science-of-scienceScience of science analysis covering disruption index, team size effects, sleeping beauty detection, and knowledge recombination with citation data.
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name: science-of-science description: Science of science analysis covering disruption index, team size effects, sleeping beauty detection, and knowledge recombination with citation data. tags:
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Trigger keywords: science of science, disruption index, CD index, sleeping beauty, knowledge recombination, novelty conventionality, team size, Matthew effect, cumulative advantage, disciplinary distance, cross-disciplinary, big team small team, Wang and Barabasi, knowledge flows, citation network, research front, knowledge base, scientific discovery patterns.
Measures whether a paper pushes science forward (disruptive) or builds on existing work (consolidating):
$\text{CD}i = \frac{n_f - n{fb}}{n_f + n_b + n_{fb}}$
where for citing papers $c$ of paper $i$:
CD = 1: fully disruptive; CD = -1: fully consolidating; CD = 0: neutral.
A sleeping beauty is a paper that has a long dormancy period before receiving high citations. Define:
Sleeping beauty score: $B \cdot L$.
For a paper's reference list, compute the conventionality (journals that are typically cited together) and novelty (atypical combinations):
$z\text{-score}{ij} = \frac{\text{co-citation}{ij} - \mu_{ij}}{\sigma_{ij}}$
where $\mu_{ij}$ is the expected co-citation count under a random reference model (Monte Carlo permutations of reference lists).
$\frac{dc_i(t)}{dt} = \alpha c_i(t)^\beta + \gamma$
preferential attachment: papers with more citations attract even more citations. Estimated via power regression on citation velocity.
pip install pandas>=2.0 numpy>=1.24 scipy>=1.11 networkx>=3.2 \
scikit-learn>=1.3 matplotlib>=3.7
import numpy as np
import pandas as pd
import networkx as nx
import matplotlib.pyplot as plt
print("Science of science environment ready")
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
# -----------------------------------------------------------------
# Simulate a citation graph for 300 papers
# Each paper has references; later papers cite earlier papers
# -----------------------------------------------------------------
np.random.seed(42)
n_papers = 300
paper_ids = list(range(n_papers))
years = np.sort(np.random.randint(2000, 2023, n_papers))
# Build reference lists: each paper cites 5-15 older papers
references = {}
for i in range(n_papers):
older = [j for j in range(i) if years[j] >= years[i] - 10]
if len(older) >= 3:
n_refs = min(np.random.randint(5, 16), len(older))
references[i] = list(np.random.choice(older, n_refs, replace=False))
else:
references[i] = older
# Who cites whom
cited_by = {i: [] for i in paper_ids}
for citer, refs in references.items():
for cited in refs:
cited_by[cited].append(citer)
# -----------------------------------------------------------------
# Compute CD index for each paper
# -----------------------------------------------------------------
def compute_cd_index(focal, references, cited_by):
"""Compute Funk & Owen-Smith CD disruption index.
Args:
focal: focal paper ID
references: dict {paper: [references]}
cited_by: dict {paper: [papers that cite it]}
Returns:
CD index in [-1, 1]
"""
focal_refs = set(references.get(focal, []))
citers_of_focal = set(cited_by.get(focal, []))
if not citers_of_focal:
return np.nan
n_f = 0 # cite focal, not focal's refs
n_fb = 0 # cite focal AND at least one focal ref
n_b = 0 # cite focal's refs but not focal
# For n_f and n_fb: iterate over papers that cite focal
for c in citers_of_focal:
c_refs = set(references.get(c, []))
cites_focal_refs = bool(c_refs & focal_refs)
if cites_focal_refs:
n_fb += 1
else:
n_f += 1
# For n_b: iterate over citers of focal's references who don't cite focal
citers_of_refs = set()
for ref in focal_refs:
