45:
741:
2626:
2073:
2840:
3320:
Katz centrality can also be used in estimating the relative status or influence of actors in a social network. The work presented in shows the case study of applying a dynamic version of the Katz centrality to data from
Twitter and focuses on particular brands which have stable discussion leaders.
1809:
An extension of this framework allows for the walks to be computed in a dynamical setting. By taking a time dependent series of network adjacency snapshots of the transient edges, the dependency for walks to contribute towards a cumulative effect is presented. The arrow of time is preserved so that
748:
Katz centrality computes the relative influence of a node within a network by measuring the number of the immediate neighbors (first degree nodes) and also all other nodes in the network that connect to the node under consideration through these immediate neighbors. Connections made with distant
3332:
in a neural network. The temporal extension of the Katz centrality is applied to fMRI data obtained from a musical learning experiment in where data is collected from the subjects before and after the learning process. The results show that the changes to the network structure over the musical
2409:
1921:
1959:
1617:
2398:
1359:
2657:
3751:
Mantzaris, Alexander V.; Danielle S. Bassett; Nicholas F. Wymbs; Ernesto
Estrada; Mason A. Porter; Peter J. Mucha; Scott T. Grafton; Desmond J. Higham (2013). "Dynamic network centrality summarizes learning in the human brain".
2983:
3221:
2135:
2176:
3665:
Laflin, Peter; Mantzaris, Alexander V; Ainley, Fiona; Otley, Amanda; Grindrod, Peter; Higham, Desmond J (2013). "Discovering and validating influence in a dynamic online social network".
1186:
1758:
1667:
2621:{\displaystyle {\hat {\mathcal {Q}}}^{}={\frac {{\hat {\mathcal {Q}}}^{}\left(I-\alpha A^{}\right)^{-1}}{\left\|{\hat {\mathcal {Q}}}^{}\left(I-\alpha A^{}\right)^{-1}\right\|}}.}
2068:{\displaystyle \left(A^{}\right)_{ij}={\begin{cases}1&{\text{there is an edge from node }}i{\text{ to node }}j{\text{ at time }}t_{k}\\0&{\text{otherwise}}\end{cases}}}
1819:
1804:
1228:
999:
3321:
The application allows for a comparison of the methodology with that of human experts in the field and how the results are in agreement with a panel of social media experts.
960:
561:
921:
1517:
882:
2280:
1139:
843:
814:
3300:
3135:
3106:
2835:{\displaystyle C_{n}^{\mathrm {broadcast} }:=\sum _{k=1}^{N}{\mathcal {Q}}_{nk}\quad \mathrm {and} \quad C_{n}^{\mathrm {receive} }:=\sum _{k=1}^{N}{\mathcal {Q}}_{kn}}
1236:
2212:
1051:
3053:
2885:
1501:
787:
767:
3271:
3013:
1951:
1702:
1421:
1394:
1094:
3244:
3077:
3033:
2649:
2272:
2252:
2232:
1640:
1481:
1461:
1441:
668:
2893:
3333:
exposure created in each session a quantification of the cross communicability that produced clusters in line with the success of learning.
551:
280:
744:
A simple social network: the nodes represent people or actors and the edges between nodes represent some relationship between actors
3146:
3867:
625:
208:
3872:
521:
506:
884:. Since Jose connects to John indirectly through Bob, the weight assigned to this connection (composed of two links) will be
511:
661:
620:
137:
3310:
Katz centrality can be used to compute centrality in directed networks such as citation networks and the World Wide Web.
501:
2081:
698:
466:
310:
257:
72:
496:
3313:
Katz centrality is more suitable in the analysis of directed acyclic graphs where traditionally used measures like
2140:
630:
536:
531:
295:
193:
132:
3789:
Park, Juyong; Newman, M. E. J. (31 October 2005). "A network-based ranking system for
American college football".
3862:
218:
516:
3435:
Junker, B. H., & Schreiber, F. (2008). Analysis of
Biological Networks. Hoboken, NJ: John Wiley & Sons.
654:
556:
23:
3336:
A generalized form of Katz centrality can be used as an intuitive ranking system for sports teams, such as in
1069:
indicate the presence (or absence) of links between two nodes through intermediaries. For instance, in matrix
3531:
1144:
702:
637:
456:
223:
157:
112:
1711:
1645:
3625:
3314:
1916:{\displaystyle \left\{A^{}\in \mathbb {R} ^{N\times N}\right\}\qquad {\text{for}}\quad k=0,1,2,\ldots ,M,}
729:
713:) between a pair of actors, Katz centrality measures influence by taking into account the total number of
541:
526:
441:
923:. Similarly, the weight assigned to the connection between Agneta and John through Aziz and Jane will be
845:. The weight assigned to each link that connects John with his immediate neighbors Jane and Bob will be
642:
461:
431:
320:
275:
1767:
1612:{\displaystyle {\overrightarrow {C}}_{\mathrm {Katz} }=((I-\alpha A^{T})^{-1}-I){\overrightarrow {I}}}
1195:
965:
962:
and the weight assigned to the connection between Agneta and John through Diego, Jose and Bob will be
3508:
3461:
2393:{\displaystyle {\mathcal {Q}}=\left(I-\alpha A^{}\right)^{-1}\cdots \left(I-\alpha A^{}\right)^{-1}.}
926:
409:
290:
3630:
3446:
2005:
887:
848:
714:
446:
315:
305:
300:
152:
97:
87:
3616:
P. Bonacich, P. Lloyd (2001). "Eigenvector-like measures of centrality for asymmetric relations".
1503:
has to be chosen such that it is smaller than the reciprocal of the absolute value of the largest
1354:{\displaystyle C_{\mathrm {Katz} }(i)=\sum _{k=1}^{\infty }\sum _{j=1}^{n}\alpha ^{k}(A^{k})_{ji}}
1099:
819:
For example, in the figure on the right, assume that John's centrality is being measured and that
3824:
3798:
3761:
3682:
3598:
3563:
3368:
Hanneman, R. A., & Riddle, M. (2005). Introduction to Social
Network Methods. Retrieved from
822:
792:
414:
285:
238:
213:
102:
92:
3276:
3111:
3082:
705:
in 1953 and is used to measure the relative degree of influence of an actor (or node) within a
3816:
3733:
3477:
2181:
1705:
1020:
582:
248:
198:
107:
82:
3038:
2857:
1486:
772:
752:
3842:
3808:
3771:
3723:
3713:
3674:
3635:
3590:
3555:
3526:
3516:
3469:
3404:
3394:
3359:
Katz, L. (1953). A New Status Index
Derived from Sociometric Analysis. Psychometrika, 39–43.
3337:
2852:
1761:
1014:
341:
330:
228:
188:
172:
3812:
3249:
2991:
1929:
1680:
1399:
1367:
1072:
3343:
Alpha centrality is implemented in igraph library for network analysis and visualization.
3056:
577:
358:
233:
142:
77:
31:
3512:
3465:
1810:
the contribution of activity is asymmetric in the direction of information propagation.
3655:
Newman, M. E. (2010). Networks: An
Introduction. New York, NY: Oxford University Press.
3229:
3226:
which is essentially identical to Katz centrality. More precisely, the score of a node
3062:
3018:
2634:
2257:
2237:
2217:
1625:
1466:
1446:
1426:
1141:, it indicates that node 2 and node 12 are connected through some walk of length 3. If
706:
587:
393:
368:
363:
337:
326:
203:
167:
162:
122:
60:
3639:
3856:
3567:
769:. Each path or connection between a pair of nodes is assigned a weight determined by
546:
451:
436:
378:
127:
117:
3828:
3702:"From Structure to Activity: Using Centrality Measures to Predict Neuronal Activity"
3686:
3325:
682:
491:
388:
243:
3055:
is a nonnegative attenuation factor which must be smaller than the inverse of the
1511:. In this case the following expression can be used to calculate Katz centrality:
3581:
Charles H. Hubbell (1965). "An input-output approach to clique identification".
709:. Unlike typical centrality measures which consider only the shortest path (the
44:
3473:
3328:, it is found that Katz centrality correlates with the relative firing rate of
740:
3718:
3701:
3678:
3445:
Grindrod, Peter; Parsons, Mark C; Higham, Desmond J; Estrada, Ernesto (2011).
1504:
694:
426:
383:
373:
3820:
3423:
Aggarwal, C. C. (2011). Social
Network Data Analysis. New York, NY: Springer.
3775:
3369:
591:
147:
3737:
3521:
3496:
3481:
2274:. The form for the dynamic communicability between participating nodes is:
1364:
Note that the above definition uses the fact that the element at location
3546:
Leo Katz (1953). "A new status index derived from sociometric analysis".
725:
710:
3803:
3728:
3399:
3382:
3602:
3559:
3409:
3329:
3140:
Half a century later, Bonacich and Lloyd defined alpha centrality as:
2978:{\displaystyle {\vec {x}}=(I-\alpha A^{T})^{-1}{\vec {e}}-{\vec {e}}\,}
3497:"Evolving graphs: Dynamical models, inverse problems and propagation"
721:
3594:
2651:
can 'broadcast' and 'receive' dynamic messages across the network:
3766:
2631:
Therefore, centrality measures that quantify how effectively node
739:
2214:
is a weighted count of the number of dynamic walks of length
3216:{\displaystyle {\vec {x}}=(I-\alpha A^{T})^{-1}{\vec {e}}\,}
2818:
2729:
2535:
2451:
2418:
2286:
2061:
749:
neighbors are, however, penalized by an attenuation factor
16:
Measure of centrality in a network based on nodal influence
3302:
is constant the order induced on the nodes is identical.
