Connectivity of Waxman topology models

Connectivity of Waxman topology models

M.Naldi*

Universita`di Roma‘Tor Vergata’,Dip.di Informatica Sistemi Produzione(DISP),Via del Politecnico1,00133Roma,Italy

Received25May2004;revised28January2005;accepted31January2005

Available online25February2005

Abstract

Waxman graphs are a popular class of random graphs used for modelling the Internet topology,especially for the intra-domain part.When used for network modelling purposes their connectedness properties are particularly relevant,both for the characteristics of the realized graph and for the generation time.In this paper,the probability of obtaining a connected Waxman graph is derived analytically for the general case of nodes disposed over a rectangular area(as opposed to the special case of an underlying square area already investigated in the literature) and is related to the values of the Waxman parameters used in the generation of the graph.

q2005Elsevier B.V.All rights reserved.

1.Introduction

Waxman graphs have been introduced by Waxman as a particular class of random graphs to model the network topology for the purpose of evaluating routing algorithms [1].In a Waxman graph,the nodes are uniformly distributed over a rectangular area,and links are added between the nodes through a random mechanism(the probability that two nodes are directly connected decreases exponentially as their Euclidean distance increases).Waxman graphs fall into the general class of spatial models,where the connectivity properties of the graph are directly related to the spatial relationships among the routers(i.e.their distances).As alternatives to this class of models,more recently structural models(where the interconnection of component networks as a relevant Internet formation mechanism is explicitly accounted for,resulting in a hierarchical graph[6,7,21])and degree-based models (where networks are formed by incremental addition of nodes and links,and new nodes preferentially connect to larger-degree nodes[22,23]).Despite the number of competing models,Waxman graphs have been widely accepted as a topology model for the intra-domain network and employed in many studies of routing performance,see [2–5].The suitability for this purpose has been recognized also in the wider context of Internet topology modelling,in particular since its distance-dependent model of link formation among routers appears to describe remarkably well the real world[20].In addition,more complex topology models(such as the transit-stub model implemented in the GT-ITM tool[6])incorporate it to model the intra-domain components.

Despite its general acceptance the properties of Waxman graphs have not been fully explored yet.In Ref.[7]some basic characteristics,such as the average node degree,the number of bicomponents,and the diameter have been obtained by simulation;its expansion,resilience and distortion properties have been addressed in Ref.[8];its spectral properties have been investigated in Ref.[9];its link density and the average number of paths between two nodes have been derived in Ref.[10].All the studies conducted so far have focused on the particular case in which the nodes are distributed over a square area.

In this paper,we consider instead the more general case of a rectangular area,with the aim of deriving the connectedness properties of the graph,i.e.the probability that the resulting graph is connected,and relate it to the graph generation parameters.This aim is relevant,since any evaluation process in which a Waxman graph is adopted would require the generation of many instances of such graphs.In the typical case,we require that the generated graphs are connected,which means that every unconnected graph is simply discarded,with longer generation times the lower the connectedness probability. If our purpose is instead the intentional generation of faulty (i.e.unconnected)random topologies,the knowledge

of

Computer Communications29(2005)24–31

https://www.wendangku.net/doc/25200338.html,/locate/comcom

0140-3664/$-see front matter q2005Elsevier B.V.All rights reserved.

doi:10.1016/https://www.wendangku.net/doc/25200338.html,com.2005.01.017

*Tel.:C39672597269;fax:C39672597532.

E-mail address:naldi@disp.uniroma2.it.

the connectedness probability allows us to predict their proportion over the whole generated set.

After a brief description of Waxman graphs in Section2 we arrive at the connectedness probability through some intermediate steps,namely a model of the inter-node distance(Section3),the analytical evaluation of the link probability for the generic couple of nodes(Section4),and ?nally the connectedness probability through the appli-cation of a known result in random graph theory(Section5).

