Generating social networks with static and dynamic utility-maximization approaches

Aldric Labarthe; Yann Kerzreho

REVIEW 4 major objections 6 minor 46 references

Generating social networks with static and dynamic utility-maximization approaches

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Every social network can be regenerated as a utility-maximizing graph.

desk verdict Ambitious micro-founded network generator with a real but repairable proof gap in its inverse theorem and a circular validation; the agent-based half is solid. read the letter →

arxiv 2411.16464 v1 pith:MXVZLJFQ submitted 2024-11-25 math.PR cs.SIphysics.soc-ph

Aldric Labarthe , Yann Kerzreho This is my paper · ORCID

classification math.PRcs.SIphysics.soc-ph MSC 91D3005C8291B16

keywords socialnetworkgenerationdot-productgraphsutilitymaximizationagent-basedmodellatentspacepairwisestabilityimperfectinformationreproduction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that human social networks can be modeled as the outcome of cost-benefit optimization: each potential relationship has a weight chosen to maximize a utility function, and the paper builds a network generator on this premise. The main result is that every weighted, undirected graph with no isolated vertices or loops is the unique solution of a social welfare maximization problem for a suitably recovered compatibility list, so any empirical network can in principle be regenerated and used to produce statistically similar synthetic networks. The same micro-foundations are implemented in an agent-based simulator with bounded information, allowing the authors to study imperfect information; their case study shows that limiting agents' information lowers total utility and degrades clustering, density, and degree. A sympathetic reader would care because the framework promises a single interpretable mechanism, individual utility maximization, that reconciles exact reproduction, flexible synthetic generation, and behavioral analysis.

What carries the argument

The load-bearing object is the ALKY utility function $U(\alpha,c)=\sum_j[\kappa\alpha_j c_j-\alpha_j^\gamma/(1-\alpha_j^\gamma)]-(\sum_j\alpha_j)^\delta$, with $\kappa,\gamma,\delta>1$, together with compatibility $c_{ij}=\langle P_i,P_j\rangle$ given by dot products in a latent Euclidean space. Each edge weight appears in exactly two agents' utilities; at a social optimum the marginal utilities of the two endpoints cancel, and because the marginal terms are bijective in compatibility, each $c_{ij}$ is uniquely determined by the target weights. This inverse step is Theorem 2 in the main text and Theorem 9 in the appendix. The same utility is then used in a round-by-round agent-based simulation where agents perceive only their graph-theoretic scope and update weights by gradient steps with reciprocity.

What would settle it

Take a small graph, say three vertices with equal positive weights, solve the first-order conditions of the social optimization problem to obtain the unique compatibility list, and form the symmetric matrix with those off-diagonal compatibilities and positive diagonal entries. If that matrix has a negative eigenvalue, no three Euclidean vectors can have those dot products, so the latent-vector representation used by the generator does not exist for this graph. More directly, any symmetric positive matrix with a negative eigenvalue, for instance all diagonal entries 1 and all off-diagonal entries 2, is already a counterexample to the claim that every compatibility list is realizable.

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Extended reading notes

Core claim

The paper's central claim is that any weighted undirected graph without isolated vertices or loops admits a compatibility list such that the graph's edge weights are the unique maximizer of the sum of agents' ALKY utilities. The construction works by solving the first-order conditions of the social optimization problem for the compatibility of each pair, using the fact that the utility's marginal terms are bijective in compatibility, and then representing those compatibilities as dot products of latent Euclidean vectors. Once the compatibility list is recovered, the authors regenerate the original graph exactly and generate variants by adding noise to the latent vectors rather than to the compatibilities. They also prove an invisible-hand theorem: when every agent faces the same compatibility column up to permutation, the decentralized individual optimization problem has the same edge weights as the social optimum, so the agent-based model can be compared with the analytical optimum and used to measure the cost of frictions such as limited information.

Load-bearing premise

The generator depends on assuming that any recovered set of pairwise compatibility scores can be produced by actual positions of individuals in Euclidean space, and the proof only establishes this for part of the scores, not the full list.

