing causal information with removing non-directed edges, but she herself admits that we may have directed edges without causal information (Ross, 2021). Thus, her objection should raise some skepticism. The asymmetries of the directed graph could be sufficient in absence of causal information.

There's a burden here to show that there is a non-causal topological explanation available. And as Jones (2014) suggests, there is plausibly a non-causal explanation. We may reconstruct from Kitano and Oda (2006) a deductive argument where formal notions of node redundancy are defined by paths between nodes, and a notion of fragility or vulnerability where if there is no path from initial nodes to final nodes in the network were a node missing, then it is fragile (Jones, 2014).6 This is, of course, just a rough sketch of what Jones outlines, but it is more than enough to raise skepticism about Ross' point. Notably, the formal notion of a path, which depends on the aforementioned formal asymmetries, is involved in this non-causal topological explanation. And if this is the case, she has not established whether the causal information encoded in the graph is really doing the explanatory work.

So, Ross' argument is ineffective as it cannot adjudicate whether causal or mathematical topological properties are doing the explanatory work. Replacing directed edges with non-directed ones also removes relevant formal asymmetries.

6. What Is the Dependency Relation in Topological Cases?

Ross hedges at the end when tasked with "considering the dependency relation in topological cases." To complete her account, the dependency relation between the causal topology of the bow-tie must relate empirically to the "higher-level explanatory target of interest," or the structure realized by the biological pathway (Ross, 2021: 9817). Establishing this is extraordinarily important, so it's a shame that we are left with just "good reason" to think the dependency relation is empirical (Ross, 2021: 9817).

Ross says, "the main reason for this is that this relation [between causal topology and the higher-level explanatory target of interest] involves a significant amount of empirical information" (Ross, 2021: 9817), and that "causal topological structures that figure in the explanans contain significant empirical information—each causal link specifies information about a different-making relationship between properties and the world" (Ross, 2021: 9817). However, these are two distinct claims: the first concerns the relationship between the explanans and the explanandum, while the other concerns the information encoded in a directed graph.

But, while we may grant that causal information is encoded in the explanans, this itself is not sufficient to show that the dependency relation is empirical. A straightforward example of this is a covering law explanation.7 Premises in a deductive argument can encode causal information, yet the deduction from those premises is not an empirical relation. The reason is obvious: the causal information in the premises is simply not relevant to the formal deduction (the formal deduction depends on syntax, not the content). This suggests a prima facie necessary condition for the empirical dependence relation: that the encoded causal information is relevant to the explanation.

However, spelling out explanatory relevance is tricky business. Prima facie, relevance is a graded notion (it admits of degrees), and what is relevant depends on a case-by-case basis (perhaps best understood as depending on particular explanatory-why questions, or depending on certain contexts). Furthermore, as I argued in the last section, Ross struggles to determine whether causal or non-causal properties are doing the explanatory work in biological pathways with bow-tie topologies. And if something doesn't do explanatory work, how is it explanatorily relevant?

To anticipate objections, Ross can plausibly respond that while we can reconstruct a non-causal topological story, this does not adequately characterize scientific reasoning and imagination. Biologists frequently use causal topologies to proffer explicitly causal arguments. And after all, Jones only provides a reconstruction of a possible deductive argument that Oda and Kitano could make (Jones, 2014). This is a well-taken objection. But Ross still has not provided an account of what that looks like. In the next section, I would like to elaborate on what Ross thinks is going on when we use directed graphs in reasoning.

7. How Do Graphs and Representations Play a Role in Scientific Reasoning?

In her discussion of the dependency relation, Ross suggests a reason why there's an empirical relation between topological structure and the explanatory outcome. Specifically, that "in considering different causal connections, we imagine how these differences matter for the causal processes that propagate along these connections, and, ultimately, for the explanatory target of interest" (Ross, 2021: 9817). In this, imagination plays a role in how these causal patterns relate to the explanandum, and she is broadly discussing how we reason with causal topologies (Ross, 2021). In this section, I suggest that providing a story about how causal topological structure interacts with other accounts (e.g., mechanistic or non-causal topological) is a path forward.

First: what exact role does a directed network graph play in imagination and reasoning about causation? Consider the following:

Assume we can think about abstract causal patterns non-topologically. So, as I write, I can imagine abstract causal structures and what they are instantiated in. For instance, I can think about how the causal chains in freight yards might branch and converge (Tieri et al., 2010). I can imagine the possible empirical consequences of branching and converging, and check whether they obtain. But to aid in my reasoning, I could also turn to my whiteboard and draw it. But, at what point did my imagining about causal patterns become topological?

There are two plausible ways to respond to the previous paragraph. (1) Perhaps employing a graph didn't merely lessen my cognitive load. Instead, my reasoning with a graph played a distinctive role by supplying topological information not already present in my thinking. Alternatively, (2) reasoning about abstract causal patterns has always been topological.

