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REVIEW 3 major objections 4 minor 84 references

This book argues that the global statistical regularities of real networks emerge from purely local growth rules, and pins the mechanism to one identity: reaching a neighbor of a random node is already preferential attachment.

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2026-08-04 19:38 UTC pith:6F7SHTGY

load-bearing objection A serious book with a solid core and a fragile empirical keystone: the community-count exponent needs the nested re-run before it can carry the roughness reading. the 3 major comments →

arxiv 2608.01853 v1 pith:6F7SHTGY submitted 2026-08-03 physics.soc-ph cond-mat.dis-nncs.SI

Local network growth: How simple rules drive network complexity

Alexei Vazquez This is my paper
classification physics.soc-ph cond-mat.dis-nncs.SI MSC 05C8291D30 PACS 89.75.Fb89.75.Hc
keywords local network growthpreferential attachmentscale-free networksclustering hierarchydegree correlationscommunity structureshortest path multiplicityRamsey community number
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The book's thesis is that the recurring regularities of real networks—broad, often scale-free degree distributions; a clustering hierarchy; systematic degree correlations; communities; and redundant shortest paths—are not imposed by any global design rule but emerge from purely local growth: each new node acts only on a node it has reached and that node's immediate neighbors. The load-bearing identity is elementary: the probability that a particular node is the neighbor of a randomly chosen node is k_i / Σ_j k_j, which is exactly the linear preferential-attachment probability of the standard global model. From this the book derives, for surfing, friends-of-friends, gene duplication, and duplication–split rules, the whole package of network signatures, with the sign of degree correlations set by which neighbor is chosen. It then shows that communities are an inevitable by-product of locality rather than of node heterogeneity: the evidence for a split grows linearly with the number of links while the cost of naming the split grows only logarithmically. A sympathetic reader would take from this that preferential attachment is a macroscopic perception, not a mechanism, and that network complexity is self-organized—with the book's own caveat that the measured 'super-fractal' community-count exponent (B ~ n^0.61) sits on an instrument ceiling of order √n and is not certified beyond it.

Core claim

The central claim is that a growth rule taking one step outward from a randomly chosen node—to one of its neighbors—attaches preferentially by degree, because a well-connected node belongs to more neighborhoods and is more likely to be hit. Search, triadic closure, and duplication are shown to be instantiations of this single local step, delivering the power-law degree distribution, the inverse-degree clustering hierarchy, and the degree correlations as a package: reaching outward gives disassortative hubs, closing triangles within a neighborhood gives assortative ones. Past a 'Ramsey community number' of a few hundred nodes, communities are all but certain under any local rule with short cy

What carries the argument

The central object is the neighbor-of-a-random-node identity: if a growth rule reaches node i by picking a node at random and stepping to one of its neighbors, the probability of hitting i is k_i / Σ_j k_j—exactly the linear preferential-attachment probability of the standard model. The book shows that random-walk search, triadic closure, and duplication are all instantiations of this one local step, and that the same step generates triangles and degree correlations as a package. The second piece of machinery is the evidence ratio of the degree-corrected block model used as a closed-form formula, which turns the Ramsey community number into an exact crossing between an extensive log-evidence

Load-bearing premise

The load-bearing empirical premise is that real networks carry more communities than self-similarity predicts—B ~ n^β with β ≈ 0.61—and the book itself states that the flat block model used for the count 'cannot report more than of order sqrt(n) groups' and 'is not able to report an asymptotic exponent above one half'; if the instrument ceiling, rather than the networks, sets the counts of the largest systems, the super-fractal finding and the roughness reading built on it co

What would settle it

Re-run the Chapter 6 community census on the same ~100 networks and growing systems with a nested block model that lifts the √n ceiling, as the book itself recommends. If the fitted exponent falls to 1/2 or below, or if the largest networks' community counts stop growing once the ceiling is removed, then β ≈ 0.61 is an instrument artifact and the roughness interpretation α ≈ 0.65 fails with it; the converse—an exponent that survives or rises toward one—would confirm the book's reading.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • If locality is the common cause, then no global information needs to be assumed to explain scale-free structure; standard preferential attachment is an effective description, not a mechanism, and models that use it are coarse-grained accounts of a local process.
  • The sign of degree correlations becomes a diagnostic of the microscopic move: search and duplication produce disassortative networks, triadic closure produces assortative ones, so one framework covers technological, biological, and social networks with a single dial.
  • Communities are a generic consequence of local growth: past a few hundred nodes, any local rule with short cycles—even a triangle-free duplication–split rule—yields communities with near-certainty, while degree-preserving shuffles and global-attachment growth do not.
  • The number of communities grows as a power of network size (B ~ n^β, β≈0.61 across close to a hundred real networks; β≈0.56–0.60 for the book's local rules), and local growth boosts the multiplicity of shortest paths, tying route redundancy to the same community structure.
  • In duplication–split schedules, the critical path becomes a vanishing fraction of large projects and delay risk is governed by network-wide percolation with threshold 1/⟨k⟩—so project performance is a property of the grown activity network, not of any single path.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The √n ceiling coincidence implies that the community-count exponent measured with a flat model cannot, by itself, distinguish a network property from a detector limit; the nested-model re-run the book calls 'the obvious next measurement' is the decisive experiment. If the exponent survives above 1/2, the roughness reading α=1/β−1≈0.65 becomes a genuine statement about a non-smooth limit; if it co
  • The neighbor-of-a-random-node identity suggests a design rule for the whole space of scale-free mechanisms: any growth rule whose acceptance kernel is a one-step random walk on the existing graph will display effective preferential attachment. The observable that distinguishes mechanisms is therefore not the degree exponent but the correlation sign and the clustering level, which is a wider design
  • If motifs and communities are two vocabularies for the same structure, then a testable extension is to track the combined density of motifs-plus-communities on growing real systems (the Internet, co-authorship) and ask whether it is constant while the motif/community balance shifts with instrument—a quantity the book argues survives changes of instrument but does not itself measure over time on re

