Kamran Behnia

On the origin and the amplitude of T-square resistivity in Fermi liquids

In 1937, Baber, Landau and Pomeranchuk postulated that collisions between electrons generates a contribution to the electric resistivity of metals with a distinct T 2 temperature dependence. The amplitude of this term is small in common metals, but dominant in metals hosting either heavy carriers or a low concentration of them. The link between the temperature dependence and the size of the scattering phase space is straightforward, but not the microscopic source of dissipation. To explain how electron-electron collisions lead to momentum leak, Umklapp events or multiple electron reservoirs have been invoked. This interpretation is challenged by a number of experimental observations: the persistence of T-square resistivity in dilute metals (in which the two mechanisms are irrelevant), the successful extension of Kadowaki-Woods scaling to dilute metals, and the observation of a size-dependent T-square thermal resistivity (T /κ) and its Wiedemann-Franz (WF) correlation with T-square electrical resistivity. This paper argues that much insight is provided by the case of normal liquid 3 He where the T-square temperature dependence of energy and momentum diffusivity is driven by fermion-fermion collisions. The amplitude of T-square resistivity in 3 He and in metals share a common scaling. Thus, the ubiquitous T-square electrical resistivity ultimately stems from the Fermi-liquid temperature dependence of momentum diffusivity.

The contribution of electron-electron scattering to the temperature dependence of electrical conductivity in conducting solids has a long history, which begins in the late 1930s with its theoretical identification by Baber [1] and (contemporaneously and independently) by Landau and Pomeranchuk [1,2]. The concept began to attract significant interest several decades later [3,4] with the accumulation of experimental data. In common metals, this effect is not easy to observe and remains small compared with the contribution due to electron-phonon scattering. The discovery of metals hosting strongly-correlated electrons changed the experimental landscape in 1980s. In heavy-fermion systems [5] electron-electron scattering dominates the temperature dependence of resistivity at cryogenic temperatures. Kadowaki and Woods [6] (following an idea first put forward by Rice about transition metals [3]) observed that the prefactor of T-square resistivity scales with the square of the electronic specific heat. This scaling, based on the argument that both quantities are amplified by enhancement in the density of states proved influential during decades of experimental research on heavy fermion metals (for an early example see [7]). It was also used as a road-map to track the emergence of what was dubbed 'Non-Fermi-liquid behavior' in strongly-correlated-electron systems [8]. The microscopic origin of dissipation did not attract much debate. It was assumed that Umklapp events allowed by the large Fermi surface of the metallic solids of interest are what drive the T-square resistivity [9].

Recently, a modified version of the Rice or Kadowaki-Woods scaling was proposed to explain the amplitude of the T-square prefactor in dilute metals, which are zerotemperature conductors hosting a very low concentration of mobile electrons [10][11][12]. These are either stoichiometric semi-metals (such as bismuth) or semiconductors sufficiently doped to be on the metallic side of the metalinsulator transition (such as oxygen-reduced strontium titanate). In their case, the relevant scaling is between the prefactor of T-square resistivity and the inverse of the square of the Fermi energy (and not the specific heat which is contaminated by the unusually low carrier density).

Two new experimental developments have accompanied the renewal of interest for this theme (see ref. [13] for a review of possible origins of T-square resistivity prior to them). The first is the observation that T-square resistivity persists even when the microscopic origin of the dissipation mechanism cannot be identified [10,12]. The second is the observation of T-square thermal resistivity in semi-metals close to the ballistic regime [14,15]. They raise a number of questions: Is the T-square resistivity in dilute metallic strontium titanate [10,[16][17][18] due to electron-electron scattering ? If yes, given the absence of Umklapp events and multiple pockets, what causes it? If not, what kind of electron-phonon scattering mechanism can generate a T-square resistivity [19]? A similar question was raised long ago in the case of bismuth. The observed T-square resistivity to was attributed to scattering between electrons residing in different pockets [20]. But this was subsequently contested by other authors [21,22]. It is raised again by the observation of T-square resistivity in another dilute metallic solid, namely Bi 2 O 2 Se [12], implying that SrTiO 3 [10] is not a lonely anomaly. More fundamentally and beyond these specific cases, What is the reason behind the success of scaling à la Kadowaki-Woods across a wide variety of families of materials (provided that one takes in to account the carrier density)? How does this fit with the assumption that the microscopic mechanism generating momentum decay from electron-electron collisions varies from one case to the other?

The aim of the present paper is to argue in favor of a route from electron-electron scattering to T-square resistivity in metals starting from our understanding of arXiv:2112.11092v3 [cond-mat.str-el] 26 Feb 2022 thermal transport in liquid 3 He [23-25, 27? ]. It has been known for decades that, well below the degeneracy temperature of ∼ 1.5K, collisions between fermions in liquid 3 He generate a thermal conductivity which decreases with the inverse of temperature [28][29][30][31]. This is strictly equivalent to the T-square thermal resistivity of electrons, as defined by metal physicists. This picture has no need for Umklapp events, or a bottleneck provided by multiple fermionic reservoirs. Its relevance to electrons in metals was experimentally demonstrated in the case of semi-metallic antimony [15]. Its potentially broader relevance would solve a number of pending puzzles in other specific materials.

I will begin by a short historical account of experimental and theoretical investigations of T-square electrical resistivity. Then, I will recall (much less frequent) studies detecting the thermal counterpart of T-square resistivity. The next section is devoted to normal liquid 3 He. Greywell's extended study of the evolution of specific heat [32] and thermal conductivity [32] with pressure is reviewed. It remains the most detailed experimental verification of the theory of momentum and energy diffusivity in Fermi liquids. By comparing electrons in metals (in presence of lattice and its disorder) and fermions in 3 He (in absence of both) a common scaling emerges. This leaves little doubt regarding the main driver of T-square thermal resistivity. The neglected role of viscosity of the electron liquid [33,[START_REF] Giuliani | Quantum Theory of the Electron Liquid[END_REF] in the emergence of T-square electrical resistivity is highlighted by this analogy.

I. Ubiquity and scalability of T-square resistivity in Fermi liquids

Fig. 1 shows the low-temperature resistivity of four different metals. Bismuth and graphite are elemental semimetals. Sr 2 RuO 4 and UPt 3 are correlated metals each with an unconventional superconducting ground state. Low-temperature resistivity of these four solids quadratically increases with temperature, following :

ρ = ρ 0 + AT 2 (1)
The first term on the right side is residual resistivity, ρ 0 , attributed to impurity scattering. The second term, A (in ohm.m.K -2 ), is attributed to electron-electron scattering. It will be the focus of this paper. The lowtemperature resistivity of these four solids lacks any trace of T 5 resistivity caused by electron-phonon scattering in the Bloch-Grüneisen picture [START_REF] Ziman | Principles of the Theory of Solids[END_REF]. This is because A is large enough to impede its observation. In contrast, A is orders of magnitude smaller in noble or alkali metals and temperature-dependent resistivity is dominantly T 5 .

What sets the amplitude of A in a given metal? Does it correlate with ρ 0 ? Numerous experiments point to a negative answer to the latter question and indicate that A is an intrinsic property of a Fermi liquid. In many heavy-fermion metals, the quality of crystals have improved over the years, pulling down ρ 0 by more than one order of magnitude, yet A is similar in clean and dirty crystals. In URu 2 Si 2 , for example,in 1986, Palstra el al. [START_REF] Palstra | [END_REF] found ρ 0 = 33µΩ.cm and A = 0.1µΩ.cmK -2 . A quarter of century later, working on much cleaner crystals, Matsuda et al. [39] reported ρ 0 = 1.05µΩ.cm and A = 0.099µΩ.cmK -2 . The same is true for UPt 3 , where the magnitude of A remains the same in crystals in which ρ 0 differe by one order of magnitude [40]. One can go also along the opposite direction by introducing disorder through electronic irradiation, as in the case of metallic strontium titanate, where a twofold enhancement in ρ 0 leaves A unchanged [41].

The intrinsic nature of A is backed by the success of scaling approaches, which was initiated by Rice in 1968 [3]. He demonstrated that A scales with the square of the electronic T-linear specific heat γ (whose unit is J.K -1 .mol -2 ) in seven elemental transition metals. 18 years later, Kadowaki and Woods (KW) [6] applied this scaling to the newly-discovered Heavy-fermion metals and found that a similar scaling works there too. They also noticed a difference in the amplitude of the ratio. In heavy fermions, it is ( A γ 2 = 10 -7 Ω.m( mol.K J ) 2 ). This is more than one order of magnitude larger than what Rice had found in transition metals ( A γ 2 = 4 × 10 -9 Ω.m( mol.K J ) 2 ). Theses historical plots are reproduced in Fig. 2.

This difference between strongly-correlated and weakly-correlated systems became fuzzier as the data became more abundant. In 2003, Tsuiji et al. [42] showed that most Yb-based heavy fermions are closer to the Rice scaling line than to the KW scaling line. Nevertheless, this scaling approach is impressively effective over eight to nine orders of magnitude. It implied that one could predict the rough amplitude of one measurable quantity (namely A), thanks to knowing the amplitude of another measurable quantity (namely γ).

The Pauli exclusion principle is the ultimate reason behind both the temperature dependence of electronic specific heat and electron-electron contribution to the resistivity. Electrons giving rise to both properties are those confined to a thermal window near the Fermi energy. This leads to the following expression for the electron specific heat (per volume):

C e T = γ = π 2 2 nk 2 B E F ( 2
)
Here n is the carrier density. Now, the phase space for collisions between two electrons is proportional to

( k B T E F ) 2 . Since A is expressed in Ωm K -2
, it can be written as [10]:

A = e 2 ( k B E F ) 2 quad (3)
1 0 3 1 0 4 1 0 2 1 0 5 1 0 -3 1 0 -2 1 0 -1 1 0 0 1 0 1 1 0 -2 1 0 -1 1 0 0 1 0 1 1 0 -3 1 0 -2 1 0 -1 1 0 0 1 0 1 W A l M o O s P d A l W M o P d S b B i A ( n Ω c m K -2 ) E F / K B ( K ) B i C ( g r a p h i t e ) O s b ) S b C ( g r a p h i t e ) A ( n Ω c m K -2 ) γ ( m J / K 2 m o l ) a )
FIG. 3. The proper scaling: a) A vs. γ in several metallic elements including low-density semi-metals. The blue solid line represents A ∝ γ 2 . b) A of the same metallic elements vs. the Fermi energy, EF , in kelvin. The red solid line represents

A ∝ E -2 F .
Here, and e are the Planck constant (divided by 2π) and the fundamental charge. The material-dependent parameter quad has the dimension of a length and condenses all the microscopic details of scattering.

