Spectral properties and the accuracy of mean-field approaches for epidemics on correlated networks

TL;DR

This paper compares stochastic simulations and mean-field theories for epidemic thresholds on correlated networks, revealing how spectral properties influence the accuracy of these theoretical approaches.

Contribution

It provides a detailed analysis of the spectral factors affecting mean-field predictions for epidemic thresholds on correlated networks with power-law degree distributions.

Findings

QMF and PQMF accurately predict thresholds for γ<5/2

Correlations do not alter the threshold vanishing phenomenon

Spectral properties explain the accuracy variations of mean-field theories

Abstract

We present a comparison between stochastic simulations and mean-field theories for the epidemic threshold of the susceptible-infected-susceptible (SIS) model on correlated networks (both assortative and disassortative) with power-law degree distribution $P(k)\sim k^{-\gamma}$. We confirm the vanishing of the threshold regardless of the correlation pattern and the degree exponent $\gamma$. Thresholds determined numerically are compared with quenched mean-field (QMF) and pair quenched mean-field (PQMF) theories. Correlations do not change the overall picture: QMF and PQMF provide estimates that are asymptotically correct for large size for $\gamma<5/2$, while they only capture the vanishing of the threshold for $\gamma>5/2$, failing to reproduce quantitatively how this occurs. For a given size, PQMF is more accurate. We relate the variations in the accuracy of QMF and PQMF predictions with changes in the spectral properties (spectral gap and localization) of standard and modified adjacency matrices, which rule the epidemic prevalence near the transition point, depending on the theoretical framework. We also show that, for $\gamma<5/2$, while QMF provides an estimate of the epidemic threshold that is asymptotically exact, it fails to reproduce the singularity of the prevalence around the transition.

Tables

Table 1. Table 1: Properties of the set with 99 networks of distinct types. We show the network size N 𝑁 N , the average degree ⟨ k ⟩ delimited-⟨⟩ 𝑘 \langle{k}\rangle , the Pearson coefficient P 𝑃 P , the IPRs of both A i ​ j subscript 𝐴 𝑖 𝑗 A_{ij} and critical B i ​ j subscript 𝐵 𝑖 𝑗 B_{ij} matrices, the spectral gap of A i ​ j subscript 𝐴 𝑖 𝑗 A_{ij} , the thresholds of simulations ( λ c subscript 𝜆 𝑐 \lambda_{c} ), QMF ( λ c QMF superscript subscript 𝜆 𝑐 QMF \lambda_{c}^{\text{QMF}} ) and PQMF ( λ c PQMF superscript subscript 𝜆 𝑐 PQMF \lambda_{c}^{\text{PQMF}} ) theories.

