Ugo Authesserre
email: authesserre@chimie.ups-tlse.fr

Sophie Hameury
email: sophie.hameury@univ-poitiers.fr

Aymeric Dajnak
email: dajnak@chimie.ups-tlse.fr

Nathalie Saffon-Merceron

Antoine Baceiredo
email: baceiredo@chimie.ups-tlse.fr

David Madec
email: madec@chimie.ups-tlse.fr

Eddy Maerten
email: maerten@chimie.ups-tlse.fr

Complexes of Dichlorogermylene with Phosphine/Sulfoxide-Supported Carbone as Ligand †

Keywords: carbone, ligand, germylene, coordination, ylide

Due to their remarkable electronic features, recent years have witnessed the emergence of carbones L 2 C, which consist in two donating L ligands coordinating a central carbon atom bearing two lone pairs. In this context, the phosphine/sulfoxide-supported carbone 4 exhibits a strong nucleophilic character, and here, we describe its ability to coordinate dichlorogermylene. Two original stable coordination complexes were obtained and fully characterized in solution and in the solid state by NMR spectroscopy and X-ray diffraction analysis, respectively. At 60 • C, in the presence of 4, the Ge(II)-complex 5 undergoes a slow isomerization that transforms the bis-ylide ligand into an yldiide.

Introduction

The rapid development of homogeneous catalysis in the last decades is highly related to the intensive research that was accomplished toward ligand design. Due to their lone pair(s) and their related nucleophilic character, oxygen-, nitrogen-, sulfur-and phosphoruscontaining ligands have been dominating in the field for several decades [START_REF] Mortreux | Industrial Applications of Homogeneous Catalysis[END_REF]. In the late 1980s, the discovery of the first stable carbenes represented the milestone leading to the emergence of carbon-based ligands (I, Figure 1) [START_REF] Igau | Analogous α,α'-Bis-Carbenoid Triply Bonded Species: Synthesis of a Stable λ 3 -Phosphinocarbene-λ 5 -Phosphaacetylene[END_REF][START_REF] Arduengo | A Stable Crystalline Carbene[END_REF]. Indeed, the development of these molecules, containing a divalent C(II) atom bearing a vacant orbital and a lone pair, exhibiting a high σ-donation and a strong binding ability toward transition metals, render them essential tools for catalysis. The corresponding organometallic complexes have been proven to be particularly robust and efficient and offering in numerous catalytic processes a larger scope of reaction [START_REF] Glorius | N-Heterocyclic Carbenes in Transition Metal Catalysis[END_REF][START_REF] Nolan | N-Heterocyclic Carbenes in Synthesis[END_REF][START_REF] Herrmann | N-Heterocyclic Carbenes: A New Concept in Organometallic Catalysis[END_REF][START_REF] César | Chiral N-heterocyclic carbenes as stereodirecting ligands in asymmetric catalysis[END_REF]. The related carbon(0) species (II), also named carbones, bearing two lone pairs on the central carbon atom are a new emerging class of η 1 -carbon ligands. Even though carbodiphosphoranes (II, L, L = PPh 3 ) were discovered in the 1960s by Ramirez [START_REF] Ramirez | Hexaphenylcarbodiphosphorane, (C 6 H 5 ) 3 PCP(C 6 H 5 ) 3[END_REF], these species were at first regarded as two cumulated phosphorus ylide functions on a central carbon atom. It was only in 2006, after the theoretical investigations by Frenking et al. [START_REF] Tonner | Carbodiphosphoranes: The Chemistry of Divalent Carbon(0)[END_REF][START_REF] Tonner | NHC) 2 : Divalent Carbon(0) Compounds with N-Heterocyclic Carbene Ligands-Theoretical Evidence for a Class of Molecules with Promising Chemical Properties[END_REF][START_REF] Tonner | Divalent Carbon(0) Chemistry, Part 1: Parent Compounds[END_REF][START_REF] Tonner | Divalent Carbon(0) Chemistry, Part 2: Protonation and Complexes with Main Group and Transition Metal Lewis Acids[END_REF], that these molecules were considered as a carbon atom in the zerooxidation state stabilized by two L-phosphine ligands, in agreement with the description initially used by Kaska in 1973 [START_REF] Kaska | Transition metal complexes of hexaphenylcarbodiphosphorane[END_REF]. Since then, this family of ligands has considerably grown, leading to a large structural diversity of carbones II and a better understanding of their behavior [START_REF] Fujii | Synthesis and structure of (MeN)Ph2S=C=SPh2(NMe)[END_REF][START_REF] Alcarazo | Coordination chemistry at carbon[END_REF][START_REF] Dellus | An isolable mixed P,S-bis(ylide) as an asymmetric carbon atom source[END_REF][START_REF] Alcarazo | Chiral heterobimetallic complexes of carbodiphosphoranes and phosphinidene-carbene adducts[END_REF][START_REF] Schmidbaur | Coordination Chemistry at Carbon: The Patchwork Family Comprising (Ph 3 P) 2 C, (Ph 3 P)C(C 2 H 4 ), and (C 2 H 4 ) 2 C[END_REF][START_REF] Morosaki | Structures, and Reactivities of Two Chalcogen-Stabilized Carbones[END_REF][START_REF] Morosaki | Structure, and Reactivities of Iminosulfane-and Phosphane-Stabilized Carbones Exhibiting Four-Electron Donor Ability[END_REF][START_REF] Chen | Expanding the Ligand Framework Diversity of Carbodicarbenes and Direct Detection of Boron Activation in the Methylation of Amines with CO 2[END_REF][START_REF] Troadec | Donor-Stabilized Silylene/Phosphine-Supported Carbon(0) Center with High Electron Density[END_REF][START_REF] Corberan | First Cyclic Carbodiphosphoranes of Copper(I) and Gold(I) and Their Application in The Catalytic Cleavage of X-H bonds (X = N and O)[END_REF]. Naturally, owing to the existence of two lone pairs, they are strong σand π-donors (two-or four-electron donating ability); they have been used as original ligands for the preparation of organometallic complexes with interesting applications in catalysis [START_REF] Goldfogel | Intermolecular hydroamination of 1,3-dienes catalyzed by bis(phosphine)carbodicarbenerhodium complexes[END_REF][START_REF] Hsu | Synthesis and isolation of an acyclic tridentate bis(pyridine)carbodicarbene and studies on its structural implications and reactivities[END_REF][START_REF] Roberts | Acid Activation of Carbodicarbene Catalysts for Rh-Catalyzed Hydroarylation of Dienes[END_REF][START_REF] Pranckevicius | Cyclic Bent Allene Hydrido-Carbonyl Complexes of Ruthenium: Highly Active Catalysts for Hydrogenation of Olefins[END_REF][START_REF] Marcum | Chiral Pincer Carbodicarbene Ligands for Enantioselective Rhodium-Catalyzed Hydroarylation of Terminal and Internal 1,3-Dienes with Indoles[END_REF][START_REF] Liberman-Martin | Ruthenium Olefin Metathesis Catalysts Featuring a Labile Carbodicarbene Ligand[END_REF]. The strong donor ability of carbones II also enables the synthesis and isolation of novel reactive species. For example, Alcarazo et al. took advantage of the two available lone pairs of carbodiphosphoranes to stabilize reactive molecules such as dihydrido borenium cation III [START_REF] Inés | Structure, and Reactivity of a Dihydrido Borenium Cation[END_REF]. In the same vein, several groups have used carbones or strong σ-donating ligands to prepare dichlorogermylene adducts giving access to germyliumylidenes (IV) or germylones [START_REF] Xiong | A cyclic Germadicarbene ("Germylone") from Germyliumylidene[END_REF][START_REF] Li | Acyclic Germylones: Congeners of Allenes with a CentralGermanium[END_REF][START_REF] Khan | Stabilization of a two-Coordinate [GeCl]+ Cation by Simultaneous s and p Donation from a Monodentate Carbodiphosphorane[END_REF][START_REF] Roy | Accessing Low-Valent Inorganic Cations by Using an Extremely Bulky N-Heterocyclic Carbene[END_REF][START_REF] Chu | A coordination Compound of Ge0 Stabilized by a Diiminopyridine Ligand[END_REF][START_REF] Shan | An N-Heterocyclic Silylene-Stabilized Digermanium(0) Complex[END_REF][START_REF] Zhou | A Bis(silylenyl)pyridine Zero-Valent Germanium Complex and Its Remarkable Reactivity[END_REF][START_REF] Nguyen | Ge(0) Compound Stabilized by a Diimino-Carbene Ligand: Synthesis and Ambiphilic Reactivity[END_REF]. In this context, we report here the coordination ability of a phosphine/sulfoxide-supported carbone 4 [START_REF] Gonzalez | Phosphine/Sulfoxide-Supported Carbon(0) Complex[END_REF] towards dichlorogermylene.

