Multi-Semi-Directed Networks

• Multi-semi-directed networks are mixed phylogenetic graphs obtained by semi-deorienting multi-rooted networks, ensuring that directional reticulation features are preserved.
• They satisfy strict graph-theoretic conditions, including specific degree constraints, absence of semi-directed cycles, and no non-trivial edge-paths between reticulations.
• These networks extend beyond phylogenetics to dynamic systems, statistical physics, and topological deep learning, supporting advanced network reconstruction and analysis.

Multi-semi-directed networks are mixed graphs that can be obtained from directed phylogenetic networks that may have more than one root. In that explicit sense, they generalize semi-directed networks, which correspond to the one-root case, and they retain direction exactly where de-orientation does not erase reticulation structure. The concept is most sharply formalized in recent phylogenetics, where it answers the previously implicit question of when a mixed graph is the semi-deorientation of some rooted or multi-rooted network; in adjacent literatures, closely related constructions appear under different names for partially directed growth, multilayer directionality, or mixed directed/undirected dynamics (Holtgrefe et al., 24 Jul 2025).

1. Terminology, scope, and basic construction

In the phylogenetic setting, the underlying object is a mixed graph $G=(V,E,A)$, where $V$ is a nonempty vertex set, $E$ is a set of undirected edges $\{u,v\}$, and $A$ is a set of directed arcs $(u,v)$. For a vertex $v$, the indegree $d^-_G(v)$ counts incoming arcs, the outdegree $d^+_G(v)$ counts outgoing arcs, $d^e_G(v)$ counts incident undirected edges, and the total degree is $V$0. A vertex is a reticulation if $V$1, a leaf if either $V$2 or $V$3, and a root if $V$4 (Holtgrefe et al., 24 Jul 2025).

A multi-rooted network on a finite leaf set $V$5 is a directed acyclic mixed graph with $V$6, satisfying $V$7 for all non-root vertices and $V$8 for all vertices. Its semi-deorientation is obtained by replacing each arc $V$9 by an undirected edge $E$0 whenever $E$1, and then suppressing any vertex $E$2 with $E$3. A $E$4-semi-directed network is the semi-deorientation of some $E$5-rooted network; a multi-semi-directed network is a $E$6-semi-directed network for some $E$7; and a semi-directed network is the special case $E$8 (Holtgrefe et al., 24 Jul 2025).

This formalism refines earlier semi-directed phylogenetic usage. A semi-directed binary phylogenetic network can also be described as the result of deleting the root of a rooted binary phylogenetic network, suppressing the resulting indegree-0 outdegree-2 vertex, and undirecting all tree edges while keeping edges directed into reticulations. In that representation, semi-directed networks sit between rooted directed and unrooted undirected phylogenetic networks (Linz et al., 2023).

2. Explicit graph-theoretic characterization

A central result is that multi-semi-directedness is no longer merely an existential rooted-origin property. It admits explicit intrinsic characterizations. One characterization states that a mixed graph $E$9 is a multi-semi-directed network if and only if three conditions hold: $\{u,v\}$0 and $\{u,v\}$1 for all $\{u,v\}$2; $\{u,v\}$3 contains no semi-directed cycle; and $\{u,v\}$4 contains no non-trivial edge-path between two reticulations. An equivalent formulation replaces the last two conditions by requiring that each cycle contains at least one sink and each sink component is a pendant subtree (Holtgrefe et al., 24 Jul 2025).

The one-root case is stricter. A mixed graph is semi-directed, rather than merely multi-semi-directed, if and only if the same degree condition holds, each cycle contains at least one sink, and the graph contains a $\{u,v\}$5-path between each pair of vertices. Here a semi-directed path is a path whose arcs, if present, are traversed forward, while a $\{u,v\}$6-path is a path $\{u,v\}$7 such that both $\{u,v\}$8 and $\{u,v\}$9 are semi-directed paths. This condition forces all rootings to collapse to the one-root regime (Holtgrefe et al., 24 Jul 2025).

