Complex and Social Network Analysis in Python
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The document discusses complex social networks, defining them as structures comprising actors and relationships, characterized by non-trivial topological features not found in simple networks. It elaborates on various network analysis tools, measures, and models, including Dijkstra's algorithm for shortest path calculations and concepts like degree distribution, clustering coefficients, and network resilience. Additionally, it provides empirical results on different online social networks' characteristics and the implications of network structures on connectivity and information flow.
![⎡A k ⎤
⎣ ⎦ij number of paths of length k from i to j
3 All paths of length 3 starting
[A ]ii
and ending in i → triangles](https://image.slidesharecdn.com/csnap-120714113205-phpapp01/85/Complex-and-Social-Network-Analysis-in-Python-18-320.jpg)
![Dijkstra Algorithm (single source shortest path)
from heapq import heappush, heappop
# based on recipe 119466
def dijkstra_shortest_path(graph, source):
distances = {}
predecessors = {}
seen = {source: 0}
priority_queue = [(0, source)]
while priority_queue:
v_dist, v = heappop(priority_queue)
distances[v] = v_dist
for w in graph[v]:
vw_dist = distances[v] + 1
if w not in seen or vw_dist < seen[w]:
seen[w] = vw_dist
heappush(priority_queue,(vw_dist,w))
predecessors[w] = v
return distances, predecessors
O(m· pushQ + n· ex-minQ)=O(m log n + n log n)](https://image.slidesharecdn.com/csnap-120714113205-phpapp01/85/Complex-and-Social-Network-Analysis-in-Python-36-320.jpg)
![Dijkstra Algorithm (single source shortest path)
from heapq import heappush, heappop
# based on recipe 119466
def dijkstra_shortest_path(graph, source):
distances = {}
predecessors = {}
seen = {source: 0}
priority_queue = [(0, source)]
while priority_queue:
v_dist, v = heappop(priority_queue)
distances[v] = v_dist
for w in graph[v]:
vw_dist = distances[v] + 1
if w not in seen or vw_dist < seen[w]:
seen[w] = vw_dist
heappush(priority_queue,(vw_dist,w))
predecessors[w] = v
return distances, predecessors
O(m· pushQ + n· ex-minQ)=O(m log n + n log n)](https://image.slidesharecdn.com/csnap-120714113205-phpapp01/85/Complex-and-Social-Network-Analysis-in-Python-37-320.jpg)
![Dijkstra Algorithm (single source shortest path)
from heapq import heappush, heappop
# based on recipe 119466
def dijkstra_shortest_path(graph, source):
distances = {}
predecessors = {}
seen = {source: 0}
priority_queue = [(0, source)]
while priority_queue:
v_dist, v = heappop(priority_queue)
distances[v] = v_dist
for w in graph[v]:
vw_dist = distances[v] + 1
if w not in seen or vw_dist < seen[w]:
seen[w] = vw_dist
heappush(priority_queue,(vw_dist,w))
predecessors[w] = v
return distances, predecessors
O(m· pushQ + n· ex-minQ)=O(m log n + n log n)](https://image.slidesharecdn.com/csnap-120714113205-phpapp01/85/Complex-and-Social-Network-Analysis-in-Python-38-320.jpg)
![class AdjacencyListGraph(object):
def __init__(self):
self.node = {}
self.adj = {}
def add_node(self, node, **attrs):
if node not in self.adj:
self.adj[node] = {}
self.node[node] = attrs
else: # update attr even if node already exists
self.node[node].update(attrs)
def add_edge(self, u, v, **attrs):
if u not in self.adj:
self.adj[u] = {} self.node u
self.node[u] = {}
if v not in self.adj:
self.adj[v] = {} self.adj u
self.node[v] = {}
k1 k2 ...
datadict=self.adj[u].get(v,{}) attribute
datadict.update(attrs) v1 attribute
self.adj[u][v] = datadict v2 e1 e2 ...
