15.network_science




网络科学简介



王成军

wangchengjun@nju.edu.cn

计算传播网 http://computational-communication.com

FROM SADDAM HUSSEIN TO NETWORK THEORY

A SIMPLE STORY (1) The fate of Saddam and network science

  • Invasion that started in March 19, 2003. Many of the regime's high ranking officials, including Saddam Hussein, avoided capture.
  • Hussein was last spotted kissing a baby in Baghdad in April 2003, and then his trace went cold.

  • Designed a deck of cards, each card engraved with the images of the 55 most wanted.

    • It worked: by May 1, 2003, 15 men on the cards were captured, and by the end of the month another 12 were under custody.
    • Yet, the ace of spades, i.e. Hussein himself, remained at large.

The capture of Saddam Hussein:

  • shows the strong predictive power of networks.

  • underlies the need to obtain accurate maps of the networks we aim to study; and the often heroic difficulties we encounter during the mapping process.

  • demonstrates the remarkable stability of these networks: The capture of Hussein was not based on fresh intelligence, but rather on his pre-invasion social links, unearthed from old photos stacked in his family album.

  • shows that the choice of network we focus on makes a huge difference: the hierarchical tree, that captured the official organization of the Iraqi government, was of no use when it came to Saddam Hussein's whereabouts.

A SIMPLE STORY (2): August 15, 2003 blackout.

VULNERABILITY DUE TO INTERCONNECTIVITY

  • The 2003 blackout is a typical example of a cascading failure.
  • 1997, when the International Monetary Fund pressured the central banks of several Pacific nations to limit their credit.
  • 2009-2011 financial melt-down

An important theme of this class:

  • we must understand how network structure affects the robustness of a complex system.

  • develop quantitative tools to assess the interplay between network structure and the dynamical processes on the networks, and their impact on failures.

  • We will learn that failures reality failures follow reproducible laws, that can be quantified and even predicted using the tools of network science.

NETWORKS AT THE HEART OF COMPLEX SYSTEMS

Complex

[adj., v. kuh m-pleks, kom-pleks; n. kom-pleks] –adjective

    1. composed of many interconnected parts; compound; composite: a complex highway system.
    1. characterized by a very complicated or involved arrangement of parts, units, etc.: complex machinery.
    1. so complicated or intricate as to be hard to understand or deal with: a complex problem.
             Source: Dictionary.com

Complexity

a scientific theory which asserts that some systems display behavioral phenomena that are completely inexplicable by any conventional analysis of the systems’ constituent parts. These phenomena, commonly referred to as emergent behaviour, seem to occur in many complex systems involving living organisms, such as a stock market or the human brain.   Source: John L. Casti, Encyclopædia Britannica  

COMPLEX SYSTEMS

  • society
  • brain
  • market
  • cell

Stephen Hawking: I think the next century will be the century of complexity.

Behind each complex system there is a network, that defines the interactions between the component.

  • Social graph
  • Organization
  • Brain
  • finantial network
  • business
  • Internet
  • Genes

Behind each system studied in complexity there is an intricate wiring diagram, or a network, that defines the interactions between the component.   

We will never understand complex system unless we map out and understand the networks behind them.

  

TWO FORCES HELPED THE EMERGENCE OF NETWORK SCIENCE

THE HISTORY OF NETWORK ANALYSIS

  • Graph theory: 1735, Euler

  • Social Network Research: 1930s, Moreno

  • Communication networks/internet: 1960s

  • Ecological Networks: May, 1979.

While the study of networks has a long history from graph theory to sociology, the modern chapter of network science emerged only during the first decade of the 21st century, following the publication of two seminal papers in 1998 [2] and 1999 [3]. The explosive interest in network science is well documented by the citation pattern of two classic network papers, the 1959 paper by Paul Erdos and Alfréd Rényi that marks the beginning of the study of random networks in graph theory [4] and the 1973 paper by Mark Granovetter, the most cited social network paper [5]. Both papers were hardly or only moderately cited before 2000. The explosive growth of citations to these papers in the 21st century documents the emergence of network science, drawing a new, interdisciplinary audience to these classic publications.