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript is the third edition of a book-length argument that the macroscopic regularities of real networks—scale-free degree distributions, clustering hierarchy, degree correlations, communities, and shortest-path multiplicity—are not imposed by global preferential attachment but emerge from purely local growth rules in which each newcomer acts only on a reached node and its immediate neighbors. The book develops mean-field theories for search/random-walk, triadic closure, gene duplication, and duplication–split growth; gives exact block-model evidence calculations for the ring, diamond lattice, and pseudofractal web; and reports an empirical census of roughly one hundred networks whose community counts scale as B ~ n^β with β≈0.61. It then interprets β>1/2 as evidence for a non-smooth network limit with roughness α≈0.65. The final chapters discuss nested block models, graphon limits, and the distinction between network structure and detector resolution.

Significance. If the central thesis holds, the book offers a genuine unification: a single local mechanism explains multiple global network regularities, and it provides a concrete microscopic alternative to preferential attachment. The strengths are substantial and deserve explicit credit. The exact evidence-ratio computations for the ring are parameter-free and checkable; I verified that log R ≈ (c−1)n ln 2 matches the stated exact formula for c=1..4. The diamond-lattice renormalization flow and the closed-form Ramsey number r_κ(b,s;q) are elegant, and the pseudofractal comparison between plain and degree-corrected block models is a persuasive warning about null models. The book is also honest in repeatedly flagging the dependence of community numbers on the detection instrument. However, the empirical pillar of the third edition—the super-fractal exponent β>1/2—is built on an estimator whose resolution ceiling has the same functional form as the null hypothesis being tested. That issue is load-bearing, because the roughness reading α=1/β−1 is derived directly from the same exponent.

major comments (3)
  1. [§6.12, Eq. (6.50); §7.5] The central empirical claim B ∝ n^β with β≈0.61 is measured with a flat degree-corrected SBM, and §7.5 states that this estimator 'cannot report more than of order √n groups' and 'is not able to report an asymptotic exponent above one half.' Since the fractal benchmark being tested is itself B∼√n, a fit over four decades cannot certify an asymptotic exponent above 1/2; the largest networks may be pinned at the detector ceiling. The manuscript is also internally inconsistent about whether the decisive nested re-run has been done: the preface says lifting the ceiling moves the exponent 'from two thirds to one', §6.12 says refitting real networks raises β from 0.63 to 0.80, and §7.5 calls the nested re-run 'the obvious next measurement.' These cannot all be true. The book must either supply the nested-model census for the real networks with full reporting, or explicitly retract the asymptot
  2. [§7.4–7.5, Eq. (7.6)] The roughness inversion α = 1/β − 1 ≈ 0.65 is directly built on the same measured β. If the flat-SBM ceiling, rather than the networks, sets the community counts at large n, then α is not a property of the network limit but of the estimator. Equation (7.4), α = k* ∝ n^{1/(1+α)}, is derived under a smoothness model for the graphon and a specific risk decomposition; inverting a method-limited exponent to infer network roughness is not valid unless the estimator is known to resolve the true partition. The claim that real networks are 'rough, not smooth' therefore needs an independent estimator (e.g., the nested model) before it can be accepted, or it must be presented as a conditional statement.
  3. [§4.6, Eq. (4.12)] The unifying mathematical assertion—'the chance that a particular node i is the neighbor of a randomly chosen node is k_i / Σ_j k_j'—is not true for the operation those words describe. If one picks a uniformly random node and then one of its neighbors uniformly, the probability of landing on i is (1/n) Σ_{u∼i} 1/k_u, which is not proportional to k_i (a star is a counterexample). The identity k_i/Σ_j k_j corresponds to choosing a uniformly random edge endpoint, or to the stationary distribution of a degree-biased walk. The book's own search model (§2.4) gives an affine rate A(k) = const + (q_e ν_a/ν_s) k, not exact proportional attachment. The statement should be corrected and its domain of validity stated; as written, it overstates the exactness of the local-to-global equivalence.
minor comments (4)
  1. [§6.12, Fig. 6.11] The scatter plot of ~100 networks has no per-point error bars and no reporting of the fitted slope's robustness to removing the largest or smallest networks. Since the conclusion is an exponent above 1/2, a sensitivity analysis (e.g., jackknife or removal of the top decade) would be valuable.
  2. [Ch. 7.5, with §6.12] The three statements about nested re-runs (preface: 'from two thirds to one'; §6.12: 0.63 to 0.80; §7.5: 'obvious next measurement') should be reconciled in one place, with the actual number and estimator specified, rather than appearing in separate chapters.
  3. [References] Several load-bearing results are cited as 'Vazquez (2026a–e)' or 'Vazquez (2025)' with no bibliographic details. For a self-contained book, these should be listed fully or the results derived in the text; otherwise readers cannot verify the exact claims.
  4. [Ch. 1, Table 1.1] The corrected clustering coefficient in Eq. (1.8) is undefined when ω_i=0; the table would benefit from stating how such nodes are handled in the averages, especially for low-degree nodes in sparse networks.

