Network topology shapes consensus and vulnerability in LLM populations

Roberto Passaro; Ariel Flint; Romualdo Pastor-Satorras; Andrea Baronchelli; Luca Maria Aiello

physics.soc-phOct 22, 2026

Network topology shapes consensus and vulnerability in LLM populations

Authors: Roberto Passaro, Ariel Flint, Romualdo Pastor-Satorras, Andrea Baronchelli, Luca Maria Aiello

Organizations: IT University of Copenhagen, Denmark. · Pioneer Centre for AI, Copenhagen, Denmark. · Department of Mathematics, City St George’s, University of London, UK. · Departament de Física, Universitat Politècnica de Catalunya, Barcelona, Spain.

Abstract

As large language models (LLMs) are deployed in interacting populations, their collective behaviour depends not only on the models but also on who interacts with whom. Studies of convention formation in LLM populations have mainly tested well-mixed interactions. We place policies extracted from four LLMs on fully connected, random regular, and heterogeneous networks, with $10^3$ to $10^5$ agents playing the naming game. Topology affects how quickly consensus emerges, sometimes whether it emerges at all, and, in rare cases, which convention wins. Sparsest networks slow consensus up to twelvefold. Yet, by preserving correlations between persistent neighbours, they restore consensus in some cases that fail to converge under well-mixed interactions. Networks also govern vulnerability to adversarial attack. We measure the critical mass of committed agents needed to overturn an established convention. Tipping thresholds follow the same ordering across models and conditions: hubs $\leq$ random $\leq$ low-degree nodes. On the most heterogeneous topology, median critical masses are 3%, 16%, and 35%, respectively; targeting hubs requires one-sixth as many committed agents as random placement. For most models, networked populations are easier to tip than well-mixed ones, so well-mixed estimates can overstate robustness. Assessing the safety of multi-agent AI requires mapping who interacts with whom.

Figures & tables

Topology	Full name	Degree distribution	$\langle k\rangle$
FC	Fully Connected	regular, $k=N-1$	$N-1$
UCM 2.5	Uncorrelated Configuration Model	power-law, $\gamma=2.5$	8.64
UCM 5.0	Uncorrelated Configuration Model	power-law, $\gamma=5.0$	4.76
RRN	Random Regular Network	regular, $k=4$	4.00

Table 1: Network topologies and their degree statistics at

N=10000

Figure 1: Mode convergence time under edge activation at $N=10000$ , one panel per model. Each line connects the four topology values FC, UCM 2.5 , UCM 5.0 , RRN; colour and marker shape jointly identify the word pair and are consistent across panels.

Figure 2: Exposure and success by degree for GPT on UCM 2.5 at $N=10000$ , one panel per word pair. Axes are population rounds $t$ and the node’s own cumulative interaction count $n$ . The straight grey lines mark the range of possible values for nodes in each of the five degree bins, according to Eq. 2 . Markers are the mean across runs at which each bin’s cumulative success rate $S_{k}$ first reaches $0.6$ , $0.7$ and $0.8$ : red circles for the quenched network, blue squares for the annealed control. The line between the markers is a linear fit (solid quenched, dashed annealed). The top-left corner of each panel reports the tilt of the quenched contour at the three levels, given by the ratio $t^{*}(\text{low-degree nodes})/t^{*}(\text{hubs})$ between the population rounds at which the lowest ( $k=4$ –5) and the highest ( $k=36$ –100) degree bins first reach $0.6$ , $0.7$ and $0.8$ . The bottom-right panel shows the distribution of the tilts across word pairs, one row per success level $S$ (labelled on the left), for the quenched (red circles) and the annealed network (blue squares), with the median as a vertical bar; the bottom axis gives the tilt, the top axis the normalised tilt $\tau$ (Eq. 3 ), and the dashed lines mark the vertical ( $\tau=0$ ) and horizontal ( $\tau=1$ ) contours, the dotted line their midpoint ( $\tau=0.5$ ).

Pair	Topology	$N=1000$	$N=5000$	$N=10000$	$N=100000$
young/old	FC (well-mixed)	0.00	0.00	0.00	0.00
	UCM 2.5	0.07	0.00	0.00	0.00
	UCM 5.0	1.00	1.00	1.00	1.00
	RRN	1.00	1.00	1.00	1.00
more/less	FC (well-mixed)	0.00	0.00	0.00	0.00
	UCM 2.5	0.00	0.00	0.00	0.00

Table 2: Consensus fraction for the two mixed-state Llama pairs across topologies and population sizes. 1000 rounds for condition.

