{"id":"eda9de60-74a6-48bc-9c7e-00213c67d852","arxiv_id":"1908.09472","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Directed social influence weights and confirmation bias parameters can be exactly recovered from full noiseless opinion trajectories when information sources are controlled, and approximately recovered when the bias model is unknown.","lead":"This paper gives a method to recover the influence network among people from their recorded opinion changes, when people also respond to outside information sources and have confirmation bias. It matters because exact network recovery is a step toward measuring echo chambers and information spread in online social settings.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Invalid step in Theorem 2's necessity proof: setting \\hat A = -\\hat W is not admissible because \\hat A is diagonal; the iff characterization is unproven, though Theorem 3's sufficient construction stands.","rationale":"I read the paper's central algorithmic claim charitably: under the stated noiseless, controlled-source, piecewise-linear model, the identity WP = Q and the inversion W = QP^{-1} when P is full rank are correct, and the numerical demonstration is consistent. The model assumptions (s_i = x_i(0), u_d = 0, no noise) are acknowledged and are not, by themselves, a defect. The load-bearing problem I find is internal: the proof of the claimed necessary condition in Theorem 2 contains a mathematically inadmissible step. The theorem is used to justify the 'if and only if' characterization in the abstract, even though Theorem 3's sufficient rank condition does not rely on it. Because the flaw is in a proof of necessity rather than in the construction, the appropriate disposition is conditional acceptance pending a correct proof or a weakened claim. I do not see grounds for rejection: the constructive exact-inference result appears sound.","tokens_in":20312,"tokens_out":42827,"duration_ms":444634,"concrete_test":"Analytically re-derive the necessity proof with the admissible diagonal choice \\hat A_{ii} = -(\\hat W x(0))_i / x_i(0) and check whether the resulting \\tilde A has the form (16) when x_i(0) = 0 for a non-follower. As a computational companion, enumerate 2-agent instances of Problem I with one follower x_1(0) != 0, one non-follower x_2(0) = 0, and rank(P) = 1, and test whether any feasible alternative (A', W') with W' of the form (16) reproduces x(0), x(1), x(2), x(3). Finding such an alternative supports Theorem 2; finding none would disprove the stated necessity.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Appendix F derives (W - \\tilde W)W^k Lx(0) = 0 from \\tilde WP = WP, then says 'we can set \\hat A = -\\hat W' to obtain x(0) in ker(\\hat A + \\hat W). But \\hat A = \\tilde A - A is diagonal by (14b) and (18b), while \\hat W = \\tilde W - W is generally non-diagonal, so \\hat A = -\\hat W is not an admissible choice. The correct condition (\\hat A + \\hat W)x(0) = 0 would require a diagonal \\hat A with entries -((\\hat W x(0))_i / x_i(0)), which demands x_i(0) != 0 for every row; condition (19) only guarantees this for followers of the source. No argument shows that this diagonal choice is compatible with the structural form (16). Since this step is the bridge from non-uniqueness of \\tilde WP = Q to non-solvability, the necessity half of Theorem 2, and hence the paper's claimed necessary-and-sufficient characterization, is not established by the appendix. The sufficient half used in Theorem 3 (rank(P)=n implies W = QP^{-1}) is independent of this step and appears sound.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies exact and approximate inference of a directed social network's topology and of confirmation-bias parameters from noiseless full-state observations of a discrete-time opinion dynamics model. Under a piecewise-linear bias model and zeroed information sources, the nonlinear dynamics are rewritten as x(k+1)=A x(0)+W x(k), and the authors derive the correlation identity W P = Q from the state-difference matrices P and Q. They state necessary and sufficient conditions for exact inference (Theorems 1 and 2), give the closed-form recovery W=Q P^{-1} when rank(P)=n (Theorem 3), treat the no-bias case (Problem II), and, for an unknown bias model, prove that the rows of the influence matrix corresponding to non-followers of the information source can still be recovered exactly (Theorem 4, Algorithm 1). The theoretical results are supplemented by simulations on a 12-node example and on Krackhardt's advice network.","tokens_in":20580,"tokens_out":23893,"duration_ms":248040,"significance":"If the gaps identified below are repaired, this is a valuable contribution: it provides a simple, linear-algebraic identification procedure