Results and Discussion

Synthesis

For the preparation of the phosphine/sulfoxide-supported carbone 4, we followed the previously described synthesis [START_REF] Gonzalez | Phosphine/Sulfoxide-Supported Carbon(0) Complex[END_REF] but several practical aspects were improved. Indeed, after the complete oxidation of methyldiphenylsulfonium, acidification and filtration of the precipitates (carboxylic acids), the expected methyldiphenylsulfoxonium salt 1 was extracted from the aqueous solution by liquid/liquid extraction using dichloromethane as a solvent (Scheme 1). This extraction avoids the possible thermal degradation of the sulfoxonium salt during the evaporation of water under reduced pressure (if prolonged heating above 50 • C is performed). The yield of this step was improved to 53%, after two successive recrystallizations. The coupling reaction between sulfoxonium salt 1 and chlorophosphonium 2 in the presence of two equivalents of lithium diisopropylamide (LDA) was also improved in terms of reaction time (Scheme 1). It was found that heating the reaction mixture up to 60 • C considerably sped up the reaction since a full conversion was reached in 24 h instead of 96 h at room temperature. Protonated precursor 3 was obtained as a white powder upon concentration (70% yield). The final deprotonation was performed in THF solution at RT with potassium hydride (KH) leading to the selective formation of 4, which was isolated in 69% yield. Scheme 1. Synthetic path of phosphine/sulfoxide-supported carbone 4.

Dichlorogermylene Coordination

Previous experimental results and DFT calculations have already established that phosphine/sulfoxide-supported carbone 4 exhibits a strong nucleophilic character [START_REF] Gonzalez | Phosphine/Sulfoxide-Supported Carbon(0) Complex[END_REF]. The potential usefulness of 4 as a ligand towards transition metals was demonstrated by selective reactions with several organometallic complexes [Au(I), Rh(I)] [START_REF] Gonzalez | Phosphine/Sulfoxide-Supported Carbon(0) Complex[END_REF]. Therefore, its coordinating ability toward GeCl 2 /dioxane should be an interesting approach to access original low-valent germanium derivatives.

Ligand 4 reacts immediately with one equivalent of GeCl 2 dioxane leading to the selective formation of complex 5 which has been isolated as colorless crystals from a saturated solution of CH 2 Cl 2 /pentane (yield 80%, Scheme 2). In comparison with the hexaphenylcarbodiphosphorane analogue [START_REF] Khan | Stabilization of a two-Coordinate [GeCl]+ Cation by Simultaneous s and p Donation from a Monodentate Carbodiphosphorane[END_REF], complex 5 exhibits a good solubility in common organic solvent and could be fully characterized by NMR spectroscopy. The 31 P NMR spectrum displays a signal at lower field (δ = 49.5 ppm) compared to that of free ligand 4 (δ = 29.0 ppm), reminiscent of the protonated precursor 3 (δ = 45.0 ppm). In the 13 C NMR spectrum, the central carbon atom appears as a doublet at δ = 47.0 ppm (J PC = 77.3 Hz). The addition of a second equivalent of GeCl 2 •dioxane to 5, or the direct use of two equivalents of GeCl 2 •dioxane with 4, leads to the quantitative formation of bis- germylene complex 6, which has been isolated in the crystalline form from C 6 D 6 solution (yield 75%). It has been fully characterized by NMR spectroscopy, and particularly, the 31 P NMR spectrum indicates a signal at δ = 51.2 ppm and the central carbon atom appears in 13 C NMR spectrum as a doublet at δ = 46.6 ppm (J PC = 79.1 Hz). Isolated in the crystalline form, the molecular structures of complexes 5 and 6 were confirmed by X-ray diffraction analysis (Figure 2, see Supplementary Materials). The selected geometrical parameters for experimental structures can be found in Table 1.

