Consider an electrical circuit: a network made of wires joining resistors in parallel and in sequence, with some portion hidden inside an opaque box. It is not always possible to determine that portion by testing the visible leads. However, we prove here that if the hidden portion has a particular form made of connected cycles, and we can test the resistance between all the pairs of leads, then the lengths and connected structure of the cycles in the circuit are uniquely determined. The mathematics used to recover that circuit is more typically found in work on phylogenetic networks.

Modeling heredity as the flow of genetic information suggests that mutations in DNA might be analogous to resistance in an electrical circuit. The weights of edges in a phylogenetic network can represent genetic distances: if we have the genomes of the two endpoints of an edge then we can use a model of mutation rates to calculate a real number distance. For several edges that form the unique path between two taxon-labeled leaves, the total distance is the sum of those edge weights. Paths between leaves are only unique if the network is a tree. When paths between are not unique, one option is to take the distance to be that of the minimum length path. This option may correspond to a parsimonious approach—assuming the least complicated history. This minimum path length distance is studied for instance in Forcey and Scalzo (2020a).

Instead, however, a greater weight of an edge could represent a greater loss of information. Dividing and rejoining of edges illustrates events such as speciation, recombination, or hybridization. If the genetic information of an ancestor genome can be shared among descendants, and then collaboratively recovered upon hybridization, then a different metric than minimum distance may be appropriate. Here we consider weighted phylogenetic networks with the resistance distance, or resistance metric. The distance between two leaves of the network is found by considering the edge weights as electrical resistance, obeying Ohm's law. The metric resistance distance for all nodes (not only leaves) of a graph is introduced in Klein and Randić (1993), and studied closely in subsequent papers such as Yang and Klein (2015) and Yang and Klein (2019). To study graphs, the resistance of each edge is often assumed to have unit value, but the definitions allow any weight. We review the definitions in section 2.2.3.

In Curtis et al. (1998) and Curtis and Morrow (1990, 1991), the authors study circular planar graphs with boundary nodes that are analogous to the leaves of our phylogenetic networks. They consider resistance values (or conductivity) on the edges. They prove that complete information about the linear map which transforms electric current values at each boundary node to electric current values at all the edges can be used in some cases to recover the resistance values. In our applications there is no way to know the complete map of boundary currents to edge currents. However, we seek only to recover the graphical structure of the network, not the original edge weights.

In Ejov et al. (2019), the authors consider the entire set of resistance distances (again using unit values for edges), between any pair of nodes (not only leaves.) They show that using this metric is useful for discovering Hamiltonian cycles via algorithms for the Traveling Salesman problem. There is a close connection to our applications, since the algorithm neighbor-net can be used as a greedy approach to the Traveling Salesman problem as shown in Levy and Pachter (2011).

We start by reviewing some equations from electric circuit theory.

The main result in this section is that the resistance metric is Kalmanson for 1-nested phylogenetic networks, and that the unique associated split network has the same exterior form as the original 1-nested phylogenetic network. First we show that dNR obeys the Kalmanson condition: there exists a circular ordering of [n] such that for all i < j < k < l in that ordering,

Theorem 3.1. Given a 1-nested phylogenetic network N with positive weighted edges and n leaves, the resistance metric on its leaves is Kalmanson.

Proof: The cyclic order that we need to exist in order to demonstrate the Kalmanson property is found by choosing any cyclic order of [n] consistent with N. That is, we choose an outer planar drawing of N and use the induced cyclic order of the leaves arranged around the exterior of that drawing.

Begin by noting that for each pair of the four leaves i, j, k, l there is a sub-graph, called the pairwise circuit, for instance P_ik, made of all the edges which are part of any path between those two leaves. The pairwise circuit will contain perhaps some cycles—it will in fact be a series of cycles connected by paths. We are especially interested in the intersection I of the two “crossing” pair circuits, I = P_ik ∩ P_jl. There are three basic cases to consider.