Network	$N$	$⟨k⟩$	$P$	IPR${}_{\text{A}}$	IPR${}_{\text{B}}$	$\Delta {\Lambda }_{\text{A}}^{1,2}$	${\lambda }_c$	${\lambda }_c^{\text{QMF}}$	${\lambda }_c^{\text{PQMF}}$
Karate club	$34$	$4.59$	$-0.476$	$0.0730$	$0.0649$	$1.75$	$0.235$	$0.149$	$0.181$
Radoslaw Email	$167$	$38.9$	$-0.295$	$0.0133$	$0.0132$	$45.2$	$0.0191$	$0.0165$	$0.0168$
Spanish B	$12,643$	$8.70$	$-0.290$	$0.0246$	$0.0174$	$47.5$	$0.0105$	$0.00897$	$0.00951$
Spanish A	$11,558$	$7.45$	$-0.282$	$0.0190$	$0.0150$	$57.5$	$0.0113$	$0.00985$	$0.0103$
US Air Transportation	$500$	$11.9$	$-0.268$	$0.0176$	$0.0173$	$29.3$	$0.0251$	$0.0208$	$0.0214$
Little Rock Lake	$183$	$26.6$	$-0.266$	$0.0148$	$0.0145$	$14.6$	$0.0291$	$0.0242$	$0.0249$
Japanese	$2,698$	$5.93$	$-0.259$	$0.0296$	$0.0214$	$21.7$	$0.0281$	$0.0233$	$0.0250$
English	$7,377$	$12.0$	$-0.237$	$0.0120$	$0.0103$	$65.3$	$0.0101$	$0.00914$	$0.0094$
French	$8,308$	$5.74$	$-0.233$	$0.0351$	$0.0200$	$26.3$	$0.0197$	$0.0165$	$0.0179$
Jung	$6,120$	$16.4$	$-0.233$	$0.0478$	$0.0335$	$46.9$	$0.00810$	$0.00703$	$0.00743$
JDK	$6,434$	$16.7$	$-0.223$	$0.0484$	$0.0341$	$47.6$	$0.00810$	$0.00696$	$0.00737$
Political blogs	$1,222$	$27.4$	$-0.221$	$0.00701$	$0.00687$	$14.1$	$0.0153$	$0.0135$	$0.0137$
Internet	$22,963$	$4.22$	$-0.198$	$0.0146$	$0.0116$	$18.4$	$0.0165$	$0.0140$	$0.0148$
AS Caida	$26,475$	$4.03$	$-0.195$	$0.0240$	$0.0140$	$18.5$	$0.0173$	$0.0144$	$0.0157$
EU email	$224,832$	$3.02$	$-0.189$	$0.00340$	$0.00328$	$15.2$	$0.0107$	$0.00975$	$0.0101$
UC Irvine	$1,893$	$14.6$	$-0.188$	$0.00643$	$0.00608$	$28.6$	$0.0233$	$0.0208$	$0.0214$
Linux, mailing list	$24,567$	$12.9$	$-0.185$	$0.00395$	$0.00386$	$147$	$0.00490$	$0.00448$	$0.00452$
AS Oregon	$6,474$	$3.88$	$-0.182$	$0.0868$	$0.0429$	$19.1$	$0.0281$	$0.0216$	$0.0249$
Linux, soft.	$30,817$	$13.8$	$-0.175$	$0.0256$	$0.0197$	$94.1$	$0.00670$	$0.00585$	$0.00616$
Gnutella, Aug. 25, 2002	$22,663$	$4.83$	$-0.173$	$0.000815$	$0.000464$	$1.79$	$0.108$	$0.0916$	$0.104$
Les Miserables	$77$	$6.60$	$-0.165$	$0.0492$	$0.0482$	$3.05$	$0.123$	$0.0833$	$0.0919$
Petster-cats	$148,826$	$73.2$	$-0.164$	$0.00687$	$0.00635$	$405$	$0.000900$	$0.000847$	$0.000855$
C. Elegans, neural	$297$	$14.5$	$-0.163$	$0.0189$	$0.0176$	$10.1$	$0.0511$	$0.0410$	$0.0434$
Libimseti	$220,970$	$156$	$-0.139$	$0.000406$	$0.000398$	$348$	$0.00110$	$0.00106$	$0.00106$
David Copperfield	$112$	$7.59$	$-0.129$	$0.0473$	$0.0397$	$7.57$	$0.103$	$0.0760$	$0.0844$
Political books	$105$	$8.40$	$-0.128$	$0.0444$	$0.0419$	$0.313$	$0.133$	$0.0838$	$0.0927$
Google	$15,763$	$18.9$	$-0.122$	$0.0430$	$0.0303$	$65.1$	$0.00670$	$0.00575$	$0.00608$
Social 3	$32$	$5.00$	$-0.119$	$0.0665$	$0.0568$	$2.16$	$0.265$	$0.167$	$0.205$
Euron	$33,696$	$10.7$	$-0.116$	$0.00379$	$0.00361$	$43.9$	$0.00910$	$0.00844$	$0.00859$
Web Stanford	$255,265$	$15.2$	$-0.116$	$0.0245$	$0.0230$	$117$	$0.00250$	$0.00223$	$0.0023$
Bay Wet	$128$	$32.4$	$-0.112$	$0.0151$	$0.0147$	$25.6$	$0.0301$	$0.0252$	$0.0259$
Bay Dry	$128$	$32.9$	$-0.104$	$0.0148$	$0.0145$	$25.7$	$0.0301$	$0.0249$	$0.0256$
Gnutella, Aug. 30, 2002	$36,646$	$4.82$	$-0.104$	$0.000672$	$0.000604$	$2.19$	$0.0897$	$0.0773$	$0.0856$
Gnutella, Aug. 31, 2002	$62,561$	$4.73$	$-0.0927$	$0.000921$	$0.000731$	$1.85$	$0.0881$	$0.0759$	$0.0844$
Petster-hamster	$1,788$	$14.0$	$-0.0889$	$0.0100$	$0.00938$	$21.6$	$0.0249$	$0.0217$	$0.0223$
Petster-dogs	$426,485$	$40.1$	$-0.0884$	$0.00176$	$0.00157$	$300$	$0.00140$	$0.00135$	$0.00136$
Network Science	$379$	$4.82$	$-0.0817$	$0.0794$	$0.0705$	$2.21$	$0.230$	$0.0964$	$0.110$
AS Skitter	$1,694,616$	$13.1$	$-0.0814$	$0.00746$	$0.00709$	$251$	$0.00160$	$0.00149$	$0.00151$
Slashdot zoo	$79,116$	$11.8$	$-0.0746$	$0.00229$	$0.00218$	$42.8$	$0.00830$	$0.00767$	$0.00778$
Wikipedia, edits	$113,123$	$35.8$	$-0.0651$	$0.00295$	$0.00266$	$169$	$0.0027$	$0.00253$	$0.00255$
CiteSeer	$365,154$	$9.43$	$-0.0632$	$0.0177$	$0.0110$	$4.54$	$0.0202$	$0.0172$	$0.0183$
Cora	$23,166$	$7.70$	$-0.0553$	$0.0100$	$0.00898$	$3.66$	$0.0381$	$0.0317$	$0.0334$
Thesaurus	$23,132$	$25.7$	$-0.0477$	$0.0017$	$0.00156$	$44.7$	$0.0105$	$0.0100$	$0.0102$
DBLP, citations	$12,495$	$7.93$	$-0.0461$	$0.0282$	$0.0174$	$12.0$	$0.0277$	$0.0234$	$0.0251$
Dolphins	$62$	$5.13$	$-0.0436$	$0.0526$	$0.0493$	$1.26$	$0.231$	$0.139$	$0.164$
DBpedia	$3,915,921$	$6.42$	$-0.0427$	$0.201$	$0.0840$	$153$	$0.00190$	$0.00140$	$0.00180$
Wikipedia, pages	$2,070,367$	$40.9$	$-0.0418$	$0.00477$	$0.00294$	$194$	$0.00130$	$0.00124$	$0.00127$
Epinions	$75,877$	$10.7$	$-0.0406$	$0.00219$	$0.00211$	$79.8$	$0.00570$	$0.00543$	$0.00548$
Slashdot	$51,083$	$4.56$	$-0.0347$	$0.144$	$0.0347$	$20.2$	$0.0219$	$0.0170$	$0.0201$
Hep-Th, citations	$27,400$	$25.7$	$-0.0305$	$0.00931$	$0.00705$	$20.3$	$0.00990$	$0.00899$	$0.00922$
S 838	$512$	$3.20$	$-0.0300$	$0.179$	$0.0340$	$0.889$	$0.382$	$0.200$	$0.297$
Gowalla	$196,591$	$9.67$	$-0.0293$	$0.0180$	$0.00764$	$60.0$	$0.00650$	$0.00585$	$0.00609$
Amazon, Mar. 12, 2003	$400,727$	$11.7$	$-0.0203$	$0.118$	$0.0381$	$7.97$	$0.0273$	$0.0178$	$0.0227$
Amazon, Jun. 6, 2003	$403,364$	$12.1$	$-0.0176$	$0.0891$	$0.0279$	$7.87$	$0.0252$	$0.0175$	$0.0219$
Amazon, May. 5, 2003	$410,236$	$11.9$	$-0.0169$	$0.0843$	$0.0309$	$7.79$	$0.0249$	$0.0172$	$0.0214$
Air traffic	$1,226$	$3.93$	$-0.0152$	$0.0191$	$0.0154$	$1.38$	$0.152$	$0.109$	$0.127$
Gnutella, Aug. 4, 2002	$10,876$	$7.35$	$-0.0132$	$0.00469$	$0.00377$	$4.26$	$0.0685$	$0.0586$	$0.0637$
Gnutella, Aug. 24, 2002	$26,498$	$4.93$	$-0.00778$	$0.214$	$0.0800$	$8.68$	$0.0865$	$0.0511$	$0.0700$
Hep-Ph, citations	$34,401$	$24.5$	$-0.00644$	$0.00421$	$0.00323$	$3.61$	$0.0143$	$0.0131$	$0.0133$
S 420	$252$	$3.17$	$-0.00591$	$0.0542$	$0.0172$	$0.398$	$0.400$	$0.229$	$0.320$
Amazon, May. 2, 2003	$262,111$	$6.87$	$-0.00248$	$0.106$	$0.0114$	$0.771$	$0.0605$	$0.0425$	$0.0508$
S 208	$122$	$3.10$	$-0.00201$	$0.0419$	$0.0301$	$0.475$	$0.437$	$0.244$	$0.338$
Digg	$29,652$	$5.72$	$0.00265$	$0.00457$	$0.00346$	$12.5$	$0.0369$	$0.0324$	$0.0348$
US Power grid	$4,941$	$2.67$	$0.00346$	$0.0409$	$0.0386$	$0.874$	$0.396$	$0.134$	$0.157$
Gnutella, Aug. 5, 2002	$8,842$	$7.20$	$0.0146$	$0.00931$	$0.00855$	$7.68$	$0.0505$	$0.0425$	$0.0453$
Jazz	$198$	$27.7$	$0.0202$	$0.0143$	$0.0141$	$12.6$	$0.0301$	$0.0250$	$0.0257$
Gnutella, Aug. 9, 2002	$8,104$	$6.42$	$0.0331$	$0.00782$	$0.00737$	$13.1$	$0.0409$	$0.0351$	$0.0370$
Gnutella, Aug. 8, 2002	$6,299$	$6.60$	$0.0355$	$0.00795$	$0.00752$	$13.7$	$0.0413$	$0.0352$	$0.0371$
LiveJournal	$5,189,808$	$18.8$	$0.0394$	$0.00157$	$0.00157$	$42.7$	$0.00240$	$0.00186$	$0.00186$
High school, 2012	$180$	$24.7$	$0.0464$	$0.0102$	$0.0101$	$4.49$	$0.0401$	$0.0332$	$0.0344$
Open flights	$2,905$	$10.8$	$0.0489$	$0.00963$	$0.00942$	$20.6$	$0.0181$	$0.0159$	$0.0162$
Gnutella, Aug. 6, 2002	$8,717$	$7.23$	$0.0516$	$0.0103$	$0.00957$	$3.20$	$0.0545$	$0.0447$	$0.0478$
URV email	$1,133$	$9.62$	$0.0782$	$0.00956$	$0.00865$	$3.78$	$0.0581$	$0.0482$	$0.0512$
High school, 2011	$126$	$27.1$	$0.0829$	$0.0173$	$0.0171$	$11.6$	$0.0361$	$0.0294$	$0.0304$
DBLP, collaborations	$1,137,114$	$8.83$	$0.0964$	$0.00797$	$0.00840$	$0.0594$	$0.0113$	$0.00847$	$0.00855$
MathSciNet	$332,689$	$4.93$	$0.103$	$0.0110$	$0.0103$	$1.56$	$0.0347$	$0.0277$	$0.0291$
Social 1	$67$	$4.24$	$0.103$	$0.0486$	$0.0418$	$0.975$	$0.292$	$0.179$	$0.223$
Cond-Mat, 1993-2003	$21,363$	$8.55$	$0.125$	$0.0103$	$0.00947$	$7.41$	$0.0309$	$0.0264$	$0.0275$
Protein 1	$95$	$4.48$	$0.129$	$0.0723$	$0.0670$	$0.314$	$0.384$	$0.187$	$0.232$
Cond-Mat, 1995-1999	$13,861$	$6.44$	$0.157$	$0.0163$	$0.0146$	$3.34$	$0.0509$	$0.0400$	$0.0424$
College football	$115$	$10.7$	$0.162$	$0.00977$	$0.00967$	$1.50$	$0.124$	$0.0928$	$0.102$
Cond-Mat, 1995-2003	$27,519$	$8.44$	$0.166$	$0.00917$	$0.00847$	$6.09$	$0.0293$	$0.0248$	$0.0258$
US Patents	$3,764,117$	$8.77$	$0.168$	$0.0103$	$0.0100$	$8.05$	$0.0113$	$0.00885$	$0.00899$
Facebook links	$63,392$	$25.8$	$0.177$	$0.00143$	$0.00140$	$25.8$	$0.00810$	$0.00754$	$0.00762$
Cond-Mat, 1995-2005	$36,458$	$9.42$	$0.177$	$0.00814$	$0.00761$	$12.7$	$0.0223$	$0.0195$	$0.0201$
Hep-Th, 1995-1999	$5,835$	$4.74$	$0.185$	$0.0523$	$0.0523$	$3.70$	$0.0913$	$0.0554$	$0.0587$
AstroPhys, 1993-2003	$17,903$	$22.0$	$0.201$	$0.00447$	$0.00432$	$18.9$	$0.0117$	$0.0106$	$0.0108$
Protein 2	$53$	$4.64$	$0.209$	$0.0536$	$0.0500$	$0.722$	$0.305$	$0.172$	$0.210$
Facebook wall	$43,953$	$8.30$	$0.216$	$0.00229$	$0.00214$	$7.86$	$0.0277$	$0.0252$	$0.0261$
Dublin	$410$	$13.5$	$0.226$	$0.0263$	$0.0261$	$3.62$	$0.0601$	$0.0428$	$0.0448$
Actor coll. net.	$374,511$	$80.2$	$0.226$	$0.000600$	$0.000599$	$429$	$0.00120$	$0.00118$	$0.00118$
Astrophysics	$14,845$	$16.1$	$0.228$	$0.00504$	$0.00494$	$5.64$	$0.0155$	$0.0135$	$0.0138$
PGP	$10,680$	$4.55$	$0.238$	$0.0166$	$0.0163$	$4.25$	$0.0301$	$0.0236$	$0.0243$
Hep-Th, 1993-2003	$8,638$	$5.74$	$0.239$	$0.0312$	$0.0312$	$8.03$	$0.0669$	$0.0322$	$0.0333$
Reactome	$5,973$	$48.8$	$0.241$	$0.00414$	$0.00413$	$27.2$	$0.00550$	$0.00481$	$0.00483$
Flickr	$105,722$	$43.8$	$0.247$	$0.00105$	$0.00105$	$101$	$0.00170$	$0.00162$	$0.00163$
E. Coli, transcription	$97$	$4.37$	$0.412$	$0.0854$	$0.0807$	$0.327$	$0.328$	$0.153$	$0.184$
Hep-Ph, 1993-2003	$11,204$	$21.0$	$0.630$	$0.00389$	$0.00389$	$153$	$0.00450$	$0.00408$	$0.0041$
GR-QC, 1993-2003	$4,158$	$6.46$	$0.639$	$0.0209$	$0.0209$	$7.49$	$0.0273$	$0.0219$	$0.0225$

