Semi-Directed Networks

• Semi-directed networks are mixed graphs with both directed and undirected edges that capture locally oriented reticulations in settings where the global root may be uncertain.
• They enable reconstruction of evolutionary and structural patterns via quarnets and blob tree decompositions, supporting robust algorithmic comparisons.
• Their versatile framework applies to preferential-attachment models, statistical physics, and mixed linear dynamic systems, bridging interdisciplinary research.

Semi-directed networks, also written semidirected networks, are mixed graphs containing both directed and undirected connections. In contemporary phylogenetics, they are used to represent evolutionary histories when reticulation events retain an identifiable orientation but the global root is uncertain, so only arcs entering reticulations remain directed while the remaining edges are left undirected. Closely related mixed-graph formulations define such objects through the existence of one or more rooted partners and study when node types, directed-path structure, or induced subnetworks are invariant across rootings. The same term also appears in other areas, notably preferential-attachment network models with limited directedness and mixed linear dynamic networks combining diffusive couplings with directed signal-flow links (Huber et al., 2024, Maxfield et al., 2024, Holtgrefe et al., 24 Jul 2025, Sumour et al., 2012, M. et al., 16 Apr 2026).

1. Terminological scope and domain-specific usage

The label “semi-directed network” is not attached to a single universal formalism. Instead, it denotes a family of partially oriented network models whose precise meaning depends on the application domain.

Domain	Core meaning	Representative result
Phylogenetics	Mixed graph with reticulation directions retained and other edges undirected	Quarnets encode all binary level-2 semi-directed networks, but not level-3 (Huber et al., 2024)
Semidirected phylogenetic comparison	Network admitting rooted partners with stable directed-path structure	Edge-based dissimilarity extends Robinson–Foulds distance (Maxfield et al., 2024)
Complex networks	Preferential-attachment network with limited directedness	Degree exponent becomes non-universal in $m$ (Sumour et al., 2012)
Statistical physics on SDBA graphs	Partially reciprocal influence network for spin dynamics	Relaxation follows a Vogel–Fulcher law (Sumour et al., 2016)
Control and system identification	Mixed dynamic network with undirected diffusive and directed interconnections	Consistent identification algorithm for all dynamics (M. et al., 16 Apr 2026)

In phylogenetics, the mixed-graph viewpoint is central because root placement is often difficult to determine from data, whereas the local orientation of reticulation edges may still be meaningful. In network science and control, by contrast, the mixed structure reflects partial reciprocity, asymmetric influence, or the coexistence of physical variable-sharing and directed information flow (Huber et al., 2024, Maxfield et al., 2024, M. et al., 16 Apr 2026).

2. Formal graph-theoretic models in phylogenetics

A common phylogenetic construction starts from a rooted directed acyclic phylogenetic network and forms an underlying semi-directed network by replacing all arcs by edges except those entering reticulations, then suppressing the root when applicable. In the notation of one formulation, if $N_d$ is a directed network on leaf set $X$, then the associated semi-directed network is denoted $N=\underline{N_d}$; different rootings can yield the same underlying semi-directed network, and the construction is well-defined (Huber et al., 2024).

A complementary formulation defines a semidirected graph as a tuple $N=(V,E_U\sqcup E_D)$ with undirected edges $E_U$ and directed edges $E_D$, and studies those semidirected acyclic graphs that admit a rooted partner. In this setting, hybrid edges and the partition of vertices into tree and hybrid nodes are stable across all rooted partners, which makes the mixed graph a rooting-invariant object rather than merely a rooted network with deleted directions (Maxfield et al., 2024).

Recent work has given explicit recognition criteria for these mixed graphs. For multi-semi-directed phylogenetic networks, one characterization requires: degree constraints $d(v)\neq 2$ and $d^{-}(v)\in\{0,d(v)-1\}$ for all vertices, absence of semi-directed cycles, and absence of non-trivial edge-paths between reticulations. For single-root semi-directed networks, an additional wedge-path condition is imposed: there must be a wedge-path between each pair of vertices. Equivalent formulations are also given in terms of sinks on cycles and pendant sink components (Holtgrefe et al., 24 Jul 2025).

Several structural notions recur across these formulations. A blob is a maximal 2-edge-connected subgraph in the mixed-graph sense, and the blob tree $B(N)$ is obtained by contracting each blob to a single vertex. The level of a network is the maximum number of reticulations in any blob; level-$N_d$0 means every blob contains at most $N_d$1 reticulations, and strict level-$N_d$2 means that some blob contains exactly $N_d$3 reticulations (Huber et al., 2024). In the semidirected-rooted-partner framework, the undirected part induces a forest, root components are maximal undirected components under the reachability preorder, and completion directs all undirected edges in the directed part in the unique way shared by all rooted partners (Maxfield et al., 2024).