Circularity Check

1 steps flagged

Core local-rule derivations are parameter-free and non-circular; the third edition's empirical super-fractal claim, however, rests on load-bearing self-citations and a detector ceiling that coincides with the null, so the 'roughness' reading is not established.

specific steps
  1. self citation load bearing [Ch 6.12, 'How many communities do real networks have?'; Ch 7.5, 'Five meanings of one number']
    "Following Vazquez (2026a), we can now say that it does... refitting the real networks of Figure 6.11 the same way raises the exponent of Eq. (6.50) from 0.63 to 0.80... Only the nested block model, which lifts the ceiling to order n/ln n, can settle that, and re-running the census with it is the obvious next measurement."

    The empirical pillar of the third edition—the super-fractal exponent β≈0.61 and the claim that lifting the flat-SBM ceiling moves it to 0.80–1—is attributed to the author's own cited papers (Vazquez 2026a and related self-citations) rather than to reproducible analysis in the book. Ch 6.12 reports the nested rerun as already done, while Ch 7.5 calls it 'the obvious next measurement,' so the load-bearing measurement is not independently established; the claim is supported mainly by an unverified self-citation chain.

full rationale

Most of the book's derivations are self-contained. The Ch2–4 mean-field calculations (Eqs 2.4, 2.7, 2.9, 3.8–3.9, 4.7, 4.9, 4.11) solve explicit balance equations and are checked by simulation; they do not fit the target exponents and then reuse them. Ch5 tests the duplication–split rule by fitting q from each of two degree distributions and checking their agreement and the critical-path scaling, a legitimate consistency test. Ch6's exact solvable cases (ring, diamond, pseudofractal) are pencil-and-paper results (Eqs 6.8, 6.12, 6.21–6.32, 6.36–6.39) with no circularity. The flagged problem is the empirical super-fractal pillar. Ch6.12 measures B∝n^β with the flat degree-corrected SBM, whose resolution ceiling is ~√n; Ch7.5 explicitly states this estimator 'cannot report more than of order √n groups' and 'is not able to report an asymptotic exponent above one half.' Since the fractal null is also √n, the measured β≈0.61 and the derived roughness α=1/β−1 (Ch7.4) cannot be certified as network properties rather than detector behavior. This is a serious limitation, and the book's own statements are inconsistent (Ch6.12 reports the nested rerun done; Ch7.5 calls it 'the obvious next measurement'). The empirical results are also cited to the author's own prior papers rather than reproduced. None of this makes the central local-rule derivations circular, but it means the third edition's headline empirical claim is not independently established.

Axiom & Free-Parameter Ledger

9 free parameters · 10 axioms · 4 invented entities

The central derivations rest on the mean-field approximation (acknowledged to bias the exponents), uniform node arrival times, the PageRank equation, and specific prior choices (Beta(alpha,alpha)) in the evidence ratios, where the ring calculation shows the verdict for c = 1 depends on alpha (Eq. 6.9). The book's readings also depend on the sparse-regime classification of real networks, the generalization from the ring to 'networks with more links than nodes', and the single-rule-per-network attribution. Free dials (q_e, q_v, u, q, q_e_retention) set the predicted exponents, and several are back-fit to the data they then 'predict'. The invented entities are bookkeeping devices and detector-relative definitions, with no independent falsifiable handles outside the book's own derivations.

