Unitary t-designs mimic the Haar-measure up to t-th moments. It is known that Clifford operations can implement at most 3-designs. In this work, we quantify the non-Clifford resources required to break this barrier. We find that it suffices to inject t-design. Strikingly, the number of non-Clifford gates required is independent of the system size – asymptotically, the density of non-Clifford gates is allowed to tend to zero. We also derive novel bounds on the convergence time of random Clifford circuits to the t-th moment of the uniform distribution on the Clifford group. Our proofs exploit a recently developed variant of Schur-Weyl duality for the Clifford group, as well as bounds on restricted spectral gaps of averaging operators.Many quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full  The notion of a unitaryt-design captures an efficiently realizable version of uniform randomness [t-design if it matches the uniform Haar measure up to t-th moments.Random vectors and unitaries are ubiquitous in protocols and arguments of quantum information and many-body physics. In quantum information, a paradigmatic example is the protocol –3, whichprotocol . In thesndomness –7. More decoupling procedures, a primitive in quantum Shannon theory [Applications abound. The randomness provided by designs is used to foil attackers in quantum cryptography protocols –10. It gn theory . In quann theory , 20. Unin theory , 21–23.t-designs [n qubits with t-design. In practice, it is often desirable to find more structured implementations. Designs consisting of Clifford operations would be particular attractive from various points of view: (i) Because the Clifford unitaries form a finite group, elements can be represented exactly using a small number (fault-tolerant architectures [magic-state distillation) carries a significant overhead. The difference is so stark that in this context, Clifford operations are often considered to be a free resource, and the complexity of a circuit is measured solely in terms of the number of non-Clifford gates [The multitude of applications motivates the search for efficient constructions of unitary -designs –28. In p-designs  show thatectures , 30, Clird gates , 32.t-design for any local gate set that generates an exact unitary design of order The Clifford group is known to form a unitary ment}t=3 –35, but ment}t>3 –37. In fment}t>3 . Moreovement}t>3 , 39 togeHow many non-Clifford gates are required to generate an approximate unitaryt-design? A direct application of the random circuit model of Ref. [system size-independent number of This leads us to the central question underlying this work:  of Ref.  yields aWe establish this main result for two different circuit models Fig. . In Sectt-th tensor power of the Clifford group, described by a variant of Schur-Weyl duality developed in a sequence of recent works [t-design from a representation theoretical perspective – is independent of the system size. Refs. [sphericalt-designs that consist of a system size-independent number of Clifford orbits. It has been left as an open problem whether these ideas can be generalized from spherical designs to the more complex notion of unitary designs, and whether the construction can be made efficient [The key to this scaling lies in the structure of the commutant of the nt works , 41–43. e. Refs. , 42 haveT gate to a random Clifford circuit has dramatic effects on the entanglement spectrum. A relation to t-designs was suspected. Our result provides a rigorous understanding of this observation.Finally, we note that in Ref. , it has U(d). The measure U chosen according to t-th moment operator associated with t-design by the diamond norm distance of its moment operator to the moment operator of the Haar measure U(d).To state our results precisely, we need to formalize the relevant notion of approximation, as well as the circuit model used. Let ing Ref. , we quanApproximate unitary design). Let t-design if. Let K-interleaved Clifford circuit of depthk is the random circuit acting on n qubits described by the probability distribution. Let t and n. In this sense, our construction can be seen as a classical-quantum hybrid construction of unitary designs: The scaling is significantly improved by outsourcing as many tasks as possible to a classical computer. A construction in which all parts of the random unitary are local random circuits is considered in Corollary We give the proofs of this theorem in Sect. dom bits  and then)) gates . Combinementclass2pt{minim to Ref.  in the dK-interleaved Clifford circuits are sufficient to approximate t-designs. This intuition is supported by the numerical results of Ref. [T-gate has dramatic effects on the entanglement spectrum of a quantum circuit.For designs generated from general random local circuits, numerical results suggest that convergence is much faster in practice than indicated by the proven bounds . We expe of Ref. , which sT-gates can be efficiently simulated [T-gates. Combined with our result, this implies that for fixed additive errors O(nt) suffices to approximate t-designs [It is moreover noteworthy that circuits with few imulated –51. The -designs , 52. If t-th moment operator of the Haar measure and the commutant of the diagonal action of the Clifford group. The latter was proven to be spanned by representations of so-called stochastic Lagrangian sub-spaces in Ref. [t-design in diamond norm.For the proof of Theorem  in Ref. . In part in Ref.  on restrrelative approximate designs:Moreover, the bound for Theorem Relativeapproximatet-design). We call a probability t-design if(K-interleaved Clifford circuits as relative t-designs). There are constants K-interleaved Clifford circuit is a relative t-design in depth (O(nt) at least until Hence, if we drop the system-size independence, we can achieve a scaling of K-interleaved Clifford circuits to be the more relevant case, the same method of proof works for Haar-interleaved Clifford circuits. Here, we draw not from the gate set U(2). The advantage is that we obtain explicit constants for the depth, while the depth in the K-interleaved setting has to depend on a constant (as K might be arbitrarily close to the identity).While we believe the setting of t-designs). For k form an additive t-design for all . Let n qubits. A local random Clifford circuit of depth m is the random circuits described by the probability measure . Let relativeapproximate Cliffordt-design if. Let The proof of the theorem is given in Sect. k-times alternatingly applies a local random Clifford circuit of depth m, and a unitary drawn from the probability measure We can combine this result with the bounds obtained in Sect. t-design for all . Let t-groups and a criterion for the universality of finite gate sets [The complete argument for the corollary is given at the end of Sect. ate sets , 39, 55.Hilbert-Schmidt inner productadjoint action of an invertible operator Given a  Hilbert space p-normsA. In particular, we use the trace normFrobenius or Hilbert-Schmidt normspectral normdiamond normWe consistently reserve the notation e useful 12\documen-qubit Clifford groupU(1) yielding a finite group with minimal center H and the phase gate S.In this section, we review some of the machinery developed in Ref. . Recall t-th diagonal representation of the Clifford group, defined ast copies of an n-qubit system. However, it will turn out that the operators commuting with this representation naturally factorize with respect to a different tensor structure on this Hilbert space, namely stochastic Lagrangian sub-spaces [For this work, the b-spaces :Stochastic Lagrangian sub-spaces). Consider the quadratic form T is totally t .T has dimension stochastic: T is  developed in Ref. [Since the representation in question is fixed throughout this paper, we will simplify the notation from now on and write Stochastic orthogonal group). Consider the quadratic form stochastic orthogonal groupO with entries in . A defect subspace is a subspace q, that is, that  to keep with the notation of that reference.The quadratic form g., Ref. , App. A guage of , all q-iference,  works siN, it holds that Calderbank-Shor-Sloane (CSS) codesFor any defect subspace  of Ref. , Sect. 4. Considr(T) operators with right defect subspace given by N, i.e., with N we obtain the full set Recall from Ref.  that ther(T)). Suppose  operators which we define asIn the following, we will often work with a normalized version of the t-design. This is made precise by the following two theorems which rely on two distinct proof strategies and come with different trade-offs.In this section, we give a bound on the number of non-Clifford gates needed to leverage the Clifford group to an approximate unitary K-interleaved Clifford circuit with depth k acting on n qubits is an additive t-design for all (Unitary designs with few non-Clifford gates). Let K-interleaved Clifford circuit has an associated probability measure A. Thus, we can rewrite the difference of moment operators asRecall from Def. Non-vanishing probability of applying the identity). We apply K, R(K) is Hermitian. The non-vanishing probability of applying K and T-gate.. Let The second lemma is of a more technical nature.(Diamond norm bound). Consider non-orthogonal. Thus, the projectors do not possess a natural expansion in this basis and we can not directly use the above bounds. However, we can write it explicitly in a suitable orthonormal basis of the commutant obtained by the Gram-Schmidt procedure from the basis The difficulty of using these results to bound the differencet! spaces (Properties of the constructed basis). Let t by their leading order term in the following. Specifically, the bounds in Lemma t.We believe that the explicit bounds in Lemma i and j do not correspond to permutations and hence n.Notice that from , we haveing Eqs.  and 33)37\documeentclass1pt{minimar, i. Moreover, we have used the triangle inequality,We provide two bounds for l changes. Moreover, we will treat differently the cases The second bound follows from Eqs.  and 33)33), and =i using . Moreoveain from  and 27)\documentto bound . Eqs. . Let t-th moment operator was defined asspectral norm asThe proof of Theorem t-designs). Suppose that t-design.. Let We will prove Proposition k-step random walk t-design, if we choose First, note that entclass1pt{minimaG and write G is closed under taking inverses, the moment operator r-th largest eigenvalue of a normal operator X. Hence, we findFor the sake of readibility, let us from now on drop the dependence on entclass1pt{minimanerators . Equivalt-designs in depth To this end, consider the linear operator  By Ref. , Lem. 1,entclass1pt{minimalocal Hamiltonians with vanishing ground state energy:Therefore, we consider the more relevant case when . Let from Eq. . It hold(Ground spaces). The Hamiltonians frustration-free  Let the Hamiltonian s in Eq.  and assut. To this end, we make again use of the averaging operator G|n) and d is the diameter of the associated Cayley graph (given in Ref. [We can now combine the bound in  with any in Ref.  as \docuc(G). We note that the applicability of Ref. [Since the representation  of Ref. , Cor. 1 We can combine Theorem t-design for all . Let m local random walk on the Clifford group (i)q prime and orIf (ii)eitherorIf (iii)If Then, one of the following cases apply: t-design.Note that a finitely generated infinite subgroup of a. Finite case. In the classification in Ref. [t-groups was shown for  in Ref. , the nonH is either one of finitely many exceptional cases or H is isomorphic to the Clifford group ii), the This establishes the finite version of .The former, the Lie-type cases, happen in dimensions cument}$$^n\pm 1/2b. Infinite case. Define the commutant for a set rom Ref.  applied . Consider r(T) is described by a stochastic orthogonal and a defect space First, recall that ral norm . Given smentclass2pt{minimframe operator associated to the basis Next, we define a Clifford frame operator). We define the Clifford frame operator of the basis . To der(Overlap of stochastic Lagrangian sub-spaces). We have n qubits. Since the operators r(T) are entry-wise non-negative, we have Denote by cp. Thm. . We obtaLet synthesis operator of the frame as the mapDefine the K be a single qubit gate which is not contained in the Clifford group. Then, there is a constant (Overlap bound). Let r(T) of the commutant of tensor powers of the Clifford group either belong to the commutant of the powers of the unitary group, or else are far away from it.The proof of Lemma t and for all t-th moment operator of the single-qubit unitary group (Haar symmetrization). For all s in Eq.  and \docT that attains this bound.The proof is given in Sect. The second ingredient to Lemma  that the representation .g. Ref. ). Hence,s in Eq. . CombiniQuantum gates with algebraic entries). If we restrict to gates K that have only algebraic entries, we can apply the result from Ref. [T-gate and for essentially all gates that might be used in practical implementations. Here, we have chosen the more general approach.