According to these equations, γ ∝ E -1 F and A ∝ E -2 F . As a consequence, the A∝ γ 2 , provided that quad does not drastically vary from one metal to another. At the same time, the scaling should collapse when the comparison is made between metals with widely different carrier densities. As seen in Fig. 3a, this is indeed the case. As soon as one adds bismuth (Bi), graphite (C) and antimony (Sb), to Rice's plot, the scaling does not work. In these semi-metals, a single electron is shared by thousands of atoms and the magnitude of γ is pulled down by the low carrier density. As a consequence, A decreases with γ.

In dilute metallic strontium titanate, where carrier density can be tuned by many orders of magnitude, the simple KW scaling between A and γ does not hold either [43]. The electronic specific heat with doping depends both on the effective mass, which increases by a factor of ≈ 2, but much more on the carrier density, which changes by several orders of magnitude. The Sommerfeld coefficient, γ is compatible with the experimental quantum oscillations [44] and calculated band structure [17]. The magnitude of A matches what was calculated by Hussey [45]. Nevertheless, A decreases with γ as it does when comparing Bi, graphite and Sb in Fig. 3a.

To include dilute metals in this scaling scheme, it is necessary to replace γ 2 by E 2 F as the horizontal axis [10]. The Fermi energy is a quantity accessible to the experimentalist. It can be quantified by measuring the Fermi radius and the effective mass of electrons, or by a combined knowledge of γ and n. It is true that a multi-band metal has several Fermi energies. In semi-metals, the Fermi energy of electrons and holes differ, because of the mismatch in their masses. However, since we are dealing with orders of magnitude, the average Fermi energy can do the job.

Fig. 3b, is a plot of A vs. E 2 F for the same elements plotted in Fig. 3a. They are listed together with their relevant specific properties in table I. The scaling operates across more than five orders of magnitude of A. This scaling was first put forward in ref. [10] and was extended in ref. [11] and ref. [12].

The simple KW scaling (i.e A/γ 2 ) for correlated metals fails also for dilute metals. A revealing case is CeNiSn, a low-density Kondo semi-metal [46]. The prefactor of its T-square resistivity is A = 2.2µΩcmK -2 , two orders of magnitude larger than what is expected according to its electronic specific heat, which is γ = 40mJ.mol -1 K -2 [47] combined with the KW A γ 2 universal ratio. However, this discrepancy evaporates if one extracts the Fermi energy from the slope of the Seebeck coefficient [48], as noticed by Kuitra et al. [49]. The latter is indeed adequately large in CeNiSn [50].

Hussey [45] was the first to put his finger on the inadequacy of the simple KW scaling. He was partly motivated by the case of Na 0.7 CoO 2 , in which an unexpectedly large A/γ 2 was resolved [51]. He proposed a modified version of the KW scaling in which A scales with the electronic specific heat per volume and not per mole. However, finding a general expression for A remains a challenge, since it is transport coefficient and cannot be derived out without specific and contestable assumptions about scattering details. An alternative amendment to the KW scaling [52] consisted in abandoning the idea of scaling A with a quantity directly accessible to the experiment.

Empirically, however, Eq. 3 gives a reasonable account of A in numerous metals over eight orders of magnitude [12]. The list includes numerous newly discovered semimetals, such as WTe 2 [53], which have attracted attention for other reasons. The material-dependent quad resides in the range of 1-40 nm [12]. Each family of metal clusters along specific values of quad . In elemental metals of Fig. 3, quad ≈ 2nm. In layered metals, such as Sr 2 RuO 4 [36] or La 1.7 Sr 0.3 Cu0 4 [54], the out-ofplane resistivity is orders of magnitude larger the inplane resistivity. As a consequence, A and quad are also anisotropic. In these cases, our focus will be the in-plane transport.

This brings us to the physical significance of this phenomenological length scale. Mott [START_REF] Mott | Metal-Insulator Transitions[END_REF] argued that A is proportional to the collision cross section of the two electrons, σ cs . Therefore:

quad ∝ k F σ cs (4)
The Fermi wavevector, k F , and the collision crosssection σ cs are expected to evolve in opposite directions with the evolution of carrier density. In a dilute metal, the Fermi wave-vector is short and the collision cross section (in real space) is large. The opposite is true in a dense metal. We note that quad in Bi and Al are roughly similar. We will come back to this question in section 5.

A fundamental question is the following: Why should electron-electron collision decay the charge flow ? Answers to this question fall in to two categories.

First: If the wave-vector of one of the two electrons after the collision is larger than the width of the Brillouin zone, then a finite amount of momentum (equal to G, where G is a unit vector of the reciprocal lattice) will be lost. This Umklapp option was what Landau and Pomeranchuk had it in mind in postulating T-square resistivity [2]. Umklapp events require a sufficiently large Fermi surface, which is the case of all metals, put under scrutiny by Rice [3] and by Kadowaki and Woods [6]. In the case of heavy-fermion metals the role of Umklapp events in generating T 2 -ressitivity was explicitly recognized by Maebashi and Fukuyama [9].

Second: If there are two distinct electron reservoirs, then momentum exchange between these two reservoirs may constitute a bottleneck for momentum loss. This was the scenario explicitly put forward by Baber [1] and was invoked by Hartman [20] in the case of bismuth in which the Fermi pockets are too small to allow Umklapp events.

However, none of these two options work for dilute metallic strontium titanate [10], which motivated a search for alternative mechanisms of T-square resistivity [19] generated by electron-phonon and not electronelectron scattering. It is true that strontium titanate is a non-trivial solid with peculiar phonon modes [START_REF] Yamada | [END_REF]. However, it is not the only case of T-square resistivity in absence of Umklapp or multiple pockets. Bi 2 O 2 Se [12] is yet another. Moreover, specifically tailored scenarios fail to explain the success of the scaling approach across many orders of magnitude irrespective of the presence or absence of Umklapp events.

In search of an alternative solution, let us turn our attention to heat transport.

II. The Wiedemann-Franz law and the T-square thermal resistivity

In the zero temperature limit, thermal conductivity, like specific heat and in contrast to electrical conductivity, vanishes. Disorder, or finite size lead to a finite residual electrical resistivity, ρ 0 and a residual T-linear thermal conductivity, κ0

T . Their ratio obeys the Wiedeamnn-Franz law :

ρ 0 κ 0 T = π 2 3 k 2 B e 2 (5)
L 0 = π 2 3 k 2 B
e 2 is known as the Sommerfeld value. The validity of this equality has been experimentally verified in all bulk metals hitherto investigated.

The fact that the phase space for electron-electron scattering grows quadratically with temperature has a signature in thermal transport. It leads to a thermal conductivity to the inverse of the temperature. In order to make a parallel with electrical resistivity, it is convenient to define thermal resistivity as W T = T κ and then, one has:

W T = W T 0 + BT 2 (6)
The T-square prefactor, B, is the thermal counterpart of A in Eq. 1. Several experiments have quantified B in a number of metals [14,15,67,[70][71][72][73][START_REF] Uher | Thermal conductivity of metals[END_REF][START_REF] Paglione | [END_REF][76][77]. These studies remain much less numerous than those devoted to charge transport. A selection of their findings is shown in Fig. 4.

These experiments find that L 0 W T 0 = ρ 0 , but BL 0 > A. In other words, The WF law is valid in the zerotemperature limit, but not at finite temperature.

Historically, this finite-temperature deviation from the WF correlation has been understood as a specific manifestation of the different responses of momentum and energy flow to inelastic collisions [START_REF] Ziman | Principles of the Theory of Solids[END_REF]. The starting point is a distinction between 'vertical' and 'horizontal' collision events, A 'horizontal event', in which the loss of energy is concomitant with a loss of momentum, would affect both electric and thermal conductivity. However, one can conceive 'vertical' events, where a 'hot' electron loses the energy it had to occupy a state slightly above the Fermi energy, without significant modification of its momentum vector. Such events, which require finite temperature, have little effect on electric resistivity, but drastically reduce the thermal resistivity [START_REF] Ziman | Principles of the Theory of Solids[END_REF].

The distinction between 'vertical' and 'horizontal' is also relevant to electron-phonon scattering. The resistivity of noble metals is dominated by electron-phonon scattering and follows a T 5 temperature dependence. Their Lorenz number becomes significantly lower than the Sommerfeld value at intermediate temperature range [78,79]. The WF law is eventually recovered above the Debye temperature, when scattering becomes elastic.

There is an important difference between being scattered by phonons and being scattered by other electrons, however. In the case of e-ph scattering, the temperature dependence of the phase space differs for energy and momentum diffusion. Thermal resistivity follows T 3 , the temperature dependence of the phonon population. Electrical resistivity, on the other hand, follows T 5 . This is because the fraction of the phonons capable of inducing 'horizontal' events shrinks by an additional factor of T 2 . The T 5 e-ph electrical resistivity is the low-temperature asymptotic behavior expected in the Bloch-Grünesisen model [START_REF] Ziman | Principles of the Theory of Solids[END_REF], successfully applied to the experimental data in numerous elemental metals. The T 3 e-ph thermal resistivity (W T ∝ T 3 , i.e. W ∝ T 2 ) have been experimentally resolved in several elemental metals [80,81] as well as in the semi-metallic WP 2 [14].

In contrast, the phase space for e-e scattering follows T 2 for both thermal and electric resistivity. The differentiation between 'vertical' and 'horizontal' scattering would only lead to a smaller prefactor for electrical resistivity. Herring argued that this difference is bounded to a value less than 2, that is BL 0 /A < 2. Li and Maslov, on the other hand, argued that in a compensated semimetal BL 0 /A can become arbitrarily small, if the screening length is long enough compared to the Fermi wavelength [82]. These theoretical proposals have not been confirmed by the experimental data. The bound proposed by Herring is violated in several cases, such as W (BL 0 /A ≈ 6) [71] and WP 2 (BL 0 /A ≈ 5) [14]. Moreover, the largest reported deviation from the WF law occurs in Al (BL 0 /A ≈ 10) [67], which is not a compensated semi-metal.

T-square thermal resistivity, in contrast with T-square electric resistivity, does not require Umklapp events or multiple reservoirs. Normal e-e scattering inside a Fermi pocket can generate T-square resistivity. This is another dichotomy, distinct from horizontal and vertical events and capable of playing a role in the observed downward deviation from the Wiedemann-Franz law. Let us consider the case of thermal transport in a liquid of neutral fermions without the annoying presence of a lattice.