Case 1: The intersection I is a single cycle. Here the four leaves i, j, k, l have pairwise circuits that reach the cycle I at four different nodes. Notice that any of the two pairwise circuits summed in the Kalmanson condition will include all four of the smaller pairwise circuits from each of the four leaves i, j, k, l to the node of I closest to that respective leaf. We will call those closest nodes v_i, v_j, v_k, v_l. The three sums in the Kalmanson condition all share some terms in common: those which come from the weighted edges in pairwise paths between the four leaves and the respective nodes v_i, v_j, v_k, v_l. Discarding these common terms, we are left with terms that come from the weighted edges in I. Thus, the only differences between the three sums in the Kalmanson condition arise from the different contributions of the cycle I. We denote by a, b, c, d the cumulative edge weights between the four nodes v_i, v_j, v_k, v_l, following the cyclic order. For instance, in Figure 11, a is the weight of the edge between v_l and v_i and b is the sum of the weights on edges of I between the nodes v_i and v_j.

Thus, we can write the sums explicitly:

After discarding the common terms, we consider just the remaining sums of fractions. All the edge weights are positive, and the denominators of all three are the same. Clearly the third sum, when expanded, has a numerator larger than either of the first two.

Case 2: The intersection I is a series of cycles containing at least two cycles. In this case there are two possible ways that the inequalities are satisfied, depending on which pair of consecutive leaves (i, j or j, k) reach the same end of I, that is, have their attaching nodes (v_i, v_j or v_k) in I at the same end of I. In Figure 12, below we choose i, j to do so, on the left-hand cycle, but the other option is similar. Checking this case can be done visually for the equality: dNR(i,k)+dNR(j,l)=dNR(i,l)+dNR(j,k) since the two sums end up using precisely the same effective resistances. That is, both dNR(i,k)+dNR(j,l) and dNR(i,l)+dNR(j,k) have all terms in common: both the portions from the paths outside of I as in case 1, and the summands contributed by I, which are the terms:

The inequality dNR(i,k)+dNR(j,l)>dNR(i,j)+dNR(k,l) (for the subcase where again i, j reach the same end of I) is easily checked. Here, after discarding the terms in common, the larger sum contains more terms than the smaller (from the parts of I not in the pairwise circuits for i, j and k, l). As well, when the smaller sum has terms with denominator matching a term in the larger, the numerator is indeed larger in the latter. For instance, in Figure 12, after discarding the common terms contributed by the paths outside of I, the sum dNR(i,j)+dNR(k,l) has the sum contributed by I:

The numerator here is exceeded by the sum contributed by I in dNR(i,k)+dNR(j,l) as just listed above. Finally, notice that there are sub-cases of Case 2 in which the smaller sum will have fewer or no terms at all contributed by I; these occur when I includes a path at one end or at both ends. See Figure 13 for example.

Case 3: The intersection I is a path. In this case it is quickly verified that the Kalmanson inequality is satisfied as an equality. See Figure 14 for example.

The fact that effective resistance distance is a Kalmanson metric immediately suggests that it would be a good candidate for modeling weighted phylogenetic networks. First there is the intuition from experience that if two pathways of heredity exist, the ancestor individual or species will have more in common with the extant individual or species. Thus, mutations in the genetic code play the role of resistors to the flow of information.

Secondly, Kalmanson metrics are known to be the only example for which each metric is represented uniquely by a circular split system, as seen in Lemma 2.15. In the case of the resistance distance, the associated unique split network has an additional advantage: it is guaranteed to represent faithfully every split displayed by the original 1-nested network.

Theorem 3.2. Given a 1-nested phylogenetic network N with positive weighted edges and n leaves, and letting dNR be the resistance metric on the n leaves, then the unique associated split network Rw(N)=N(dNR) displays precisely the same splits as displayed by N.

Proof: A split A|B can be displayed by N in three possible ways: either it is displayed by a single bridge e with weight w(e), by a pair of edges both in the same cycle c with respective weights a_c and x_c, or in more than one way. Let the weight of a specific display of a split in N be w(e) in the first case and (a_cx_c)/z_c in the second case, where z_c is the sum of all the weights in the cycle. We claim: if the split A|B in Σ(N¯) is assigned the sum of the weights of all distinct displays of that split as displayed in N, then the resulting distance metric d from the weighted split network thus constructed is indeed dNR. Therefore, we will conclude, since Theorem 2.1 shows that dNR is Kalmanson, that the weighted split network thus constructed is equal to the unique split network corresponding to dNR, as found for instance by the algorithm neighbor-net.