(rom Ref.  and saveImplications for quantum information processing). Theorem ( in Ref. . Extendit! spaces (Properties of the constructed basis). Let n elements with no fixed points is known from Ref. [D stands for “derangement” as permutations without fixed points are sometimes called. Then, the number of permutations having exactly k fixed points is k points times the number p will shortly become useful. Note that we have for any The form of  is up torom Ref.  to be113k fixed points. For any such permutation, the product of overlaps collapses to only We start by bounding the diagonal coefficients sing Eq.  with \doused Eq.  in the l use Eq.  as beforNote that we can alternatively bound ith Eqs.  and . For all s in Eq.  and \docr(T), we can turn the statement  of T has components of equal Hamming weight.For an analysis of the tightness of the bound, see “Appendix C”. Recall thattatement  from oneT with trivial  and non-trivial defect spaces.We will bound the probability in slightly different ways for spaces a. Case I: trivial defect sub-spaces In this case, O. The next proposition treats a slightly more general situation.O has a column of Hamming weight r. Then the probability that O preserves the Hamming weight of a vector y chosen uniformly at random from (Hamming bound). Let r. Orthogonal stochastic matrices O satisfy r that can appear is The bound in Eq.  decreaseO does not preserve weights for bothy and O changes the Hamming weight.The proof idea is as follows: For each r entries of Assume without loss of generality that the first y be a uniformly distributed random vector on Oy, and O(y) by flipping the first r components. This operation changes the Hamming weight by r components of O(y) equals r, this condition cannot be met, and correspondingly Let r, this probability becomesIn case of odd b. Case II: non-trivial defect sub-spaces We now turn to Lagrangians T with a non-trivial defect subspace.x is chosen uniformly at random from T be a stochastic Lagrangian with non-trivial defect sub-spaces. Then, for an element  drawn uniformly from T, we have(Defect Hamming bound). Let N. Permuting coordinates of x is drawn uniformly from G has a column, say the first, with G have Hamming weight 1.  is distributed uniformly in T. By the first cited fact, x is distributed uniformly in x, y), for each We have proven the first advertised claim. It implies the second one, as argued next. Let t-designs). Suppose that t-design.. This implies thatThis follows similar to Ref. [ition in  is equivsymmetric subspaceSince  in Ref. , Lem. 4,r(T).In order to prove Lemma There is a constant There are Denote the ground state projector of Then, it holds thatWhile conditions 1) and 2) are merely translation-invariance with finit range of interactions and frustration-freeness in disguise, the third condition is highly non-trivial and involves knowledge of the ground-space structure. Usually, finding the ground space in a basis can be just as hard as computing the spectral gap in the first place. Fortunately, the ground space structure of the Hamiltonians . Let the Hamiltonian s in Eq.  and assuThe first condition immediately follows from the fact that we consider a translation-invariant 2-local Hamiltonian. It is fulfilled for any choice of q due to translation-invariance. Thus, we can set The second condition follows again for all l. We have to bound the quantity k is large enough. Concretely, we have by Lem. The third condition requires a calculation and a non-trivial choice of mentclasspt{minimamentclasspt{minimaHence, for the choices We make use of the Nachtergaele lemma. We have to verify the three conditions of Lemma t-designs. This is surprising, conceptually interesting and practically relevant: After all, it is the main objective in quantum gate synthesis to minimize the number of non-Clifford gates in a circuit implementation of a given unitary. There are multiple open questions and ways to continue this work:n is near to optimal, the scaling in t can probably be improved.Similar to the result in Ref. , the scaAnother natural open question is whether the condition mentclasspt{minimaOur result holds for additive errors in the diamond norm. For relative errors, our bounds can be used to obtain a quadratic advantage in the number of non-Clifford gates in Corollary dits for arbitrary d, giving rise to analogous conclusions concerning an independence of the system size for additive errors in the diamond norm.We strongly expect that the results can be generalized to quWe hope the present work stimulates such endeavors.We have found that a number of non-Clifford gates independent of the system size suffices to generate


As phylogenetic networks become more widely studied and the networks grow larger, it may be useful to “simplify” such networks into especially tractable networks. Recent results have found methods to simplify networks into normal networks. By definition, normal networks contain no redundant arcs. Nevertheless, there may be redundant arcs in networks where speciation events involving allopolyploidy occur. It is therefore desirable to find a different tractable class of networks that may contain redundant arcs. This paper proposes distinct-cluster tree-child networks as such a class, here abbreviated as DCTC networks. They are shown to have a number of useful properties, such as quadratic growth of the number of vertices with the number of leaves. A DCTC network is shown to be essentially a normal network to which some redundant arcs may have been added without losing the tree-child property. Every phylogenetic network can be simplified into a DCTC network depending only on the structure of the original network. There is always a CSD map from the original network to the resulting DCTC network. As a result, the simplified network can readily be interpreted via a “wired lift” in which the original network is redrawn with each arc represented in one of two ways. A (rooted) phylogenetic tree is a tree in which the vertices correspond to biological species, the leaves are extant species, and the branchings correspond to speciation events, usually by mutation. Recently there has been increased interest in speciation events such as hybridization and lateral gene transfer which are not modeled well using trees  such that there is another directed path from u to v that does not include . More details are given in Sect. There are various interesting classes of networks that have been investigated. Tree-child networks  so they have “distinct clusters,” abbreviated DC.The network is tree-child, abbreviated TC.Since they have both properties, they are called distinct-cluster tree-child networks, or, more briefly, DCTC networks.Roughly, DCTC networks are defined by two properties: N has additional interesting properties which in many ways resemble those of normal networks. In particular, a DCTC network satisfies the following: (3)N has n leaves. Then the number of vertices of N is at most Suppose (4)Every vertex is visible.(5)The number of hybrid vertices is at most (6)v with out-degree at least 2, there exist two distinct leaves x, y such that v is the most recent common ancestor of x and y.For every vertex Property (4) is true of all tree-child networks . This w Willson . , if N has n leaves, then N. The construction resembles that in Willson . An X-network V, A) where V is a finite set of vertices and A is a finite set of arcs. There are no directed cycles, there are no loops , and there is at most one arc  for in-degree of a vertex v in N, denoted indeg(v) or indeg , is the number of arcs , i.e., the number of parents of v. The out-degree of a vertex v, denoted outdeg(v) , is the number of arcs  , i.e., the number of children of v. Here root; it is the only vertex with in-degree 0. A leaf is a vertex Briefly, suppose X-networks except that cycles are possible. If that occurs we will explicitly specify that the network is not necessarily acyclic.Occasionally we may have to deal with p or p(x). The arc of form x-arc. If x is not specified, any such arc will be called an X-arc.Except where specified otherwise, we assume that each leaf Note that we make no assumption about the network being binary. A vertex may have in-degree greater than 2 or out-degree greater than 2 or both.path in N from a to b is a sequence N are thus directed. If there is a path from u to v, then we write V. For every vertex v, it is true that A x and y are distinct vertices. A vertex u is a common ancestor of x and y if v is a most recent common ancestor of x and y, denoted v is a common ancestor of x and y and, in addition, for every common ancestor u of x and y, x and y exists, it is unique.Suppose v is hybrid or reticulate if u of v is a tree-child if v, u) is the only arc coming into u. A vertex v is trivial if A vertex a, b) is redundant or a short-cut if there exists a path a, b).An arc . We call it the cluster of v. Note that cl(v) is nonempty. It is immediate that if N.For each X-networks which will be of interest:There are several types of X-network is successively cluster-distinct (SCD)  satisfies that either (i)(ii)Thus, successive vertices have different clusters except possibly for the arc An  Willson  if each  Willson  because X-network N is tree-child  is the power set of X;the cluster map N has no redundant arcs; andNote that because of (3), any regular network is SCD.An X-network N is normal  and X-networks, X, it is a metric on the collection of regular X-networks  map (Willson digraph map).For each arc For each u, v in V such that For each N. Thus, in the undirected graph Und(N) of N, if W and edge set For each Note that if N and Let  Willson  \documenX-networks. A CSD-map leaf-preserving if for each (7)V such that (8)x-arc x-arc the If Let  Willson  that theV. A wired lift of f (or of N) is a pair For each arc u, v) representspre-arc of For every arc Call the members of representative arcs since each represents an arc of Let V. Thus for all Note that the collection of all N and X-networks and Willson  Suppose N. An example is shown in Fig. N except that each arc may be solid or dashed. Suppose u, v) a solid arrow if N and each vertex of The wired lift mentclass2pt{minim Willson , the wir Willson , while tX-networks, with f. If u and v are in V, we say there is an allowed step from u to v if either Let generalized path or g-path in N from a to b is a sequence A N denote the initial X-network and N. Because of the dashed arcs, we see that N, there is clearly no path from 12 to 1. Note also that all children of 12 in N are hybrid. But In Fig. mentclass2pt{minimmentclass2pt{minimX-networks, with g-path in N from a to b.Willson  Let \docX-network. A subset strongly closed if it satisfies the following: Suppose there are vertices V with k such that K.Let X-network and (i)(ii)K is strongly closed, and(iii)for every strongly closed Thus, K is the unique minimal strongly closed subset of A containing D.Willson  Let \docK of the theorem is denoted K(D) and called the strong closure of D.The subset The following theorem summarizes the fundamental construction  Willson . While ( Willson .X-network and X-arc. There is a uniquely determined X-network There is a projection map For each arc