III. Energy and momentum diffusivity in 3 He

Soon after the conception of Landau's Fermi liquid theory [83], Abrisokov and Khalatnikov wrote a paper on the kinetic properties of a Fermi liquid with the case of 3 He in mind [23]. In this paper, they derived expressions for the temperature dependence of viscosity and thermal conductivity.

Kinematic viscosity is the diffusion constant for momentum and thermal diffusivity is the diffusion constant for energy. In the simple kinetic theory for classical gases, there is a simple expression for diffusivity :

D = 1 3 τ v 2 m (7)
Here, v m is the mean molecular velocity and τ , the mean scattering time. In a degenerate fermionic fluid, v m becomes the Fermi velocity and τ is the inverse of the fermion-fermion scattering rate, which follows T -2 . Thus, both the viscosity and the thermal diffusivity are expected to decrease quadratically with temperature. Now, thermal conductivity is proportional to the product of (energy) diffusivity and the specific heat and the latter is T-linear. Therefore, one expects a thermal conductivity following T -1 .

The temperature dependence obtained by such simplistic arguments are confirmed by the outcome of the elaborate calculations performed by Abriokosov and Khatalnikov [23]. They quantified the expected prefactors for

Sb

Jaoui 2021

 L 0 WT , L 0 WT (ncm) T 2 (K 2 )
FIG. 4. T-square thermal resistivity in several metals: Low-temperature electrical resistivity, ρ and thermal resistivity normalized by the Sommerfeld value, L0W T (in order to be expressed in the same units), for Ni [70], Al [67], W [71], Sb [14], CeRhIn5 [START_REF] Paglione | [END_REF] and WP2 [15]. In all cases, the two data sets join as the temperature tends to zero. In all cases, the slope of the thermal resistivity is higher.

the T -2 viscosity and the T -1 thermal conductivity. Brooker and Sykes [25? ] revisited the subject a decade later and found that the previous result for theoretical thermal conductivity was too large by a factor of 2. Moreover, they performed the first comparison between theory and experiment and found [START_REF] Brooker | [END_REF] their theoretical κT to be still 1.5 times larger than the experimentally-measured κT reported by Wheatley and his collaborators [28,29].

A year after, Dy and Pethick [85] carried out a new set of calculations (assuming negligible Landau parameters when > 2) and found a closer agreement with the experimental data. An extensive study of thermal conductivity in 3 He was published by Greywall in 1984 [30], a year after his study of specific heat [32]. They document in detail how entropy transport evolves with the strength of correlations in this emblematic Fermi liquid.

The phase diagram of 3 He is shown in Fig. 5a. The superfluid transition occurs below 3 mK. The normal liquid is a degenerate liquid of fermions below 1 K. Under a pressure of ≈3 MPa (30 atmospheres), interaction between fermions becomes strong enough in order to solidify the liquid. Greywall measured the thermal conductivity κ of the liquid down to ∼ 7 mK at different pressures, ranging from vacuum up to the threshold of solidification. He found that, in the first approximation, κ is proportional to the inverse of temperature, as expected and quantified the product of thermal conductivity and temperature, κT , at different pressures. Now, since W T ≡ T κ , the expression κ ∝ T -1 is strictly equivalent to W T ∝ T 2 . Fig. 5b shows W T extracted from Greywall's κ(T ) data at two different pressures (or molar volumes) as a function of T 2 . One can see that at low temperature, thermal resistivity displays a T-square temperature dependence. At high pressure, when interactions are stronger, the slope (i.e. B of Eq.6) is larger. Moreover, when the prefactor is larger, the temperature below which the T-square behavior is visible shifts to lower temperatures. These features do not surprise those familiar with the Kadowaki-Woods ratio in heavyelectron metals. For example, near the quantum critical point of CeCoIn 5 [START_REF] Bianchi | [END_REF]88] and YRh 2 Si 2 [89,90], the electronic specific heat and the T-square prefactor increase and the domain of validity of the T-square resistivity shrinks.

With increasing pressure, interaction between 3 He atoms intensify and above a threshold pressure, the liquid solidifies. This striking version of pressure-tuned correlation inspired P. W. Anderson to state that "the Mott insulator is a form of quantum solid, and the melting transition in 3 He is our best example of a Mott transition." [91].

Fig. 6 shows how the specific heat and the fermionfermion scattering (which is proportional to (κT ) -1 ) evolve with pressure. One can see that both smoothly enhance as the solidification approaches. Red circles in Fig. 6 are values expected by theory [85] which are close to the experimentally-measured data.

Calkoen and van Weert [31] observed that the magnitude of (κT ) -1 [30] allows to extract a Landau parameter, which is in excellent agreement with the one extracted from the specific heat [32]. They argued that at zero temperature limit, one has:

κT | 0 = 5 18π 3 p 3 F v 2 F A 2 (8)
Here, p F and v F are Fermi momentum and Fermi velocity. The parameter A (not to be mistaken with the prefactor of T-square resistivity) has the dimensions of the Planck constant and was identified as a combination of Landau Fermi-liquid parameters [31]. Extracting the Fermi temperature from the specific heat per volume, one can monitor the evolution of (κT ) -1 and A = A with the Fermi temperature. As seen in Fig. 6c-d [30] -0.391 Wheatley (1975) (p=0) [92] 1.24 0.51 Alvesalo et al. (1975) [93] 0.65 0.23 Bertinatet al. (1974) [94] 1.29 0.524 Black et al. (1971) [95] 1.58 -TABLE II. The rate of fermion-fermion scattering extracted from measurements of viscosity and from measurements of thermal conductivity as reported by several authors. The time scale associated with momentum diffusion is 2-3 times longer than the scattering time associated with energy diffusion. Compare this with what is seen in metals (Fig. 4) and the dichotomy between vertical and horizontal events.

, (κT ) -1 steadily Reference τ 0 η T 2 (ps.K 2 ) τ 0 κ T 2 (ps.K 2 ) Greywall (1984) (p=0)
increases with the Fermi energy. This is concomitant with the evolution of dimensionless A , intimately linked with the Landau parameter F a 0 [31]. Thus the amplitude and the pressure dependence of κT in 3 He are well understood. The prefactor of T-square thermal resistivity, B, is the inverse of κT and therefore, Eq.8 can be rewritten as:

B ≡ (κT ) -1 = 9π 3 10 A 2 E 2 F k F (9)
Here, A is replaced by dimensionless A and the presence of the Fermi energy E F = 1 2 p F v F is explicit. Several studies have been devoted to measuring the temperature dependence of viscosityη. They have confirmed that the expected T-square decrease in the amplitude of viscosity (η ∝ T -2 ). These studies allow quantifying the fermion-fermion scattering rate giving rise to the temperature-dependent viscosity (τ 0 η T 2 ). Several authors have compared it then with the one extracted from thermal conductivity(τ 0 κ T 2 ). A summary of these numbers reported by various authors is given in table 2. One can see that the scattering time for momentum diffusivity is 2-3 times longer than the scattering time for energy diffusivity. Like in the case of metals, and by a comparable factor, thermal transport is punished more than momentum transport by fermion fermion scattering.

Normal liquid 3 He demonstrates not only the persistence of T-square thermal resistivity in a single fermionic reservoir without Umklapp scattering, but also the scaling between its amplitude and the square of the Fermi energy. Let us now check the relevance of these facts to metals. [30]. correlated (W [71] and WP 2 [14]) and three strongly correlated (UPt 3 [73], CeRhIn 5 [START_REF] Paglione | [END_REF] and CeCoIn 5 [76]) metals. Note that CeCoIn 5 is not a Fermi liquid at zero magnetic field. Its resistivity displays a T-square resistivity only in presence of a finite magnetic field restoring the Fermi-liquid behavior [START_REF] Bianchi | [END_REF]88]. The plot compares the amplitude of B in these metals and B ≡ (κT ) -1 in 3 He, discussed in the previous section, as a function of their γ. The data points scatter close to the dashed line which represents a γ 2.15 slope in the log-log scale. Note that 3 He data points evolves slightly faster, but reasonably close to the dashed line.

IV. A common thread

Up to now, the amplitude of T-square thermal resistivity in metals has been analyzed starting from T-square electrical resistivity. It has been taken for granted that the latter cannot arise by normal (i.e not Umklapp), intra-pocket electron-electron scattering. Fig. 7a which focuses on thermal transport, provides an alternative and a short-cut root. Normal momentum-conserving between fermionic quasi-particles generates T-square thermal resistivity in metals as it does in 3 He.

The idea that normal electron-electron scattering plays a role in generating B has found support by the results of recent experiments on thermal transport in bulk antimony (Sb). In this semi-metal, electrons are quasiballistic with a mean-free-path approaching (and scaling) with the sample size. Jaoui et al. [15] quantified the prefactors of the thermal (B) and the electric (A) T-square resistivities and found that the BL 0 /A ratio evolves with the change in residual resistivity (or sample size). The cleaner the sample, the larger was the B/A ratio and the deviation from the WF law [15]. This behavior finds a natural explanation in the hydrodynamic scenario of heat transport as the driver of the deviation from the WF law at the onset of ballistic regime. Indeed, according to a theoretical work by Principi and Vignale [96] the WF ratio between the thermal and the electric conductivity is reduced by a 1 + τ 0 /τ ee factor, where τ 0 is the momentum relaxing time and τ ee is the electron-electron scattering time. In Sb, at the onset of ballistic regime (∼ 10K), these two time scales are comparable in amplitude. In cleaner samples, τ 0 is larger and the deviation from the WF law is more pronounced. Thus attributing a role to normal electron-electron collisions in generating B provides a satisfactory explanation to the evolution of the ratio of the two prefactors with sample size. In contrast, the traditional picture invoking horizontal and vertical events fails to explain why the deviation between thermal and electrical resistivity grows with sample size.. According to Principi and Vignale [96], the WF law is recovered when τ 0 τ ee . In most metals, this condition is easily satisfied in a reasonable temperature range thanks to unavoidable disorder. In all metals, the condition will be satisfied at sufficiently low temperature, because as T → 0, τ -1 ee will diverge and τ -1 0 will stay finite. Therefore, the T-square thermal resistivity should have an electrical counterpart of the amplitude set of ∼ L 0 B and T-square electrical resistivity is expected even without Umklapp (or other momentum-relaxing) collision events between electrons. The combination of Normal electron-electron scattering and disorder will suffice for this. FIG. 6. Tuning correlation with pressurea) T-linear Specific heat as a function of pressure [32]. b) (κT ) -1 plotted against pressure, according to experiment (black solid circles) [30] and theory (red solid circles) [85]. c) The same quantity plotted as a function of the Fermi energy at each pressure. d) The dimensionless parameter A (see text) as a function of the Fermi energy. In these panels, (κT ) -1 and T-linear specific heat are values reported by Greywall who extrapolating the data to extract zero-temperature limit.

