{"id":"38ca39cd-4676-48a5-a783-853e4678b309","arxiv_id":"2502.07624","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A core-periphery profiling method that ranks nodes by coreness then k-core centrality identifies denser cores and often finds maximum cliques.","lead":"This paper proposes a new way to find dense core-periphery structures in networks by ranking nodes first by k-core coreness and then by centrality within the k-core. The method often finds denser cores than the classic rich-core method, and in many graphs it finds the maximum clique.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The density advantage over rich-core may come entirely from the coreness pre-sort; without a max-k-core-shell baseline, the r* boundary and centrality tie-breaker are not shown to do the work.","rationale":"The paper's headline contribution is a new ranking (coreness then within-k-core centrality) combined with the rich-core d+ boundary. The empirical claim is that this produces denser cores than rich-core. But the density measure phi rewards small size, and the MCC ranking makes the top of the list a k-core shell, which is already a dense nested subgraph by construction. For seven networks the method simply returns the maximum k-core shell, where phi=1 is forced. Thus the observed improvement may be a property of k-core decomposition, not of the proposed r* selection or centrality tie-break. This is not an accusation of error, but a missing control that is needed to attribute the result. The reader flagged the unproven r* justification; I agree, but the more decisive issue is empirical: the proposed mechanism is confounded with the maximum k-core baseline. The multiplex aggregation in Eq. (2) has an additional issue of averaging integer coreness values across layers, but this is secondary to the single-layer claim. A simple ablation on the same 12 networks would settle the attribution. I keep the CONDITIONAL verdict because the empirical numbers are plausible and the method may be useful, but the paper should be required to add the baseline comparison and, preferably, code and error bars before acceptance.","tokens_in":11113,"tokens_out":11326,"duration_ms":122122,"concrete_test":"Run the same 12 networks with a baseline that returns the highest k-core shell (all nodes with coreness equal to Mcore), and a second baseline that returns the top-Mcore nodes sorted by degree or eigenvector centrality; report phi and clique size for each. If the max-k-core baseline matches or beats MCC on phi in at least 10/12 networks and finds the same cliques, the r* and centrality steps are not doing the work; if MCC clearly outperforms on networks such as Dolphins, Facebook, or Email, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Table I, for 7 of the 12 networks under MCC-E the core size is exactly Mcore or Mcore+1 (Karate 5/4, Les Miserables 10/9, Jazz 30/29, Netscience 20/19, DBLP 115/114, Citeseer 87/86, In-2004 489/488), and in all of these the density is phi=1.0. This is the behavior of the highest k-core shell, not of the d+ maximization or of the centrality tie-breaker: a k-core shell with k+1 nodes is necessarily a clique, so any method that returns that shell will report phi=1. For Email, MCC-E gives Vcore=184 and phi=0.0816, which is not denser than rich-core (0.0825), showing the method is not uniformly 'dense'. The paper never ablates the two components: (i) ranking by coreness and taking the maximum k-core shell, versus (ii) the d+ turning point r* and the within-core centrality ordering. Without that ablation, the central claim that MCC improves on rich-core cannot be attributed to the proposed mechanism. The Discussion's random-walk justification (Eq. 5) is inherited from [21] and assumes alpha_c is increasing, which is not guaranteed when a newly added node has d+=0.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an improved method for profiling core-periphery structure, called MCC-E and MCC-D, in which nodes are ranked first by k-core coreness and then by a centrality measure (eigenvector or degree) within the k-core. The core boundary is set at the rank r* that maximizes the number of links to higher-ranked nodes, d+(r), following the rich-core framework of Ma and Mondragón. The authors report that on twelve real-world single-layer networks the identified cores are generally denser than those from the rich-core method, that the top-ranked nodes lying exactly on the upper bound d+(r)=r-1 plus a greedy expansion yield cliques that match the maximum clique in 8 of 12 networks, and that a multiplex extension gives dense multiplex cores on five datasets. The paper includes complexity analysis and discussion of the random-walk interpretation of the core boundary.","tokens_in":11344,"tokens_out":3889,"duration_ms":39514,"significance":"If the claims hold, the method is a practically valuable, parameter-free heuristic for dense core and approximate maximum-clique detection in large networks, with linear-logarithmic complexity. The paper's strengths include the clarity of the algorithmic definitions, the breadth of the empirical validation across networks spanning several orders of magnitude in size, and the explicit comparison with the established rich-core method. The multiplex extension addresses a timely problem. However, the central claim that the new ranking and boundary procedure, rather than the coreness pre-sort alone, produces the density improvement is not yet supported by an ablation study, and the theoretical justification for the boundary r* remains heuristic and depends on an unverified monotonicity condition.","major_comments":[{"comment":"The reported MCC-E cores coincide