Editorial extensions

If this is right

Any empirical network meeting the loose conditions can be cloned exactly by solving the social optimization problem, and clones can be varied by perturbing latent vectors.
All generated optimal graphs satisfy pairwise stability, linking the macro-level optimum to standard micro-level equilibrium notions.
Under symmetric compatibility structures, the agent-based decentralized process converges to the same edge weights as the social optimum, so the analytical generator serves as a benchmark for the simulator.
Bounded information reduces average utility, clustering, average degree, and density, with less clustered networks more resilient to frictions.
The framework opens a route to estimate utility parameters or compatibilities from real graphs, since the inverse problem is well-posed for the ALKY family.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the Gram-factorization gap in Proposition 1 is closed, the result would establish a general equivalence between arbitrary weighted graphs and utility-maximizing equilibria, letting network scientists reinterpret observed graphs as revealed preference data.
The contrast between noise on compatibilities and noise on latent vectors suggests a diagnostic rule for other generators: perturb the latent cause, not the derived feature, to vary outputs continuously.
The scope parameter behaves like an information budget, so a natural extension is to estimate the effective scope from real networks by comparing observed clustering with the agent-based model's predicted clustering.
The invisible-hand theorem suggests that in populations with heterogeneous compatibility columns, the gap between individual and social optima should widen, and the agent-based model should show larger optimality costs than the near-zero values reported here.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The genuinely new piece is the ALKY utility and the inverse claim (Theorem 2): any weighted undirected graph without isolated vertices can be made the unique optimum of their social optimization problem by fitting a compatibility list, and then perturbing latent vectors gives similar synthetic networks. The agent-based version with scope-limited agents is a separate, more self-contained contribution; the random-effects regressions are competently done and the effects of limited information are plausible.

What the paper does well: the framework is interpretable, the two generators are clearly separated, and the authors are honest about the substitution and information assumptions. The invisible-hand Theorem 4 gives a clean condition under which decentralized and social optima coincide. That is a useful result for people building micro-founded network models.

The soft spots are real. Proposition 1, which bridges the compatibility list to the latent-space dot-product representation, has a genuine proof gap. The proof isolates a positive definite principal submatrix and concludes K ≥ I suffices, but that does not establish the full matrix is a Gram matrix, and the argument sets η_i = ⟨P_i, P_i⟩ before any P_i is shown to exist. The stress-test note is correct: the statement is probably salvageable (choose diagonal entries large, Cholesky with K = I), but as written Theorem 2 is not established. Lemma 2's concavity proof also has an unhandled singular case in the induction; the claim is plausible but the proof needs repair.

The validation also overreaches. The similarity generator first fits a compatibility list that makes the original graph the exact optimum, then perturbs. So reproduction is guaranteed by construction, and the Train Bombing example shows Gini and average distance deviating substantially under the best settings, with no baseline generator compared. The “high fidelity” claim should be toned down.

The citation pattern is fine; the relevant RGG and Jackson–Wolinsky literature is cited. Code is claimed but no repository URL or hash appears in the preprint.

Bottom line: the framework is promising and likely repairable, but the central existence theorem and the validation need correction. The ABM part stands on its own. Worth a serious referee, but the referee should ask for the proof repair and a more honest validation. I would not cite it until the proof is fixed.

Referee Report

4 major / 6 minor

Summary. The paper introduces a utility-maximization framework for generating weighted undirected social networks. Each individual is represented by a latent vector in a Euclidean space, and pairwise compatibility is the dot product of the two latent vectors. The ALKY utility function makes each agent's benefit from an edge depend on that compatibility, with convex and substitutable costs. The Social Optimization Problem (SOP) maximizes the sum of all agents' utilities over edge weights, and the paper claims that every graph without isolated vertices or loops can be made the unique SOP optimum for a suitably chosen compatibility list (Theorem 2 / Theorem 9). The corresponding static generator first recovers a compatibility list from the empirical graph, obtains latent vectors by a Gram factorization, and then perturbs those vectors to produce synthetic networks. A second, agent-based generator implements individual optimization with local scope, reciprocity, and memory, and the paper reports a case study on the effects of imperfect information. Empirical results are shown for the Moreno train-bombing network and for 50 synthetic compatibility structures.