K(D) is the strong closure of D, then If X-network, Suppose Willson  Suppose D consists of a list of arcs in N. For each arc N to a point. If these contractions result in any directed cycles, we contract the arcs in any such cycle. The result is N, it is a kind of quotient graph of N.The idea of  Willson  to be a Parts (1), (2), and (3) are directly from Willson . We provA that starts with K(D). An easy inductive argument shows that each Let Now we may complete the proof of Part (4). Suppose D and X-network, then all the identifications in X-network that contracts the members of D.Note that (4) shows that if As an application, there is the following result:N be an acyclic X-network. There is an X-network There is a leaf-preserving CSD map Willson  Let N bea, b) such that b is a leaf to ensure that no leaf becomes hybrid.Essentially, the computation of X-network N is distinct-cluster or DC if, whenever u and u and v is a leaf p(x) where p(x) is hybrid with out-degree 1. Thus the only way to have p(x) is hybrid with out-degree 1. More briefly, the only way to have x-arc p(x) is hybrid with out-degree 1. It is immediate that if p(x) is hybrid with out-degree 1, then An tree-child path if, for A path The following theorem restates some results in Cardona et al. .X-network that is tree-child. v to some leaf Given any vertex Assume there is a tree-child path Suppose X-network that is both SCD and tree-child. Then N is distinct-cluster (DC).Suppose u and u is a leaf, say v is a leaf. So we may assume that neither u nor v is a leaf. There is a tree-child path u to a leaf N is tree-child, and hence v to v is a vertex of P or else N is SCD. Thus Suppose X-network N a DCTCX-network if it is acyclic, distinct-cluster (DC), and tree-child (TC).We will call an X-network that is SCD and tree-child is a DCTC X-network.An M is DC but not TC since 9 has no tree-child. N is TC but not DC since In Fig. n leaves has at most Any tree-child network with a et al. . The folee Steel .X-network with n leaves. Then the number of hybrid vertices is at most Suppose N is TC, the root N is DC, c of c is a tree-child. We cannot have c and Since c to the leaf c to c. Hence, c to c lies on the path from c lies on a cycle, contradicting that N is acyclic.Choose a tree-path from Suppose the hybrid vertices are X-network. A vertex v is visible  if therentclass1pt{minimav is not visible, then the evolutionary history of gene flow in the corresponding phylogenetic network may have bypassed v and the presence of v could have no genetic impact on any of the leaves. Hence, visibility of all vertices is a desirable property in a phylogenetic X-network.As is pointed out in Francis et al.  if a verX-network. Then every vertex v is visible.Suppose u is visible since it is on every path from the root to x is a strict descendant of u.” This result is shown in Cardona et al. , where iX-network or a normal X-network. Every Let X-network and every normal X-network is tree-child. Every DCTC X-network. Suppose u is a leaf u is hybrid with out-degree 1; oru has out-degree at least 2, and for each child c of u, Suppose c be the unique child of u. Then N is DC by Theorem c is a child of u, since N is DC it follows If m be the maximal in-degree of a hybrid vertex and n be the number of leaves. Then Let X there is a metric  on the class of tree-child phylogenetic X-networks.For each The following facts about tree-child networks, proved in Cardona et al. , necessaX-network. u and v satisfy v is hybrid with out-degree 1, so Suppose w distinct from u and v such that u, v) is redundant.Suppose Let v to u to N is CD this means that (1) By Theorem u to w and a path from w to v. Hence,  is redundant. (2) By Part (1) there is a path from N is a DCTC X-network and If M that is DCTC but not normal, because  is redundant. On the right N is normal but not DCTC because X-network there is a closely related normal X-network.The property of being DCTC is closely related to being normal, but not the same. Figure The operator S If X-network, let S(N) denote the result of contracting every arc  such that w is a leaf. The operator S was briefly introduced in Willson ( Willson . Some baX-network. X-network except that a leaf may be hybrid.S(N) contains no vertices with out-degree one.N is normal, then S(N) is both regular and normal.If Let (4)(5)For all (6)Suppose D may contain X-arcs. (2) is immediate by the construction. (3) follows from Willson ((1) follows naturally, with leaves possibly being hybrid because  Willson , and (4)u, w), and every nontrivial path starting at u must pass through w since For (5), if X-network, let R(N) denote the result of removing all redundant arcs from N. More explicitly, let A that are not redundant in N; then X-network. For more details, see Willson (If  Willson .S will be to compute S(R(N)) where N is an acyclic X-network. Note that R(N) then has no redundant arcs, and often the result can be simplified. The simplification might be performed by the operator S. If N is DCTC, we show below that S(R(N)) is normal.An important use of S(R(N)). Start with the DCTC X-network N. Remove the two redundant arcs from A to form S(R(N)) the leaf  is hybrid. In S(R(N)) the map Figure N is a DCTC X-network, then S(R(N)) is a normal network. This relationship shows a close relationship between DCTC networks and normal networks. It will be the basis for an expanded analysis in Sect. The following theorem shows that if X-network. Then R(N) is a normal X-network.u, For each vertex u, w) in R(N) either satisfies p(x) is hybrid with out-degree one.Any arc (S(R(N)) is a normal X-network, possibly having some leaves that are hybrid, and containing no vertex with out-degree one. No two vertices have the same cluster.Suppose R(N) contains no redundant arcs, so we must only show R(N) is TC. Since N is tree-child, each vertex u that is not a leaf has a tree-child w. The arc  cannot be redundant. Otherwise, if  is redundant in N, then by redundancy there is a lengthening path starting at u and ending at w but not including the arc . Hence, u, w) remains an arc in R(N), so w is a tree-child of u in R(N).(1) It is immediate that N from u to u to R(N). Hence, u to R(N) is also such a path in N, proving Part (2).(2) Note that u, w) is an arc of R(N) then N is DC, (3) If (R(N) is normal, Part (4) follows from Theorem Since X-network with n leaves is quadratic in n. We also study when a vertex is the In this section, we prove that the number of vertices in a DCTC post-hybrid if p(x) is hybrid with out-degree one. Equivalently, x is post-hybrid iff N is a DCTC X-network, let N is understood) denote the number of Call X-network N with Figure X-network and N. Then Suppose N is CD, there is exactly one vertex for each member of Cl(N) except for the vertices p(x) for p(x) is hybrid with out-degree one. For such x, there are two vertices p(x) and (1) Since N to obtain R(N). By definition, R(N) contains no redundant arcs. Clearly R(N) is tree-child since no arc w of u is redundant; otherwise there would be a lengthening path from u to w which does not contain , hence another parent of w. It follows that R(N) is normal.(2) Remove all redundant arcs of cl(u) for any vertex u. Hence, R(N) remains CD. For every arc  such that cl(w) is not u, w) with S(R(N)) which will have exactly c(N) vertices, one for each cluster. But S(R(N)) is normal and has no vertices of out-degree one. By a result in Willson (X includes the root as well as the leaves.)Moreover, the removal of redundant arcs does not change  Willson , it has  Willson  differs (3) follows immediately.n leaves. Suppose p(x) is hybrid with out-degree 1. Yet N is tree-child and there must be a tree-path from the root p(x) is hybrid, this is not possible. Hence, (4) Clearly (5) N with N. On the other hand, Part (5) will be improved in Theorem Figure N is tree-child, then it is immediate that for any vertex u there is a tree-child path from u to a leaf.If most recent common ancestorx and y may sometimes be traced back to The notion of the entclass1pt{minimaX-network that every vertex with out-degree at least two has the form The following theorem shows in a DCTC X-network. Suppose There exists j such that For every Assume u. Assume u to (3)(4)If (5)Suppose i that u is hybrid with out-degree one. But the latter contradicts that u has out-degree at least 2. Hence Part (1) is true.(1) Assume for each Part (2) follows since in the proof of Part (1) the choice of u.The second half of (3) is immediate since i and s such that i is minimal satisfying this condition. If u to u and i, contradicting that u. If i, contradicting that u, which is not possible since Next I claim that there are no i, i, s satisfying For (4) suppose w has the parents Consider the case where x cannot be post-hybrid, but it is possible that y is post-hybrid.Note in the proof that x, y, it does not say that for all u is a common ancestor of 2 and 3 and v of 2 and 3 we must have N, but neither is an ancestor of the other.While Theorem entclass1pt{minimaA related observation in Fig. For Fig. X-network. x and y is not post-hybrid.If u is hybrid.If p(x) that are hybrid with out-degree one, both having cluster The only vertices which do not have the form Suppose u has out-degree one, let c be its unique child. Then N is DC, there exists u is hybrid, proving Part (2). Then Part (3) is immediate. (1) follows from the theorem. If N is DCTC, we also have the following result involving u satisfies If X-network. For each If Let u is a common ancestor of cl(u). Conversely, if For (1), it is immediate that for u is hybrid. Thus, For (2), if In Fig. The next result makes use of most recent common ancestors to find an upper bound on X-network, Suppose P denote the set of post-hybrid members of X, so Let p(x). It follows that there is no tree-child path from p(x) to If u to u :We now obtain our best upper bound for X-network and Suppose n the function v(N) is an integer, For fixed X-network in Fig. For the DCTC We show in Theorem X-networks Several different DCTC L, M, M allows the redundant arc . The redundant arc  in N did not require a new vertex. In general, adding a redundant arc between two existing vertices in a DCTC X-network N to create M does not change the set of clusters and hence yields Their differences will involve redundant arcs. For example, consider Fig. X is a nonempty set. Let X such that for each X-network which we shall denote X, and the map Suppose i et al.  construcX-network (possibly having hybrid leaves).X-network N (possibly having hybrid leaves) with cluster set Cl(N) is regular iff N is isomorphic with Reg(Cl(N)).An Figure Following are some properties of i et al. : \documeX-network. Then S(R(N)) is X-isomorphic with Reg(Cl(N)).X-networks and X-isomorphic.Suppose Suppose S(R(N)) is normal and contains no vertices with out-degree one. By results in Willson (S(R(N)) is X-isomorphic with Reg(Cl(S(R(N)))). But Cl(S(R(N))) (1) By Theorem 4.11 Willson , this ne Willson  we know  Willson , provingX-isomorphic with (2) From Part (1), for X-network N, all DCTC X-networks M such that u of M that is hybrid, then there exists a unique X-network. Next recursively adjoin redundant arcs to a, b) where a and b are vertices of a, b) where a is a vertex of b is a new vertex in the middle of what was the arc Theorem N, we compute Cl(N) and then Reg(Cl(N)). Suppose we want to reconstruct N. Let N. We could continue to get another DCTC X-network with the same clusters as N by adjoining new redundant arcs For example, consider Fig. N is a DCTC X-network with n leaves, then the number of vertices is at most n leaves and Theorem  Bickner  of interr3 and the child r3m1 is the child of r3 that is an ancestor of 3.We start with r4 with a tree-child path of new vertices r4, r4m1, r4m2, 4. Note r4 also has tree-child r3; r4m1 also has child r3m1, and r4m2 also has child p2. Note that r3m1 has become hybrid with tree-child 3, while r3 still has the tree child p1. Since p2 was already hybrid in r3m1 is the only new hybrid. Every non-leaf vertex of v of To obtain rnm1 but rn still has the tree-child rn. All non-leaf vertices of Given v of For each vertex We have