HELIUM THREE

In this picture, the (relative) success of the A vs. E -2 F scaling across different families of metals is a consequence of the approximate validity of the WF law, which implies A ≈ π 2 3 ( k B e ) 2 B. This, combined with Eq. 3 and Eq.9, yields the following expression for the phenomenological quad : quad ≈ 3π 5 10

A 2 k F (10)
In the weakly-interacting metals considered by Rice [3] quad ≈ 1.6 nm and in the strongly-correlated metals scrutinized by Kadowaki and Woods quad ≈ 40 nm [10,12]. Many other metals fall somewhere in between. According to Eq. 10, the magnitude of quad in 3 He Mpa. Fig. 7b) plots L 0 B in 3 He and A in lightly-doped strontium titanate as a function of the Fermi energy. The two curves follow a similar trend.

V. Concluding remarks

Thus, when the thermal transport becomes the departing point, our puzzles find solutions. The link between the magnitude of A and the Fermi energy becomes understandable. The persistence of T-square in dilute metals becomes unsurprising.

The temperature dependence of resistivity in SrTiO 3-δ does not reduce to this issue. Well below the degeneracy temperature, the exponent of resistivity in this metal is close to two, but it smoothly evolves with warming [97,98]. In order to explain resistivity in a wide temperature range, one needs to invoke coupling between electrons and transverse optical phonons as recently shown by two independent theoretical studies [19,99] Nazaryan and Feigel'man [99], in contrast to Kumar et al. [19] , incorporated in their calculations the evolution of the effective mass with temperature found by experiment [98] and gave an account of both resistivity and thermopower in a wide temperature range, starting from the minimum energy of phonons. Coupling with phonons cannot explain the persistence of T-square resistivity below this temperature range, which unavoidably leads to electronelectron scattering. The present approach not only provides an explanation for this persistence, but also for its amplitude.

Note that electronic thermal conductivity is orders of magnitude lower than the total thermal conductivity in SrTiO 3-δ [100]. To check the validity of the WF law one needs a way to separate the two. This has been only done in the zero-temperature limit [101]. Quantifying the electronic thermal resistivity and quantifying B in this system remains an experimental challenge.

The case of graphene is beyond the scope of the present paper. In graphite, T-square resistivity emerges below 6 K [21]. An investigation of resistivity in graphene, focused on the relevance of Bloch-Grüneisen resistivity [102] did not explore such low temperatures. Theory expects the phase space of e-e scattering to follow a T 2 lnT temperature dependence [4,103] in two dimensions. Recently, T 2 resistivity was observed in a Moiré superlattice of graphene on top of boron-nitride [104] and in twisted bilayer graphene [105]. Comparing the amplitude of Tsquare in these studies with what has been observed in three-dimensional solids emerges as a future task.

The distinction between e-ph and e-e scattering is not impermeable. There are cases where one suspects that scattering between electrons is accompanied by exchange of phonons [106][107][108]. It has been suggested that phonon exchange amplifies the magnitude of A in Al [106] or in A15 compounds [4]. It remains a task for theory to sort out the link between phonon exchange and the electronelectron cross-section.

The interplay between disorder and electronic interaction is yet to be sorted out. The recovery of the WF law in the picture put forward by Principi and Vignale [96] requires a hierarchy between the two scattering time: τ 0 < τ ee . Upon warming, τ ee becomes steadily shorter and eventually this inequality is no more satisfied. As seen in Fig. 1, the T-square behavior seen in several metals extends to temperatures above AT 2 ∼ ρ 0 . This implies a crucial missing component.

The analogy between the thermal resistivity of metals and 3 He is beyond dispute. But how relevant is momentum diffusivity to electrical resistivity of metals? The usual answer to this question is negative. The thermal current is accompanied by a gradient of thermal energy, but the drift velocity is identical along the charge current. In presence of ohmic response, the electron viscosity [109] of a metallic Fermi liquid has seldom been considered relevant to diffusive electric transport. However, Landauer [110,111] contested the assumption that current flows homogeneously in a macroscopic sample hosting defects. He postulated that, within a screening distance of each scattering center, electric field and charge current are both inhomogeneous.

Let us end by noting a number of studies, which may prove relevant to the subject matter of the present paper. Einzel and Parpia have already examined the link between Drude conductivity in metals and the Poiseuille flow in normal liquid 3 He in presence of aerogel [112]. Nazraov et al. have shown that dynamical electronelectron interactions, known to govern the conductivity of nanoscale junctions [113], play a role in setting the zerotemperature resistivity of a bulk metal [114]. Evidence for inhomogeneous charge flow is provided by experiments imaging the flow of current in a two-dimensional electron gas (2DEG) [115], which have found that it follows branching strands driven by ripples in the energy landscape.

Thus, the viscosity of the electron liquid may be more accessible than previously thought, and the 'hydraulic analogy' [116] deeper than a mere metaphor.

  On the origin and the amplitude of T-square resistivity in Fermi liquids Kamran Behnia LPEM (CNRS-Sorbonne University), ESPCI Paris, PSL University, 75005 Paris, France (Dated: March 1, 2022)

FIG. 1 .FIG. 2 .

 12 FIG. 1. Examples of T-square resistivity: Resistivity as function of T 2 in bismuth [20](a), in graphite [21] (b), in UPt3 [35] (c) and in Sr2RuO4 [36] (d).

Fig. 7a isFIG. 5 .

 5 Fig.7ais a plot à la Kadowaki-Woods (or Rice), exclusively focused on thermal transport. Each data point represents a metallic solid in which the thermal resistivity prefactor B has been quantified. It includes two weakly

FIG. 7 .

 7 FIG. 7. Continuity between 3 He and metals: a) The magnitude of T-square thermal resistivity, B, in 3 He and in metals where the T-square thermal resistivity has been measured plotted against the T-linear specific heat, γ. The dashed line, which is a guide to eye has a slope of 2.15. b) The T-square thermal resistivity prefactor multiplied by the Sommerfeld value (L0B) in 3 He and the T-square electrical resistivity prefactor (A) in lightly-doped single-band strontium titanate as a function of the Fermi temperature.

TABLE I .

 I ElementCarrier density (m 3 ) γ (mJ.mol -1 .K -2 ) EF (K) A (nΩ.cmK -2 ) ref.The experimentally resolved A in a number of elements together with a number of relevant physical properties. In semi-metals (Bi, graphite, Sb, Mo and W), the carrier density is the sum of the density of electrons and holes and the Fermi energy is the average of the Fermi energy of the holes and electrons.

	Bi	6×10 23	0.0085	220	12	[20, 55-57]
	C(graphite)	6×10 24	0.0138	315	5	[21, 58, 59]
	Sb	1.1×10 26	0.119	1030	0.6	[15, 55-57]
	Mo	2.5×10 28	1.9	2.5 × 10 4	1.5 × 10 -3	[60, 61]
	W	1.4×10 28	0.84	4.5 × 10 4	7 × 10 -4	[61-63]
	Pd	5×10 28	9.43	4 × 10 3	0.034	[3, 64]
	Al	6×10 28	1.37	4.5 × 10 4	5.3 × 10 4	[65-67]

Acknowledgements-I am grateful to Benoît Fauqué, Mikhail Fei'gelman, Alexandre Jaoui, Xiao Lin, Dmitrii Maslov, Vladimir Mineev, and Giovanni Vignale for their valuable comments. This work was supported by the Agence Nationale de la Recherche (ANR-18-CE92-0020-01; ANR-19-CE30-0014-04).


Bara Diop
email: diop.bara@ugb.edu.sn

Cheikh Talibouya Diop

Lamine Diop
email: lamine.diop@univ-tours.fr

Ghislain Atemezing

Gaoussou Camara

Bruce Watson

A Semantic Measure for Outlier Detection in Knowledge Graph

Keywords: Knowledge graph, Pattern Mining, Itemset, Outlier Detection Entities

Nowadays, there is a growing interest in data mining and information retrieval applications from Knowledge Graphs (KG). However, the latter (KG) suffers from several data quality problems such as accuracy, completeness, and different kinds of errors. In DBpedia, there are several issues related to data quality. Among them, we focus on the following: several entities are in classes they do not belong to. For instance, the query to get all the entities of the class Person also returns group entities, whereas these should be in the class Group. We call such entities "outliers."

The discovery of such outliers is crucial for class learning and understanding. This paper proposes a new outlier detection method that finds these entities. We define a semantic measure that favors the real entities of the class (inliers) with positive values while penalizing outliers with negative values and improving it with the discovery of frequent and rare itemsets. Our measure outperforms FPOF (Frequent Pattern Outlier Factor) ones. Experiments show the efficiency of our approach.