Significance. If the inverse-design theorem can be rigorously established, the central idea is valuable: rather than fitting a generative process to summary statistics, one solves for a compatibility structure that makes the target graph the exact equilibrium of a microfounded model, then perturbs in latent space. The release of code and data and the careful random-effects analysis of the agent-based model are strengths. However, the proof of the main representation theorem is incomplete as written, and the empirical evidence for high-fidelity reproduction is limited to a single network and is partly circular. These issues are substantial but appear repairable, so the paper merits a major revision rather than rejection.

major comments (4)

[Appendix A, Lemma 1 and Lemma 2] The proof that every compatibility list can be realized as a Gram matrix is invalid as written. The argument constructs a matrix \tilde C with unspecified diagonal entries \eta_i, considers a maximal positive-definite principal submatrix \tilde C_{\bar L}, and concludes that K \ge I \ge \bar l suffices. A maximal positive-definite principal submatrix can have size strictly smaller than I, and the proof never establishes that the full matrix \tilde C is positive semidefinite. The argument is also circular: it sets \eta_i = \langle P_i, P_i \rangle before any P_i has been shown to exist. Since Theorem 2 / Theorem 9 constructs a compatibility list for an arbitrary graph, and the generator requires latent vectors P_i whose dot products reproduce that list, Proposition 1 is load-bearing for the paper's central claim. The statement is likely salvageable, for instance by choosing the diagonal entries \eta_i large enough to make \tilde C strictly diagonally dominant and then applying Cholesky factorization with K = I, but the proof as printed does not establish the factorization.
[Appendix A, Lemma 1 and Lemma 2] The concavity proofs behind the uniqueness theorems contain genuine mathematical errors. In Lemma 1, the determinant reduction to a diagonal matrix with entries h(c_i,\alpha_i) is false for a matrix of the form D + s 11^T: for example, in dimension two the determinant is h_1 h_2 + s(h_1 + h_2), not h_1 h_2, so the printed determinant identity is not valid. Moreover, the generic utility class in Definition 6 does not imply h < 0, so the conclusion of strict concavity does not follow from the stated assumptions. In Lemma 2, the claim that when the assumption \nexists (i,j) with s_i(\alpha_i)=0 and s_j(\alpha_j)=0 fails, 'H_I will have at least one column and row that is full of 0' is false: for I=2 and s_1=s_2=0, the Hessian is the 1x1 matrix h_1+h_2, which is nonzero under the stated sign conditions. These errors matter because Theorems 5 and 6, and hence the uniqueness used throughout the paper, rely on strict concavity. For the ALKY utility and the SOP feasible set, a correct proof is available (each s_i is strictly negative away from \alpha_i=0, so the Hessian is negative definite), but the manuscript should give a correct and explicit proof.
[Section III.A, Table I and Figure 1] The empirical validation of the static generator is weaker than the text claims and is partly circular. The method first invokes Theorem 2 to solve for the compatibility list that makes the original graph the exact optimum of the SOP, then perturbs the latent vectors. Regenerating the original graph is therefore guaranteed by construction, and the reported synthetic networks are local perturbations of a fitted compatibility structure rather than an independent test of the model's generative mechanism. Furthermore, the demonstration uses a single network, the Moreno train-bombing graph, with no comparison to a baseline generator or a null model. To support the statement that the generator reproduces real networks 'with high fidelity' (Introduction and Conclusion), the authors should either present out-of-sample validation (for example, fit the compatibility structure on a training subgraph and evaluate the held-out structure) or explicitly reframe the results as a sensitivity analysis of the optimum to latent-space noise.
[Section II.B and Theorem 2] The claim that the model can reproduce 'every empirical network' is stated more broadly than what is actually proved. Theorem 2 requires the existence of an individuals set I, which by Definition 7 includes a Euclidean space and latent vectors P_i; the existence of such P_i for an arbitrary compatibility list is exactly the content of Proposition 1, whose proof is incomplete. In addition, Theorem 2's compatibility list C is constructed from an arbitrary weight list \alpha in [0,1)^{n_N}, but the paper does not discuss the computational or numerical stability of solving for P_i when the number of nodes is large. The authors should state the precise assumptions under which the full construction is valid and acknowledged that the 'every network' claim is conditional on a corrected Proposition 1.

minor comments (6)

[Definition 8] The definition of an individual graph has mismatched parentheses and writes \alpha and C as vectors of (R^*_+)^{n_I} and (R_+)^{n_I}, but \alpha contains zero entries in the feasible set; the notation should be cleaned up.
[Table I] The row labelled 'Vertices' reports 486 for a network described as having 64 individuals; this appears to be the sum of degrees rather than the number of vertices. The quantities 'Del. Vertices' and 'New Vertices' are also undefined and should be defined in the table note.
[General] Figure 1 is referenced in Section II.B but no figure appears in the manuscript; please include the figure or remove the reference.
[Code and data availability] The code availability section says the implementation is on a GitHub repository but does not give a URL, repository name, or version identifier; please provide a working link and commit hash.
[Definition 3] The set-builder notation in the ALKY utility definition is not well formed; it should be written as a set of functions parameterized by \kappa,\gamma,\delta.
[Section II.D] The sentence 'the model has two degree of freedom' is unclear; the relationship between utility parameters and the compatibility list should be stated more precisely.