citers_of_refs.update(cited_by.get(ref, []))
citers_of_refs -= citers_of_focal # exclude those who also cite focal
n_b = len(citers_of_refs)
denom = n_f + n_fb + n_b
if denom == 0:
return np.nan
return (n_f - n_fb) / denom
# Compute for all papers (subset for speed)
cd_scores = []
for pid in range(min(200, n_papers)):
cd = compute_cd_index(pid, references, cited_by)
cd_scores.append({"paper_id": pid, "year": years[pid],
"cd_index": cd,
"n_citations": len(cited_by[pid]),
"n_references": len(references[pid])})
cd_df = pd.DataFrame(cd_scores).dropna()
print("=== Disruption Index Summary ===")
print(cd_df["cd_index"].describe().round(3))
print(f"\nDisruptive papers (CD > 0.5): {(cd_df['cd_index'] > 0.5).sum()}")
print(f"Consolidating papers (CD < -0.5): {(cd_df['cd_index'] < -0.5).sum()}")
# Most disruptive papers
top_disruptive = cd_df.nlargest(5, "cd_index")[["paper_id", "year", "cd_index", "n_citations"]]
print("\nTop 5 Most Disruptive Papers:")
print(top_disruptive.round(3).to_string(index=False))
# CD index over time trend
annual_cd = cd_df.groupby("year")["cd_index"].agg(["mean", "std"]).reset_index()
print("\nCD Index Trend Over Time:")
print(annual_cd.round(3).to_string(index=False))
# -----------------------------------------------------------------
# Visualization
# -----------------------------------------------------------------
fig, axes = plt.subplots(1, 3, figsize=(15, 5))
# CD distribution
axes[0].hist(cd_df["cd_index"], bins=25, color="steelblue", edgecolor="black")
axes[0].axvline(0, color="red", ls="--", label="Neutral")
axes[0].axvline(cd_df["cd_index"].mean(), color="orange", ls="-",
label=f"Mean={cd_df['cd_index'].mean():.2f}")
axes[0].set_xlabel("CD Index"); axes[0].set_ylabel("Frequency")
axes[0].set_title("Disruption Index Distribution")
axes[0].legend()
# CD vs. citations scatter
axes[1].scatter(cd_df["cd_index"], np.log1p(cd_df["n_citations"]),
alpha=0.4, s=20, color="steelblue")
axes[1].set_xlabel("CD Index"); axes[1].set_ylabel("ln(Citations + 1)")
axes[1].set_title("Disruption vs. Citation Impact")
# Annual CD trend
axes[2].plot(annual_cd["year"], annual_cd["mean"], "o-", color="steelblue")
axes[2].fill_between(annual_cd["year"],
annual_cd["mean"] - annual_cd["std"],
annual_cd["mean"] + annual_cd["std"],
alpha=0.2, color="steelblue")
axes[2].axhline(0, color="red", ls="--")
axes[2].set_xlabel("Year"); axes[2].set_ylabel("Mean CD Index")
axes[2].set_title("Disruption Trend Over Time")
plt.tight_layout()
plt.savefig("disruption_index.png", dpi=150, bbox_inches="tight")
plt.close()
print("\nFigure saved: disruption_index.png")
import numpy as np
import pandas as pd
import statsmodels.api as sm
import matplotlib.pyplot as plt
from scipy.stats import spearmanr
# -----------------------------------------------------------------
# Simulate paper dataset with team size metadata
# Based on Wu, Wang, and Evans (2019) findings
# -----------------------------------------------------------------
np.random.seed(42)
n_papers = 2000
team_sizes = np.random.choice(range(1, 20), n_papers,
p=np.exp(-0.15 * np.arange(19)) /
np.exp(-0.15 * np.arange(19)).sum())
# Large teams tend to have higher average citations (incremental work)
# Small teams more likely to be disruptive but higher variance
base_cit = 10
citations = np.array([
int(np.random.lognormal(
np.log(base_cit) + 0.03 * ts, # slight positive team effect on mean
max(1.5 - 0.05 * ts, 0.5) # smaller variance for larger teams
))
for ts in team_sizes
])
# Disruption index (anticorrelated with team size)
disruption = np.array([
np.random.normal(0.3 - 0.02 * ts, 0.3)
for ts in team_sizes
])
disruption = np.clip(disruption, -1, 1)
# Novel work indicator (1 = novel combination)
novelty = np.where(np.random.random(n_papers) < np.exp(-0.1 * team_sizes) * 0.4, 1, 0)
df_team = pd.DataFrame({
"team_size": team_sizes,
"citations": citations,
"disruption": disruption,