self.adj[v][u] = datadict attribute
v3 attribute
f1 f2 ...
attribute
attribute](https://image.slidesharecdn.com/csnap-120714113205-phpapp01/85/Complex-and-Social-Network-Analysis-in-Python-45-320.jpg)
![Network Destruction Process
def attack(graph, centrality_metric):
graph = graph.copy()
steps = 0
ranks = centrality_metric(graph)
nodes = sorted(graph.nodes(), key=lambda n: ranks[n])
while nx.is_connected(graph):
graph.remove_node(nodes.pop())
steps += 1
else:
return steps
Power-Law Cluster Random
Random Attack 220 10
PageRank driven 19 149
Betweenness driven 22 157
Degree 19 265](https://image.slidesharecdn.com/csnap-120714113205-phpapp01/85/Complex-and-Social-Network-Analysis-in-Python-51-320.jpg)
![References
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[13] Milgram, S. 1967. The small world problem. Psychology Today. 1, 1, 61–67.
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Proceedings of the 8th ACM SIGCOMM conference on Internet measurement. 43–56.
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Proceedings of the 16th annual ACM-SIAM Symposium on Discrete Algorithms. 311–320.
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Proceedings of the 3rd conference on Online social networks. 1–8.
[19] Watts, D. J. and Strogatz, S. 1998. Collective dynamics of small-world networks. Nature. 393, 6684, 440–442.](https://image.slidesharecdn.com/csnap-120714113205-phpapp01/85/Complex-and-Social-Network-Analysis-in-Python-59-320.jpg)
Complex and Social Network Analysis in Python
- 3. A social network is a structure composed by actors and their relationships Actor: person, organization, role ... Relationship: friendship, knowledge... A social networking system is system allowing users to: • construct a profile which represents them in the system; • create a list of users with whom they share a connection • navigate their list of connections and that of their friends (Boyd, 2008) So, what is a complex network? A complex network is a network with non-trivial topological features— features that do not occur in simple networks such as lattices or random graphs but often occur in real graphs. (Wikipedia). Foggy.
- 4. COMPLEX NETWORKS A complex network is a network with non-trivial topological features—features that do not occur in simple networks such as lattices or random graphs but often occur in real graphs. • Non-trivial topological features (what are topological features?) • Simple networks: lattices, regular or random graphs • Real graphs • Are social networks complex networks?
- 5. TOOLS • • • Matplotlib • IPython • NetworkX
- 6. BASIC NOTATION Adjacency Matrix Network ⎧1 if (i,j) ∈E G = (V, E) E ⊂ V 2 A ij = ⎨ ⎩0 otherwise {(x, x) x ∈V } ∩ E = ∅ Directed Network Undirected Network k = ∑ A ji i in A symmetric x ki = ∑ A ji = ∑ A ij j ki out = ∑ A ij j j k in = 4 x kx = 3 out j ki = kiin + kiout
- 7. BASIC NOTATION Adjacency Matrix Assume the Network network ⎧1 if (i,j) ∈E G = (V, E) E ⊂ V 2 A ij = ⎨ connected! ⎩0 otherwise {(x, x) x ∈V } ∩ E = ∅ Directed Network Undirected Network k = ∑ A ji i in A symmetric x ki = ∑ A ji = ∑ A ij j ki out = ∑ A ij j j k in = 4 x kx = 3 out j ki = kiin + kiout
- 8. Path p = v0 ,…,vk (vi−1 ,vi ) ∈E Path Length: length( p) Set of paths from i to j: Paths(i, j) j i Shortest path length: ( L(i, j) = min {length( p) p ∈Paths(i, j)} ) Shortest/Geodesic path:
- 10. Average geodesic distance (i) = (n − 1)−1 ∑ L (i, j) k∈ {i} V Average shortest path length = n −1 ∑ (i) i∈V Characteristic path length ( CPL = median {(i) i ∈V } )
- 11. ⎡ A k ⎤ number of paths of length k from i to j ⎣ ⎦ij
- 12. ⎡ A k ⎤ number of paths of length k from i to j ⎣ ⎦ij number of paths!