free parameters (9)
  • q_e (surfer link-following patience, Ch 2) = inferred > 0.5 from measured gamma ~ 2.1
    Sets the in-degree exponent gamma = 1 + 1/q_e, the clustering amplitude 2(1+q_e)/k, and the correlation sign; back-fit to the data it then 'predicts' (Ch 2.5).
  • q_v and nu_s/nu_a (Ch 2) = none
    Probability a visit creates a link and the surfer-to-new-page arrival ratio; enter the constant term of A(k) and the parameter a in Eq. (2.6), otherwise unconstrained.
  • u (triangle-closing probability, Ch 3) = none
    Single dial of the DEB-style simulation; maps to mu_1 = u/(1-u) and sets the exponent gamma(u) of Eq. (3.10); not fitted to social data in the text shown.
  • q_v (self-interaction probability, Ch 4) = none
    Sets the clustering amplitude 2q_v/k in Eq. (4.9); free parameter.
  • q_e (retention probability, Ch 4) = none
    Sets the density transition at q_e = 1/2 and the multifractal spectrum Eq. (4.11); free parameter.
  • q (duplication probability, Ch 5) = 0.1 to 0.5 across 77 projects, most above 0.2
    Exponent of the predecessor/successor distributions is 1/q; fitted to real schedules, after which the model 'predicts' the exponent it was fit to (Ch 5.3, 5.5).
  • epsilon (certainty threshold for r_k) = 0.05
    Chosen in Eq. (6.2); r_k scales with log((1-eps)/eps), so the threshold value matters for the reported numbers.
  • alpha (Beta prior parameter) = 1 in the ring calculation
    Prior on block link probabilities, Eq. (6.3); the ring knife-edge decay exponent includes 2alpha + 3 (Eq. 6.9), so the qualitative verdict for c = 1 depends on this prior choice.
  • L and W (bubble model dials) = scanned L = 1..9, W = 1..3
    Swept to show r_k grows with cycle length and diverges beyond W = 2; model dials, not data-derived.
axioms (10)
  • domain assumption Mean-field approximation: fluctuating local quantities are replaced by their averages in the growth equations, and the resulting exponents are treated as model predictions.
    Invoked in the boxes of Chapters 2-4 (balance condition leading to Eq. (2.2), rate equations Eq. (3.1), duplication equation Eq. (4.1)); the book concedes the predicted exponent is 'a slight overestimate, the price of the mean-field approximation' (Ch 2.6).
  • domain assumption Nodes arrive at a constant rate, so birth times are uniform, P(n_i = n) = 1/n, used to convert growth laws into degree distributions.
    Ch 3.3 (Eq. 3.7) and Ch 4.2 (Eq. 4.6).
  • standard math The stationary visiting probability of a random walk with restarts satisfies the PageRank equation (Eq. 2.1).
    Ch 2.2, attributed to Brin and Page (1998); used as the seed relation v_i proportional to k_in.
  • ad hoc to paper The evidence-ratio calculation for whether a network is split uses Beta(alpha,alpha) priors on link probabilities and a Beta prior on the label proportion (Eq. 6.3), with alpha = 1 in the ring analysis.
    The ring's knife-edge result depends on these priors: Eq. (6.9) has decay exponent 2alpha + 3, so the prior sets the 'price' of labeling; Eq. (6.10) special-cases alpha = 1 to obtain the binomial coefficient.
  • domain assumption Community existence is defined detector-relative: r_k is the size where a chosen detection method reports more than one community with 95% probability (Eqs. 6.1-6.2).
    Ch 6.2: 'r_k is a property of the network and of the rule used to interrogate it'; the book later shows r_k varies by a factor of 25 between plain and degree-corrected models and that a motif vocabulary can remove communities entirely (Ch 6.9, Ch 8-9).
  • standard math The degree-corrected block model with Gamma(alpha,alpha) priors on block affinities is the null that already knows the degrees (Eq. 6.19).
    Ch 6.8, following Karrer and Newman (2011) and Peixoto; used to distinguish hubs from genuine community structure.
  • domain assumption Real networks are in the sparse, fixed-mean-degree regime, so dense or L_p graphon limits do not apply and finite-size computation is the only instrument.
    Ch 7.1: fitted delta = -0.04 for the Internet over 733 snapshots; cond-mat densifies with delta = +0.36. The chapter's readings depend on this regime classification.
  • domain assumption A network is broken into communities precisely when it carries more links than nodes (extensive cyclomatic number), generalized from the ring to arbitrary networks.
    Ch 6.7-6.8: log R ~ (m-n) ln 2 is derived for a single lattice family and then asserted generally: 'A tree has none, and is never split'.
  • domain assumption Each network family is governed predominantly by one local rule: search for web and citations, triadic closure for social, duplication for PPI, duplication-split for schedules.
    Ch 1.5, 'each built around one concrete local rule and a family of real networks it illuminates', and the Ch 4.5 table; mixture effects and overlapping mechanisms are not treated, so attributions are underdetermined.
  • domain assumption Static networks from a public repository, sampled for size range, are representative for estimating the community-count scaling law.
    Ch 6.12: 'chosen only to span the widest possible range of sizes... with nothing else in common'; the bootstrap CI treats the networks as exchangeable draws from a common population.
invented entities (4)
  • potential link (p-state) no independent evidence
    purpose: Bookkeeping device for 'friend of a friend'; a pair sharing a common neighbor but not yet linked, enabling the three-state rate equations (Eqs. 3.1-3.4).
    A modeling construct; its testable content is confined to the derived degree and clustering distributions.
  • Ramsey community number r_k no independent evidence
    purpose: Defines the minimum size at which a detection method reports communities with 95% probability; turns 'do communities emerge?' into a finite-vs-infinite question.
    Explicitly detector-relative (Ch 6.2); the same network yields r_k from 42 to 1095 depending on the null model (Ch 6.9).
  • community fixed point (K,0) no independent evidence
    purpose: Limit of the renormalization flow of block densities (u_in, u_cross) for the diamond lattice, giving log R/m -> ln K per link.
    Verified on the constructed diamond and pseudofractal lattices only; asserted as a general mechanism for why evidence for partitions is extensive.
  • density of communities as the growth limit no independent evidence
    purpose: Replaces 'shape' as what locally grown networks converge to: so many communities per node, constant in size, set by the growth rule.
    The book proposes this as the third edition's resolution, but its own text reports the decisive nested census as both completed (preface: exponent 'from two thirds to one') and pending (Ch 7.5: 're-running the census with it is the obvious next measurement').