Equations

$${\kappa }_{\text{nn}}(k)=\sum _{k^{\prime }=k_{\text{min}}}^{k_{\text{max}}}{k}^{\prime }P(k^{\prime }|k),$$

$$P(k^{\prime }|k)=P_{\text{e}}(k^{\prime })=k^{\prime }P(k^{\prime })/⟨k⟩$$

$$P_{\text{link}}(q^{\prime },q)=\frac{f(q^{\prime },q)}{f_{\text{max}}},$$

$$f(q,q^{\prime })=1+\frac{({\kappa }_{\text{nn}}(q)-⟨k{⟩}_{\text{e}})({\kappa }_{\text{nn}}(q^{\prime })-⟨k{⟩}_{\text{e}})}{⟨k{\kappa }_{\text{nn}}{⟩}_{\text{e}}-⟨k{⟩}_{\text{e}}^2},$$

$$\psi =N\frac{⟨{\rho }^2⟩-⟨\rho ⟩}{⟨\rho ⟩}.$$

$$\frac{d{\rho }_k}{dt}=-{\rho }_k+(1-{\rho }_k)\lambda \sum _{l}kP(l|k){\rho }_l,$$

$${\lambda }_c{{\rm Y}}^{(1)}=1,$$

$$C_{kl}=\frac{klP(l)}{⟨k⟩}f(l,k).$$

$$\frac{d{\rho }_i}{dt}=-{\rho }_i+\lambda (1-{\rho }_i)\sum _{j=1}^N{A}_{ij}{\rho }_j,$$

$${\lambda }_{\text{c}}^{\text{QMF}}{\Lambda }^{(1)}=1$$

$${\rho }_i=\frac{\lambda {\sum }_j{A}_{ij}{\rho }_j}{1+\lambda {\sum }_j{A}_{ij}{\rho }_j}.$$

$$\rho \simeq a_1(\lambda {\Lambda }^{(1)}-1)$$

$$a_1(N)=\frac{{\sum }_{i=1}^N{v}_i^{(1)}}{N{\sum }_{i=1}^N{\left[v_i^{(1)}\right]}^3}.$$