Let's start with (2). Let's suppose that my reasoning about abstract causal patterns has always been topological. Perhaps I have a mental representation—in this case, a mental image—of a system that encodes a causal topology from which I can draw certain inferences. But if this is the case, there is nothing distinctive about employing directed graph representations. They merely play a role to lessen or optimize cognitive work that could already have been done.

Ross settles for (2). For example, she holds that many abstract causal structures (such as branches, cycles, and lines) are causal topological (Ross, 2021). If this is the case, she is discussing a rather broad category of how we reason. If this is the case, I think that I'm warranted in asking about how to characterize scientific reasoning and imagination. This is to push her for more.

But is it fair to ask this of Ross? Insofar as she just shows the possibility of causal topological explanation, this may be uncharitable to raise. For instance, why couldn't I raise the same issue for mechanistic or non-causal topological information in various representations? Certainly, I can make mental representations and reason mechanistically.

However, contrast (2) with Bechtel (2019). He discusses how network representations can be used to generate mechanistic insight. Therefore, he is highlighting how graphs and representations may play a distinctive role in epistemic practices, or (1). In his example, high-throughput data is used to generate large network representations. The nodes represent genes, and the edges represent epistatic interactions between them. But in order to glean any mechanistic insight from these networks, they must be interpreted. Researchers reinterpret the initial messy "hairball" representation by applying clustering algorithms to identify highly connected nodes and reorganizing the nodes (Bechtel 2019). Then, they annotate the representation with hierarchical annotations (such as the Gene Ontology), which are well-suited to identify latent mechanisms and sub-mechanisms. This highlights a collaborative interaction between network and mechanistic accounts, which appears as a happy pluralism.8

Given her aforementioned pluralist commitments, how do causal topologies interact with other accounts? Bechtel tells an interesting story about network representations and their mechanistic interpretation to support valuable collaborations between the two. Ross lacks an analogous story. Showing how causal topologies might collaborate with others would certainly help her case. The successful demonstration of this would lend credence to the existence of "causal topologies." So, I offer this as a positive suggestion and path forward.

8. Conclusion

There is a fair deal to be dissatisfied with Ross. "How do we make sense of 'causal topology?'", "What is the dependence relation?" and "What is really doing the explanatory work?" She leaves many gaps and much unsaid. On one hand, she may leave room for a "promising area of future work." To that extent, I offered a positive suggestion about next steps. On the other hand, we might worry that there's no strong motivation to blur the line between causal and topological explanation.

References

Bechtel, W. (2020). Hierarchy and levels: Analysing networks to study mechanisms in molecular biology. Philosophical Transactions of the Royal Society B: Biological Sciences, 375(1796), 20190320. https://doi.org/10.1098/rstb.2019.0320

Behrens, T. E. J., & Sporns, O. (2011). Human connectomics. Current Opinion in Neurobiology, 22(1), 1–10.

Burnston, D. C. (2025). The Bechtel approach to the scope and limits of mechanistic explanation. Biological Theory, 1–13.

Craver, C. F. (2007). Explaining the brain: Mechanisms and the mosaic unity of neuroscience. Clarendon Press.

Craver, C. F. (2016). The explanatory power of network models. Philosophy of Science, 83(5), 698–709. https://doi.org/10.1086/687856

Duke University. (n.d.). Section III.1 [PDF]. Retrieved May 2, 2026, from https://courses.cs.duke.edu/fall06/cps296.1/Lectures/sec-III-1.pdf

Euler, L. (1956). The seven bridges of Königsberg. In J. R. Newman (Ed.), The world of mathematics (Vol. 1). Simon and Schuster.

Flagg, L. A., West, M., Stringer, K. L., & Borch, C. (2022). Connectedness (mathematics). In Research starters. EBSCO. https://www.ebsco.com/research-starters/mathematics/connectedness-mathematics

Fornito, A., Zalesky, A., & Bullmore, E. (2016). Fundamentals of brain network analysis. Academic Press.

Glymour, C. (1998). Learning causes: Psychological explanations of causal explanation. Minds and Machines, 8(1), 39–60.

Hempel, C. (1965). Aspects of scientific explanation and other essays in the philosophy of science. The Free Press.

Huneman, P. (2018). Diversifying the picture of explanations in biological sciences: Ways of combining topology with mechanisms. Synthese, 195(1), 115–146.

Jones, N. (2014). Bowtie structures, pathway diagrams, and topological explanation. Erkenntnis, 79(5), 1135–1155.

Kitano, H., & Oda, K. (2006). Robustness trade-offs and host-microbial symbiosis in the immune system. Molecular Systems Biology, 2, 1–10.