proved the following:n leaves which is both DCTC and normal and which has For X-network with n leaves has at most 3 vertices, where n is needed.If N is an acyclic X-network there is an X-network N, it follows that Recall from Theorem X-network. Then X-network containing no redundant arcs.Let v is not a leafFrom Willson , the firu, v) is contracted in the formation of If w be another child of u beside v. Since N is normal, we may assume at least one of u. Choose a tree-path from v to the leaf x and from w to the leaf y. Note that If  Willson , \documep(x) for some p(x) be denoted u(x).By Theorem 4.3 of Willson  \documenu(x), p(x)) to suppress the trivial vertex p(x). Thus, Now Theorem u, v] the tree-child of v becomes a tree-child for . And whenever we contracted (u(x), p(x)) into , u(x), p(x)]. Hence N contained no redundant arcs, the same is true about But it is clearly tree-child as well since whenever we contracted The operatorS is described in Sect. X-network N, let v is not a leafu(x), p(x)) where, for p(x) is a trivial vertex with unique parent u(x) having out-degree at least 2.N is normal, then The proof of Theorem X-network. Then X-network, but contains no redundant arcs.Let We will illustrate the use of Corollary X-network N, this section shows how to produce a uniquely determined X-network which is DCTC and which we denote Given an  Willson  used to  Willson  involvesX-network. A vertex v of N is a tree-child obstacle or (in this section) an obstacle if v is not a leaf; andv is hybrid. Hence, whenever  is an arc, u has a parent other than v.every child of An X-network N is tree-child obstacle-free if it contains no tree-child obstacle.Let N is an X-network that is tree-child obstacle-free. Then N is a tree-child X-network.Suppose v that is not a leaf, there is an arc  with c is a tree-child of v. Hence, N is tree-child. By hypothesis, for every vertex X-network N, suppose we seek a related DCTC X-network. Our strategy will be to compute Given an Just as for pre-normal obstacles in Willson , there aN be an X-network. Suppose c is a tree-child obstacle. An allowable 1-fold parent chain of c is a path c, c is of type 1 if c has an allowable 1-fold parent chain. If c has type 1, and Let c is a tree-child obstacle. An allowable k-fold parent chain for c is a path c is of typek if c is not of type c has an allowable k-fold parent chain.In this situation, for each such allowable k-fold parent chain writeSuppose X-network every tree-child obstacle has a type. Figure X-network N in which 6 is a tree-child obstacle that has no type. Nevertheless, in an SCD X-network, the next result shows that every tree-child obstacle has a type.It is false that in every N be an SCD X-network. Then every tree-child obstacle c has a unique type.Let N is SCD, every child c of d of d with maximal cl(d), then d has no parent other than d is a tree-child of N is SCD and no child of X, It is clear that the type, if it exists, is unique. The root c which has maximal length k. Write it as k-fold parent chain of c since c has type Consider a path from  Willson , every aThe next result shows a way to remove a single obstacle of type 1.N is an X-network and c is a type 1 tree-child obstacle with allowable 1-fold parent chain p, c such that p has tree-child d and is not an obstacle.Suppose p, c) is not redundant, D is strongly closed and from Willson  into a vertex . In p, c], d). If q is any parent of d in p, c], then q was a parent of d in N as well, contradicting that d was a tree-child of p. Since  is a non-redundant arc in the X-network N and p and c. Let a, b) is a non-redundant arc in N and b is a tree-child of a. Assume Suppose (X-network N: The procedure is similar to that in Willson (These results lead to a procedure for finding a DCTC network closely related to a given  Willson  for findProcedure DCTC.Input An X-network N.Output An X-network and an integer.1. Let 2. If 3. Let 4. If c in 5. For each tree-child obstacle k.(5a). Compute the type (5b). Initialize k-fold parent chain c, let (5c). For each allowable k-fold parent chains for c.(5d). Let c.6. Let 7. Let 8. If i.9. Output X-network N will be denoted height of N is itself DCTC, and the height is 1 only if The network which is output from procedure DCTC applied to the X-network r and an X-network The procedure terminates and outputs an integer N and not any arbitrary choices.There is a leaf-preserving CSD map N.Let Suppose we apply the procedure DCTC to an c, then by Theorem 7.2 c has a type, so step (5a) can be carried out. Hence, the procedure is well-defined. Every time, the procedure enters step 5, the network has at least one obstacle, so the set D in step 6 is nonempty. Then at least one arc of N is finite, it follows that the procedure terminates.(1) By construction, for (2) It is immediate that the output (3) is immediate since no choices are made between different obstacles or allowable parent chains for an obstacle.N to (4) is immediate if (5) Just as in Willson , one cank-fold parent chain c and let (5c’). Select one allowable C indicates the choice of each parent chain when there are more than one possible. As in Willson  for vi is the number of visible vertices. In a tree tc is the number of non-leaf vertices with no tree-child. In a tree, tc.0N, for the specified network M, When several networks are being analyzed related to the network For example, in Fig. Let N, two of them using real data.This section contains three detailed examples of the calculation of N as described in  in u and v provided there is a path in N from u to v and in addition there is no visible vertex w distinct from u and v such that there are paths from u to w and from w to v. It is proved in Francis et al. =\psi =\psi , and noN had none. It also contains the most hybrid vertices among the three DCTC networks, just one more than Of the three DCTC networks, N, From the mentclass2pt{minimN in the sense of N was simplified. Moreover, We see that Viola and present a phylogenetic network N with 16 leaves and with 21 proposed polyploid speciations in their Fig. Marcussen et al. in Marcussen et al.  study thn et al. . Here weN has been published in Francis et al.  for It is interesting that N have been reduced to 10 reticulations in N but some of the hybrids of N have been merged. Hybrids 6, 13, 14, 57, 59 in N disappeared since they were merged with their parental line. For any DCTC X-network M, From the table, the 21 hybrids of x, y); for example, In x, y) of distinct leaves, Moreover . In their Additional File 4, Figure S7–9a is the cluster network for their dataset 9, constructed using all fragments passing the SH test against 100 random trees by support at least 15%. Let N be the network of their Figure S7–9a. The wired lift of N.Here is another example with real data. Kamneva et al.  study alThere are 13 leaves with Drymocallis being the outgroup that roots the network. X-networks. There is, however, no CSD map from N to It is clear that N, while N. The other hybrid 25 of N was merged with its parental line and descendant 28 into .From Table N in terms of the distance Note that N, we have seen that Given a general network D consists of the arcs of some allowable parent chains. We here consider these two operations separately.The construction of N on the left with N, there is a “ladder” structure involving 4, 5, 6, 7, and 8. Note that N may be correct in the sense that there might have been several stages of contribution to the genome of 9, nothing really identifies the relevant species. The result would be indistinguishable from the result if the species 5, 6, 7, and 8 were permuted. It can be argued that the data do not really support a network with distinct vertices 5, 6, 7, and 8; the simplification of N into To study the first operation, consider Fig. M of Fig. M, M is DCTC, so The fact that these identifications occur in N on the left, there is another “ladder” N. As for Fig. N.Similar considerations apply to a more complicated situation as in Fig. N is the Viola dataset for Example Less extreme “ladders” can occur. Figure entclass1pt{minimaa in N with a unique child b which is not a leaf, so a and b are not distinguishable from the data; these include N, such arcs  may merely indicate the distinction between a hybrid vertex a and the next descendant b where another speciation event occurs. If b is a tree-child, as in the case p(x)) and these can be mentally retained. When b is hybrid, as for example in the case a and b are parts of another ambiguous ladder, justifying the identification of 47 and 48.Figure N in the formation of We see that in general the merging of arcs from D consists of the arcs of some allowable parent chains to make the network tree-child) is different. Figure N on the left and also N, N and N is assumed to describe the genetic history, a mutation from 5 which is absent in 1 and 4 but present in 2 and 3 should be more common than if instead The interpretation of the second operation  Fig.  which ar(N) Fig.  5 has beFragaria dataset N of Kamneva et al. (For the a et al.  and ExamN so that every vertex becomes visible. As a result, every vertex has at least one leaf on which its genetic influence is important. Changes from In general,


Prioritization of self-related information (e.g. self-face) may be driven by its extreme familiarity. Nevertheless, the findings of numerous behavioral studies reported a self-preference for initially unfamiliar information, arbitrarily associated with the self. In the current study, we investigated the neural underpinnings of extremely familiar stimuli  and stimuli newly assigned to one’s own person and to a close-other (abstract shapes). Control conditions consisted of unknown faces and unknown abstract shapes. Reaction times (RTs) to the self-face were shorter than to close-other’s and unknown faces, whereas no RTs differences were observed for shapes. P3 amplitude to the self-face was larger than to close-other’s and unknown faces. Nonparametric cluster-based permutation tests showed significant clusters for the self-face vs. other  faces. However, in the case of shapes P3 amplitudes to the self-assigned shape and to the shape assigned to a close-other were similar, and both were larger than P3 to unknown shapes. No cluster was detected for the self-assigned shape when compared with the shape assigned to the close-other. Thus, our findings revealed preferential attentional processing of the self-face and the similar allocation of attentional resources to shapes assigned to the self and a close-other. In order to ensure our adaptive functioning in complex social environments, only some pieces of incoming information are selected for further processing. Such selection is often guided by the self-relevance of the information . To starImportantly, in the cited studies self-related stimuli were represented by highly familiar items like one’s own name or face. The daily exposure to one’s own face and name across the whole lifespan determines their extreme familiarity in comparison with all other faces and names. Hence, it cannot be clearly differentiated whether the observed effects were caused by the self-relevance or familiarity of those stimuli .To control for the confounding effects of familiarity, Sui and colleagues  introducAll in all, a brief association of a neutral shape with the self seems to increase the salience of those stimuli and is sufficient to elicit the self-prioritization effect . Self-prHowever, it is worth noting that in trials with matching pairs of self-associated shape and self-label, participants were processing both self-associated arbitrary stimuli and familiar verbal labels with an established meaning. Therefore, the self-advantage may be caused by the high familiarity of the self-label and not by the self-association of the shape. This fact led some studies to test self-prioritization effects in experimental paradigms with new self- vs. other-associated stimuli only. For example, Most studies investigating the self-prioritization of information that was newly assigned to one’s self vs. another person were based on behavioral measures . In contrast, studies