Circularity Check

2 steps flagged · score 6.0 of 10

The static generator's claimed 'reproduction' reduces to an inverse fit by construction, and Proposition 1's proof assumes the latent vectors it must construct.

fitted input called prediction [Section II.B, paragraph after Theorem 2 and Figure 1]
"we first use Theorem 2 to recover the compatibility matrix associated with a given tuple of parameters (that can be freely chosen) from the real adjacency list. With this tuple of parameters and this compatibility list, the model is always able to regenerate cloned networks (as the solution of the SOP is unique)."

The pipeline solves the inverse problem: Theorem 9's proof constructs C so that (∇O)(α)=0 for the target weight list, with 'each c_{i,j} is determined by a single equation', and then concludes that α is the unique SOP solution 'by construction of C'. The claim that the generator regenerates the original graph therefore restates the fit rather than testing the model. The 'similar networks' are then obtained by adding white noise to the fitted P matrix and re-solving the same fitted objective, so their closeness to the original is inherited from the construction, not an independent prediction.
other [Appendix A, Proposition 1]
"We first build a compatibility matrix \tilde C ∈ M(I, I) from the compatibility list C (and denoting P_i the characteristic vector in E of the individual i): (\tilde C)_{i,j} = c_{i,j} = ⟨P_i, P_j⟩ if i ≠ j, η_i else."

The proposition is supposed to prove that an arbitrary compatibility list C can be realized as pairwise dot products ⟨P_i,P_j⟩ of latent vectors. The proof opens by assuming exactly such vectors, writing the off-diagonal entries as c_{i,j}=⟨P_i,P_j⟩, and later sets η_i=⟨P_i,P_i⟩. Existence of the P_i is the conclusion, not a premise. The subsequent maximal-positive-definite-submatrix argument also never supplies a factorization of the full matrix. Since this proposition is the stated justification for recovering a P matrix from the fitted C (main text: 'A proof of existence of such a P Matrix is provided in the supplementary materials'), the latent-space realization step is not established as written.

full rationale

The agent-based simulations in Section III.B are forward simulations on independently generated compatibility structures and do not inherit the inverse-fit circularity; they provide independent content. The static generator, however, rests on an inverse-optimality construction in which the target graph is forced to be the unique maximizer by solving for C, and the 'similar networks' are perturbations of the fitted latent vectors. That reduction makes the regeneration claim true by construction rather than by empirical confirmation. Additionally, Proposition 1, the bridge from the fitted compatibility list to the latent vectors P_i, is argued circularly by presupposing the P_i. There are no load-bearing self-citations in the paper. Overall, the central reproduction claim partially reduces to its own inputs, while the ABM component and the mathematical uniqueness theorems retain independent content; hence a moderate circularity score.

Assumptions & free parameters 4 free parameters · 4 assumptions · 2 invented entities

The model's generative flexibility comes from one free compatibility parameter per edge plus hand-chosen utility and simulation parameters. The central existence theorem rests on an unproven but likely repairable Gram-matrix claim.

free parameters (4)

Utility parameters kappa, gamma, delta = kappa=10, gamma=9, delta=2
Fixed by hand for all simulations; the paper claims they can be absorbed into the compatibility list, but this is not demonstrated for the ABM dynamics.
Noise standard deviation sigma for perturbation of characteristic vectors = 0.1 (selected as best among 0.05, 0.1, 0.2)
Used in the similar-network generation; the value 0.1 is chosen post hoc because it gives the closest benchmark statistics.
Latent dimension K = number of individuals (rule of thumb)
The paper states that setting the latent dimension to the number of individuals can be taken as a rule of thumb (Section II.B).
ABM hyperparameters lambda, Omega, omega, varpi = 0.1, 0.1, 0.05, 0.0001
Learning rate, max step, minimum weight, and minimum gradient threshold used in Algorithms 3 and 4; chosen by hand without sensitivity analysis.