- 13. ⎡ A k ⎤ number of paths of length k from i to j ⎣ ⎦ij number of paths! degree
- 14. CLUSTERING COEFFICIENT () −1 ki Local Clustering Coefficient Ci = 2 T (i) T(i): # distinct triangles with i as vertex Clustering Coefficient 1 C = ∑ Ci n i∈V • Measure of transitivity x • High CC → “resilient” network • Counting triangles 1 ( ) Δ(G) = ∑ N(i) = trace A 3 6 i∈V
- 15. CLUSTERING COEFFICIENT () −1 ki Local Clustering Coefficient Ci = 2 T (i) T(i): # distinct triangles with i as vertex Clustering Coefficient 1 C = ∑ Ci kx = 5 n i∈V ✓ ◆ 1 kx = 10 2 • Measure of transitivity • High CC → “resilient” network • Counting triangles 1 ( ) Δ(G) = ∑ N(i) = trace A 3 6 i∈V
- 16. CLUSTERING COEFFICIENT () −1 ki Local Clustering Coefficient Ci = 2 T (i) T(i): # distinct triangles with i as vertex Clustering Coefficient 1 C = ∑ Ci T (x) = 3 n i∈V • Measure of transitivity • High CC → “resilient” network • Counting triangles 1 ( ) Δ(G) = ∑ N(i) = trace A 3 6 i∈V
- 17. CLUSTERING COEFFICIENT () −1 ki Local Clustering Coefficient Ci = 2 T (i) T(i): # distinct triangles with i as vertex Clustering Coefficient 1 C = ∑ Ci n i∈V x • Measure of transitivity −1 High CC → “resilient” network ⎛ kx ⎞ • T (x) = 3 = 10 • Counting triangles ⎝2 ⎠ 1 ( ) Δ(G) = ∑ N(i) = trace A 3 kx = 5 Cx = 0.3 i∈V 6
- 18. ⎡A k ⎤ ⎣ ⎦ij number of paths of length k from i to j 3 All paths of length 3 starting [A ]ii and ending in i → triangles
- 19. DEGREE DISTRIBUTION • Every “node-wise” property can be studies as an average, but it is most interesting to study the whole distribution. • One of the most interesting is the “degree distribution” px = # {i ki = x } 1 n
- 20. Many networks have power-law degree distribution. pk ∝ k −γ γ >1 • Citation networks • Biological networks k r =? • WWW graph • Internet graph • Social Networks Power-Law: ! gamma=3 1000000 log-log plot 100000 10000 1000 100 10 1 0.1 1 10 100 1000
- 21. Generated with random generator 80-20 Law Few nodes account for the vast majority of links Most nodes have very few links This points towards the idea that we have a core with a fringe of nodes with few connections. ... and it it proved that implies super-short diameter
- 22. HIGH LEVEL STRUCTURE Isle Isolated community Core
- 23. CONNECTED COMPONENTS Most features are computed on the core Undirected: strongly connected = weakly connected Directed/Undirected Directed: strongly connected != weakly connected The adjacency matrix is primitive iff the network is connected
- 24. Facebook Hugs Degree Distribution 10000000 1000000 100000 10000 1000 100 10 1 1 10 100 1000