for a directed opinion-formation model with confirmation bias, and it identifies exactly which parts of the topology remain identifiable when the bias model is unknown. The appendices contain detailed proofs, the core identity (22) is correct for the noiseless model, and the numerical experiments are consistent with the theory. The paper's main strengths are the explicit treatment of confirmation bias and the partial-recovery guarantee for non-followers in the model-agnostic setting. However, the advertised necessary-and-sufficient characterization currently outruns the proofs, and one auxiliary lemma has a sign error; these issues are local and repairable.","major_comments":[{"comment":"","section":"Section III-A, Theorems 1–3, Eq. (19)"},{"comment":"","section":"Appendix F, Theorem 2 necessity proof"},{"comment":"","section":"Appendix H, Lemma 3, Eq. (41)"}],"minor_comments":[{"comment":"The symbol B is used both for the bipartite edge set in the model and for the weighted adjacency matrix in the simulation; please rename one of them.","section":"Section II-A and Section VI-A"},{"comment":"References [1] and [41] appear to be the same Allerton paper and should be consolidated.","section":"References [1] and [41]"},{"comment":"'two following two conditions' should read 'the following two conditions'.","section":"Definition 1"},{"comment":"The time index k in (29) is unspecified; state explicitly that the formula holds for any k≥0.","section":"Equation (29)"},{"comment":"The displayed equality in (59) mixes A^l and W^l typographically; the intended identity is <tilde>W W^l L x(0)=0, l=0,…,m−1, and should be written uniformly.","section":"Appendix A, Eq. (59)"}],"recommendation":"major_revision","confidential_remarks":"The paper is close to publishable after a major revision. The core recovery idea and the partial-inference result for non-followers are valuable, and the experiments are consistent with the theory. I would ask the authors to correct the initial-condition condition to x_i(0)≠0 for all i, repair the necessity proof of Theorem 2, and fix the sign in Lemma 3 before acceptance. The relationship to the authors' own prior work [39] and the Allerton precursor is disclosed, so there is no novelty concern on that ground."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper does something real. It gives an exact reconstruction method for directed influence weights and per-node confirmation-bias parameters from observed opinion trajectories under the authors' linearized bias model. The important modeling step is Section III: turning the state-dependent dynamics (1) into the LTI form x(k+1)=Ax(0)+Wx(k), where the bias parameters hide in the diagonal of W. The correlation identity WP=Q then yields a clean sufficient algorithm: if rank(P)=n and followers have nonzero initial opinions, W=QP^{-1} recovers topology and the gamma_i. Lemma 1/2 and the proof of Theorem 1 are correct as far as I can tell; Theorem 3 is a genuine sufficient-condition result, and the simulations confirm it. This goes beyond [17], which handles undirected consensus without bias, and the joint inference of topology plus bias parameters is the genuinely new part.\n\nThe soft spot is the claimed necessity half of Theorem 2. In Appendix F, after deriving (W-tilde W)W^k Lx(0)=0, the proof says 'we can set hat A = -hat W.' That step is not admissible: hat A is diagonal by (14b) and (16a), while hat W is generally not, so hat A=-hat W can hold only in trivial cases. To make the argument go through you would need to show a diagonal hat A exists with (hat A+hat W)x(0)=0, which requires additional conditions on rows where x_i(0)=0. As written, the necessary-and-sufficient characterization is not established; the paper would be safer stating Theorem 2 as a sufficiency result, or fixing the proof. This does not undermine the main reconstruction method, which relies on the sufficient direction and Lemma 2, but it does mean the 'iff' language overstates what is proved.\n\nThe other limitations are more contextual than fatal. The method needs noise-free full-state measurements, the ability to set all information sources to zero, and the specific piecewise-linear bias form; the authors acknowledge most of this. The approximate algorithm in Problem III is heuristic, and honest about it. The solvability conditions ignore nonnegativity of weights and bias parameters, so in principle the inferred W can have negative entries; the algorithm thresholds them in the approximate case. The self-citations are to the model the paper builds on; the inference results are derived, not restated.