THE EMERGENCE OF NETWORK SCIENCE

  • Movie Actor Network, 1998;
  • World Wide Web, 1999.
  • C elegans neural wiring diagram 1990
  • Citation Network, 1998
  • Metabolic Network, 2000;
  • PPI network, 2001

The universality of network characteristics:

The architecture of networks emerging in various domains of science, nature, and technology are more similar to each other than one would have expected.

THE CHARACTERISTICS OF NETWORK SCIENCE

  • Interdisciplinary
  • Empirical
  • Quantitative and Mathematical
  • Computational

THE IMPACT OF NETWORK SCIENCE

Google

Market Cap(2010 Jan 1): $189 billion

Cisco Systems

networking gear Market cap (Jan 1, 2919): $112 billion

Facebook

market cap: $50 billion

Health: From drug design to metabolic engineering.

The human genome project, completed in 2001, offered the first comprehensive list of all human genes. Yet, to fully understand how our cells function, and the origin of disease, we need accurate maps that tell us how these genes and other cellular components interact with each other.

Security: Fighting Terrorism.

Terrorism is one of the maladies of the 21st century, absorbing significant resources to combat it worldwide. Network thinking is increasingly present in the arsenal of various law enforcement agencies in charge of limiting terrorist activities. It is used to disrupt the financial network of terrorist organizations, to map terrorist networks, and to uncover the role of their members and their capabilities. While much of the work in this area is classified, several success stories have surfaced. Examples include the use of social networks to capture Saddam Hussein or the capture of the individuals behind the March 11, 2004 Madrid train bombings through the examination of the mobile call net- work.

Epidemics: From forecasting to halting deadly viruses.

While the H1N1 pandemic was not as devastating as it was feared at the beginning of the outbreak in 2009, it gained a special role in the history of epidemics: it was the first pandemic whose course and time evolution was accurately predicted months before the pandemic reached its peak (Image 1.9) [14]. This was possible thanks to fundamental advances in understanding the role of networks in the spread of viruses. Indeed, before 2000 epidemic modeling was dominated by compartment models, assuming that everyone can infect everyone else one word the same socio-physical compartment. The emergence of a network-based framework has fundamentally changed this, offering a new level of predictability in epidemic phenomena. Today epidemic prediction is one of the most active applications of network science. It is the source several fundamental results, covered in this book, that are used to predict the spread of both biological and electronic viruses. The impact of these advances are felt beyond biological viruses. In January 2010 network science tools have predicted the conditions necessary for the emergence of viruses spreading through mobile phones. The first major mobile epidemic outbreak that started in the fall of 2010 in China, infecting over 300,000 phones each day, closely followed the predicted scenario.

Brain Research: Mapping neural network.

The human brain, consisting of hundreds of billions of interlinked neurons, is one of the least understood net- works from the perspective of network science. The reason is simple: we lack maps telling us which neurons link to each other. The only fully mapped neural map available for research is that of the C.Elegans worm, with only 300 neurons. Should detailed maps of mammalian brains become available, brain research could become the most prolific application area of network science. Driven by the potential impact of such maps, in 2010 the National Institutes of Health has initiated the Connectome project, aimed at developing the technologies that could provide an accurate neuron-level map of mammalian brains.

The Bridges of Konigsberg

Can one walk across the seven bridges and never cross the same bridge twice and get back to the starting place?

Can one walk across the seven bridges and never cross the same bridge twice and get back to the starting place?

1735: Euler’s theorem:

  • If a graph has more than two nodes of odd degree, there is no path.
  • If a graph is connected and has no odd degree nodes, it has at least one path.

COMPONENTS OF A COMPLEX SYSTEM

Networks and graphs

  • components: nodes, vertices N
  • interactions: links, edges L
  • system: network, graph (N,L)

network often refers to real systems

  • www,
  • social network
  • metabolic network.