with the highest k-core shell in 7 of 12 networks (Karate 5/4, Les Misérables 10/9, Jazz 30/29, Netscience 20/19, DBLP 115/114, Citeseer 87/86, In-2004 489/488), all with density φ=1.0. Since a subgraph of size k+1 within a k-core is necessarily a clique, these results could be produced by the coreness pre-sort alone, without the d+ turning-point rule or the within-core centrality tie-breaker. The paper does not ablate these components (e.g., by comparing against simply taking the maximum k-core shell, or by randomizing the within-shell order), so the central claim that MCC improves on rich-core cannot be attributed to the proposed mechanism.","section":"Section V, Eq. (5)"},{"comment":"The random-walk justification assumes that the persistence probability α_c is increasing in the core size c, but when a newly added node has d+=0, the numerator of Eq. (5) is unchanged while the denominator increases, so α_c can decrease. The statement that the boundary corresponds to α''_c=0, and hence to an extremum of d+(r), relies on a monotonicity condition that is not established for the coreness-based ordering. This weakens the theoretical grounding of the central boundary rule r*.","section":"Section IV, Eqs. (2)-(3)"},{"comment":"The multiplex extension aggregates layer-wise coreness and centrality into a single richness value per node [Eq. (2)], but computes d+ layer by layer by counting neighbors with higher richness [Eq. (3)]. The paper acknowledges that the two conventions differ when richness values are tied, but it does not analyze how tie-breaking affects the multiplex core boundary or the densities reported in Table II. In addition, the equal-weight aggregation in Eq. (2) is an untested assumption; without sensitivity analysis over layer-weighting schemes, the multiplex contribution is less supported than the single-layer results.","section":"Section III.B, Fig. 3"},{"comment":"The clique identification is a greedy expansion whose behavior is illustrated only by the specific examples in Fig. 3. The headline empirical claim that the identified clique equals the maximum clique in 8 of 12 networks would be stronger if the paper reported the number of expansion steps per network and compared against a simpler baseline, such as the clique obtained from the maximum k-core shell without any centrality ordering. As written, the reader cannot tell how much of the clique-finding success comes from the proposed ranking versus from the k-core decomposition itself.","section":"Section III.B, Fig. 3"}],"minor_comments":[{"comment":"The footnote suggests weighting richness by maximum coreness and maximum centrality when layers differ in link density, but it is not stated whether this weighting is used in the experiments in Table II; please clarify.","section":"Table I"},{"comment":"The rich-core column reports clique sizes Sc, but the text does not state explicitly whether the same greedy expansion procedure was applied to the rich-core ranking for all networks; a sentence in the caption or in Section III.B would remove ambiguity.","section":"Section II"},{"comment":"The degree centrality within the k-core is defined as ed/(en-1), but the normalization of eigenvector centrality is not specified. Since the ranking depends on the centrality values only through their order, a brief note on normalization would improve reproducibility.","section":"Section III.B"},{"comment":"Because tied richness values are broken randomly, the results in Table I and Fig. 2(e) may vary across realizations. Reporting the variability (e.g., across several random tie-breaking runs) for at least a few networks would help gauge the stability of the reported density and clique sizes.","section":"Section V"},{"comment":"The notation α''_c in Eq. (5) uses a continuous second derivative while c is an integer core size; a discrete formulation or an explicit statement of the continuous approximation would avoid confusion.","section":"Section V"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the proposed method is simple and practical. The main concern is attribution: much of the reported advantage over rich-core appears to stem from the coreness pre-sort, which is not a new ingredient. An ablation study is essential before publication. The theoretical justification for the boundary r* also needs to be made rigorous or explicitly qualified. I recommend major revision rather than rejection, as the empirical findings are potentially useful and the issues are addressable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the density gain over rich-core mostly comes from ranking by coreness, not from the d+(r) boundary or the centrality tie-breaker, and the paper never shows otherwise. The clique-finding offshoot is the most genuinely new and useful piece.\n\nThe method is a simple, plausible extension of the rich-core idea: rank nodes by coreness, break ties by a within-k-core centrality, then cut at the rank that maximizes links to higher-ranked nodes. The empirical presentation is clear, with 12 networks, and the authors show several cases where their core is a perfect clique. The clique heuristic—nodes on the d+ = r-1 line plus greedy expansion—is clever, and matching the maximum clique in 8 of 12 networks is a nice result. The multiplex extension and the O(E+V log V) complexity claim are also fine.