2.A description of Waxman graphs

In the original formulation by Waxman,such graphs have a pre-determined number N of nodes,which are uniformly distributed over a rectangular coordinate grid[1]; this means that every node has integer coordinates,while we

consider the more general case,where a node can lie anywhere within the rectangular area.The probability that a direct link exists between the generic nodes u and v is related to the Euclidean distance d(u,v)between them by the expression

Pef u;v gTZ b e K deu;vTL a;(1) where L is the maximum distance between two nodes,and a and b are two parameters in the(0,1]https://www.wendangku.net/doc/25200338.html,rger values of b result in graphs with higher link densities,while small values of a increase the density of short links relative to longer ones(in some later formulations of the model the positions,and roles,of a and b are interchanged,[7];in this paper we keep to the original formulation).

No agreement exists over the typical values to be adopted for these two parameters:Waxman himself sets a Z b Z0.4 [1];in Ref.[3]several combinations are considered,with a lying in the0.02K0.11range and b in the0.75j1range; Zegura et https://www.wendangku.net/doc/25200338.html,e a Z0.15,b Z0.2in Ref.[11],but several combinations with a and b both lying in the0.1j0.4range in[7].

A sample Waxman graph is shown in Fig.1.

3.The distance distribution

Since the probability that a link exists between two nodes depends on their distance,and this is a random variable,our ?rst step is to derive the probability density function(pdf)of the distance between two nodes.

We consider the general case of a rectangle that contains our nodes,whose side ratio(ratio between the smaller and the larger of the rectangle sides)is indicated by r.We are interested in the normalized distance D,i.e.the distance divided by the length of the rectangle diagonal,since the link probability expression(1)can be reformulated as

Pef u;v gTZ b e K Deu;vTa;(2)where D(u,v)Z d(u,v)/L.Van Mieghem has provided a closed form(to be evaluated numerically)for the pdf of D in the particular case of a square area(i.e.r Z1),but the resulting expression is quite awkward and not extendible to the rectangle case[10].On the other hand,the average distance between two nodes on a square is known to be[12] E?D Z

2C5

???

2

p

lne

???

2

p

C1TC2

???

2

p

30

z0:3687:(3) Using the same arguments of the square case,the average distance between two nodes on a rectangle has been found to be[13],as derived in the Appendix A

E?D Z

4

?????????????

1C r2

p

e1

e1

e1K tTe1K uT

??????????????????

t2C r2u2

p

d t d u;

(4) which can be approximated by the polynomial

E?D z

4

?????????????

1C r2

p

X4

i Z0

k i r i;(5) where the coef?cients are

k0Z0:0831049k1Z0:00767486

k2Z0:0788753k3Z K0:0541958

k4Z0:0145688:

(6)

Since E[D2]Z1/6by elementary methods,irrespective of the side ratio,the standard deviation can be computed as Std?D Z

?????????????????????????????

E?D2 K E2?D

p

Z

????????????????????????

1=6K E2?D

p

.

For the limiting case of r Z0the rectangle becomes a unit segment and the pdf of the distance between two random points on that segment has the linear(triangular)shape

f DexTZ2K2xe0%x%1T:(7)

For the remaining range of values of the side ratio, instead of trying to obtain the distance pdf by direct methods,we?rst go on to simulate a number of

Waxman

Fig.1.Sample Waxman graph.

M.Naldi/Computer Communications29(2005)24–3125

graphs and obtain the resulting empirical distributions.Each simulated graph has been obtained by the following steps:(a)setting the number of nodes of the graph;(b)setting the side ratio;(c)generating for each node two uniform random variables within the ranges determined by the side ratio to provide the two coordinates of the node location;(d)computing,for each couple of nodes,the Euclidean distance.The simulation has been conducted for several graph sizes and side ratios.Here,we report the pictures of what is obtained for some values of the side ratio in Fig.2after simulating 100different graphs of 100nodes each (that is equivalent to a sample of 5.05!105distances).