I INTRODUCTION

Nowadays, the use of different Knowledge Graphs is becoming more and more important. Knowledge Graph is defined as follows: "A knowledge graph (i) mainly describes real world entities and their interrelations, organized in a graph, (ii) defines possible classes and relations of entities in a schema, (iii) allows for potentially interrelating arbitrary entities with each other and (iv) covers various topical domains" [START_REF] Paulheim | Knowledge graph refinement: A survey of approaches and evaluation methods[END_REF]. Knowledge Graphs are constructed from various data sources using various ways. For example, DBpedia is built from the free encyclopedia Wikipedia. In the Semantic Web, the term Knowledge Graph is often used to designate the Knowledge Graphs of the Semantic Web [START_REF] Ristoski | Exploiting semantic web knowledge graphs in data mining[END_REF]. In these, Semantic Web principles are used to represent and publish publicly available data as Linked Open Data. Most Semantic Knowledge Graphs (KG) are defined as classes (types) hierarchy where entities may belong to several classes. For instance, in figure 2, we can see that entity Bob_M arley belongs to classes Artist, P erson, and Agent. Hence, the type prediction problem can be reformulated as a hierarchical multi-label classification one [START_REF] Melo | Type Prediction in RDF Knowledge Bases Using Hierarchical Multilabel Classification[END_REF]. In this context, one solution for entity type prediction is the local classifier per node approach, which consists of training each node of the class hierarchy separately and determining the overall type. However, it has been observed that type information and, more generally, KG suffer from several issues related to data quality such as accuracy, like typing error [START_REF] Caminhas | Detecting and Correcting Typing Errors in DBpedia[END_REF], completeness, like schema completeness and property completeness, and different kinds of errors. In [START_REF] Issa | Linked data quality : completeness and conciseness[END_REF], a systematic literature review on Linked Data completeness and an approach for Linked Data completeness assessment have been proposed. Among them, we focus on the following: several entities are in classes they do not belong to. For instance, let us consider the following query (figure 1) to get all the entities of the class P erson. It also returns group entities, whereas these should be in the class Group (see table 1). We call such entities "outliers." Discovering such outliers is very important for correct class learning and understanding. Outlier detection methods on semantic web data have been proposed to find these outliers [START_REF] Diop | Pattern Sampling in Distributed Databases[END_REF][START_REF] Diop | Pattern on demand in transactional distributed databases[END_REF]. These methods use outlier factors based on frequent pattern discovery methods that have been proposed before [START_REF] He | FP-outlier: Frequent pattern based outlier detection[END_REF][START_REF] Ren | An Approach for Analyzing Infrequent Software Faults Based on Outlier Detection[END_REF]. The idea behind this is that transactions that contain more frequent patterns will have a big value of FPOF measure [START_REF] He | FP-outlier: Frequent pattern based outlier detection[END_REF] and are unlikely to be outliers. In contrast, transactions with small FPOF values are likely to be outliers. Let us consider the following three entities of class Artist: https://dbpedia.org/resource/Masaba_Gupta, https://dbpedia.org/resource/Bertram_Goodman, and https://dbpedia.org/resource/Mystik. Masaba Gupta and Bertram Goodman are real artists, while Mystik is an outlier because it is a song. However, as we will see in example 4, FPOF measure gives a value higher for Mystik than for Masaba Gupta and Bertram Goodman. FPOF does not consider semantics aspects of the entities, but these are very important in KG and specifically in DBpedia. The KG data is based on a well-designed schema which is the ontology. The application of the FPOF measure on semantic web data while completely ignoring the semantics behind the ontology has further motivated our work. In a knowledge graph, ontology plays a very important role. It is a reference schema for organizing data while defining the concepts and relationships between these concepts. Among its components, we have the classes, the attributes or properties, the relations, etc. The role of the ontology here is to provide the semantics behind the properties describing the entities. We propose a semantic measure based on properties (domain and range properties).

SELECT distinct ?entity WHERE {

?entity a dbo:Person. } Table 1: Some entities of the result of the query of Figure 1.

The main contributions of our paper are as follows :

• We define a semantic measure that favors the real entities of the class (inliers) with positive values while penalizing outliers with negative values and improving it with the discovery of frequent and rare itemsets. • We propose a generic algorithm for outlier detection based on the semantic measure in a knowledge graph. This algorithm can be used for frequent or rare patterns. • We present a set of experiments on DBpedia classes showing that our method outperforms FPOF like ones. The rest of the paper is organized as follows. Section II discusses related work. In section III, we propose the basic definitions and problem reformulation. Section IV proposes our measure and presents a generic algorithm for outlier detection in a knowledge graph while presenting a theoretical analysis. Section V presents the results of our experiments, and section VI concludes the paper.

Thing

Activity

Game

Sales

Agent

Person

Athlete Artist

Bob_M arley

II RELATED WORK

In this section, we will present the different dimensions of linked data quality and the outlier detection methods.

Linked Data quality.

[25] cites three approaches that are used to construct a Knowledge Graph namely manual approaches, like Freebase [START_REF] Bollacker | Freebase: A Shared Database of Structured General Human Knowledge[END_REF] and Cyc [START_REF] Matuszek | An Introduction to the Syntax and Content of Cyc[END_REF], cooperative approaches, like DBpedia [START_REF] Lehmann | DBpedia -A Large-scale, Multilingual Knowledge Base Extracted from Wikipedia[END_REF], and Yago [START_REF] Suchanek | YAGO: a core of semantic knowledge[END_REF] and automatic approaches, like NELL [START_REF] Carlson | Toward an Architecture for Never-Ending Language Learning[END_REF]. The manual approaches denote that the construction of the Knowledge Graph is done completely by a human, the automatic ones indicate that supervised or unsupervised learning techniques have been used and the cooperative ones indicate that most of the tasks are done by a human [START_REF] Al-Aswadi | Recent trends in knowledge graphs: theory and practice[END_REF]. For the data quality of the Knowledge Graph, manual approaches tend to be more accurate compared to other approaches [START_REF] Caminhas | Detecting and Correcting Typing Errors in DBpedia[END_REF]. With the extensive growth of data and the many domains they covered it is impossible to build or maintain a knowledge graph by manual methods, automatic methods seem more suitable although they are more likely to contain several errors. For example on DBpedia several types of errors such as typing errors, incorrect numerical data, etc. have been noted. In [START_REF] Wand | Anchoring data quality dimensions in ontological foundations[END_REF] the dimensions of data quality were classified into four categories:

• Intrinsic Category which allows to assess the validity and consistent of the data. In this category, we have Accuracy, Consistency and Trustworthiness. • Contextual category : "highlights the requirement that data quality must be considered within the context of the task at hand" [START_REF] Wand | Anchoring data quality dimensions in ontological foundations[END_REF]. In this category we have Relevancy, Completeness, and Timeliness • Representational Category : "A concept that data quality is related to the "format of the data (concise and consistent representation) and meaning of data (interpretability and ease of understanding)" [START_REF] Wand | Anchoring data quality dimensions in ontological foundations[END_REF]. • Accessibility Category : "Accessibility dimensions are about how easily accessible and secure data is, such as availability and security." [START_REF] Issa | Linked data quality : completeness and conciseness[END_REF] One of the solutions to the problem of data quality in KG is the language SHACL (Shapes Constraint Language) 1 . SHACL is a W3C Recommendation that defines a language for validating RDF graphs against a set of conditions. For its use, a good understanding of the ontology is necessary to define these conditions. Another problem is the scalability since each KG has its ontology.

It is almost impossible to define a generic approach to assessing data quality across all dimensions, and one dataset may be suitable for one purpose but not for another [START_REF] Issa | Linked data quality : completeness and conciseness[END_REF]. Depending on the application, it is imperative to address particular data quality issues. For class learning and understanding, the problem we face is misclassified entities (Accuracy dimension). It is very important to have suitable parameters to find misclassified entities in a given class.

Outlier detection. One solution for solving this problem of misclassified entities is outlier detection. Outlier detection has been the topic of several surveys and reviews. In these surveys, different classifications methodologies are presented. More cited categories are the following: Nearest Neighbour Based Outlier Detection techniques, Distance-Based Techniques, Density-Based Techniques, Cluster-Based Techniques, Statistical Approach Based Techniques. Distance-Based [START_REF] Knorr | Algorithms for Mining Distance-Based Outliers in Large Datasets[END_REF] and Nearest Neighbour Based Techniques [START_REF] Ramaswamy | Efficient Algorithms for Mining Outliers from Large Data Sets[END_REF] rely on the notion of distance. A distance-based outlier in a dataset D is a data object with a given percentage of the objects in D having a distance of more than d min away from it [START_REF] Ren | An Approach for Analyzing Infrequent Software Faults Based on Outlier Detection[END_REF]. Nearest Neighbour based outlier detection techniques require a distance or similarity measure between two data points. If a point x has a short distance to its k neighbors, it is considered as normal otherwise it is considered as outlier. Density-based techniques measure density of a point x within a small region by counting number of points within a neighborhood region. Breunig et al. [START_REF] Breunig | LOF: Identifying Density-Based Local Outliers[END_REF] introduced the concept of Local Outlier Factor (LOF), a score which is assigned to every point based on its local density. All data points are sorted in decreasing order of LOF value. Points with high scores are detected as outliers. In Cluster-Based Outlier detection techniques, a cluster represents a collection of data objects similar to one another within the same cluster and dissimilar to the objects in other clusters. Inliers correspond to data in a cluster while outliers do not belong to any cluster. Statistical approach based technique assumes a distribution or probability model for the given data and then identifies outliers with respect to the model using a discordancy test [START_REF] Mohemad | A Comparative Study for Outlier Detection Techniques in Data Mining[END_REF]. Many of these techniques suffer from high dimensional space curse and high computational cost [START_REF] Said | Outlier Detection Scoring Measurements Based on Frequent Pattern Technique[END_REF]. More recently, a new trend has appeared, using frequent pattern technique. Several measures were consequently proposed. FPOF (Frequent Pattern Outlier Factor) has been presented by [START_REF] He | FP-outlier: Frequent pattern based outlier detection[END_REF]. The idea behind this measure is that transactions that contain more frequent patterns will have a big value of FPOF measure and are unlikely to be outliers. In contrast, transactions with small FPOF values are likely to be outliers. [START_REF] Ren | An Approach for Analyzing Infrequent Software Faults Based on Outlier Detection[END_REF] proposed another measure WCFPOF (Weighted Closed Frequent Pattern Outlier Factor) in order to overcome the drawbacks of FPOF. According to this measure, transactions that contain more closed frequent patterns are more likely to be inliers and those that contain less closed frequent patterns are likely to be outliers. Another approach for overcoming the drawbacks of FPOF measurement was formulated by [START_REF] Zhang | An Improved Method of Outlier Detection Based on Frequent Pattern[END_REF]. According to this one, transactions that contain longer frequent pattern (i.e. longer superset) are more likely to be inliers because they contain more subset frequent patterns, while transactions that contain short frequent patterns are likely to be outliers. By approximating the Frequent Pattern Outlier Factor (FPOF) with sampled patterns, [START_REF] Diop | Pattern on demand in transactional distributed databases[END_REF] proposes a method for detecting misclassified entities. The observation is the same for these measures: they do not consider semantics.

III PRELIMINARIES AND PROBLEM REFORMULATION Knowledge graph.