$$Y_4=\sum _{i=1}^N{\left[v_i^{(1)}\right]}^4.$$

$$\frac{d{\rho }_i}{dt}=-{\rho }_i+\lambda \sum _j{\varphi }_{ij}{A}_{ij}$$

$$\frac{d{\varphi }_{ij}}{dt}$$

$${\varphi }_{ij}\approx \frac{(2+\lambda ){\rho }_j-\lambda {\rho }_i}{2+2\lambda }-{\rho }_i{\rho }_j+\mathcal{O}({\rho }^3,\lambda {\rho }^2),$$

$${\rho }_i=\frac{\lambda {\sum }_i{B}_{ij}(\lambda ){\rho }_j}{1+\lambda {\sum }_i{B}_{ij}(\lambda ){\rho }_j},$$

$$B_{ij}=\frac{2+\lambda }{2\lambda +2}\frac{A_{ij}}{1+\frac{{\lambda }^2{k}_i}{2\lambda +2}}\simeq \frac{A_{ij}}{1+\frac{{\lambda }^2{k}_i}{2}},$$

$${\lambda }_{\text{c}}^{\text{PQMF}}{\Omega }^{(1)}({\lambda }_{\text{c}}^{\text{PQMF}})=1$$

$$\epsilon =\frac{{\lambda }_c-{\lambda }_c^{\text{MF}}}{{\lambda }_c},$$

$$P=\frac{{\sum }_{ij}\left(A_{ij}-\frac{k_i{k}_j}{N⟨k⟩}\right)k_i{k}_j}{{\sum }_{ij}\left(k_i{\delta }_{ij}-\frac{k_i{k}_j}{N⟨k⟩}\right)k_i{k}_j}.$$

Full text

I Introduction

Metabolic chains of protein interactions Vázquez et al. (2002), collaborations among scientists, co-starring in a movie Newman (2010), or person-to-person contacts Cattuto et al. (2010), are all examples of interacting systems that can be modeled using complex networks Newman (2010). A large number of networks representing real systems show a heavy-tailed degree distribution described by a power-law, $P(k)\sim k^{-\gamma}$ Barabási and Albert (1999); Albert and Barabási (2002), usually with strong levels of correlations Pastor-Satorras et al. (2001); Newman (2002). Degree correlations are encoded in the conditional probability $P(k^{\prime}|k)$ that a vertex of degree $k$ is connected to a vertex of degree $k^{\prime}$ Pastor-Satorras et al. (2001). Technological networks, such as the Internet, show in general disassortative mixing Pastor-Satorras et al. (2001); Newman (2002), i.e., vertices of large degree tend to be connected with those of small degree, and vice-versa. Assortative mixing occurs in social networks, where connections preferentially occur among vertices exhibiting similar degree. Since uncorrelated networks usually simplify theoretical approaches, they are typical benchmarks for the investigation of dynamical processes on networks and have been considered in many studies Barrat et al. (2008); Pastor-Satorras et al. (2015); Wang et al. (2017). However, the ubiquitousness of correlations in real networks naturally calls for the investigation of the effect of correlated interaction patterns. While the effects of degree correlations have been considered for several dynamical processes Boguñá et al. (2003); Eguíluz and Klemm (2002); Warren et al. (2002); Mieghem et al. (2010); Goltsev et al. (2008); Kenah and Miller (2011), a full understanding of their effects on the performance of theoretical approaches is still missing.

A basic approach to investigate dynamical processes on networks is the heterogeneous mean-field (HMF) theory, in which degree heterogeneity and correlations are taken into account through the distributions $P(k)$ and $P(k|k^{\prime})$, respectively Pastor-Satorras and Vespignani (2001); Pastor-Satorras et al. (2015); Barrat et al. (2008); Barabási and Pósfai (2016). A more refined approach is provided by the quenched mean-field theory (QMF) Wang et al. (2003); Chakrabarti et al. (2008); Castellano and Pastor-Satorras (2010), which considers the full topology as described by the unweighted adjacency matrix (defined as $A_{ij}=1$ if vertices $i$ and $j$ are connected and $A_{ij}=0$ otherwise) and thus takes into account the detailed connectivity structure.

A crucial question in this context is the ability of theories to accurately predict the epidemic threshold of the susceptible-infected-susceptible (SIS) dynamics, the most basic epidemic process with an absorbing-state phase-transition Castellano and Pastor-Satorras (2010); Ferreira et al. (2012); Goltsev et al. (2012); Lee et al. (2013); Boguñá et al. (2013); Mata and Ferreira (2013, 2015); Castellano and Pastor-Satorras (2017). For random uncorrelated networks, such as those created according to the uncorrelated configuration model Catanzaro et al. (2005), when $\gamma<5/2$ the two theories tend to agree, predicting a vanishing threshold as the network size diverges Ferreira et al. (2012). For $\gamma>3$ instead, QMF theory correctly predicts again the asymptotic vanishing of the epidemic threshold Chatterjee and Durrett (2009), while HMF fails, predicting the existence of a finite threshold. In spite of being qualitatively correct, QMF theory is however not able to accurately predict the effective finite-size epidemic threshold in this case Boguñá et al. (2013). A further quantitative improvement of the QMF theory has been achieved in Ref. Mata and Ferreira (2013) [hereafter called of pair QMF (PQMF) theory] by means of the explicit inclusion of pairwise dynamical correlations Gleeson (2011, 2013); Cator and Mieghem (2012)

In this work, we investigate the ability of the aforementioned approaches (HMF, QMF, PQMF) to quantitatively predict the value of the epidemic threshold for both uncorrelated and correlated networks generated using the Weber-Porto model Weber and Porto (2007) and for real-world topologies. We find that correlations do not change qualitatively the scenario found on uncorrelated networks. The epidemic threshold vanishes asymptotically with the system size for both assortative and disassortative correlations. For $\gamma<5/2$, both QMF and PQMF seem to provide an asymptotically exact estimate of the numerical threshold, while they are only qualitatively correct for $\gamma>5/2$. As in the case of uncorrelated networks Mata and Ferreira (2013), PQMF outperforms the other theories. The amplitude of the discrepancies between numerics and theory is correlated with violations of the assumptions underlying them, revealing that both theories tend to be more accurate if the principal eigenvector of the (effective) adjacency matrix is not strongly localized or the spectral gap is large. The same scenario is found to hold when SIS dynamics is considered on a set of real-world topologies. In addition, we analyze the singularity of the prevalence near the transition point through the critical exponent $\beta$, defined as $\rho\sim(\lambda-\lambda_{c})^{\beta}$. Interestingly, we find that for $\gamma<5/2$ even if QMF provides an asymptotically exact estimate of the position of the epidemic threshold, the QMF prediction for the prevalence exponent, $\beta^{\text{(QMF)}}=1$ Mieghem (2012); Goltsev et al. (2012), is correct only not too close to the transition.

The rest of the paper is organized as follows. Section II describes the models used to generate correlated heavy-tailed networks, the implementation of the SIS model, and the theoretical approaches. The comparison between simulations and theory on synthetic and real networks in presented in Sec. III. Our conclusions are drawn in Sec. IV. Appendix A, summarizing the properties of the investigated real networks, complements the paper.