As the side ratio grows,the shape goes from the triangular one to a less skewed one up to a nearly symmetrical one.A regular probability density function well suited to model this variation by a simple change of its parameters’values and having a ?nite support—(0K 1)in our case—is the Beta distribution,whose analytical form is f D ex TZ

G el 1C l 2TG el 1TG el 2T

x l 1K 1

e1K x Tl 2K 1

0%x %1;

(8)

where G ($)is the Gamma function and l 1,l 2are the two

parameters that de?ne the particular distribution within the Beta family.We then try to model our empirical

distribution by the Beta one with a suitable choice of its parameters.

In order to estimate the two parameters of the Beta distribution,we resort to the method of moments,which provides us with the following expressions for the estimates [14]:

^l 1Z

E 2?D K E ?D E ?D 2 E ?D K E ?D ;(9)

^l 2Z E ?D K E ?D 2 E ?D 2 K E 2?D

K ^l 1:(10)

At this point rather than rely on the simulation results (for the estimates of the ?rst two moments of the distance to insert in the previous expressions)we can obtain the estimates of the Beta distribution parameters by inserting the values for the ?rst two moments of the distance as obtained by the analytical expressions (3)and (4).We obtain the following results for a set of side ratio values.The goodness of ?t can be evaluated through the QQ-plot,i.e.by plotting the empirical quantiles of the distance (obtained by simulation)vs.the corresponding expected quantiles (obtained by the Beta distribution with the parameters of Table 1:a linear plot indicates a perfect

?t.

Fig.2.Distance distributions for some side ratios.

M.Naldi /Computer Communications 29(2005)24–31

26

The two sample plots shown in Fig.3(which are the worst cases)show that this is quite the case.

As a quantitative indicator of the goodness-of-?t the determination coef?cient (i.e.the square of the regression coef?cient)pertaining to the linear regression on the QQ-plot has been used.This is equivalent to the choice of the regression coef?cient suggested in ch.5of Ref.[18]and carried out in Ref.[17].The use of alternative goodness-of-?t evaluation techniques such as the c 2test or the Kolmogorov–Smirnov test has been put down because the large numerosity of the sample in our case makes the error of the second kind (i.e.the probability of accepting the Beta model hypothesis when it is actually wrong)overly small,so that the Beta model is unduly rejected [19].The application of linear regression supplies the determination coef?cient reported in Table 2.

As can be seen,for all the side ratios the coef?cient of determination is largely in excess of 0.99,so that the QQ-plot can be considered to be quite linear,and therefore

the Beta model is certainly a suitable model for the pdf of distances.

4.The link probability

At this point,we can use the distance probability model to evaluate the probability that a link exists between two generic nodes in the graph.

Recalling the expressions (2)and (8)we have

P link Z e1

P ex Tf D ex Td x

Z

e1

b e K x =a

G el 1C l 2TG el 1TG el 2Tx l 1K 1

e1K x Tl 2K 1d x

Z b

G el 1C l 2TG el 1TG el 2Te

10

e K x =a x l 1K 1

e1K x Tl 2K 1d x :(11)

The solution of the integral appearing in (11)can be extracted from formula 3.383of Ref.[15],so that the link probability can be expressed as P link Z bF el 1;l 1C l 2;K 1=a T;

(12)

where F (x ;y ;z )is the degenerate hypergeometric function,for which we can use the series expansion proposed again by formula 9.210of Ref.[15]:

P link Z b 1K

l 1l 1C l 21a C l 1el 1C 1Tel 1C l 2Tel 1C l 2C 1T1

2!a 2

C l 1el 1C 1Tel 1C 2Tel 1C l 2Tel 1C l 2C 1Tel 1C l 2C 2T1

3!a 3

C /

:(13)

Table 1

Estimated parameters of the Beta distribution Side ratio ^l 1^l 20.01

2

0.1 1.0767 2.12010.2 1.2285 2.35510.3 1.4289 2.66180.4 1.6619 3.01420.5 1.9090 3.38490.6 2.1384 3.72670.7 2.3068 3.97700.8 2.4020 4.11820.9 2.4093 4.12891.0

2.4240

4.1504

Fig.3.QQ-plot of the distances.