Language: (Network, node, link)

graph: mathematical representation of a network

  • web graph,
  • social graph (a Facebook term)

Language: (Graph, vertex, edge)

CHOOSING A PROPER REPRESENTATION

The choice of the proper network representation determines our ability to use network theory successfully.

In some cases there is a unique, unambiguous representation. In other cases, the representation is by no means unique.   For example, the way we assign the links between a group of individuals will determine the nature of the question we can study.

If you connect individuals that work with each other, you will explore the professional network.

http://www.theyrule.net

If you connect those that have a romantic and sexual relationship, you will be exploring the sexual networks.

If you connect individuals based on their first name (all Peters connected to each other), you will be exploring what?

It is a network, nevertheless.

UNDIRECTED VS. DIRECTED NETWORKS

Undirected

Links: undirected

  • coauthorship
  • actor network
  • protein interactions

In [174]:
%matplotlib inline
import matplotlib.pyplot as plt

import networkx as nx
Gu = nx.Graph()
for i, j in [(1, 2), (1, 4), (4, 2), (4, 3)]:
    Gu.add_edge(i,j)
nx.draw(Gu, with_labels = True)


Directed

Links: directed

  • urls on the www
  • phone callss
  • metabolic reactions

In [173]:
import networkx as nx
Gd = nx.DiGraph()
for i, j in [(1, 2), (1, 4), (4, 2), (4, 3)]:
    Gd.add_edge(i,j)
nx.draw(Gd, with_labels = True)


Degree, Average Degree and Degree Distribution


In [175]:
nx.draw(Gu, with_labels = True)


Undirected network:

Node degree: the number of links connected to the node.

$k_1 = k_2 = 2, k_3 = 3, k_4 = 1$


In [177]:
nx.draw(Gd, with_labels = True)


Directed network

In directed networks we can define an in-degree and out-degree. The (total) degree is the sum of in-and out-degree.

$k_3^{in} = 2, k_3^{out} = 1, k_3 = 3$

Source: a node with $k^{in}= 0$; Sink: a node with $k^{out}= 0$.

For a sample of N values: $x_1, x_2, ..., x_N$:

Average(mean):

$<x> = \frac{x_1 +x_2 + ...+x_N}{N} = \frac{1}{N}\sum_{i = 1}^{N} x_i$

For a sample of N values: $x_1, x_2, ..., x_N$:

The nth moment:

$<x^n> = \frac{x_1^n +x_2^n + ...+x_N^n}{N} = \frac{1}{N}\sum_{i = 1}^{N} x_i^n$

For a sample of N values: $x_1, x_2, ..., x_N$:

Standard deviation:

$\sigma_x = \sqrt{\frac{1}{N}\sum_{i = 1}^{N} (x_i - <x>)^2}$


In [20]:
import numpy as np
x = [1, 1, 1, 2, 2, 3]
np.mean(x), np.sum(x), np.std(x)


Out[20]:
(1.6666666666666667, 10, 0.7453559924999299)

For a sample of N values: $x_1, x_2, ..., x_N$:

Distribution of x:

$p_x = \frac{The \: frequency \: of \: x}{The\: Number \:of\: Observations}$

其中,$p_x 满足 \sum_i p_x = 1$


In [22]:
plt.hist(x)
plt.show()



In [113]:
from collections import defaultdict, Counter
freq = defaultdict(int)
for i in x:
    freq[i] +=1

freq


Out[113]:
defaultdict(int, {1: 3, 2: 2, 3: 1})

In [37]:
freq_sum = np.sum(freq.values())
freq_sum


Out[37]:
6

In [38]:
px = [float(i)/freq_sum for i in freq.values()]
px


Out[38]:
[0.5, 0.3333333333333333, 0.16666666666666666]

In [41]:
plt.plot(freq.keys(), px, 'r-o')
plt.show()