First we check that the claim holds. Consider the pairwise circuit P_ij in N for a given pair i, j of leaves. It will be a series of paths and cycles, as seen for example in Figure 15. Thus each cycle c in P_ij will be split into two circuit-parallel paths p_c and q_c of respective lengths p, q. Both paths begin and end at the two nodes where that cycle is attached to the rest of the series. Now the resistance distance dNR(i,j) will be the sum of the weights of the (non-circuit-parallel) paths, and of the effective resistances of the circuit-parallel paths. Specifically, every weighted edge of P_ij not in a cycle will contribute its weight to the sum, and every weighted edge in a cycle of P_ij will appear in one of two factors in the numerator of the term giving the effective resistance from those circuit-parallel paths. We see that

where c₁, …, c_p and c_p+1, …, c_p+q are the weights of the circuit-parallel paths of cycle c ∈ P_ij, with z_c = c₁ + ⋯ + c_p+q being the total weight of c. That is, we expand the numerator of each term from a cycle. Now, the distance metric corresponding to the weighted split network we constructed using Σ(N¯) has distance

Now splits in N, and thus in Σ(N¯), which separate leaves i, j are precisely those displayed by a bridge in P_ij or by a pair of circuit-parallel edges in a cycle of P_ij. Thus, using the weights for splits (as stated above):

in the split metric, gives us the desired claim: d=dNR.

Then we conclude that since the weighted circular split network associated to the original Kalmanson metric dNR is the unique such network where the split metric equals the original Kalmanson metric, then N(dNR) will have precisely the splits of N and thus of Σ(N¯).

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Remark: The fact that we can take a weighted 1-nested phylogenetic network N and build a weighted circular split network s which has the same metric, ds=dNR, implies another proof that the resistance distance is Kalmanson. Since the circular split network is planar, and the split metric on it is the same as the minimum path network on it, that metric is guaranteed to be Kalmanson. However, our original proof has the advantage that we see which of the inequalities are strict, and which are actually equalities.

The first important implication of these theorems is that the resistance distance on any 1-nested phylogenetic network N is precisely represented by a unique circular split network N(dNR). Exactly all the splits displayed by the original N are present in N(dNR). Thus the function L applied to the unweighted version of N(dNR) returns the unweighted version of N itself.

Theorem 3.3. Given weighted 1-nested N, we have that N(dNR)¯=Σ(N¯). Thus, L(N(dNR))¯=N¯.

Proof: The first equality follows directly from Theorem 3.2, since neighbor-net is guaranteed to output the splits of the unique circular split network associated to the Kalmanson metric given by the resistance distance, which is indeed all the splits displayed by the network N. Then from Forcey and Scalzo (2020a), we have the second equality since L ○ Σ is shown there to be the identity map.

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The first application implied by this result is that when using neighbor-net on a measured distance matrix, if we assume that it reflects a resistance distance, we can always recover the form of the original network. The weights of splits in the result of neighbor net are interesting, they are in fact terms in the expansion of the calculated resistance distance. However, the first advantage we see is that the original unweighted phylogenetic network can be directly recovered by taking the exterior of the result of neighbor-net.

As an alternative to neighbor-net, there are polytopes which can serve as the domain for linear programming that finds the best-fit 1-nested phylogenetic network for a measured distance matrix.

In Durell and Forcey (2020), the authors describe a new family of polytopes. This family lies between the Symmetric Traveling Salesman Polytope [STSP(n)] and the Balanced Minimum Evolution Polytope [BME(n)]. Our polytopes are called the level-1 network polytopes BME(n, k) for 0 ≤ k ≤ n − 3. All have dimension (n2)−n. In Forcey and Scalzo (2020a), we looked at implications of the Galois connections studied there for these polytopes, especially using S_w, the function based on minimum path distance. It turns out that if we assume an input distance metric represents the resistance distance on a 1-nested phylogenetic network N, then the result of neighbor-net or of linear programming on a BME polytope is a network accurately showing all the splits of N. Also, neighbor-net is statistically consistent, as shown in Bryant et al. (2007). Therefore, as a measured set of pairwise distances approach the resistance distance of N, the output of neighbor-net will approach the faithfully phylogenetic circular split network N. This is in contrast to minimum path distance where some genetic connections are assumed to be negligible, and then are lost in the output of neighbor-net. However, the theorems about minimum path distance, specifically Theorems 8, 9, and 11 of Durell and Forcey (2020), play an important role in the proof of Theorem 4.5 here. Here we repeat some of the same introductory definitions and remarks and then extend the results to resistance distance.