"novel": novelty,
"year": np.random.randint(2010, 2023, n_papers),
})
# -----------------------------------------------------------------
# Analysis: team size effects
# -----------------------------------------------------------------
# OLS: citations ~ log(team_size) + year
df_team["ln_team"] = np.log(df_team["team_size"])
df_team["ln_cit"] = np.log1p(df_team["citations"])
X = sm.add_constant(df_team[["ln_team", "year"]])
model = sm.OLS(df_team["ln_cit"], X).fit(cov_type="HC3")
print("=== Citations ~ Team Size (OLS) ===")
print(model.summary().tables[1])
# Spearman correlation: team size vs. disruption
r_d, p_d = spearmanr(df_team["team_size"], df_team["disruption"])
print(f"\nSpearman(team_size, disruption): r={r_d:.3f}, p={p_d:.4f}")
# Binned analysis: team size categories
df_team["team_cat"] = pd.cut(df_team["team_size"],
bins=[0, 3, 7, 12, 20],
labels=["Solo/Duo (1-3)", "Small (4-7)",
"Medium (8-12)", "Large (13-20)"])
team_summary = df_team.groupby("team_cat", observed=True).agg(
n_papers=("citations", "count"),
mean_cit=("citations", "mean"),
median_cit=("citations", "median"),
mean_disruption=("disruption", "mean"),
novel_rate=("novel", "mean"),
).reset_index()
print("\n=== Team Size Category Analysis ===")
print(team_summary.round(3).to_string(index=False))
# -----------------------------------------------------------------
# Visualization
# -----------------------------------------------------------------
fig, axes = plt.subplots(2, 2, figsize=(12, 10))
# Citations by team size bin (boxplot)
grouped_cit = [df_team[df_team["team_cat"] == cat]["citations"].values
for cat in df_team["team_cat"].cat.categories]
axes[0, 0].boxplot(grouped_cit, labels=df_team["team_cat"].cat.categories,
showfliers=False, patch_artist=True)
axes[0, 0].set_title("Citation Counts by Team Size")
axes[0, 0].set_ylabel("Citations")
axes[0, 0].set_xticklabels(df_team["team_cat"].cat.categories,
rotation=15, ha="right")
# Disruption by team size bin (violin plot alternative)
disruption_groups = [df_team[df_team["team_cat"] == cat]["disruption"].values
for cat in df_team["team_cat"].cat.categories]
axes[0, 1].boxplot(disruption_groups, labels=df_team["team_cat"].cat.categories,
showfliers=False, patch_artist=True)
axes[0, 1].axhline(0, color="red", ls="--")
axes[0, 1].set_title(f"Disruption Index by Team Size\n(r={r_d:.2f})")
axes[0, 1].set_ylabel("CD Index")
axes[0, 1].set_xticklabels(df_team["team_cat"].cat.categories,
rotation=15, ha="right")
# Scatter: team size vs. disruption
axes[1, 0].scatter(df_team["team_size"], df_team["disruption"],
alpha=0.2, s=10, c="steelblue")
# Running mean
ts_sorted = df_team.sort_values("team_size")
running_mean = ts_sorted.groupby("team_size")["disruption"].mean()
axes[1, 0].plot(running_mean.index, running_mean.values, "r-", lw=2)
axes[1, 0].set_xlabel("Team Size"); axes[1, 0].set_ylabel("CD Index")
axes[1, 0].set_title("Team Size vs. Disruption")
# Novelty rate by team size
axes[1, 1].bar(team_summary["team_cat"],
team_summary["novel_rate"] * 100,
color="green", edgecolor="black", alpha=0.7)
axes[1, 1].set_title("Novel Combination Rate by Team Size")
axes[1, 1].set_ylabel("Novel Papers (%)")
axes[1, 1].set_xticklabels(team_summary["team_cat"], rotation=15, ha="right")
plt.tight_layout()
plt.savefig("team_size_effects.png", dpi=150, bbox_inches="tight")
plt.close()
print("\nFigure saved: team_size_effects.png")
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
# -----------------------------------------------------------------
# Simulate annual citation histories for 200 papers
# Most have normal decay; a few are sleeping beauties
# -----------------------------------------------------------------
np.random.seed(42)
n_papers_sb = 200
max_age = 25 # years since publication
def simulate_normal_citation_history(peak_age=3, peak_cit=50, decay=0.3, n_years=25):
"""Simulate a normal citation trajectory (rise then fall)."""