- 25. Facebook Hugs Degree Distribution 10000000 1000000 100000 10000 1000 100 10 1 1 10 100 1000 For small k power-laws do not hold
- 26. Facebook Hugs Degree Distribution 10000000 1000000 100000 10000 1000 For large k we have 100 statistical fluctuations 10 1 1 10 100 1000 For small k power-laws do not hold
- 27. Facebook Hugs Degree Distribution 10000000 Nodes: 1322631 Edges: 1555597 m/n: 1.17 CPL: 11.74 1000000 Clustering Coefficient: 0.0527 Number of Components: 18987 100000 Isles: 0 10000 Largest Component Size: 1169456 1000 For large k we have 100 statistical fluctuations 10 1 1 10 100 1000 For small k power-laws do not hold
- 28. Facebook Hugs Degree Distribution 10000000 Nodes: 1322631 Edges: 1555597 m/n: 1.17 CPL: 11.74 1000000 Clustering Coefficient: 0.0527 Number of Components: 18987 100000 Isles: 0 10000 Largest Component Size: 1169456 1000 For large k we have 100 statistical fluctuations 10 1 1 10 100 1000 For small k power-laws do not hold Moreover, many distributions are wrongly identified as PLs
- 29. Online Social Networks OSN Refs. Users Links <k> C CPL d γ r Club Nexus Adamic et al 2.5 K 10 K 8.2 0.2 4 13 n.a. n.a. Cyworld Ahn et al 12 M 191 M 31.6 0.2 3.2 16 -0.13 Cyworld T Ahn et al 92 K 0.7 M 15.3 0.3 7.2 n.a. n.a. 0.43 LiveJournal Mislove et al 5 M 77 M 17 0.3 5.9 20 0.18 Flickr Mislove et al 1.8 M 22 M 12.2 0.3 5.7 27 0.20 Twitter Kwak et al 41 M 1700 M n.a. n.a. 4 4.1 n.a. Orkut Mislove et al 3 M 223 M 106 0.2 4.3 9 1.5 0.07 Orkut Ahn et al 100 K 1.5 M 30.2 0.3 3.8 n.a. 3.7 0.31 Youtube Mislove et al 1.1 M 5 M 4.29 0.1 5.1 21 -0.03 Facebook Gjoka et al 1 M n.a. n.a. 0.2 n.a. n.a. 0.23 FB H Nazir et al 51 K 116 K n.a. 0.4 n.a. 29 n.a. FB GL Nazir et al 277 K 600 K n.a. 0.3 n.a. 45 n.a. BrightKite Scellato et al 54 K 213 K 7.88 0.2 4.7 n.a. n.a. FourSquare Scellato et al 58 K 351 K 12 0.3 4.6 n.a. n.a. LiveJournal Scellato et al 993 K 29.6 M 29.9 0.2 4.9 n.a. n.a. Twitter Java et al 87 K 829 K 18.9 0.1 n.a. 6 0.59 Twitter Scellato et al 409 K 183 M 447 0.2 2.8 n.a. n.a.
- 30. Erdös-Rényi Random Graphs Connectedness p Threshold log n / n G(n, p) p G(n, m) p p p p Ensembles of Graphs p p When describe values of p properties, we actually the p Pr(Aij = 1) = p expected value of the property
- 31. Erdös-Rényi Random Graphs Connectedness p Threshold log n / n G(n, p) p G(n, m) p p p p Ensembles of Graphs p p When describe values of p properties, we actually the p Pr(Aij = 1) = p expected value of the property d := d = ∑ Pr(G)⋅ d(G) ∝ log n Pr(G) = p m (1− p) () n 2 −m G log k −1 C = k (n − 1) = p ⎛ n⎞ m =⎜ ⎟ p ⎝ 2⎠ ⎛ n − 1⎞ k k k pk = ⎜ p (1− p)n−1−k n→∞ pk = e − k ⎝k ⎟ ⎠ k!