As expected, the S1-C1 (1.650 Å) and P1-C1 (1.725 Å) bond lengths in 5 are very similar to those observed in the protonated precursor 3 (1.653 Å and 1.719 Å respectively), because of their similar environments. The repulsion between the p π -lone pair at the carbon and the one at the Ge atom explains the long C1-Ge1 bond length (2.071 Å). It is slightly longer than the one observed by Alcarazo et al. with carbodiphosphoranes (2.063 Å) but much longer than a typical C-Ge bond (1.95 Å) [START_REF] Meiners | Reaction of a diarylgermylene with a phosphaalkyne: Formation of a germadiphosphacyclobutene with an exocyclic C=Ge double bond[END_REF]. The P1-C1-S1 angle decreased significantly upon complexation with GeCl 2 (from 121 • in 3 or 4 to 113.7 • in 5). The introduction of a third heteroatom (Ge) around the carbon, which is less electronegative than P and S, influences the atomic orbital distribution with a pronounced s character toward the C-Ge bond and increased p-character in the C-P and C-S bonds. This phenomenon together with the covalent radius of the germanium atom justify the narrowing of the P1-C1-S1 angle in 5 [START_REF] Alabugin | Hybridization Trends for Main Group Elements and Expanding the Bent's Rule beyond Carbon: More than Electronegativity[END_REF]. Nevertheless, the C1 atom environment remains almost planar (∑ • = 357.5 • ). In 6, with the coordination of the second GeCl 2 unit, the previously discussed repulsion disappears, leading to a shortening of the C1-Ge1 bond length to 1.980 Å. The other bond lengths and angles (P1-C1, C1-S1, PCS) remain almost unchanged (Table 1). The dative nature of the Ge1-Ge2 bond is confirmed by its length (2.582 Å), way longer than classical Ge-Ge σ-bonds (2.40-2.50 Å) [START_REF] Amadoruge | Singly Bonded Catenated Germanes: Eighty Years of Progress[END_REF]. The same tendency was observed for carbodiphosphoranes [START_REF] Khan | Stabilization of a two-Coordinate [GeCl]+ Cation by Simultaneous s and p Donation from a Monodentate Carbodiphosphorane[END_REF] or NHC [START_REF] Al-Rafia | Controlled Growth of Dichlorogermanium Oligomers from Lewis Basic Hosts[END_REF]. Contrary to the carbodiphosphorane analogues, chloride abstraction from 5 using AlCl 3 or KB(C 6 F 5 ) 4 in order to prepare germyliumylidene derivatives only led to complex mixtures. Nevertheless, when 5 is treated in the presence of one equivalent of 4 at 60 • C for 60 h, we observed the gradual consumption of 5 with the concomitant formation of a new compound that exhibits a signal at δ = 56.3 ppm in 31 P NMR (this reaction does not occur in the absence of 4) (Scheme 3). In fact, the process is base-catalyzed, but with 10 mol % of 4, the reaction time is slower and needs 90 h [Note: catalytic amounts (15 mol %) of alternative Lewis bases such as DMAP or Et 3 N can be used but the reactions are less selective, see Supplementary Materials for more details]. The structure of the new product 7, determined by X-ray diffraction analysis (Figure 3), involves the 1,3-migration of a phenyl group from the sulfur to the germanium atom. Considering the ligand moiety, this isomerization transforms a bis-ylide (carbone) to an original yldiide [START_REF] Fustier-Boutignon | Geminal Dianions Stabilized by Main Group Elements[END_REF]. Unfortunately, because of the presence of 4 in the media, product 7 could not be isolated in pure form for complete characterization despite several attempts.

Conclusions

In summary, the excellent coordination ability of the phosphine/sulfoxide-supported carbone ligand 4 allows the preparation of Ge(II)-complex 5. The strong donation of 4 results in an enriched germylene 5, which becomes sufficiently nucleophilic to coordinate a second GeCl 2 unit. The two original and stable coordination complexes 5 and 6 were fully characterized by NMR spectroscopy and X-ray diffraction analysis. Interestingly, the Ge(II)-complex 5 shows an original isomerization in the presence of 4 that transforms the bis-ylide ligand into an yldiide, thanks to a phenyl migration. Efforts are currently underway to extend the diversity of organometallic complexes that can be obtained with 4 to consider its application in catalysis.

Materials and Methods

General Comments

All manipulations were performed under an inert atmosphere of argon by using standard Schlenk techniques or high-pressure NMR tube techniques. Dry and oxygen-free solvents were used. 1 H, 13 C, 19 F and 31 P NMR spectra were recorded on Brucker Avance II 300 MHz, Avance III HD 400 MHz and Avance I and II 500 MHz spectrometers (Brucker, Karlsruhe, Germany). Chemical shifts are expressed in parts per million with residual solvent signals as internal reference ( 1 H and 13 C{ 1 H}). 19 F and 31 P NMR chemical shifts were reported in ppm relative to CFCl 3 and 85% H 3 PO 4 , respectively. The following abbreviations and their combinations are used: br, broad; s, singlet; d, doublet; t, triplet; q, quartet; hept, heptuplet; m, multiplet. 1 H and 13 C resonance signals were attributed by means of 2D COSY, HSQC and HMBC experiments. Mass spectra were recorded on a Hewlett Packard 5989A spectrometer (Hewlett-Packard, Palo Alto, CA, USA). All commercially available reagents were used without further purification otherwise noted. Preparation of diphenylsulfonium, 2 and 4 were prepared following previously reported procedures [START_REF] Gonzalez | Phosphine/Sulfoxide-Supported Carbon(0) Complex[END_REF].

Synthesis

Diphenylmethylsulfoxonium triflate 1: A suspension of diphenylmethylsulfonium triflate (14.96 g, 42.7 mmol, 1 eq.), Na 2 CO 3 (13.58 g, 128.1 mmol, 3 eq.) and metachloroperbenzoic acid (MCPBA) (22.13 g, 128.1 mmol, 3 eq.) in water (400 mL) was stirred at RT for three days. Na 2 CO 3 (4.53 g, 42.7 mmol, 1 eq.) and MCPBA (7.28 g, 42.7 mmol, 1 eq.) were added and the mixture was stirred at RT for one additional day. A 37% aqueous solution of HCl was added to the solution until pH = 1. White solid was filtrated off and washed with an aqueous solution of HCl at pH = 1 (3 × 20 mL). The product was then extracted with CH 2 Cl 2 (3 × 20 mL). The solution was dried then the solvent was removed under reduced pressure. The residue was purified by successive crystallizations in acetone/pentane, yielding 1 as colorless crystals (8.30 g, 22.6 mmol, 53%). 1 Protonated phosphine/sulfoxide precursor 3: In situ prepared LDA (18.2 mmol, 2 eq.) in THF (20 mL) was added dropwise to a suspension of 1 (4 g, 10.9 mmol, 1.2 eq.) in THF (100 mL) at -80 • C and the reaction was stirred at this temperature for 2 h. A suspension of 2 (2.92 g, 9.1 mmol, 1 eq.) in THF (50 mL) was then added dropwise to the solution at -80 • C. The solution was allowed to warm up to RT then heated at 60 • C for two days. The solution was concentrated (2/3 of the solvent removed) and a white precipitate was observed. The solvent was filtered off and the white solid was washed with small volumes of THF (4 × 2 mL). The remaining solid was extracted in CH 2 Cl 2 then dried under a high vacuum. Ylide 3 was obtained as a white powder (3.9 g, 6.37 mmol, 70%). 31 Phosphine/sulfoxide-carbone-GeCl 2 complex 5: To a solution of 4 (37.1 mg, 0.08 mmol, 1 eq.) in C 6 D 6 (0.6 mL), GeCl 2 •dioxane (9 mg, 0.08 mmol, 1 eq.) was added. The reaction is complete after a few minutes. C 6 D 6 was removed under reduced pressure and the residue afforded crystals (38.9 mg, 0.064 mmol, 80% yield) from a saturated solution of CH 2 Cl 2 /pentane. 31

Figure 1 .

 1 Figure 1. NHC (I), carbones (II), dihydrido borenium (III) and germyliumylidene (IV) stabilized by carbodiphosphoranes and phosphine/sulfoxide-supported carbone 4 (for a better readability, formal charges in (III,IV) and 4 were omitted).

Scheme 2 .