assumptions (4)

domain assumption ALKY utility functions belong to the admissible utility class U_I.
Definition 3 and Theorem 10 assert this; the class U_I is designed so that concavity and separability hold, which is a modeling postulate.
ad hoc to paper Any positive compatibility list can be realized by Euclidean dot products in dimension K at least I.
Proposition 1 asserts this but its proof only exhibits a positive definite submatrix, not a factorization of the full matrix; the claim needs a correct proof or a restriction on compatibility lists.
domain assumption All individuals share the same ALKY utility function in simulations.
Section II.D fixes homogeneous utility with gamma=9, delta=2, kappa=10; Theorem 4's equivalence also requires identical utilities.
domain assumption The compatibility matrix columns are symmetric and every individual has positive compatibility with every other.
Definition 2 and the no-orthogonality condition in Definition 1; positive c_ij are required for the utility's linear benefit term to favor links.

invented entities (2)

Latent characteristic vectors P_i
purpose: Each individual is represented by a vector; dot products define pairwise compatibilities c_ij.
Recovered from the target graph by inverting the model (Theorem 2), not observed in data; no falsifiable handle outside the model.
ALKY utility function
purpose: Parametric utility governing edge-weight choices with substitution effects.
A new functional form introduced by the authors; its parameters are not estimated from behavioral data.

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Cite this review

Pith. "Pith review of Generating social networks with static and dynamic utility-maximization approaches." pith.science (2026). https://pith.science/paper/MXVZLJFQ

@misc{pith2026241116464,
  author       = {Pith},
  title        = {Pith review of: Generating social networks with static and dynamic utility-maximization approaches},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MXVZLJFQ}},
  note         = {Machine review of arXiv:2411.16464}
}

read the original abstract

In this paper, we introduce a conceptual framework that model human social networks as an undirected dot-product graph of independent individuals. Their relationships are only determined by a cost-benefit analysis, i.e. by maximizing an objective function at the scale of the individual or of the whole network. On this framework, we build a new artificial network generator in two versions. The first fits within the tradition of artificial network generators by being able to generate similar networks from empirical data. The second relaxes the computational efficiency constraint and implements the same micro-based decision algorithm, but in agent-based simulations with time and fully independent agents. This latter version enables social scientists to perform an in-depth analysis of the consequences of behavioral constraints affecting individuals on the network they form. This point is illustrated by a case study of imperfect information.

Figures

Figures reproduced from arXiv: 2411.16464 by the authors.

**Figure 1.** Methodology to generate similar networks or perfect clones of a real network We name P Matrix in figure 1 the matrix that is formed by all individuals’ characteristic vectors: the ith row of the P Matrix is the characteristic vector of the individual i (in E). Several linear algebra methods can be used to recover candidates for these characteristic vectors. A proof of existence of such a P Matrix is provided 1 This… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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For a given I ∈ N∗, a utility function U (α, C) is an element of UI , the set of applications U : RI−1 × (R∗ +)I−1 − →R with [0, 1) ⊆ R ⊆R, and RI−1 convex

F undamental definitions Definition 6 (Utility function) . For a given I ∈ N∗, a utility function U (α, C) is an element of UI , the set of applications U : RI−1 × (R∗ +)I−1 − →R with [0, 1) ⊆ R ⊆R, and RI−1 convex. Every element U : (α, C) 7− →R of UI verifies:
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[40]

∀i, ∂U ∂αi = H(ci, αi) + S(α) 3

with H bijective and increasing in its second component, ∃ ( H : R∗ + × R − →R, H ∈ C 1 S : RI−1 − →R, S ∈ C 1 s.t. ∀i, ∂U ∂αi = H(ci, αi) + S(α) 3. ∃ ( h : R∗ + × R − →R, h ∈ C 0 s : RI−1 − →R, s ∈ C 0 s.t. ∀j ̸= k, ∂2U ∂αjαk = s(α), ∂2U ∂α2 j = h(cj, αj) + s(α) with   ...
[41]

For a given individuals graph GN (I , α, C), we define for all individual i ∈ J1..N K the Individual Optimization Problem (IOP) as: ( argmaxαi Ui(αi, Ci) s.t

A network optimization problem Definition 9 (Individual Optimization Problem) . For a given individuals graph GN (I , α, C), we define for all individual i ∈ J1..N K the Individual Optimization Problem (IOP) as: ( argmaxαi Ui(αi, Ci) s.t. αi ∈ [0, 1)I−1 (A1) Theorem 5 (Unicity...
[42]