investigating the neural correlates of such information processing are rather rare . One of In yet another fMRI study , particiFurther, in an event-related potential (ERP) study three unfamiliar faces were identified with the verbal labels “You,” “Friend,” and “Stranger” instead of shapes . AfterwaIn the current ERP study, we investigated the neural underpinnings of highly familiar and new information that was arbitrarily assigned to the self and to a close-other. We were interested in whether previously irrelevant, abstract information that was newly associated with the self would benefit from preferential processing as is the case for well-known self-referential information, and whether self-prioritization effects would be comparable in both cases. Therefore, we directly compared the processing of two types of stimuli: extremely familiar  stimuli, and stimuli that were newly assigned to one’s own person and to a close-other (an abstract shape). The control conditions consisted of unknown faces and unknown abstract shapes. We decided to present participants with abstract shapes alone, without labels. The reason for doing so was twofold. First, this approach avoids the aforementioned controversies regarding familiar labels. Second, as there was no need to use any labels in the case of faces, this approach (i.e. avoiding labels) guaranteed similar visual stimulation in both cases.Participants were tasked with indicating whether presented stimuli  were familiar or unknown. Prior to the EEG recording session, arbitrarily selected shapes were associated with the self and a close-other (i.e. one shape for each person). The close-other was freely chosen by each participant as representative of the most significant person in their life at the time of experimentation. This operationalization of a close-other was used in several previous studies on the topic of self-prioritization . It is wIn this study, the P3 results obtained for faces served as a kind of reference for ERP results for shapes. We expected to observe enhanced P3 associated with self-face processing when compared to close-other and unknown face processing. We aimed at testing whether information newly assigned to the self and a close-other would lead to an analogous pattern of findings. As far as behavioral indices of self-prioritization are also concerned, we were interested in whether behavioral findings would be comparable for faces and shapes.Moreover, the distinct spatial patterns of activity elicited by faces and shapes were also tested with nonparametric cluster-based permutation tests . This meM = 27.594; SD = 3.131). The Edinburgh Handedness Inventory  were tested in the study, ranging in age between 21 and 34 years old (nventory  indicatenventory .www.nencki.edu.pl) developed at the Nencki Institute in cooperation with the University of Warsaw. SONAR-II is dedicated to the asymptomatic population of people who do not meet the criteria for SARS-CoV-2 testing but who may come into contact with infected people.An additional present-day restriction was a negative test for the SARS-CoV-2 virus. As all participants were PhD students and employees at the Nencki Institute, they took part in the SONAR-II project . All participants provided written informed consent prior to the study and received financial compensation for their participation.We used two different types of stimuli in this study: (1) faces and (2) shapes. The set of stimuli was individually tailored to each participant. Faces belonged to three categories: self-face, close-other’s face, and unknown faces. Participants freely chose the close-other according to their subjective high level of closeness and importance. We did not predefine the type of relationship between the participant and their close-other in order to avoid a spuriously close relation. The only restriction placed on the selection of the close-other was that they had to be of the same gender as the participant. Twenty-two participants chose their friend, eight their sibling, and two their partner. The face of each participant and their close-other was photographed  prior to the study. The photographs of eight unknown neutral faces were taken from the Chicago Face Database , gender 2. Shape assignment to each experimental “condition”  was pseudo-random on the group level; e.g. a self-assigned shape in a given set of stimuli was assigned to a close-other or unknown condition in some other set of stimuli. Faces and shapes were presented against a black background. The second type of stimuli consisted of abstract shapes. In previous studies on the processing of new information assigned to the self and others, simple geometric figures  were typically used. As the number of shapes was supposed to be equal to the total number of faces  we generated 10 different abstract shapes. We aimed at equalizing low-level physical features of faces and shapes. Thus, each shape’s area was equal to face oval’s area, i.e. 43.12 cmParticipants were seated in a comfortable chair in a dimly lit and sound-attenuated room, 57 cm from the computer monitor . Subsequently, during the electrode cap placement and adjustment of EEG electrode impedances, they were primed for the task: two different shapes, one assigned to the participant and the second to their close-other, were presented on the monitor and participants were required to associate those geometric shapes with the self and chosen close-other. The learning phase lasted on average 23.53 min (SD = 5.900). Just before the beginning of the task, participants were asked to draw the assigned shapes. This was done to check the efficiency of learning.After electrode cap placement , the participants used an adjustable chinrest to maintain a stable head position. Presentation software  was used for stimuli presentation. Participants performed a recognition task—if they recognized a presented face or shape (i.e. representing the participant or their close-other), they were asked to push the response button assigned to “YES.” If that was not the case, they were asked to press the button defined as “NO.” The assignment of “YES” and “NO” buttons was counterbalanced across the participants.After reading the instructions displayed on the screen, the subjects confirmed they understood the task and initiated the experiment by pressing a response button. Trials with faces and shapes were inter-mixed (in one session) and their order was pseudo-random with respect to the type of stimulus  and the experimental condition . Each trial started with a blank screen, presented for 1500 ms. Next, a white fixation cross  was centrally displayed for 100 ms and followed by a blank screen which lasted either (randomly) 100, 200, 300, or 400 ms. Afterwards, a face or a shape was presented for 500 ms. Regardless of which stimulus was shown, participants were asked to push the appropriate response button as quickly as possible. Next, a blank screen was shown and lasted until a response was made. The procedure structure is presented in To account for possible fatigue during the experiment, a break was planned in the middle of experimental session. It was terminated after one minute, unless the participant ended it earlier. Participants needed 24 min on average to complete the whole experiment.The electroencephalogram (EEG) was continuously recorded with 62 Ag-AgCl electrically shielded electrodes mounted on an elastic cap  and positioned according to the extended 10–20 system, with two additional electrodes placed on the left and right earlobes. The EEG signal was recorded using the BrainAmp MR plus amplifier  and digitized at a 500-Hz sampling rate, using BrainVision Recorder software . EEG electrode impedances were kept below 10 kΩ. The EEG signal was recorded against an average of all channels calculated by the amplifier hardware.Responses were scored as correct if the appropriate button was pressed within 100–1000 ms of stimulus onset. In order to conduct statistical analyses of behavioral and ERP data in a consistent manner, similarly to our analyses of ERP components, only every other trial with an unfamiliar stimulus was included in the analyses. This was done because the number of trials with unfamiliar faces (80) and shapes (80) was doubled in comparison to the number of trials with familiar faces and shapes in each familiar  condition (40). Statistical analyses were performed using JASP software . Mean ac10 between 1 and 3 implies anecdotal evidence in favor of H1, between 3 and 10—moderate evidence, between 10 and 30—strong evidence, between 30 and 100—very strong, and higher than 100—extreme evidence. As far as low values of BF10 are concerned, a BF10 between 0.33 and 1 indicates anecdotal evidence in favor of H0, between 0.1 and 0.33—moderate evidence, and between 0.03 and 0.1—strong evidence of the absence of an effect. Finally, a BF10 between 0.01 and 0.03 and lower than 0.01 indicates very strong and extreme evidence for the absence of an effect, respectively.The traditional null-hypothesis significance-testing approach was complemented with Bayesian analysis methods and Bayes factors (BFs) were computed using JASP software . BFs werOff-line analysis of the EEG was performed using BrainVisionAnalyzer software . The 62 channels were re-referenced off-line to the algebraic average of the left and right earlobes, notch filtered at 50 Hz, and band-pass- filtered from 0.1 to 30 Hz using a Butterworth filter. The next step in data pre-processing was the correction of ocular artifacts using Independent Component Analysis (ICA) . After eAfterward, the EEG signal was segmented into 1200 ms epochs, from −200 ms before to 1000 ms after stimulus onset. The subsequent automatic artifact rejection procedure allowed only trials, which fulfilled the following requirements: the maximum permitted voltage step per sampling point was 50 μV, the maximum permitted absolute difference between two values in the 200-ms-long segment was 100 μV, the minimal and maximal allowed amplitudes were −150 μV and 150 μV, and the lowest permitted activity in the 100 ms interval was 0.5 μV.Trials with correct responses were subsequently analyzed. In the case of unfamiliar stimuli, only every other trial was included in the analyses. This was done in order to keep a similar signal-to-noise ratio (SNR) for each experimental condition (defined by the type of stimulus and type of face). It should be reminded that the total number of trials with unfamiliar faces and shapes (80 for each of them) was twice as large as the total number of trials with familiar faces and shapes in the single “self” and “close-other” conditions (40 for each type of stimulus). The mean number of segments averaged afterwards for each experimental condition was as follows: self-face—37.031 (SD = 2.800), shape assigned to self—35.250 (SD = 3.298), close-other’s face—36.938 (SD = 2.449), shape assigned to close-other −35.031 (SD = 3.441), unknown other face—36.344 (SD = 2.868), and unknown shape—35.938 (SD = 3.816). In the final stage of pre-processing, the epochs were baseline-corrected by subtracting the mean of the pre-stimulus period.Selection of electrodes for ERP analysis was orthogonal to potential differences between experimental conditions . Therefohttp://statcheck.io/index.php). A two-way repeated measure ANOVA was performed with “type of stimulus”  and “condition”  as within-subject factors. All effects with more than one degree of freedom in the numerator were adjusted for violations of sphericity . The reported results were cross-checked with Statcheck  was used in all Bayesian tests. BF10 were interpreted according to The traditional null hypothesis significance testing approach was complemented with Bayesian analysis methods. To test whether the self-face and other faces, as well as the self-assigned shape and other shapes, were characterized by similar levels of neural activity, BFs were computed using JASP software . The maiCluster-based permutation tests were used here as an exploratory analysis procedure, as they efficiently handle the multiple comparisons problem in high-dimensional magnetoencephalographic and EEG data . In