3. Structural invariants, classes, and internal geometry

The blob decomposition is the coarse structural invariant that organizes most of the modern theory. For semi-directed phylogenetic networks, the blob tree is always an undirected phylogenetic tree, and the level bound localizes combinatorial complexity inside blobs. This supports a division between global tree-like structure and local reticulate structure, which is exploited both in reconstruction and in comparison algorithms (Huber et al., 2024).

Several special classes from rooted-network theory have now been transferred to the semi-directed setting. In one semidirected framework, weakly tree-child means that at least one rooted partner is tree-child, while strongly tree-child means that all rooted partners are tree-child; for complete $N_d$4-networks this admits a linear-time characterization using the directed part, trivial root components, and distinguished sets $N_d$5 inside root components (Maxfield et al., 2024). In the explicit mixed-graph characterization framework, a vertex is an omnian if $N_d$6 and $N_d$7, and a multi-semi-directed network is strongly tree-child if and only if it has no omnians. The same paper also gives criteria for weakly tree-child, strongly tree-based, weakly orchard, strongly orchard, weakly forest-based, and weakly tree-based networks, using Hall-type conditions, cherry-picking reductions, HGT-consistent labellings on binary resolutions, and path-system decompositions (Holtgrefe et al., 24 Jul 2025).

The geometry of spaces of semi-directed phylogenetic networks has likewise been developed through rearrangement moves. The cut edge transfer (CET) move prunes across a cut edge and reattaches elsewhere, subject to a rooted-partner compatibility condition that guarantees the result is again semi-directed. For fixed leaf set $N_d$8 and reticulation number $N_d$9, the space of semi-directed level-1 networks is connected under CET when $X$0, and weakly connected when $X$1; in both cases the diameter is $X$2. With the additional moves CET$X$3 and CET$X$4, which respectively add and delete reticulations, the space of all semi-directed phylogenetic networks on a fixed leaf set is connected under extended CET (Linz et al., 2023).

4. Reconstruction from quarnets and small subnetworks

A major line of work studies whether semi-directed phylogenetic networks are determined by their induced subnetworks on four leaves. These subnetworks are called quarnets. For $X$5 with $X$6, the restriction is written $X$7, and the set of all quarnets is

$X$8

A class is encoded by quarnets if equality of quarnet sets forces isomorphism of the full networks on the same leaf set (Huber et al., 2024).

For semi-directed level-1 phylogenetic networks, two constructive reconstructions from complete quarnet sets were established earlier. The sequential algorithm starts from a single quarnet and adds leaves one at a time via uniquely determined attachment moves, while the cherry-blob algorithm identifies exterior structures such as tree cherries, reticulation cherries, and larger exterior blobs directly from quarnets. For binary semi-directed level-1 networks, both procedures recover the unique parent network from its complete quarnet set (Huebler et al., 2019).

Subsequent work reduced the amount of required four-leaf information. An $X$9-time algorithm reconstructs binary $N=\underline{N_d}$0-leaf semi-directed level-1 networks from direct access to all quarnets while using only an asymptotically optimal $N=\underline{N_d}$1 of them; a related $N=\underline{N_d}$2-time algorithm reconstructs the tree-of-blobs of any binary semi-directed network of unbounded level from $N=\underline{N_d}$3 splits of its quarnets. When the network contains no triangles, reconstruction can be carried out using only four-cycle quarnets together with the splits of the other quarnets (Frohn et al., 2024).

For bounded higher level, the current dividing line is now sharp in one important case. Semi-directed binary level-2 phylogenetic networks with at least four leaves are encoded by quarnets, but semi-directed binary level-3 networks are not: there exist non-isomorphic level-3 examples with identical quarnet sets. At the same time, the blob tree is always encoded by the quarnets of a semi-directed binary network, via a characterization of CE-splits through 4-leaf restrictions. These results are explicitly linked to statistical consistency arguments for network-reconstruction programs, including the Squirrel software tool under development (Huber et al., 2024).

5. Distances, representations, and algorithmic comparison

Comparison of semidirected networks has led to a distinct line of theory centered on rooting-invariant representations. For a leaf-labelled semidirected network, node $N=\underline{N_d}$4-vectors count directed paths from a vertex to each labelled leaf in a rooted partner; stability theorems show that these vectors are independent of the rooted partner in the directed part, and directional $N=\underline{N_d}$5-vectors are similarly well-defined for edges in root components once a local orientation is fixed. This yields an edge-based $N=\underline{N_d}$6-representation $N=\underline{N_d}$7 for complete networks, and the associated dissimilarity

$N=\underline{N_d}$8

is defined as the symmetric-difference cardinality of the corresponding multisets (Maxfield et al., 2024).