 2 Scheme 2. Coordination of phosphine/sulfoxide-supported carbone 4 with one or two equivalents of germylene dichloride.

Figure 2 .

 2 Figure 2. Molecular structures of 5 (left) and 6 (right). Ellipsoids are drawn at the 30% probability level; hydrogen atoms are omitted for clarity. Selected bond lengths (Å) and angles ( • ): 5: C1-Ge1 2.071(2), C1-P1 1.725(2), C1-S1 1.650(2), P1-C10 1.802(3), S1-O1 1.457(2), Ge1-Cl2 2.299(1), Ge1-Cl1 2.331(1), S1-C22 1.780(3), S1-C16 1.784(3). P1-C1-S1 113.73(14), P1-C1-Ge1 128.12(13), Ge1-C1-S1 115.65(13), C1-Ge1-Cl2 96.82(7), Cl2-Ge1-Cl1 95.37(3), Cl1-Ge1-C1 98.81(7); 6: C1-Ge1 1.980(2), C1-P1 1.748(2), C1-S1 1.665(2), Ge1-Ge2 2.582(1), Ge1-Cl1 2.185(1), Ge1-Cl2 2.245(1), Ge2-Cl4 2.267(1), Ge2-Cl3 2.275(1), P1-C10 1.792(2), S1-O1 1.455(1), S1-C22 1.778(2), S1-C16 1.782(2), P1-C1-S1 112.40(10), P1-C1-Ge1 127.01(9), Ge1-C1-S1 117.30(9), C1-Ge1-Cl1 105.44(5), Cl1-Ge1-Cl2 101.33(2), Cl2-Ge1-Ge2 107.25(2), Cl1-Ge1-Ge2 109.25(2), Ge1-Ge2-Cl4 92.09(2), Cl4-Ge2-Cl3 98.84(3), Ge1-Ge2-Cl3 89.41(2).

Scheme 3 .

 3 Scheme 3. Evolution of 5 in presence of 4 upon heating at 60 • C.

Figure 3 .

 3 Figure 3. Molecular structure of 7. Ellipsoids are drawn at the 30% probability level; hydrogen atoms are omitted for clarity. Selected bond lengths (Å) and angles ( • ): P1-C1 1.707(3), C1-S1 1.744(3), C1-Ge1 1.891(3), Ge1-C22 1.927(3), Ge1-Cl2 2.172(1), Ge1-Cl1 2.173(1), P1-C10 1.805(3), S1-O1 1.499(2), S1-C16 1.800(4), P1-C1-S1 114.48(18), S1-C1-Ge1 120.89(17), Ge1-C1-P1 122.37(19), C1-Ge1-C22 119.52(14), C1-Ge1-Cl2 111.27(10), C22-Ge1-Cl2 104.77(10), C1-Ge1-Cl1 111.72(10), C22-Ge1-Cl1 106.65(11), Cl2-Ge1-Cl1 101.15(4), O1-S1-C1 112.64(15), O1-S1-C16 105.28(16), C1-S1-C16 100.04(16).

  P{1 H} NMR (202 MHz, 298 K, CDCl 3 ): δ = 45.0 ppm (s).1 H NMR (300 MHz, 298 K, CDCl 3 ): δ = 0.76 (d, J HH = 6.5 Hz, 12H, CH 3iPr ), 3.08-3.24 (m, 4H, CH 2 ), 3.40 (hept, J HH = 6.5 Hz, 1H, CH iPr ), 3.43 (hept, J HH = 6.5 Hz, 1H, CH iPr ), 3.80 (d, J PH = 13.7 Hz, 1H, PCHS), 7.57 (m, 9H, CH ar ), 8.05-8.16 (m, 2H, CH ar ), 8.18-8.25 (m, 4H, CH ar ).13 C{ 1 H} NMR (75 MHz, 298 K, CDCl 3 ): δ = 19.9-20.1 (m, CH 3iPr ), 31.6 (d, J CP = 137.2 Hz, PCHS), 39.1 (d, J PC = 8.6 Hz, CH 2 ), 45.1 (d, J PC = 5.7 Hz, CH iPr ), 121.2 (q, J CF = 318.9 Hz, CF 3 ), 126.7 (d, J PC = 121.1 Hz, C ipso P side ), 127.1 (s, CH ar ), 130.0 (d, J PC = 13.7 Hz, CH ar ), 130.4 (s, CH ar ), 132.8 (d, J PC = 11.7 Hz, CH ar ), 134.4 (s, CH ar ), 134.5 (d, J PC = 3.1 Hz, CH ar ), 140.4 ppm (d, J PC = 2.4 Hz, C ipso S side ).19 F NMR (282 MHz, 298 K CDCl 3 ): δ = -78.1 (s). HRMS (ES +): m/z [M] + calculated for C 27 H 34 ON 2 PS = 465.2130, found = 465.2133. The elemental analysis calculated (%) for C 28 H 34 F 3 N 2 O 4 PS 2 : C 54.71, H 5.58, N 4.56; found: C 54.63, H 5.47, N 4.51.

  P {1 H} NMR (202 MHz, 298 K, C 6 D 6 ): δ = 49.5 ppm (s).31 P {1 H} NMR (162 MHz, 298 K, CD 2 Cl 2 ): δ = 50.9 ppm (s).1 H NMR (500 MHz, 298 K, CD 2 Cl 2 ): δ = 0.70 (d, J HH = 6.5 Hz, 6H, CH 3iPr ), 0.81 (d, J HH = 6.5 Hz, 6H, CH 3iPr ), 3.08-3.18 (m, 2H, CH 2 ), 3.19-3.28 (m, 2H, CH 2 ), 3.58-3.69 (m, 2H, CH iPr ), 7.59-7.71 (m, 9H, CH ar ), 8.07-8.14 (m, 6H, CH ar ).13 C{ 1 H} NMR (126 MHz, 298 K, CD 2 Cl 2 ): δ = 19.9 (m, CH 3iPr ), 39.3 (d, J PC = 8.9 Hz, CH 2 ), 46.0 (d, J PC = 4.6 Hz, CH iPr ), 47.0 (d, J PC = 77.3 Hz, PCS), 129.1 (s, CH ar ) 129.9 (d, J PC = 125.5 Hz, C ipso P side ), 130.1 (d, J PC = 13.9 Hz CH ar ), 130.3 (s, CH ar ) 132.9 (d, J PC = 11.8 Hz, CH ar ), 134.2 (d, J PC = 3.1 Hz, CH ar ), 134.7(s, CH ar ), 140.7 (d, J PC = 3.1 Hz, C ipso S side ). HRMS (DCI-CH 4 ): m/z [M + H] + calcd for C 27 H 34 Cl 2 GeN 2 OPS: 609.0713; found: 609.0715.

Table 1 .

 1 Selected geometrical parameters for 3, 4, 5 and 6 1 .

		3	4	5	6
	P1-C1	1.719(15)	1.800(1)	1.725(2)	1.748(2)
	S1-C1	1.653(13)	1.593(1)	1.650(2)	1.665(2)
	P1-C1-S1	120.98(9)	120.74(8)	113.73(14)	112.40(10)
	C1-Ge1	-	-	2.071(2)	1.980(2)
	Ge1-Ge2	-	-	-	2.582(1)

1

Bond lengths in Å and angle in deg.