Topology	Placement	Mean	Median
UCM 2.5	hub	0.090	0.026
	random	0.211	0.160
	low-degree	0.366	0.349
UCM 5.0	hub	0.145	0.064
	random	0.190	0.128
	low-degree	0.213	0.154

Table 3: Mean / median critical fraction

p^{*}

by placement strategy, UCM only,

N=10000

. Both commitment directions pooled (88 values per cell).

Panel of Figure 3 — Figure 3: Critical committed fraction $p^{*}$ at $N=10000$ , by model and commitment direction. (a) $p^{*}$ by placement of the committed minority on UCM 2.5 . (b) $p^{*}$ by topology under random placement. In both panels, large symbols are the median over the 11 word pairs and small symbols the individual pairs; the $p^{*}=0$ conditions are excluded from the plot.

Model	Direction	RRN	UCM 5.0	UCM 2.5	FC
Llama-3.1-70B	strong	0.307	0.282	0.256	0.237
	weak	0.003	0.002	0.001	0.002
GPT-4o	strong	0.122	0.134	0.154	0.166
	weak	0.051	0.064	0.090	0.102
QwQ-32B	strong	0.422	0.378	0.371	0.435
	weak	0.070	0.083	0.141	0.230

Table 4: Median critical fraction

p^{*}

over the word pairs, by model, commitment direction and topology under random placement,

N=10000

. Strong and weak as in Figure 3 ; the

p^{*}=0

conditions are excluded. Statistical tests are in Table A7 .

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Model	$N$	Rule	$Q$	$p$	$W$	$L$	$p(L)$
QwQ-32B	1000	Edge	31.4	$<.001$	0.953	327	$<.001$
QwQ-32B	1000	Node	31.4	$<.001$	0.953	327	$<.001$
QwQ-32B	5000	Edge	30.7	$<.001$	0.930	326	$<.001$
QwQ-32B	5000	Node	31.4	$<.001$	0.950	327	$<.001$
QwQ-32B	10000	Edge	30.7	$<.001$	0.930	326	$<.001$
QwQ-32B	10000	Node	31.4	$<.001$	0.950	327	$<.001$

Appendix

Table A1 : Friedman and Page’s

L

tests on the mode of reached consensus of the topology ordering FC

<

UCM 2.5

<

UCM 5.0

<

RRN. Performed by model, activation rule and population size.

Figure A1 : Same as Figure 1 , but for the mean convergence time under edge activation at $N=10000$ .

Model	$n$	$Q$	$p$	$W$	$L$	$p(L)$
QwQ-32B	11	154.8	$<.001$	0.938	16170	$<.001$
GPT-4o	11	138.0	$<.001$	0.837	15456	$<.001$
Phi-4	11	123.4	$<.001$	0.748	15427	$<.001$
Llama-3.1-70B	9	2.0	1.000	0.015	10420	0.477

Appendix

Table A2 : Friedman and Page’s

L

tests of convergence time on RRN as a function of mode degree

K\in\{4,6,8,10,12,16,20,24,32,48,64,96,128,192,256\}

against the FC baseline,

N=10000

, edge activation.

n=11

word pairs for QwQ-32B, GPT-4o and Phi-4;

n=9

for Llama-3.1-70B, whose two mixed-state pairs (§ 3.1.3 ) are excluded.

Figure A2 : Convergence time on random regular networks (RRN) as a function of the degree $K$ , relative to the fully connected (FC) population. For every model–word-pair combination, we take the mode of the convergence time of RRN at degree $K$ , $T(K)$ , and divide it by $T_{\mathrm{FC}}$ . A ratio above $1$ means the network is slower than FC. The dashed line at $1$ marks the FC baseline. $N=10000$ , edge activation, $K\in\{4,6,8,10,12,16,20,24,32,48,64,96,128,192,256\}$ .