ages = np.arange(n_years)
cit = peak_cit * np.exp(-0.5 * ((ages - peak_age) / decay)**2)
cit = np.maximum(cit + np.random.normal(0, 2, n_years), 0).astype(int)
return cit
def simulate_sleeping_beauty(dormancy=15, awakening_year=18, n_years=25):
"""Simulate a sleeping beauty: dormant then sudden awakening."""
cit = np.zeros(n_years, dtype=int)
cit[:dormancy] = np.random.randint(0, 3, dormancy) # near-zero before awakening
wake_age = min(awakening_year, n_years - 1)
for t in range(wake_age, n_years):
cit[t] = int(5 * np.exp(0.4 * (t - wake_age)) + np.random.normal(0, 3))
return np.maximum(cit, 0)
# Generate citation histories
cit_histories = []
for i in range(n_papers_sb):
if i < 5: # first 5 are sleeping beauties
cit_histories.append(simulate_sleeping_beauty(
dormancy=np.random.randint(10, 18),
awakening_year=np.random.randint(15, 22)
))
else:
cit_histories.append(simulate_normal_citation_history(
peak_age=np.random.randint(2, 7),
peak_cit=np.random.randint(5, 100),
decay=np.random.uniform(0.2, 0.8)
))
cit_matrix = np.array(cit_histories)
# -----------------------------------------------------------------
# Sleeping beauty score: B * L
# B = (c_max - c_t0) / (t_max - t0)
# L = dormancy length
# -----------------------------------------------------------------
def sleeping_beauty_score(cit_history, dormancy_threshold=5):
"""Compute sleeping beauty score for a citation trajectory.
Args:
cit_history: array of annual citations
dormancy_threshold: max citations during sleeping period
Returns:
score, beauty_length, awakening_year
"""
n = len(cit_history)
c_max = cit_history.max()
t_max = int(np.argmax(cit_history))
c_0 = cit_history[0]
# Find dormancy: longest initial period with citations <= threshold
dormancy_end = 0
for t in range(n):
if cit_history[t] <= dormancy_threshold:
dormancy_end = t
else:
if t > dormancy_end + 1:
break
beauty_length = dormancy_end
if t_max <= beauty_length or (t_max - 0) == 0:
return 0.0, 0, t_max
B = (c_max - c_0) / max(t_max, 1)
score = B * beauty_length
return score, beauty_length, t_max
sb_scores = []
for i, hist in enumerate(cit_histories):
score, length, awaken = sleeping_beauty_score(hist)
total_cit = hist.sum()
sb_scores.append({
"paper_id": i,
"is_sleeping_beauty": i < 5,
"sb_score": score,
"dormancy_length": length,
"awakening_year": awaken,
"total_citations": total_cit,
"peak_citations": hist.max(),
})
sb_df = pd.DataFrame(sb_scores)
print("=== Sleeping Beauty Detection ===")
print(f"Top 10 SB scores:")
print(sb_df.nlargest(10, "sb_score")[
["paper_id", "is_sleeping_beauty", "sb_score", "dormancy_length",
"total_citations"]].round(2).to_string(index=False))
# How many true sleeping beauties in top 10?
top10_sb = sb_df.nlargest(10, "sb_score")["is_sleeping_beauty"].sum()
print(f"\nTrue sleeping beauties in top 10: {top10_sb}/5")
# -----------------------------------------------------------------
# Visualization
# -----------------------------------------------------------------
fig, axes = plt.subplots(1, 2, figsize=(12, 5))
# Citation histories
colors_sb = {True: "red", False: "steelblue"}
for i in range(n_papers_sb):
is_sb = sb_df.loc[i, "is_sleeping_beauty"]
alpha = 0.7 if is_sb else 0.05
lw = 2 if is_sb else 0.5
axes[0].plot(cit_histories[i], color=colors_sb[is_sb],
alpha=alpha, lw=lw)
axes[0].set_xlabel("Years Since Publication"); axes[0].set_ylabel("Annual Citations")
axes[0].set_title("Citation Trajectories\n(red = sleeping beauties)")
# SB score distribution
axes[1].scatter(sb_df["dormancy_length"],
sb_df["peak_citations"],
c=sb_df["sb_score"], cmap="hot", s=30, alpha=0.7)
sb_true = sb_df[sb_df["is_sleeping_beauty"]]
axes[1].scatter(sb_true["dormancy_length"],
sb_true["peak_citations"],
c="red", s=100, marker="*", zorder=5, label="True SBs")
axes[1].set_xlabel("Dormancy Length (years)")
axes[1].set_ylabel("Peak Citations")
axes[1].set_title("Sleeping Beauty Space")
axes[1].legend()
plt.tight_layout()
plt.savefig("sleeping_beauties.png", dpi=150, bbox_inches="tight")
plt.close()
print("\nFigure saved: sleeping_beauties.png")
import numpy as np
import pandas as pd
from scipy.stats import zscore
def compute_recombination_novelty(reference_lists, journal_assignments,
n_permutations=100):
"""Compute median z-score of journal pair co-citations.