  H NMR (300 MHz, 298 K, Acetone-d 6 ): δ = 4.86 (s, 3H, CH 3 ), 7.91-7.98 (m, 4H, CH ar ), 8.04-8.11 (m, 2H, CH para ), 8.31-8.37 (m, 4H, CH ar ). 13 C{ 1 H} NMR (75 MHz, 298 K, Acetone-d 6 ): δ = 39.7 (s, CH 3 ), 128.4 (s, CH ar ), 132.0 (s, CH ar ), 132.9 (s, C ipso ), 138.1 (s, CH para ). The signal of the triflate could not be detected. 19 F NMR (282 MHz, 298 K, DMSO-d 6 ) δ = -77.7 (s). Mp = 157 • C. HRMS (ES +): m/z [M] + calculated for C 13 H 13 OS = 217.0687, found = 217.0690.

Funding: This research was funded by The Agence Nationale de la Recherche, ANR-19-CE07-0013.

Institutional Review Board Statement: Not applicable.

Data Availability Statement: Not applicable.

Acknowledgments:

The Agence Nationale de la Recherche (ANR-16-CE07-0018-01) and the Ministère de l'Enseignement Supérieur et de la Recherche are gratefully acknowledged for Ph.D. grant to U.A. The authors would like to thank the CNRS and the Université de Toulouse, UPS for financial support. The authors would like to thank the reviewers for all their useful and helpful comments.

Phosphine/sulfoxide-carbone-GeCl 2-GeCl 2 complex 6: To a solution of 4 (37.1 mg, 0.08 mmol, 1 eq.) in C 6 D 6 (0.6 mL), GeCl 2 •dioxane (18 mg, 0.16 mmol, 2 eq.) was added. Product 6 directly crystallized in C 6 D 6 (45.0 mg, 0.06 mmol, 75% yield). 31 P { 1 H} NMR (202 MHz, 298 K, C 6 D 6 ): δ = 51.2 ppm (s). 31 P { 1 H} NMR (202 MHz, 298 K, CD 2 Cl 2 ): δ = 51.2 ppm (s). 1 H NMR (300 MHz, 298 K, CD 2 Cl 2 ): δ = 0.70 (d, J HH = 6.4 Hz, 6H, CH 3iPr ), 0.82 (d, J HH = 6.5 Hz, 6H, CH 3iPr ), 3.09-3.18 (m, 2H, CH 2 ), 3.18-3.28 (m, 2H, CH 2 ), 3.60-3.70 (m, 2H, CH iPr ), 7.60-7.71 (m, 9H, CH ar ), 8.06-8.13 (m, 3H, 6CH ar ). 13 Isomer 7: To a solution of 5 (48.6 mg, 0.08 mmol, 1 eq.) in C 6 D 6 (0.6 mL), one equivalent of 4, 37.1 mg, 0.08 mmol, 1 eq.) was added. The reaction was heated at 60 • C until complete consumption of 5 (after 60 h). Crystals of 7 could be obtained from the crude mixture. Unfortunately, despite several attempts, 7 was only analyzed by spectroscopy as a mixture. The spectroscopic data were extracted from a mixture containing 7 and 4 in a 51% to 45% ratio (4% of protonated phosphine/sulfoxide precursor 3). The reaction can also be performed with 10 mol% of 4 (3.7 mg, 0.008 mmol). 31

X-ray Data

The data of the structures for 5, 6 and 7 were collected at 193 K on a Bruker-AXS APEX II CCD Quazar diffractometer [START_REF] César | Chiral N-heterocyclic carbenes as stereodirecting ligands in asymmetric catalysis[END_REF] equipped with a 30 W air-cooled microfocus source or on a Brucker-AXS D8-Venture diffractometer (5 and 6) equipped with a CMOS area detector with MoKα radiation (wavelength = 0.71073 Å) by using phi-and omega-scans. The data were integrated with SAINT, and an empirical absorption correction with SADABS was applied [START_REF]SADABS. Program for Data Correction, Version 2016/2[END_REF]. The structures were solved using an intrinsic phasing method (ShelXT) [START_REF] Sheldrick | SHELXT-Integrated space-group and crystal-structure determination[END_REF] and refined using the least-squares method on F 2 (ShelXL-2014) [START_REF] Sheldrick | Crystal structure refinement with SHELXL[END_REF]. All non-H atoms were treated anisotropically. All H atoms attached to C atoms were fixed geometrically and treated as riding on their parent atoms with C-H = 0.95 Å (aromatic), 0.98 Å (CH 3 ), 0.99 Å (CH 2 ) or 1.0 Å (CH) with U iso (H) = 1.2U eq (CH, CH 2 ) or U iso (H) = 1.5U eq (CH 3 ).

Supplementary crystallographic data for CCDC-2068304 (5), CCDC-2068305 (6), CCDC-2068306 [START_REF] César | Chiral N-heterocyclic carbenes as stereodirecting ligands in asymmetric catalysis[END_REF] can be obtained free of charge from The Cambridge Crystallographic Data Centre via https://www.ccdc.cam.ac.uk/structures/ accessed on 25 March 2021.

Supplementary Materials:

The following are available online. NMR spectra and crystallographic data.

Conflicts of Interest:

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Sample Availability: Samples of the compounds are not available from the authors.


Vineeth S Varma

Bikash Adhikari

Irinel-Constantin Morȃrescu

Elena Panteley

Optimal campaign strategy for social media marketing with a contrarian population

We address formally the problem of opinion dynamics when the social network composed of conformists and contrarians are not only influenced by their neighbors, but also by an external influential entity referred to as a marketer. The population of contrarians tries to have an opinion that is the opposite of the opinion held by the conformists. The influential entity tries to sway the overall opinion as close as possible to the desired opinion by using a specific influence budget. The main technical issue addressed is finding how the marketer should allocate its budget among the agents such that the agents' opinion will be as close as possible to the desired opinion while taking into account the behavior of the contrarian population. Our main results show that the marketer has to prioritize certain agents over others based on their initial condition, their influence power in the social graph and the population class they belong to. Numerical examples illustrate the analysis.

Introduction

During the last decade, social networks gained increasing importance in our daily life. Consequently, more and more companies are using digital social networks to promote specific goods and/or ideas. This motivated the scientific community to give more attention to the analysis of opinion dynamics in social networks. This is a challenging task since human behavior is very different from one individual to another and the interactions in the network can change over time. Various mathematical models [START_REF] Degroot | Reaching a consensus[END_REF][START_REF] Friedkin | Social influence and opinions[END_REF][START_REF] Deffuant | Mixing beliefs among interacting agents[END_REF][START_REF] Hegselmann | Opinion dynamics and bounded confidence models, analysis, and simulation[END_REF][START_REF] Altafini | Consensus problems on networks with antagonistic interactions[END_REF][START_REF] Chowdhury | Continuous opinions and discrete actions in social networks: a multi-agent system approach[END_REF] have been proposed in order to capture different features of this complex dynamics. Empirical models based on in vitro and in vivo experiments have also been developed [START_REF] Davis | chap. Group decision making and quantitative judgments : A consensus model[END_REF]16,[START_REF] Kerckhove | Modelling influence and opinion evolution in online collective behaviour[END_REF].