Definition 4.1. For a binary, 1-nested phylogenetic network N, (weighted or unweighted) the vector x(N) is defined to have lexicographically ordered components x_ij(N) for each unordered pair of distinct leaves i, j ∈ [n] as follows:

where k is the number of bridges in N and b_ij is the number of bridges traversed in a path from i to j. For example, see Figure 16.

The convex hull of all the x(N) such that binary N has n leaves and k nontrivial bridges is the level-1 network polytope BME(n, k). As shown in Durell and Forcey (2020), the vertices of BME(n, k) are precisely the vectors x(N) for N binary with n leaves and k nontrivial bridges. In light of Theorems 3.1 and 3.2, we can now characterize the vertices in terms of resistance distance:

Theorem 4.2. Every 1-nested phylogenetic network found as an image L[Rw(N)¯] gives rise to a face of BME(n, k) for some k. In particular, the vertices of the polytopes BME(n, k) correspond to images L[Rw(N)¯] which exhibit k non-trivial bridges, for weighted 1-nested networks N with n leaves and such that any node not in a cycle has degree three.

Proof: The image R_w(N) will faithfully represent all splits, as seen in Theorem 3.2. Thus, Rw(N)¯ will be faithfully phylogenetic, in the range of Σ. Specifically, the function R_w will introduce bridges that separate all cycles, thus insuring that any node in a cycle will have degree three. Therefore, if the non-cycle nodes of N are degree three, L[Rw(N)¯] will be a binary unweighted 1-nested phylogenetic network.

Also as shown in Durell and Forcey (2020) and repeated in Forcey and Scalzo (2020a), an equivalent definition of the vector x(N) is the vector sum of the vertices of the STSP(n) which correspond to cyclic orders consistent with N. Recall that the vertices of STSP(n) are the incidence vectors x(c) for each cyclic order c of n, where the i, j component is 1 for i and j adjacent in the order c, 0 otherwise. This equivalent definition for binary 1-nested phylogenetic networks may also be applied to any 1-nested phylogenetic network:

Lemma 4.3. For a 1-nested phylogenetic network N, the vector x(N) is equal to ∑cx(c) where the sum is over all cyclic orders c of [n] consistent with N.

We point out, for the sake of attribution, that for phylogenetic trees t (with nodes of any degree), Lemma 4.3 with N = t gives a formula for x(t) that agrees with the definition of the coefficient n_t in Semple and Steel (2004), in the proof of Theorem 4.2 of that paper.

In Forcey and Scalzo (2020a), it is shown that the minimum path distance vector for a 1-nested phylogenetic network may be seen as a linear functional, and that it is minimized over the BME(n, k) polytope. Specifically,

Theorem 4.4. Given any weighted 1-nested phylogenetic network N with n leaves, the product x(N^)·dN is minimized over BME(n, k) precisely for the unweighted binary 1-nested networks N^ with k bridges such that Sw(N)¯≤Σ(N^).

Here N^ is used to denote a variable binary 1-nested phylogenetic network, taking values from the set of networks which refine Sw(N)¯. By this refinement we mean taking values from the set of networks with a superset of the set of splits displayed by Sw(N)¯. Now we can extend that result to resistance distances. In fact it becomes stronger: binary networks can be directly recovered even when they have long edges, since the action of R_w preserves all splits. Precisely, we have:

Theorem 4.5. The minimum of x(N)·dNR is achieved at the face of BME(n, k) with vertices x(N^), for unweighted binary networks N^ with k bridges such that N^ refines N¯.