\n\nWho should read this: people working on network identification from opinion dynamics, especially control theorists. It deserves a serious referee; the Theorem 2 gap needs to be fixed or the claim softened, but the sufficient construction and the problem setup are worth publishing.","headline":"Solid sufficient-condition result for exact topology and bias inference; the advertised iff characterization has a real proof gap in Theorem 2, but the main reconstruction method survives.","tokens_in":21058,"tokens_out":7549,"would_cite":true,"duration_ms":74678,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91D30","93B30","93C55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a directed social network's influence weights and each agent's confirmation-bias parameters can be recovered exactly from observed opinions as $\\mathbf{W}=\\mathbf{Q}\\mathbf{P}^{-1}$, and characterizes precisely when…","keywords":["network topology inference","confirmation bias","opinion dynamics","directed social networks","cyber-social networks","exact inference","correlation matrix identity","piecewise linear bias"],"falsifier":"Generate a trajectory from (13) with random initial opinions that keep every $x_i(0)$ nonzero and make $P$ full rank; compute $\\tilde{W}=QP^{-1}$ and compare with the planted $W$. The first case in which the error exceeds floating-point precision would refute Theorem 3, since no such case should exist if the proof is correct.","tokens_in":20139,"feed_emoji":"🕸️","tokens_out":8560,"duration_ms":83919,"temperature":0.7,"pith_summary":"This paper sets out to show that the full influence structure of a directed social network can be recovered exactly from observed opinion trajectories, even when agents are biased toward information that matches their prior opinions. Under a piecewise-linear model of confirmation bias, the unknown influence matrix turns out to solve a single linear identity $\\mathbf{W}P=Q$ built from differences of measured opinions, so the topology and bias parameters follow from one matrix inversion. The paper also identifies the necessary and sufficient conditions under which this exact recovery is possible, and proves that when the bias model is unknown and sources are uncontrolled, exact recovery is still possible for agents who do not follow any information source. A sympathetic reader should care because the result makes a previously qualitative notion, confirmation bias, directly measurable from observable opinion data, and because it marks the boundary between what is and is not inferable in such models.","feed_headline":"Inverting one correlation matrix reveals a network's hidden links","feed_subtitle":"It also recovers each user's confirmation-bias strength from observed opinions.","key_machinery":"The load-bearing object is the state-difference correlation identity $\\mathbf{W}P=Q$ for the linear time-invariant representation $x(k+1)=Ax(0)+Wx(k)$, which follows because consecutive opinion differences satisfy $x(k+1)-x(k)=W^k L x(0)$ with $L=W+A-I$. The known measurement matrices $P$ and $Q$ are computed by summing outer products of successive differences, so when $P$ is invertible the unknown matrix $\\mathbf{W}$ is simply $\\mathbf{Q}\\mathbf{P}^{-1}$. The proof of exactness grounds uniqueness failure in the condition $x(0)\\in L^{-1}\\ker(\\hat O)\\cap\\ker(\\hat A+\\hat W)$, then uses the full-rank condition on the matrix $[Lx(0), WLx(0), \\ldots, W^{n-1}Lx(0)]$ to force $\\hat W=0$ and hence $\\hat A=0$.","core_discovery":"The central claim is Theorem 3: if every follower of an information source has nonzero initial opinion and the matrix $P=\\sum_{k=0}^{m-1}(x(k+1)-x(k))(x(k+1)-x(k))^\\top$ has full rank $n$, then the weighted influence matrix is exactly $\\mathbf{W}=\\mathbf{Q}\\mathbf{P}^{-1}$ with $Q=\\sum_{k=0}^{m-1}(x(k+2)-x(k+1))(x(k+1)-x(k))^\\top$. The off-diagonal entries of $\\mathbf{W}$ are the social influence weights $w_{i,j}$; the diagonal entries give the confirmation-bias slope $\\gamma_i = [\\mathbf{W}]_{i,i}/x_i(0)$; and the intercept $\\beta_i$ is recovered from the formula in (29). The paper also proves necessity: exact inference is solvable if and only if the linear equation $\\tilde{W}P=Q$ has a unique solution, and it gives the practical sufficient check rank($P$)=$n$. In the no-bias case with uncontrolled sources, the same identity recovers $\\mathbf{W}$ when $P$ is full rank, although with multiple sources the source-to-agent weights cannot be separated. In the unknown-bias case, exactness is retained only for the incoming links of non-followers.","pith_inferences":["If measurement noise is present, the identity $\\tilde{W}=QP^{-1}$ becomes a noisy linear solve rather than an exact reconstruction; a natural extension, not explored in the paper, is a regularized total-least-squares estimator that exploits the known structure of $W$ (zero diagonal, nonnegative off-diagonals).","Because the rank