The root structure is itself determined by the mixed graph. If $A$0 is a $A$1-semi-directed network, then its reticulation number is

$A$2

Equivalently, if $A$3 denotes the set of reticulations, then all rootings of a multi-semi-directed network have exactly

$A$4

roots. The same framework also characterizes all possible root locations: admissible roots are confined to source components, and each source component must contain exactly one chosen root location, either at a vertex, along an edge, or along an outgoing arc (Holtgrefe et al., 24 Jul 2025).

3. Structural subclasses inherited from rooted-network theory

Once multi-semi-directed networks are characterized intrinsically, several rooted-network classes can be transported into the mixed setting. For tree-child structure, the decisive notion is the omnian, defined in the mixed graph as a vertex $A$5 with $A$6 and $A$7. A multi-semi-directed network is strongly tree-child if and only if it has no omnians. Weak tree-childness is less restrictive: in the $A$8-semi-directed case it requires $A$9, $(u,v)$0 for all $(u,v)$1, and no non-trivial edge-path between any two vertices in $(u,v)$2, where $(u,v)$3 is the set of omnians and $(u,v)$4 the reticulations (Holtgrefe et al., 24 Jul 2025).

Tree-basedness also admits mixed-graph criteria. For a semi-directed network with omnian set $(u,v)$5, define

$(u,v)$6

for $(u,v)$7. The network is strongly tree-based if and only if for each $(u,v)$8,

$(u,v)$9

This is a Hall-type condition on reticulation children of omnians. Weak forest-basedness is characterized by a collection $v$0 of semi-directed paths satisfying path-partition conditions denoted (P1)–(P4); in the one-root case, weak tree-basedness is equivalent to the existence of such a collection satisfying (P1)–(P3) (Holtgrefe et al., 24 Jul 2025).

Orchard structure extends as well. Weak orchardness for a multi-semi-directed network is equivalent to several conditions, including the existence of a sequence of cherry and reticulated-cherry reductions reducing the network to a forest of isolated taxon vertices and taxon-taxon edges, and the existence of a rooting with a binary resolution admitting an HGT-consistent labelling. Strong orchardness, in the semi-directed case, is characterized through strong cherry-picking sequences and the behavior of source-component reticulated cherries (“scr-cherries”) under reduction (Holtgrefe et al., 24 Jul 2025).

4. Search spaces and reconstruction theory

The geometry of semi-directed network spaces has been developed most fully for level-1 phylogenetic networks. The cut edge transfer move (CET) moves a subnetwork attached by a cut edge to a new attachment position while preserving semi-directedness. For fixed reticulation number $v$1, the space of semi-directed level-1 networks on a fixed leaf set $v$2 is connected under CET when $v$3, and weakly connected under CET when $v$4. Allowing the additional moves CET$v$5 and CET$v$6, which add and delete reticulations, connects the full space of semi-directed level-1 networks on $v$7, and in fact the space of all semi-directed phylogenetic networks on $v$8 (Linz et al., 2023).

Reconstruction from local 4-leaf subnetworks, or quarnets, provides a parallel line of development. An earlier result for semi-directed level-1 networks showed that a complete set of quarnets uniquely determines the parent network, and supplied two reconstruction algorithms: a sequential method based on leaf attachments and a cherry-blob method based on recursively identifying exterior structures (Huebler et al., 2019).

Subsequent work sharpened the identifiability frontier. Semi-directed binary level-2 phylogenetic networks are encoded by their quarnets, whereas this fails for level-3: there exist non-isomorphic level-3 networks with identical quarnet sets. Even in that negative regime, the blob tree $v$9, obtained by contracting each blob to a single vertex, is always encoded by quarnets (Huber et al., 2024).

Algorithmically, binary $d^-_G(v)$0-leaf semi-directed level-1 networks can be reconstructed in $d^-_G(v)$1 time from quarnets while using only an asymptotically optimal $d^-_G(v)$2 of them. At the coarser level of blob structure, an $d^-_G(v)$3-time algorithm reconstructs the tree-of-blobs of any binary $d^-_G(v)$4-leaf semi-directed network with unbounded level from $d^-_G(v)$5 splits of its quarnets. For triangle-free level-1 networks, the full reconstruction relies only on $d^-_G(v)$6 quarnet-splits together with $d^-_G(v)$7 four-cycle quarnets (Frohn et al., 2024).