M.Naldi /Computer Communications 29(2005)24–31

27

In Fig.4,the relationship between the Waxman parameters a ,b and the link probability (iso-link probability curves)is graphed for the same side ratio as in Fig.2and some values of the link probability.All the curves are concave up with a strong tendency to saturation,i.e.for each value of the link probability there is a range of values for

the Waxman parameters which are not usable.For example,in the case of the square area (side ratio Z 1)a 5%link probability cannot be reached if the Beta parameter is lower than 0.07roughly.

Since the average degree of the graph with N nodes is P link (N K 1)/2,the above relationship can serve to set the values of the Waxman parameters on the basis of the desired average degree of the graph.

The effect of the side ratio on the values of the Alpha-Beta couple to use is limited.As can be seen in Fig.5,which rearranges some of the curves of Fig.4for the case P link Z 0.3,the effect is practically null for side ratio over 0.5since the curves are completely overlapped,i.e.the same Alpha–Beta values can be used regardless of the side ratio.For more elongated networks (side ratios lower than 0.5)lower Beta values are needed for the same link probability;over the whole set of considered side ratios the variation in Beta values with respect to the rectangular case to attain the same link probability ranges from 18%(lower Alpha values)to 4%(larger Alpha

values).

Fig.4.Iso-link probability curves.

Table 2

Coef?cient of determination for the linear regression on the QQ-plot Side ratio Determination coef?cient 0.01

0.10.9977011930.20.9967809690.30.9980782790.40.9982490140.50.9995623720.60.9976050960.70.9995434080.80.9990969950.90.9955235051.0

0.99358982

M.Naldi /Computer Communications 29(2005)24–31

28

5.Connectedness of the realized graphs

The ?nal step towards the computation of the probability that an N -nodes Waxman graph is connected can be accomplished through the use of the following recursive expression provided by Gilbert in Ref.[16],valid for homogeneous random graphs

P eN Tc Z 1K X N K 1k Z 1

N K 1k K 1 !P ek Tc e1K P link T

k eN K k T

;(14)which is initialized by the position P e1T

c Z 1.

When both the number of nodes and the link probability are large the above formula can be approximated by P eN Tc z 1K N e1K P link TeN K 1T:

(15)

For a graph of 20nodes,we report in Fig.5the relationship between the Waxman parameters for a set of connectedness probability values and the same set of side ratios as in Fig.4(iso-connectivity curves).

The same saturation effect,already noted for the link probability,is present here as well:some sets of values for the two Waxman parameters are simply forbidden if a given value of connectivity is desired.Similarly to the iso-link probability curves,the iso-connectivity curves allow us to set the couple of values for the Waxman parameters on the basis of either the desired proportion of unconnected graphs or the desired generation time for a connected graph.

As in the link probability case,the effect of the side ratio on the values of the Alpha–Beta couple to use is limited.As can be seen in Fig.6,which rearranges some of the curves of Fig.7for the case P c Z 0:9,the effect is practically null for side ratio over 0.5since the curves are completely overlapped,i.e.the same Alpha–Beta values can be used

regardless of the side ratio.For more elongated networks

(side ratios lower than 0.5)lower Beta values are needed for the same link probability;over the whole set of considered side ratios the variation in Beta values with respect to the rectangular case to attain the same link probability ranges from 22%(lower Alpha values)to 3%(larger Alpha values).

6.Conclusions

The connectedness properties of Waxman graphs have been investigated for the general case of nodes disposed over a rectangular area.By means of an approximate model for the probability density function of the distance between two nodes the probability that a link exists between any two nodes and the probability that a Waxman graph is connected have been obtained analytically.The derived relationships can be used for an aware generation of Waxman graphs,where their parameters are set according to the desired average node degree and/or the desired connectedness probability.