geneWe directly compared: self-face vs. close-other’s face, self-face vs. unknown faces, close-other’s face vs. unknown faces, self-shape vs. close-other’s shape, self-shape vs. unknown shapes and close-other’s shape vs. unknown shapes. As clustering in both space and time was used, such an analysis procedure revealed differences in the spatial distributions of activity as a function of time between the tested conditions.F = 28.758, P < 0.001, η2 = 0.141) and “condition”  = 4.022, P = 0.023, η2 = 0.028), as well as a significant two-way interaction: “condition” × “type of stimulus”  = 10.689, P < 0.001, η2 = 0.119). The significance of the “type of stimulus” factor indicated a significantly higher accuracy score in the case of faces in comparison with shapes (see P = 0.020), whereas other differences were non-significant .A repeated-measures ANOVA conducted on the mean number of correct responses revealed the significant main effects of “type of stimulus” , the self-face and unknown faces , and the close-other’s face vs. unknown faces . Significant differences in accuracy rates were present between the self-assigned shape and unknown shapes  and between the close-other assigned shape and unknown shapes , whereas the self-assigned shape and the close-other assigned shape did not differ .Post-hoc tests of the “condition” × “type of stimulus” interaction revealed non-significant differences in accuracy scores between the self-face and the close-other’s face , but it was similar for faces and shapes in the case of the “unknown” condition .Moreover, post hoc tests of the “condition” × “type of stimulus” interaction also showed that the number of correct responses to faces was significantly higher than to shapes in the “self” and “close-other” conditions  = 25.374, P < 0.001, η2 = 0.159) and a significant two-way interaction: “condition” × “type of stimulus”  = 36.036, P < 0.001, η2 = 0.230). Post hoc tests of the “condition” factor showed that RTs in the “self” condition were substantially shorter than in the “close-other” (P < 0.001) and “unknown” conditions (P < 0.001). However, this pattern of findings was driven mainly by RTs to faces. Post hoc tests of the “condition” × “type of stimulus” interaction revealed significantly shorter RTs to the self-face than to the close-other’s face  and unknown faces , as well as shorter RTs to the close-other’s face than to unknown faces . In contrast, in the case of shapes, all differences between conditions were non-significant .A repeated-measures ANOVA conducted on mean RTs revealed the significant main effect of “condition” . The opposite effect, i.e. longer RTs, was observed for unknown faces when compared to unknown shapes , but no significant differences were found in the case of the “close-other” condition .Post hoc tests of the “condition” × “type of stimulus” interaction also showed that RTs to the self-face were significantly shorter than to the self-assigned shape  = 27.004, P < 0.001, η2 = 0.466), “condition”  = 32.288, P < 0.001, η2  = 0.510), and a significant 2-way “condition” × “type of stimulus” interaction  = 15.514, P < 0.001. η2 = 0.334). The significance of the “type of stimulus” factor indicated that the P300 amplitude for faces was significantly higher than for shapes  as well as for stimuli associated with the close-other (P < 0.001). The difference between stimuli associated with self and close-other was non-significant (P = 0.221).Post-hoc analyses for the “condition” factor showed that the unknown stimuli evoked significantly lower P3 than stimuli associated with the self . This was also the case for the close-other condition: P3 to the close-other’s face was larger than P3 to the shape assigned to a close-other . Such an effect was not observed for unknown stimuli, as the difference between unknown faces and unknown shapes was not significant . Moreover, P3 amplitude was significantly increased for self-face in comparison to close-other’s face  as well as in comparison to unknown faces , and for close-other’s face compared to unknown faces . P3 amplitudes to the self-assigned shape and the close-other assigned shape did not differ . However, unknown shapes were associated with lower P3 amplitude than the shape assigned to the self  and to the close-other .Post hoc tests performed for the two-way “condition” × “type of stimulus” interaction revealed that the self-face was associated with significantly higher P3 than the shape assigned to the self  are processed preferentially . RTs analyses revealed that all self-related stimuli were characterized by shorter RTs in comparison to the other stimuli (i.e. referring to the close-other and unknown people). Additionally, it should be highlighted that this result was primarily driven by the RTs to faces—significant differences were observed between self-face and other faces, but not between shapes. Moreover, RTs to the self-face were significantly shorter than to the self-assigned shape. Such faster reactions to faces than to shapes were not observed in the case of the “close-other” and “unknown” conditions.Based on numerous studies showing behavioral indices of the self-prioritization effect for new information that is arbitrarily associated with the self , one migTo start with, one of the differences refers to the presence vs. absence of labels. Specifically, in the matching trials of the self-prioritization task used in previous studies, participants were processing not only self-associated arbitrary stimuli but also familiar verbal labels with a pre-experimentally established meaning. Therefore, the self-advantage may be caused by the familiarity of the labels, rather than the self-association of the shape. In contrast, the present study did not present labels with shapes. However, it should be noted that the self-prioritization effect in RTs was also observed in the absence of any pre-experimentally familiar stimuli related to the self . Thus, tCrucially, the behavioral tasks performed by participants were different. While in previous studies, it was the perceptual matching task, in the current study it was the discrimination of familiar vs. unfamiliar stimuli. Thus a question may arise whether participants ignored self- and close-other associations when responding to the shapes and simply re-conceptualized those shapes as simply representing the category “familiar.” Such strategy was fully efficient in successfully accomplishing the task and accuracy rates seemed to support this view as they did not differ for self- and close-other assigned shapes.It is worth noting that the self-relevance facilitated stimulus processing only when task sets drew attention to previously formed shape-label associations . CompareEven so, we found faster reactions to one’s own face than to other faces. Our RTs results are in line with the findings reported in numerous studies, typically reporting shorter RTs to the self vs. other faces . In a reOn the neural level, the ERP findings differed for faces and shapes. First of all, amplitudes of P3 to faces were—in general—higher than amplitudes of P3 to shapes. As the P3 component is linked to the cognitive evaluation of stimulus significance , this fiHowever, the pattern of findings was different in the case of shapes. P3 to the self-assigned shape and P3 to the close-other assigned shape were similar, and both were larger than P3 to unknown shapes. Moreover, nonparametric cluster-based permutation tests showed significant clusters for comparisons of “self vs. ‘unknown’ and ‘close-other” vs. “unknown” conditions, but no significant cluster was detected for the self-assigned shape when compared to the close-other assigned shape. The latter is in line with the lack of differences in P3 amplitude between the self- and close-other conditions.Due to methodological differences, it is rather difficult to directly compare our P3 findings for the self-assigned shape to previous ERP findings on the processing of newly acquired self-related information . In WoźnHowever, our P3 results to faces corroborate the findings of previous studies reporting enhanced P3 to the self-face in comparison with other (either familiar or unfamiliar) faces . Such anIn the present study, one of the main differences between the processing of faces and shapes referred to the relation between the “self” and “close-other” conditions. While those two conditions differed in the case of faces, they did not differ in the case of shapes, as indicated by significant differences found both in the neural underpinnings and in RTs for faces and lack of such differences for shapes. The most obvious explanation of this dissociation refers to the familiarity of processed information. Specifically, the self-advantage found for the self-face vs. close-other’s face comparison was not observed if levels of familiarity of information referring to the self and the close-other were strictly equalized, as it was done for shapes. Thus, our P3 findings for the “self” and “close-other” conditions may be driven by the higher pre-experimental familiarity of one’s own face than the close-other’s face.Moreover, this pattern of P3 findings may also be interpreted in reference to the attentional processing of information related to the self and close-other. Specifically, it has been proposed that the mechanisms boosting the prioritized processing of self-relevant information could be driven by automatic capture of attention and prioritized allocation of attention to self-related stimuli review . Indeed,In general, interpretations of P3 findings referring to attentional processes are in line with the notion that P3 reflects stimulus processing only, i.e. with the view that P3 is a signature of a comprehensive evaluation of incoming stimuli . This evMoreover, one may view P3 findings reported in the current study in the light of the locus coeruleus-noradrenergic (LC-NE) system activity. The pivotal role of the LC-NE system in regulating task engagement is well documented . ThroughSo far, we viewed our P3 findings in the light of attentional mechanisms. However, this view may be complemented by the interpretation of P3 as reflecting processing of stimuli that are highly arousing in nature . These tAs the P3 component reflects the cognitive evaluation of stimulus significance , differeThe limitations of the current study are as follows. Shapes were arbitrarily assigned to one’s own person and to the chosen close-other, and their processing was compared to unknown shapes. As those familiar conditions (the self and the close-other) are personally relevant, it is a matter of debate whether similar—or rather dissimilar—patterns of behavioral and electrophysiological findings would be observed for shapes assigned to famous people. Therefore, the inclusion of such an additional experimental condition would provide a more global view on the processing of newly acquired information referring to the self and others. Moreover, our study did not provide an answer to the question of whether the self-relevance of newly acquired information triggers the self-representation in the brain, similarly to highly familiar self-referential information (self-face). In order to adequately relate to this issue, some source analyses  should be done. However, due to rather a low number of experimental trials, such analyses—in the case of our present study—would be not very reliable. Future EEG studies may investigate whether newly learned and long-term established self-related information are represented in the same (or overlapping) neural network in the brain.In conclusion, P3 and permutation test results revealed a clear self-advantage in the case of faces, i.e. significant differences between the processing of the self-face and other faces . These findings may be viewed in the light of preferential attention allocation to highly familiar and well-established self-referential information. However, the processing of new information arbitrarily assigned to one’s own person and the close-other did not differ. We propose that this effect is mainly driven by similar attentional biases to self- and close-other assigned shapes.