Meghanathan, N. (2015). Small-world networks (CSC 641, Module 6) [Lecture slides]. Jackson State University. https://www.jsums.edu/nmeghanathan/files/2015/08/CSC641-Fall2015-Module-6-Small-World-Networks-reduced.pdf

Niss, K., Hu, J. X., Gomez-Casado, C., Joeris, T., Agace, W. W., Belling, K. G., et al. (2018). Bow-tie motifs enable protein multifunctionality by connecting cellular processes in the interactome. bioRxiv. https://doi.org/10.1101/495895

NRICH. (n.d.). The bridges of Konigsberg. Retrieved May 2, 2026, from https://nrich.maths.org/problems/bridges-konigsberg

Palumbo, M. C., Farina, L., Colosimo, A., Giuliani, A., Tun, K., & Dhar, P. K. (2006). Networks everywhere? Some general implications of an emergent metaphor. Current Bioinformatics, 1(2), 219–234.

Pincock, C. (2007). A role for mathematics in the physical sciences. Noûs, 42(2), 253–275.

Putnam, H. (1975). Philosophy and our mental life. In Mind, language and reality: Philosophical papers. Cambridge University Press.

Reutlinger, A., & Andersen, H. (2016). Abstract versus causal explanations? International Studies in the Philosophy of Science, 30(2), 129–146.

Reutlinger, A. (2018). Extending the counterfactual theory of explanation. In A. Reutlinger & J. Saatsi (Eds.), Explanation beyond causation: Philosophical perspectives on non-causal explanations (pp. 74–95). Oxford University Press.

Ross, L. N. (2021). Distinguishing topological and causal explanation. Synthese, 198(10), 9803–9820.

Salmon, W. (1984). Scientific explanation and the causal structure of the world. Princeton University Press.

Tieri, P., Grignolio, A., Zaikin, A., Mishto, M., Franceschi, C., Remondini, D., et al. (2010). Network, degeneracy and bow tie: Integrating paradigms and architectures to grasp the complexity of the immune system. Theoretical Biology and Medical Modelling, 7, 1–16.

University of Waterloo. (n.d.). What is topology? Retrieved May 2, 2026, from https://uwaterloo.ca/pure-mathematics/about-pure-math/what-is-pure-math/what-is-topology

Watts, D., & Strogatz, S. (1998). Collective dynamics of small-world networks. Nature, 393, 440–442. https://doi.org/10.1038/30918

Woodward, J. (2003). Making things happen: A theory of causal explanation. Oxford University Press.

Notes

1 An important note: the mathematical study of topology is the study of spatial properties preserved under continuous deformation, such as stretching, bending, and reflecting (and, importantly, not deformations like tearing or gluing). This is to explore the mathematics of shape. We may apply topology to graphs, and we treat graphs as a simplicial complex, where nodes are treated as 0-simplices and edges are 1-simplices. By treating graphs as such, this allows us to use topology to explore graphs. However, this is not the sense of topology used by philosophers in this debate. Really, they are discussing graph theory, but I opt to use the already existing language in this debate.

2 See Craver (2016) for dissent that these are explanations without any "ontic commitments" (Craver, 2016: 701).

3 In more detail, small-world networks are networks where the mean distance between any two random nodes "grows proportionally to the logarithm of the number of nodes in the network" (Meghanathan, 2015).

4 We should note some tension between explanations being abstract, but nonetheless providing a causal explanation. Reutlinger and Anderson argue that the received view is that explanations are non-causal in virtue of being abstract, yet abstractness is not sufficient for an explanation being non-causal (Reutlinger and Anderson, 2016). They argue that abstract explanations do identity causes in many theories of causation, and that many abstractions are explanatory in many causal theories of explanations.

5 It is important to note that the correspondence between mathematical/empirical dependency relations and non-causal/causal explanations is not defended in Ross (2021). But it is an open possibility and taken quite seriously (Woodward 2003; Jansson and Saatsi 2017; Reutlinger 2016).

6 Jones does provide a plausible reconstruction of the argument, even if it doesn't track with the way Oda and Kitano used their graph of the biological pathway (Jones, 2014).

7 To note: Hempel (1965) required that premises in a covering law explanation needed empirical content. Empirical content is not causal, but premises can include causal content. This can be trivial, like in the sense a sentence is about a cause.

8 This might sound like an odd suggestion, given that Bechtel is often interpreted as a "mechanistic imperialist" or "mechanista" (Mitchell, 2026). For a dissenting opinion that Bechtel is best interpreted as an "integrative pluralist," see Burnston (2025). I, at least, did feel Bechtel aimed to show how "different kinds of explanatory models… mutually contribute to explanation" (Burnston, 2025: 11).