Args:
reference_lists: list of lists; each inner list = paper's references (paper IDs)
journal_assignments: dict {paper_id: journal}
n_permutations: Monte Carlo permutations
Returns:
per-paper novelty score (negative z-score = novel)
"""
from collections import Counter
# Observed co-citation counts by journal pair
observed_pairs = Counter()
for refs in reference_lists:
journals = [journal_assignments.get(r, "Unknown") for r in refs]
journals = [j for j in journals if j != "Unknown"]
journals = list(set(journals)) # unique journals in reference list
for i in range(len(journals)):
for j in range(i + 1, len(journals)):
key = tuple(sorted([journals[i], journals[j]]))
observed_pairs[key] += 1
# Null distribution via reference list permutations
all_refs_flat = [r for refs in reference_lists for r in refs]
null_pair_counts = []
for _ in range(n_permutations):
# Shuffle journal assignments
shuffled = {pid: journal_assignments.get(pid, "Unknown")
for pid in np.random.permutation(list(journal_assignments.keys()))}
perm_pairs = Counter()
for refs in reference_lists:
journals = [shuffled.get(r, "Unknown") for r in refs]
journals = [j for j in journals if j != "Unknown"]
journals = list(set(journals))
for i in range(len(journals)):
for j in range(i + 1, len(journals)):
key = tuple(sorted([journals[i], journals[j]]))
perm_pairs[key] += 1
null_pair_counts.append(perm_pairs)
# Z-scores for each journal pair
all_pairs = set(observed_pairs.keys())
z_scores = {}
for pair in all_pairs:
obs = observed_pairs[pair]
null_vals = [null[pair] for null in null_pair_counts]
mu = np.mean(null_vals)
sigma = np.std(null_vals)
z_scores[pair] = (obs - mu) / max(sigma, 0.1)
# Paper-level median z-score
paper_novelty = []
for refs in reference_lists:
journals = [journal_assignments.get(r, "Unknown") for r in refs]
journals = [j for j in journals if j != "Unknown"]
journals = list(set(journals))
paper_z = []
for i in range(len(journals)):
for j in range(i + 1, len(journals)):
key = tuple(sorted([journals[i], journals[j]]))
if key in z_scores:
paper_z.append(z_scores[key])
paper_novelty.append(np.median(paper_z) if paper_z else 0.0)
return paper_novelty, z_scores
# Demonstrate
np.random.seed(42)
n_jrnl_pool = 20
journals = [f"J{i:02d}" for i in range(n_jrnl_pool)]
n_papers_r = 50
ref_lists = [list(np.random.choice(range(100), size=np.random.randint(5, 15),
replace=False))
for _ in range(n_papers_r)]
j_assign = {i: journals[i % n_jrnl_pool] for i in range(100)}
paper_novelty, pair_z = compute_recombination_novelty(ref_lists, j_assign,
n_permutations=20)
print(f"Mean paper novelty (median z): {np.mean(paper_novelty):.3f}")
print(f"Novel papers (z < -0.5): {sum(z < -0.5 for z in paper_novelty)}/{n_papers_r}")
import numpy as np
import pandas as pd
import scipy.optimize as opt
def estimate_matthew_effect(citation_histories, time_window=5):
"""Estimate preferential attachment exponent.
Fit: delta_c ~ c(t)^beta using log-linear regression.