\n\nThe soft spot is attribution. For 7 of the 12 networks, the MCC-E core is exactly the maximum k-core shell or shell plus one; a shell of size k+1 is necessarily a clique, so phi=1 follows automatically. The paper never compares against the trivial baseline of taking the top k-core shell, so the d+ turning point and the centrality ranking are not shown to be doing anything. Email is a counterexample where MCC-E is slightly worse than rich-core. The Discussion's random-walk justification is inherited from the rich-core paper and assumes alpha_c is increasing, which is not guaranteed when a new node has d+=0. These issues do not invalidate the heuristic, but they mean the core-density claim is overstated as a story about the proposed mechanism.\n\nTwo smaller things. First, the maximum-clique sizes for In-2004 and Wiki-Talk look suspicious: Bron-Kerbosch on those graphs is not feasible, so either an approximation was used or the numbers are wrong; the paper should say. Second, there is no code or data, which is a real gap for a method paper.\n\nWho is this for? A practitioner who wants a fast, parameter-free dense-core and clique heuristic will get something out of it. The theory is thin and the mechanism unproven, but the clique approximation is worth knowing. It deserves a serious referee: the paper needs an ablation, a max-k-core baseline comparison, and reproducible code before acceptance. My own verdict would be conditional, not outright accept.","headline":"The density gain over rich-core mostly comes from the coreness pre-sort rather than the proposed d+ boundary, but the clique heuristic is a genuinely useful practical offshoot.","tokens_in":11893,"tokens_out":3035,"would_cite":false,"duration_ms":29732,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["89.75.Hc"],"model":"deepseek-v4-flash","headline":"Ranking nodes by coreness then in-core centrality yields denser cores and, often, the maximum clique.","keywords":["core-periphery structure","k-core decomposition","rich-core method","network core density","clique detection","multiplex networks","node centrality","complex networks"],"falsifier":"Build a network whose largest clique consists of low-coreness nodes attached to a sparse periphery, run the proposed MCC-E method, and check whether the expanded clique is smaller than the maximum clique; a failure on such a graph would show the heuristic is not universal. A second check is to search real networks for cases where the MCC core density falls below the rich-core density, which would contradict the paper's general-density claim.","tokens_in":10867,"feed_emoji":"🕸️","tokens_out":12826,"duration_ms":98231,"temperature":0.7,"pith_summary":"The paper aims to make core-periphery profiling produce structurally cohesive cores without adding parameters. It proposes ranking nodes first by their $k$-core coreness and then by a centrality measure computed inside that core, and taking the rank at which the number of links to higher-ranked nodes is maximal as the core boundary. Across twelve real-world networks, this approach generally yields smaller and denser cores than the degree-based rich-core method, with several cores becoming fully connected. The same ordering also gives a direct route to clique detection: the top-ranked nodes that sit exactly on the line $d^+(r)=r-1$ form a clique, and after expansion this clique matches the true maximum clique in eight of the twelve networks. The method extends to multiplex networks, where layer-averaged rankings identify dense multiplex cores.","feed_headline":"Coreness-first ranking finds denser cores and near-max cliques","feed_subtitle":"A parameter-free, large-scale method that also approximates the NP-hard maximum clique.","key_machinery":"The load-bearing object is the $d^+(r)$ curve together with its upper bound. For each node ordered by richness, $d^+(r)$ is the number of links to nodes ranked higher; the turning point $r^*$ where this count is maximal defines the core-periphery boundary, with all earlier nodes in the core. The curve's upper bound, $d^+(r)=r-1$, captures the fact that the node at rank $r$ can connect to at most all $r-1$ nodes before it, so any sequence of top-ranked nodes lying exactly on that bound must be mutually connected and hence a clique. The ordering that makes the construction work is the richness tuple $\\mu_i=(k_i, e_{m_i})$, which puts structurally deep nodes ahead of merely high-degree ones; the paper explains the boundary choice through random-walk persistence: as the core expands, the probability that a random walker remains inside it grows fastest around the inflection of $d^+(r)$, where the second derivative vanishes.","core_discovery":"The central claim is that ordering nodes by coreness before any centrality measure makes the core boundary found by the rich-core construction sharper and the resulting core genuinely dense. Concretely, each node receives a richness tuple $\\mu_i=(k_i, e_{m_i})$, where $k_i$ is its $k$-core coreness and $e_{m_i}$ is its degree or eigenvector centrality within the $k$-core; the variants MCC-E and MCC-D use eigenvector and degree centrality respectively. Nodes are sorted by this tuple and, reading down the ranking, $d^+(r)$ counts how many links the node at rank $r$ sends to earlier nodes. The boundary of the core is the rank $r^*$ at which $d^+(r)$ reaches its maximum, and the paper shows that on the twelve test networks these cores are denser, and usually smaller, than the rich-core baseline. Because every node at rank $r$ can have at most $r-1$ links to earlier nodes, the curve $d^+(r)=r-1$ is an upper bound, and consecutive top-ranked nodes that lie exactly on it are pairwise connected and therefore form a clique; expanding that seed clique by adding later nodes connected to all of its members yields a clique that equals the maximum clique in eight networks and falls only slightly short in the remaining four.","pith_inferences":["A testable extension the paper leaves open is whether the choice of centrality inside