5. Related interpretations outside phylogenetics

Outside phylogenetics, the expression “multi-semi-directed” is often only implicit, but several constructions are closely related. In statistical physics, semi-directed Barabási–Albert networks modify preferential attachment through asymmetric Kertész-list updates. Two variants are defined, SDBA1 and SDBA2. In both, each new node selects $d^-_G(v)$8 older nodes preferentially, but the new node is added to the Kertész list only once rather than $d^-_G(v)$9 times, and only one selected old node receives reciprocal influence. The degree distribution follows

$d^+_G(v)$0

with $d^+_G(v)$1 depending on $d^+_G(v)$2, decreasing from $d^+_G(v)$3 toward $d^+_G(v)$4 as $d^+_G(v)$5 increases. The term “multi-semi-directed” is not used there, but this suggests a multi-link operational meaning: each new node makes multiple partially directed attachments under a modified preferential-attachment rule (Sumour et al., 2016).

A related, construction-level version appears in “Non-Universality in Semi-Directed Barabasi-Albert Networks,” where semi-directedness is defined by adding the $d^+_G(v)$6 chosen old nodes plus only once the new node to the Kertész list. In that model, the BA exponent is no longer universal: the paper states that $d^+_G(v)$7 decreases from $d^+_G(v)$8 to $d^+_G(v)$9 for increasing $d^e_G(v)$0. Here again, a plausible implication is that “multi-semi-directed” may describe a family of multi-link, asymmetrically reinforced preferential-attachment networks rather than a formally named class (Sumour et al., 2012).

Multilayer analogues also exist. Directed multiplex networks are multiplex systems in which at least one layer is directed; this directly includes one directed plus one undirected layer. For diffusion modeled by continuous-time random walks, the appropriate supra-Laplacian is

$d^e_G(v)$1

and convergence is controlled by the real part of the smallest nonzero eigenvalue. Such systems can exhibit an optimal intermediate coupling $d^e_G(v)$2, rather than fastest diffusion only in the fully coupled limit. This is not the same notion as phylogenetic multi-semi-directedness, but it is a mathematically adjacent model of mixed directionality across layers (Tejedor et al., 2017).

6. Dynamic-network and learning generalizations

In systems and control, the closest analogue is the mixed linear dynamic network: a network with both undirected diffusive couplings and additional directed dynamic links. The mixed model is

$d^e_G(v)$3

where $d^e_G(v)$4 is symmetric and encodes the undirected diffusively coupled part, while $d^e_G(v)$5 is hollow and encodes the directed links. For this class, one can derive dynamic-network models, formulate global network-identifiability conditions from $d^e_G(v)$6 and $d^e_G(v)$7, and construct a tractable two-step identification algorithm that yields consistent estimates under the stated structural and excitation assumptions (M. et al., 16 Apr 2026).

A different generalization appears in topological deep learning. Semi-Simplicial Neural Networks operate on semi-simplicial sets, where higher-order simplices are ordered rather than unordered, and face maps induce asymmetric relations between simplices. In that setting, a layer takes the form

$d^e_G(v)$8

or, with routing,

$d^e_G(v)$9

These models are designed for directed higher-order motifs rather than mixed graphs in the phylogenetic sense, but they show that “semi-” and “directed” can also be combined at the level of ordered multi-way relations. In the reported experiments, SSNs are strictly more expressive than standard graph and TDL models, and on brain-dynamics classification they outperform the second-best model by up to $V$00 and message-passing GNNs by up to $V$01 in accuracy (Lecha et al., 23 May 2025).

Taken together, these strands show that multi-semi-directed networks are not a single transdisciplinary object. In the strict present-day sense, they are mixed phylogenetic graphs arising as semi-deorientations of multi-rooted networks. More broadly, the literature points to a family resemblance: multiple partially directed attachments in growth models, directed-plus-undirected layers in multiplex diffusion, mixed dynamic interconnections in control, and ordered higher-order directionality in topological learning. The strongest formal definition remains the phylogenetic one, but the surrounding research suggests a wider methodological role for mixed directionality whenever neither fully directed nor fully undirected structure is adequate (Holtgrefe et al., 24 Jul 2025).