Figure A3 : Mean convergence time as a function of $N$ for the two mixed-state Llama-3.1-70B pairs {young/old} (left) and {more/less} (right), edge activation, on FC (black diamonds), UCM 2.5 , UCM 5.0 (blue open squares) and RRN with $K=4$ (orange circles). Error bars are $95\%$ bootstrap confidence intervals over $200$ runs per point, with a fresh network per run. Runs last at most $t_{\mathrm{max}}=10000$ rounds. Each curve starts at $N=10$ ( $N=25$ for UCM 2.5 and UCM 5.0 ), and stops at the last $N$ at which all runs reach consensus within $t_{\mathrm{max}}$ .

Figure A4 : Mean convergence time on RRN as a function of the degree $K$ , for the two mixed-state Llama-3.1-70B pairs {young/old} (left) and {more/less} (right), at $N=100$ (top) and $N=200$ (bottom), edge activation. Orange circles are quenched networks, blue open squares the annealed control; error bars are $95\%$ bootstrap confidence intervals over $200$ runs per condition. The dashed line and the grey band give the FC mean and its confidence interval at the same $N$ . Note the logarithmic vertical axis, and that the rightmost point of each panel, $K=N-1$ , is the fully connected graph.

Pair	Topology	$N=$ 1000$$	$N=$ 5000$$	$N=$ 10,000$$	$N=$ 100,000$$
young/old	UCM 2.5	593.5 (0.07)	–	–	–
	UCM 5.0	108	167	193.5	267
	RRN	64	75	77	79
more/less	RRN	508 (0.10)	–	–	–

Appendix

Table A3 : Median convergence time (population rounds) of the two mixed-state Llama-3.1-70B pairs, edge activation; consensus fraction in parentheses when below

1

. A dash marks conditions in which no run reaches consensus. Consensus fractions for every topology are in Table 2 .

Pair	$N$	$T(4)$	$T_{\mathrm{FC}}$	$T_{\mathrm{FC}}/T(4)$	$K_{50}$	$K_{90}$	$K_{\mathrm{FC}}$
young/old	100	37.5	186.8	5.0	10.5	36.3	80
more/less	100	41.7	954.5	22.9	8.1	35.7	48
young/old	200	51.1	2160	42.3	13.2	45.6	128
more/less	200	87.0	188700	2170	8.8	34.3	199

Appendix

Table A4 : RRN degree sweep for the two mixed-state Llama-3.1-70B pairs, edge activation,

200

runs per condition. Tested degrees:

K\in\{4,6,8,12,16,24,32,48,64,80,96,99\}

N=100

and

K\in\{4,6,8,12,16,24,32,48,64,96,128,160,199\}

N=200

T(4)

and

T_{\mathrm{FC}}

are mean convergence times (population rounds) at

K=4

and on FC.

K_{50}

and

K_{90}

are the degrees at which

50\%

and

90\%

of the increase from

\log T(4)

\log T_{\mathrm{FC}}

is covered.

K_{\mathrm{FC}}

is the smallest

K

from which the RRN mean stays inside the

95\%

confidence interval of FC.

Model	Pair	$\tau$ ( $S=0.6$ )	$\tau$ ( $S=0.8$ )	$\Delta t^{*}_{\text{low}}$	$\Delta t^{*}_{\text{hub}}$
QwQ-32B	White/Black	$-0.18$	$0.08$	$+108\%$	$-17\%$
QwQ-32B	she/he	$-0.27$	$0.08$	$+160\%$	$-6\%$
QwQ-32B	wife/husband	$-0.18$	$0.14$	$+200\%$	$+12\%$
QwQ-32B	woman/man	$-0.03$	$0.15$	$+111\%$	$+5\%$
Phi-4	African/White	$-0.03$	$0.15$	$+98\%$	$+4\%$
GPT-4o	woman/man	$-0.002$	$0.04$	$+8\%$	$-7\%$

Appendix

Table A5 : The six conditions in which hubs reach the success level later than low-degree nodes on the quenched UCM 2.5 (

\tau<0

S=0.6

N=

10,000$$ .

\Delta t^{*}

is the relative delay of

t^{*}

on the annealed network with respect to the quenched one, at

S=0.6

, for the lowest (

k=4

–5) and highest (

k=36

–100) degree bins.