Average Degree

Undirected

$<k> = \frac{1}{N} \sum_{i = 1}^{N} k_i = \frac{2L}{N}$

Directed

$<k^{in}> = \frac{1}{N} \sum_{i=1}^N k_i^{in}= <k^{out}> = \frac{1}{N} \sum_{i=1}^N k_i^{out} = \frac{L}{N}$

Degree distribution

P(k): probability that a randomly selected node has degree k

$N_k = The \:number\: of \:nodes\:with \:degree\: k$

$P(k) = \frac{N_k}{N}$

Adjacency matrix

$A_{ij} =1$ if there is a link between node i and j

$A_{ij} =0$ if there is no link between node i and j


In [182]:
plt.figure(1)
plt.subplot(121)
pos = nx.spring_layout(Gu)  #定义一个布局,此处采用了spring布局方式
nx.draw(Gu, pos, with_labels = True)
plt.subplot(122)
nx.draw(Gd, pos, with_labels = True)


Undirected

$A_{ij} =1$ if there is a link between node i and j

$A_{ij} =0$ if there is no link between node i and j

$A_{ij}=\begin{bmatrix} 0&1 &0 &1 \\ 1&0 &0 &1 \\ 0 &0 &0 &1 \\ 1&1 &1 & 0 \end{bmatrix}$

Undirected

$A_{ij} = A_{ji}, \: A_{ii} = 0$

$k_i = \sum_{j=1}^N A_{ij}, \: k_j = \sum_{i=1}^N A_{ij} $

$ L = \frac{1}{2}\sum_{i=1}^N k_i = \frac{1}{2}\sum_{ij}^N A_{ij} $

Directed

$A_{ij} =1$ if there is a link between node i and j

$A_{ij} =0$ if there is no link between node i and j

$A_{ij}=\begin{bmatrix} 0&0 &0 &0 \\ 1&0 &0 &1 \\ 0 &0 &0 &1 \\ 1&0 &0 & 0 \end{bmatrix}$

Note that for a directed graph the matrix is not symmetric.

Directed

$A_{ij} \neq A_{ji}, \: A_{ii} = 0$

$k_i^{in} = \sum_{j=1}^N A_{ij}, \: k_j^{out} = \sum_{i=1}^N A_{ij} $

$ L = \sum_{i=1}^N k_i^{in} = \sum_{j=1}^N k_j^{out}= \frac{1}{2}\sum_{i,j}^N A_{ij} $

WEIGHTED AND UNWEIGHTED NETWORKS

$A_{ij} = W_{ij}$

BIPARTITE NETWORKS

bipartite graph (or bigraph) is a graph whose nodes can be divided into two disjoint sets U and V such that every link connects a node in U to one in V; that is, U and V are independent sets.

  • Hits algorithm
  • recommendation system

Ingredient-Flavor Bipartite Network

Path

A path is a sequence of nodes in which each node is adjacent to the next one

- In a directed network, the path can follow only the direction of an arrow. 

Distance

The distance (shortest path, geodesic path) between two nodes is defined as the number of edges along the shortest path connecting them.

- *If the two nodes are disconnected, the distance is infinity.

Diameter:

Diameter dmax is the maximum distance between any pair of nodes in the graph.

Shortest Path

The path with the shortest length between two nodes (distance).

Average path length/distance, $<d>$

The average of the shortest paths for all pairs of nodes.

  • for a connected graph: where $d_{ij}$ is the distance from node i to node j

$<d> = \frac{1}{2 L_{max} }\sum_{i, j \neq i} d_{ij}$

  • In an undirected graph $d_{ij} =d_{ji}$ , so we only need to count them once:

$<d> = \frac{1}{L_{max} }\sum_{i, j > i} d_{ij}$

Cycle

A path with the same start and end node.

CONNECTEDNESS

Connected (undirected) graph:

any two vertices can be joined by a path. A disconnected graph is made up by two or more connected components.

  • Largest Component: Giant Component
  • The rest: Isolates

Bridge:

if we erase it, the graph becomes disconnected.