3079:. The original definition by Katz used a constant vector
3791:
Journal of
Statistical Mechanics: Theory and Experiment
3279:
3252:
3232:
3149:
3114:
3085:
3065:
3041:
3021:
2994:
2896:
2860:
2660:
2637:
2412:
2283:
2260:
2240:
2220:
2184:
2143:
2084:
1962:
1932:
1822:
1770:
1714:
1683:
1648:
1628:
1520:
1489:
1469:
1449:
1429:
1402:
1370:
1239:
1198:
1147:
1102:
1075:
1023:
968:
929:
890:
851:
825:
795:
775:
755:
3294:
3265:
3238:
3215:
3129:
3100:
3071:
3047:
3027:
3007:
2977:
2879:
2834:
2643:
2620:
2392:
2266:
2246:
2226:
2206:
2170:
2130:{\displaystyle t_{0}<t_{1}<\cdots <t_{M}}
2129:
2067:
1945:
1915:
1798:
1752:
1696:
1661:
1634:
1611:
1495:
1475:
1455:
1435:
1415:
1388:
1353:
1222:
1180:
1133:
1088:
1045:
993:
954:
915:
876:
837:
808:
781:
761:
3532:20.500.11820/9ccf649d-eee7-44ec-9d3b-c34411bd3a6f
3700:Fletcher, Jack McKay; Wennekers, Thomas (2017).
2137:are ordered but not necessarily equally spaced.
1926:representing the adjacency matrix at each time
3651:
3649:
2171:{\displaystyle Q\in \mathbb {R} ^{N\times N}}
662:
8:
3431:
3429:
3108:. Hubbell introduced the usage of a general
1677:is the number of nodes) consisting of ones.
1057:are variables that take a value 1 if a node
3495:Peter Grindrod; Desmond J. Higham. (2010).
1017:of a network under consideration. Elements
3447:"Communicability across evolving networks"
669:
655:
480:
264:
18:
3802:
3765:
3727:
3717:
3629:
3530:
3520:
3408:
3398:
3370:http://faculty.ucr.edu/~hanneman/nettext/
3281:
3280:
3278:
3257:
3251:
3231:
3212:
3201:
3200:
3191:
3181:
3151:
3150:
3148:
3116:
3115:
3113:
3087:
3086:
3084:
3064:
3040:
3020:
3015:is the external importance given to node
2999:
2993:
2974:
2963:
2962:
2948:
2947:
2938:
2928:
2898:
2897:
2895:
2887:, Katz centrality is defined as follows:
2865:
2859:
2823:
2817:
2816:
2809:
2798:
2766:
2765:
2760:
2744:
2734:
2728:
2727:
2720:
2709:
2671:
2670:
2665:
2659:
2636:
2599:
2582:
2545:
2534:
2532:
2531:
2515:
2498:
2461:
2450:
2448:
2447:
2443:
2428:
2417:
2415:
2414:
2411:
2378:
2361:
2330:
2313:
2285:
2284:
2282:
2259:
2239:
2219:
2195:
2183:
2156:
2152:
2151:
2142:
2121:
2102:
2089:
2083:
2053:
2038:
2029:
2021:
2013:
2000:
1988:
1972:
1961:
1937:
1931:
1871:
1853:
1849:
1848:
1832:
1821:
1787:
1769:
1741:
1731:
1713:
1688:
1682:
1649:
1647:
1627:
1599:
1581:
1571:
1533:
1532:
1522:
1519:
1488:
1468:
1448:
1428:
1407:
1401:
1369:
1342:
1332:
1319:
1309:
1298:
1288:
1277:
1245:
1244:
1238:
1197:
1153:
1152:
1146:
1110:
1101:
1080:
1074:
1031:
1022:
979:
967:
940:
928:
901:
889:
862:
850:
824:
800:
794:
774:
754:
3706:International Journal of Neural Systems
3352:
1188:denotes Katz centrality of a node
603:
569:
483:
474:
405:
350:
267:
256:
180:
56:
30:
1483:. The value of the attenuation factor
1181:{\displaystyle C_{\mathrm {Katz} }(i)}
1753:{\displaystyle (I-\alpha A^{T})^{-1}}
1662:{\displaystyle {\overrightarrow {I}}}
7:
1813:Network producing data of the form:
3667:Social Network Analysis and Mining
2785:
2782:
2779:
2776:
2773:
2770:
2767:
2751:
2748:
2745:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
1543:
1540:
1537:
1534:
1289:
1255:
1252:
1249:
1246:
1163:
1160:
1157:
1154:
789:and the distance between nodes as
14:
1443:degree connections between nodes
3813:10.1088/1742-5468/2005/10/P10014
2015:there is an edge from node
1799:{\displaystyle (I-\alpha A^{T})}
1223:{\displaystyle \alpha \in (0,1)}
994:{\displaystyle (0.5)^{4}=0.0625}
43:
2755:
2743:
1876:
1870:
1065:and 0 otherwise. The powers of
955:{\displaystyle (0.5)^{3}=0.125}
3843:"Welcome to igraph's new home"
3286:
3206:
3188:
3165:
3156:
3121:
3092:
2968:
2953:
2935:
2912:
2903:
2609:
2589:
2583:
2558:
2546:
2539:
2526:
2505:
2499:
2474:
2462:
2455:
2435:
2429:
2422:
2368:
2362:
2320:
2314:
2192:
2185:
1979:
1973:
1839:
1833:
1793:
1771:
1738:
1715:
1596:
1578:
1555:
1552:
1383:
1371:
1339:
1325:
1267:
1261:
1217:
1205:
1175:
1169:
1122:
1103:
1040:
1024:
976:
969:
937:
930:
916:{\displaystyle (0.5)^{2}=0.25}
898:
891:
859:
852:
1:
3640:10.1016/S0378-8733(01)00038-7
1423:reflects the total number of
877:{\displaystyle (0.5)^{1}=0.5}
2403:This can be normalized via:
1134:{\displaystyle (a_{2,12})=1}
3754:Journal of Complex Networks
838:{\displaystyle \alpha =0.5}
809:{\displaystyle \alpha ^{d}}
3889:
3673:(4). Springer: 1311–1323.