II Models and Methods

II.1 Weber-Porto configuration model

The degree correlations encoded in the conditional probability $P(k^{\prime}|k)$ can be more easily interpreted by the simple metrics of the average degree of the nearest-neighbors as a function of the vertex degree Pastor-Satorras et al. (2001), defined as

[TABLE]

where $k_{\text{min}}$ and $k_{\text{max}}$ are the lower and upper cutoffs of the degree distribution. If $\kappa_{\text{nn}}(k)$ increases or decreases with $k$, the networks are assortative or disassortative, respectively. In the case of uncorrelated networks we have Pastor-Satorras et al. (2001)

[TABLE]

which implies that $\kappa_{\text{nn}}=\langle{k^{2}}\rangle/\langle{k}\rangle=\langle{k}\rangle_{\text{e}}$ does not depend on $k$. We use here the edge distribution average $\langle{A(k)}\rangle_{\text{e}}=\sum_{k}A(k)P_{\text{e}}(k)$ where $P_{\text{e}}(k)$ is the probability that an edge ends on a vertex of degree $k$.

We are interested in heavy-tailed networks with degree distribution $P(k)\sim k^{-\gamma}$ and correlation given by $\kappa_{\text{nn}}(k)\sim k^{\alpha}$. These networks can be generated using an algorithm proposed by Weber and Porto Weber and Porto (2007), hereafter called Weber-Porto configuration model (WPCM). The degree of each vertex is drawn according to the degree distribution $P(k)$ and initially each node has $k$ unconnected stubs. Two stubs are randomly chosen and connected with probability

[TABLE]

where $q$ and $q^{\prime}$ are the respective degrees of the chosen vertices and $f_{\text{max}}$ is the maximum value of

[TABLE]

computed over the whole network. Self- and multiple connections are forbidden. In the absence of degree correlations, we have $\kappa_{\text{nn}}=\langle{k}\rangle_{\text{e}}$, implying $f(q,q^{\prime})=1$ and $P_{\text{link}}=1$. See Ref. Weber and Porto (2007) for more details.

Figure 1 shows $\kappa_{\text{nn}}$ as a function of $k$ for networks obtained with the WPCM algorithm Weber and Porto (2007) using different values of $\gamma$ and $\alpha$, with lowest degree $k_{\text{min}}=3$. We adopt different upper cutoffs for the degree distribution. For $\gamma<3$, the structural cutoff $k_{\text{max}}=2\sqrt{N}$ Boguñá et al. (2004) is used, while for $\gamma>3$, a rigid cutoff is determined by the condition $NP(k_{\text{max}})=1$ Dorogovtsev and Mendes (2002). The first choice allows to enhance the effects of hubs and to approach faster the thermodynamic limit while fulfilling the criterion $k_{\text{max}}<\sqrt{\langle{k}\rangle N}$ necessary to produce uncorrelated networks in the case $\alpha=0$ Boguñá et al. (2004). The second choice is justified by numerical reasons explained in Sec. II.2. The predetermined scaling law $\kappa_{\text{nn}}(k)\sim k^{\alpha}$ is very well reproduced. Small deviation for positive or negative $\alpha$ are due to the network finite size that prevents $\kappa_{\text{nn}}$ from decaying or increasing indefinitely with $k$. The range of the power-law behavior is extended as the network size increases.

II.2 SIS simulations

In the SIS model, each edge of an infected vertex transmits the epidemics with rate $\lambda$, while infected nodes recover spontaneously with constant rate $\mu$. The latter is fixed to $\mu=1$ without loss of generality. The model can be simulated with the optimized Gillespie scheme proposed in Ref. Ferreira et al. (2012). See also Ref. Cota and Ferreira (2017) for more details.

We consider quasi-stationary simulations de Oliveira and Dickman (2005) in which the dynamics returns to a previously visited active configuration whenever the absorbing state, consisting of all vertices susceptible, is visited. This strategy permits to circumvent the difficulties of dealing with the absorbing-state, which is the only true stationary state for any finite-size networks. More details can be found in Refs. Cota and Ferreira (2017); Sander et al. (2016).

The effective transition point $\lambda_{c}(N)$, above which the epidemic remains in an active phase for very long periods can be estimated using the position of the maximum of the dynamical susceptibility Ferreira et al. (2012)

[TABLE]

The choice of structural (for $\gamma<3$) and rigid (for $\gamma>3$) upper cutoffs allows the determination of the epidemic threshold unambiguously, avoiding multiple peaks and the smearing of the transition that can appear for SIS on power-law degree distribution networks, specially with large values of $\gamma$ Ferreira et al. (2012); Mata and Ferreira (2015).

II.3 Mean-field theories for correlated networks

In this subsection we summarize the predictions of the theoretical approaches that will be compared to numerical simulations in Sec. III. For QMF and PQMF approaches, the equations for uncorrelated and correlated networks are formally the same: correlations have only the effect of modifying the entries of the adjacency matrix $A_{ij}$.

II.3.1 Correlated Heterogeneous Mean-Field theory

HMF theory takes into account nearest-neighbors correlations by the explicit consideration of the conditional probability $P(k^{\prime}|k)$. The HMF equation for the density of infected vertices with degree $k$, $\rho_{k}$, is given by Boguñá and Pastor-Satorras (2002)

[TABLE]

which yields an epidemic threshold given by

[TABLE]

where $\Upsilon^{(1)}$ is the largest eigenvalue of the connectivity matrix $C_{kl}=kP(l|k)$. For WPCM networks we have, $P(l|k)=P_{\text{e}}(l)f(l,k)$, therefore

[TABLE]

In the absence of correlations, $C_{kl}=\frac{klP(l)}{\langle k\rangle}$, implying that $\lambda_{c}=\frac{\langle{k}\rangle}{\langle{k^{2}}\rangle}$ Boguñá and Pastor-Satorras (2002); Boguñá et al. (2003). It has been shown Boguñá et al. (2003) that the HMF threshold vanishes for scale-free networks with $2<\gamma<3$ in the thermodynamic limit, irrespective of degree correlations.

II.3.2 Quenched Mean-Field theory

According to the QMF theory, which neglects pairwise dynamical correlations, the evolution of the probability $\rho_{i}$ that a vertex $i$ is infected is given by Chakrabarti et al. (2008)

[TABLE]

where $N$ is the network size. The epidemic threshold is given by

[TABLE]

where $\Lambda^{(1)}$ is the largest eigenvalue (LEV) of the adjacency matrix $A_{ij}$. In the steady state we have

[TABLE]

Using Eq. (11), Goltsev et al. Goltsev et al. (2012) have shown that $\rho_{i}\sim v^{(1)}_{i}$for $\lambda\gtrsim\lambda_{c}^{\text{QMF}}$, where $\{v^{(1)}_{i}\}$ is the principal eigenvector (PEV) corresponding to the LEV of $A_{ij}$, $\sum_{i}A_{ij}v^{(1)}_{j}=\Lambda^{(1)}v^{(1)}_{i}$. So, the order parameter $\rho=\sum_{i}\rho_{i}/N$ of the QMF theory vanishes at $\lambda_{c}^{\text{QMF}}$ as

[TABLE]

where

[TABLE]

This same result was obtained independently in Ref. Mieghem (2012).