condition depends on the trajectory exploring enough independent directions, agents whose opinions converge quickly will produce near-singular $P$; in practice, active probing or deliberately varied source opinions may be needed to satisfy the condition.","The same difference-correlation construction applies to any linear time-invariant-plus-drift process anchored at initial states, so the technique may transfer to other network-reconstruction problems, such as epidemic spreading or financial contagion, whenever the drift term is known to equal the initial state.","The paper's Problem III result suggests a practical litmus test: exact row recovery under an unknown bias model can certify which agents are not followers of any information source, without knowing the bias mechanism."],"forward_implications":["When the conditions of Theorem 3 hold, a single matrix inversion recovers the entire directed topology and all bias parameters; no node-by-node probing or external stimulation is needed.","The necessity result implies a clean boundary: if any follower starts with opinion zero, or if the accumulated difference data are rank-deficient, no exact inference method can succeed for this model.","In the no-bias setting, exact topology inference remains possible even when information sources are uncontrollable, as long as their opinions are known and $P$ is full rank.","In the unknown-bias setting, the algorithm is guaranteed exact for the incoming weights of non-followers, which could be used to identify which agents are not exposed to information sources.","As a corollary of the recovered matrices, the steady-state opinions are predicted by $x^*=(I-W)^{-1}Ax(0)$, so the same data yield the long-run consensus value."],"supporting_citations":[{"why":"Supplies the cyber-social opinion dynamics model with confirmation bias that all three inference problems are built on.","marker":"[39]"},{"why":"Provides the prior exact-inference method via constrained Lyapunov equations whose failure on directed topologies motivates the new identity-based approach.","marker":"[17]"},{"why":"Reports the preliminary topology-inference analysis that this paper extends to joint inference with confirmation bias.","marker":"[41]"},{"why":"Supports the assumption that each individual's innate opinion equals her initial opinion, which is used to write the dynamics in linear form.","marker":"[43]"},{"why":"Supplies the real-world advice-network dataset used in the numerical study of approximate inference.","marker":"[44]"}],"fun_headline_variants":["Exact network inference from opinions: one matrix inverse","Full-rank condition enables exact topology and bias recovery","From opinion states to network weights: a sharp formula","Recover cyber-social network topology and bias from states"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The exact-recovery claim holds only if the observed opinions are generated exactly by the stated model with piecewise-linear confirmation bias, innate opinions equal to initial opinions, and (in Problem I) information sources that can be silenced; any noise, model mismatch, or uncontrollable source breaks the identity that makes the inversion exact.","fun_headline_variants_meta":{"raw":{"variants":["Exact network inference from opinions: one matrix inverse","Full-rank condition enables exact topology and bias recovery","From opinion states to network weights: a sharp formula","Recover cyber-social network topology and bias from states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000207,"raw_usage":{"total_tokens":1417,"prompt_tokens":982,"completion_tokens":435,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":598,"completion_tokens_details":{"reasoning_tokens":373}},"tokens_in":598,"tokens_out":435,"duration_ms":5135,"temperature":1.0,"reasoning_tokens":373,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:12:00.126584+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate a trajectory from (13) with random initial opinions that keep every $x_i(0)$ nonzero and make $P$ full rank; compute $\\tilde{W}=QP^{-1}$ and compare with the planted $W$. The first case in which the error exceeds floating-point precision would refute Theorem 3, since no such case should exist if the proof is correct.","supporting_citations":[{"cited_title":"On network topology inference of social networks,","cited_arxiv_id":null,"evidence_quote":"Reports the preliminary topology-inference analysis that this paper extends to joint inference with confirmation bias."},{"cited_title":"Debia sing social wisdom,","cited_arxiv_id":null,"evidence_quote":"Supports the assumption that each individual's innate opinion equals her initial opinion, which is used to write the dynamics in linear form."}],"review_version":1}