The adjacency matrix of a network with several components can be written in a block-diagonal form, so that nonzero elements are confined to squares, with all other elements being zero:

Strongly connected directed graph:

has a path from each node to every other node and vice versa (e.g. AB path and BA path).

Weakly connected directed graph:

it is connected if we disregard the edge directions.

Strongly connected components can be identified, but not every node is partof a nontrivial strongly connected component.

  • In-component: nodes that can reach the scc,
  • Out-component: nodes that can be reached from the scc.

Clustering coefficient

Clustering coefficient:

what fraction of your neighbors are connected? Watts & Strogatz, Nature 1998.

  • Node i with degree $k_i$
  • $C_i$ in [0,1]

$C_i = \frac{2e_i}{k_i(k_i -1)}$

Global Clustering Coefficient:

$C = \frac{3 \times \mbox{number of triangles}}{\mbox{number of connected triplets of vertices}} = \frac{\mbox{number of closed triplets}}{\mbox{number of connected triplets of vertices}}$


In [215]:
G1 = nx.complete_graph(4)
pos = nx.spring_layout(G1)  #定义一个布局,此处采用了spring布局方式
nx.draw(G1, pos = pos, with_labels = True)



In [206]:
print(nx.transitivity(G1))


1.0

In [216]:
G2 = nx.Graph()
for i, j in [(1, 2), (1, 3), (1, 0), (3, 0)]:
    G2.add_edge(i,j)
nx.draw(G2,pos = pos, with_labels = True)



In [198]:
print(nx.transitivity(G2))


0.6

In [219]:
G3 = nx.Graph()
for i, j in [(1, 2), (1, 3), (1, 0)]:
    G3.add_edge(i,j)
nx.draw(G3, pos =pos, with_labels = True)



In [200]:
print(nx.transitivity(G3))


0.0

THREE CENTRAL QUANTITIES IN NETWORK SCIENCE

  • A. Degree distribution: $p_k$
  • B. Path length: $<d>$
  • C. Clustering coefficient: $C_i$

Typical Network Science Research

  • Discovering, Modeling, Verification
    • WATTSDJ,STROGATZSH.Collective dynamics of‘small-world’ networks. Nature, 1998, 393(6684): 440–442.
    • BARABÁSI A-L, ALBERT R. Emergence of scaling in random networks. Science, 1999, 286(5439): 509-512.

Typical Math Style

Fan Chung & Linyuan Lu, The average distance in random graphs with given expected degree,. PNAS, 19, 15879-15882 (2002).

Typical Physical Style

A.-L.Barabási,R.Albert,H.Jeong Mean-field theory for scale-free random networks. Physica A 272, 173–187 (1999).

Typical Computer Science Style

  • Community detection
  • Link prediction
  • Recommendation algorithms

Typical control sytle

Controllability of Complex Networks

Liu Y Y, Slotine J J, Barabási A L. Nature, 2011, 473(7346): 167-173.

The random network model

Erdös-Rényi model (1960)

Definition: A random graph is a graph of N nodes where each pair of nodes is connected by probability p.

G(N, L) Model

N lableled nodes are connected with L randomly placed links. Erdös-Rényi(1959)

G(N, p) Model

Each pair of N labeled nodes is connected with probability p. Gilbert (1959)


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In [144]:
import networkx as nx
import matplotlib.pyplot as plt
BA= nx.random_graphs.barabasi_albert_graph(500,1)  #生成n=20、m=1的BA无标度网络
pos = nx.spring_layout(BA)          #定义一个布局,此处采用了spring布局方式
nx.draw(BA,pos,with_labels=False,node_size = 30)  #绘制图形
plt.show()



In [145]:
nx.degree_histogram(BA)[:3]


Out[145]:
[0, 343, 76]

In [146]:
BA.degree().items()[:3]


Out[146]:
[(0, 6), (1, 27), (2, 13)]

In [147]:
plt.hist(BA.degree().values())
plt.show()