A knowledge graph is an RDF dataset described by a set of triples (subject, predicate, object). It can be represented as a tuple K = (T , A) where T is called a Tbox formed by names and assertions about concepts (or classes) and roles (or predicates), and A is called an ABox formed by assertions about individuals called entities and facts. This paper focuses on DBpedia [START_REF] Bizer | DBpedia -A crystallization point for the Web of Data[END_REF] TBox and its entities. For example, (dbo : Artist, rdf s : subClassOf, dbo : P erson) is an assertion in DBpedia TBox which means that the concept Artist is a subclass of the concept P erson (or P erson is a superclass of Artist). In other words, all artists are also persons. For example, (Bob_M arley, rdf :type, dbo:Artist) means that Bob_M arley is an artist, then an entity of the class Artist. The triple (Bob_Marley, spouse, Rita_Marley) is an assertion in DBpedia ABox which means that Bob_M arley has spouse Rita_M arley. Given a class C ∈ T and an entity e ∈ A, (e, rdf : type, C) implies that e is an instance of C. To better formalize and define all these notions and the others we need to formulate our problem, we use the Description Logics (DL) [START_REF] Baader | The Description Logic Handbook: Theory, Implementation, and Applications[END_REF] formal notations. In that case, the assertion (dbo:Artist, rdf s:subClassOf, dbo:P erson) is denoted by dbo : Artist ⊑ dbo : P erson which means that dbo:Artist is subsumed by dbo:P erson. dbo: Artist(Bop_M arley) denotes that Bop_M arley is an instance (or entity) of the class Artist. The relation spouse(Bob_M arley, Rita_M arley) materializes that the predicate spouse links the subject Bob_M arley to the object Rita_M arley. We distinguish two types of predicates from these examples: outgoing and incoming. If we have the triple (X, P, Y ), then we say that P is an outgoing predicate for X and an incoming predicate for Y whatever the type of X and Y . Then, there is a case where an outgoing predicate P is specific to subjects that instantiate a class C. In that case, we say that C is the domain of P , denoted by ∃P.⊤ ⊑ C. Another case is when an incoming predicate is specific to objects of a class C. Here, we say that C is the range of the predicate P , denoted by ⊤ ⊑ ∀P.C. The disjointness of two classes C 1 and C 2 is denoted by C 1 ⊑⌝C 2 . For example, according to the ontology of DBpedia, it should not have entities belonging to both the class P erson and the class Organization, then we have P erson ⊑⌝Organization. There are several types of knowledge graphs, but for this work, we are interested in these where ABox and TBox are available and accessible by SPARQL queries like DBpedia.

In the following, we show the transformation from RDF data to the transactional database as done in [START_REF] Diop | Pattern on demand in transactional distributed databases[END_REF].

Transactional database. Now, we will show how to represent a transaction from the predicates that describe an entity. Since we are interested in predicates having domains, we consider the transactional database defined on the set of items I = {P : (∃C ∈ T )(∃P.⊤ ⊑ C)}. An itemset (or pattern), denoted by φ, is a non empty subset of I. Formally, we have φ ⊆ I. The set of all patterns that can be generated from I is called the pattern language L = 2 I \ ∅. In this paper, a transaction is a couple (e, I e ) ∈ A × L where I e is the set of all predicates describing the entity e that appears in I. Formally, we have I e = {P ∈ I : (∃C ∈ T )(C(e))(∃P.⊤ ⊑ C)}. In the following, we denote such transaction as I e . So in our context, a transactional database D C is a multi-set of transactions defined in I where all items (predicates) of I describes an entity of class C. It means that we are making a restriction in class C in which we look for whether it contains outliers or not. For example, to find outliers in the class Artist, we only consider entities that instantiate Artist.

Example 1:

For instance, D Artist in Table 2 is a toy dataset of 20 transactions from the class Artist of DBpedia 2 (entities are prefixed by dbr(http://dbpedia.org/resource/) and predicates by dbo(http://dbpedia.org/ontology/)). For instance, to obtain the transaction I Bob_M arley , we run the following SPARQL query on the Dbpedia endpoint (https://dbpedia.org/sparql/): SELECT DISTINCT ?P WHERE { <http://dbpedia.org/resource/Bob_Marley> ?P ?object. ?P rdfs:domain ?C. } D Artist is built from the set of 38 items I = {deathPlace, deathDate, birthPlace, birthDate, deathCause, partner, relative, child, parent, spouse, birthYear, deathYear, birth-Name, nationality, field, training, residence, restingPlacePosition, restingPlace, cinematography, director, producer, starring, writer, runtime, Work/runtime, artist, endingTheme, network, openingTheme, previousWork, starring, subsequentWork, completionDate, numberOfEpisodes, bandMember, formerBandMember, hometown}. Table 3 gives the domain and the range (if it exists) of the set of items (predicates) in I.

The transaction I Bob_M arley = {deathPlace, deathDate, birthPlace, birthDate, deathCause, partner, relative, child, parent, spouse, birthYear, deathYear} contains 12 items. so, there are 2 12 -1 patterns that appear in L(D Artist ). For example, φ = {spouse, birthY ear} is one among them. This pattern also belongs to the transaction I Hank_W illiams . 2 Access in 18/10/2021 We already remarked that, semantically, some of these entities such as Twice_as_Nice, Children_of_Eve, Mystik, and The_Lead, are not artists, and yet they are instantiated in the Artist class. We call them outliers while the others are well instantiated in Artist; they are inliers.

Definition 1: Outlier, Inlier Given two disjoint classes C and C ′ (C ⊑⌝C ′ ) and an entity e. If e is instantiated in C but really defined as an entity of C ′ , then e is an outlier. However, if really e is an instance of C, then it is called an inlier. In this paper, we focus on two interestingness measures frequency and rare. So, given an interestingness measure m it is possible to extract a set of interesting patterns. We denote by q(•), such that q(•) ∈ {true, f alse}, the constraint that a pattern of D C should respect according to the interestingness measure m. Therefore, on the one hand, the set of all frequent patterns given a minimum threshold α can be formulated by Pset(D C , freq α ) = {φ ∈ L(D C ) : q(freq(φ, D C ) ≥ α) = true}. On the other hand, the set of all rare patterns given a minimum threshold α can be formulated by

Pset(D C , rare α ) = {φ ∈ L(D C ) : q(0 < freq(φ, D C ) < α × |D C |) = true}.
Now we present FPOF in the knowledge graph as done in [START_REF] Diop | Pattern on demand in transactional distributed databases[END_REF] to detect outliers. FPOF : Frequent Pattern Based Outlier Factor. As it was introduced in Section II, FPOF is based on frequent patterns. The main idea behind this approach for transactional databases is that inliers will contain most of the frequent patterns while most of the patterns that outliers contain will have a low frequency. So, they formulate the following metric. Based on the frequency of the itemsets, the FPOF measure gives a higher score to the entity M ystik (which is an outlier) than to entities M asaba_Gupta and Bertram_Goodman (which are inliers). The FPOF measure is based only on the frequency of items that describe transactions while ignoring the semantics.

In knowledge graphs, the frequency of items only does not allow to find the outliers of a set because semantics are decisive for entity definition. For example, birthN ame is a predicate that defines a person, which is less frequent than the predicate abstract, which is generally used by the entities of all classes.

The problem that we want to solve in this paper can be formulated as follows:

Given a class C of knowledge graph K = (T , A), let D C be its transactional database, and Pset(D C , m α ) a set of interesting patterns built with an interestingness measure m, Q1: which metric can be used to favor the inliers and penalize the outliers? Q2: how to benefit from the semantic of the TBox T to improve the metrics? The set of all frequent (rare) patterns given a minimum threshold α k

The number of super-classes C

The set of concepts C ≤k (C)

The set of concepts containing C, its super-classes at a level at most equal to k and its sub-classes.

SCℓ

The set of super-classes SCℓ ≤k (C)

The set of super-classes of C which are located at a level less than or equal to k with respect to C. setOut

The set of outliers setIn

The set of inliers

IV ONTOLOGY AND PATTERN-BASED OUTLIER DETECTION IN KNOWLEDGE GRAPH

This section shows how to take advantage of semantic relationships that arise in ontology to improve the mined pattern according to a measure of interest. However, let us introduce the idea behind the rare patterns for detecting outliers.

In a similar way to the FPOF metric, the rare patterns can be used to detect outliers from the transactional database. A naive method is to sum, for any transaction, the frequency of the rare patterns it contains. In that case, the outliers will have high scores while the inliers will have low scores. But, the main problem with this basis metric follows from the fact that many inliers will have null scores. This is why in this paper, we will introduce semantics on metrics, particularly on items forming patterns that are nothing other than properties that appear in the ontology. It is important to note that all the properties that we use to make up the patterns have domains, and some of them also have ranges. In the rest of this paper, the metrics as well as the algorithm that we are going to introduce are valid for both rare and frequent patterns.

Intuition behind this method.

Our approach is based on the fact that a property P with domain C must necessarily be an outgoing property of an entity of the class C when it is used. It is very important to note that P can also be an outgoing property of any entity of a subclass of C (C ′ ⊑ C) or of a super-class of C located at a certain level k ≥ 0. We say that a superclass C ′ of the class C is at level k with refer to C if there are k -

1 classes {C 1 , ..., C k-1 }, with C i ̸ = C j for all i ̸ = j, in T such that C ⊑ C 1 ⊑ ... ⊑ C k-1 ⊑ C ′ .
Indeed, the fact of favoring all the super-classes of C will distort the calculations because there will be outliers who will benefit from it. For example, Artist and Athlete are two classes in the same level that have super-classes Person (of level k = 1), Agent (of level k = 2) and Thing (of level k = 3). When searching for outliers in Artist, it is not interesting to consider properties that have Thing as its domain, because the latter is the parent class of all classes in DBpedia. However, properties that have Agent as their domain are only interesting if the outliers are in a twin class C ′ (Activity for instance) or one of sub-classes of C ′ (Game, Sales, ...). The same is true for properties that have Person as their domain, if some outliers really belong in class Organization, then they are useful for detecting outliers. If, on the other hand, the real class of the outliers is Athlete, then the properties having Person as their domain are not interesting to detect the outliers. So, the properties which can allow us to verify if an entity is an outlier or not, are not obvious. It is why retrieving these properties is a real problem of efficiency for our method because we do not know apriori at what level k ≥ 0 we must stop to detect outliers since we ignore their real classes. To solve this problem, we propose to vary the value of k to find the most relevant properties to detect outliers. In practice, the value of k is small. Therefore, we denote by SCℓ ≤k (C) = {C ′ ∈ T : (∃i, 1

≤ i ≤ k)(C i ∈ T )(C ⊑ C ′ ⊑ C i )
} the set of super-classes of C which are located at a level less than or equal to k with respect to C. So,

SCℓ ≤∞ (C) = {C ′ ∈ T : C ⊑ C ′ }.
Let us denote by C ≤k (C) the set of concepts containing C, its super-classes at a level at most equal to k and its sub-classes:

C ≤k (C) = {C} ∪ {C ′ ∈ T : C ′ ⊑ C} ∪ {C ′ : C ′ ∈ SCℓ ≤k (C)}.
The semantic judgment we have on a property P that appears in a pattern φ can be formulated as follows:

• (a) If a property P is interesting enough to well describe an inlier e of a class C in a certain level k then it must necessarily have a domain which is part of C ≤k (C). The set of predicates in the pattern φ that meet this intuition is defined by

{P ∈ φ : (∃C ′ ∈ C ≤k (C))(∃P.⊤ ⊑ C ′ )}.
• (b) An entity found in class C which has an outgoing property P whose domain does not belong to C ≤∞ (C) is likely to be poorly described, and looks like an outlier if most of the properties it contains have this tendency. The set of predicates in the pattern φ that meet this intuition is defined by {P ∈ φ : (∄C ′ ∈ C ≤∞ (C))(∃P.⊤ ⊑ C ′ )}. • (c) If a property P that matches the intuition in (a) additionally has a range that is in C ≤k (C), then it is determinant for class C. The set of predicates in the pattern φ that meet this intuition is defined by {P ∈ φ :

(∃C ′ ∈ C ≤k (C))(∃P.⊤ ⊑ C ′ )(∃C ′′ ∈ C ≤k (C))(⊤ ⊑ P.C ′′ )}.
• (d) If a property P that matches the intuition in (b) additionally has a range that is part of C ≤k (C), then the use of P is ambiguous. The set of predicates in the pattern φ that meet this intuition is defined by {P ∈ φ :

(∄C ′ ∈ C ≤∞ (C))(∃P.⊤ ⊑ C ′ )(∃C ′′ ∈ C ≤k (C))(⊤ ⊑ ∀P.C ′′ )}.
Based on these intuitions, we can now define our algorithm which combines the mined patterns and the semantic relationships from ontology to detect outliers.