pith-pipeline@v1.3.0-daily-deepseek · 7627 in / 8155 out tokens · 366577 ms · 2026-08-04T19:38:01.467071+00:00 · methodology

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Cite this review

Pith. "Pith review of Local network growth: How simple rules drive network complexity." pith.science (2026). https://pith.science/paper/6F7SHTGY

@misc{pith2026260801853,
  author       = {Pith},
  title        = {Pith review of: Local network growth: How simple rules drive network complexity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6F7SHTGY}},
  note         = {Machine review of arXiv:2608.01853}
}
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read the original abstract

The Internet, a living cell, a circle of friends, a billion-dollar construction project: these systems share almost nothing -- yet, drawn as networks, they look astonishingly alike. Each has a few giant hubs among a multitude of sparsely connected nodes, short paths between any two parts, dense local clustering, communities, and many redundant routes. For two decades such patterns have been credited to "preferential attachment," the rich getting richer -- a rule that, taken literally, asks every newcomer to survey the whole network before it links. This book makes a simpler case, and defends it one mechanism at a time: the global regularities of real networks are not imposed from above but emerge from purely local rules, in which each new node acts only on a node it has reached and that node's immediate neighbours. A surfer following links, a friend introducing a friend, a gene copied with its connections -- none consults the network as a whole, yet each builds, in the aggregate, the full and unmistakable signature of a real complex system. Written for the curious reader as much as the specialist, with the ideas told in plain language and the mathematics set aside in boxes that can be skipped, it shows how citation graphs, the web, social ties, protein interactions, and project schedules all grow themselves from the same handful of local rules -- one local decision at a time.

Figures

Figures reproduced from arXiv: 2608.01853 by Alexei Vazquez.