This dissimilarity extends the Robinson–Foulds distance in both classical tree regimes. For unrooted trees it reduces to symmetric difference on splits; for rooted trees it reduces to symmetric difference on rooted clusters. On complete strongly tree-child $N=\underline{N_d}$9-networks it is a true metric, because equality of edge-based $N=(V,E_U\sqcup E_D)$0-representations implies phylogenetic isomorphism. The representation can be computed in $N=(V,E_U\sqcup E_D)$1 time, and the dissimilarity can be computed in $N=(V,E_U\sqcup E_D)$2 time after sorting. Under mild bounds on hybrid indegree and the number of root components, this becomes near-quadratic in the number of leaves (Maxfield et al., 2024).

The rearrangement geometry supplied by CET complements this metric viewpoint. CET furnishes connected search spaces for semi-directed level-1 networks with fixed reticulation number, CET$N=(V,E_U\sqcup E_D)$3/CET$N=(V,E_U\sqcup E_D)$4 allow movement between reticulation tiers, and CET1 provides a local variant analogous to NNI. These results justify CET-based proposal mechanisms for hill-climbing and MCMC over semi-directed network spaces, particularly in the level-1 regime (Linz et al., 2023).

6. Uses beyond phylogenetics

In complex-network growth models, “semi-directed” refers to a preferential-attachment construction in which directionality is introduced through asymmetric list updates or partial reciprocity. In the semi-directed Barabási–Albert model of one study, a new node chooses $N=(V,E_U\sqcup E_D)$5 older targets, the targets are each appended once to the Kertész list, and the new node is appended only once rather than $N=(V,E_U\sqcup E_D)$6 times. The resulting in-degree distribution obeys

$N=(V,E_U\sqcup E_D)$7

for the case $N=(V,E_U\sqcup E_D)$8, so the exponent decreases from $N=(V,E_U\sqcup E_D)$9 at $E_U$0 toward $E_U$1 as $E_U$2 increases, breaking the usual $E_U$3 universality of undirected BA growth (Sumour et al., 2012).

A related statistical-physics literature studies Ising and Potts models on semi-directed BA networks with asymmetric influence neighborhoods. Two variants, SDBA1 and SDBA2, differ in whether a new node influences all selected targets or is influenced by them, with one reverse link in each case. Monte Carlo simulations report that magnetization decays after a characteristic time $E_U$4 obeying a Vogel–Fulcher–Tammann form

$E_U$5

with $E_U$6, rather than a conventional equilibrium ferromagnetic transition. The papers interpret this as “unusual ferromagnetism” or metastable, glass-like relaxation on partially directed scale-free topologies (Sumour et al., 2016).

In systems and control, mixed linear dynamic networks provide a different meaning again. Here the undirected part represents diffusive, Laplacian-structured couplings dictated by physical variable-sharing, while the directed part represents input-output signal flow such as digital control or nonsymmetric components. The model

$E_U$7

combines a symmetric polynomial matrix $E_U$8 for the undirected diffusive part with a hollow proper rational matrix $E_U$9 for directed modules. For this mixed setting, identifiability conditions are given in terms of left coprimeness, a diagonal block condition after permutation, the presence of at least one excitation, a linear parameter constraint, known zero structure of $E_D$0, and the exclusion of bidirectional directed links between a pair of nodes. A two-step procedure—first estimating a polynomial surrogate model by convex constrained least squares, then mapping back to the original mixed model by linear least-squares relations—yields consistent estimates of all dynamics under the stated assumptions (M. et al., 16 Apr 2026).

7. Open problems and current research directions

Several active problems remain open. For quarnet-based reconstruction, the maximal subclasses of semi-directed binary level-3 networks that are still encoded by quarnets are not known; one conjecture isolates the known counterexample family and its leaf-insertion variants as the essential obstruction. Further directions include inference rules for semi-directed quarnets, polynomial-time reconstruction of level-2 networks from complete quarnet sets, and robustness to sparse or noisy four-leaf input (Huber et al., 2024).

For graph-theoretic characterization and class theory, open questions include extending the strong orchard characterization from semi-directed to multi-semi-directed networks, finding a direct path-partition characterization of strongly forest-based networks, characterizing weakly orchard networks directly through HGT-consistent labellings on the semi-directed graph itself, and developing efficient decision procedures for strong orchard status without enumerating rootings (Holtgrefe et al., 24 Jul 2025).

For comparison and search spaces, the edge-based $E_D$1-dissimilarity is known to be a metric on complete strongly tree-child semidirected networks, but extending metricity beyond this class remains unresolved. Weighted or continuous variants that incorporate branch lengths or inheritance probabilities also remain open (Maxfield et al., 2024). In rearrangement theory, higher-level analogues of CET connectivity, exact diameter bounds, and the computational complexity of CET distance are open problems (Linz et al., 2023). For few-quarnet reconstruction, the gap between the $E_D$2 upper bound and $E_D$3 lower bound for blobtree reconstruction in unbounded-level networks remains, as do extensions of the canonical reconstruction framework to level-2 and to noisy quarnet data (Frohn et al., 2024).