Some mathematical model target consensus as collective asymptotic behavior of the network [START_REF] Degroot | Reaching a consensus[END_REF][START_REF] Friedkin | Social influence and opinions[END_REF] while some others lead to the network clustering [START_REF] Hegselmann | Opinion dynamics and bounded confidence models, analysis, and simulation[END_REF][START_REF] Altafini | Consensus problems on networks with antagonistic interactions[END_REF][START_REF] Morȃrescu | Opinion dynamics with decaying confidence: Application to community detection in graphs[END_REF]. In order to enforce consensus some recent studies propose the control of one or few agents (see [START_REF] Caponigro | Sparse feedback stabilization of multi-agent dynamics[END_REF][START_REF] Dietrich | Control via leadership of opinion dynamics with state and time-dependent interactions[END_REF]). Besides these methods of controlling opinion dynamics towards consensus, we also find few attempts to control the discrete-time dynamics of Supported by CEFIPRA under the project 6001-1/2019 opinions such that, as many as possible reach a certain set after a finite number of influences [START_REF] Hegselmann | Optimal opinion control : The campaign problem[END_REF]. In very recent work ( [START_REF] Morȃrescu | Majority influence process in group judgment : Test of the social judgment scheme model in a group polarization context[END_REF]) the authors developed a formal method for the optimal space-time allocation of a budget allowing an influencer to bring the consensus value of the network as close as possible to a desired value.

It is noteworthy that most of the proposed mathematical models consider that the opinions of two interacting individuals are approaching one to another. Although this behavior seems to be sociologically accepted, one can also provide real-life situations in which interactions are antagonistic. In this case, the distance between the opinions of two interacting individuals will increase (see [START_REF] Altafini | Consensus problems on networks with antagonistic interactions[END_REF][START_REF] Hendrickx | A lifting approach to models of opinion dynamics with antagonisms[END_REF][START_REF] Proskurnikov | Opinion dynamics in social networks with hostile camps: Consensus vs. polarization[END_REF]). To the best of our knowledge, there exists no attempt to control this kind of opinion dynamics, and only analysis of the asymptotic behavior is reported.

In this paper, we consider the challenging problem that requires to minimize the distance between the average of opinions and a desired value using a given control/marketing budget over a social network split into two groups. Basically, this social network with the contrarian population represents a model for real cases such as supporters of competing teams, parties, etc. On top of this assumption on the network structure, we also assume that the maximal marketing influence cannot instantaneously make the opinion of one individual to be equal with the desired value.

To provide a mathematical model we consider that the opinion dynamics is fast enough such that we can assume it evolves in continuous time and we want to design a marketing strategy that minimizes the distance between the average of the opinions and the desired value after a campaign with certain budget constraints. This results in a linear-impulsive closed-loop dynamics in which the jumps are controlled by the influencer. Our main result shows that the optimal control strategy consists of influencing as much as possible the most central/popular (see [START_REF] Bonacich | Eigenvector-like measures of centrality for asymmetric relations[END_REF] for a formal definition of centrality) individuals of the network.

It is worth highlighting that in this study we do not control the state of the influencing entity which is assumed to be constant. Instead, we control the influence weight that the marketer can have on different individuals of the social network. It was shown in [START_REF] Morȃrescu | Majority influence process in group judgment : Test of the social judgment scheme model in a group polarization context[END_REF] that this approach allows highlighting the effectiveness of targeted marketing with respect to broadcasting strategies when budget constraints are present. Notation. Let R := (-∞, ∞), R ≥0 := [0, ∞), and Z ≥0 := {0, 1, 2, . . .}. We use E for the expectation taken over the relevant stochastic variables and 1(•) for the indicator function, taking the value 1 when the condition is satisfied and 0 otherwise.

Problem statement

We consider a social network populated by agents belonging to the set V := {1, 2, . . . , N } with the connections given by a i,j indicating the influence of agent j on agent i. All agents belong to the class of conformists denoted by the set V + or to the class of contrarians denoted by V -. We use c n ∈ {1, -1} to denote the agent class with c n = 1 for all n ∈ V + and c n = -1 for all n ∈ V -. These two sets are non-intersecting and may be interpreted as two hostile camps as considered in [START_REF] Proskurnikov | Opinion dynamics in social networks with hostile camps: Consensus vs. polarization[END_REF].

The opinion of all agents belongs to the interval [-1, 1] and we denote the collection of all opinions by x ∈ [-1, 1] N . The internal opinion dynamics of the network is given by ẋi =

j∈V a i,j x j -|a i,j |x i (1)
for all i ∈ V. The above dynamics is similar to a standard consensus model as in [START_REF] Deffuant | Mixing beliefs among interacting agents[END_REF] but a i,j may be positive or negative depending on the interaction type. In this work, we focus on antagonistic interaction between agents that belong to different classes and standard consensus interactions among agents of the same class, i.e., contrarians or conformist. Basically, this implies that a i,j = c i c j |a i,j |.

As a result of this extra term, agents i and j will have a consensus only if i, j belong to the same class. If they belong to different classes, agent i will try to have an opinion in opposition to agent j. We can write the overall dynamics for all agents as ẋ = -Lx

where L is a Laplacian-like matrix with

L i,j = k∈V |a i,k | for all i = j ∈ V, -a i,j for all i = j ∈ V (3)
The influence of the external entity (marketer) with a desired opinion d is limited to a campaign occurring at t = 0. We model this influence through an impulsive control/jump at t = 0. Due to the contrarian population, the marketer may also desire to bring some agents closer to the opinion -d but has must select the best agents due to budget constraints. We, therefore, model the impulsive control as follows

x n (0

+ ) = (1 -|u n |)x n (0) + u n d (4)
for all n ∈ V with u n ∈ [-ū, ū], with ū denoting the control saturation and

n∈V |u n | ≤ B is the budget constraint.
The main objective of this paper is to provide an optimal control u such that the distance between the average opinion of all agents and the desired opinion d ∈ {-1, 1}, given by

1 N x N -d is minimized asymptotically. Given x 0 ∈ [-1, 1] N , the optimization problem can be stated formally as follows Minimize u J(u) := lim t→∞ 1 N x(t) N -d where ẋ(t) = -Lx(t), x(0) = x 0 ∀t > 0, under (4) with u n ∈ [-ū, ū] ∀n ∈ V and n∈V |u n | ≤ B. (5)
3 Analysis

To begin our analysis, we first characterize the asymptotic opinions of the agents under the dynamics [START_REF] Bonacich | Eigenvector-like measures of centrality for asymmetric relations[END_REF]. For this purpose, we rely on results established in [START_REF] Proskurnikov | Opinion dynamics in social networks with hostile camps: Consensus vs. polarization[END_REF].