3474:10.1103/PhysRevE.83.046120
3295:{\displaystyle {\vec {e}}}
3130:{\displaystyle {\vec {e}}}
3101:{\displaystyle {\vec {e}}}
717:between a pair of actors.
693:of a node is a measure of
3719:10.1142/S0129065717500137
3679:10.1007/s13278-013-0143-7
522:Exponential random (ERGM)
189:Informational (computing)
2207:{\displaystyle (Q)_{ij}}
1642:is the identity matrix,
1046:{\displaystyle (a_{ij})}
1005:Mathematical formulation
209:Scientific collaboration
3868:Social network analysis
3048:{\displaystyle \alpha }
2880:{\displaystyle A_{i,j}}
1496:{\displaystyle \alpha }
782:{\displaystyle \alpha }
762:{\displaystyle \alpha }
701:. It was introduced by
638:Category:Network theory
158:Preferential attachment
3873:Algebraic graph theory
3522:10.1098/rspa.2009.0456
3317:are rendered useless.
3315:eigenvector centrality
3296:
3267:
3240:
3217:
3131:
3102:
3073:
3049:
3029:
3009:
2979:
2881:
2836:
2814:
2725:
2645:
2622:
2394:
2268:
2248:
2228:
2208:
2172:
2131:
2069:
1947:
1917:
1800:
1754:
1698:
1663:
1636:
1613:
1497:
1477:
1457:
1437:
1417:
1390:
1355:
1314:
1293:
1224:
1192:, then, given a value
1182:
1135:
1090:
1047:
995:
956:
917:
878:
839:
810:
783:
763:
745:
730:eigenvector centrality
527:Random geometric (RGG)
3776:10.1093/comnet/cnt001
3297:
3268:
3266:{\displaystyle e_{j}}
3241:
3218:
3132:
3103:
3074:
3050:
3030:
3010:
3008:{\displaystyle e_{j}}
2980:
2882:
2837:
2794:
2705:
2646:
2623:
2395:
2269:
2249:
2229:
2209:
2173:
2132:
2070:
1948:
1946:{\displaystyle t_{k}}
1918:
1801:
1755:
1699:
1697:{\displaystyle A^{T}}
1664:
1637:
1614:
1498:
1478:
1458:
1438:
1418:
1416:{\displaystyle A^{k}}
1391:
1389:{\displaystyle (i,j)}
1356:
1294:
1273:
1225:
1183:
1136:
1091:
1089:{\displaystyle A^{3}}
1061:is connected to node
1048:
996:
957:
918:
879:
840:
811:
784:
764:
743:
643:Category:Graph theory
3277:
3250:
3230:
3147:
3112:
3083:
3063:
3039:
3019:
2992:
2894:
2858:
2658:
2635:
2410:
2281:
2258:
2238:
2218:
2182:
2141:
2082:
1960:
1930:
1820:
1768:
1712:
1681:
1669:is a vector of size
1646:
1626:
1518:
1487:
1467:
1447:
1427:
1400:
1368:
1237:
1196:
1145:
1100:
1073:
1021:
966:
927:
888:
849:
823:
793:
773:
753:
3513:2010RSPSA.466..753G
3466:2011PhRvE..83d6120G
3400:10.1017/nws.2016.21
3246:differs exactly by
2851:Given a graph with
2790:
2701:
2031: at time
2023: to node
447:Degree distribution
98:Community structure
3560:10.1007/BF02289026
3460:(4). APS: 046120.
3383:"Spectral ranking"
3381:Vigna, S. (2016).
3292:
3263:
3236:
3213:
3127:
3098:
3069:
3045:
3025:
3005:
2975:
2877:
2832:
2756:
2661:
2641:
2618:
2390:
2264:
2244:
2224:
2204:
2168:
2127:
2065:
2060:
1943:
1913:
1796:
1750:
1694:
1659:
1632:
1609:
1493:
1473:
1453:
1433:
1413:
1386:
1351:
1230:, mathematically:
1220:
1178:
1131:
1086:
1043:
991:
952:
913:
874:
835:
806:
779:
759:
746:
631:Network scientists
557:Soft configuration
3507:(2115): 753–770.