In [148]:
def plotDegreeDistributionLongTail(G):
    degs = defaultdict(int)
    for i in G.degree().values(): degs[i]+=1
    items = sorted ( degs.items () )
    x, y = np.array(items).T
    plt.plot(x, y, 'b-o')
    plt.legend(['Degree'])
    plt.xlabel('$K$', fontsize = 20)
    plt.ylabel('$P_K$', fontsize = 20)
    plt.title('$Degree\,Distribution$', fontsize = 20)
    plt.show()  
    
plotDegreeDistributionLongTail(BA)



In [150]:
def plotDegreeDistribution(G):
    degs = defaultdict(int)
    for i in G.degree().values(): degs[i]+=1
    items = sorted ( degs.items () )
    x, y = np.array(items).T
    plt.plot(x, y, 'b-o')
    plt.xscale('log')
    plt.yscale('log')
    plt.legend(['Degree'])
    plt.xlabel('$K$', fontsize = 20)
    plt.ylabel('$P_K$', fontsize = 20)
    plt.title('$Degree\,Distribution$', fontsize = 20)
    plt.show()   
    
plotDegreeDistribution(BA)



In [192]:
BA= nx.random_graphs.barabasi_albert_graph(50000,10)  #生成n=50000、m=1的BA无标度网络
plotDegreeDistribution(BA)


无标度的意义

度分布的n阶矩被定义为:

$ <k^n> = \sum_{k_{min}}^{k_{max}}k^np_k = \int_{k_{min}}^{k_{max}}k^np_kdk $ (1)

低阶矩具有明确的统计意义:

  • n = 1的时候,一阶矩是$<k^{}>$,即平均度。
  • n = 2的时候,二阶矩是$<k^2>$,可以帮助计算方差 $ \delta^2 = <k^2> - <k^{}>^{2} $,测量了度的离散程度(the spread in the degrees)。
  • n = 3的时候,三阶矩是$<k^3>$, 决定了度分布的偏度(skewness),测量了$p_k$围绕着 的对称性。

对于无标度网络而言,满足幂律分布:

$p(k) = Ck^{-\gamma}$ (2)

由公式(1)和(2)可以得到:

$ <k^n> = \int_{k_{min}}^{k_{max}}k^np_kdk = C \frac{k_{max}^{n- \gamma +1} - k_{min}^{n - \gamma +1}}{n - \gamma + 1} $ (3)

可以使用wolframalpha的积分计算器积分来进行简单验证,例如x^(n-r)dx从10到100积分 网页链接:[http://www.wolframalpha.com/input/?i=integrate+x%5E%28n-r%29+dx+from+10+to+100][1]

显然:

  • 当$n - \gamma +1 <= 0$时,随着$k_{max}增加,$$k_{max}^{n- \gamma +1} \rightarrow 0$。所有满足$n <= \gamma -1$的n阶矩都是有限的。
  • 当$n - \gamma +1 >0 $时,随着$k_{max}增加,$$k_{max}^{n- \gamma +1} \rightarrow \infty$。所有满足$n > \gamma -1$的n阶矩都是无极限的。

对于无标度网络而言,一般幂参数$2 < \gamma < 3$,所以:

  • 对于n = 1的情况,即一阶矩平均度$<k^{}>$是有限的。

  • 但对于n >= 2的情况,即$k^2$或$k^3$是无极限的。二阶和高阶矩无穷大是“无标度”的来源

一言以蔽之,在无标度的意义是,在网络中随机抽取一个节点的度可以显著的不同于平均度$<k>$

上图最为直接的描绘出了这种特点,即与正态分布等相比,无标度网络下降的慢

  • 对于任何指数类型的分布,如泊松分布或高斯分布,随机选取一个节点的度在平均度附近,因此平均度就是这些网络的尺度

  • 对于一个幂律分布而言,因为二阶矩发散,在网络中随机抽取一个节点的度可以显著的不同于平均度$<k^{}>$, 因此平均度不再是网络的尺度,称之为无标度。


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