∀i ∈ J1..N K, Ci ∈ S (¯c), with S (¯c) the symmetric group of ¯c

∃¯c ∈ RN s.t. ∀i ∈ J1..N K, Ci ∈ S (¯c), with S (¯c) the symmetric group of ¯c
[43]

Moreover, α∗∗ the solution of the associated SOP, verifies: ∀(i, j) ∈ J1..N K2, α∗∗ i,j = (αmax {i,j})∗ min {i,j} Proof (Theorem 8)

there exists (H, S) two functions satisfying conditions of Definition 6 such that ∀i ∈ J1..N K, (I )i = (Pi, Ui), ∀j ∈ J1..N − 1K, ∂Ui ∂αi j = H(ci j, αi j) + S(αi) Then there exists ˆα ∈ RnN such that ∀i ∈ J1..N K, (αi)∗ ∈ S ( ˆα). Moreover, α∗∗ the solution of the associated...
[44]

a unique compatibility list ¯C such that α is the unique unconstrained solution of the Social Optimization Problem for the individuals graph G(I , α,¯C)
[45]

C = ( C ∈ (R∗ +)nI , c ij ∈ ( {¯cij} if αij > 0 (0, ¯cij] if αij = 0 ) Proof (Theorem 9)

a set of compatibility lists C such that ∀C ∈ C, α is the unique solution of the Social Optimization Problem for the individuals graph G(I , α, C). C = ( C ∈ (R∗ +)nI , c ij ∈ ( {¯cij} if αij > 0 (0, ¯cij] if αij = 0 ) Proof (Theorem 9). Let’s take I ∈ (E, ⟨., .⟩) × UN . For p...

1996
[46]

for all (i, j) such that αi,j is a non-null component of α, we have ∀k ∈ {i, j}, U k(α, C) ≥ Uk(α, C)|αi,j =0
[47]

Every graph G(I , α, C) that is solution of the Social Optimization Problem (Definition 10) is pair- wise stable

for all (i, j) such that αi,j is null component of α, we have ∀a ∈ (0, 1), ∀k ∈ {i, j}, U k(α, C) ≥ Uk(α, C)|αi,j =a Proposition 2. Every graph G(I , α, C) that is solution of the Social Optimization Problem (Definition 10) is pair- wise stable. Proof (Proposition 2). We prove...
[48]

Alky utility function Definition 12 (ALKY utility function). We consider the set U ALKY I of applications U : [0, 1)I−1 × (R∗ +)I−1 − →R: U ALKY I =    κ >1, γ >1, δ >1, U: (α, c) 7− → I−1X j=1 " καjcj − αγ j 1 − αγ j # −   I−1X j=1 αj   δ   16 Remark 5. Analytic...

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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a set of compatibility lists C such that ∀C ∈ C, α is the unique solution of the Social Optimization Problem for the individuals graph G(I , α, C). C = ( C ∈ (R∗ +)nI , c ij ∈ ( {¯cij} if αij > 0 (0, ¯cij] if αij = 0 ) Moreover, this property will allow us to derive mul- tiple similar synthetic graphs from any real graph, that will share some overall benc...

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In addition, all generated networks verify pairwise stability [21] (see the supplementary materials for a proof)

Using Theorem 2, we know that every network that is associated with a weighted and undirected graph, with- out any isolated vertex nor loop, can be generated using 3 our model. In addition, all generated networks verify pairwise stability [21] (see the supplementary materials for a proof). Theorem 2. For any individuals set I ∈ (E, ⟨., .⟩)×UN , and weight...

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but that are the result of a decentralized optimization problem. C. Decentralized graph generation: an agent-based model To tackle these two limitations, we introduce an agent-based model that tries to implement the Individ- ual Optimization Problem (Definition 5) with temporal dynamics and limited information. Definition 5 (Individual Optimization Proble...

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Moreover, α∗∗ the solution of the associated SOP, verifies: ∀(i, j) ∈ J1..N K2, α∗∗ i,j = (αmax {i,j})∗ min {i,j} D

there exists an ALKY utility function such that ∀i ∈ J1..N K, (I )i = (Pi, Ui), Ui = U 2 Then there exists ˆα ∈ RnN such that ∀i ∈ J1..N K, (αi)∗ ∈ S ( ˆα). Moreover, α∗∗ the solution of the associated SOP, verifies: ∀(i, j) ∈ J1..N K2, α∗∗ i,j = (αmax {i,j})∗ min {i,j} D. Simulation methodology Hereinafter, we only use homogeneous individuals sets, where...

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