the $k$-core matters systematically; betweenness or local clustering may outperform degree and eigenvector centrality on networks with bottlenecks.","The clique heuristic should break down when the true maximum clique has low coreness, since such nodes would be ranked late; planting a large clique weakly attached to a sparse periphery is a direct way to probe that limit.","The random-walk story for $r^*$ is a heuristic; comparing the core boundary against the maximum of the persistence-probability second derivative on synthetic planted-core networks would show how often the two coincide.","Because the method is built on static pairwise links, a time-resolved or group-interaction version is a natural next step for dynamic and higher-order interaction data."],"forward_implications":["Core-periphery profiling becomes one parameter-free pass: $k$-core decomposition, one centrality calculation, and a scan for the maximum of $d^+(r)$.","Because the resulting cores are smaller and denser, studies of influence, rich-club ordering, or contagion will be working with a more tightly interconnected set of nodes.","The clique-identification procedure gives a fast approximation to the NP-hard maximum-clique problem, matching it exactly in eight of the twelve tested networks.","On multiplex networks, the layer-averaged richness ranking finds denser multiplex cores than the existing multiplex rich-core baseline on all five datasets tested."],"supporting_citations":[{"why":"This reference defines the rich-core method that this paper improves, ranking nodes by degree and taking the maximum of links to higher-ranked nodes as the core boundary.","marker":"[21]"},{"why":"This reference supplies the k-core decomposition that produces the coreness value used as the first component of each node's richness.","marker":"[25]"},{"why":"This reference provides the multiplex rich-core baseline and the per-layer counting of links to richer nodes that the multiplex extension adapts.","marker":"[24]"},{"why":"This reference supplies the maximum-clique enumeration algorithm used to compute the ground-truth clique sizes.","marker":"[36]"},{"why":"This reference gives the improved worst-case enumeration algorithm used for the same maximum-clique ground truth.","marker":"[37]"},{"why":"This reference is the additional maximal-clique reporting source behind the maximum-clique ground truth.","marker":"[38]"}],"fun_headline_variants":["Coreness-first ranking yields denser cores and near-max cliques","Denser cores and near-max cliques by sorting coreness first","Parameter-free core profiling finds dense cores and near-max cliques","Coreness-first ordering sharpens cores and approximates max cliques"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the rank at which the number of links to higher-ranked nodes peaks marks the true core boundary once nodes are ordered by coreness; the paper justifies this with a heuristic random-walk argument rather than a proof that the chosen core is the densest possible one.","fun_headline_variants_meta":{"raw":{"variants":["Coreness-first ranking yields denser cores and near-max cliques","Denser cores and near-max cliques by sorting coreness first","Parameter-free core profiling finds dense cores and near-max cliques","Coreness-first ordering sharpens cores and approximates max cliques"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000353,"raw_usage":{"total_tokens":1952,"prompt_tokens":1008,"completion_tokens":944,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":624,"completion_tokens_details":{"reasoning_tokens":870}},"tokens_in":624,"tokens_out":944,"duration_ms":6572,"temperature":1.0,"reasoning_tokens":870,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T12:05:39.387353+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Build a network whose largest clique consists of low-coreness nodes attached to a sparse periphery, run the proposed MCC-E method, and check whether the expanded clique is smaller than the maximum clique; a failure on such a graph would show the heuristic is not universal. A second check is to search real networks for cases where the MCC core density falls below the rich-core density, which would contradict the paper's general-density claim.","supporting_citations":[{"cited_title":"Community detection in networks: A user guide,","cited_arxiv_id":null,"evidence_quote":"This reference defines the rich-core method that this paper improves, ranking nodes by degree and taking the maximum of links to higher-ranked nodes as the core boundary."},{"cited_title":"Core-periphery organization of complex networks,","cited_arxiv_id":null,"evidence_quote":"This reference supplies the k-core decomposition that produces the coreness value used as the first component of each node's richness."},{"cited_title":"Models of core/periphery structures,","cited_arxiv_id":null,"evidence_quote":"This reference provides the multiplex rich-core baseline and the per-layer counting of links to richer nodes that the multiplex extension adapts."},{"cited_title":"Finding multiple core-periphery pairs in networks,","cited_arxiv_id":null,"evidence_quote":"This reference supplies the maximum-clique enumeration algorithm used to compute the ground-truth clique sizes."},{"cited_title":"Identification of core-periphery structure in networks,","cited_arxiv_id":null,"evidence_quote":"This reference gives the improved worst-case enumeration algorithm used for the same maximum-clique ground truth."},{"cited_title":"Rich-cores in networks,","cited_arxiv_id":null,"evidence_quote":"This reference is the additional maximal-clique reporting source behind the maximum-clique ground truth."}],"review_version":1}