Within the QMF framework, Equation (12) works well, close to the threshold $\lambda_{c}^{\text{QMF}}$, under the hypothesis that the network presents a spectral gap, i.e., the second largest eigenvalue of $A_{ij}$ is much smaller than the first, $\Lambda^{(1)}\gg\Lambda^{(2)}$ . According to Eqs. (12) and (13), the QMF theory predicts the existence of an endemic state, with a finite fraction of infected vertices above the threshold $\lambda_{c}^{\text{QMF}}=1/\Lambda^{(1)}$, only if $a_{1}\sim\mathcal{O}(1)$, which occurs when the PEV is delocalized. Localization can be quantified by the inverse participation ratio (IPR) for the normalized PEV Goltsev et al. (2012), defined as

[TABLE]

If the PEV is delocalized then $Y_{4}\sim N^{-1}$, while $Y_{4}\sim\mathcal{O}(1)$ if the PEV is localized on a finite number of vertices, but weaker forms of localization can be observed Pastor-Satorras and Castellano (2016).

For random uncorrelated power-law networks the PEV is always localized Pastor-Satorras and Castellano (2016). For $\gamma<5/2$ it is (weakly) localized on a subextensive set of nodes coinciding with the maximum $K$-core, a subgraph of strongly mutually interconnected nodes with degree larger than or equal to $K$ Seidman (1983). In such a case $Y_{4}\sim N^{(\gamma-3)/2}$. For $\gamma>5/2$ it is instead strongly localized on the largest hub plus its nearest neighbors and $Y_{4}\sim\mathcal{O}(1)$ Pastor-Satorras and Castellano (2018). Hence within QMF theory the threshold separates the absorbing-phase from an active but strictly nonendemic state. However this does not imply that QMF predictions are necessarily flawed. Equation (9) factorizes the state of nearest neighbors and thus neglects dynamical correlations among them. These dynamical correlations actually transmit the infection from the localized PEV to the rest of the network, and thus may in principle transform the active but localized state just above $\lambda_{\text{c}}^{\text{QMF}}$ into a full-fledged endemic state Boguñá et al. (2013); Castellano and Pastor-Satorras (2019).

II.3.3 Pair Quenched Mean-Field theory

An improvement with respect to QMF theory is obtained by taking into account some dynamical correlations using the pairwise approximation developed in Ref. Mata and Ferreira (2013), where all derivation details can be found. Consider the probability $\phi_{ij}$ that a vertex $i$ is susceptible and a neighbor $j$ is infected. The dynamical system to be solved is

[TABLE]

and

[TABLE]

where $\omega_{ij}=1-\phi_{ij}-\rho_{i}$.

Here, we develop a bit further the theory to analyze the steady state near the critical point. Keeping only leading terms up to second order in $\rho_{i}$ in Eq. (16) we obtain

[TABLE]

where we kept only leading order in $\lambda\approx\lambda_{c}\ll 1$ Mata and Ferreira (2013) for quadratic terms in $\rho_{i}$. Plugging Eq. (17) in Eq. (15) with $d\rho_{i}/dt=0$, we obtain

[TABLE]

where

[TABLE]

is an effective, weighted adjacency matrix. The last passage in Eq. (19) assumes $\lambda\ll 1$.

Equation (18) has exactly the same form of the stationary $\rho_{i}$ in Eq. (11), obtained for QMF theory, replacing $A_{ij}$ by $B_{ij}$. Therefore, all the spectral analysis described in subsection II.3.2 found for QMF theory can be extended to the PQMF case with the replacement of spectral properties of $A_{ij}$ by those of $B_{ij}$. For example, the epidemic threshold is given by

[TABLE]

where $\Omega^{(1)}$ is the largest eigenvalue of $B_{ij}$. One can check that this result is exactly the same presented in Ref. Mata and Ferreira (2013) expressed in a different way. For $\lambda\gtrsim\lambda_{\text{c}}^{\text{PQMF}}$ we have that $\rho_{i}\sim w_{i}^{(1)}$, where $\{w_{i}^{(1)}\}$ is the PEV of $B_{ij}(\lambda_{\text{c}}^{\text{PQMF}})$ and $\rho\simeq b_{1}(\lambda\Omega^{(1)}(\lambda_{\text{c}}^{\text{PQMF}})-1)$ where $b_{1}(N)$ has the same form of Eq. (13) replacing $v_{i}$ by $w_{i}$. So, the IPR of $\{w_{i}^{(1)}\}$, denoted by $Y_{4}[B_{ij}]$, allows to quantify the localization in the PQMF theory.

III Results

III.1 Accuracy of the theoretical estimates for the epidemic threshold

Figure 2 shows the dependence of the epidemic threshold as a function of the network size obtained in simulations with different values of $\gamma$ and $\alpha$. We concentrate for the moment on two values of $\gamma$, representative of the cases $\gamma<5/2$ and $\gamma>3$, for which the physical mechanisms underlying the epidemic transition are clear Castellano and Pastor-Satorras (2012). Later we will discuss the case $5/2<\gamma<3$, whose interpretation is hampered by extremely long crossover phenomena in the spectral properties. As we can see from this figure, all thresholds vanish as $N$ diverges, regardless of the correlation level ($\alpha$) and heterogeneity ($\gamma$). Compared to the uncorrelated case, assortative networks ($\alpha>0$) have a smaller threshold, while the threshold is larger for $\alpha<0$, i.e., disassortative mixing, in agreement with the behavior of the LEV of the adjacency matrix Mieghem et al. (2010); Goltsev et al. (2012). In the case $\gamma>3$, this phenomenology can be qualitatively explained by considering the mechanism of long-range mutual reinfection of hubs Boguñá et al. (2013); Ferreira et al. (2016); Castellano and Pastor-Satorras (2019), which triggers the epidemic transition. According to this mechanism, the subgraph consisting of the hub plus its nearest-neighbors can sustain in isolation an active state for times long enough to permit the activation of other hubs, even if they are not directly connected. This mechanism is at work independent of degree correlations, as long as distances among hubs increase slowly enough with network size. In assortative networks, communication among hubs is enhanced since they have larger probability to be closer; for disassortative topology the converse is true and larger values of $\lambda$ are needed to trigger the transition.

The accuracy of HMF theory is tested with respect to simulations in Fig. 3. For $\gamma=2.3$, we see a non-negligible asymptotic discrepancy between HMF and simulations in the case of correlated networks. Interestingly HMF appears to overestimate the threshold for disassortative networks, while it underestimates it for assortative ones. For larger values of $\gamma$ the discrepancy is conspicuous and the epidemic threshold is significantly overestimated, as can be seen in the insets of Fig. 3.

Comparisons between QMF and PQMF theories and simulations are shown in Figs. 4(a) and 5(a) in the range of network size $10^{3}\leq N\leq 10^{8}$. For $\gamma=2.3$, both QMF and PQMF theories appear to converge asymptotically to the epidemic threshold observed in simulations. PQMF displays a faster convergence than QMF, this effect being enhanced for smaller values of $\alpha$. For $\gamma=3.5$, the predictions of PQMF and QMF theories succeed, qualitatively, in predicting that the threshold approaches zero in the thermodynamic limit even in the presence of correlations. However, the theoretical threshold estimates depart from simulation results leading to decreasing ratios $\lambda_{c}^{MF}/\lambda_{c}$ in the large network limit. We expect this ratio to decrease asymptotically as $1/\ln(k_{\text{max}})$  Castellano and Pastor-Satorras (2019), in agreement with recent rigorous results Huang and Durrett (2018). Again, PQMF theory performs better than QMF. In this case, the improvement of PQMF over QMF grows with $\alpha$.