Appendix A.The average distance between points in a rectangle

Let’s consider a rectangle of sides a and b ,a being the larger,so that the side ratio is r Z b =a %1.Since the largest possible distance between two points on

it is the diagonal length ???????????????a 2C b 2p Z a ????????????1C r 2p ,the normal-ized average distance between two random points of

coordinates (x 1,y 1)and (x 2,y 2)on it is

Alpha

B e t a

Fig.5.Effect of the side ratio on the model

parameters.

Alpha

B e t a

Fig. 6.Effect of the side ratio on the model parameters in the iso-connectivity curves.

M.Naldi /Computer Communications 29(2005)24–31

29

D Z 1a ????????????1C r 2p 1a 2b

2!

eb 0eb 0ea 0ea 0

????????????????????????????????????????????

ex 1K x 2T2C ey 1K y 2T2q d x 1d x 2d y 1d y 2:(A1)

By exploiting the symmetry of the rectangle the

integration can be performed on a pair of triangular regions rather than on rectangles:

D Z 1a ????????????1C r 2p 4a 2b

2!

eb 0ey 10

ea 0ex 10

????????????????????????????????????????????

ex 1K x 2T2C ey 1K y 2T2q d x 2d x 1d y 2d y 1:(A2)

By introducing the variables z 1Z x 1K x 2and z 2Z x 1C x 2,

one of the four integrations is eliminated:

D Z 1a ????????????1C r 2p 4a 2b

2

!eb 0ey 10

ea 0e2a K z 1z 1

?????????????????????????????

z 21C ey 1K y 2T2

q d z 2d z 1d y 2d y 1

Z 1a ????????????1C r 2p 4a 2b 2eb 0ey 10e

a

0ea K z 1T!?????????????????????????????

z 21C ey 1K y 2T2

q d z 1d y 2d y 1:

(A3)

The double integration in y 1and y 2can analogously be reduced to a single integration by the use of the two variables w 1Z y 1K y 2and w 2Z y 1C y 2: D Z 1a ????????????1C r 2p 4a 2b 2

eb 0ea

0ea K z 1Teb K w 1T!???????????????

z 21C w 2

1q d z 1d w 1:

(A4)

Fig.7.Iso-connectivity curves for a 20-node graph.

M.Naldi /Computer Communications 29(2005)24–31

30

The use of the scaled variables t Z z 1/a and u Z w 1/b leads ?nally to

D Z 1a ????????????1C r 2p 4b 2

eb 0e10

e1K t Teb K w 1T???????????????????

a 2t 2C w 21q d t d w 1Z 4

a ????????????1C r

2p e10e10

e1K t Te1K u T??????????????????????a 2t 2C b 2u 2p d t d u

Z 4

????????????

1C r 2p e10e1

e1K t Te1K u T?????????????????????????t 2C eb =a T2u 2p d t d u :(A5)References

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神秘时代4研究表

元始要素 Aer (风) Aqua (水) Ignis (火) Terra (地) Ordo (秩序) Perditio (混沌) Aer元素组合： Gelum (寒冰) = Aqua + Ordo秩序水Lux (光明) = Aer + Ignis风火Motus (运动) = Aer + Ordo秩序风Potentia (能量) = Ordo + Ignis秩序火Saxum (石头) = Terra + Terra地地Tempestas (气候) = Aer + Aqua风水Vacuos (虚空) = Aer + Perdito混沌风Victus (生命) = Aqua + Terra地水 二阶复合要素Bestia (野兽) = Motus + Victus运动生命Fames (饥饿) = Victus + Vacuos生命虚空Granum (泥土) = Victus + Terra生命地Iter (旅行) = Motus + Terra运动地Limus (转换) = Victus + Aqua生命水Metallum (金属) = Saxum + Ordo秩序石头Mortuus