"""
pairs = []
for hist in citation_histories:
for t in range(len(hist) - time_window):
c_t = hist[t]
delta_c = hist[t + time_window] - hist[t]
if c_t > 0 and delta_c > 0:
pairs.append((np.log(c_t), np.log(delta_c)))
if len(pairs) < 10:
return None
ln_c, ln_delta = zip(*pairs)
ln_c = np.array(ln_c); ln_delta = np.array(ln_delta)
# OLS
A = np.column_stack([np.ones_like(ln_c), ln_c])
result = np.linalg.lstsq(A, ln_delta, rcond=None)
alpha, beta = result[0]
print(f"Matthew effect: ln(Δc) = {alpha:.3f} + {beta:.3f}·ln(c)")
print(f"Beta > 1: super-linear preferential attachment = {beta > 1}")
return {"alpha": alpha, "beta": beta, "n_pairs": len(pairs)}
# Use cit_histories from sleeping beauty step
matthew = estimate_matthew_effect(cit_histories[:50], time_window=3)
| Problem | Cause | Fix |
|---|---|---|
| CD = NaN for many papers | No citing papers (zero forward citations) | Expected for recent papers; filter to age ≥ 5 years |
| All CD ≈ 0 | Very sparse citation graph | Need longer time window; check graph has sufficient edges |
| Sleeping beauty false positives | Threshold too low | Tune dormancy_threshold to 10% of field median citations |
| Recombination z-score undefined | Only 1 journal in references | Skip single-journal papers for pair analysis |
| Matthew effect beta > 2 | Outlier highly-cited papers | Winsorize citations at 99th percentile |
| Team size analysis confounded by field | Different publishing norms | Include field fixed effects in regression |
import numpy as np
import pandas as pd
def rao_stirling_diversity(field_shares):
"""Compute Rao-Stirling diversity: sum_ij d_ij * p_i * p_j.
Args:
field_shares: array of proportions (sum to 1)
Returns:
diversity index
"""
n = len(field_shares)
# Assume all fields equally distant (d_ij = 1 for i≠j)
rs = 0.0
for i in range(n):
for j in range(i + 1, n):
rs += 2 * field_shares[i] * field_shares[j] # d_ij = 1
return rs
np.random.seed(42)
n_pap = 200
n_fields = 8
ref_field_distributions = np.random.dirichlet(np.ones(n_fields) * 0.5, n_pap)
diversity = np.array([rao_stirling_diversity(shares)
for shares in ref_field_distributions])
print(f"Mean Rao-Stirling diversity: {diversity.mean():.3f}")
print(f"High diversity papers (RS > 0.8): {(diversity > 0.8).sum()}/{n_pap}")
import numpy as np
import pandas as pd
import networkx as nx
def knowledge_flow_network(citation_df, source_field_col="source_field",
target_field_col="target_field"):
"""Build directed knowledge flow network between disciplines.
Args:
citation_df: DataFrame with citing and cited paper fields
Returns:
G: weighted directed NetworkX graph
"""
G = nx.DiGraph()
flow_counts = citation_df.groupby(
[source_field_col, target_field_col]).size().reset_index(name="weight")
for _, row in flow_counts.iterrows():
src, tgt, w = row[source_field_col], row[target_field_col], row["weight"]
if src != tgt:
if G.has_edge(src, tgt):
G[src][tgt]["weight"] += w
else:
G.add_edge(src, tgt, weight=w)
return G
fields = ["CS", "Physics", "Biology", "Chemistry", "Math", "Medicine"]
np.random.seed(42)
n_cit = 500
# Citations from papers in one field to papers in another
cit_df = pd.DataFrame({
"source_field": np.random.choice(fields, n_cit,
p=[0.30, 0.20, 0.20, 0.15, 0.10, 0.05]),
"target_field": np.random.choice(fields, n_cit),
})
cit_df = cit_df[cit_df["source_field"] != cit_df["target_field"]]
G_flow = knowledge_flow_network(cit_df)
print(f"Knowledge flow network: {G_flow.number_of_nodes()} fields, "
f"{G_flow.number_of_edges()} directed edges")
# Identify biggest knowledge importers (in-degree weighted)
in_strength = {n: sum(d["weight"] for _, _, d in G_flow.in_edges(n, data=True))
for n in G_flow.nodes()}
print("\nKnowledge importers (in-strength):")
for f, s in sorted(in_strength.items(), key=lambda x: x[1], reverse=True):
print(f" {f}: {s}")