Figure 1.1
Figure 1.1. Figure 1.1: Degree. A node’s degree is simply its number of links. Drawn here with node size growing with degree: a few hubs (such as the central one, k = 7) carry many links, while most nodes have only one or two. That imbalance — many small nodes, a few giant hubs — is the hallmark of a scale-free network. is a number, the exponent, typically between 2 and 3 for real networks. A power law is just a rule that says … view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Clustering coefficient. The clustering of the central node (gray) is the fraction of its neighbor pairs that are themselves linked. (a) Four of the six neighbor pairs are linked (the bold links), so c ≈ 0.67 — a cliquish neighborhood. (b) No neighbors are linked to each other, so c = 0. 1998) is that ⟨c⟩ in real networks is far larger than you would get by wiring the same nodes up at random — the quantit… view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: Distance. The distance between two nodes is the number of links on the shortest route between them. The shortest route from A to B here is three hops (in bold); the lower route is longer. In real networks these shortest distances stay small even as the network grows — the small-world effect. et al., 2001): ⟨knn⟩(k) = X k′ k ′ p(k ′ | k). (1.3) Here p(k ′ | k) is read “the probability that a link from a d… view at source ↗
Figure 1.4
Figure 1.4. Figure 1.4: Degree correlations. (a) In an assortative network, hubs tend to link to other hubs (“birds of a feather”). (b) In a disassortative one, hubs avoid each other and link instead to low￾degree nodes (“opposites attract”). Social networks lean assortative; technological and biological ones, disassortative. to normalize) gives ⟨knn⟩unc = P k′ k ′ · k ′ p(k ′ ) P k′ k ′p(k ′) = ⟨k 2 ⟩ ⟨k⟩ , (1.4) where ⟨k⟩ is … view at source ↗
Figure 1.5
Figure 1.5. Figure 1.5: Clustering hierarchy. Low-degree nodes sit inside tight clusters where everyone is connected (high clustering, the dark triangles), while a hub bridges many such clusters. The hub’s own neighbors — one drawn from each cluster — do not know each other, so the hub itself has low clustering. Cliquishness therefore falls as degree rises. their degree k and averaging the clustering coefficient within each gro… view at source ↗
Figure 1.6
Figure 1.6. Figure 1.6: Why the standard clustering coefficient misleads. The hub’s neighbors all have degree two, and each spends its spare link on another neighbor of the hub. They are therefore as intercon￾nected as their degrees allow — no rewiring could add a single link among them. Yet the standard coefficient, Eq. (1.2), divides by all [PITH_FULL_IMAGE:figures/full_fig_p021_1_6.png] view at source ↗
Figure 1.7
Figure 1.7. Figure 1.7: The clustering hierarchy, and what is left of it. Four real networks — the Internet at the autonomous-system level, the Gnutella file-sharing overlay, the yeast protein interaction net￾work, and cond-mat co-authorship — drawn from the Netzschleuder repository (Section 1.4). (a) The standard clustering coefficient falls with degree in every network, roughly as k −1 (dashed guide): the hierarchy of Eq. (1.… view at source ↗
Figure 1.8
Figure 1.8. Figure 1.8: What the data demand. The same four networks, from the Netzschleuder repository through graph-tool. (a) The degree distributions are broad and hub-dominated; the Internet spans four decades of degree. Gnutella’s protocol caps its peers, so its tail is the shortest. (b) The average neighbor degree, where the two families part: falling for the Internet, Gnutella and the yeast proteins — disassortative — bu… view at source ↗
Figure 1.7
Figure 1.7. Figure 1.7: to write with one another, and the network is assortative (Newman, 2002). The single number r in [PITH_FULL_IMAGE:figures/full_fig_p028_1_7.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Surfing as a random walk. From the page it is on, the searcher follows one of the outgoing links with probability qe (solid arrows) or, with probability 1 − qe, jumps to a page chosen at random (dashed arrow). A page reached along many routes — like the highlighted hub — is visited most often; this visiting frequency is the page’s PageRank, and it grows in proportion to the number of links that point at … view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Two ways to search, when a new node i joins. (a) In the random-walk model, i links to a randomly chosen node j and then takes a single step to one of j’s neighbors, linking to it as well. (b) In the recursive-search model, i follows all of j’s links, and then their links in turn, linking to every node its widening search reaches. New links are drawn in bold, pre-existing links thin and gray. 2.6 Two ways… view at source ↗
Figure 3
Figure 3. Figure 3: shows this elementary step. Notice that it is a thor [PITH_FULL_IMAGE:figures/full_fig_p041_3.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Triadic closure. Two people (bottom) who already share a common friend (top) form a potential link — a friend-of-a￾friend pair, drawn dashed. Closing the triangle turns that potential link into a real one, the elementary step by which the friends-of￾friends rule builds social structure. generates the whole suite of social-network signatures at once: an effective “rich get richer” and hence a power-law de… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Duplication and divergence. (a) A gene duplication adds a copy i ′ (gray) wired to every interaction partner of the ancestor i, plus a link between the twins when the protein is self-interacting. (b) Divergence then prunes the now-redundant interactions (dotted), each partner kept by only one twin, so the copies drift apart. steps are thoroughly local — the copy needs to know only its parent and its pare… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Refining an activity. A generic activity is replaced either by (a) two copies that run in parallel, both inheriting the predecessor (left, gray) and successor (right, gray) of the parent — duplication, which widens the schedule — or by (b) two specialized activities that run in sequence — a split, which lengthens it. generic work is parallelized into many simultaneous strands, and the schedule fans out i… view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: Communities emerge from local rules. (a) A network grown by a local rule settles into dense communities (two here, one dark gray, one white) joined by only a few links. (b) Shuffling the links at random while keeping every node’s number of connections destroys the communities — so the structure was the work of the rule, not of the degrees alone. and Bergstrom, 2008). Both, crucially, report no communitie… view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: Communities in a local-search network. An in￾stance of the local-search model LS(n=50, ℓ=1). The stochastic block model resolves two communities, shown dark gray and white; they emerge even though the rule treats every node alike and never mentions groups. have the emergent communities property; the controls — reshuffling a network’s links, and growing one with no locality at all — do not. Holding the fo… view at source ↗