3454:Physical Review E
3289:
3239:{\displaystyle j}
3209:
3159:
3124:
3095:
3072:{\displaystyle A}
3028:{\displaystyle j}
2971:
2956:
2906:
2644:{\displaystyle n}
2613:
2542:
2458:
2425:
2267:{\displaystyle j}
2247:{\displaystyle i}
2227:{\displaystyle w}
2056:
2032:
2024:
2016:
1874:
1706:transposed matrix
1657:
1635:{\displaystyle I}
1607:
1530:
1476:{\displaystyle j}
1456:{\displaystyle i}
1436:{\displaystyle k}
720:It is similar to
679:
678:
599:
598:
507:Bianconi–Barabási
401:
400:
219:Artificial neural
194:Telecommunication
3880:
3863:Graph invariants
3847:
3846:
3839:
3833:
3832:
3806:
3786:
3780:
3779:
3769:
3748:
3742:
3741:
3731:
3721:
3697:
3691:
3690:
3662:
3656:
3653:
3644:
3643:
3633:
3613:
3607:
3606:
3578:
3572:
3571:
3543:
3537:
3536:
3534:
3524:
3492:
3486:
3485:
3451:
3442:
3436:
3433:
3424:
3421:
3415:
3414:
3412:
3402:
3378:
3372:
3366:
3360:
3357:
3338:college football
3301:
3299:
3298:
3293:
3291:
3290:
3282:
3272:
3270:
3269:
3264:
3262:
3261:
3245:
3243:
3242:
3237:
3222:
3220:
3219:
3214:
3211:
3210:
3202:
3199:
3198:
3186:
3185:
3161:
3160:
3152:
3136:
3134:
3133:
3128:
3126:
3125:
3117:
3107:
3105:
3104:
3099:
3097:
3096:
3088:
3078:
3076:
3075:
3070:
3054:
3052:
3051:
3046:
3034:
3032:
3031:
3026:
3014:
3012:
3011:
3006:
3004:
3003:
2984:
2982:
2981:
2976:
2973:
2972:
2964:
2958:
2957:
2949:
2946:
2945:
2933:
2932:
2908:
2907:
2899:
2886:
2884:
2883:
2878:
2876:
2875:
2853:adjacency matrix
2847:Alpha centrality
2841:
2839:
2838:
2833:
2831:
2830:
2822:
2821:
2813:
2808:
2789:
2788:
2764:
2754:
2742:
2741:
2733:
2732:
2724:
2719:
2700:
2699:
2669:
2650:
2648:
2647:
2642:
2627:
2625:
2624:
2619:
2614:
2612:
2608:
2607:
2606:
2598:
2594:
2593:
2592:
2562:
2561:
2544:
2543:
2538:
2533:
2524:
2523:
2522:
2514:
2510:
2509:
2508:
2478:
2477:
2460:
2459:
2454:
2449:
2444:
2439:
2438:
2427:
2426:
2421:
2416:
2399:
2397:
2396:
2391:
2386:
2385:
2377:
2373:
2372:
2371:
2338:
2337:
2329:
2325:
2324:
2323:
2290:
2289:
2273:
2271:
2270:
2265:
2253:
2251:
2250:
2245:
2233:
2231:
2230:
2225:
2213:
2211:
2210:
2205:
2203:
2202:
2177:
2175:
2174:
2169:
2167:
2166:
2155:
2136:
2134:
2133:
2128:
2126:
2125:
2107:
2106:
2094:
2093:
2078:The time points
2074:
2072:
2071:
2066:
2064:
2063:
2057:
2054:
2043:
2042:
2033:
2030:
2025:
2022:
2017:
2014:
1996:
1995:
1987:
1983:
1982:
1952:
1950:
1949:
1944:
1942:
1941:
1922:
1920:
1919:
1914:
1875:
1872:
1869:
1865:
1864:
1863:
1852:
1843:
1842:
1805:
1803:
1802:
1797:
1792:
1791:
1762:matrix inversion
1759:
1757:
1756:
1751:
1749:
1748:
1736:
1735:
1703:
1701:
1700:
1695:
1693:
1692:
1668:
1666:
1665:
1660:
1658:
1650:
1641:
1639:
1638:
1633:
1618:
1616:
1615:
1610:
1608:
1600:
1589:
1588:
1576:
1575:
1548:
1547:
1546:
1531:
1523:
1502:
1500:
1499:
1494:
1482:
1480:
1479:
1474:
1462:
1460:
1459:
1454:
1442:
1440:
1439:
1434:
1422:
1420:
1419:
1414:
1412:
1411:
1395:
1393:
1392:
1387:
1360:
1358:
1357:
1352:
1350:
1349:
1337:
1336:
1324:
1323:
1313:
1308:
1292:
1287:
1260:
1259:
1258:
1229:
1227:
1226:
1221:
1187:
1185:
1184:
1179:
1168:
1167:
1166:
1140:
1138:
1137:
1132:
1121:
1120:
1095:
1093:
1092:
1087:
1085:
1084:
1052:
1050:
1049:
1044:
1039:
1038:
1015:adjacency matrix
1000:
998:
997:
992:
984:
983:
961:
959:
958:
953:
945:
944:
922:
920:
919:
914:
906:
905:
883:
881:
880:
875:
867:
866:
844:
842:
841:
836:
815:
813:
812:
807:
805:
804:
788:
786:
785:
780:
768:
766:
765:
760:
691:alpha centrality
671:
664:
657:
542:Stochastic block
532:Hyperbolic (HGN)
481:
344:
333:
265:
173:Social influence
47:
19:
3888:
3887:
3883:
3882:
3881:
3879:
3878:
3877:
3853:
3852:
3851:
3850:
3841:
3840:
3836:
3804:physics/0505169
3788:
3787:
3783:
3750:
3749:
3745:
3699:
3698:
3694:
3664:
3663:
3659:
3654:
3647:
3631:10.1.1.226.2113
3618:Social Networks
3615:
3614:
3610:
3595:10.2307/2785990
3580:
3579:
3575:
3545:
3544:
3540:
3501:Proc. R. Soc. A
3494:
3493:
3489:
3449:
3444:
3443:
3439:
3434:
3427:
3422:
3418:
3387:Network Science
3380:
3379:
3375:
3367:
3363:
3358:
3354:
3349:
3308:
3275:
3274:
3253:
3248:
3247:
3228:
3227:
3187:
3177:
3145:
3144:
3110:
3109:
3081:
3080:
3061:
3060:
3057:spectral radius
3037:
3036:
3017:
3016:
2995:
2990:
2989:
2934:
2924:
2892:
2891:
2861:
2856:
2855:
2849:
2815:
2726:
2656:
2655:
2633:
2632:
2578:
2568:
2564:
2563:
2530:
2529:
2525:
2494:
2484:
2480:
2479:
2446:
2445:
2413:
2408:
2407:
2357:
2347:
2343:
2342:
2309:
2299:
2295:
2294:
2279:
2278:
2256:
2255:
2236:
2235:
2216:
2215:
2191:
2180:
2179:
2150:
2139:
2138:
2117:
2098:
2085:
2080:
2079:
2059:
2058:
2051:
2045:
2044:
2034:
2011:
2001:
1968:
1964:
1963:
1958:
1957:
1933:
1928:
1927:
1847:
1828:
1827:
1823:
1818:
1817:
1783:
1766:
1765:
1737:
1727:
1710:
1709:
1684:
1679:
1678:
1644:
1643:
1624:
1623:
1577:
1567:
1521:
1516:
1515:
1485:
1484:
1465:
1464:
1445:
1444:
1425:
1424:
1403:
1398:
1397:
1366:
1365:
1338:
1328:
1315:
1240:
1235:
1234:
1194:
1193:
1148:
1143:
1142:
1106:
1098:
1097:
1076:
1071:
1070:
1027:
1019:
1018:
1007:
975:
964:
963:
936:
925:
924:
897:
886:
885:
858:
847:
846:
821:
820:
796:
791:
790:
771:
770:
751:
750:
738:
687:Katz centrality
675:
613:
578:Boolean network
552:Maximum entropy
502:Barabási–Albert
419:
336:
325:
113:Controllability
78:Complex network
65:
52:
51:
50:
49:
48:
32:Network science
17:
12:
11:
5:
3886:
3884:
3876:
3875:
3870:
3865:
3855:
3854:
3849:
3848:
3834:
3797:(10): P10014.
3781:
3743:
3712:(2): 1750013.
3692:
3657:
3645:
3624:(3): 191–201.
3608:
3589:(4): 377–399.
3573:
3538:
3487:
3437:
3425:
3416:
3393:(4): 433–445.
3373:
3361:
3351:
3350:
3348:
3345:
3307:
3304:
3288:
3285:
3260:
3256:
3235:
3224:
3223:
3208:
3205:
3197:
3194:
3190:
3184:
3180:
3176:
3173:
3170:
3167:
3164:
3158:
3155:
3123:
3120:
3094:
3091:
3068:
3044:
3024:
3002:
2998:
2986:
2985:
2970:
2967:
2961:
2955:
2952:
2944:
2941:
2937:
2931:
2927:
2923:
2920:
2917:
2914:
2911:
2905:
2902:
2874:
2871:
2868:
2864:
2848:
2845:
2844:
2843:
2829:
2826:
2820:
2812:
2807:
2804:
2801:
2797:
2793:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2763:
2759:
2753:
2750:
2747:
2740:
2737:
2731:
2723:
2718:
2715:
2712:
2708:
2704:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2668:
2664:
2640:
2629:
2628:
2617:
2611:
2605:
2602:
2597:
2591:
2588:
2585:
2581:
2577:
2574:
2571:
2567:
2560:
2557:
2554:
2551:
2548:
2541:
2537:
2528:
2521:
2518:
2513:
2507:
2504:
2501:
2497:
2493:
2490:
2487:
2483:
2476:
2473:
2470:
2467:
2464:
2457:
2453:
2442:
2437:
2434:
2431:
2424:
2420:
2401:
2400:
2389:
2384:
2381:
2376:
2370:
2367:
2364:
2360:
2356:
2353:
2350:
2346:
2341:
2336:
2333:
2328:
2322:
2319:
2316:
2312:
2308:
2305:
2302:
2298:
2293:
2288:
2263:
2243:
2223:
2201:
2198:
2194:
2190:
2187:
2165:
2162:
2159:
2154:
2149:
2146:
2124:
2120:
2116:
2113:
2110:
2105:
2101:
2097:
2092:
2088:
2076:
2075:
2062:
2052:
2050:
2047:
2046:
2041:
2037:
2028:
2020:
2012:
2010:
2007:
2006:
2004:
1999:
1994:
1991:
1986:
1981:
1978:
1975:
1971:
1967:
1940:
1936:
1924:
1923:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1868:
1862:
1859:
1856:
1851:
1846:
1841:
1838:
1835:
1831:
1826:
1795:
1790:
1786:
1782:
1779:
1776:
1773:
1747:
1744:
1740:
1734:
1730:
1726:
1723:
1720:
1717:
1691:
1687:
1656:
1653:
1631:
1620:
1619:
1606:
1603:
1598:
1595:
1592:
1587:
1584:
1580:
1574:
1570:
1566:
1563:
1560:
1557:
1554:
1551:
1545:
1542:
1539:
1536:
1529:
1526:
1492:
1472:
1452:
1432:
1410:
1406:
1385:
1382:
1379:
1376:
1373:
1362:
1361:
1348:
1345:
1341:
1335:
1331:
1327:
1322:
1318:
1312:
1307:
1304:
1301:
1297:
1291:
1286:
1283:
1280:
1276:
1272:
1269:
1266:
1263:
1257:
1254:
1251:
1248:
1243:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1177:
1174:
1171:
1165:
1162:
1159:
1156:
1151:
1130:
1127:
1124:
1119:
1116:
1113:
1109:
1105:
1083:
1079:
1042:
1037:
1034:
1030:
1026:
1006:
1003:
990:
987:
982:
978:
974:
971:
951:
948:
943:
939:
935:
932:
912:
909:
904:
900:
896:
893:
873:
870:
865:
861:
857:
854:
834:
831:
828:
803:
799:
778:
758:
737:
734:
707:social network
677:
676:
674:
673:
666:
659:
651:
648:
647:
646:
645:
640:
634:
633:
628:
623:
615:
614:
612:
611:
608:
604:
601:
600:
597:
596:
595:
594:
585:
580:
572:
571:
567:
566:
565:
564:
559:
554:
549:
544:
539:
534:
529:
524:
519:
517:Watts–Strogatz
514:
509:
504:
499:
494:
486:
485:
477:
476:
472:
471:
470:
469:
464:
459:
454:
449:
444:
439:
434:
429:
421:
420:
418:
417:
412:
406:
403:
402:
399:
398:
397:
396:
391:
386:
381:
376:
371:
366:
361:
353:
352:
348:
347:
346:
345:
338:Incidence list
334:
327:Adjacency list
323:
318:
313:
308:
303:
298:
296:Data structure
293:
288:
283:
278:
270:
269:
261:
260:
254:
253:
252:
251:
246:
241:
236:
231:
226:
224:Interdependent
221:
216:
211:
206:
201:
196:
191:
183:
182:
178:
177:
176:
175:
170:
168:Network effect
165:
163:Balance theory
160:
155:
150:
145:
140:
135:
130:
125:
123:Social capital
120:
115:
110:
105:
100:
95:
90:
85:
80:
75:
67:
66:
64:
63:
57:
54:
53:
42:
41:
40:
39:
38:
35:
34:
28:
27:
15:
13:
10:
9:
6:
4:
3:
2:
3885:
3874:
3871:
3869:
3866:
3864:
3861:
3860:
3858:
3844:
3838:
3835:
3830:
3826:
3822:
3818:
3814:
3810:
3805:
3800:
3796:
3792:
3785:
3782:
3777:
3773:
3768:
3763:
3759:
3755:
3747:
3744:
3739:
3735:
3730:
3725:
3720:
3715:
3711:
3707:
3703:
3696:
3693:
3688:
3684:
3680:
3676:
3672:
3668:
3661:
3658:
3652:
3650:
3646:
3641:
3637:
3632:
3627:
3623:
3619:
3612:
3609:
3604:
3600:
3596:
3592:
3588:
3584:
3577:
3574:
3569:
3565:
3561:
3557:
3553:
3549:
3548:Psychometrika
3542:
3539:
3533:
3528:
3523:
3518:
3514:
3510:
3506:
3502:
3498:
3491:
3488:
3483:
3479:
3475:
3471:
3467:
3463:
3459:
3455:
3448:
3441:
3438:
3432:
3430:
3426:
3420:
3417:
3411:
3406:
3401:
3396:
3392:
3388:
3384:
3377:
3374:
3371:
3365:
3362:
3356:
3353:
3346:
3344:
3341:
3339:
3334:
3331:
3327:
3322:
3318:
3316:
3311:
3305:
3303:
3283:
3258:
3254:
3233:
3203:
3195:
3192:
3182:
3178:
3174:
3171:
3168:
3162:
3153:
3143:
3142:
3141:
3138:
3118:
3089:
3066:
3058:
3042:
3022:
3000:
2996:
2965:
2959:
2950:
2942:
2939:
2929:
2925:
2921:
2918:
2915:
2909:
2900:
2890:
2889:
2888:
2872:
2869:
2866:
2862:
2854:
2846:
2827:
2824:
2810:
2805:
2802:
2799:
2795:
2791:
2761:
2757:
2738:
2735:
2721:
2716:
2713:
2710:
2706:
2702:
2666:
2662:
2654:
2653:
2652:
2638:
2615:
2603:
2600:
2595:
2586:
2579:
2575:
2572:
2569:
2565:
2555:
2552:
2549:
2519:
2516:
2511:
2502:
2495:
2491:
2488:
2485:
2481:
2471:
2468:
2465:
2440:
2432:
2406:
2405:
2404:
2387:
2382:
2379:
2374:
2365:
2358:
2354:
2351:
2348:
2344:
2339:
2334:
2331:
2326:
2317:
2310:
2306:
2303:
2300:
2296:
2291:
2277:
2276:
2275:
2261:
2241:
2221:
2199:
2196:
2188:
2163:
2160:
2157:
2147:
2144:
2122:
2118:
2114:
2111:
2108:
2103:
2099:
2095:
2090:
2086:
2048:
2039:
2035:
2026:
2018:
2008:
2002:
1997:
1992:
1989:
1984:
1976:
1969:
1965:
1956:
1955:
1954:
1938:
1934:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1866:
1860:
1857:
1854:
1844:
1836:
1829:
1824:
1816:
1815:
1814:
1811:
1807:
1788:
1784:
1780:
1777:
1774:
1763:
1745:
1742:
1732:
1728:
1724:
1721:
1718:
1707:
1689:
1685:
1676:
1672:
1654:
1651:
1629:
1604:
1601:
1593:
1590:
1585:
1582:
1572:
1568:
1564:
1561:
1558:
1549:
1527:
1524:
1514:
1513:
1512:
1510:
1506:
1490:
1470:
1450:
1430:
1408:
1404:
1380:
1377:
1374:
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563:
562:LFR Benchmark
560:
558:
555:
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550:
548:
547:Blockmodeling
545:
543:
540:
538:
535:
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530:
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523:
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518:
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512:Fitness model
510:
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452:Assortativity
450:
448:
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185:
184:
181:Network types
179:
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156:
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139:
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128:Link analysis
126:
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118:Graph drawing
116:
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686:
683:graph theory
680:
537:Hierarchical
492:Random graph
340: /
329: /
311:Neighborhood
153:Transitivity
133:Optimization
3410:2434/527942
736:Measurement
728:and to the
583:agent based
497:Erdős–Rényi
138:Reciprocity
103:Percolation
88:Small-world
3857:Categories
3583:Sociometry
3347:References
2234:from node
2178:for which
1505:eigenvalue
695:centrality
610:Categories
467:Efficiency
462:Modularity
442:Clustering
427:Centrality
415:Algorithms
239:Dependency
214:Biological
93:Scale-free
3821:1742-5468
3767:1207.5047
3626:CiteSeerX
3568:121768822
3287:→
3207:→
3193:−
3175:α
3172:−
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2969:→
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2796:∑
2707:∑
2601:−
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1296:∑
1290:∞
1275:∑
1203:∈
1200:α
827:α
798:α
777:α
757:α
359:Bipartite
281:Component
199:Transport
148:Homophily
108:Evolution
83:Contagion
3829:15120571
3738:28076982
3482:21599253
3273:, so if
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2254:to node
1760:denotes
726:PageRank
711:geodesic
703:Leo Katz
626:Software
588:Epidemic
570:Dynamics
484:Topology
457:Distance
394:Weighted
369:Directed
364:Complete
268:Features
229:Semantic
24:a series
22:Part of
3687:7125694
3603:2785990
3509:Bibcode
3462:Bibcode
3330:neurons
1013:be the
699:network
410:Metrics
379:Labeled
249:on-Chip
234:Spatial
143:Closure
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3035:, and
2988:where
989:0.0625
722:Google
685:, the
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475:Models
432:Degree
389:Random
342:matrix
331:matrix
321:Vertex
276:Clique
258:Graphs
204:Social
61:Theory
3825:S2CID
3799:arXiv
3762:arXiv
3683:S2CID
3599:JSTOR
3564:S2CID
3450:(PDF)
1622:Here
950:0.125
715:walks
697:in a
607:Lists
437:Motif
384:Multi
374:Hyper
351:Types
291:Cycle
73:Graph
3817:ISSN
3795:2005
3734:PMID
3478:PMID
2115:<
2109:<
2096:<
1463:and
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911:0.25
316:Path
306:Loop
301:Edge
244:Flow
3809:doi
3772:doi
3724:hdl
3714:doi
3675:doi
3636:doi
3591:doi
3556:doi
3527:hdl
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3470:doi
3405:hdl
3395:doi
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