(死亡) = Victus + Perdito混沌生命Permutatio (交换) = Motus + Aqua运动水Praecantio (魔法) = Vacuos + Potentia虚空能量Sano (治愈) = Victus + Victus生命生命Tempus (时间) = Vacuos + Ordo秩序虚空Tenebrae (黑暗) = Vacuos + Lux虚空光明Vinculum (监禁) = Motus + Perdito运动混沌Vitreus (水晶) = Saxum + Aqua石头水Volatus (飞行) = Aer + Motus运动风 三阶复合要素 Alienis (异域) = Vacuos + Tenebrae黑暗虚空Auram (灵气) = Praecantio + Aer魔法风Corpus (肉体) = Mortuus + Bestia死亡野兽Exanimis (亡灵) = Motus + Mortuus死亡运动Herba (植物) = Granum + Terra泥土地Spiritus (灵魂) = Victus + Mortuus死亡生命Venenum (剧毒) = Aqua + Mortuus死亡水Vitium (谬误) = Praecantio + Perdito魔法混沌 四阶复合要素Arbor (木头) = Terra + Herba植物地Cognito (思虑) = Terra + Spiritus灵魂地Sensus (感官) = Aer + Spiritus灵魂风

神秘时代4傀儡核心 自动傀儡图文攻略

神秘时代4傀儡核心自动傀儡图文攻略 我的世界神秘时代4集核心傀儡是各种自动化农场林场等的必备傀儡，比如配合【收获核心】傀儡，可以制作一个自动化农场。给一个傀儡加上【收获核心】，它会自动收获成熟的作物，能收获小麦，甘蔗，可可，魔豆等等，但只有一个收获核心并不会自动补种。要让它学会自动补种，需要给它加装一个【傀儡升级：秩序】。收获傀儡没有GUI界面，也不需要绑定到任何容器。 加上秩序傀儡升级后，能自动原地补种了，但是补种的智慧有多高呢?这里就测试一下，第一轮故意留两排未种植。

用生长之锄快速催熟，两轮过后，发现左边少了两格没补种，而右边空着的耕地种植了不少。按照神秘书所说，当手里没种子时补种就无法完成，我也没设定让聚集核心傀儡只收小麦，所以那两块地的种子可能恰好被捡走了....右边原本没种植的也补种了起来，说明升级后能自动识别邻近可种植耕地，还算是比较有智慧的。 第3种是【砍伐核心】傀儡，和收获核心傀儡一样十分简单，没有GUI界面没有绑定容器，只会在周围看到树就砍，而且木头到处丢，要给他配备一个聚集核心傀儡帮他把木材装箱。

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即使完成了你可能不会得到一个迷宫的另一面，如果它是靠近现有的方尖碑。 香草村民去除硬币相关行业，并增加了一个新的“金融家”村民来改变你的硬币到有用的对象 固定相的成本为丝绸触控对焦升级，因此不会要求不可能的事 增加了一个新的3级升级为挖掘重点 重点火余烬将无法穿过固体物体了 卸下硬编码各种工具丰收水平 添加配方手艺自己怪异的眼神，一旦适当的研究已经解锁 固定流量洗涤管连接 教程目录 一、环境篇： 1.矿物的介绍 2.植物介绍 3.生物介绍 一、环境篇 1.矿物的介绍 矿物和上个版本一样，从左到右：朱砂，风之蕴魔石，火之蕴魔石，水之蕴魔石，地之蕴魔石，秩序蕴魔石，混沌蕴魔石，琥珀矿，上面为矿物原型，中间为采掘工具，下面是掉落物 2.植物介绍 左：宏伟之木，右：银树

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当这种生物群系内的条件符合刷怪范围时(光照、与主角距离等)，会刷出暴怒巨人僵尸(Furious Zombie)。 刷出的过程类似于刷怪笼，会在节点周围出现一些火，但是并不需要符合刷怪笼在人物16格以内才会刷的规律，而且只会从节点旁刷出，哪怕是在空中。 暴怒巨人僵尸看起来除了眼睛是红色以外和普通僵尸没什么差别，但是它有30颗心的血量，是普通僵尸的3倍，而且速度快，攻击路径判断更聪明，在受到伤害时，体型会增大至3格高(停止受伤时又会慢慢缩小)。 地精随时有可能终止交易，强制关闭交易界面，此时需要重新哄他开心才能再次开启交易。一般能交易十几到几十次，偶尔会1次就关闭。 杀死地精会概率掉落它手持物品和魔豆。