Proposition 1. Given the opinions x(0 + ) formed after the campaign, we have

lim t→∞ x(t) = cv x(0 + ) ( 6
)
where v is the left eigen-vector of the matrix L associated with eigenvalue 0 with

n∈V |v n | = 1 and sign(v n ) = sign(c n ).
Proof. First, we use Lemma 1 in [START_REF] Proskurnikov | Opinion dynamics in social networks with hostile camps: Consensus vs. polarization[END_REF] to conclude that if (2) establishes modular consensus (consensus in absolute value of each x n , then lim t→∞ x(t) = ρv x(0 + ) [START_REF] Degroot | Reaching a consensus[END_REF] where ρ n ∈ {1, -1} for all n ∈ V. By construction in (3), we have that a i,j is positive for i = j only when they both belong to the same set V + or V -and is negative otherwise. This implies that V + or V -form hostile camps as defined in [START_REF] Proskurnikov | Opinion dynamics in social networks with hostile camps: Consensus vs. polarization[END_REF], allowing us to apply Lemma 2 from [START_REF] Proskurnikov | Opinion dynamics in social networks with hostile camps: Consensus vs. polarization[END_REF]. This states that modular consensus is established for quasi-strongly connected graphs with ρ i = -ρ j for all i, j in opposite camps. Since we pick sign(v n ) = sign(c n ), we have that ρ n = c n .

Proposition 1 allows us to study the impact of the control on the cost function in 5. We provide the optimal control u which will minimize this cost in the following. We define a sorting index

γ n := |v n ||c n d -x n (0)| (8) Theorem 1. If card(V + ) = card(V -)
, the final cost is invariant to the control and J(u) = 1 for all u. If card(V + ) > card(V -) and the desired opinion be d, then the optimal control u * is given by

u * o(n) =    c o(n) ū if n ≤ B ū Bc o(n) -c o(n) ū B ū if n = B ū + 1 0 otherwise (9)
where o :

V → V is a bijection such that γ o(1) ≥ γ o(2) ≥ • • • ≥ γ o(N ) . When card(V + ) < card(V -)
, setting -d as the desired opinion and applying u * minimizes the cost J(u).

Proof : First, we rewrite the minimization (5) problem as follows

Minimize u lim t→∞ 1 N x(t) N -d where ẋ(t) = -Lx(t), ∀t > 0, x n (0 + ) = (1 -u + n -u - n )x n (0) + u + n d -u - n d, ∀n ∈ V u + n , u - n ∈ [0, ū], such that u + n u - n = 0, ∀n ∈ V, n∈V u + n + u - n ≤ B. (10)
in order to separate the positive and negative control action. Note that we have that lim

t→∞ x(t) = cv x(0 + ) ( 11
)
since the dynamics in (0, ∞) is given by ẋ = -Lx and using [START_REF] Altafini | Consensus problems on networks with antagonistic interactions[END_REF]. Denoting x * := v x(0 + ), which is a scalar and belongs to the interval [-1, 1], we have that the final cost is given by

J = n∈V c n N x * -d = card(V + ) -card(V -) N x * -d . (12)
We use

G := card(V + ) -card(V -) N . Therefore, 1. if card(V + ) = card(V -), we have J = d for any x * , 2. if card(V + ) > card(V -), we have J = |Gx * -d|, 3. and if card(V + ) < card(V -), J = |Gx * + d|.
This implies that minimizing the final cost J is equivalent of minimizing (x * -d) 2 for case 2 and (x * + d) 2 for case 3. We have

∂((d -x * ) 2 ) ∂u + n = -2(d -x * )v n , ∂((d -x * ) 2 ) ∂u - n = 2(d -x * )v n , ∂((d + x * ) 2 ) ∂u + n = 2(d + x * )v n , ∂((d + x * ) 2 ) ∂u - n = -2(d + x * )v n . (13)
In the two cases cases, 2(d -x * ) and 2(d + x * ) are respectively positive functions which hold true for all n. Applying Lemma 1 (provided in the Appendix) for each case, we have the result by setting u n = u + n -u - n . Theorem 1 provides the control strategy to be implemented in order to minimize the cost J(u), which implies minimizing the distance between the average final opinion and the desired opinion. When the number of contrarians and conformists are the same, the opinions are polarised around 0 in a symmetric fashion. Therefore, the final average opinion is always 0, leading to a fixed cost of J(u) = |d|.

Nominally, we assume that the set of contrarians is a minority, i.e., card(V + ) > card(V -). In this case, we select the most influential agents based on the index γ which depends on the vector centrality v as well as the distance to the desired opinion. However, since the contrarians oppose the conformists, the optimal strategy is to push the contrarians closer to -d and the conformists closer to d. Since the conformists are a majority, the minimal cost is when all conformists are at d and all contrarians at -d. The control in Theorem 1 precisely achieves this, i.e. it brings the conformists as close as possible to d and the contrarians as close as possible to -d, under the budget constraints.

Note that when card(V -) = 0, i.e. there are no antagonistic relations, the optimal control u * in Theorem 1 matches the result in [START_REF] Morȃrescu | Majority influence process in group judgment : Test of the social judgment scheme model in a group polarization context[END_REF] (which considers a i,j ≥ 0 for all i, j) as expected. When the population of contrarians are larger, i.e. card(V + ) < card(V -), the minimal cost occurs when all conformists are at -d and all contrarians at d. This is achieved by simply setting the new desired opinion to be -d and applying the control in Theorem 1.

To demonstrate and compare the numerical results the following budget allocation strategies are adopted: Strategy 1: Optimal Budget Allocation, where the budget is allocated to agents according to Theorem 1. Strategy 2: Uniform Budget Allocation, where budget is allocated uniformly to all the agents with negative control for contrarians, i.e. u i = c i B N . Strategy 3: Positive Budget Allocation, where budget is allocated uniformly and an identical control is applied to all the agents, i.e. u i = B N .

Strategy 3 corresponds to traditional advertising campaigns through Television or radio broadcasts as the same advertising action is applied to all agents regardless of their individual characteristics. The network structure we consider is a strongly connected directed graph with 10 nodes (N=10) and the number of conformists is greater than that of the contrarians (|V | + > |V | -). The strongly connected graph is represented in Figure 1. The initial opinions are uniformly chosen between [-1, 1] and represented in Table 1. Fig. 2: Opinion dynamics with the control applied impulsively at t = 0, red and cyan plots represent conformists and contrarians.

In Figure 1, the red and cyan nodes represent the conformist and the contrarian agents, respectively and the size of the nodes represents the agent's centrality.

The centralities of all the agents are also represented in Table 1. As stated in Section 2, positive and negative values of c represents conformists and contrarians respectively. Consider that the budget B = 4, the maximum control that can be allocated to each agent ū = 0.7, and opinions before the campaign as given in Table 1. When no control action is applied, we see that J(0) = 0.8877. Using strategy 1, budget allocation is performed based on the influence power of an agent γ (represented in Table 1) while satisfying the budget constraints.