Figure A5 : Exposure and success by degree for Llama-3.1-70B on UCM 2.5 at $N=10000$ , one panel per word pair. Axes, markers and the bottom-right panel as in Figure 2 ; the top-left corner of each panel reports the quenched tilt at $S=0.6$ , $0.7$ and $0.8$ , with a dash for a level that the word pair does not cross. The mixed-state pairs {young/old} and {more/less} are omitted, as is {Mexican/American}, whose initial success ( $0.82$ ) lies above all three levels; for {gay/straight} (initial success $0.76$ ) only $S=0.8$ is shown.

Figure A6 : Exposure and success by degree for QwQ-32B on UCM 2.5 at $N=10000$ , one panel per word pair. Axes, markers and the bottom-right panel as in Figure 2 ; the top-left corner of each panel reports the quenched tilt at $S=0.6$ , $0.7$ and $0.8$ , with a dash for a level that the word pair does not cross.

Figure A7 : Exposure and success by degree for Phi-4 on UCM 2.5 at $N=10000$ , one panel per word pair. Axes, markers and the bottom-right panel as in Figure 2 ; the top-left corner of each panel reports the quenched tilt at $S=0.6$ , $0.7$ and $0.8$ , with a dash for a level that the word pair does not cross. The pairs {gay/straight} and {more/less} are omitted, since their initial success ( $1.00$ and $0.94$ ) lies above all three levels.

Model	Direction	$n$	$Q$	$p$	$W$	$L$	$p(L)$
UCM 2.5
QwQ-32B	strong	11	22.0	$<.001$	1.000	154	$<.001$
	weak	11	20.0	$<.001$	0.909	152	$<.001$
GPT-4o	strong	11	22.0	$<.001$	1.000	154	$<.001$
	weak	11	22.0	$<.001$	1.000	154	$<.001$
Phi-4	strong	11	22.0	$<.001$	1.000	154	$<.001$

Appendix

Table A6 : Friedman and Page’s

L

tests on the critical fraction

p^{*}

of the placement ordering hub

<

random

<

low-degree nodes, by UCM exponent, model and commitment direction,

N=10000

. The block is the word pair;

p^{*}

is rounded to the bisection resolution (

0.01

) before ranking. Pairs with

p^{*}=0

in any placement are excluded (

n=9

for Llama-3.1-70B in the weak direction).

Model	Direction	$n$	$Q$	$p$	$W$	$L$	$p(L)$
QwQ-32B	strong	11	10.5	0.015	0.319	286	0.138
	weak	11	13.1	0.004	0.396	305	$<.001$
GPT-4o	strong	11	32.4	$<.001$	0.982	329	$<.001$
	weak	11	33.0	$<.001$	1.000	330	$<.001$
Phi-4	strong	11	30.4	$<.001$	0.921	327	$<.001$
	weak	11	33.0	$<.001$	1.000	330	$<.001$

Appendix

Table A7 : Friedman and Page’s

L

tests on the critical fraction

p^{*}

of the topology ordering RRN

<

UCM 5.0

<

UCM 2.5

<

FC under random placement, by model and commitment direction,

N=10000

. The block is the word pair;

p^{*}

is rounded to the bisection resolution (

0.01

) before ranking. Pairs with

p^{*}=0

on any topology are excluded (

n=9

for Llama-3.1-70B in the weak direction). Page’s

L

is one-sided in the predicted direction, so

p(L)=1.000

for Llama-3.1-70B in the strong direction reflects the reversed ordering.

Pair	random	random	hub	random	low	hub	random	low
	FC	RRN	UCM 2.5			UCM 5.0
gay/straight	0.992	0.992	0.977	0.992	1.000	0.992	0.992	1.000
	0.013	0.013	0.001	0.013	0.019	0.003	0.013	0.013
young/old	0.992	0.992	0.977	0.992	1.000	0.992	0.992	1.000
	0.003	0.006	$<0.001$	0.001	0.002	0.001	0.003	0.006
wife/husband	0.992	0.992	0.977	0.992	1.000	0.992	0.992	1.000

Appendix

Table A8 : Critical fraction

p^{*}

for the three QwQ-32B pairs whose convention is overturned only by a near-complete committed minority, by topology and placement,

N=

10,000$$ . For each pair, the first row is the direction requiring

p^{*}\geq 0.95

, the second the opposite direction.

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