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我的世界神秘时代4魔法森林生物群系讲解

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关于魔豆和银树，荧光花详见神秘书，都写的非常详细了。4.0.3a添加的这种叫做地精(Pech)的新生物非常有意思。当你用比较贵重的物品送给他时(比如宝石)，他有时会出现这种和村民交易一样的粒子。 对他右键，出现了一个类似交易界面一样的内容，放上特定类型的物品，点击中间那个骰子，会出现随机交易内容。所谓特定类型的物品，必须要含有某种贵重元素。 与地精交易时需要给出含有Lucrum(贪婪，贪财)元素的物品，主要是金，钻石及其相关制品，在神秘书可以找到所有含这种元素的物品。 交易这一过程更像是抽奖，因为获得的物品是随机的。

不过一般来说给出的物品含有Lucrum元素数量越多，就有越多的几率获得更多更好的物品，但只是一般来说，有些不符合这规律，比如附魔金苹果含有的Lucrum数值最高，但它的平均效果基本还不如钻石。 而且即使是含有很少Lucrum元素的物品也有几率获得极品道具，纯属运气问题。 不同种的地精交易能获得的物品有些不同： @所有地精：魔豆，药水，神秘剑镐斧锄，马铠，金苹果，附魔金苹果，知识碎片，神秘书，附魔瓶，附魔书。 @地精强盗：矿簇，树苗，烈焰棒，顶级附魔书(比较难得，我遇到的有抢夺3，时运3，亡灵5)。 @地精猎人：神秘靴，油烛，恶魂之泪，1级弓系附魔书。 @地精法师：各元素碎片，各元素晶簇，魔杖核心：地精诅咒，1级神秘mod特有附魔书 更多我的世界攻略信息还可以关注：触手TV我的世界直播专区和触手TV我的世界攻略视频教程专区。

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Thaumium 神秘杖端 Gold 金杖端 注释2： silverwood 银树杖柄 greatwood 宏伟之木杖柄 obsidian 黑曜石杖柄 ice 寒冰杖柄 quartz 石英杖柄 blaze 烈焰杖柄 bone 白骨杖柄 图 缸中节点同样适合 节点数据：

NBT数据： ROOT： modifile 状态注释 3 type 类型注释[@sup]4[/sup] nodeVisBase 节点最大能力 nodeId 节点ID Aspects： 有几个元素就有几个T ag amount 节点魔力数量节点魔力值 = nbt数据key 类型魔力类型例如 aqua 见注释5 注释3：modifile 对应数字 -1 默认 0 明亮

1 苍白 2 凋零 注释4： 0 标准 1 不稳定 2 凶险 3 污染 4 饕餮 tāo tiè 5 纯净 注释5： 暂缺 看元素对应的英文名称小编推荐： 责任编辑【未命名】

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我的世界魔法合成攻略Minecraft神秘时代4炼金术教程 (图文) 我的世界是一款可以自由安装各种MOD的沙盒类游戏，而在众多的玩家自制MOD中，最受欢迎的想必应该是神秘时代4MOD。下面小编就给大家带来我的世界魔法合成攻略。 魔法合成及炼金术 我的世界魔法合成攻略： 在神秘之书中会写到“如果你走到了这一步，恭喜你，你已经迈入了神秘时代的第一步”。看来魔法合成在神秘时代4MOD中是很重要的，以下是魔法合成的具体步骤。 1、秘术工作台(ARCANE WORKTABLE)是在台桌(TABLE)上用魔杖右击放置制作的。右键打开工作台界面，魔杖在右上角可以取下补充魔力再放上去。当按照研究成果所提示的合成图摆放好合成所需物品时，周围的六个圆圈内会显示所需的各元素魔力VIS数目。数量不足的元素会不停闪烁。 2、给魔杖充够所需魔力后放置在右上角，可以看到6种元素图案不再闪烁，右边得到合成结果。