From the assumption that the residual seismic data (the difference between the modeled and observed data) obeys the Rényi α-Gaussian probability distribution, we introduce an outlier-resistant criterion to deal with erratic measures in the FWI context, in which the classical FWI based on l2-norm is a particular case. The new misfit function arises from the probabilistic maximum-likelihood method associated with the α-Gaussian distribution. The PGF technique works on the forward modeling process by dividing the computational domain into outside target area and target area, where the wave equation is solved only once on the outside target (before FWI). During the FWI processing, Green’s functions related only to the target area are computed instead of the entire computational domain, saving computational efforts. We show the effectiveness of our proposed approach by considering two distinct realistic P-wave velocity models, in which the first one is inspired in the Kwanza Basin in Angola and the second in a region of great economic interest in the Brazilian pre-salt field. We call our proposal by the abbreviation α-PGF-FWI. The results reveal that the α-PGF-FWI is robust against additive Gaussian noise and non-Gaussian noise with outliers in the limit α → 2/3, being α the Rényi entropic index.The estimation of physical parameters from data analyses is a crucial process for the description and modeling of many complex systems. Based on Rényi  Full-waveform inversion (FWI) is a powerful methodology to estimate subsurface physical parameters by exploring the complete waveforms recorded in a seismic survey , 2. FromDespite the FWI potentials, it is inherently an ill-posed problem in the sense of Hadamard, which means that at least one of the following features is violated: the solution (i) exists, (ii) is unique; and (iii) depends continuously on the observed data. The characteristic (ii) is commonly violated in FWI because in a typical geophysical survey there is information only on the positions associated with the seismic receivers that cover a small area of the region of interest, which leads the FWI to solve an inconsistent and overdetermined system of equations. Regarding the characteristic (iii), the FWI solution may be unstable, as a small variation in the noise level of the observed data can lead to discontinuous changes in the reconstructed subsurface model . Furtherl1-norm of the error. Such criterion is based on the assumption that the errors obey the Laplace distribution. Its success is associated with the long tails of the Laplace distribution ,–66,α-entgiven by :Zα=1-απx are independent and identically distributed by the α-Gaussian distribution  is formulated as the following minimization task:Aψs(ω) = Ss(ω). Thus, the solution of the optimization problem formulated in )]+subject α-adjoint equation,α → 1.Computing the stationary point of )†Λs(ω),∂Lα).We noticSs(ω)).2We noticthat Eqs , respectthat Eqs . In thisgrangian , we obtaα-misfit function is given by:α-misfit function is computed by solving the wave equation and correlating it with the solution Λ of the wave equation in α → 1.Thus, the function  is a parα-misfit function mitigates the effects of non-Gaussian errors, specially the outliers. It is possible to see such behavior through the analysis of the α-adjoint-source , we notice that the adjoint wavefield tends to zero (Λs → 0), and therefore ∇mϕα(m)→0 in that case. Indeed, the α-misfit function magnifies small errors and suppresses large ones, since the α-adjoint-source −ds,r for small errors and inverse of the errors Δds,r, Λs ∝ 1/Δds,r for large ones. The classical approach (α → 1), on the other hand, linearly suppresses small errors and magnifies large errors , which explain the sensitivity of this approach to non-Gaussian errors [In contrast to the classical approach, the t-source  is propon errors .A is given by the inverse of the Green’s function G−1, ψs is the discretized pressure wavefield, and h is the grid spacing (in meters), Fω is the amplitude of the Fourier coefficients of the seismic source and Us is a column vector with zero elements everywhere except on the source position xs representing δ(x−xs).As discussed earlier, solving the wave equation is a critical issue of the FWI technique. In this way, the patched Green’s function (PGF) technique appears as a powerful alternative to reduce the computational cost of imaging problems based on wave equation. In the PGF framework, the solution of the frequency-domain wave equation  and at the position of sources (s) and receivers (r). The subscripts e and i refer, respectively, to the elements of the outer edge and the inner edge of the target region, connected through the employment of the potential Vi,e. Eqs -1ges,Gts=gtiViVieGes.where tht and the blue squares represent the area outside the target region t′, in which h is the grid spacing. The PGF method is performed in two steps: First, we find the element that connects the edges of the target area e and i  algorithm [l-BFGS optimizer fetches an informative  minimum of the misfit function using its gradient ∇mϕ(m). In this way, the l-BFGS computes an approximation of the inverse Hessian matrix H−1 from previous gradients by imposing a secant condition [β along the descent-direction search of the gradient, which satisfies the Wolfe conditions [μp is the peak frequency.To illustrate how the lgorithm  to solvelgorithm . The l-Bondition . Moreovenditions . In addinditions , which ind and 8th order approximations for time and space, respectively, in a regular grid with 5m spacing with a convolution-perfectly matched layer (C-PML) [nd and 4th order approximations for time and space, respectively, in a regular grid with 10m spacing with perfectly matched layer (PML) absorbing layers [m spacing with C-PML absorbing boundaries [To avoid the inversion crime in all numerical experiments, we generated the observed dataset in the time domain by employing the finite difference method with 2 (C-PML) , 75. In g layers  for the undaries .α-FWI regarding erratic data (outliers), we consider a 2D acoustic velocity model which is widely used in geophysical imaging tests, named the Marmousi model. Such a model presents a complex velocity geometry, as depicted in m in-depth with 183 equally spaced seismic sources spaced 48m apart, from 216m to 8952m, in which we employ a Ricker wavelet with μp = 5Hz as the seismic source. Furthermore, we consider 23 equally spaced receivers located every 408m, from 96m to 9072m, deployed at the ocean floor (240m in-depth) to simulate an ocean bottom nodes (OBN) acquisition, which is a marine type seismic acquisition often used in the last years. The recording time was 5s. To analyze the robustness of the dB and 60dB, respectively, as depicted in SNR = 70dB and that only one of the seismic traces from each receiver is contaminated by outliers  and Gaussian noise to simulate a realistic circumstance like to Ref. . In the α → 1), and the last seven ones are based on the α-misfit function with α = 0.35, 0.45, 0.55, 0.65, 0.75, 0.85, and 0.95. In all numerical experiments, we inverted for 200 l-BFGS iterations from the above-described geometry seismic acquisition and the initial model depicted in For each noisy-scenario, we carried out eight data inversions, in which the first one refers to the classical approach  the Pearson product-moment correlation coefficient ; and (ii) the normalized root-mean-square (NRMS). The Pearson’s R measures the linear correlation between the reconstructed model and the true model, in which it varies between −1 and 1, inclusive. In this regard, R = 1 means a strong correlation between the two models and, in our case, R-values far from 1 mean low correlation. The NRMS varies from 0 (perfect model) to ∞ (bad model) and is defined as:ctrue is true model and cFWI corresponds to the FWI result. The statistical measures for the noiseless data case (first scenario) are summarized in R ≥ 0.8, according to the strength-scale suggested by Ref. [α = 0.55 and α = 0.65 have the highest Pearson’s R and the lowest NRMS value. by Ref. ), althouα-FWI results are very satisfactory as the resulting models are close to the true model. In fact, regardless of the α-value, the reconstructed P-wave models present the main structures of the Marmousi model despite the imprint of the noise. Again, case α = 0.55 has a higher correlation and lower NRMS error compared to the true model, as summarized in Figs α → 1) completely fails to reconstruct a P-wave velocity model, as depicted in α-value decreases, which means a greater deviation from the Gaussian statistics, the resulting model is more suitable , the misfit function decay is more accentuated  and α = 0.35 cases to compare our proposal with the classical approach. We notice that the water layer in all P-wave velocity models are assumed to be known and, therefore, are kept constant during the inversion process.In this section, we explore the potential of the α-PGF-FWI. In the first one, we consider the noiseless data set generated by the acquisition geometry described previously. In the second one, we consider the same data set, but polluted by non-Gaussian noise. The non-Gaussian perturbations were generated through a Student’s t-distribution with three degrees of freedom. We solve the forward problem by solving the 2-D frequency-domain acoustic wave equation using the finite-difference method, in which we consider a 9-point stencil with a C-PML boundary condition [The first two experiments consist of imaging the pre-salt region, which remains a great challenge, using the ondition , 75 in okm/s), and a linear velocity increasing, from 5.4km in-depth, rang the second frequencies group. Again, the resulting model is used as the initial model for the third group. For each frequency group, we compute 20 l-BFGS iterations.We discretize the Helmholtz operator on a regular grid with a spacing of 25 m. In the inversion process, we consider a multiscale approach , 84 in wkm to 8.5km in-depth and horizontally from 2.5km to 17.5km, as well-demarcated in Figs α-PGF-FWI results for the noiseless and non-Gaussian noise cases, respectively, in which the left column refers to the classical approach (α → 1) and the right column refers to our proposal with α = 0.35. We notice in The target region consists of a rectangular area, mostly composed of the Brazilian pre-salt, which comprises a region from 4.5α-PGF-FWI with α = 0.35 decays more accentuated . It is worth mentioning that if the observed data does not contain noise or is polluted by Gaussian noise, α-PGF-FWI and classical PGF-FWI have similar performances. However, the α-PGF-FWI is robust to non-Gaussian noise, while the classical PGF-FWI estimates biased models in these circumstances. The reason is that large errors are better attenuated in the adjoint-source of our proposal compared to the classical approach, leading to a better weighting of waveforms crossing the less illuminated regions of the model. Indeed, the classical misfit function treats all residuals equally, while the proposed misfit function weights residual data according to their amplitude, giving less importance to large errors in the dataset.We notice that the α-PGF-FWI is a powerful methodology to deal with non-Gaussian errors, which may become a valuable tool in geophysical imaging problems, especially when the data is polluted by large errors. It is also worth noting that our proposal can be applied to any problem of estimating physical parameters linked to partial differential equations similar to the Helmholtz equation, such as in biomedical imaging issues [The numerical results show that the g issues .Plots and numerical simulations were done, respectively, with MATLAB R2016b and Julia Language, on a computer hosting a quad-core (Intel Xeon E5-1620 v3) processor at 3.50GHz and 256GB RAM memories.