III.2 Relation with spectral properties

What is the origin of the discrepancies between theoretical predictions and numerical results observed in Section III.1?

In this subsection we investigate which spectral feature is correlated with the performance of the theoretical approaches depends. We consider both QMF and PQMF theories, testing their accuracy against the spectral properties of adjacency matrices $A_{ij}$ and $B_{ij}$, respectively.

Let us consider first the case $\gamma=3.5$. The real threshold is not the QMF one because the PEV is localized. As pointed out in Ref. Goltsev et al. (2012) this in principle implies that the actual threshold coincides with the inverse of the largest eigenvalue corresponding to a delocalized PEV, coinciding with the HMF threshold $\lambda_{c}^{\text{HMF}}=\langle{k}\rangle/\langle{k^{2}}\rangle$. But actually the QMF approach neglects dynamical correlations, which have the effect of allowing mutual reinfection events among different hubs in the network. In this way an endemic global state can be established thanks to the long-range interactions among localized states Castellano and Pastor-Satorras (2019) setting the actual threshold to an intermediate value: $\lambda_{c}^{QMF}<\lambda_{c}<\lambda_{c}^{HMF}$. With this picture in mind we can predict that, if the localization is stronger (higher values of the IPR $Y_{4}$), the actual threshold will be farther from $\lambda_{c}^{QMF}$ and thus the performance (accuracy) of the QMF approach will be reduced.

We plot the dependence of $Y_{4}$ on the system size $N$ for $\gamma=3.5$ in Fig. 5(b): The IPR of $A_{ij}$ converges to a finite value in the thermodynamic limit, irrespective of the correlation degree, representing a PEV localized on a finite set of vertices Pastor-Satorras and Castellano (2016); Goltsev et al. (2012). The saturation with size occurs earlier for disassortative and later for assortative correlations, compared to the uncorrelated case. In general, for a given size $N$, $Y_{4}$ is larger for smaller $\alpha$. As expected, a better QMF performance occurs for smaller $Y_{4}$.

The IPR analysis for the PQMF theory, involving $B_{ij}$, has a qualitatively similar behavior of QMF, but presenting lower values for the IPR. Hence, the PQMF steady state solution is less localized than that of the QMF theory. Correspondingly, the PQMF performance is better than the QMF performance. We also calculate, in Figure 5(c), the dependence of the spectral gap on the system size, both for the adjacency matrix $A_{ij}$ (involved in QMF) and $B_{ij}$ (entering in PQMF). The spectral gap is defined as the difference $\Lambda^{(1)}-\Lambda^{(2)}$ between the largest and second largest positive eigenvalues of the adjacency matrices. The gap of the adjacency matrix $A_{ij}$ is small and it decreases as $N$ grows, as predicted by Ref. Chung et al. (2003). The gap is smaller for smaller $\alpha$. The dependence of spectral gap of $B_{ij}$ on size is qualitatively similar to the gap of $A_{ij}$.

Notice that, while the amplitude of the spectral gap matters for the validity of the QMF prediction for the prevalence above the critical point [Eq. (12)], it does not play any role in the determination of $\lambda_{\text{c}}^{\text{QMF}}$. Therefore there is no conceptual reason for expecting a correlation between QMF performance and spectral gap size. We find numerically such a correlation in Fig. 5, but we cannot attribute a causal meaning to it.

Let us consider now $\gamma=2.3$. In this case the physical mechanism underlying the epidemic transition is different, as it does not involve the interaction between distant hubs, rather the extension of activity from the max K-core to the rest of the network. The connection between QMF performance and localization is not easily predictable.

As shown in Fig. 4(b), the IPR for $\gamma=2.3$ follows a power-law $Y_{4}\sim N^{-\nu}$ with $\nu\approx(3-\gamma)/2$, which corresponds to the IPR localized in the maximum $K$-core of the network Pastor-Satorras and Castellano (2016). Correlations leave the scaling exponent unchanged, altering only the prefactor, the smaller $\alpha$ the smaller the IPR. This means that the PEV is still localized on a sub-extensive fraction of nodes. However, since $Y_{4}$ increases with $\alpha$, the PEV is more localized for positive than for negative $\alpha$. The same is true for the matrix $B_{ij}$ of the PQMF theory. Interestingly, the effect on the performance of the theoretical approaches is opposite. QMF works better for larger $Y_{4}$, PQMF works better for smaller $Y_{4}$. We have no simple interpretation for this result.

Figure 4(c) shows the spectral gap for the WPCM networks with $\gamma=2.3$. In this case the gap increases with network size and it is smaller for smaller $\alpha$. This is true also for the spectral gap of PQMF. Finally, let us observe that there is almost no difference between the spectral properties of $A_{ij}$ and $B_{ij}$ for $\gamma=2.3$. This is indeed not surprising for $\alpha=0$ since the term $\lambda^{2}k_{i}$ in the denominator of Eq. (19) is asymptotically negligible, because $\lambda_{c}^{2}k_{\text{max}}\sim k_{\text{max}}^{2\gamma-5}\rightarrow 0$ as $N\rightarrow\infty$ for $\gamma<5/2$.

III.3 The intermediate case $2.5<\gamma<3$

As for the other values of $\gamma$, in this range the vanishing of the threshold with $N$ is observed regardless of the correlation pattern. The localization phenomenon of the PEV in the case $5/2<\gamma<3$ is asymptotically analogous to the case $\gamma>3$. However, very strong crossover effects are observed in this case, because of the presence of a localization process on the max K-core (as for $\gamma<5/2$) competing with the localization around the hub Pastor-Satorras and Castellano (2016). As a consequence, already in the uncorrelated case, the PEV gets strongly localized around the largest hub only for very large values of $N$. Correlations further complicate the picture: Panel (b) of Fig. 6 shows that disassortative correlations accelerate the convergence to the final localized state. For $\alpha>0$ instead, $Y_{4}$ is a decreasing function of $N$. The upward bend of the curve hints at an incipient crossover, but one cannot exclude that the asymptotic behavior is different for $\alpha>0$. A similar pattern is observed for what concerns the spectral gap (Fig. 6(c)).

With regard to the performance of the theoretical approaches, for negative or zero correlations the scenario perfectly matches what happens for $\gamma>3$: all theories somehow fail in capturing the way the threshold vanishes, with PQMF being less inaccurate than the others. In the case $\alpha=0.2$ numerical results seem to suggest that both theories describe quite well how the threshold changes with the system size. However, the large crossover effects mentioned above do not allow to draw any firm conclusion.

We can summarize our findings by stating that the performance in predicting the behavior of epidemic threshold of the QMF and PQMF theories on WPCM networks is correlated with the size of the spectral gap and the IPR of PEV of the respective $A_{ij}$ and $B_{ij}$ matrices that rule the prevalence near to the transition point. A large spectral gap or a low IPR lead to a good performance of the mean-field theories while the converse, small gap or large IPR, lead to deviations from the theoretical predictions. QMF seems to be more correlated with the spectral gap while PQMF with the IPR, at least in the regime where the gap is significant and the theories are accurate.