Proof: We claim that x(N)·dNR is the same as x(N)·dN′ for N′=Lw(Rw(N)). That is because the leaves which are adjacent in some circular order consistent with N and thus in R_w(N) have distance between them which is the sum of the splits that separate them. Since those leaves are adjacent, the shortest path of splits between them will lie on the exterior of R_w(N). In fact, for adjacent i, j an edge of a cycle on the path between them with weight a, contributes a(b+c+d+...)a+b+c+d+... to dNR(i,j), where the other edges of that cycle have weights b, c, d, .... Bridges e between them contribute their weights w(e). These values are the same as those for the splits displayed between i, j, seen in the proof of Theorem 3.2. Therefore:

We know from Theorems 8, 9, and 11 of Durell and Forcey (2020) that for any weighted 1-nested phylogenetic network M with n leaves, the product x(M^)·dM is minimized over BME(n, k) precisely for binary networks M^ with k bridges such that M¯≤M^.

Thus, in our case we have x(N^)·dN′ is minimized over BME(n, k) precisely for the unweighted binary networks N^ with k bridges such that N′¯≤N^. Here, (N′)¯=N¯, since Lw(Rw(N))¯=N¯. The inequality here is refinement.

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For example compare Figures 10, 16. It is easily checked that although dN′≠dNR, we have x(N)·dN′=x(N)·dNR=51.4

The implication then is that using either linear programming on BME(n, 0) or neighbor-net, assuming that the resistance metric is valid, the resulting split network gives the true exterior form of the original 1-nested phylogenetic network.

In this section, we examine functions between 1-nested and 2-nested networks, and circular split networks. We point out how well the various distance measurement distinguish or do not distinguish between network types, via examples. Then we make some conjectures based on observations.

A question is raised about the mathematics which precedes the work described in this paper: what sort of measurement should actually yield the experimental resistance distances in a real example? What should play the role of attaching the ohmmeter to pairs of wires? Usually, DNA sequences of length m are aligned (a multi-step problem of its own) and then the number of disagreeing sites is counted. Let p be the proportion of disagreements to the length m of the sequence: p = (m − c)/m where c is the number of correct, matching sites. Then there is a selection of mutation models, such as the simplest Jukes-Cantor model, which predict a distance D which is the expected total number of mutations. Experimentally we find that distance D as a function of the observed disagreements. Alternately we could choose D from the list of evolutionary models: for instance

for Kimura's two parameter model. Or, alignment-free models such as the k-mer distance measures as described in Allman et al. (2017).

Here, we would want a distance D = R which is summed when in sequence but obeys the Ohm equations. The answer will depend both on the model of mutation we choose and the model of recombination we choose. For instance, D=-34ln(1-43p) for the Jukes-Cantor model, as described in Jukes and Cantor (1969). Rewriting using p = (m − c)/c we have:

D has the graph in Figure 23. The c-axis is explained by the fact that in the Jukes-Cantor model, mutations of the 4 nucleotides A, G, T, C can replace any letter with another—including a self replacement. This implies that the smallest number of matching sites is m4, while the largest is m. We can use D for the resistance distance only if there is experimental evidence that for circuit-parallel paths we have D = D₁D₂/(D₁ + D₂), where D₁(c₁) and D₂(c₂) are the distances for each path, in expected numbers of mutations as a function of correct matching sites. There are certainly some features of D that look promising, including the shape of its graph: resistance typically ranges from 0 to infinity. Assuming that the formula for D over the circuit-parallel paths does hold, when one of the circuit-parallel resistances is infinite: say D₁ → ∞; then we see that D → D₂. Similarly, as c₁ → m/4, we have that c, the number of correct sites after recombination, approaches c₂.

When both branches have the same distance D₁ = D₂, and it obeys Ohm's law, we see the total resistance D = D₁/2. Using the formula for D(c) and D₁(c₁) and solving for c we get the following function, graphed in Figure 24:

Thus, as a first check the geneticist could compare two genomes and their hybrid genome with a common ancestor. When the two are close to the same distance from the common ancestor (both have c₁ matching sites), then the pair (c₁, c) for c the number of matches between the hybrid and the common ancestor might fit the parabola as seen in Figure 24. If that fit is achieved, then it would be reasonable to apply the theorems of this paper.

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author/s.

SF did the majority of research and writing. DS: student researcher, contributed most of section 5.1, 2-nested networks, from his thesis, especially the combinatorics. All authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.