Figure 6.3
Figure 6.3. Figure 6.3: Communities in a duplication–split network. An instance of the duplication–split model DS(n=50, q=0.3). Two communities (dark gray and white) again emerge — even though this rule builds no triangles at all, so the network is far less cliquish than the local-search one of [PITH_FULL_IMAGE:figures/full_fig_p073_6_3.png] view at source ↗
Figure 6.4
Figure 6.4. Figure 6.4: Cutting the ring. The block model splits the ring into two contiguous arcs (dark and white), preferring this to no split at all once the ring is large enough. But every node looks the same, so the cut (dashed) could sit between any pair of beads and score exactly the same: the model says the ring must be cut, without saying where. The two arcs are real communities in the way a magnet has a real direction… view at source ↗
Figure 6.5
Figure 6.5. Figure 6.5: The ring wants to be broken. (a) The probability that a block model prefers two arcs to no split. The plain cycle (c = 1) is never split; the next-nearest-neighbor ring (c = 2) is split beyond rκ ≃ 34 nodes (dash-dotted line; dotted line marks 99% certainty). (b) The evidence for the split against the prediction log R ≃ (c − 1) n ln 2 (lines), Eq. (6.12), for c = 1, . . . , 4. The cycle lies exactly on z… view at source ↗
Figure 6.6
Figure 6.6. Figure 6.6: The diamond’s two communities. The b = s = 2 diamond lattice at generation t = 2. Its two bundles — the left half (open circles) and the right half (gray) — are the two communities; they touch only at the poles A and B (black), so the only links crossing between them are the few at top and bottom. As the lattice grows, that seam becomes an ever fainter share of each bundle’s interior, which is why the sp… view at source ↗
Figure 6.7
Figure 6.7. Figure 6.7: The Ramsey number as a crossing. (a) Evidence for the split on the diamond lattice against size. rκ is simply where the curve clears the threshold ln[q/(1 − q)] (dotted, q = 0.99): at n = 44 for a degree-corrected block model (dashed line) but only at n = 684 for a plain one (dash-dotted). (b) The densities of Eq. (6.22) flow to the community fixed point (K, 0): the interiors settle at K = 2 times chance… view at source ↗
Figure 6.8
Figure 6.8. Figure 6.8: Building the pseudofractal web. The deterministic rule, shown for the first three generations. Start from a triangle (t = 0); at each step, every existing link gives birth to a new node joined to both its ends, and the old links are kept. The three corners of the original triangle (black) never stop gaining neighbors and become the towering hubs; each later node is fainter the later it was born. The math… view at source ↗
Figure 6.9
Figure 6.9. Figure 6.9: Communities, or just degrees? The pseudofractal web under a plain block model (open circles) and one corrected for the degree sequence (filled squares). The corrected model is certain of the recursive communities from n = 42; the plain one denies them until n = 1095. The degree gradient, far from manufacturing the split, delays the detector that cannot see past it. local structure has none — but it does … view at source ↗
Figure 6.10
Figure 6.10. Figure 6.10: The modular limit: the caveman graph. A ring of tight cliques (here six cliques of four nodes), each joined to its neighbors by a single bridge (dashed). Each clique is so much denser inside than out that it is unmistakably one community, so the number of communities simply counts the cliques and grows in proportion to size, B ∝ n (β = 1). The self-similar networks of Section 6.10 sit at the opposite li… view at source ↗
Figure 6.11
Figure 6.11. Figure 6.11: The number of communities grows as a power law of size. Number of communities B against number of nodes n for close to a hundred real networks (log–log), each counted with the degree-corrected block model. The solid line is the fitted power law B ∝ n 0.61; the dashed line is the fractal √ n. Denser networks (darker fill) sit above the line, sparse ones below; geographic, spatially embedded networks (squ… view at source ↗
Figure 6.12
Figure 6.12. Figure 6.12: The power law holds within a single growing system. Number of communities B against size n (log–log) as three systems grow: (a) the Internet at the autonomous-system level, from daily snapshots; (b) protein interactomes across organisms; (c) the cond-mat co-authorship network, accumulated year by year to nearly 400,000 authors. Each is a power law (solid line, fitted exponent) above the fractal √ n (das… view at source ↗
Figure 6.13
Figure 6.13. Figure 6.13: Local rules reproduce the super-fractal exponent. Number of communities against size for the two local-growth models grown to 105 nodes: (a) triadic closure and (b) duplication–split. Each follows a power law (solid line) with an exponent above the fractal √ n (dashed), in the same super-fractal range as the real networks of [PITH_FULL_IMAGE:figures/full_fig_p104_6_13.png] view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: The limit picture, under two scalings. Adjacency matrices of the same two-block structure at n = 60, 240, 960, nodes ordered by block. (a) Dense: the link probabilities are held fixed, and the picture sharpens onto a step function — the graphon. (b) Sparse: the mean degree is held fixed instead, so the probabilities fall as 1/n. The structure is identical, and just as detectable, but the picture fades to… view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: How many blocks can be seen. (a) The risk of a k￾block description, Eq. (7.3), for three network sizes, each normalized to its own minimum (dots). The optimum moves right as √ n. (b) That optimum against size, with the community counts of the real networks of the last chapter and their fitted power law. The vertical placement of the k ∗ line depends on the graphon’s smoothness constant and carries no mea… view at source ↗