III.4 Real networks

We extend our analysis to a set of 99 real-world networks encompassing a broad range of origins, sizes and topological features, see Appendix A. The spectral gap and IPR of matrices $A_{ij}$ and $B_{ij}$ are compared in the scatter plots shown in Figs. 7(a) and (b). We see that the spectral gap is almost the same for both adjacency matrices while the IPR extracted from $B_{ij}$ is smaller than the one extracted from $A_{ij}$, in particular in the range of large IPR values. This shows that the PQMF matrices $B_{ij}$ are less localized than the matrix $A_{ij}$, relevant for QMF theory. The relative errors between QMF or PQMF mean-field theories and simulations, defined as

[TABLE]

are compared in the scatter plot shown in Fig. 7(c). As in the case random networks, PQMF outperforms QMF theory for all investigated networks.

On this set of networks, we test the relation observed for synthetic correlated networks, connecting qualitatively the accuracy of QMF and PQMF threshold predictions with the properties of the adjacency matrices (spectral gap and IPR), respectively, and with the Pearson coefficient $P$, measuring network topological correlations. $P$ is defined as Newman (2010)

[TABLE]

The Pearson coefficient lays in the interval $-1<P<1$, being negative for disassortative, null for uncorrelated, and positive for assortative networks. The analyses for QMF are shown in the scatter plots of the relative error $\varepsilon$ against the corresponding topological properties in Figs. 7(d)-(e). Qualitatively similar patterns obtained for PQMF are not shown. We can see that in real networks, the correlation between the performance of the theoretical prediction and the spectral gap is on average the same as the one observed for the WPCM: A larger spectral gap is associated to a larger accuracy. The inverse correlation with the IPR is again preserved: A smaller $Y_{4}$ corresponds to a more accurate prediction. We do not find instead a significant correlation with the Pearson coefficient. Statistical analyses were performed using the correlation coefficients obtained from either power-law, in the case of spectral gap and IPR, or from exponential, in the case of Pearson coefficient, regressions of the scatter plots. We obtain strong statistical correlations with $|r|\gtrsim 0.70$ (p-value $<10^{-5}$) for both QMF and PQMF using either IPR or spectral gap of the corresponding matrices. Values $r\lesssim 0.2$ (p-value $>0.05$) for correlation with the Pearson coefficient of the network confirms no significant statistical correlations.

III.5 Epidemic prevalence near to the epidemic threshold

Figure 2 shows that the QMF prediction for the epidemic threshold tends to the same limit of numerical simulations both for uncorrelated and correlated networks for $\gamma=2.3$. This observation naturally leads to wonder whether QMF is asymptotically an exact description for SIS dynamics on random networks with $\gamma<5/2$. In order to answer this question we test the exactness of the other prediction of the QMF theory, Eq. (12), stating that the fraction of infected individuals decays to zero linearly as the threshold is approached from above. Numerical results, for the case of uncorrelated networks $\alpha=0$, are shown in Fig. 8, where the density and the infection rates are rescaled to conform to Eq. (12). We can clearly see the existence of two scaling regimes. For $\lambda\Lambda_{1}-1\ll 1$ the density scales with an exponent larger than the prediction $\beta^{\text{(QMF)}}=1$. The observed exponent is consistent with the exact result of Ref. Mountford et al. (2013) $\beta=1/(3-\gamma)$ which is also (probably accidentally) the value predicted by HMF theory Pastor-Satorras and Vespignani (2002). This exponent is observed in a regime very close to the transition, where the system is kept asymptotically active only by virtue of the QS method. We performed a non-perturbative analysis by integrating the QMF equations using a fourth order Runge-Kutta method for $\lambda>\frac{1}{\Lambda^{(1)}}$ for $N=10^{7}$. A comparison with simulation results confirms that the QMF theory correctly predicts the linear behavior of the prevalence $\rho$ around the epidemic transition, but only sufficiently far from it. In the immediate neighborhood of the threshold the decay is more rapid.

IV Conclusions

The determination of the epidemic threshold in models of disease spreading in complex topologies is a nontrivial problem in network science. Several theoretical approaches have been proposed, applying approximations with different levels of stringency, that provide contrasting predictions on the epidemic threshold. Among the main theoretical approaches at the mean-field level we can consider, in decreasing order of approximation, the heterogeneous mean-field theory (HMF), neglecting dynamical correlations and the actual pattern of connections in the network (preserving only its statistical properties); the quenched mean-field theory (QMF), also neglecting dynamical correlations but keeping the network structure; and the pair quenched mean-field theory (PQMF) that incorporates dynamical correlations between pairs of connected nodes. In this paper we have presented a comparison of the predictions of these three approximate theories for the case of the susceptible-infected-susceptible (SIS) epidemic model, focusing on the case of networks with a power-law degree distribution and degree correlations, representative of many real networked systems.

Comparing the predictions with actual stochastic simulations of the SIS process, we observe that, independently of the degree of correlations, the predictions of PQMF are more accurate than those of QMF, while both outperform HMF, which fails to predict the vanishing threshold observed for a degree exponent $\gamma>3$. While overall PQMF is more accurate than QMF, the two approximations show different levels of accuracy when compared in networks with different levels of correlations. Thus, for the case of synthetic networks generated with the Weber-Porto algorithm Weber and Porto (2007), we observe that, for fixed network size and degree heterogeneity, QMF predictions are more accurate in assortative networks than in disassortative ones. On the other hand, PQMF is increasingly accurate in the presence of disassortative correlations for small degree exponent, while it is more accurate when correlations are assortative if the degree exponent is large.

We propose a criterion for the accuracy of the QMF and PQMF approaches based on the spectral properties of the networks. We observe that the accuracy is positively correlated with the amplitude of the spectral gap of the adjacency matrix and is inversely related with degree of localization of the principal eigenvalue, as measured by the inverse participation ratio. This general observation is corroborated by the analysis of a large set of real correlated networks, covering a wide range of sizes and topological features.

Additionally, we investigate the behavior of the order parameter of the transition, measured in terms of the prevalence or density of infected nodes in the steady state, for $\gamma<5/2$. We observe that, in uncorrelated synthetic networks, the linear decay (critical exponent $\beta=1$) predicted by QMF theory is observed in stochastic simulations not very close to the transition. When fluctuation effects become more important, i.e., very close to the transition, the observed exponent $\beta$ crosses over to the value $\beta=1/(3-\gamma)$, in agreement with rigorous mathematical results Mountford et al. (2013).

Acknowledgements.

This work was partially supported by the Brazilian agencies CAPES, CNPq and FAPEMIG. S.C.F. thanks the support from the program Ciência sem Fronteiras - CAPES under project No. 88881.030375/2013-01. R.P.-S. and C.C. acknowledge financial support from the Spanish MINECO, under project FIS2016-76830-C2-1-P. R. P.-S. acknowledges additional financial support from ICREA Academia, funded by the Generalitat de Catalunya. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

Appendix A Summary of real networks properties

We consider 99 real networks with diverse structural properties, based on the lists of Refs.  Radicchi (2015); Radicchi and Castellano (2015). Here we investigate their giant connected components, after symmetrizing all edges (weighted and/or directed) and avoiding multiple and self connections. The list of networks with some topological properties and epidemic (SIS) parameters is shown in Table LABEL:tab:realnetsprops. For detailed information about the original references for all the networks, please check Refs. Radicchi (2015); Radicchi and Castellano (2015).