A generic algorithm for outlier detection in knowledge graph. We propose an algorithm based on the four intuitions presented in the previous section. To be in agreement with these intuitions, we say that a predicate that verifies intuition (a) brings a bonus of 1 to the score of the entity that it describes while that which verifies (b) brings a penalty of 1 to the entity's score. On the other hand, a predicate that verifies intuition (c) brings a bonus of 2 (bonus of 1 for the domain and of 1 for the range) while a predicate that verifies intuition (d) brings a penalty of 2 for the entity it describes (penalty of 1 for the domain and 1 for the range). Thereby, given a pattern we introduce the notion of reliability, a metric to compute the total score a pattern brings to an entity for a given class.

Definition 5: Reliability

Let e be an entity of a class C described by a certain predicates that belong to I, a pattern φ ⊆ I and k a positive integer. The reliability of the pattern φ to the entity e with respect to the class C is defined as follows:

reliability(φ, e, C, k) =                          0 if φ ̸ ⊆ I e |{P ∈ φ : (∃C ′ ∈ C ≤k (C))(∃P.⊤ ⊑ C ′ )}| +|{P ∈ φ : (∃C ′ ∈ C ≤k (C))(∃P.⊤ ⊑ C ′ )(⊤ ⊑ P.C ′ )}| -|{P ∈ φ : (∄C ′ ∈ C ≤∞ (C))(∃P.⊤ ⊑ C ′ )}| -|{P ∈ φ : (∄C ′ ∈ C ≤∞ (C))(∃P.⊤ ⊑ C ′ ) (∃C ′′ ∈ C ≤k (C))(⊤ ⊑ ∀P.C ′′ )}| otherwise (1)
It is clear that the reliability of a pattern is the sum of the contribution of all the predicates it contains. Now, we can present the algorithm exploring this metric to capture the outliers that disturb the veracity of all the instances in a given class. Let us recall that the goal of this paper is not to find frequent or rare patterns, we just reuse the existing methods.

Example 5:

Let us consider this two patterns φ 1 = {birthDate, birthN ame} and φ 2 = {bandM ember, hometown}. We are now going to compute the reliability of the transactions (Robin_Harris, I Robin_Harris ) and (7icons, I 7icons ) with refer to the class Artist and with different values of k. We have : reliability(φ 1 , Robin_Harris, Artist, 0) = 0 because none of the properties in φ 1 has a domain that belongs in C ≤0 (Artist) = {Artist} but in C ≤∞ (Artist). These predicates are neutral since they have a domain that is a superclass of Artist. However, reliability(φ 1 , Robin_Harris, Artist, 1) = (+1) + (+1) = +2 because birthDate and birthName have a domain that belongs in C ≤1 (Artist) = {Artist, P erson} then (+1). reliability(φ 2 , 7icons, Artist, 0) = -1 because, in one hand, the domain of the property bandMember doesn't belong in C ≤∞ (Artist) while its range doesn't belong in C ≤0 (Artist). In other hand, the domain of the property hometown belongs in the set C ≤∞ (Artist) but not in C ≤0 (Artist). Let us now compute reliability(φ 2 , 7icons, Artist, 1). We know that the range of bandMember belongs in C ≤1 (Artist) while its domain doesn't belong in C ≤∞ (Artist). So, this property gives a score of ((-1)+(-1)) to the entity 7icons. The property hometown gives the same score like in k = 0. So, we have reliability(φ 2 , 7icons, Artist, 1) = (-1) + (-1) = -2. setIn ← setIn ∪ {(e, score(e, C, k))} 9: return setOut Algorithm 1, named ONTOPOD, computes and returns a set of entities likely to be outliers. It takes as input the transactional database obtained from the entities of a class C of a given knowledge graph K = (T , A), an interestingness measure m α , with α the minimum threshold, and a positive integer k for the maximum level of superclasses to be considered for C. In the end, it returns all the entities that have a negative score, and therefore likely to be outliers. Then, it computes the score of each entity e according to its transaction I e and the extracted patterns Pset(D C , m α ) if it is not empty (line 4). Finally, all the entities having a negative score are retrieved and stored in the setOut variable (line 6) which is finally returned at output (line 9). Interestingly, it is also possible to easily flip inliers (those with positive scores) in the setIn variable (line 8). We maintain the idea that the method fails to judge entities that have a score equal to zero (0). We specifically use three variants of ONTOPOD. The first is ONTOPOD-N o which means that Pset(D C , m α ) is empty. The second is ONTOPOD -Freq that corresponds to the case where the interestingness measure is the frequency, Pset(D C , freq α ). The last is ONTOPOD -Rare which means that the rare patterns are used Pset(D C , rare α ).

Example 6:

Table 6 gives the scores of ONTOPOD applied in the dataset of Table 7 that corresponds to the Artist class by promoting the super-class at level 1 (Person) (m α = no means that there is no interestingness measure that has been provided). As we can see it, our method manages to find outliers that the FPOF method failed to judge (Masaba_Gupta, Bertram_Goodman, 7icons). It is also important to note that without an interestingness measure, certain entities risk to have a score equal to zero (Brett_Newski), hence the interest in finding frequent or rare patterns for deciding. Moreover, with frequent patterns, ONTOPOD finds all the outliers while with rare patterns, it adds an inlier (Brett_Newski) to the outliers.

Theoretical analysis of Algorithm 1.

In this section, we give the theoretical analysis of our algorithm ONTOPOD. As we said earlier, we do not consider the complexity of pattern mining, which depends on the used pattern mining algorithm. However, the complexity of our method also depends on the size of the set of mined patterns.

V EXPERIMENTS

We will show in this section the effectiveness of our method by presenting the dataset used and the different measures to evaluate our measure.

Datasets. We use two DBpedia datasets for our experiments: online and benchmark datasets [START_REF] Ofer | DBPedia Classes : Hierarchical Taxonomy of Wikipedia article classes[END_REF]. The benchmark dataset lacks information on the properties of the entities, which is why we use the online dataset.

DBpedia Online: DBpedia Online is a set of human-understandable online RDF schema descriptions accessible to applications. The data is organized in a hierarchical structure. In this dataset, we will extract, for a class C, its entities (name, properties, domain, and range of properties), its super-classes, and sub-classes. As already stated these classes contain outliers. We will use the benchmark dataset to test the performance of our methods.

Benchmark dataset: This benchmark dataset is a reference database containing 342,781 data instances that can be used for hierarchical classification tasks. It has 3 levels l1, l2, l3, with respectively 9, 70, and 219 classes. We will focus on some classes of level 2. This dataset contains the real classes of the entities but does not cover the data of the online database.

Extracting information from DBpedia online. We have chosen to work with some DBpedia classes. These are the classes «Animal,» «Artist,» «Athlete,» «Company,» «EducationalInstitution,» «Group,» «Politician,» and «NaturalPlace.» We first choose a class C of these classes for the online extraction and then use SPARQL queries to extract its entities, super-classes, and sub-classes. The entities of the sub-classes of C are also part of the entities of C. The properties of each entity and the domains and ranges are also extracted. For instance, for the Artist class, the following SPARQL queries were executed: After the extraction of the online information, for a class C, we keep only the resources having at least two properties that are in the gold standard since it is these resources whose real class is known.

In Table 7, we give the number of entities, the number of outliers, and the number of inliers in each class. A set of properties describes each entity. Frequent and rare patterns extraction and Evaluation metrics.

To detect frequent and rare patterns, we used the f pgrowth [START_REF] Han | Mining Frequent Patterns without Candidate Generation[END_REF] algorithm to find all supports of the patterns in the database. Then we separated the database into frequent and rare patterns using the average support as a separation threshold.

For the evaluation of the methods, we used three metrics: the inliers rate IR, the outliers rate OR and the harmonic mean of the IR and the OR called F-score. For each method B, we calculate IR(B), OR(B) and F-score(B).

IR(B) = ni B ni T OR(B) = no B no T F -socre = 2 * IR * OR IR + OR
Where ni B is the number of inliers found by the method B, and ni T is the total number of inliers in the database, and no B is the number of outliers found by the method B and no T is the total number of outliers in the database.

There are two sets of experiments conducted in each class to evaluate the performance of the methods. The first one is the variation of the f-score of the methods according to the value of k and the second one is the outlier rate and the f-score of the methods for the best value of k.

F-scores evolution according to k.

Figure 3 shows the evolution of the F-scores of the methods according to the value of k. We notice that, in general, the best values of the Fscore correspond to k = 0. The figures show that our method outperforms the FPOF method. For some classes, the FPOF method cannot even find outliers. Most outliers are entities of superclasses, and their properties are very frequent.

VI CONCLUSION

This paper proposed a generic semantic measure of outlier detection that favors some entities and disfavors others from the semantics behind the ontology (especially the properties and class hierarchy). We have tested our measure on a gold standard dataset of DBpedia. The experiments showed the effectiveness of our approach. In semantic web knowledge graphs, ontology plays a vital role that should not be neglected. Our measure is based on the domains and ranges of properties, the hierarchy of classes, the frequency of properties which are excellent parameters to find the entities outliers. If there are instances classified in class C that have (frequent) properties which have for a domain (and range) the class C (or its super-classes), we can reasonably suppose that this increases the precision of their membership in C. It is clear that if the ontology is not complete (i.e., it does not contain all the classes and properties of the data) or the real entities are poorly described (for instance, they have few properties), the performance of our measure decreases.