Figure 7.3
Figure 7.3. Figure 7.3: The same network, divided by the two priors. An instance of LS(n=200, ℓ=1), laid out once and coloured twice. The flat model resolves four groups; the nested model splits the larger of them and resolves five, and it does so while spending twelve bits fewer on the description as a whole — which is the sense in which the finer division is not merely permitted but preferred. At n = 100 the two return the sa… view at source ↗
Figure 7.4
Figure 7.4. Figure 7.4: How many communities, under each prior. The mean number of communities against size for four growth rules, averaged over twenty networks at each size, with the best power law fitted to each method and its exponent in the legend. The flat model comes out near its own √ n ceiling throughout; the nested model grows almost in proportion to n. What survives of Chapter 6, then, is the inequality. Every method … view at source ↗
Figure 8.1
Figure 8.1. Figure 8.1: The two halves of the prediction, on real networks. [PITH_FULL_IMAGE:figures/full_fig_p142_8_1.png] view at source ↗
Figure 8.2
Figure 8.2. Figure 8.2: The detection method is part of the answer. Every panel compares three methods on identical networks: a flat block model, a nested one, and the nested one with triangles and squares in its alphabet. (a) duplication–split, (b) local search. The communities likelihood Pκ(n), from a hundred networks at each size up to n = 200 and twenty above it, with the dotted line at the 95% threshold whose crossing defi… view at source ↗
Figure 8.3
Figure 8.3. Figure 8.3: Two rules built from squares, and only one of them says so. Panels as in [PITH_FULL_IMAGE:figures/full_fig_p159_8_3.png] view at source ↗
Figure 9.1
Figure 9.1. Figure 9.1: Two-colouring the L + 2 cycle. It closes for even L. For odd L, or for any L once a diagonal is added, one link is left with both ends the same colour. 9.1 A dial for the shortest cycle What L controls is the shortest cycle the rule ever closes, and through it the whole cycle spectrum. Two faces sharing a link close a cycle of length a + b − 2, so the lengths present are generated by the motif and its gl… view at source ↗
Figure 9.2
Figure 9.2. Figure 9.2: The alternation, and the diagonal that breaks it. Both panels plot the community count under the motif alphabet divided by the count the nested block model returns on the same networks: one where the communities survive, falling away where they are carried off. (a) L = 1 to 5, odd L filled and even L open. (b) BB(n, 2) against two variants that each add one link per step, of which only the diagonal destr… view at source ↗
Figure 9.3
Figure 9.3. Figure 9.3: Meso-structure, counted without regard to which vocabulary names it. (a) motifs planted, κ0, and (b) motifs plus communities, κ0 + κ, against size, with a line of slope one for the eye. Every rule is parallel to it in both panels, while κ alone runs from n 0.25 to n 0.93 across the same five rules [PITH_FULL_IMAGE:figures/full_fig_p185_9_3.png] view at source ↗
Figure 10.1
Figure 10.1. Figure 10.1: Multiplicity on a ring. (a) On the plain cycle, the two arcs between antipodal nodes tie, so both are shortest (µ = 2); every other pair has one arc strictly shorter, hence a unique shortest path. (b) On the next-nearest-neighbor ring, a step may cover one node or two, so for endpoints an odd number of steps apart there are several ways to combine the steps into a shortest path — here the two two-hop ro… view at source ↗
Figure 10.2
Figure 10.2. Figure 10.2: The random baseline. Average shortest path multi￾plicity ⟨µ⟩ against network size n for randomized networks (starred), which keep each node’s degree but destroy all local structure. (a) LS(n, 1)∗ , degree exponent γ = 5: growth stalls, the points falling away from the fitted logarithm. (b) DS(n, 1/3)∗ , γ = 3: logarith￾mic, near the predicted slope 1/e. (c) DS(n, 2/5)∗ , γ = 5/2: the predicted power law… view at source ↗
Figure 10.3
Figure 10.3. Figure 10.3: Local rules boost the multiplicity. (a) Local search LS(n, 1) grows log-quadratically, Eq. (10.31) — while its shuffled counterpart, with identical degrees, saturates ( [PITH_FULL_IMAGE:figures/full_fig_p200_10_3.png] view at source ↗
Figure 10.4
Figure 10.4. Figure 10.4: Why squares multiply routes. The two highlighted corners are the endpoints. (a) In a triangle they are joined directly — a single shortest route. (b) In a square they are opposite corners, two steps apart by two distinct routes (drawn solid and dashed), one around each side. Local rules that close four-cycles, such as gene duplication, thread the network with these redundant shortest paths. only one sho… view at source ↗
Figure 10
Figure 10. Figure 10: makes the contrast concrete [PITH_FULL_IMAGE:figures/full_fig_p201_10.png] view at source ↗
Figure 10.5
Figure 10.5. Figure 10.5: Cycle length and randomness. (a) The bubble model BB(n, L), which closes loops of length L + 2. Even L (2 and 4) reaches a higher multiplicity than odd L (1 and 3), as the square-counting predicts — yet every L stays log-quadratic, so cycle geometry alone does not buy the exponential. (b) The determin￾istic bubble model BBD(n, 1) follows a power law n 0.10 — found exactly in Section 10.8 — falling below… view at source ↗
Figure 10.6
Figure 10.6. Figure 10.6: The diamond lattice, solved exactly. (a) The diamond lattice: every link is replaced, at every generation, by two parallel two-link paths. (b) Its average multiplicity, computed exactly, follows the stretched exponential of Eq. (10.41) across fifteen decades. With 2732 nodes, a typical pair of nodes is already joined by some 7 × 1015 equally short routes — the fastest of the deterministic constructions,… view at source ↗
Figure 10.7
Figure 10.7. Figure 10.7: Real networks. Average multiplicity over connected pairs, ⟨µ⟩c (Eq. (10.43)), against size, with the mean-field expectation c2/e for a random network of the same degrees. (a) The Internet at the autonomous-system level: a power law, falling below the random expectation — the mark of disassortative, star-like wiring. (b) Protein interaction networks of a dozen organisms: exponential growth, the duplicati… view at source ↗

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