- 32. p Watts-Strogatz Model In the modified model, we only add the edges. 3(κ − 2) ki = κ + si C→ 4(κ − 1) + 8κ p + 4κ p 2 Edges in the lattice # added log(npκ ) ≈ shortcuts κ p 2
- 33. Strogatz-Watts Model - 10000 nodes k = 4 1 CPL(p)/CPL(0) C(p)/C(0) 0.8 CPL(p)/CPL(0) 0.6 C(p)/C(0) 0.4 0.2 0 0 0.2 0.4 p 0.6 0.8 1 Short CPL Large Clustering Coefficient Threshold Threshold
- 34. Barabási-Albert Model Connectedness log n Threshold log log n BARABASI-ALBERT-MODEL(G,M0,STEPS) FOR K FROM 1 TO STEPS N0 ← NEW-NODE(G) ADD-NODE(G,N0) A ← MAKE-ARRAY() FOR N IN NODES(G) PUSH(A, N) pk ∝ x −3 FOR J IN DEGREE(N) PUSH(A, N) log n FOR J FROM 1 TO M ≈ log log n N ← RANDOM-CHOICE(A) ADD-LINK (N0, N) C ≈ n −3/4 Scale-free entails short CPL Transitivity disappears with network size No analytical proof available
- 35. ANALYSIS • There are many network features we can study • Let’s discuss some algorithms for the ones we studied so-far • Also consider the size of the networks ( > 1M nodes ), so algorithmic costs can become an issue
- 36. Dijkstra Algorithm (single source shortest path) from heapq import heappush, heappop # based on recipe 119466 def dijkstra_shortest_path(graph, source): distances = {} predecessors = {} seen = {source: 0} priority_queue = [(0, source)] while priority_queue: v_dist, v = heappop(priority_queue) distances[v] = v_dist for w in graph[v]: vw_dist = distances[v] + 1 if w not in seen or vw_dist < seen[w]: seen[w] = vw_dist heappush(priority_queue,(vw_dist,w)) predecessors[w] = v return distances, predecessors O(m· pushQ + n· ex-minQ)=O(m log n + n log n)
- 37. Dijkstra Algorithm (single source shortest path) from heapq import heappush, heappop # based on recipe 119466 def dijkstra_shortest_path(graph, source): distances = {} predecessors = {} seen = {source: 0} priority_queue = [(0, source)] while priority_queue: v_dist, v = heappop(priority_queue) distances[v] = v_dist for w in graph[v]: vw_dist = distances[v] + 1 if w not in seen or vw_dist < seen[w]: seen[w] = vw_dist heappush(priority_queue,(vw_dist,w)) predecessors[w] = v return distances, predecessors O(m· pushQ + n· ex-minQ)=O(m log n + n log n)
- 38. Dijkstra Algorithm (single source shortest path) from heapq import heappush, heappop # based on recipe 119466 def dijkstra_shortest_path(graph, source): distances = {} predecessors = {} seen = {source: 0} priority_queue = [(0, source)] while priority_queue: v_dist, v = heappop(priority_queue) distances[v] = v_dist for w in graph[v]: vw_dist = distances[v] + 1 if w not in seen or vw_dist < seen[w]: seen[w] = vw_dist heappush(priority_queue,(vw_dist,w)) predecessors[w] = v return distances, predecessors O(m· pushQ + n· ex-minQ)=O(m log n + n log n)
- 39. Computational Complexity of ASPL: All pairs shortest path matrix based (parallelizable): ( ) Θ n 3 All pairs shortest path Dijkstra w. Fibonacci Heaps: O ( n log n + nm ) 2 Computing the CPL q#S elements are ≤ than x µ(a) = M 1 (a) x = M q (S) and (1-q)#S are > than x 2 q#S(1-δ) elements are ≤ than x M 1 (a) x ∈Lqδ (S) 3 or (1-q)#S(1-δ) are > than x Huber Method L 1 , 1 (a) 2 5 2 2 (1 − δ ) 2 Let R a random sample of S such that #R=s, then s = 2 ln q δ2 Mq(R) ∈ Lqδ(S) with probability p = 1-ε.