The budget of [0.7, 0.7, 0.7, 0.7, 0.7, 0.5] is allocated to the agents [START_REF] Dietrich | Control via leadership of opinion dynamics with state and time-dependent interactions[END_REF][START_REF] Caponigro | Sparse feedback stabilization of multi-agent dynamics[END_REF][START_REF] Hegselmann | Opinion dynamics and bounded confidence models, analysis, and simulation[END_REF][START_REF] Friedkin | Social influence and opinions[END_REF][START_REF] Davis | chap. Group decision making and quantitative judgments : A consensus model[END_REF][START_REF] Chowdhury | Continuous opinions and discrete actions in social networks: a multi-agent system approach[END_REF] and the resultant cost J(u * ) = 0.7331. To better understand the advantage Table 1: Data of the designed marketing strategy (strategy 1), the results obtained are compared with the ones obtained from strategy 2 and 3. Using strategy 2, the total available budget B = 4 is allocated uniformly among N agents and the resultant cost is 0.7728. Next, with strategy 3, the positive budget is uniformly allocated to all the agents and the cost obtained is 0.8646. Finally, we consider a large-scale network with 300 nodes, formed by 279 conformists and 21 contrarians. The graph is constructed in the following manner, among the 300 agents, 25 nodes are taken to be central and influence 30 to 60 agents, while the remaining 275 have 1 to 30 agents they influence. The initial opinion of the agents are generated by the formula x i (0) = -1 + 2i N . As seen in Figure 3, the cost with no control (or when budget B = 0) is the same for all strategies i.e., J(0) = 1. It is clear from the plot that the cost is smaller using Strategy 1 (based on Theorem 1) for any budget between 0 and N ū = 210 compared to the other strategies. Note that when B = N ū, there is enough budget to allocate the maximum control to all agents, making Strategy 1 and 2 equivalent. However, for strategy 3, due to positive budget allocation even to contrarians, the cost is significantly higher (J = 0.48) even when B = N ū = 210.

Conclusion

In this work, we have shown how an optimal campaign strategy can be designed for the control of opinion dynamics over a social network. The main novelty of this work, with respect to previous works on control of opinion dynamics is that we consider the presence of contrarians in the network, which have an antagonistic relationship with the other agents. The external entity or the marketer wants to bring the average opinion of all agents as close as possible to a desired opinion d. Interestingly, we see that the optimal strategy involves bringing the conformists closer to d and the contrarians closer to -d, and allocating the available budget among the best agents sorted according to a centrality measure and the distance of their initial opinions to d or -d. The future directions to this work would involve considering multiple-campaigns or a continuous control as well as the presence of a competing marketer, resulting in a non-cooperative game.

A Appendix

Lemma 1. Given an optimization problem (OP) under the following standard form minimize

y∈R N C(y + , y -) subject to y s i -ȳ ≤ 0, ∀i ∈ {1, ..., N }, s ∈ {+, -} -y s i ≤ 0, ∀i ∈ {1, ..., N }, s ∈ {+, -} -B + N i=1 y + i + y - i ≤ 0 (14)
where N ∈ N, N ≥ 1, ȳ < 1, B ≥ 0 and C(y) is a decreasing convex function in y i such that the following condition holds. For all i ∈ {1, . . . , N }, ∃g(y) ≥ 0 such that

∂C(y) ∂y

+ i = -k i g(y) and ∂C(y) ∂y - i = k i g(y)
for some k i ∈ R.

Then an optimal solution y * to this OP is given by water-filling as follows

y + * o(i) =        ȳ if i ≤ B ȳ and k o(i) > 0 B -ȳ B ȳ if i = B ȳ + 1 and k o(i) > 0 0 otherwise (15)
and Proof: Note that all the constraint functions of the considered OP are affine, which corresponds to sufficient conditions for applying KKT conditions. Since the OP is convex, KKT conditions are necessary and sufficient for optimality. By denoting the Lagrangian by

y - * o(i) =        ȳ if i ≤ B ȳ and k o(i) < 0 B -ȳ B ȳ if i = B ȳ + 1 and k o(i) < 0 0 otherwise (
(y, λ + , λ+ , λ, λ -, λ-) = C(y) +   N i=1 s∈{+,-} λs i (y s i -ȳ) -λ s i y s i   + λ + -B + N i=1 y + i + y - i . (17)
Let us assume that ∂C(y) ∂y i = k i g(y), in this case the first necessary and sufficient condition for optimality can be simplified to write

k i g(y ) = λ + i -λ+ i + λ + λ - i -λ- i (18)
which must hold for all i ∈ {1, . . . , N }. The primal feasibility conditions write 0 ≤ y s i ≤ ȳ ∀i ∈ {1, . . . , N }, s ∈ {+, -}, and

N i=1 y + i + y - i ≤ B. (19)
All the KKT multipliers must satisfy the dual feasibility conditions: λ s i ≥ 0, λs i ≥ 0, λ s ≥ 0 for all i ∈ {1, . . . , N }, s ∈ {+, -}. At last, the complementary slackness conditions are given by λs i (y s i -ȳ) = 0, λ s i y i = 0,

λ N i=1 y + i + y - i -B = 0.
Since g(y ) is identical for all i ∈ {1, . . . , N } and s ∈ {+, -} and is nonnegative, we must have λ s i , λs i and λ * chosen so that (18) holds. We get y + i y - i = 0, ∀i ∈ {1, ..., N }. Take y from (16). Set λ s j = λs j = 0 for j = O β k ū and s = + if k j > 0 and s = -if k j < 0 as it is the only component with a non-saturated solution. For any n, y + * n = 0 if k n < 0 and y - * n = 0 if k n > 0 can be imposed as the left hand side of (18) will be negative and positive correspondingly and letting λ s n = 0 will be possible.

For any i such that O(i) < j, we have k i ≥ k j and this can be satisfied by setting y sign(ki) i = ȳ and having λsign(ki) i > 0 and λ s i = 0. On the other hand, for any i such that O(i) > j, we set y * i = 0 and the KKT conditions are satisfied if λs i = 0 and λ s i > 0. The solution from (16) can also be verified to satisfy (19) and therefore, we have it satisfying all the KKT conditions.

Fig. 1 :

 1 Fig. 1: Red and cyan nodes represent conformists and contrarians, respectively with the size scaled based on vector centrality v.

  075 0.054 0.224 0.095 -0.111 0.052 -0.108 0.160 0.057 -0.064 Gamma (γ) 0.015 0.016 0.112 0.067 0.078 0.052 0.065 0.128 0.085 0.101

Fig. 3 :

 3 Fig. 3: Cost vs Budget.

  16) where o : {1, . . . , N } → {1, . . . , N } represents an ordering function which can be any bijection for Case 2 and, one satisfying |k o(1) | ≥ |k o(2) | ≥ • • • ≥ |k o(N ) | for Case 1.