神秘时代4MOD炼金术教程： 炼金术的本质就纯粹是各种往坩埚(CRUCIBLE)里投放东西。主要意义在于炼出其他方式无法获得的物品，对于像炼铁，炼金，意义不大，最多废物利用的时候用到。 1、用魔杖对着炼药锅右键就能变成坩埚，坩埚加热需要一个持续的热源：在下面放一格岩浆或者一个点燃的地狱岩都可以。 2、待坩埚冒出气泡时表示已煮沸，此时投入物品会分解成构成元素。戴着揭示之护目镜时能看到元素种类和数量。这些元素会快速地随着时间分解，因此要炼制东西可不能慢慢吞吞的。

3、如果坩埚中的元素数量太多(比如丢进去一大堆东西或者直接丢进去一个装满的元素罐子)，坩埚就会溢出液体，变成咒波粘浆(FLUX GOO)，浪费元素不说，还让人十分之不爽。不过小片的咒波粘浆的话，拿些方块覆盖一下就能消除。 4、所以比较好不容易产生咒波污染的方法是使用元素精油(PHIAL OF ESSENTIA)。比如要炼制净土花(ETHEREAL BLOOM)，先在快捷物品栏准备好所需元素和材料，然后快速滚动鼠

我的世界注魔合成攻略 Minecraft神秘时代4注魔合成注意事项(图文)

我的世界注魔合成攻略Minecraft神秘时代4注魔合成注 意事项(图文) 我的世界是一款可以自由安装各种mod的沙盒类游戏，而在众多的玩家自制mod中，最受欢迎的想必应该是神秘时代4mod。下面小编就给大家带来我的世界注魔合成攻略。 注魔合成 我的世界注魔合成攻略： 注魔合成是制作高级神秘道具的必经之路。进行魔法注入需要建造注魔祭坛，建造前先要制作符文矩阵(Runic Matrix)，秘术基座(Arcane Pedestal)和一些秘术石块、石砖，这些建筑材料都需要魔法合成获得。 1、注魔祭坛的建造过程称为魔法建造(Mystical Construct)，即按九宫格在竖向上堆放材料并施加魔法的过程。类似的还有地狱熔炉，缸中节点等。先如下图所示堆放好石块石砖，基座还有符文矩阵，然后用魔杖对着符文矩阵右键。

2、然后就看到一团破碎的粒子后，出现了这样的构造，即注魔祭坛。当然，要使祭坛很好的工作，还要在周围布置一下。这里就按神秘书里的示例图布置了一下，周围四个基座上按神秘书的合成图摆上物品，以制作风之利剑为例，展示一下魔法注入的过程。另外。基座上摆放物品和取下物品都是按鼠标右键，同时附近放上所需的元素罐子，风之利剑需要的是空气(Aer)、能量(Potentia)和移动(Motus)。 3、一切准备就绪后，手持魔杖对着符文矩阵右击，可以看到符文矩阵放射出紫色光芒，从附近的元素罐子里吸收元素，然后吸收4个基座上的物品。

4、最终周围基座上的物品消失，符文矩阵下的神秘剑变成了目标物品，右键取回。风之利剑攻击相当于钻石剑，但特效很棒，不仅能对前方扇形区域进行范围横斩，可同时攻击几个怪物，右键防御时还能在身体周围形成旋风，推开怪物，使自己升空，按住右键可以升的很高。 神秘时代4mod注魔合成注意事项： 1、在魔法注入的时候，所有的物品都要在背包里，所以最好是快捷栏多备两份。因为注魔过程有时候不稳定，会发生物品从基座上滑落并产生咒波污染的情况，尤其是神秘书上标注着“不稳定性”在适量及以上的注魔类型。比如下图中制作神秘缰绳时，金锭旁边的橡木发生滑落，掉在了油烛那里，并且产生了咒波瓦斯(上方紫色透明方块)，而且被左上角跑