- 42. 2 2 (1 − δ ) 2 s = 2 ln q δ 2
- 43. HOW ABOUT THE MEMORY? • Different representations • Different trade-offs (space/time) • How easy is metadata to manipulate • Disk/RAM
- 44. POPULAR REPRESENTATIONS • Adjacency List • Incidence List • Adjacency Matrix (using sparse matrices) • Incidence Matrix (using sparse matrices)
- 45. class AdjacencyListGraph(object): def __init__(self): self.node = {} self.adj = {} def add_node(self, node, **attrs): if node not in self.adj: self.adj[node] = {} self.node[node] = attrs else: # update attr even if node already exists self.node[node].update(attrs) def add_edge(self, u, v, **attrs): if u not in self.adj: self.adj[u] = {} self.node u self.node[u] = {} if v not in self.adj: self.adj[v] = {} self.adj u self.node[v] = {} k1 k2 ... datadict=self.adj[u].get(v,{}) attribute datadict.update(attrs) v1 attribute self.adj[u][v] = datadict v2 e1 e2 ... self.adj[v][u] = datadict attribute v3 attribute f1 f2 ... attribute attribute
- 46. 32 bit variant Nodes Edges NX bytes Sparse bytes Dense bytes 100 733 198000 7734 10000 500 3922 1037976 41224 250000 1000 19518 4621688 199184 1000000 2000 96941 20927728 977414 4000000 5000 487686 108248888 4896864 25000000 10000 987274 221310552 9912744 100000000 20000 1986718 443525880 19947184 400000000
- 47. DISK BASED SOLUTIONS • HDF5 (Pytables, h5py) • Map-Reduce (Hadoop) • Graph DBs (Riak, Neo4J, Allegro) • “NoSQL graphs” (Mongo, ...) • SQL DBs (PostgreSQL)
- 48. How about social networking applications? profile posts name, email, ... post connections post post post When a user opens his page, his contacts profiles are read to get their posts. A user that has many friends (high degree) has his profile read more often. The degree distribution becomes the profile accesses distribution. Very good for caching! And what about the clustering coefficient? Friends of friends tend to be friends...
- 49. X Minimize: (G) = (loc(v) loc(u)) Auv v,u2V 2 unfortunately NP complete “location of v profile” We can however use community detection to improve locality. We define a cluster to be a subgraph with some cohesion. Different cluster definition exist, with different trade-offs. But profiles in a cluster tend to be accessed together, so that actually we can store the information “close” (disk-layout, sharding) We can also study the correlations between geography and clustering and hopefully use that info. In general, for SNS knowing the network structure gives insight on how to optimize stuff.
- 50. NETWORK PROCESSES • Study of processes that occur on real networks • “Network destruction”: models malfunctions in the network • “Idea/Disease” diffusion over networks • Link prediction
- 51. Network Destruction Process def attack(graph, centrality_metric): graph = graph.copy() steps = 0 ranks = centrality_metric(graph) nodes = sorted(graph.nodes(), key=lambda n: ranks[n]) while nx.is_connected(graph): graph.remove_node(nodes.pop()) steps += 1 else: return steps Power-Law Cluster Random Random Attack 220 10 PageRank driven 19 149 Betweenness driven 22 157 Degree 19 265
- 56. Thanks for your kind attention.
- 57. Massachussets Boston Nebraska Omaha Wichita 6 Degrees Kansas Milgram’s Experiment • Random people from Omaha & Wichita were asked to send a postcard to a person in Boston: • Write the name on the postcard • Forward the message only to people personally known that was more likely to know the target
- 58. Massachussets 1st run: 64/296 arrived, most Boston delivered to him by 2 men Nebraska 2nd run: 24/160 arrived, 2/3 delivered by “Mr. Jacobs” Omaha 2 ≤ hops ≤ 10; µ=5.x Wichita 6 Degrees CPL, hubs, ... Kansas ... and Kleinberg’s Intuition Milgram’s Experiment • Random people from Omaha & Wichita were asked to send a postcard to a person in Boston: • Write the name on the postcard • Forward the message only to people personally known that was more likely to know the target
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