In our future work, we will add a preprocessing step using this outlier detection measure in the automatic classification process of DBpedia resources. Indeed, to build the learning base, it is necessary to select positive and negative examples. The measure will allow us to find true positive examples. Finally, we will apply this method to other Knowledge Graphs like Yago and Wikidata.

Figure 1 :

 1 Figure 1: Query to get all the entities of the class P erson.

Figure 2 :

 2 Figure 2: A sub-part of the DBpedia hierarchy

Definition 2 :Definition 3 : 3 :

 233 Frequency of a pattern Given a transactional database D C = {(e 1 , I e 1 ), • • • , (e n , I en )} of a class C and a pattern φ ∈ L(D C ). The frequency of φ is the number of transactions of D containing φ. Formally, freq(φ, D C ) = |{(e i , I e i ) ∈ D C : φ ⊆ I e i }|. Example 2: The frequency of φ = {spouse, birthY ear} in D Artist is 2 because only I Bob_M arley and I Hank_W illiams contain φ. The frequency of φ 1 = {spouse} is freq(φ 1 , D Artist ) = |{M asaba_Gupta, Bob_M arley, Hank_W illiams}| = 3. According to the frequency, one can judge a pattern as frequent or rare (not frequent) in a dataset. Frequent and Rare pattern Let us consider a dataset D C of class C, a minimum support threshold α ∈ [0, 1], and φ a pattern of L(D C ). We say that : • φ is frequent in D C if and only if freq(φ, D C ) ≥ α × |D C |. • φ is rare in D C if and only if 0 < freq(φ, D C ) < α × |D C |. Example If we consider a minimum threshold α = 4/20, then φ = {spouse, birthY ear} is a rare pattern in D Artist because freq(φ, D Artist ) = 2 20 < 4 20 but the pattern φ ′ = {birthDate, birthN ame} is frequent in D Artist because freq(φ, D Artist ) = 11 20 > 4 20 .

Definition 4 : 4 :

 44 Given a transactional database D C = {(e 1 , I e 1 ), ..., (e n , I en )} and a minimum frequency threshold α > 0, the frequent pattern outlier factor of the transaction (e, I e ) denoted by fpof (e, D C ) is defined as follows: fpof (e, D C ) = 1 |Pset(D C , freq α )| × φ∈Pset(D C ,freqα)∧φ⊆I e freq(φ, D C ) |D C | Example Let us consider the database D Artist shown in

Algorithm 1

 1 ONTOPOD (Ontology and Pattern-based Outlier Detection) Input: A set of patterns Pset(D C , m α ) obtained from the transactional database D C of a class C of a knowledge graph K = (T , A) and an interestingness measure m α with α ∈ [0, 1] and a positive integer k Output: A set of entities weigted by their scores setOut (for outliers ) and setIn (for inliers) obtained from D C 1: setOut ← ∅ ▷ The set of outliers 2: setIn ← ∅ ▷ The set of inliers 3: for (e, I e ) ∈ D C do 4: score(e, C, k) ← reliability(I e , e, C, k)+ φ∈Pset(D C ,mα)∧φ⊂I e reliability(φ, e, C, k) 5: if score(e, C, k) < 0 then 6: setOut ← setOut ∪ {(e, score(e, C, k))}

  Generally, to compute the score of an transaction (e, I e ), we first compute the reliability of the itemset I e with refer to the corresponding entity e in O(|I|). Then, we compute the reliability of each pattern φ in Pset(D C , m α ) to a transaction (e, I e ) with a complexity in O(|I| × |Pset(D C , m α )|). Thereby, the complexity of ONTOPOD is in O(|I| × |D C |) + O(|I| × |D C | × |Pset(D C , m α )|). So, the complexity of ONTOPOD is generally in O(|I| × |D C | × (1 + |Pset(D C , m α )|)). In the case of ONTOPOD -N o, where the set of patterns Pset(D C , m α ) is empty, the complexity is only in O(|I| × |D C |).

Figure 3 :

 3 Figure 3: F-score of methods for each class

Figure 4

 4 shows the performance of the methods on the different classes.

  Figure 4(b) is the result of figure 3 for the maximum f-score value (which generally corresponds to k = 0).

  Figure 4(a) shows the outlier rate for each method in each class. For the classes Company, EducationalInstitution, Politician, and NaturalPlace our method greatly outperforms the FPOF method. The outliers have frequent properties that are why the FPOF method fails to find them.

Figure 4 :

 4 Figure 4: Outlier rate and F-score per class

Table 2 :

 2 D Artist : an example of a transactional database from the class Artist of DBpedia

	entity	Itemset (set of predicates)
	Bob_Marley	{deathPlace, deathDate, birthPlace, birthDate, deathCause, partner, relative,
		child, parent, spouse, birthYear, deathYear}
	Omar_Kiam	{birthDate, birthName, birthPlace, deathDate, deathPlace, nationality}
	Bertram_Goodman {field, training}
	Giuliana_Camerino {birthDate, birthName, birthPlace, deathDate, deathPlace, nationality}
	Twice_as_Nice	{cinematography, director, producer, starring, writer, runtime, Work/runtime}
	Robin_Harris	{birthDate, birthName, birthPlace, deathDate, deathPlace, nationality}
	Brett_Newski	{birthDate, hometown}
	LaWanda_Page	{birthDate, birthName, birthPlace, deathDate, deathPlace, nationality}
	Frank_Suero	{birthDate, birthName, birthPlace, deathDate, deathPlace, nationality}
	Josephus_Thimister {birthDate, birthName, birthPlace, deathDate, deathPlace, nationality}
	Sid_James	{birthDate, birthName, birthPlace, deathDate, deathPlace, nationality}
	Masaba_Gupta	{residence, spouse}
	Hank_Williams	{deathPlace, deathDate, birthPlace, birthDate, restingPlacePosition, death-
		Cause, relative, restingPlace, spouse, birthName, birthYear, deathYear}
	Children_of_Eve	{cinematography, director, producer, writer, runtime, Work/runtime}
	Greg_Giraldo	{birthDate, birthName, birthPlace, deathDate, deathPlace, nationality}
	Mystik	{artist, previousWork, producer, writer, runtime, Work/runtime}
	Jerry_Clower	{birthDate, birthName, birthPlace, deathDate, deathPlace, nationality}
	Mackenzie_Taylor {birthDate, birthName, birthPlace, deathDate, deathPlace, nationality}
	The_Lead	{endingTheme, network, openingTheme, previousWork, starring, subsequent-
		Work, completionDate, numberOfEpisodes, runtime, Work/runtime}
	7icons	{bandMember, formerBandMember, hometown}

Table 3 :

 3

	P	dom(P ) range(P )	P	dom(P )		range(P )
	deathPlace	Person	Place	birthYear	Person		gYear
	deathDate	Person	date	deathYear	Person		gYear
	birthPlace	Person	Place	birthName	Person		langString
	birthDate	Person	date	nationality	Person		Country
	deathCause Person		field	Artist
	partner	Person	Person	training	Artist	EducationalInstitution
	relative	Person	Person	residence	Person		Place
	child	Person	Person	restingPlacePosition Person		SpatialThing
	parent	Person	Person	restingPlace	Person		Place
	spouse	Person	Person	cinematography	Film		Person
	P director producer starring writer runtime Work/runtime artist endingTheme network	dom(P ) Film Work Work Work Work Work MusicalWork TShow Bcaster	range(P ) Person Agent Actor Person double minute Agent Work Bcaster	P openingTheme previousWork subsequentWork completionDate numberOfEpisodes bandMember formerBandMember hometown	dom(P ) TelevisionShow Work Work Work TShow Band Band Agent	range(P ) Work Work Work date nonNegInt Person Person Settlement

Domain and range (optional) of the predicates of our dataset D Artist (EduIns: EducationalInstitution, TShow: TelevisionShow, Bcaster: Broadcaster, nonNegInt: nonNegativeInteger)

Table 2 .

 2 Each entity is described by a set of properties. For example, the entity Sid_James is described by the properties birthDate, birthName, deathDate, birthPlace, and nationality. Considering the minsup 0.2 (4/20), the fpof values are calculated in Table4(column 3).

Table 4 :

 4 D Artist : an example of transactional database of the class Artist of DBpedia with fpof values

	Artist	Properties

Table 5 :

 5 Notations

	Symbol	Definition
	e	An entity
	I	The set of items of the database
	I e	All the items of the entity e
	φ	A pattern of items (a non empty subset of I )
	A	The ABox
	T	The TBox
	K	The Knowledge Graph
	C	A class
	D C	A transactional database from the class C
	m	An interestingness measure
	α	A minimum frequency threshold
	Pset(D

C , m α )

Table 6 :

 6 D Artist : an example of transactional database from the Artist class of DBpedia with fpof and ONTOPOD values ONTOPOD with Pset(D Artist , m α ) and k = 1 Artist FPOF m α = no m α = freq 0.2 m α = rare 0.2

	Sid_James	0,5129	6	198	6
	Mackenzie_Taylor	0,5129	6	198	6
	Giuliana_Camerino 0,5129	6	198	6
	Jerry_Clower	0,5129	6	198	6
	Robin_Harris	0,5129	6	198	6
	Greg_Giraldo	0,5129	6	198	6
	LaWanda_Page	0,5129	6	198	6
	Frank_Suero	0,5129	6	198	6
	Josephus_Thimister 0,5129	6	198	6
	Omar_Kiam	0,5129	6	198	6
	Hank_Williams	0,2704	14	94	28606
	Bob_Marley	0,1371	17	49	34801
	Brett_Newski	0,0098	1	2	2
	Children_of_Eve	0,0090	-9	-13	-293
	Mystik	0,0091	-7	-11	-227
	Twice_as_Nice	0,0091	-11	-15	-711
	The_Lead	0,0091	-11	-15	-5639
	Masaba_Gupta	0	3	3	9
	Bertram_Goodman	0	2	2	6
	7icons	0	-4	-4	-20

Table 7 :

 7 Number of outlier and inlier entities in classes

	D	|Entities|	|Outliers|	|Inliers|
	Animal	1991	20	1971
	Artist	2484	96	2388
	Athlete	3040	16	3024
	Company	7848	1013	6835
	EducationalInstitution	3477	15	3462
	Group	1519	386	1133
	P olitician	6920	210	6710
	N aturalP lace	8195	12	8183

https://www.w3.org/TR/shacl/

A ACKNOWLEDGEMENTS

This research was supported by the Partnership for skills in Applied Sciences, Engineering and Technology (PASET) -Regional Scholarship and Innovation Fund (RSIF).