大规模语义网络结构：统计分析与模型

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"这篇论文深入探讨了语义网络分析，主要关注三种类型的语义网络：单词联想、WordNet和Roget's Thesaurus的大型结构。通过对这些网络的统计分析，作者揭示了它们具有小世界结构，即稀疏连接、短平均路径长度以及强烈的局部聚类特性。此外，节点连接数的分布遵循幂律，表明存在一种无标度的连接模式，大多数节点拥有相对较少的连接，而少数节点作为中心节点拥有大量连接，形成了所谓的‘枢纽’。这种规律性也在其他复杂的自然网络如万维网中被发现，但与基于继承的传统语义组织模型并不一致。" 在本文中，语义网络分析是一个关键概念，它涉及到如何理解和建模词汇之间以及概念之间的关系。通过分析不同类型的语义网络，我们可以了解到这些网络在大规模结构上的共性。例如，"小世界结构"是指尽管网络中的节点通常连接稀疏，但任意两个节点之间的平均距离（路径长度）却相对较短，这意味着信息可以在网络中快速传播。这种结构在现实世界中的许多网络中都得到了验证，包括社会网络和生物网络。 "强局部聚类"是指网络中的节点往往倾向于与其邻近的节点形成紧密的集群，这反映了语义上的相似性和相关性。例如，在WordNet中，同义词和相关概念往往紧密相连，形成了紧密的簇。 "幂律分布"则揭示了节点连接数的不均匀性，大多数节点有少量链接，而少数节点（称为“中心节点”或“枢纽”）具有大量的连接。这种分布模式在现实世界的复杂网络中非常常见，比如互联网上的网页链接分布。在语义网络中，这些中心节点可能代表一些核心概念或高频词汇，它们连接着大量其他概念，构成了网络的主要脉络。论文还指出，这些观察到的规律与传统的基于继承的语义组织模型不一致。传统模型往往假设语义关系是自上而下的层次结构，而实际的语义网络则展现出更加复杂和非线性的拓扑结构。这一发现对于理解人类认知、自然语言处理以及知识图谱的构建具有重要意义，因为它提供了一种更为精确的语义关系建模方式。

tions in a network, it is possible for two networks to have the same number of connections but dif

ferent clustering coefficients(see Fig. 2).Finally, notethat because the definitionsforT

and k

are

independent of whether the connections are based on edges or arcs, the clustering coefficients for

a directed network and the corresponding undirected network are equal.

The degree distribution P(k) represents the probability that a randomly chosen node will

have degree k (i.e., will have k neighbors). For directed networks, we will concentrate on the

distribution of in-degrees, although one can also look at the out-degree distribution. We esti

mate these distributions based on the relative frequencies of node degrees found throughout the

network. The most straightforward feature of P(k) is the expected value <k> under P(k). This

quantity, estimated by simply averaging the degree of all nodes over the network, ranges be

tween 0 and n (for a fully connected network of n nodes) and represents the mean density of

connections in the network. More information can be gleaned by plotting the full distribution

P(k) as a function of k, using either a bar histogram (for small networks, as in Fig. 2), a binned

scatter plot (for large networks, as in Fig. 5), or a smooth curve (for theoretical models, as later

illustrated in Fig. 3). As we explain in the next section, the shapes of these plots provide char

acteristic signatures for different kinds of network structure and different processes of network

growth.

Fig. 2 shows these statistical measures for two different networks with 10 nodes and 15

edges. These examples illustrate how networks equal in size and density of connections may

differ significantly in their other structural features. Fig. 2 also illustrates two general proper-

ties of random graphs—graphs that are generated by placing an edge between any two nodes

with some constant probability p independent of the existence of any other edge (a model intro-

duced by Erdös and Réyni, 1960). First, for fixed n and <k>, high values of C tend to imply

high values of L and D. Second, the degree distribution P(k) is approximately bell shaped, with

an exponential tail for high values of k.

Although these two properties hold reliably for ran-

dom graphs, they do not hold for many important natural networks, including semantic net-

works in natural language. We next turn to a detailed discussion of the small-world and

scale-free structures that do characterize natural semantic networks. Both of these structures

can be thought of in terms of how they contrast with random graphs: Small-world structures

are essentially defined by the combination of high values of C together with low values of L

and D, whereas scale-free structures are characterized by non-bell-shaped degree distributions,

with power-law (rather than exponential) tails.

3. Small-world and scale-free network structures

Interest in the small-world phenomenon originated with the classic experiments of Milgram

(1967) on social networks. Milgram’s results suggested that any two people in the United

States were, on average, separated by only a small number of acquaintances or friends (popu

larly known as “six degrees of separation”). Although the finding of very short distances be

tween random pairs of nodes in a large, sparsely connected network may seem surprising, it

does not necessarily indicate any interesting structure. This phenomenon occurs even in the

random graphs described previously, where each pair of nodes is joined by an edge with proba

bility p. When p is sufficiently high, the whole network becomes connected, and the average

distance L grows logarithmically with n, the size of the network (Erdös & Réyni, 1960).

M. Steyvers, J. B. Tenenbaum/Cognitive Science 29 (2005) 47

Watts and Strogatz (1998) sparked renewed interest in the mathematical basis of the

small-world phenomenon with their study of several naturally occurring networks: the power

grid of the western United States, the collaboration network of international film actors, and

the neural network of the worm Caenorhabditis elegans. They showed that although random

graphs with comparable size n and mean connectivity <k> describe very well the short path

lengths found in these networks, they also exhibit clustering coefficients C, that are orders of

magnitude smaller than those observed in the real networks. In other words, natural small-

world networks somehow produce much shorter internode distances than would be expected in

equally dense random graphs, given how likely it is that the neighbors of a node are also one

another’s neighbors (see Note 4). For the remainder of this article, we use the term small-world

structure to refer to this combination of short average path lengths L, and relatively high clus

tering coefficients C (by comparison with equally dense random graphs).

Small-world structures have since been found in many other networks (reviewed in Strogatz,

2001), including the WorldWideWeb (WWW; Adamic, 1999; Albert,Jeong, & Barabási, 1999),

networks of scientific collaborators (Newman, 2001), and metabolic networks in biology

(Jeong, Tombor, Albert, Oltval, & Barabási, 2000). Watts and Strogatz (1998) proposed a simple

abstract model for the formation of small-world structures, in which a small number of the con-

nections in a low-dimensional regular lattice are replaced with connections between random

pairs of nodes. The local neighborhood structure of the lattice leads to high clustering, whereas

the long-range random connections lead to very short average path lengths.

Amaral, Scala, Barthélémy, and Stanley (2000) distinguished between different classes of

small-world networks by measuring the degree distribution P(k). In one class of networks,

such as C. elegans and the U. S. power grid, the degree distribution decays exponentially. This

behavior is well described by random graph models or variants of the Watts and Strogatz

(1998) model. In other systems, such as the WWW or metabolic networks, the degree distribu-

tion follows a power law (Barabási & Albert, 1999),

for values of γ typically between 2 and 4. Fig. 3 illustrates the difference between power-law

and exponential degree distributions. Intuitively, a power-law distribution implies that a small

but significant number of nodes are connected to a very large number of other nodes, whereas

48 M. Steyvers, J. B. Tenenbaum/Cognitive Science 29 (2005)

() (2

)

Pk k

γ-

Fig. 3. Two different kinds of degree distributions that may be observed in complex networks. (a) A power-law distribu

tion (dashed) and anexponential distribution(solid), shownwiththe same linear scales as the histograms inFigure2b.(b)

When plotted with log-log scaling, the same distributions are more clearly differentiated, particularly in their tails.

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