The Eating Disorder Examination Questionnaire (EDE-Q) is a frequently used scale to evaluate eating behaviors and attitudes. In recent years, its use has increased due to the fact that the use of short forms is more practical. The aim of this study was to validate the short Turkish form of the Eating Disorder Examination Questionnaire (EDE-Q) including 13 items.The study included 924 adults at a mean age of 30.3 ± 10.93 years. EDEQ-13 was translated and adapted to Turkish according to the Beaton Guidelines. The Eating Attitudes Test-26 (EAT-26), the Satisfaction with Life Scale (SWLS), and the Body Appreciation Scale (BAS) were used to analyze their relationships to EDE-Q-13.In this study, the rate of the total variance explained by the factors of EDE-Q-13 according to the Explanatory Factor Analysis (EFA) results of the scale was 83.54%. The Cronbach's alpha value of the scale was 0.89, and the Cronbach's alpha values of the 5 subscales were calculated in the range of 0.75–0.94. The criterion validity analysis showed an acceptable correlation between EDE-Q-13 and EAT-26, SWLS, and BAS. The confirmatory factor analysis (CFA) revealed that the model had fit values that were acceptable or good.Both EFA and CFA results showed that it is appropriate to use the Turkish EDE-Q-13. EDE-Q-13 was significantly correlated with eating disturbances, body appreciation, and life satisfaction. In conclusion, the Turkish version of EDE-Q-13 possesses high levels of validity and reliability. The rate of eating disorders has been increasing in recent years. Although there are many scales in the evaluation of eating disorders, one of the most frequently used scales is the Eating Disorder Examination Questionnaire (EDE-Q). In recent years, its use has increased due to the fact that the use of short forms is more practical. The aim of this study was to validate the short Turkish form of EDE-Q. The study included 924 adults. Our results showed that it is appropriate to use the Turkish EDE-Q-13. EDE-Q-13 was significantly correlated with eating disturbances, body appreciation, and life satisfaction. In conclusion, the Turkish version of EDE-Q-13 possesses high levels of validity and reliability. Eating disorders are a group of chronic psychiatric diseases characterized by severe disturbances in eating behaviors originating from the interactions of genetic, biological, psychological, and socio-cultural factors . AlthougVarious scales are used in the evaluation of the variety and severity of eating disorder symptoms, and the most widely used instrument among these scales is the Eating Disorder Examination Questionnaire (EDE-Q), which is a 28-item self-report instrument developed by Fairburn and Beglin . EDE-Q hShorter versions of EDE-Q have been developed because the 4-factor structure of the original questionnaire was not confirmed. It was also not economical in terms of time due to the large number of items , 21. TheConsidering that there is an alarming increase in eating disorders in Turkey, especially in young people , it is iAll participants were Turkish-speaking and Turkish national adults living in Turkey. Non-Turkish individuals, non-adult individuals, and those who did not agree to participate were not included in the study. Simple random sampling was used to select participants. Before the study, a power analysis was performed using the G*Power software to determine the number of individuals to be included in the sample. Based on alpha (α) = 0.05, power (1 − β) = 0.80, and the considered effect size, the minimum required sample size was calculated as 560 participants, and it was decided that at least 616 individuals should be recruited with the possibility of 10% data loss. A questionnaire was used as the data collection tool. The questionnaire was disseminated via e-mail or WhatsApp to the individuals who agreed to participate in the study. The participants were able to fill out the online questionnaire after they agreed to participate in the study.2 , and 52.5% had a normal body weight according to the BMI classification. The test–retest reliability coefficient of the scale was found to be 0.93.The study included 924 participants  between the ages of 18 and 64. Their mean age was 30.3 ± 10.93 years, 42.0% of the participants were students, and 20.3% worked in the private sector. The mean BMI of the participants was 24.3 ± 4.51 kg/mEDE-Q was developed by Fairburn and Beglin  to evaluEDE-Q-13 consists of 13 items and 5 subscales, namely Eating Restraint (ER), Shape and Weight Over-Evaluation (SWO), Body Dissatisfaction (BD), Bingeing, and Purging . Its valThe Eating Attitudes Test—Short Form (EAT-26) was developed by Garner et al. . ErgüneyIn 1985, Diener et al.  developeThe Body Appreciation Scale (BAS) was developed by Avalos et al.  to measuThe validity and reliability study of the scale in Turkish was performed by Bakalım et al. . The Tur2. According to the World Health Organization (WHO) [2 (underweight), 18.50–24.99 kg/m2 , 25.00–29.99 kg/m2 (pre-obese), and ≥ 30.00 kg/m2 (obese).The height and body weight values of the participants were recorded based on self-report. Body Mass Index (BMI) values were calculated with the following equation: 'BMI = Body weight (kg)/(height (m))on (WHO) , BMI is The demographic characteristics of the participants, their age, height, and body weight were recorded based on self-report. Several scales with established validity and reliability in Turkish were used to determine the criterion validity of the scale, including the Eating Attitudes Test-Short Form (EAT-26) , 28, theFor the validity and reliability study of the Turkish EDE-Q-13, permission was obtained from Lev-Ari  via e-map-value smaller than 0.05 was considered statistically significant.The data obtained in this study were analyzed using the SPSS  for Windows 25.0 program. Descriptive statistical methods were used for data analysis. To determine the normality of the distribution of the data, Skewness and Kurtosis Tests and Z-values were used. The Cronbach’s alpha value was calculated using the SPSS program to determine internal consistency indicating reliability. A value of Cronbach's alpha should be at least 0.60 to be acceptable, and a good value is considered to be 0.70 or above . A p-val2/df ≤ 3.0, RMSEA ≤ 0.05, 0.90 ≤ GFI, 0.95 ≤ AGFI, 0.95 ≤ CFI, 0.95 ≤ NFI and 0.95 ≤ TLI indicate good fit, and 3 ≤ χ2/df ≤ 5, 0.05 ≤ RMSEA ≤ 0.08, 0.80 ≤ GFI ≤ 0.90, 0.85 ≤ AGFI ≤ 0.95, 0.85 ≤ CFI ≤ 0.95, 0.80 ≤ NFI ≤ 0.95, and 0.80 ≤ TLI ≤ 0.95 indicate acceptable fit according to conventions for Model Confirmatory Factor Analysis Fit Indices [Explanatory Factor Analysis (EFA) and Confirmatory Factor Analysis (CFA) were performed to assess the reliability and construct validity of the adapted scale. The CFA in this study was performed using the AMOS-24 software. The following parameters were examined within the scope of CFA: Kaiser–Meyer–Olkin (KMO) value and multiple fit indices including Root Mean Square Error of Approximation (RMSEA), Goodness-of-Fit Index (GFI), Adjusted Goodness-of-Fit Index (AGFI), Comparative Fit Index (CFI), Normed Fit Index (NFI), Tucker–Lewis Index (TLI). In CFA, χ Indices .2(78) = 8694.05; KMO = 0.84 p < 0.01). Accordingly, it was assumed that the data had multivariate normal distribution. The total variance explained by the factors was 83.54%.The results of the EFA and Cronbach’s alpha internal consistency coefficient of EDE-Q-13 are presented in Table A Cronbach’s alpha coefficient above 0.70 is considered sufficient for reliability. In this study, as seen in Table 2/df = 4.90, and AGFI = 0.93.The model goodness-of-fit statistics of the scale were found as follows: GFI = 0.95, CFI = 0.97, NFI = 0.96, and TLI = 0.96. These statistics were found to show good fit based on recommended thresholds. The other indices reached acceptable fit levels as: RMSEA = 0.06, χThe t-statistics for the items showed that all items were significant. The accepted limit for factor loading values was determined as 0.40. In the analyses, no item with a score below 0.40 was identified, and the factor loadings were within acceptable limits. The CFA results of EDE-Q-13 are shown in Fig. Correlations between the EDE-Q-13 subscales and EAT-26, BAS, SWLS, and BMI are given in Table This study was conducted to establish the validity and reliability of the Turkish version of EDE-Q-13, which was originally developed by Lev-Ari . The relThe results of the present study support the original 13-item and 5-factor structure  of EDE-QCFA values and construct validity of short versions of EDE-Q are very important. In this study, according to recommended thresholds in the literature, the goodness-of-fit index values of the scale were determined to indicate a good or acceptable fit. In contrast, for EDE-Q-8  and EDE-This study examined the properties of EDE-Q-13 in terms of its evaluations of eating disorder pathologies as measured by EAT-26. There was a significant positive correlation between the EDE-Q-13 scores of the participants and their EAT-26  scores, and this result was in line with the results of previous studies , 40. TheThis study had some limitations. First, the assessment of all scales was based on self-report. Second, the body weights and heights of the individuals were also recorded based on their self-reports. Third, the sample was limited to some demographic characteristics. Although the aim was to include equal numbers of male and female participants, the majority of the individuals who agreed to participate in the study consisted of female individuals. Additionally, EDE-Q was validated in clinical and non-clinical samples . Due to Despite some limitations, the validity and reliability of a short form of EDE-Q were demonstrated for the first time in a Turkish sample with this study. In the present study, it was determined that the Turkish version of EDE-Q-13 possesses high levels of validity and reliability. Both EFA and CFA results showed that it is appropriate to use the Turkish version of EDE-Q-13. EDE-Q-13 takes less time because it is approximately 50% shorter than the original EDE-Q, thereby reducing the burden on both the researcher and the participants. Therefore, it may also prevent the participants from withdrawing due to the time burden. It was also thought that EDE-Q-13 has an advantage because it includes Bingeing and Purging items as a Likert-type scale, which are not available in other short versions, but they were asked in an open-ended form in the original scale .Scales that are used in the identification of eating disorders, whose prevalence has increased in recent years, are important in preventing the risk of eating disorders. In this study, it was determined that the Turkish version of EDE-Q-13 showed high levels of validity and reliability. The Turkish version of EDE-Q-13 is considered particularly useful for epidemiological studies and large-scale surveys, as it is short, user-friendly, and reliable for assessing eating disorder symptoms, investigating risk factors, and examining the psychological and physical consequences of eating disorders in the Turkish population.